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Technology Development of Fast-Response Aerodynamic Pressure Probes

Technology Development of Fast-Response Aerodynamic Pressure Probes International Journal of Turbomachinery Propulsion and Power Article Technology Development of Fast-Response Aerodynamic Pressure Probes Paolo Gaetani and Giacomo Persico * Laboratorio di Fluidodinamica delle Macchine, Dipartimento di Energia, Politecnico di Milano Via Lambruschini 4, I-20158 Milano, Italy; paolo.gaetani@polimi.it * Correspondence: giacomo.persico@polimi.it; Tel.: +39-022-399-8605 Received: 11 February 2020; Accepted: 29 March 2020; Published: 12 April 2020 Abstract: This paper presents and discusses the recent developments on the Fast-Response Aerodynamic Pressure Probe (FRAPP) technology at the Laboratorio di Fluidodinamica delle Macchine (LFM) of the Politecnico di Milano. First, the di erent geometries developed and tested at LFM are presented and critically discussed: the paper refers to single-sensor or two-sensor probes applied as virtual 2D or 3D probes for phase-resolved measurements. The static calibration of the sensors inserted inside the head of the probes is discussed, also taking into account for the temperature field of application: in this context, a novel calibration procedure is discussed and the new manufacturing process is presented. The dynamic calibration is reconsidered in view of the 15-years’ experience, including the extension to probes operating at di erent temperature and pressure levels with respect to calibration. As for the probe aerodynamics, the calibration coecients are discussed and the most reliable set here is evidenced. A novel procedure for the quantification of the measurement uncertainty, recently developed and based on the Montecarlo methodology, is introduced and discussed in the paper. Keywords: FRAPP; pressure sensor; temperature correction; transfer function; aerodynamics; uncertainty quantification 1. Introduction Measuring the unsteady flow downstream of turbomachinery rotors evolved from being a ‘niche’ research activity in the nineties to becoming a common practice the scientific studies of present-day turbomachinery [1–5], with also relevant examples of industrial applications [6–8]. Such evolution was supported by the technical development of instrumentation technology, of novel data-reduction methods, and by the practical experience of the experimentalists. A key contribution to this development came from one specific measurement technology, i.e., the Fast Response Aerodynamic Pressure Probe (FRAPP), which has undoubted advantages with respect to other intrusive or non-intrusive techniques in terms of rigidity, reliability, promptness. Last but not least, it provides total and static pressure measurements, which can be used for the evaluation of the blade-row and stage performance. Thanks to the very high temporal resolution, these probes also allowed to investigate experimentally the complex flow phenomena connected to unsteady blade row interaction [9–13]. The FRAPP concept comes from the combination of fast-response pressure transducers, typically of the piezoresistive kind, with aerodynamic directional pressure probes. The transducers can be flush-mounted on the probe head [14], enhancing the frequency response despite fragility; in other examples [6,15–17], the researchers preferred to embed the transducers within the probe head to enhance the probe strength. Excellent reviews on the early stages of FRAPP development can be found in [18–20]. However, since then, further relevant improvements were made on the basic technology and some of them are reported in the very recent review proposed in [21]. Int. J. Turbomach. Propuls. Power 2020, 5, 6; doi:10.3390/ijtpp5020006 www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2020, 5, 6 2 of 16 A specific version of FRAPP technology has been the object of research and development at Politecnico di Milano since 1998. The probe concept, that is alternative to those presented by the authors listed above, was first proposed in [22], and further elaborated in [23]. With the aim of minimizing the probe blockage while maximizing the instrumentation reliability, an optimal configuration was identified by using single- or two-sensor probes operated as virtual three- of four-sensor probes, and by adopting commercial transducers that only need to be mounted and glued within the probe head. This design implies the adoption of a relatively large line-cavity system connecting the pressure tap on the probe head to the sensor, strongly influencing the probe dynamic response. However, dedicated computational studies and the set-up of a novel dynamic calibration facility [24] has ultimately led to the development of FRAPPs featuring a dynamic response of the order of 100 kHz after digital compensation with the experimental transfer function. This paper proposes a review of the most relevant advances in FRAPP technology conceived and applied at Politecnico di Milano in the last decade, in terms of high-temperature applications, dynamic identification and uncertainty quantification, analyzing systematically probes for two-dimensional and three-dimensional measurements. 2. FRAPP Description Before discussing technological aspects and metrological issues, this section proposes a review of the current probes’ shape configurations and their implication on the measurement capability. Two probes were considered, one cylindrical for two-dimensional flow measurements and one spherical for three-dimensional flow measurements. Even though the probes share several technical features, they are discussed separately in the following section. 2.1. Cylindrical Probe The cylindrical shape makes the probe inherently suitable for measuring the flow direction in a plane normal to the cylinder axis (called the “yaw” angle in this paper), along with the total and static pressure. Conversely, it has a low sensitivity to the flow components parallel to the axis and hence, to the related flow angle, called the “pitch” in this paper, so that it can be considered insensitive for pitch angle values within10 , as resulted from a dedicated test campaign. As the probe embeds a single sensor for miniaturization, it needs three pressure readings measured at di erent rotations around the probe axis to virtually simulate the operation of a three-sensor probe. This prevents real-time unsteady measurements to be performed; however, unless operation instabilities are of concern, in case of unsteady turbomachinery flows, this is not a severe limitation as one is typically interested in the periodic component of the flow unsteadiness. The unsteady periodic component can be extracted by means of ensemble averages locked on the rotor wheel, using a key-phasor signal. The virtual operation prevents from achieving direct turbulence measurements, even if an estimate of the turbulence intensity is possible considering the signal acquired by the probe at the angular position aligned to the phase-averaged flow direction when the unresolved flow angle fluctuations are suciently low (9 ), as discussed in [25]. As well visible in Figure 1, which reports a drawing of the cylindrical FRAPP, the probe size and shape are determined by the sensor characteristics. The cylindrical shape was selected to miniaturize as much as possible the probe, thus reducing the probe blockage. The probe is, in practice, manufactured around one of the smallest transducers commercially available, which simply needs to be inserted within the cylinder and glued with a proper material. In this way, the size of the probe can be minimized down to about 2 mm. The smooth shape of the probe, combined with its miniaturization, guarantees optimal probe aerodynamics in terms of dynamic errors, according to dedicated studies performed at the early stages of FRAPP technology development [26]. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 3 of 16 As well visible in Figure 1, which reports a drawing of the cylindrical FRAPP, the probe size and shape are determined by the sensor characteristics. The cylindrical shape was selected to miniaturize as much as possible the probe, thus reducing the probe blockage. The probe is, in practice, manufactured around one of the smallest transducers commercially available, which simply needs to be inserted within the cylinder and glued with a proper material. In this way, the size of the probe can be minimized down to about 2 mm. The smooth shape of the probe, combined with its Int. J. Turbomach. Propuls. Power 2020, 5, 6 3 of 16 miniaturization, guarantees optimal probe aerodynamics in terms of dynamic errors, according to dedicated studies performed at the early stages of FRAPP technology development [26]. Figure 1. Representation of the cylindrical Fast-Response Aerodynamic Pressure Probe (FRAPP). Figure 1. Representation of the cylindrical Fast-Response Aerodynamic Pressure Probe (FRAPP). The probe design concept allows to deal with a relatively high temperature. Present-day The probe design concept allows to deal with a relatively high temperature. Present-day piezoelectric transducers can operate up to about 550 K and epoxy resins are commercially available piezoelectric transducers can operate up to about 550 K and epoxy resins are commercially available for temperatures greater than 600 K. The combination of these two elements allows manufacturing for temperatures greater than 600 K. The combination of these two elements allows manufacturing high-temperature FRAPP in a straightforward way and without external cooling—a topic that, however, high-temperature FRAPP in a straightforward way and without external cooling—a topic that, has been the object of dedicated investigations [27] that proved to be successful at the cost of an increase however, has been the object of dedicated investigations [27] that proved to be successful at the cost of probe blockage. of an increase of probe blockage. A further specific aspect of the present design is related to the installation of the sensor inside the A further specific aspect of the present design is related to the installation of the sensor inside probe head and to the subsequent line-cavity system connecting the pressure tap on the probe head to the probe head and to the subsequent line-cavity system connecting the pressure tap on the probe the sensor. The topic is discussed in detail in [23], in which several analytical and numerical techniques head to the sensor. The topic is discussed in detail in [23], in which several analytical and numerical are applied and compared to optimize the shape of the internal cavities in order to maximize the probe techniques are applied and compared to optimize the shape of the internal cavities in order to promptness. By virtue of this study, the promptness of the FRAPP resulted of about 80 kHz. Such value maximize the probe promptness. By virtue of this study, the promptness of the FRAPP resulted of was—and still is—suciently high to match the specifications of all the FRAPP applications considered about 80 kHz. Such value was—and still is—sufficiently high to match the specifications of all the by the authors in their experience. FRAPP applications considered by the authors in their experience. 2.2. Spherical Probe 2.2 Spherical Probe To overcome limitations in measurement capability of cylindrical probes, an alternative To overcome limitations in measurement capability of cylindrical probes, an alternative configuration suitable for unsteady 3D measurements was developed at Politecnico di Milano, configuration suitable for unsteady 3D measurements was developed at Politecnico di Milano, featuring a spherical head shape and named sFRAPP. featuring a spherical head shape and named sFRAPP. In order to enhance the sensitivity to the flow components parallel to the probe axis, the probe In order to enhance the sensitivity to the flow components parallel to the probe axis, the probe head features a spherical shape with two pressure taps. The combination of the geometrical constraints head features a spherical shape with two pressure taps. The combination of the geometrical imposed by the transducers as well as by the miniaturization of external and internal dimensions led constraints imposed by the transducers as well as by the miniaturization of external and internal to a head diameter of 3.8 mm and to a diameter of 0.3 mm for the two pneumatic lines feeding the dimensions led to a head diameter of 3.8 mm and to a diameter of 0.3 mm for the two pneumatic lines cavities facing the sensors. feeding the cavities facing the sensors. One pressure tap was drilled on the probe tip, with an inclination of 60 with respect to the probe One pressure tap was drilled on the probe tip, with an inclination of 60° with respect to the probe axis, and it was employed only for measurement of the pitch angle. The second pressure tap was axis, and it was employed only for measurement of the pitch angle. The second pressure tap was drilled on the equatorial plane with an inclination of 90 with respect to the probe axis: it can be drilled on the equatorial plane with an inclination of 90° with respect to the probe axis: it can be aligned to the ‘pitch’ tap as shown in Figure 2 or it can be rotated with respect to that by 180 around aligned to the ‘pitch’ tap as shown in Figure 2 or it can be rotated with respect to that by 180° around the probe stem. the probe stem. Int. J. Turbomach. Propuls. Power 2020, 5, 6 4 of 16 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 4 of 16 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 4 of 16 Figure 2. Representation of the spherical FRAPP with aligned taps. Figure 2. Representation of the spherical FRAPP with aligned taps. A preliminary analysis of the sFRAPP is reported in [28], and its first application downstream A preliminary analysis of the sFRAPP is reported in [28], and its first application downstream Figure 2. Representation of the spherical FRAPP with aligned taps. of a turbine stage is documented in [29]. The probe operating mechanism is still based on multiple of a turbine stage is documented in [29]. The probe operating mechanism is still based on multiple pressure readings taken at different rotations of the probe around its own stem: the combined use of pressure readings taken at di erent rotations of the probe around its own stem: the combined use of A preliminary analysis of the sFRAPP is reported in [28], and its first application downstream four pressure readings allows to measure both the flow directions, alongside total and static pressure. four pressure readings allows to measure both the flow directions, alongside total and static pressure. of a turbine stage is documented in [29]. The probe operating mechanism is still based on multiple The configuration shown in Figure 2 allows reconstructing both the flow directions with just 3 The configuration pressure read shown ings taken a in Figur t diff eerent rotati 2 allows reconstr ons of the probe aroun ucting both the d its own stem: the combined flow directions with just 3 use of rotations, rotations, while 4 rotations are required for the configuration with 2 opposed taps (a further rotation four pressure readings allows to measure both the flow directions, alongside total and static pressure. while 4 rotations are required for the configuration with 2 opposed taps (a further rotation shifted by shifted by 180° with respect to the central one of the others). However, the use of opposed taps allows The configuration shown in Figure 2 allows reconstructing both the flow directions with just 3 180 with respect to the central one of the others). However, the use of opposed taps allows reducing reducing the dimension of the internal cavities as, in this latter case, a shorter line connects the ‘pitch’ rotations, while 4 rotations are required for the configuration with 2 opposed taps (a further rotation the dimension tap to the sensor. Theoreti of the internal cacavities l estimates as, and expe in this rimental dynam latter case, a shorter ic calibrat line ions showed a connects the red‘pitch’ uction tap shifted by 180° with respect to the central one of the others). However, the use of opposed taps allows of promptness to 40 kHz, about half of the promptness of the sFRAPP with opposite taps, but still to the sensor. Theoretical estimates and experimental dynamic calibrations showed a reduction of reducing the dimension of the internal cavities as, in this latter case, a shorter line connects the ‘pitch’ very high and fully suitable for typical turbomachinery applications. promptness to 40 kHz, about half of the promptness of the sFRAPP with opposite taps, but still very tap to the sensor. Theoretical estimates and experimental dynamic calibrations showed a reduction high and fully suitable for typical turbomachinery applications. of promptness to 40 kHz, about half of the promptness of the sFRAPP with opposite taps, but still 3. Thermally Corrected Calibration of the Pressure Sensor very high and fully suitable for typical turbomachinery applications. 3. Thermally Corrected Calibration of the Pressure Sensor Due to the sensor sensitivity to the temperature, a sensor calibration in pressure and temperature is required before performing the aerodynamic and the dynamic ones. The sensor sensitivity to 3. Thermally Corrected Calibration of the Pressure Sensor Due to the sensor sensitivity to the temperature, a sensor calibration in pressure and temperature temperature is measured by applying an additional resistance (“sense resistor” in the following) on is required before performing the aerodynamic and the dynamic ones. The sensor sensitivity to Due to the sensor sensitivity to the temperature, a sensor calibration in pressure and temperature the bridge, as showed in Figure 3. The voltage drop across the sense resistor (ΔVT) is mainly a function temperatur is required e is measur before ed perfor by applying ming the an ae addi rodynam tional ic ran esistance d the dy (“se nam nse ic one resistor” s. The in sensor the following) sensitivity to on the of the current flowing across the bridge, which depends on the bridge temperature. Thus, by reading temperature is measured by applying an additional resistance (“sense resistor” in the following) on bridge, as showed in Figure 3. The voltage drop across the sense resistor (DV ) is mainly a function of the voltage difference across the bridge (ΔVP) and ΔVT, the sensor behavior can b T e fully documented. the bridge, as showed in Figure 3. The voltage drop across the sense resistor (ΔVT) is mainly a function the current flowing across the bridge, which depends on the bridge temperature. Thus, by reading the ΔV of the current flowing across the bridge, which depends on the brid T ge temperature. Thus, by reading voltage di erence across the bridge (DV ) and DV , the sensor behavior can be fully documented. P T the voltage difference across the bridge (ΔVP) and ΔVT, the sensor behavior can be fully documented. sense resistor ΔV Whaeastone ΔV sense resistor supply bridge Whaeastone ΔV supply bridge ΔV Figure 3. Electrical scheme for the pressure and temperature calibration. ΔV Figure 3. Electrical scheme for the pressure and temperature calibration. Figure 3. Electrical scheme for the pressure and temperature calibration. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 5 of 16 Int. J. Turbomach. Propuls. Power 2020, 5, 6 5 of 16 The probe temperature was set by inserting it in an oven: the insertion length was chosen to be representative of what is needed in the real application in order to minimize the effect of different The probe temperature was set by inserting it in an oven: the insertion length was chosen to be thermal conductions along the stem. representative of what is needed in the real application in order to minimize the e ect of di erent The calibration procedure was applied as follow: first, the oven temperature (Ti) was set and a thermal conductions along the stem. consistent waiting time (typically 30 min) was scheduled to bring the probe to a steady thermal The calibration procedure was applied as follow: first, the oven temperature (Ti) was set and condition. Then, a pressure ramp was applied up to the calibration range foreseen for the tests. To a consistent waiting time (typically 30 min) was scheduled to bring the probe to a steady thermal include possible hysteretic behavior in the calibration coefficients and uncertainty, the pressure ramp condition. Then, a pressure ramp was applied up to the calibration range foreseen for the tests. had both positive and negative slopes. Then, the oven temperature was modified and the procedure To include possible hysteretic behavior in the calibration coecients and uncertainty, the pressure ramp repeated: also for the temperature ramp, positive and negative slopes could be applied. had both positive and negative slopes. Then, the oven temperature was modified and the procedure For each temperature level (Ti), the pressure data were then fit by a linear function: the results repeated: also for the temperature ramp, positive and negative slopes could be applied. show the slope (KP,i), intercept (QP,i) and uncertainty (UP,i). For each temperature level (Ti), the pressure data were then fit by a linear function: the results KP,i, QP,i, and UP,i were then fit by polynomial functions (typically linear or parabolic, depending show the slope (K i), intercept (Q i) and uncertainty (U i). P, P, P, on the trend), to find the KP and QP (as function of the ΔVT). The sensor temperature Ti and ΔVT,i K i, Q i, and U i were then fit by polynomial functions (typically linear or parabolic, depending P, P, P, could also be fitted to have KT, QT. Figure 4 shows typical calibration results. on the trend), to find the K and Q (as function of the DV ). The sensor temperature Ti and DV i P P T T, could also be fitted to have K , Q . Figure 4 shows typical calibration results. T T Figure 4. Calibration coecients: top-left: pressure vs. DV at given temperature; top-right: sensor Figure 4. Calibration coefficients: top-left: pressure vs. ΔVP at given temperature; top-right: sensor Temperature vs. DV ; bottom-left: slope Kp vs. DV ; bottom-right: intercept Q vs. DV T T P T. Temperature vs. ΔVT; bottom-left: slope Kp vs ΔVT; bottom-right: intercept QP vs ∆VT. The procedure is accurate and its only critical point is a random o set on Q i due to the T, The procedure is accurate and its only critical point is a random offset on QT,i due to the sensor sensor thermal sensitivity while the K i is perfectly repeatable: this occurrence, whose magnitude T, thermal sensitivity while the KT,i is perfectly repeatable: this occurrence, whose magnitude unfortunately depends on the single transducer, requires an online check to measure it during tests. unfortunately depends on the single transducer, requires an online check to measure it during tests. Once K , Q K , and Q are found, during the probe application, the sensor pressure (P) and T T, P P Once KT, QT, KP, and QP are found, during the probe application, the sensor pressure (P) and temperature (T) can be calculated by temperature (T) can be calculated by P = K  DV + Q (1 (1) ) P = KP × ∆VP + QP P P P T = KT × ∆VT + QT (2) T = K  DV + Q (2) T T T Int. J. Turbomach. Propuls. Power 2020, 5, 6 6 of 16 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 6 of 16 Uncertainty Quantification 3.1. Uncertainty Quantification The uncertainty evaluation was made by considering the uncertainty on each measured point in pressure for a given oven (probe) temperature and the contribution due to the least square interpolation The uncertainty evaluation was made by considering the uncertainty on each measured point in among the pairs (Pi, DV i). The contribution on the single measured point takes into account the pressure for a given oven (probe) temperature and the contribution due to the least square P, standard deviation of the population, the reference transducer uncertainty, and the data acquisition interpolation among the pairs (Pi, ∆VP,i). The contribution on the single measured point takes into board analog-digital converter specifications. Gaussian distribution was considered for the first two account the standard deviation of the population, the reference transducer uncertainty, and the data quantities while a rectangular distribution was applied to the AD converter. All these contributions are acquisition board analog-digital converter specifications. Gaussian distribution was considered for considered not cross correlated, are made homogeneous in terms of units, and are considered to include the first two quantities while a rectangular distribution was applied to the AD converter. All these 95% of the Gaussian distribution data: finally, given the functional dependence f = f(x1, x2, ::: , xn), contributions are considered not cross correlated, are made homogeneous in terms of units, and are the uncertainty propagation approach is considered: considered to include 95% of the Gaussian distribution data: finally, given the functional dependence f = f(x1, x2, …, xn), the uncertainty propagation approach is considered: n 2 @ f U = u (3) @x i=1 (3) To couple the uncertainty on the single calibration point and the one due to the least square interpolation, To couplethe the uncertai highest among nty on the si the single ngle ca points libra one tio is n poi consider nt an ed. d the one due to the l Overall, for a 6-baraetransducer ast square , int the erpolat following ion, t rh esults e highe wer st a e m found ong t(Figur he sine gle po 5): no int clear s one is trends cons ar idere e visible d. Over and aldata l, for a less 6-bar than a tra 0.1% nsducer, of the tfull he follow range.ing These resurlt esults s were arfound e then (Fi applied gure 5): no c duringle the ar taer rends ar odynamic e visicalibration ble and data to le get ss tan han estimation 0.1% of th of e full range. These results are then applied during the aerodynamic calibration to get an estimation of the flow field detection uncertainty. the flow field detection uncertainty. Figure 5. Extended uncertainty for the di erent probe temperature. Figure 5. Extended uncertainty for the different probe temperature. With the aim of comparing classical uncertainty analysis with an alternative systematic approach, With the aim of comparing classical uncertainty analysis with an alternative systematic the Montecarlo methodology was also applied and the same contributions to the uncertainty were approach, the Montecarlo methodology was also applied and the same contributions to the considered. The data for each temperature level (Ti) were interpolated by a least square method by uncertainty were considered. The data for each temperature level (Ti) were interpolated by a least introducing N (a number high enough to get a statistical reliability) di erent pairs (P, DV ) chosen square method by introducing N (a number high enough to get a statistical reliability) different pairs randomly into the populations characterized by the selected distributions. The results of the N (P, ∆VP) chosen randomly into the populations characterized by the selected distributions. The results calculations were N line constants and intercepts, statistically treated to obtain mean values (K , Q ) Pi Pi of the N calculations were N line constants and intercepts, statistically treated to obtain mean values and their standard deviations. As a following step, M (a number high enough to get a statistical (KPi, QPi) and their standard deviations. As a following step, M (a number high enough to get a reliability) pairs of data belonging to the population (K , DV ) were randomly chosen according to Pi Ti statistical reliability) pairs of data belonging to the population (KPi, ∆VTi) were randomly chosen a Gaussian distribution, then averaged to get K = K (DV ) and its standard deviation; the same according to a Gaussian distribution, then averaged to get K P P T P = KP (∆VT) and its standard deviation; methodology was applied to the intercept. In this way, P = P (DV , DV ), and its standard deviation P T the same methodology was applied to the intercept. In this way, P = P (∆VP, ∆VT), and its standard were available for the application in the aerodynamic calibration. deviation were available for the application in the aerodynamic calibration. To get a proper accuracy, the Montecarlo procedure requires a huge number of iterations: to make To get a proper accuracy, the Montecarlo procedure requires a huge number of iterations: to the procedure a ordable, the Latin hypercube methodology was applied to support pairs choice, and a make the procedure affordable, the Latin hypercube methodology was applied to support pairs convergence criterion was also set on the standard deviation value. The results received by the two choice, and a convergence criterion was also set on the standard deviation value. The results received methodologies show a good consistency and were the basis for the determination of the uncertainty in by the two methodologies show a good consistency and were the basis for the determination of the the aerodynamic calibration. Figure 6 shows the results of the uncertainty calculation when reported uncertainty in the aerodynamic calibration. Figure 6 shows the results of the uncertainty calculation when reported on the same chart of the probe pressure for a Mach=0.5 test: the average uncertainty covering 95% of the samples is about 5 mbar (less than 0.1% of the transducer range), that is the same order of magnitude as the previous methodology. Int. J. Turbomach. Propuls. Power 2020, 5, 6 7 of 16 on the same chart of the probe pressure for a Mach = 0.5 test: the average uncertainty covering 95% of the samples is about 5 mbar (less than 0.1% of the transducer range), that is the same order of magnitude as the previous methodology. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 7 of 16 Figure 6. Uncertainty range for a test with a Mach number equal to 0.5. Figure 6. Uncertainty range for a test with a Mach number equal to 0.5. 4. FRAPP Dynamic Analysis 4. FRAPP Dynamic Analysis The manufacturing concept of the FRAPP developed at PoliMi implies a promptness reduction The manufacturing concept of the FRAPP developed at PoliMi implies a promptness reduction with respect to that of the sensor installed within the probe due to the line-cavity system between the with respect to that of the sensor installed within the probe due to the line-cavity system between the pressure tap and the sensor ’s sensitive element. However, a proper design of the internal cavities pressure tap and the sensor’s sensitive element. However, a proper design of the internal cavities allows to obtain a promptness of 80 kHz, as shown in [23]. allows to obtain a promptness of 80 kHz, as shown in [23]. To achieve such a promptness, the probe transfer function has to be experimentally determined To achieve such a promptness, the probe transfer function has to be experimentally determined and then applied in the data processing of measurements in test-rigs. However, the use of the transfer and then applied in the data processing of measurements in test-rigs. However, the use of the transfer function for experiments downstream of turbomachinery rotors poses specific problems due to the function for experiments downstream of turbomachinery rotors poses specific problems due to the inherent complexity of dynamic calibration tests since the operative conditions are often di erent inherent complexity of dynamic calibration tests since the operative conditions are often different between calibration and tests. These aspects are discussed in detail in the following. between calibration and tests. These aspects are discussed in detail in the following. 4.1. Time- and Frequency-Domain Identification 4.1. Time- and Frequency-Domain Identification The dynamic calibration of fast-response pressure instrumentation poses, at first, technical issues related The dynam to the gener ica calibr tion ofation inputof fast signals-respo featuring nse pres unsteady sure instrument perturbations ation poses, at suciently at first, high fr technical equency. Siren disks [30] are used to generate periodic stimulus signals while shock tubes [6,24], are used to issues related to the generation of input signals featuring unsteady perturbations at sufficiently high generate frequency. a transient Siren disks non-periodic [30] are use signal, d to ge as nerate pe the travelling riodic stim shock ulus wave sign pr als while oduces a shock tube step signals in [6, 24], very good approximation. are used to generate a transient non-periodic signal, as the travelling shock wave produces a step signal Shock in very tubes good are approx preferred imat as, ion in .a single test, the dynamic response of the probe can be achieved for the full frequency range typically of interest for turbomachinery applications (100 kHz). Considering air Shock tubes are preferred as, in a single test, the dynamic response of the probe can be achieved at for t ambient he full conditions, frequency r thea dynamic nge typic content ally of ofint a traveling erest for t shock urbom involves achinery fr appl equencies ications up ( to1the 00 or kHz der). of Consider MHz. ing By air setting at ambient conditions, the dynam proper diaphragm features, ishock c content amplitude of a traveling can be shock i selected nvolves fr in theeq range uencies of typical up to the order of MHz. By setti pressure fluctuations in turbo ng proper dia machineryp(in hrag the m fe order ature ofstenths , shock amp of bars litor ude c even an be less as selauthors ected in documented the range of tin ypical pr [7,8,11,essure fluct 12]; such perturbations uations in turbom are normally achinery (in th small e order enough of ten to not ths of bar activatesr or elevant even non-linear less as auteh oects rs docum in the dynamic ented in evolution [7, 8, 11, 1 of 2]; the such pressur perturbations are normally small enough e field within the line-cavity system. By virtue to not of act such ivatephysical relevant linearity non-lin , ea the r ef transfer fects in function the dyn determined amic evolut thr ion ough of th the e pres step-r sure esponse field w is i applicable thin the lifor ne- the cavity measur system. By ement virtu of theeperiodic of such ph fluctuations ysical linear occurring ity, the transfer within funct turbomachinery ion determ. ined through the step- response Since the is applic beginning able f ofo this r th resear e meas ch 20 urement years of ago, continuous the periodic impr fluct ovements uationshave occurr been ing made with onin the turbomachin low-pressur ery. e shock tube presented in [24], in particular on the bursting diaphragm; by using the present-day Since the beginning of this research 20 years plastic materials, shock strengths of theag oro der , continuo of 0.2–0.3 us impro bars wer vements have e obtained. They been made o exhibited n incomplete the low-pres opening sure shoc whose k tubee presented ects were in [24], investigated in particu in ldetail ar on tin he [b 31 u]; rst such ing di eap ects hragm can; b be y u pr si operly ng the present-day plastic materials, shock strengths of the order of 0.2-0.3 bars were obtained. They exhibited incomplete opening whose effects were investigated in detail in [31]; such effects can be properly handled in order to minimize their impact on the determination of the probe transfer function. Figure 7a presents a typical experimental transfer function obtained with the method proposed in [24]. The experimental trend recalls closely the one of a second-order linear system, with an evident peak at about 35 kHz representing the probe line-cavity system resonance. On the basis of Int. J. Turbomach. Propuls. Power 2020, 5, 6 8 of 16 handled Int. J. Turbo in mach. order Propuls. Power to minimize 2020, 5their , x FOR PE impact ER REon VIEW the determination of the probe transfer function. 8 of 16 Figure 7a presents a typical experimental transfer function obtained with the method proposed in [24]. the frequency and amplitude at resonance, the system identification can be done: the corresponding The experimental trend recalls closely the one of a second-order linear system, with an evident peak at linear system is also plotted in comparison to the experimental one. Differences exist but occur at a about 35 kHz representing the probe line-cavity system resonance. On the basis of the frequency and high frequency (above 40 kHz). To provide a more intuitive idea of the observed non-linearity, the amplitude at resonance, the system identification can be done: the corresponding linear system is also measured step response and the analytical response are plotted in Figure 7b. The experimental trend plotted in comparison to the experimental one. Di erences exist but occur at a high frequency (above reproduces well the one of the analytical model, suggesting that the modelling is reliable for the 40 kHz). To provide a more intuitive idea of the observed non-linearity, the measured step response whole response. The largest difference is concentrated in the first overshoot in which the experiment and the analytical response are plotted in Figure 7b. The experimental trend reproduces well the one exhibits a steeper pressure rise and a higher peak. The faster pressure rise at the beginning of the of the analytical model, suggesting that the modelling is reliable for the whole response. The largest process is clearly responsible for the non-linearity observed in the frequency domain beyond 40 kHz. di erence is concentrated in the first overshoot in which the experiment exhibits a steeper pressure rise Since full linearity is not guaranteed a priori, experimental dynamic calibration is crucial to determine and a higher peak. The faster pressure rise at the beginning of the process is clearly responsible for the the transfer function of each FRAPP manufactured in order to properly compensate the measured non-linearity observed in the frequency domain beyond 40 kHz. Since full linearity is not guaranteed a signals. priori, experimental dynamic calibration is crucial to determine the transfer function of each FRAPP manufactured in order to properly compensate the measured signals. 4.2. Pressure and Temperature Correction (a) (b) Figure 7. Frequency-domain (a) and time-domain (b) identification of a typical FRAPP. Figure 7. Frequency-domain (a) and time-domain (b) identification of a typical FRAPP. 4.2. Pressure and Temperature Correction Thanks to the good linearity exhibited by the FRAPP, the use of the transfer function to Thanks to the good linearity exhibited by the FRAPP, the use of the transfer function to dynamically dynamically compensate the pressure signals measured in turbomachinery test rigs is possible, at compensate the pressure signals measured in turbomachinery test rigs is possible, at this stage. this stage. Notwithstanding that the thermodynamic conditions of the fluid in test rigs are often Notwithstanding that the thermodynamic conditions of the fluid in test rigs are often di erent from different from those occurring in the shock tube and considering that, in general, it is not possible to those occurring in the shock tube and considering that, in general, it is not possible toreproduce such reproduce such conditions in the shock tube facility, relatively simple techniques can be proposed for conditions in the shock tube facility, relatively simple techniques can be proposed for correcting the correcting the transfer function identified in the dynamic calibration experiments. In [23], the transfer function identified in the dynamic calibration experiments. In [23], the analytical model of [32] analytical model of [32] was found to reproduce fairly well the resonance frequency of the FRAPP. was found to reproduce fairly well the resonance frequency of the FRAPP. This model, as well as the This model, as well as the other ones available in literature, show that, apart from geometrical terms, other ones available in literature, show that, apart from geometrical terms, the natural frequency and the natural frequency and the non-dimensional damping of the line-cavity system exhibit the the non-dimensional damping of the line-cavity system exhibit the following dependencies: following dependencies: ! / c / T (4) N (4) ∝ ∝ √ (5) ∝ ∝ () / / (T) (5) c P Expression 4, which relates directly the natural frequency to the sound speed, is intuitively justified as the dynamic response of the line-cavity system depends on the pressure wave propagation within the probe internal cavities. In the context of perfect gases, this property provides a first straightforward correction to the transfer function for temperature differences between Int. J. Turbomach. Propuls. Power 2020, 5, 6 9 of 16 Expression 4, which relates directly the natural frequency to the sound speed, is intuitively justified as the dynamic response of the line-cavity system depends on the pressure wave propagation within the probe internal cavities. In the context of perfect gases, this property provides a first straightforward correction to the transfer function for temperature di erences between calibration and application. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 9 of 16 Expression 5 indicates that both temperature and pressure levels have an impact on the damping and, calibration and application. Expression 5 indicates that both temperature and pressure levels have an once again, it provides a tool for correcting the transfer function identified with experiments in the impact on the damping and, once again, it provides a tool for correcting the transfer function shock tube; also, the pressure level can have an e ect and demands for corrections, even though its identified with experiments in the shock tube; also, the pressure level can have an effect and demands e ect is quantitatively lower than that of the temperature. for corrections, even though its effect is quantitatively lower than that of the temperature. Figure 8 shows the impact of combined temperature and pressure correction on the amplitude of Figure 8 shows the impact of combined temperature and pressure correction on the amplitude the transfer function determined in the shock tube (TF-EXP). For this probe, a nearly perfect dynamic of the transfer function determined in the shock tube (TF-EXP). For this probe, a nearly perfect linearity is observed up to 60 kHz, with natural frequency at about 40 kHz, as shown by the comparison dynamic linearity is observed up to 60 kHz, with natural frequency at about 40 kHz, as shown by the with the analytical second order transfer function (TF-TEO_CAL). By considering an application at the comparison with the analytical second order transfer function (TF-TEO_CAL). By considering an maximum temperature level technically available for the FRAPP (550 K), a correction to the analytical application at the maximum temperature level technically available for the FRAPP (550 K), a transfer function was applied to get the analytical transfer function in the real environment of the test correction to the analytical transfer function was applied to get the analytical transfer function in the rig (TF-TEO_RIG). The impact of the corrections is negligible up to 20 kHz, while a relevant deviation real environment of the test rig (TF-TEO_RIG). The impact of the corrections is negligible up to 20 occurs for a frequency higher than 40 kHz. kHz, while a relevant deviation occurs for a frequency higher than 40 kHz. Figure 8. Impact of temperature of tests with respect to calibration on FRAPP transfer function. Figure 8. Impact of temperature of tests with respect to calibration on FRAPP transfer function. An application at high pressure and high temperature (not reported extensively for confidentiality An application at high pressure and high temperature (not reported extensively for reasons) showed that the linearized temperature-pressure correction allowed properly compensating confidentiality reasons) showed that the linearized temperature-pressure correction allowed the pressure signals measured by the probe, thus recovering the typical power spectra of unsteady properly compensating the pressure signals measured by the probe, thus recovering the typical pressure measurements in high-speed flows. power spectra of unsteady pressure measurements in high-speed flows. 5. FRAPP Aerodynamics 5. FRAPP Aerodynamics The aerodynamic calibration was performed on a convergent nozzle whose outlet section The aerodynamic calibration was performed on a convergent nozzle whose outlet section is 50 is 50 mm 60 mm and typically allows for neglecting the blockage e ects up to Mach = 0.95. mm x 60 mm and typically allows for neglecting the blockage effects up to Mach = 0.95. Convergent Convergent–divergent nozzles are also available when supersonic calibrations are required. –divergent nozzles are also available when supersonic calibrations are required. The Reynolds–Mach The Reynolds–Mach number e ects decoupling can be also achieved by a nozzle inserted in a number effects decoupling can be also achieved by a nozzle inserted in a duct brought to a chocked duct brought to a chocked condition by a downstream throat. The outlet pressure is usually set to be condition by a downstream throat. The outlet pressure is usually set to be atmospheric and the Mach atmospheric and the Mach number is set by imposing the total pressure in the upstream reservoir. number is set by imposing the total pressure in the upstream reservoir. When the aerodynamic calibration is of concern, di erent calibration coecients can be taken into When the aerodynamic calibration is of concern, different calibration coefficients can be taken account. The advantages of di erent coecient sets may arise from a pure aerodynamic behavior or into account. The advantages of different coefficient sets may arise from a pure aerodynamic behavior from the uncertainty point of view. or from the uncertainty point of view. 5.1. Cylindrical FRAPP 5.1. Cylindrical FRAPP For the 2D Frapp probe, the following is commonly applied: For the 2D Frapp probe, the following is commonly applied: P P P P P (P + P )/2 S R L L R T C − − −( + )/2 KYaw = KP = KP = (6) T S P P P P P P (6) = T = T = T S S S − − − where PL = probe Left pressure reading, PR = probe Right pressure reading, PC = probe Central pressure reading, PT = nozzle Total pressure, and PS = nozzle Static pressure. The experimental trends of the coefficients are reported in Figure 9: when the Kyaw coefficient is zero, the aerodynamic reference direction is set. The range of monotonic trend is typically up to +/- 23°, due to the 45° of spacing between the three pressure readings (Left, Central, Right) and to the range in between the two flow separation points on a cylinder positioned in cross flow, that is around Int. J. Turbomach. Propuls. Power 2020, 5, 6 10 of 16 where P = probe Left pressure reading, P = probe Right pressure reading, P = probe Central L R C pressure reading, P = nozzle Total pressure, and P = nozzle Static pressure. T S The experimental trends of the coecients are reported in Figure 9: when the Kyaw coecient is zero, the aerodynamic reference direction is set. The range of monotonic trend is typically up to +/ 23 , due to the 45 of spacing between the three pressure readings (Left, Central, Right) and to Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 10 of 16 the range in between the two flow separation points on a cylinder positioned in cross flow, that is around +/ 67  70 . depending on the probe geometry, Reynolds and Mach numbers. The total +/- 67° ÷ 70°. depending on the probe geometry, Reynolds and Mach numbers. The total pressure pressure coecient di ers mainly for the compressibility e ect over the probe cylindrical head, with coefficient differs mainly for the compressibility effect over the probe cylindrical head, with the the Reynolds number having a lower impact in this range. Reynolds number having a lower impact in this range. Figure Figure 9. 9. 2D 2D F FRAPP RAPP calibration calibration coeffi coecients cientsfor for varying varying Mac Mach hnumber number and and yaw yaw angl angle. e. In the application phase, the real Total and Static pressures (and by these the Mach number) and In the application phase, the real Total and Static pressures (and by these the Mach number) and the flow angle are derived by an iterative procedure. First, the static pressure and the total pressure are the flow angle are derived by an iterative procedure. First, the static pressure and the total pressure chosen as the average between the left and right, and the central one, respectively; using these, the are chosen as the average between the left and right, and the central one, respectively; using these, KYaw is a calculated and the yaw angle is derived. Using the yaw angle, the new KPt and KPs are the KYaw is a calculated and the yaw angle is derived. Using the yaw angle, the new KPt and KPs calculated by making use of the calibration curves, properly interpolated in angles and Mach. At this are calculated by making use of the calibration curves, properly interpolated in angles and Mach. At point, the new Total and Static pressures are calculated and the second iteration can start. this point, the new Total and Static pressures are calculated and the second iteration can start. In case of thermal drift, the KYaw is less sensitive than KPt and KPs because the numerator is In case of thermal drift, the KYaw is less sensitive than KPt and KPs because the numerator is almost insensitive to the drift, which typically occurs as an o set. almost insensitive to the drift, which typically occurs as an offset. 5.2. sFRAPP 5.2. sFRAPP For the 3D sFRAPP several sets of coecients were considered and evaluated in calibration. For the 3D sFRAPP several sets of coefficients were considered and evaluated in calibration. Some of them are discussed separately in the following: Some of them are discussed separately in the following: 5.2.1. Set A 5.2.1. Set A: P P P P L R T P − − KYaw = ; KPitch = ; P P P P T S T S = ; ℎ = ; (7) P (P +P )/2 P P T C S R L − − KP = ; KP = T S P P P P T S T S (7) − −( + )/2 where KYaw, KP and KP depend on the yaw tap (and its virtual readings) and on the flow total T S = ; = − − and static pressures, while the KPitch depends on the Pitch tap (P ) and on the flow total and static where KYaw, KPT and KPS depend on the yaw tap (and its virtual readings) and on the flow total and pressures. None of the coecients are defined by mixing the pitch and the yaw taps and all include static pressures, while the KPitch depends on the Pitch tap (PP) and on the flow total and static P and P . The KPt is defined as usually found in the literature for multi-hole probes. KPs typically T s pressures. None of the coefficients are defined by mixing the pitch and the yaw taps and all include refer to the static pressure (P ) and to the average of the lateral holes, whose value is close to the PT and Ps. The KPt is defined as usually found in the literature for multi-hole probes. KPs typically static pressure. In this probe, the lateral holes provide the pressure readings P , P , and P ; however, L R P refer to the static pressure (PS) and to the average of the lateral holes, whose value is close to the static these lateral holes are not symmetrical with respect to the probe head (as, instead, occurs for 5-hole pressure. In this probe, the lateral holes provide the pressure readings PL, PR, and PP; however, these probes of conical/prismatic head shape), and for this reason, the average of the corresponding pressure lateral holes are not symmetrical with respect to the probe head (as, instead, occurs for 5-hole probes readings is always very di erent with respect to the actual static pressure of the flow, making the KPs of conical/prismatic head shape), and for this reason, the average of the corresponding pressure coecient not null in any condition. For this reason, only P and P are considered in the proposed L R readings is always very different with respect to the actual static pressure of the flow, making the KPs definition. As a further technical consideration, the P is measured with a di erent transducer KPs coefficient not null in any condition. For this reason, only PL and PR are considered in the with respect to P and P and thus, retaining P in the KPs definition would make the coecient L R P proposed KPs definition. As a further technical consideration, the PP is measured with a different sensitive to the di erent potential thermal drift of the two transducers. transducer with respect to PL and PR and thus, retaining PP in the KPs definition would make the coefficient sensitive to the different potential thermal drift of the two transducers. As for KYaw and KPitch, they have a definition consistent with multi-hole geometries, with KPitch being defined with only one tap (PT is a constant given the Mach number). As reported in Figure 10, the KPt and KPs coefficients are regular over the grid KYaw-KPitch and for this, they seem easily applicable. However, they exhibit an overlapping zone at the boundary of the matrix that leads to non-unique solutions during the interpolation procedure. Int. J. Turbomach. Propuls. Power 2020, 5, 6 11 of 16 As for KYaw and KPitch, they have a definition consistent with multi-hole geometries, with KPitch being defined with only one tap (P is a constant given the Mach number). As reported in Figure 10, the KPt and KPs coecients are regular over the grid KYaw-KPitch and for this, they seem easily applicable. However, they exhibit an overlapping zone at the boundary of the matrix that leads to non-unique solutions during the interpolation procedure. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 11 of 16 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 11 of 16 Figure 10. KPt and KPs on the KYaw and KPitch coefficient grid for sFRAPP (range ± 22°) f  or Figure 10. KPt and KPs on the KYaw and KPitch coecient grid for sFRAPP (range  22 ) for Figure 10. KPt and KPs on the KYaw and KPitch coefficient grid for sFRAPP (range ± 22°) for aerodynamic coefficients defined according to set A. aerodynamic coecients defined according to set A. aerodynamic coefficients defined according to set A. 5. 5.2.2. 2.2. Set Set BB : 5.2.2. Set B: P P L R It It diff di ers ersfro from m Set Set A on A only ly for: for: K =Y aw = , where , where P Pmax is the maximum value is the maximum valueof the of the max P (P +P )/2 max ( L R )/ It differs from Set A only for: = , where Pmax is the maximum value of the ( )/ parabola passing by the three points P , P , P : it is an artificial value because the pressure curve C L R parabola passing by the three points PC, PL, PR: it is an artificial value because the pressure curve parabola passing by the three points PC, PL, PR: it is an artificial value because the pressure curve around a cylinder is not a parabola, although it is similar to one. This new coecient does not su er around a cylinder is not a parabola, although it is similar to one. This new coefficient does not suffer around a cylinder is not a parabola, although it is similar to one. This new coefficient does not suffer from o set errors, being related to one transducer only and including di erences both at numerator from offset errors, being related to one transducer only and including differences both at numerator from offset errors, being related to one transducer only and including differences both at numerator and denominator. A similar choice for the KPitch cannot be applied because there are no virtual taps and denominator. A similar choice for the KPitch cannot be applied because there are no virtual taps and denominator. A similar choice for the KPitch cannot be applied because there are no virtual taps for such a quantity as result of the rotation along the probe stem. The other main advantage concerns for such a quantity as result of the rotation along the probe stem. The other main advantage concerns for such a quantity as result of the rotation along the probe stem. The other main advantage concerns the grid regularity that allows for a proficient interpolation over the whole angular range, as visible in the grid regularity that allows for a proficient interpolation over the whole angular range, as visible the grid regularity that allows for a proficient interpolation over the whole angular range, as visible Figure 11. in Figure 11. in Figure 11. Figure 11. KPt on the KYaw and KPitch coefficient grid (range +/- 22°) for aerodynamic coefficients Figure 11. KPt on the KYaw and KPitch coecient grid (range +/ 22 ) for aerodynamic coecients Figure 11. KPt on the KYaw and KPitch coefficient grid (range +/- 22°) for aerodynamic coefficients defined according to set B. defined according to set B. defined according to set B. 5.2.3. Set C 5.2.3. Set C: 5.2.3. Set C: P P C P It di ers from Set A only by the KPitch that is defined in this set as KPitch = . This choice It differs from Set A only by the KPitch that is defined in this set as ℎ = P P . This choice It differs from Set A only by the KPitch that is defined in this set as ℎ = . This choice allows for a direct link between the central reading of the yaw and the pitch tap, as the central reading allows for a direct link between the central reading of the yaw and the pitch tap, as the central reading allows for a direct link between the central reading of the yaw and the pitch tap, as the central reading for the yaw is, in any case, dependent on the pitch flow angle. This set, although seemingly smart, in for the yaw is, in any case, dependent on the pitch flow angle. This set, although seemingly smart, in ffact, or the ya connects w is, i the n atwo ny case, sensors dependent on the pi readings and, in tch fl caseoof w a an thermal gle. This drift, set, alt it h may ough seem increase ing substantially ly smart, in fact, connects the two sensors readings and, in case of a thermal drift, it may increase substantially fact, connects the two sensors readings and, in case of a thermal drift, it may increase substantially the final uncertainty. From a purely aerodynamic point of view, as shown in Figure 12, it changes the the final uncertainty. From a purely aerodynamic point of view, as shown in Figure 12, it changes the KPitch coefficient magnitude but does not fix the overlap at the grid boundary. KPitch coefficient magnitude but does not fix the overlap at the grid boundary. Int. J. Turbomach. Propuls. Power 2020, 5, 6 12 of 16 the final uncertainty. From a purely aerodynamic point of view, as shown in Figure 12, it changes the Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 12 of 16 KPitch coecient magnitude but does not fix the overlap at the grid boundary. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 12 of 16 Figure 12. Figure 12. KPt KPt on the KYaw and KPit on the KYaw and KPitch ch coef coe ficient grid cient grid (range (range +/- 22°) for aerodynamic coefficients +/ 22 ) for aerodynamic coecients Figure 12. KPt on the KYaw and KPitch coefficient grid (range +/- 22°) for aerodynamic coefficients defined defined ac accor cord ding ing to set to set C C. . defined according to set C. 5.2.4. Discussion on Coecient Sets 5.2.4. Discussion on Coefficient Sets 5.2.4. Discussion on Coefficient Sets Set A was abandoned because it does not guarantee a unique solution in the iterative procedure Set A was abandoned because it does not guarantee a unique solution in the iterative procedure Set A was abandoned because it does not guarantee a unique solution in the iterative procedure due to the overlap at the edges of the calibration matrices. Set C has the advantage of linking the due to the overlap at the edges of the calibration matrices. Set C has the advantage of linking the pitch due to the overlap at the edges of the calibration matrices. Set C has the advantage of linking the pitch pitch sensitivity to the most physical di erence that is the (P P ): notwithstanding this fact, it was C P sensitivity to the most physical difference that is the (PC - PP): notwithstanding this fact, it was sensitivity to the most physical difference that is the (PC - PP): notwithstanding this fact, it was abandoned as well, due to the mixture of two transducers readings that make its application critical in abandoned as well, due to the mixture of two transducers readings that make its application critical abandoned as well, due to the mixture of two transducers readings that make its application critical the context of possible (and maybe dicult to compensate) thermal drift. Moreover, it does not allow in the context of possible (and maybe difficult to compensate) thermal drift. Moreover, it does not in the context of possible (and maybe difficult to compensate) thermal drift. Moreover, it does not to fix the problem of the non-unique solution at the edges of the matrices. allow to fix the problem of the non-unique solution at the edges of the matrices. allow to fix the problem of the non-unique solution at the edges of the matrices. Set B was chosen for this type of probe as it provides both the advantage of keeping separate Set B was chosen for this type of probe as it provides both the advantage of keeping separate the Set B was chosen for this type of probe as it provides both the advantage of keeping separate the the two transducer readings and of the proper scaling of the angular sensitivity (P P ) to the local L R two transducer readings and of the proper scaling of the angular sensitivity (PL - PR) to the local two transducer readings and of the proper scaling of the angular sensitivity (PL - PR) to the local kinetic head measured by the probe, at that pitch position. To aid the physical understanding of this kinetic head measured by the probe, at that pitch position. To aid the physical understanding of this kinetic head measured by the probe, at that pitch position. To aid the physical understanding of this latter concept, i.e., of the artificial P , Figure 13 reports the yaw tap pressure measurements (P ) in max latter concept, i.e., of the artificial Pmax, Figure 13 reports the yaw tap pressure measurements (PC) in latter concept, i.e., of the artificial Pmax, Figure 13 reports the yaw tap pressure measurements (PC) in the yaw calibration range for the di erent pitch angles. Two observations can be drawn: the yaw calibration range for the different pitch angles. Two observations can be drawn: the yaw calibration range for the different pitch angles. Two observations can be drawn: Figure 13. Influence of flow angles on sFRAPP central pressure reading P . Figure 13. Influence of flow angles on sFRAPP central pressure reading PC. Figure 13. Influence of flow angles on sFRAPP central pressure reading PC. (a) the maximum pressure depends on the pitch angle as a consequence of the tap position with a) the maximum pressure depends on the pitch angle as a consequence of the tap position with a) the maximum pressure depends on the pitch angle as a consequence of the tap position with respect to the flow direction: therefore, the P allows for considering this di erence. max respect to the flow direction: therefore, the Pmax allows for considering this difference. respect to the flow direction: therefore, the Pmax allows for considering this difference. (b) the di erence between Pc at Yaw = 0 and Yaw =20 is slightly greater for negative pitches, where b) the difference between Pc at Yaw = 0° and Yaw =20° is slightly greater for negative pitches, also (P P ) has higher values: the Set B denominator is then able to scale properly the angular b) the differenc L R e between Pc at Yaw = 0° and Yaw =20° is slightly greater for negative pitches, sensitivity of the yaw tap. where also (PL - PR) has higher values: the Set B denominator is then able to scale properly where also (PL - PR) has higher values: the Set B denominator is then able to scale properly the angular sensitivity of the yaw tap. the angular sensitivity of the yaw tap. 5.3. Uncertainty Quantification 5.3. Uncertainty Quantification To quantify the uncertainty level in the calibration matrixes building and application, the same To quantify the uncertainty level in the calibration matrixes building and application, the same methodologies as those described in the static calibration can be applied. The results discussed in the methodologies as those described in the static calibration can be applied. The results discussed in the Int. J. Turbomach. Propuls. Power 2020, 5, 6 13 of 16 5.3. Uncertainty Quantification To quantify the uncertainty level in the calibration matrixes building and application, the same Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 13 of 16 methodologies as those described in the static calibration can be applied. The results discussed in the following were obtained for the FRAPP probe. In this work, only uncertainties in the calibration and in following were obtained for the FRAPP probe. In this work, only uncertainties in the calibration and the probe application when the probe was applied in a steady flow are considered: uncertainties due in the probe application when the probe was applied in a steady flow are considered: uncertainties to probe installation and positioning, unsteady e ects (e.g., the probe stem vortex shedding or possible due to probe installation and positioning, unsteady effects (e.g. the probe stem vortex shedding or interaction with the cascades), intrusiveness, and spatial discretisation are not considered as highly possible interaction with the cascades), intrusiveness, and spatial discretisation are not considered as dependent on the kind of positioner used and on the application foreseen for the probe: in any case, highly dependent on the kind of positioner used and on the application foreseen for the probe: in any besides the positioning, the other uncertainty sources are very dicult to be evaluated. case, besides the positioning, the other uncertainty sources are very difficult to be evaluated. As the calibration matrixes require the application of iterative procedures, the Montecarlo As the calibration matrixes require the application of iterative procedures, the Montecarlo methodology is particularly attractive for computing the uncertainty propagation. methodology is particularly attractive for computing the uncertainty propagation. The calibration coecients (KYaw, KPt, KPs) were first calculated by choosing pressure values The calibration coefficients (KYaw, KPt, KPs) were first calculated by choosing pressure values (P , P , P , P , P ) randomly in each population (for a given Mach number and angular position) L C R T S (PL, PC, PR, PT, PS) randomly in each population (for a given Mach number and angular position) according to the Gaussian distribution resulting from the static calibration (see the static calibration according to the Gaussian distribution resulting from the static calibration (see the static calibration paragraph). Then, the results of the calibration coecients were averaged and their standard deviation paragraph). Then, the results of the calibration coefficients were averaged and their standard was calculated, then they were all stored in proper calibration files. deviation was calculated, then they were all stored in proper calibration files. During the measurements campaign, to obtain the flow quantities, the measured pressures were During the measurements campaign, to obtain the flow quantities, the measured pressures were used coupled to the calibration matrixes: the measured pressures were analysed by the same procedure used coupled to the calibration matrixes: the measured pressures were analysed by the same applied in the calibration processes, leading to a gaussian distribution with their own mean value and procedure applied in the calibration processes, leading to a gaussian distribution with their own standard deviation. A number N of di erent sets of pressures and coecients were selected randomly mean value and standard deviation. A number N of different sets of pressures and coefficients were in each population and by an iterative procedure, the flow quantities calculated. Since the process selected randomly in each population and by an iterative procedure, the flow quantities calculated. is statistical, all the data were then averaged and the standard deviation was calculated. As for the Since the process is statistical, all the data were then averaged and the standard deviation was result distribution, since the input data were chosen according to a Gaussian distribution, the output calculated. As for the result distribution, since the input data were chosen according to a Gaussian ones were of the same kind. The number N of di erent sets was dynamically chosen according to distribution, the output ones were of the same kind. The number N of different sets was dynamically the convergence criterion chosen for the standard deviation change (D / < 10 ). Moreover, in i+1,i i chosen according to the convergence criterion chosen for the standard deviation change this case, the set choice was made by applying the Latin hypercube methodology in order to save -3 (Δσi+1,i /σi <10 ). Moreover, in this case, the set choice was made by applying the Latin hypercube computational time. methodology in order to save computational time. Figure 14 shows the results for di erent run and convergence criteria: since the method is based Figure 14 shows the results for different run and convergence criteria: since the method is based on statistics, di erent runs may lead to di erent results. Notwithstanding such possible variations, the on statistics, different runs may lead to different results. Notwithstanding such possible variations, di erence is almost negligible for a given yaw angle. Finally, Table 1 shows the averaged error for the difference is almost negligible for a given yaw angle. Finally, Table 1 shows the averaged error the four quantities of interest (Yaw, Mach, P , P ). Values are high typically at low and high Mach T S for the four quantities of interest (Yaw, Mach, PT, PS). Values are high typically at low and high Mach numbers. At a low Mach number, this is due to the high transducer range that makes the transducer numbers. At a low Mach number, this is due to the high transducer range that makes the transducer uncertainty quantitatively significant in the context of the measurements; at high Mach, the overspeed uncertainty quantitatively significant in the context of the measurements; at high Mach, the on the cylinder makes the flow locally supersonic and in this case, the measurements are less accurate. overspeed on the cylinder makes the flow locally supersonic and in this case, the measurements are less accurate. Figure 14. Results in terms of standard deviation for di erent runs. Figure 14. Results in terms of standard deviation for different runs. Table 1. Averaged errors ( δ ) for the Yaw, Mach, total and static pressure. Mach δYaw δM δPt % δPs % 0.25 0.63 0.002 1.10 1.50 0.35 0.50 0.004 0.92 2.90 0.45 0.62 0.001 0.56 0.76 Int. J. Turbomach. Propuls. Power 2020, 5, 6 14 of 16 Table 1. Averaged errors () for the Yaw, Mach, total and static pressure. Mach Yaw M Pt % Ps % 0.25 0.63 0.002 1.10 1.50 0.35 0.50 0.004 0.92 2.90 0.45 0.62 0.001 0.56 0.76 0.55 0.69 0.001 0.28 0.40 0.65 0.62 0.003 0.26 0.70 0.75 0.58 0.011 0.40 0.72 0.825 0.50 0.016 0.24 1.81 0.875 0.37 0.013 0.21 2.23 0.925 0.16 0.023 0.21 2.10 6. Conclusions This paper presented the most relevant developments in FRAPP technology at Politecnico di Milano in the last decade. This study considered the two most relevant probe configurations manufactured, calibrated and applied by the authors in their experience, for both 2D and 3D unsteady flow measurements in turbomachinery. Specific challenges emerged in terms of extension to (relatively) high temperature applications, simplicity of operation, improved aerodynamics and more refined uncertainty quantification, and were all acknowledged in the paper. In particular, uncooled FRAPPs were manufactured for a temperature operation of about 550 K, without changing the external/internal probe shape and size. The need for high temperature extension also triggered specific theoretical models to handle temperature-corrected static and dynamic calibration of the probes. A physical analysis of the sensor properties and of the line-cavity system provided general rules for correcting the static and dynamic pressure measurements performed during calibrations. In the frame of these activities, the static and dynamic calibration procedures were re-analyzed to investigate the global reliability of the FRAPP technology. The probe aerodynamics were also reconsidered and several sets of aerodynamic coecients were proposed for the Spherical FRAPP, which is less consolidated with respect to the cylindrical one. This analysis highlights that clear advantages can be obtained if a specific set of coecients is applied. Finally, a novel technique based on the Montecarlo approach was introduced to evaluate the uncertainty of FRAPP measurements, combining those of the sensor (based on a Montecarlo analysis of the static calibration) with the formulation of the aerodynamic coecients. Author Contributions: The two authors have given equal contributions to this paper, in terms of conceptualization, methodology, formal analysis, investigation, and writing. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest. Nomenclature c sound speed P pressure f frequency ! natural frequency non dimensional damping dynamic viscosity density T temperature U, u uncertainty standard deviation DV voltage di erence across the bridge DV voltage di erence across the sense resistor T Int. J. Turbomach. Propuls. Power 2020, 5, 6 15 of 16 K sensor calibration slope Q sensor calibration intercept KPs, KPt sensitivity coe . to static and total pressure KYaw, KPitch sensitivity coe . to the yaw, pitch angles References 1. Roduner, C.; Kupferschmied, P.; Köppel, P.; Gyarmathy, G. On the Development and Application of the Fast-Response Aerodynamic Probe System in Turbomachines—Part 2: Flow, Surge, and Stall in a Centrifugal Compressor. J. Turbomach. 2000, 122, 517–526. [CrossRef] 2. Gaetani, P.; Persico, G.; Osnaghi, C. E ects of Axial Gap on the Vane-Rotor Interaction in a Low Aspect Ratio Turbine Stage. J. Propuls. Power 2010, 26, 325–334. [CrossRef] 3. Gaetani, P.; Persico, G.; Mora, A.; Dossena, V.; Osnaghi, C. Impeller Vaned Di user Interaction in a Centrifugal Compressor at O -Design Conditions. J. Turbomach. 2012, 134, 061034. [CrossRef] 4. Lengani, D.; Paradiso, B.; Marn, A.; Goettlich, E. Identification of spinning mode in the unsteady flow field of a low pressure turbine. J. Turbomach. 2012, 134, 051032. [CrossRef] 5. Gatti, G.; Gaetani, P.; Paradiso, B.; Dossena, V.; Bellucci, J.; Arcangeli, L. An experimental study of the aerodynamic forcing function in a 1.5 steam turbine stage. J. Eng. Gas Turbines Power 2017, 139, 052503. [CrossRef] 6. Brouckaert, J.F. Fast response aerodynamic probes for measurements in turbomachines. Proc. Inst. Mech. Eng. Part A 2007, 221, 811–813. [CrossRef] 7. Toni, L.; Ballarini, V.; Cioncolini, S.; Gaetani, P.; Persico, G. Unsteady Flow Field Measurements in an Industrial Centrifugal Compressor. In Proceedings of the 39th Turbomachinery Symposium, Houston, TX, USA, 4–7 October 2010. 8. Guidotti, E.; Tapinassi, L.; Toni, L.; Bianchi, L.; Gaetani, P.; Persico, G. Experimental and Numerical Analysis of the Flow Field in the Impeller of a Centrifugal Compressor Stage at Design Point; ASME paper GT2011-45036. In Proceedings of the ASME Turbo Expo 2011, Vancouver, BC, Canada, 6–10 June 2011. 9. Miller, R.J.; Moss, R.W.; Ainsworth, R.W.; Horwood, C.K. Time-resolved vane–rotor interaction in a high-pressure turbine stage. J. Turbomach. 2003, 125, 1–13. [CrossRef] 10. Schlienger, J.; Kalfas, A.I.; Abhari, R.S. Vortex–wake–blade interaction in a shrouded axial turbine. J. Turbomach. 2005, 127, 699–707. [CrossRef] 11. Gaetani, P.; Persico, G.; Dossena, V.; Osnaghi, C. Investigation of the Flow Field in a High-Pressure Turbine Stage for Two Stator-Rotor Axial Gaps-Part II: Unsteady Flow Field. J. Turbomach. 2007, 129, 580–590. [CrossRef] 12. Paradiso, B.; Persico, G.; Gaetani, P.; Schennach, O.; Pecnik, R.; Woisetschlger, J. Blade row interaction in a one and a half stage transonic turbine focusing on three dimensional e ects: Part I—Stator-rotor interaction, ASME paper GT2008-50291. In Proceedings of the ASME Turbo Expo 2008: Power for Land, Sea and Air, Berlin, Germany, 9–13 June 2008. 13. Persico, G.; Gaetani, P.; Osnaghi, C. A Parametric Study of the Blade Row Interaction in a High Pressure Turbine Stage. J. Turbomach. 2009, 131, 031006. [CrossRef] 14. Ainsworth, R.W.; Allen, J.L.; Batt, J.J.M. The development of fast response aerodynamic probes for flow measurements in turbomachinery. J. Turbomach. 1995, 117, 625–634. [CrossRef] 15. Heneka., A. Instantaneous three-dimensional flow measurements with a four-hole wedge probe. In Proceedings of the 7th Symposium on Measuring Techniques in Turbomachines, Stockholm, Sweden, 21–23 September 1983. 16. Gossweiler, C.; Kupferschmied, P.; Gyamarthy, G. On fast-response probes, part 1: Technology, calibration and application to turbomachinery. J. Turbomach. 1995, 117, 611–617. [CrossRef] 17. Kupferschmied, P.; Koppel, P.; Roduner, C.; Gyarmathy, G. On the development and application of the FRAP­ (fast response aerodynamic probe) system for turbomachines—Part I: The measurement system. Asme J. Turbomach. 2000, 122, 505–516. [CrossRef] 18. Sieverding, C.H.; Arts, T.; Denos, R.; Brouckaert, J.F. Measurement techniques for unsteady flows in turbomachines. Exp. Fluids 2000, 28, 285–321. [CrossRef] Int. J. Turbomach. Propuls. Power 2020, 5, 6 16 of 16 19. Ainsworth, R.W.; Miller, R.J.; Moss, R.W.; Thorpe, S.J. Unsteady pressure measurements. Meas. Sci. Technol. 2000, 11, 1055–1076. [CrossRef] 20. Kupferschmied, P.; Koppel, P.; Gizzi, W.; Roduner, C.; Gyarmathy, G. Time-resolved flow measurements with a fast-response aerodynamic probes in turbomachines. Meas. Sci. Technol. 2000, 11, 1036–1054. [CrossRef] 21. Lepicovsky, J.; Simurda, D. Past developments and current advancements in unsteady pressure measurements in turbomachines. J. Turbomach. 2018, 140, 111005. [CrossRef] 22. Barigozzi, G.; Dossena, V.; Gaetani, P. Development and fiand application of a single hole fast response pressure probe. In Proceedings of the 15th Symp. Measuring Techniques in Transonic and Supersonic Flow in Cascade and Turbomachines, Florence, Italy, 21–22 September 2000. 23. Persico, G.; Gaetani, P.; Guardone, A. Design and analysis of new concept fast-response pressure probes. Meas. Sci. Technol. 2005, 16, 1741–1750. [CrossRef] 24. Persico, G.; Gaetani, P.; Guardone, A. Dynamic calibration of fast-response probes in a low pressure shock tube. Meas. Sci. Technol. 2005, 16, 1751–1759. [CrossRef] 25. Persico, G.; Gaetani, P.; Paradiso, B. Estimation of turbulence by single-sensor pressure probes. In Proceedings of the XIX Biannual Symposium on Measuring Techniques in Turbomachinery Transonic and Supersonic Flow in Cascades and Turbomachines, Rhode-St-Genese, Belgium, 7–8 April 2008. 26. Humm, H.J.; Gizzi, W.; Gyarmathy, G. Dynamic response of aerodynamic probes in fluctuating 3D flows. In Proceedings of the 12th Symp. Measuring Techniques in Transonic and Supersonic Flow in Cascade and Turbomachines, Prague, Czech Republic, 12–13 September 1994. 27. Mersinligil, M.; Brouckaert, J.F.; Desset, J. Unsteady Pressure Measurements with a Fast Response Cooled Probe in High Temperature Gas Turbine Environments. J. Eng. Gas Turbines Power 2011, 133, 081603. [CrossRef] 28. Dossena, V.; Gaetani, P.; Persico, G. Development of High Response Pressure Probes for Time-Resolved 2D and 3D Flow Measurements in Turbomachines. In Proceedings of the 17th Symposium on Measuring Techniques in Transonic and Supersonic Flow in Cascades and Turbomachines, Stockholm, Sweden, 9–10 September 2004. 29. Persico, G.; Mora, A.; Gaetani, P.; Savini, M. Unsteady Aerodynamics of a Low Aspect Ratio Turbine Stage: Modeling Issues and Flow Physics. J. Turbomach. 2012, 134, 061030. [CrossRef] 30. Sahin, ¸ F.C.; Schi mann, J. Dynamic pressure probe response tests for robust measurements in periodic flows close to probe resonating frequency. Meas. Sci. Technol. 2018, 29, 025301. [CrossRef] 31. Gaetani, P.; Guardone, A.; Persico, G. Shock tube flows past partially opened diaphragms. J. Fluid Mech. 2008, 602, 267–286. [CrossRef] 32. Hougen, J.O.; Martin, O.R.; Walsh, R.A. Dynamics of pneumatic transmission lines. Control Engineering 1963, 10, 114–117. © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY-NC-ND) license (http://creativecommons.org/licenses/by-nc-nd/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Turbomachinery, Propulsion and Power Multidisciplinary Digital Publishing Institute

Technology Development of Fast-Response Aerodynamic Pressure Probes

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International Journal of Turbomachinery Propulsion and Power Article Technology Development of Fast-Response Aerodynamic Pressure Probes Paolo Gaetani and Giacomo Persico * Laboratorio di Fluidodinamica delle Macchine, Dipartimento di Energia, Politecnico di Milano Via Lambruschini 4, I-20158 Milano, Italy; paolo.gaetani@polimi.it * Correspondence: giacomo.persico@polimi.it; Tel.: +39-022-399-8605 Received: 11 February 2020; Accepted: 29 March 2020; Published: 12 April 2020 Abstract: This paper presents and discusses the recent developments on the Fast-Response Aerodynamic Pressure Probe (FRAPP) technology at the Laboratorio di Fluidodinamica delle Macchine (LFM) of the Politecnico di Milano. First, the di erent geometries developed and tested at LFM are presented and critically discussed: the paper refers to single-sensor or two-sensor probes applied as virtual 2D or 3D probes for phase-resolved measurements. The static calibration of the sensors inserted inside the head of the probes is discussed, also taking into account for the temperature field of application: in this context, a novel calibration procedure is discussed and the new manufacturing process is presented. The dynamic calibration is reconsidered in view of the 15-years’ experience, including the extension to probes operating at di erent temperature and pressure levels with respect to calibration. As for the probe aerodynamics, the calibration coecients are discussed and the most reliable set here is evidenced. A novel procedure for the quantification of the measurement uncertainty, recently developed and based on the Montecarlo methodology, is introduced and discussed in the paper. Keywords: FRAPP; pressure sensor; temperature correction; transfer function; aerodynamics; uncertainty quantification 1. Introduction Measuring the unsteady flow downstream of turbomachinery rotors evolved from being a ‘niche’ research activity in the nineties to becoming a common practice the scientific studies of present-day turbomachinery [1–5], with also relevant examples of industrial applications [6–8]. Such evolution was supported by the technical development of instrumentation technology, of novel data-reduction methods, and by the practical experience of the experimentalists. A key contribution to this development came from one specific measurement technology, i.e., the Fast Response Aerodynamic Pressure Probe (FRAPP), which has undoubted advantages with respect to other intrusive or non-intrusive techniques in terms of rigidity, reliability, promptness. Last but not least, it provides total and static pressure measurements, which can be used for the evaluation of the blade-row and stage performance. Thanks to the very high temporal resolution, these probes also allowed to investigate experimentally the complex flow phenomena connected to unsteady blade row interaction [9–13]. The FRAPP concept comes from the combination of fast-response pressure transducers, typically of the piezoresistive kind, with aerodynamic directional pressure probes. The transducers can be flush-mounted on the probe head [14], enhancing the frequency response despite fragility; in other examples [6,15–17], the researchers preferred to embed the transducers within the probe head to enhance the probe strength. Excellent reviews on the early stages of FRAPP development can be found in [18–20]. However, since then, further relevant improvements were made on the basic technology and some of them are reported in the very recent review proposed in [21]. Int. J. Turbomach. Propuls. Power 2020, 5, 6; doi:10.3390/ijtpp5020006 www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2020, 5, 6 2 of 16 A specific version of FRAPP technology has been the object of research and development at Politecnico di Milano since 1998. The probe concept, that is alternative to those presented by the authors listed above, was first proposed in [22], and further elaborated in [23]. With the aim of minimizing the probe blockage while maximizing the instrumentation reliability, an optimal configuration was identified by using single- or two-sensor probes operated as virtual three- of four-sensor probes, and by adopting commercial transducers that only need to be mounted and glued within the probe head. This design implies the adoption of a relatively large line-cavity system connecting the pressure tap on the probe head to the sensor, strongly influencing the probe dynamic response. However, dedicated computational studies and the set-up of a novel dynamic calibration facility [24] has ultimately led to the development of FRAPPs featuring a dynamic response of the order of 100 kHz after digital compensation with the experimental transfer function. This paper proposes a review of the most relevant advances in FRAPP technology conceived and applied at Politecnico di Milano in the last decade, in terms of high-temperature applications, dynamic identification and uncertainty quantification, analyzing systematically probes for two-dimensional and three-dimensional measurements. 2. FRAPP Description Before discussing technological aspects and metrological issues, this section proposes a review of the current probes’ shape configurations and their implication on the measurement capability. Two probes were considered, one cylindrical for two-dimensional flow measurements and one spherical for three-dimensional flow measurements. Even though the probes share several technical features, they are discussed separately in the following section. 2.1. Cylindrical Probe The cylindrical shape makes the probe inherently suitable for measuring the flow direction in a plane normal to the cylinder axis (called the “yaw” angle in this paper), along with the total and static pressure. Conversely, it has a low sensitivity to the flow components parallel to the axis and hence, to the related flow angle, called the “pitch” in this paper, so that it can be considered insensitive for pitch angle values within10 , as resulted from a dedicated test campaign. As the probe embeds a single sensor for miniaturization, it needs three pressure readings measured at di erent rotations around the probe axis to virtually simulate the operation of a three-sensor probe. This prevents real-time unsteady measurements to be performed; however, unless operation instabilities are of concern, in case of unsteady turbomachinery flows, this is not a severe limitation as one is typically interested in the periodic component of the flow unsteadiness. The unsteady periodic component can be extracted by means of ensemble averages locked on the rotor wheel, using a key-phasor signal. The virtual operation prevents from achieving direct turbulence measurements, even if an estimate of the turbulence intensity is possible considering the signal acquired by the probe at the angular position aligned to the phase-averaged flow direction when the unresolved flow angle fluctuations are suciently low (9 ), as discussed in [25]. As well visible in Figure 1, which reports a drawing of the cylindrical FRAPP, the probe size and shape are determined by the sensor characteristics. The cylindrical shape was selected to miniaturize as much as possible the probe, thus reducing the probe blockage. The probe is, in practice, manufactured around one of the smallest transducers commercially available, which simply needs to be inserted within the cylinder and glued with a proper material. In this way, the size of the probe can be minimized down to about 2 mm. The smooth shape of the probe, combined with its miniaturization, guarantees optimal probe aerodynamics in terms of dynamic errors, according to dedicated studies performed at the early stages of FRAPP technology development [26]. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 3 of 16 As well visible in Figure 1, which reports a drawing of the cylindrical FRAPP, the probe size and shape are determined by the sensor characteristics. The cylindrical shape was selected to miniaturize as much as possible the probe, thus reducing the probe blockage. The probe is, in practice, manufactured around one of the smallest transducers commercially available, which simply needs to be inserted within the cylinder and glued with a proper material. In this way, the size of the probe can be minimized down to about 2 mm. The smooth shape of the probe, combined with its Int. J. Turbomach. Propuls. Power 2020, 5, 6 3 of 16 miniaturization, guarantees optimal probe aerodynamics in terms of dynamic errors, according to dedicated studies performed at the early stages of FRAPP technology development [26]. Figure 1. Representation of the cylindrical Fast-Response Aerodynamic Pressure Probe (FRAPP). Figure 1. Representation of the cylindrical Fast-Response Aerodynamic Pressure Probe (FRAPP). The probe design concept allows to deal with a relatively high temperature. Present-day The probe design concept allows to deal with a relatively high temperature. Present-day piezoelectric transducers can operate up to about 550 K and epoxy resins are commercially available piezoelectric transducers can operate up to about 550 K and epoxy resins are commercially available for temperatures greater than 600 K. The combination of these two elements allows manufacturing for temperatures greater than 600 K. The combination of these two elements allows manufacturing high-temperature FRAPP in a straightforward way and without external cooling—a topic that, however, high-temperature FRAPP in a straightforward way and without external cooling—a topic that, has been the object of dedicated investigations [27] that proved to be successful at the cost of an increase however, has been the object of dedicated investigations [27] that proved to be successful at the cost of probe blockage. of an increase of probe blockage. A further specific aspect of the present design is related to the installation of the sensor inside the A further specific aspect of the present design is related to the installation of the sensor inside probe head and to the subsequent line-cavity system connecting the pressure tap on the probe head to the probe head and to the subsequent line-cavity system connecting the pressure tap on the probe the sensor. The topic is discussed in detail in [23], in which several analytical and numerical techniques head to the sensor. The topic is discussed in detail in [23], in which several analytical and numerical are applied and compared to optimize the shape of the internal cavities in order to maximize the probe techniques are applied and compared to optimize the shape of the internal cavities in order to promptness. By virtue of this study, the promptness of the FRAPP resulted of about 80 kHz. Such value maximize the probe promptness. By virtue of this study, the promptness of the FRAPP resulted of was—and still is—suciently high to match the specifications of all the FRAPP applications considered about 80 kHz. Such value was—and still is—sufficiently high to match the specifications of all the by the authors in their experience. FRAPP applications considered by the authors in their experience. 2.2. Spherical Probe 2.2 Spherical Probe To overcome limitations in measurement capability of cylindrical probes, an alternative To overcome limitations in measurement capability of cylindrical probes, an alternative configuration suitable for unsteady 3D measurements was developed at Politecnico di Milano, configuration suitable for unsteady 3D measurements was developed at Politecnico di Milano, featuring a spherical head shape and named sFRAPP. featuring a spherical head shape and named sFRAPP. In order to enhance the sensitivity to the flow components parallel to the probe axis, the probe In order to enhance the sensitivity to the flow components parallel to the probe axis, the probe head features a spherical shape with two pressure taps. The combination of the geometrical constraints head features a spherical shape with two pressure taps. The combination of the geometrical imposed by the transducers as well as by the miniaturization of external and internal dimensions led constraints imposed by the transducers as well as by the miniaturization of external and internal to a head diameter of 3.8 mm and to a diameter of 0.3 mm for the two pneumatic lines feeding the dimensions led to a head diameter of 3.8 mm and to a diameter of 0.3 mm for the two pneumatic lines cavities facing the sensors. feeding the cavities facing the sensors. One pressure tap was drilled on the probe tip, with an inclination of 60 with respect to the probe One pressure tap was drilled on the probe tip, with an inclination of 60° with respect to the probe axis, and it was employed only for measurement of the pitch angle. The second pressure tap was axis, and it was employed only for measurement of the pitch angle. The second pressure tap was drilled on the equatorial plane with an inclination of 90 with respect to the probe axis: it can be drilled on the equatorial plane with an inclination of 90° with respect to the probe axis: it can be aligned to the ‘pitch’ tap as shown in Figure 2 or it can be rotated with respect to that by 180 around aligned to the ‘pitch’ tap as shown in Figure 2 or it can be rotated with respect to that by 180° around the probe stem. the probe stem. Int. J. Turbomach. Propuls. Power 2020, 5, 6 4 of 16 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 4 of 16 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 4 of 16 Figure 2. Representation of the spherical FRAPP with aligned taps. Figure 2. Representation of the spherical FRAPP with aligned taps. A preliminary analysis of the sFRAPP is reported in [28], and its first application downstream A preliminary analysis of the sFRAPP is reported in [28], and its first application downstream Figure 2. Representation of the spherical FRAPP with aligned taps. of a turbine stage is documented in [29]. The probe operating mechanism is still based on multiple of a turbine stage is documented in [29]. The probe operating mechanism is still based on multiple pressure readings taken at different rotations of the probe around its own stem: the combined use of pressure readings taken at di erent rotations of the probe around its own stem: the combined use of A preliminary analysis of the sFRAPP is reported in [28], and its first application downstream four pressure readings allows to measure both the flow directions, alongside total and static pressure. four pressure readings allows to measure both the flow directions, alongside total and static pressure. of a turbine stage is documented in [29]. The probe operating mechanism is still based on multiple The configuration shown in Figure 2 allows reconstructing both the flow directions with just 3 The configuration pressure read shown ings taken a in Figur t diff eerent rotati 2 allows reconstr ons of the probe aroun ucting both the d its own stem: the combined flow directions with just 3 use of rotations, rotations, while 4 rotations are required for the configuration with 2 opposed taps (a further rotation four pressure readings allows to measure both the flow directions, alongside total and static pressure. while 4 rotations are required for the configuration with 2 opposed taps (a further rotation shifted by shifted by 180° with respect to the central one of the others). However, the use of opposed taps allows The configuration shown in Figure 2 allows reconstructing both the flow directions with just 3 180 with respect to the central one of the others). However, the use of opposed taps allows reducing reducing the dimension of the internal cavities as, in this latter case, a shorter line connects the ‘pitch’ rotations, while 4 rotations are required for the configuration with 2 opposed taps (a further rotation the dimension tap to the sensor. Theoreti of the internal cacavities l estimates as, and expe in this rimental dynam latter case, a shorter ic calibrat line ions showed a connects the red‘pitch’ uction tap shifted by 180° with respect to the central one of the others). However, the use of opposed taps allows of promptness to 40 kHz, about half of the promptness of the sFRAPP with opposite taps, but still to the sensor. Theoretical estimates and experimental dynamic calibrations showed a reduction of reducing the dimension of the internal cavities as, in this latter case, a shorter line connects the ‘pitch’ very high and fully suitable for typical turbomachinery applications. promptness to 40 kHz, about half of the promptness of the sFRAPP with opposite taps, but still very tap to the sensor. Theoretical estimates and experimental dynamic calibrations showed a reduction high and fully suitable for typical turbomachinery applications. of promptness to 40 kHz, about half of the promptness of the sFRAPP with opposite taps, but still 3. Thermally Corrected Calibration of the Pressure Sensor very high and fully suitable for typical turbomachinery applications. 3. Thermally Corrected Calibration of the Pressure Sensor Due to the sensor sensitivity to the temperature, a sensor calibration in pressure and temperature is required before performing the aerodynamic and the dynamic ones. The sensor sensitivity to 3. Thermally Corrected Calibration of the Pressure Sensor Due to the sensor sensitivity to the temperature, a sensor calibration in pressure and temperature temperature is measured by applying an additional resistance (“sense resistor” in the following) on is required before performing the aerodynamic and the dynamic ones. The sensor sensitivity to Due to the sensor sensitivity to the temperature, a sensor calibration in pressure and temperature the bridge, as showed in Figure 3. The voltage drop across the sense resistor (ΔVT) is mainly a function temperatur is required e is measur before ed perfor by applying ming the an ae addi rodynam tional ic ran esistance d the dy (“se nam nse ic one resistor” s. The in sensor the following) sensitivity to on the of the current flowing across the bridge, which depends on the bridge temperature. Thus, by reading temperature is measured by applying an additional resistance (“sense resistor” in the following) on bridge, as showed in Figure 3. The voltage drop across the sense resistor (DV ) is mainly a function of the voltage difference across the bridge (ΔVP) and ΔVT, the sensor behavior can b T e fully documented. the bridge, as showed in Figure 3. The voltage drop across the sense resistor (ΔVT) is mainly a function the current flowing across the bridge, which depends on the bridge temperature. Thus, by reading the ΔV of the current flowing across the bridge, which depends on the brid T ge temperature. Thus, by reading voltage di erence across the bridge (DV ) and DV , the sensor behavior can be fully documented. P T the voltage difference across the bridge (ΔVP) and ΔVT, the sensor behavior can be fully documented. sense resistor ΔV Whaeastone ΔV sense resistor supply bridge Whaeastone ΔV supply bridge ΔV Figure 3. Electrical scheme for the pressure and temperature calibration. ΔV Figure 3. Electrical scheme for the pressure and temperature calibration. Figure 3. Electrical scheme for the pressure and temperature calibration. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 5 of 16 Int. J. Turbomach. Propuls. Power 2020, 5, 6 5 of 16 The probe temperature was set by inserting it in an oven: the insertion length was chosen to be representative of what is needed in the real application in order to minimize the effect of different The probe temperature was set by inserting it in an oven: the insertion length was chosen to be thermal conductions along the stem. representative of what is needed in the real application in order to minimize the e ect of di erent The calibration procedure was applied as follow: first, the oven temperature (Ti) was set and a thermal conductions along the stem. consistent waiting time (typically 30 min) was scheduled to bring the probe to a steady thermal The calibration procedure was applied as follow: first, the oven temperature (Ti) was set and condition. Then, a pressure ramp was applied up to the calibration range foreseen for the tests. To a consistent waiting time (typically 30 min) was scheduled to bring the probe to a steady thermal include possible hysteretic behavior in the calibration coefficients and uncertainty, the pressure ramp condition. Then, a pressure ramp was applied up to the calibration range foreseen for the tests. had both positive and negative slopes. Then, the oven temperature was modified and the procedure To include possible hysteretic behavior in the calibration coecients and uncertainty, the pressure ramp repeated: also for the temperature ramp, positive and negative slopes could be applied. had both positive and negative slopes. Then, the oven temperature was modified and the procedure For each temperature level (Ti), the pressure data were then fit by a linear function: the results repeated: also for the temperature ramp, positive and negative slopes could be applied. show the slope (KP,i), intercept (QP,i) and uncertainty (UP,i). For each temperature level (Ti), the pressure data were then fit by a linear function: the results KP,i, QP,i, and UP,i were then fit by polynomial functions (typically linear or parabolic, depending show the slope (K i), intercept (Q i) and uncertainty (U i). P, P, P, on the trend), to find the KP and QP (as function of the ΔVT). The sensor temperature Ti and ΔVT,i K i, Q i, and U i were then fit by polynomial functions (typically linear or parabolic, depending P, P, P, could also be fitted to have KT, QT. Figure 4 shows typical calibration results. on the trend), to find the K and Q (as function of the DV ). The sensor temperature Ti and DV i P P T T, could also be fitted to have K , Q . Figure 4 shows typical calibration results. T T Figure 4. Calibration coecients: top-left: pressure vs. DV at given temperature; top-right: sensor Figure 4. Calibration coefficients: top-left: pressure vs. ΔVP at given temperature; top-right: sensor Temperature vs. DV ; bottom-left: slope Kp vs. DV ; bottom-right: intercept Q vs. DV T T P T. Temperature vs. ΔVT; bottom-left: slope Kp vs ΔVT; bottom-right: intercept QP vs ∆VT. The procedure is accurate and its only critical point is a random o set on Q i due to the T, The procedure is accurate and its only critical point is a random offset on QT,i due to the sensor sensor thermal sensitivity while the K i is perfectly repeatable: this occurrence, whose magnitude T, thermal sensitivity while the KT,i is perfectly repeatable: this occurrence, whose magnitude unfortunately depends on the single transducer, requires an online check to measure it during tests. unfortunately depends on the single transducer, requires an online check to measure it during tests. Once K , Q K , and Q are found, during the probe application, the sensor pressure (P) and T T, P P Once KT, QT, KP, and QP are found, during the probe application, the sensor pressure (P) and temperature (T) can be calculated by temperature (T) can be calculated by P = K  DV + Q (1 (1) ) P = KP × ∆VP + QP P P P T = KT × ∆VT + QT (2) T = K  DV + Q (2) T T T Int. J. Turbomach. Propuls. Power 2020, 5, 6 6 of 16 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 6 of 16 Uncertainty Quantification 3.1. Uncertainty Quantification The uncertainty evaluation was made by considering the uncertainty on each measured point in pressure for a given oven (probe) temperature and the contribution due to the least square interpolation The uncertainty evaluation was made by considering the uncertainty on each measured point in among the pairs (Pi, DV i). The contribution on the single measured point takes into account the pressure for a given oven (probe) temperature and the contribution due to the least square P, standard deviation of the population, the reference transducer uncertainty, and the data acquisition interpolation among the pairs (Pi, ∆VP,i). The contribution on the single measured point takes into board analog-digital converter specifications. Gaussian distribution was considered for the first two account the standard deviation of the population, the reference transducer uncertainty, and the data quantities while a rectangular distribution was applied to the AD converter. All these contributions are acquisition board analog-digital converter specifications. Gaussian distribution was considered for considered not cross correlated, are made homogeneous in terms of units, and are considered to include the first two quantities while a rectangular distribution was applied to the AD converter. All these 95% of the Gaussian distribution data: finally, given the functional dependence f = f(x1, x2, ::: , xn), contributions are considered not cross correlated, are made homogeneous in terms of units, and are the uncertainty propagation approach is considered: considered to include 95% of the Gaussian distribution data: finally, given the functional dependence f = f(x1, x2, …, xn), the uncertainty propagation approach is considered: n 2 @ f U = u (3) @x i=1 (3) To couple the uncertainty on the single calibration point and the one due to the least square interpolation, To couplethe the uncertai highest among nty on the si the single ngle ca points libra one tio is n poi consider nt an ed. d the one due to the l Overall, for a 6-baraetransducer ast square , int the erpolat following ion, t rh esults e highe wer st a e m found ong t(Figur he sine gle po 5): no int clear s one is trends cons ar idere e visible d. Over and aldata l, for a less 6-bar than a tra 0.1% nsducer, of the tfull he follow range.ing These resurlt esults s were arfound e then (Fi applied gure 5): no c duringle the ar taer rends ar odynamic e visicalibration ble and data to le get ss tan han estimation 0.1% of th of e full range. These results are then applied during the aerodynamic calibration to get an estimation of the flow field detection uncertainty. the flow field detection uncertainty. Figure 5. Extended uncertainty for the di erent probe temperature. Figure 5. Extended uncertainty for the different probe temperature. With the aim of comparing classical uncertainty analysis with an alternative systematic approach, With the aim of comparing classical uncertainty analysis with an alternative systematic the Montecarlo methodology was also applied and the same contributions to the uncertainty were approach, the Montecarlo methodology was also applied and the same contributions to the considered. The data for each temperature level (Ti) were interpolated by a least square method by uncertainty were considered. The data for each temperature level (Ti) were interpolated by a least introducing N (a number high enough to get a statistical reliability) di erent pairs (P, DV ) chosen square method by introducing N (a number high enough to get a statistical reliability) different pairs randomly into the populations characterized by the selected distributions. The results of the N (P, ∆VP) chosen randomly into the populations characterized by the selected distributions. The results calculations were N line constants and intercepts, statistically treated to obtain mean values (K , Q ) Pi Pi of the N calculations were N line constants and intercepts, statistically treated to obtain mean values and their standard deviations. As a following step, M (a number high enough to get a statistical (KPi, QPi) and their standard deviations. As a following step, M (a number high enough to get a reliability) pairs of data belonging to the population (K , DV ) were randomly chosen according to Pi Ti statistical reliability) pairs of data belonging to the population (KPi, ∆VTi) were randomly chosen a Gaussian distribution, then averaged to get K = K (DV ) and its standard deviation; the same according to a Gaussian distribution, then averaged to get K P P T P = KP (∆VT) and its standard deviation; methodology was applied to the intercept. In this way, P = P (DV , DV ), and its standard deviation P T the same methodology was applied to the intercept. In this way, P = P (∆VP, ∆VT), and its standard were available for the application in the aerodynamic calibration. deviation were available for the application in the aerodynamic calibration. To get a proper accuracy, the Montecarlo procedure requires a huge number of iterations: to make To get a proper accuracy, the Montecarlo procedure requires a huge number of iterations: to the procedure a ordable, the Latin hypercube methodology was applied to support pairs choice, and a make the procedure affordable, the Latin hypercube methodology was applied to support pairs convergence criterion was also set on the standard deviation value. The results received by the two choice, and a convergence criterion was also set on the standard deviation value. The results received methodologies show a good consistency and were the basis for the determination of the uncertainty in by the two methodologies show a good consistency and were the basis for the determination of the the aerodynamic calibration. Figure 6 shows the results of the uncertainty calculation when reported uncertainty in the aerodynamic calibration. Figure 6 shows the results of the uncertainty calculation when reported on the same chart of the probe pressure for a Mach=0.5 test: the average uncertainty covering 95% of the samples is about 5 mbar (less than 0.1% of the transducer range), that is the same order of magnitude as the previous methodology. Int. J. Turbomach. Propuls. Power 2020, 5, 6 7 of 16 on the same chart of the probe pressure for a Mach = 0.5 test: the average uncertainty covering 95% of the samples is about 5 mbar (less than 0.1% of the transducer range), that is the same order of magnitude as the previous methodology. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 7 of 16 Figure 6. Uncertainty range for a test with a Mach number equal to 0.5. Figure 6. Uncertainty range for a test with a Mach number equal to 0.5. 4. FRAPP Dynamic Analysis 4. FRAPP Dynamic Analysis The manufacturing concept of the FRAPP developed at PoliMi implies a promptness reduction The manufacturing concept of the FRAPP developed at PoliMi implies a promptness reduction with respect to that of the sensor installed within the probe due to the line-cavity system between the with respect to that of the sensor installed within the probe due to the line-cavity system between the pressure tap and the sensor ’s sensitive element. However, a proper design of the internal cavities pressure tap and the sensor’s sensitive element. However, a proper design of the internal cavities allows to obtain a promptness of 80 kHz, as shown in [23]. allows to obtain a promptness of 80 kHz, as shown in [23]. To achieve such a promptness, the probe transfer function has to be experimentally determined To achieve such a promptness, the probe transfer function has to be experimentally determined and then applied in the data processing of measurements in test-rigs. However, the use of the transfer and then applied in the data processing of measurements in test-rigs. However, the use of the transfer function for experiments downstream of turbomachinery rotors poses specific problems due to the function for experiments downstream of turbomachinery rotors poses specific problems due to the inherent complexity of dynamic calibration tests since the operative conditions are often di erent inherent complexity of dynamic calibration tests since the operative conditions are often different between calibration and tests. These aspects are discussed in detail in the following. between calibration and tests. These aspects are discussed in detail in the following. 4.1. Time- and Frequency-Domain Identification 4.1. Time- and Frequency-Domain Identification The dynamic calibration of fast-response pressure instrumentation poses, at first, technical issues related The dynam to the gener ica calibr tion ofation inputof fast signals-respo featuring nse pres unsteady sure instrument perturbations ation poses, at suciently at first, high fr technical equency. Siren disks [30] are used to generate periodic stimulus signals while shock tubes [6,24], are used to issues related to the generation of input signals featuring unsteady perturbations at sufficiently high generate frequency. a transient Siren disks non-periodic [30] are use signal, d to ge as nerate pe the travelling riodic stim shock ulus wave sign pr als while oduces a shock tube step signals in [6, 24], very good approximation. are used to generate a transient non-periodic signal, as the travelling shock wave produces a step signal Shock in very tubes good are approx preferred imat as, ion in .a single test, the dynamic response of the probe can be achieved for the full frequency range typically of interest for turbomachinery applications (100 kHz). Considering air Shock tubes are preferred as, in a single test, the dynamic response of the probe can be achieved at for t ambient he full conditions, frequency r thea dynamic nge typic content ally of ofint a traveling erest for t shock urbom involves achinery fr appl equencies ications up ( to1the 00 or kHz der). of Consider MHz. ing By air setting at ambient conditions, the dynam proper diaphragm features, ishock c content amplitude of a traveling can be shock i selected nvolves fr in theeq range uencies of typical up to the order of MHz. By setti pressure fluctuations in turbo ng proper dia machineryp(in hrag the m fe order ature ofstenths , shock amp of bars litor ude c even an be less as selauthors ected in documented the range of tin ypical pr [7,8,11,essure fluct 12]; such perturbations uations in turbom are normally achinery (in th small e order enough of ten to not ths of bar activatesr or elevant even non-linear less as auteh oects rs docum in the dynamic ented in evolution [7, 8, 11, 1 of 2]; the such pressur perturbations are normally small enough e field within the line-cavity system. By virtue to not of act such ivatephysical relevant linearity non-lin , ea the r ef transfer fects in function the dyn determined amic evolut thr ion ough of th the e pres step-r sure esponse field w is i applicable thin the lifor ne- the cavity measur system. By ement virtu of theeperiodic of such ph fluctuations ysical linear occurring ity, the transfer within funct turbomachinery ion determ. ined through the step- response Since the is applic beginning able f ofo this r th resear e meas ch 20 urement years of ago, continuous the periodic impr fluct ovements uationshave occurr been ing made with onin the turbomachin low-pressur ery. e shock tube presented in [24], in particular on the bursting diaphragm; by using the present-day Since the beginning of this research 20 years plastic materials, shock strengths of theag oro der , continuo of 0.2–0.3 us impro bars wer vements have e obtained. They been made o exhibited n incomplete the low-pres opening sure shoc whose k tubee presented ects were in [24], investigated in particu in ldetail ar on tin he [b 31 u]; rst such ing di eap ects hragm can; b be y u pr si operly ng the present-day plastic materials, shock strengths of the order of 0.2-0.3 bars were obtained. They exhibited incomplete opening whose effects were investigated in detail in [31]; such effects can be properly handled in order to minimize their impact on the determination of the probe transfer function. Figure 7a presents a typical experimental transfer function obtained with the method proposed in [24]. The experimental trend recalls closely the one of a second-order linear system, with an evident peak at about 35 kHz representing the probe line-cavity system resonance. On the basis of Int. J. Turbomach. Propuls. Power 2020, 5, 6 8 of 16 handled Int. J. Turbo in mach. order Propuls. Power to minimize 2020, 5their , x FOR PE impact ER REon VIEW the determination of the probe transfer function. 8 of 16 Figure 7a presents a typical experimental transfer function obtained with the method proposed in [24]. the frequency and amplitude at resonance, the system identification can be done: the corresponding The experimental trend recalls closely the one of a second-order linear system, with an evident peak at linear system is also plotted in comparison to the experimental one. Differences exist but occur at a about 35 kHz representing the probe line-cavity system resonance. On the basis of the frequency and high frequency (above 40 kHz). To provide a more intuitive idea of the observed non-linearity, the amplitude at resonance, the system identification can be done: the corresponding linear system is also measured step response and the analytical response are plotted in Figure 7b. The experimental trend plotted in comparison to the experimental one. Di erences exist but occur at a high frequency (above reproduces well the one of the analytical model, suggesting that the modelling is reliable for the 40 kHz). To provide a more intuitive idea of the observed non-linearity, the measured step response whole response. The largest difference is concentrated in the first overshoot in which the experiment and the analytical response are plotted in Figure 7b. The experimental trend reproduces well the one exhibits a steeper pressure rise and a higher peak. The faster pressure rise at the beginning of the of the analytical model, suggesting that the modelling is reliable for the whole response. The largest process is clearly responsible for the non-linearity observed in the frequency domain beyond 40 kHz. di erence is concentrated in the first overshoot in which the experiment exhibits a steeper pressure rise Since full linearity is not guaranteed a priori, experimental dynamic calibration is crucial to determine and a higher peak. The faster pressure rise at the beginning of the process is clearly responsible for the the transfer function of each FRAPP manufactured in order to properly compensate the measured non-linearity observed in the frequency domain beyond 40 kHz. Since full linearity is not guaranteed a signals. priori, experimental dynamic calibration is crucial to determine the transfer function of each FRAPP manufactured in order to properly compensate the measured signals. 4.2. Pressure and Temperature Correction (a) (b) Figure 7. Frequency-domain (a) and time-domain (b) identification of a typical FRAPP. Figure 7. Frequency-domain (a) and time-domain (b) identification of a typical FRAPP. 4.2. Pressure and Temperature Correction Thanks to the good linearity exhibited by the FRAPP, the use of the transfer function to Thanks to the good linearity exhibited by the FRAPP, the use of the transfer function to dynamically dynamically compensate the pressure signals measured in turbomachinery test rigs is possible, at compensate the pressure signals measured in turbomachinery test rigs is possible, at this stage. this stage. Notwithstanding that the thermodynamic conditions of the fluid in test rigs are often Notwithstanding that the thermodynamic conditions of the fluid in test rigs are often di erent from different from those occurring in the shock tube and considering that, in general, it is not possible to those occurring in the shock tube and considering that, in general, it is not possible toreproduce such reproduce such conditions in the shock tube facility, relatively simple techniques can be proposed for conditions in the shock tube facility, relatively simple techniques can be proposed for correcting the correcting the transfer function identified in the dynamic calibration experiments. In [23], the transfer function identified in the dynamic calibration experiments. In [23], the analytical model of [32] analytical model of [32] was found to reproduce fairly well the resonance frequency of the FRAPP. was found to reproduce fairly well the resonance frequency of the FRAPP. This model, as well as the This model, as well as the other ones available in literature, show that, apart from geometrical terms, other ones available in literature, show that, apart from geometrical terms, the natural frequency and the natural frequency and the non-dimensional damping of the line-cavity system exhibit the the non-dimensional damping of the line-cavity system exhibit the following dependencies: following dependencies: ! / c / T (4) N (4) ∝ ∝ √ (5) ∝ ∝ () / / (T) (5) c P Expression 4, which relates directly the natural frequency to the sound speed, is intuitively justified as the dynamic response of the line-cavity system depends on the pressure wave propagation within the probe internal cavities. In the context of perfect gases, this property provides a first straightforward correction to the transfer function for temperature differences between Int. J. Turbomach. Propuls. Power 2020, 5, 6 9 of 16 Expression 4, which relates directly the natural frequency to the sound speed, is intuitively justified as the dynamic response of the line-cavity system depends on the pressure wave propagation within the probe internal cavities. In the context of perfect gases, this property provides a first straightforward correction to the transfer function for temperature di erences between calibration and application. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 9 of 16 Expression 5 indicates that both temperature and pressure levels have an impact on the damping and, calibration and application. Expression 5 indicates that both temperature and pressure levels have an once again, it provides a tool for correcting the transfer function identified with experiments in the impact on the damping and, once again, it provides a tool for correcting the transfer function shock tube; also, the pressure level can have an e ect and demands for corrections, even though its identified with experiments in the shock tube; also, the pressure level can have an effect and demands e ect is quantitatively lower than that of the temperature. for corrections, even though its effect is quantitatively lower than that of the temperature. Figure 8 shows the impact of combined temperature and pressure correction on the amplitude of Figure 8 shows the impact of combined temperature and pressure correction on the amplitude the transfer function determined in the shock tube (TF-EXP). For this probe, a nearly perfect dynamic of the transfer function determined in the shock tube (TF-EXP). For this probe, a nearly perfect linearity is observed up to 60 kHz, with natural frequency at about 40 kHz, as shown by the comparison dynamic linearity is observed up to 60 kHz, with natural frequency at about 40 kHz, as shown by the with the analytical second order transfer function (TF-TEO_CAL). By considering an application at the comparison with the analytical second order transfer function (TF-TEO_CAL). By considering an maximum temperature level technically available for the FRAPP (550 K), a correction to the analytical application at the maximum temperature level technically available for the FRAPP (550 K), a transfer function was applied to get the analytical transfer function in the real environment of the test correction to the analytical transfer function was applied to get the analytical transfer function in the rig (TF-TEO_RIG). The impact of the corrections is negligible up to 20 kHz, while a relevant deviation real environment of the test rig (TF-TEO_RIG). The impact of the corrections is negligible up to 20 occurs for a frequency higher than 40 kHz. kHz, while a relevant deviation occurs for a frequency higher than 40 kHz. Figure 8. Impact of temperature of tests with respect to calibration on FRAPP transfer function. Figure 8. Impact of temperature of tests with respect to calibration on FRAPP transfer function. An application at high pressure and high temperature (not reported extensively for confidentiality An application at high pressure and high temperature (not reported extensively for reasons) showed that the linearized temperature-pressure correction allowed properly compensating confidentiality reasons) showed that the linearized temperature-pressure correction allowed the pressure signals measured by the probe, thus recovering the typical power spectra of unsteady properly compensating the pressure signals measured by the probe, thus recovering the typical pressure measurements in high-speed flows. power spectra of unsteady pressure measurements in high-speed flows. 5. FRAPP Aerodynamics 5. FRAPP Aerodynamics The aerodynamic calibration was performed on a convergent nozzle whose outlet section The aerodynamic calibration was performed on a convergent nozzle whose outlet section is 50 is 50 mm 60 mm and typically allows for neglecting the blockage e ects up to Mach = 0.95. mm x 60 mm and typically allows for neglecting the blockage effects up to Mach = 0.95. Convergent Convergent–divergent nozzles are also available when supersonic calibrations are required. –divergent nozzles are also available when supersonic calibrations are required. The Reynolds–Mach The Reynolds–Mach number e ects decoupling can be also achieved by a nozzle inserted in a number effects decoupling can be also achieved by a nozzle inserted in a duct brought to a chocked duct brought to a chocked condition by a downstream throat. The outlet pressure is usually set to be condition by a downstream throat. The outlet pressure is usually set to be atmospheric and the Mach atmospheric and the Mach number is set by imposing the total pressure in the upstream reservoir. number is set by imposing the total pressure in the upstream reservoir. When the aerodynamic calibration is of concern, di erent calibration coecients can be taken into When the aerodynamic calibration is of concern, different calibration coefficients can be taken account. The advantages of di erent coecient sets may arise from a pure aerodynamic behavior or into account. The advantages of different coefficient sets may arise from a pure aerodynamic behavior from the uncertainty point of view. or from the uncertainty point of view. 5.1. Cylindrical FRAPP 5.1. Cylindrical FRAPP For the 2D Frapp probe, the following is commonly applied: For the 2D Frapp probe, the following is commonly applied: P P P P P (P + P )/2 S R L L R T C − − −( + )/2 KYaw = KP = KP = (6) T S P P P P P P (6) = T = T = T S S S − − − where PL = probe Left pressure reading, PR = probe Right pressure reading, PC = probe Central pressure reading, PT = nozzle Total pressure, and PS = nozzle Static pressure. The experimental trends of the coefficients are reported in Figure 9: when the Kyaw coefficient is zero, the aerodynamic reference direction is set. The range of monotonic trend is typically up to +/- 23°, due to the 45° of spacing between the three pressure readings (Left, Central, Right) and to the range in between the two flow separation points on a cylinder positioned in cross flow, that is around Int. J. Turbomach. Propuls. Power 2020, 5, 6 10 of 16 where P = probe Left pressure reading, P = probe Right pressure reading, P = probe Central L R C pressure reading, P = nozzle Total pressure, and P = nozzle Static pressure. T S The experimental trends of the coecients are reported in Figure 9: when the Kyaw coecient is zero, the aerodynamic reference direction is set. The range of monotonic trend is typically up to +/ 23 , due to the 45 of spacing between the three pressure readings (Left, Central, Right) and to Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 10 of 16 the range in between the two flow separation points on a cylinder positioned in cross flow, that is around +/ 67  70 . depending on the probe geometry, Reynolds and Mach numbers. The total +/- 67° ÷ 70°. depending on the probe geometry, Reynolds and Mach numbers. The total pressure pressure coecient di ers mainly for the compressibility e ect over the probe cylindrical head, with coefficient differs mainly for the compressibility effect over the probe cylindrical head, with the the Reynolds number having a lower impact in this range. Reynolds number having a lower impact in this range. Figure Figure 9. 9. 2D 2D F FRAPP RAPP calibration calibration coeffi coecients cientsfor for varying varying Mac Mach hnumber number and and yaw yaw angl angle. e. In the application phase, the real Total and Static pressures (and by these the Mach number) and In the application phase, the real Total and Static pressures (and by these the Mach number) and the flow angle are derived by an iterative procedure. First, the static pressure and the total pressure are the flow angle are derived by an iterative procedure. First, the static pressure and the total pressure chosen as the average between the left and right, and the central one, respectively; using these, the are chosen as the average between the left and right, and the central one, respectively; using these, KYaw is a calculated and the yaw angle is derived. Using the yaw angle, the new KPt and KPs are the KYaw is a calculated and the yaw angle is derived. Using the yaw angle, the new KPt and KPs calculated by making use of the calibration curves, properly interpolated in angles and Mach. At this are calculated by making use of the calibration curves, properly interpolated in angles and Mach. At point, the new Total and Static pressures are calculated and the second iteration can start. this point, the new Total and Static pressures are calculated and the second iteration can start. In case of thermal drift, the KYaw is less sensitive than KPt and KPs because the numerator is In case of thermal drift, the KYaw is less sensitive than KPt and KPs because the numerator is almost insensitive to the drift, which typically occurs as an o set. almost insensitive to the drift, which typically occurs as an offset. 5.2. sFRAPP 5.2. sFRAPP For the 3D sFRAPP several sets of coecients were considered and evaluated in calibration. For the 3D sFRAPP several sets of coefficients were considered and evaluated in calibration. Some of them are discussed separately in the following: Some of them are discussed separately in the following: 5.2.1. Set A 5.2.1. Set A: P P P P L R T P − − KYaw = ; KPitch = ; P P P P T S T S = ; ℎ = ; (7) P (P +P )/2 P P T C S R L − − KP = ; KP = T S P P P P T S T S (7) − −( + )/2 where KYaw, KP and KP depend on the yaw tap (and its virtual readings) and on the flow total T S = ; = − − and static pressures, while the KPitch depends on the Pitch tap (P ) and on the flow total and static where KYaw, KPT and KPS depend on the yaw tap (and its virtual readings) and on the flow total and pressures. None of the coecients are defined by mixing the pitch and the yaw taps and all include static pressures, while the KPitch depends on the Pitch tap (PP) and on the flow total and static P and P . The KPt is defined as usually found in the literature for multi-hole probes. KPs typically T s pressures. None of the coefficients are defined by mixing the pitch and the yaw taps and all include refer to the static pressure (P ) and to the average of the lateral holes, whose value is close to the PT and Ps. The KPt is defined as usually found in the literature for multi-hole probes. KPs typically static pressure. In this probe, the lateral holes provide the pressure readings P , P , and P ; however, L R P refer to the static pressure (PS) and to the average of the lateral holes, whose value is close to the static these lateral holes are not symmetrical with respect to the probe head (as, instead, occurs for 5-hole pressure. In this probe, the lateral holes provide the pressure readings PL, PR, and PP; however, these probes of conical/prismatic head shape), and for this reason, the average of the corresponding pressure lateral holes are not symmetrical with respect to the probe head (as, instead, occurs for 5-hole probes readings is always very di erent with respect to the actual static pressure of the flow, making the KPs of conical/prismatic head shape), and for this reason, the average of the corresponding pressure coecient not null in any condition. For this reason, only P and P are considered in the proposed L R readings is always very different with respect to the actual static pressure of the flow, making the KPs definition. As a further technical consideration, the P is measured with a di erent transducer KPs coefficient not null in any condition. For this reason, only PL and PR are considered in the with respect to P and P and thus, retaining P in the KPs definition would make the coecient L R P proposed KPs definition. As a further technical consideration, the PP is measured with a different sensitive to the di erent potential thermal drift of the two transducers. transducer with respect to PL and PR and thus, retaining PP in the KPs definition would make the coefficient sensitive to the different potential thermal drift of the two transducers. As for KYaw and KPitch, they have a definition consistent with multi-hole geometries, with KPitch being defined with only one tap (PT is a constant given the Mach number). As reported in Figure 10, the KPt and KPs coefficients are regular over the grid KYaw-KPitch and for this, they seem easily applicable. However, they exhibit an overlapping zone at the boundary of the matrix that leads to non-unique solutions during the interpolation procedure. Int. J. Turbomach. Propuls. Power 2020, 5, 6 11 of 16 As for KYaw and KPitch, they have a definition consistent with multi-hole geometries, with KPitch being defined with only one tap (P is a constant given the Mach number). As reported in Figure 10, the KPt and KPs coecients are regular over the grid KYaw-KPitch and for this, they seem easily applicable. However, they exhibit an overlapping zone at the boundary of the matrix that leads to non-unique solutions during the interpolation procedure. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 11 of 16 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 11 of 16 Figure 10. KPt and KPs on the KYaw and KPitch coefficient grid for sFRAPP (range ± 22°) f  or Figure 10. KPt and KPs on the KYaw and KPitch coecient grid for sFRAPP (range  22 ) for Figure 10. KPt and KPs on the KYaw and KPitch coefficient grid for sFRAPP (range ± 22°) for aerodynamic coefficients defined according to set A. aerodynamic coecients defined according to set A. aerodynamic coefficients defined according to set A. 5. 5.2.2. 2.2. Set Set BB : 5.2.2. Set B: P P L R It It diff di ers ersfro from m Set Set A on A only ly for: for: K =Y aw = , where , where P Pmax is the maximum value is the maximum valueof the of the max P (P +P )/2 max ( L R )/ It differs from Set A only for: = , where Pmax is the maximum value of the ( )/ parabola passing by the three points P , P , P : it is an artificial value because the pressure curve C L R parabola passing by the three points PC, PL, PR: it is an artificial value because the pressure curve parabola passing by the three points PC, PL, PR: it is an artificial value because the pressure curve around a cylinder is not a parabola, although it is similar to one. This new coecient does not su er around a cylinder is not a parabola, although it is similar to one. This new coefficient does not suffer around a cylinder is not a parabola, although it is similar to one. This new coefficient does not suffer from o set errors, being related to one transducer only and including di erences both at numerator from offset errors, being related to one transducer only and including differences both at numerator from offset errors, being related to one transducer only and including differences both at numerator and denominator. A similar choice for the KPitch cannot be applied because there are no virtual taps and denominator. A similar choice for the KPitch cannot be applied because there are no virtual taps and denominator. A similar choice for the KPitch cannot be applied because there are no virtual taps for such a quantity as result of the rotation along the probe stem. The other main advantage concerns for such a quantity as result of the rotation along the probe stem. The other main advantage concerns for such a quantity as result of the rotation along the probe stem. The other main advantage concerns the grid regularity that allows for a proficient interpolation over the whole angular range, as visible in the grid regularity that allows for a proficient interpolation over the whole angular range, as visible the grid regularity that allows for a proficient interpolation over the whole angular range, as visible Figure 11. in Figure 11. in Figure 11. Figure 11. KPt on the KYaw and KPitch coefficient grid (range +/- 22°) for aerodynamic coefficients Figure 11. KPt on the KYaw and KPitch coecient grid (range +/ 22 ) for aerodynamic coecients Figure 11. KPt on the KYaw and KPitch coefficient grid (range +/- 22°) for aerodynamic coefficients defined according to set B. defined according to set B. defined according to set B. 5.2.3. Set C 5.2.3. Set C: 5.2.3. Set C: P P C P It di ers from Set A only by the KPitch that is defined in this set as KPitch = . This choice It differs from Set A only by the KPitch that is defined in this set as ℎ = P P . This choice It differs from Set A only by the KPitch that is defined in this set as ℎ = . This choice allows for a direct link between the central reading of the yaw and the pitch tap, as the central reading allows for a direct link between the central reading of the yaw and the pitch tap, as the central reading allows for a direct link between the central reading of the yaw and the pitch tap, as the central reading for the yaw is, in any case, dependent on the pitch flow angle. This set, although seemingly smart, in for the yaw is, in any case, dependent on the pitch flow angle. This set, although seemingly smart, in ffact, or the ya connects w is, i the n atwo ny case, sensors dependent on the pi readings and, in tch fl caseoof w a an thermal gle. This drift, set, alt it h may ough seem increase ing substantially ly smart, in fact, connects the two sensors readings and, in case of a thermal drift, it may increase substantially fact, connects the two sensors readings and, in case of a thermal drift, it may increase substantially the final uncertainty. From a purely aerodynamic point of view, as shown in Figure 12, it changes the the final uncertainty. From a purely aerodynamic point of view, as shown in Figure 12, it changes the KPitch coefficient magnitude but does not fix the overlap at the grid boundary. KPitch coefficient magnitude but does not fix the overlap at the grid boundary. Int. J. Turbomach. Propuls. Power 2020, 5, 6 12 of 16 the final uncertainty. From a purely aerodynamic point of view, as shown in Figure 12, it changes the Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 12 of 16 KPitch coecient magnitude but does not fix the overlap at the grid boundary. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 12 of 16 Figure 12. Figure 12. KPt KPt on the KYaw and KPit on the KYaw and KPitch ch coef coe ficient grid cient grid (range (range +/- 22°) for aerodynamic coefficients +/ 22 ) for aerodynamic coecients Figure 12. KPt on the KYaw and KPitch coefficient grid (range +/- 22°) for aerodynamic coefficients defined defined ac accor cord ding ing to set to set C C. . defined according to set C. 5.2.4. Discussion on Coecient Sets 5.2.4. Discussion on Coefficient Sets 5.2.4. Discussion on Coefficient Sets Set A was abandoned because it does not guarantee a unique solution in the iterative procedure Set A was abandoned because it does not guarantee a unique solution in the iterative procedure Set A was abandoned because it does not guarantee a unique solution in the iterative procedure due to the overlap at the edges of the calibration matrices. Set C has the advantage of linking the due to the overlap at the edges of the calibration matrices. Set C has the advantage of linking the pitch due to the overlap at the edges of the calibration matrices. Set C has the advantage of linking the pitch pitch sensitivity to the most physical di erence that is the (P P ): notwithstanding this fact, it was C P sensitivity to the most physical difference that is the (PC - PP): notwithstanding this fact, it was sensitivity to the most physical difference that is the (PC - PP): notwithstanding this fact, it was abandoned as well, due to the mixture of two transducers readings that make its application critical in abandoned as well, due to the mixture of two transducers readings that make its application critical abandoned as well, due to the mixture of two transducers readings that make its application critical the context of possible (and maybe dicult to compensate) thermal drift. Moreover, it does not allow in the context of possible (and maybe difficult to compensate) thermal drift. Moreover, it does not in the context of possible (and maybe difficult to compensate) thermal drift. Moreover, it does not to fix the problem of the non-unique solution at the edges of the matrices. allow to fix the problem of the non-unique solution at the edges of the matrices. allow to fix the problem of the non-unique solution at the edges of the matrices. Set B was chosen for this type of probe as it provides both the advantage of keeping separate Set B was chosen for this type of probe as it provides both the advantage of keeping separate the Set B was chosen for this type of probe as it provides both the advantage of keeping separate the the two transducer readings and of the proper scaling of the angular sensitivity (P P ) to the local L R two transducer readings and of the proper scaling of the angular sensitivity (PL - PR) to the local two transducer readings and of the proper scaling of the angular sensitivity (PL - PR) to the local kinetic head measured by the probe, at that pitch position. To aid the physical understanding of this kinetic head measured by the probe, at that pitch position. To aid the physical understanding of this kinetic head measured by the probe, at that pitch position. To aid the physical understanding of this latter concept, i.e., of the artificial P , Figure 13 reports the yaw tap pressure measurements (P ) in max latter concept, i.e., of the artificial Pmax, Figure 13 reports the yaw tap pressure measurements (PC) in latter concept, i.e., of the artificial Pmax, Figure 13 reports the yaw tap pressure measurements (PC) in the yaw calibration range for the di erent pitch angles. Two observations can be drawn: the yaw calibration range for the different pitch angles. Two observations can be drawn: the yaw calibration range for the different pitch angles. Two observations can be drawn: Figure 13. Influence of flow angles on sFRAPP central pressure reading P . Figure 13. Influence of flow angles on sFRAPP central pressure reading PC. Figure 13. Influence of flow angles on sFRAPP central pressure reading PC. (a) the maximum pressure depends on the pitch angle as a consequence of the tap position with a) the maximum pressure depends on the pitch angle as a consequence of the tap position with a) the maximum pressure depends on the pitch angle as a consequence of the tap position with respect to the flow direction: therefore, the P allows for considering this di erence. max respect to the flow direction: therefore, the Pmax allows for considering this difference. respect to the flow direction: therefore, the Pmax allows for considering this difference. (b) the di erence between Pc at Yaw = 0 and Yaw =20 is slightly greater for negative pitches, where b) the difference between Pc at Yaw = 0° and Yaw =20° is slightly greater for negative pitches, also (P P ) has higher values: the Set B denominator is then able to scale properly the angular b) the differenc L R e between Pc at Yaw = 0° and Yaw =20° is slightly greater for negative pitches, sensitivity of the yaw tap. where also (PL - PR) has higher values: the Set B denominator is then able to scale properly where also (PL - PR) has higher values: the Set B denominator is then able to scale properly the angular sensitivity of the yaw tap. the angular sensitivity of the yaw tap. 5.3. Uncertainty Quantification 5.3. Uncertainty Quantification To quantify the uncertainty level in the calibration matrixes building and application, the same To quantify the uncertainty level in the calibration matrixes building and application, the same methodologies as those described in the static calibration can be applied. The results discussed in the methodologies as those described in the static calibration can be applied. The results discussed in the Int. J. Turbomach. Propuls. Power 2020, 5, 6 13 of 16 5.3. Uncertainty Quantification To quantify the uncertainty level in the calibration matrixes building and application, the same Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 13 of 16 methodologies as those described in the static calibration can be applied. The results discussed in the following were obtained for the FRAPP probe. In this work, only uncertainties in the calibration and in following were obtained for the FRAPP probe. In this work, only uncertainties in the calibration and the probe application when the probe was applied in a steady flow are considered: uncertainties due in the probe application when the probe was applied in a steady flow are considered: uncertainties to probe installation and positioning, unsteady e ects (e.g., the probe stem vortex shedding or possible due to probe installation and positioning, unsteady effects (e.g. the probe stem vortex shedding or interaction with the cascades), intrusiveness, and spatial discretisation are not considered as highly possible interaction with the cascades), intrusiveness, and spatial discretisation are not considered as dependent on the kind of positioner used and on the application foreseen for the probe: in any case, highly dependent on the kind of positioner used and on the application foreseen for the probe: in any besides the positioning, the other uncertainty sources are very dicult to be evaluated. case, besides the positioning, the other uncertainty sources are very difficult to be evaluated. As the calibration matrixes require the application of iterative procedures, the Montecarlo As the calibration matrixes require the application of iterative procedures, the Montecarlo methodology is particularly attractive for computing the uncertainty propagation. methodology is particularly attractive for computing the uncertainty propagation. The calibration coecients (KYaw, KPt, KPs) were first calculated by choosing pressure values The calibration coefficients (KYaw, KPt, KPs) were first calculated by choosing pressure values (P , P , P , P , P ) randomly in each population (for a given Mach number and angular position) L C R T S (PL, PC, PR, PT, PS) randomly in each population (for a given Mach number and angular position) according to the Gaussian distribution resulting from the static calibration (see the static calibration according to the Gaussian distribution resulting from the static calibration (see the static calibration paragraph). Then, the results of the calibration coecients were averaged and their standard deviation paragraph). Then, the results of the calibration coefficients were averaged and their standard was calculated, then they were all stored in proper calibration files. deviation was calculated, then they were all stored in proper calibration files. During the measurements campaign, to obtain the flow quantities, the measured pressures were During the measurements campaign, to obtain the flow quantities, the measured pressures were used coupled to the calibration matrixes: the measured pressures were analysed by the same procedure used coupled to the calibration matrixes: the measured pressures were analysed by the same applied in the calibration processes, leading to a gaussian distribution with their own mean value and procedure applied in the calibration processes, leading to a gaussian distribution with their own standard deviation. A number N of di erent sets of pressures and coecients were selected randomly mean value and standard deviation. A number N of different sets of pressures and coefficients were in each population and by an iterative procedure, the flow quantities calculated. Since the process selected randomly in each population and by an iterative procedure, the flow quantities calculated. is statistical, all the data were then averaged and the standard deviation was calculated. As for the Since the process is statistical, all the data were then averaged and the standard deviation was result distribution, since the input data were chosen according to a Gaussian distribution, the output calculated. As for the result distribution, since the input data were chosen according to a Gaussian ones were of the same kind. The number N of di erent sets was dynamically chosen according to distribution, the output ones were of the same kind. The number N of different sets was dynamically the convergence criterion chosen for the standard deviation change (D / < 10 ). Moreover, in i+1,i i chosen according to the convergence criterion chosen for the standard deviation change this case, the set choice was made by applying the Latin hypercube methodology in order to save -3 (Δσi+1,i /σi <10 ). Moreover, in this case, the set choice was made by applying the Latin hypercube computational time. methodology in order to save computational time. Figure 14 shows the results for di erent run and convergence criteria: since the method is based Figure 14 shows the results for different run and convergence criteria: since the method is based on statistics, di erent runs may lead to di erent results. Notwithstanding such possible variations, the on statistics, different runs may lead to different results. Notwithstanding such possible variations, di erence is almost negligible for a given yaw angle. Finally, Table 1 shows the averaged error for the difference is almost negligible for a given yaw angle. Finally, Table 1 shows the averaged error the four quantities of interest (Yaw, Mach, P , P ). Values are high typically at low and high Mach T S for the four quantities of interest (Yaw, Mach, PT, PS). Values are high typically at low and high Mach numbers. At a low Mach number, this is due to the high transducer range that makes the transducer numbers. At a low Mach number, this is due to the high transducer range that makes the transducer uncertainty quantitatively significant in the context of the measurements; at high Mach, the overspeed uncertainty quantitatively significant in the context of the measurements; at high Mach, the on the cylinder makes the flow locally supersonic and in this case, the measurements are less accurate. overspeed on the cylinder makes the flow locally supersonic and in this case, the measurements are less accurate. Figure 14. Results in terms of standard deviation for di erent runs. Figure 14. Results in terms of standard deviation for different runs. Table 1. Averaged errors ( δ ) for the Yaw, Mach, total and static pressure. Mach δYaw δM δPt % δPs % 0.25 0.63 0.002 1.10 1.50 0.35 0.50 0.004 0.92 2.90 0.45 0.62 0.001 0.56 0.76 Int. J. Turbomach. Propuls. Power 2020, 5, 6 14 of 16 Table 1. Averaged errors () for the Yaw, Mach, total and static pressure. Mach Yaw M Pt % Ps % 0.25 0.63 0.002 1.10 1.50 0.35 0.50 0.004 0.92 2.90 0.45 0.62 0.001 0.56 0.76 0.55 0.69 0.001 0.28 0.40 0.65 0.62 0.003 0.26 0.70 0.75 0.58 0.011 0.40 0.72 0.825 0.50 0.016 0.24 1.81 0.875 0.37 0.013 0.21 2.23 0.925 0.16 0.023 0.21 2.10 6. Conclusions This paper presented the most relevant developments in FRAPP technology at Politecnico di Milano in the last decade. This study considered the two most relevant probe configurations manufactured, calibrated and applied by the authors in their experience, for both 2D and 3D unsteady flow measurements in turbomachinery. Specific challenges emerged in terms of extension to (relatively) high temperature applications, simplicity of operation, improved aerodynamics and more refined uncertainty quantification, and were all acknowledged in the paper. In particular, uncooled FRAPPs were manufactured for a temperature operation of about 550 K, without changing the external/internal probe shape and size. The need for high temperature extension also triggered specific theoretical models to handle temperature-corrected static and dynamic calibration of the probes. A physical analysis of the sensor properties and of the line-cavity system provided general rules for correcting the static and dynamic pressure measurements performed during calibrations. In the frame of these activities, the static and dynamic calibration procedures were re-analyzed to investigate the global reliability of the FRAPP technology. The probe aerodynamics were also reconsidered and several sets of aerodynamic coecients were proposed for the Spherical FRAPP, which is less consolidated with respect to the cylindrical one. This analysis highlights that clear advantages can be obtained if a specific set of coecients is applied. Finally, a novel technique based on the Montecarlo approach was introduced to evaluate the uncertainty of FRAPP measurements, combining those of the sensor (based on a Montecarlo analysis of the static calibration) with the formulation of the aerodynamic coecients. Author Contributions: The two authors have given equal contributions to this paper, in terms of conceptualization, methodology, formal analysis, investigation, and writing. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest. Nomenclature c sound speed P pressure f frequency ! natural frequency non dimensional damping dynamic viscosity density T temperature U, u uncertainty standard deviation DV voltage di erence across the bridge DV voltage di erence across the sense resistor T Int. J. Turbomach. Propuls. Power 2020, 5, 6 15 of 16 K sensor calibration slope Q sensor calibration intercept KPs, KPt sensitivity coe . to static and total pressure KYaw, KPitch sensitivity coe . to the yaw, pitch angles References 1. Roduner, C.; Kupferschmied, P.; Köppel, P.; Gyarmathy, G. On the Development and Application of the Fast-Response Aerodynamic Probe System in Turbomachines—Part 2: Flow, Surge, and Stall in a Centrifugal Compressor. J. Turbomach. 2000, 122, 517–526. [CrossRef] 2. Gaetani, P.; Persico, G.; Osnaghi, C. 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International Journal of Turbomachinery, Propulsion and PowerMultidisciplinary Digital Publishing Institute

Published: Apr 12, 2020

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