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A Low Order Flow Network Model for Double-Wall Effusion Cooling Systems

A Low Order Flow Network Model for Double-Wall Effusion Cooling Systems International Journal of Turbomachinery Propulsion and Power Article A Low Order Flow Network Model for Double-Wall Effusion Cooling Systems Michael van de Noort * and Peter Ireland Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, UK; peter.ireland@eng.ox.ac.uk * Correspondence: michael.vandenoort@eng.ox.ac.uk † This paper is an extended version of our paper published in Proceedings of the European Turbomachinery Conference ETC14 2021, Paper No. 666, Gdansk, Poland, 12–16 April 2021. Abstract: The high pressure turbine nozzle guide vane of a modern aeroengine experiences large heat loads and thus requires both highly effective internal and external cooling. This can be accomplished with double-wall effusion cooling, which combines impingement, pin-fin and effusion cooling. The combination of three cooling mechanisms causes high pressure losses, increasing potential for the migration of coolant towards low pressure regions, subsequently starving effusion holes on the leading edge of coolant supply. This paper presents a low order flow network model to rapidly assess the pressure and mass flow distributions through such cooling schemes for a flexible set of geometric and flow conditions. The model is subsequently validated by a series of experiments with varying mainstream pressure gradients. Results from the model are used to indicate design parameters to reduce the effect of coolant migration, and to minimise the risk of destructive hot gas ingestion. Keywords: coolant migration; flow networks; double-wall effusion cooling Citation: van de Noort, M.; Ireland, P. 1. Introduction A Low Order Flow Network Model Studies into the thermal efficiency of gas turbines have come to the consensus that the for Double-Wall Effusion Cooling specific fuel consumption can be improved by increasing the turbine entry temperature. To Systems. Int. J. Turbomach. Propuls. maximise this benefit, modern aeroengines use TETs well in excess of the turbine compo- Power 2022, 7, 5. https://doi.org/ nents’ melting temperatures. Cooling systems are required to prevent component failure, 10.3390/ijtpp7010005 but these reduce power output by reducing the air available for combustion and causing Academic Editor: Francesco mixing losses in the turbine, which reduce aerodynamic efficiency. Double-wall effusion Martelli cooling is viewed as a potential solution to these challenges—the high porosity make-up Received: 9 December 2021 of the system brings it close to the goal of accomplishing micro-cooling. Bunker [1,2] Accepted: 31 January 2022 noted that mirco-cooling has two goals—“spread[ing] out the cooling network in a series Published: 2 February 2022 of smaller and highly distributed channels” and “bringing the cooling fluid closer to the outer surface of the airfoil”. This is accomplished by combining three commonly used Publisher’s Note: MDPI stays neutral cooling techniques—impingement cooling, pin-fin cooling and effusion (or film) cooling, with regard to jurisdictional claims in with features densely packed to spread out the coolant, and double-walled to allow coolant published maps and institutional affil- to sit close to the outer surface. iations. Figure 1 shows an example of double-wall effusion cooling being employed in a modern commercial engine turbine blade. This combination of cooling techniques leads to high convection cooling efficiency (1) and overall cooling effectiveness (2), but also Copyright: © 2022 by the authors. produces high pressure losses. T T Licensee MDPI, Basel, Switzerland. c,e c,i h = (1) conv This article is an open access article T T m c,i distributed under the terms and T T ¥ m conditions of the Creative Commons # = (2) T T c,i Attribution (CC BY-NC-ND) license (https://creativecommons.org/ licenses/by-nc-nd/4.0/). Int. J. Turbomach. Propuls. Power 2022, 7, 5. https://doi.org/10.3390/ijtpp7010005 https://www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2022, 7, 5 2 of 19 𝑇 −𝑇 Int. J. Turbomach. Propuls. Power 2022, 7, 5 2 of 18 𝜀 = (2) 𝑇 −𝑇 Figure 1. Turbine blade using double-wall effusion cooling scheme—From Murray et al., 2017 [3]. Figure 1. Turbine blade using double-wall effusion cooling scheme—From Murray et al., 2017 [3]. Used with Permission. Used with Permission. The high The high pressure pressure losses caused losses caused b by comb y combining ining thr three co ee cooling oling schemes schemecan s can make make d double- ou- ble-wall effusion cooling systems vulnerable to coolant migration—the movement of cool- wall effusion cooling systems vulnerable to coolant migration—the movement of coolant in ant in the pedestal cavity toward low pressure regions under the influence of an external the pedestal cavity toward low pressure regions under the influence of an external pressure p gradient. ressure grCoolant adient. Cmigration oolant migr can ation can be particularly be particularly concerning concerning in components in components where where the the cool coolant a supply nt suppl toy to mainstrea mainstream stag m st nation agnati pr on pressure ra essure ratio is tio low—the is low—the ma main example in examof ple of this being the Turbine’s High Pressure NGV, where the mainstream stagnation pressure is only this being the Turbine’s High Pressure NGV, where the mainstream stagnation pressure ~3% less than that of the coolant supplied from the HP Compressor. In a configuration such is only ~3% less than that of the coolant supplied from the HP Compressor. In a configu- as that shown in Figure 1, where the cavity between the walls is continuous along both ration such as that shown in Figure 1, where the cavity between the walls is continuous surfaces, the streamwise pressure gradient will lead to coolant in the cavity migrating away along both surfaces, the streamwise pressure gradient will lead to coolant in the cavity from the LE sections and toward film holes further along the PS and SS. Spanwise pressure migrating away from the LE sections and toward film holes further along the PS and SS. gradients can have similar effects, causing cooling coolant to move from the hub to the Spanwise pressure gradients can have similar effects, causing cooling coolant to move tip. Results produced by Holgate et al. [4] showed that for an engine representative NGV, from the hub to the tip. Results produced by Holgate et al. [4] showed that for an engine the external static pressure coefficient C (3) could fall from 0.2 to 1.2 along the early representative NGV, the external static pressure coeff p icient 𝐶 (3) could fall from −0.2 to suction surface alone—given the small pressure margin at the leading edge, this provides a −1.2 along the early suction surface alone—given the small pressure margin at the leading major driving force for coolant migration. edge, this provides a major driving force for coolant migration. 𝑃 −𝑃 P P ext ext, max C 𝐶 == (3) 1 (3) r U 𝜌 𝑈 ¥ ¥ throat To combat coolant migration, recent patented designs using this cooling scheme (e.g., To combat coolant migration, recent patented designs using this cooling scheme [5,6]) use walls across the span of the cavity to create separate cooling zones for the PS, LE (e.g., [5,6]) use walls across the span of the cavity to create separate cooling zones for the PS, and SS, but this reduces the effectiveness of pin-fin cooling due to reduced throughflow LE and SS, but this reduces the effectiveness of pin-fin cooling due to reduced throughflow and turbulence generation. and turbulence generation. Coolant Coolant migr migration ation can h can have ave severel severely y de detrimental trimental impact impacts s on t on the he ffilm ilm cool cooling ing e ef ffec fec- - tiveness in both the region the coolant migrates away from and the region it migrates to. tiveness in both the region the coolant migrates away from and the region it migrates to. If coolant m If coolant migration igration is h is high igh enough, th enough, the e region region los losing ing cool coolant ant c can an ess essentially entially b be e st starved, arved, lea leading ding t to o de destr struct uctive ive hot hot g gas as ingestion, ingestion,which which w willileventually l eventually cause causcomponent e componen failur t faile. ure. In the region that the coolant migrates to, a high pressure margin across the outer skin causes In the region that the coolant migrates to, a high pressure margin across the outer skin film hole flows to be ejected into the mainstream with high velocity, potentially leading to causes film hole flows to be ejected into the mainstream with high velocity, potentially jet-lift off. In both regions, film cooling effectiveness would be significantly reduced, so the component would experience much greater heat loads and would likely have a much Int. J. Turbomach. Propuls. Power 2022, 7, 5 3 of 18 shorter operating life. Predicting and controlling coolant migration is key to ensuring the proper development of a coolant film for external cooling. This paper summarizes work done to model coolant migration within double-wall effusion cooling systems under the influence of mainstream pressure gradients, done through the use of low order flow network models (LOMs). The LOM developed is a high speed, iterative solver that produces mass flow and pressure distributions throughout cooling arrays for given boundary conditions. This allowed assessment of a high number of design possibilities that would’ve been too computationally expensive to do in a time- efficient manner using CFD. 2. Related Work 2.1. Double-Wall Effusion Cooling Systems Research has been conducted into the application of double-wall effusion cooling systems since the 1970s, but their application in modern engines is limited due to issues of manufacturing difficulty and high thermomechanical stresses. As transpiration cooling research is becoming more prevalent, double-wall effusion cooling has become a topic of increased interest—the high manufactured porosity, complex internal flow field, and low film hole blowing ratio lead to such schemes being dubbed as “quasi-transpiration” cooling by Murray et al. [3], who conducted a series of computational tests to find the effects of various geometric parameters on the thermomechanical stresses developed. Increases of the passage height, pedestal diameter and hole diameters were shown to reduce the average thermomechanical stress, as did reducing the pedestal spacing. Flow travelling through arrays of pedestals can undergo high pressure loss due to the wakes generated behind the pedestals. Research by Chyu et al. [7] and Siw et al. [8] showed that staggered arrays of pedestals caused much greater pressure drops than in-line arrays. The pedestal shape can also have a large influence—cylindrical pedestals produce less pressure loss than those of square or diamond (square rotated by 45 ) shapes [7], but also produce lower HTCs. Bamba et al. [9] found experimentally that in regions where pin-fins did not signif- icantly obstruct the passage of flow from impingement to film hole entry, such as at the LE, the contribution of the pin-fins to the pressure loss was negligible. In these regions, pedestals enhance cooling only by increasing the surface area over which heat transfer can take place. The dominant form of pressure loss in these systems, however, is from impingement cooling. Andrews et al. [10] stated that “for impingement/effusion applications it is . . . valid to assume that all the pressure loss would occur at the impingement plate”. Greater pressure losses generally equate to increased cooling performance, as is the case for pin-fin arrays. Murray et al. [11] conducted a comprehensive study of the effect of the cooling scheme’s porosity on its performance. Higher porosity geometries (particularly those with high porosity in the walls) required less coolant mass flow for a given pressure drop and exhibited “relatively high cooling effectiveness” at low coolant flow rates. The au- thors suggested cooling around the blade could be optimised by varying the porosity as required—high pressure drop regions (i.e., the SS) should use low porosity cooling geometries to minimise aerodynamic losses, whilst areas such as the PS should use higher porosity geometries for increased coolant flow and better effusion cooling performance. Wambersie et al. [12] conducted experimental tests using high-porosity panels, in- spired by transpiration-cooling designs. Whilst these panels gave exceptionally high film effectiveness levels at standard blowing rates, at low flow rates the low pressure margin made the panels susceptible to mainstream flow ingestion, localised to certain regions as a result of high streamwise and spanwise pressure gradients. Int. J. Turbomach. Propuls. Power 2022, 7, 5 4 of 18 2.2. Flow Networks The models developed for this paper were based on flow network solvers developed by Rose [13], and later by Kutz and Speer [14]. In these systems, flow networks were produced by creating lists of nodes, each representing a specific point in the cooling system. Nodes are connected by links, along which fluid is allowed to flow, forming a network of connected nodes. Boundary nodes form the inlet and outlet of flow within a network. In order to solve for the flow through a network, the law of mass conservation is applied at each internal node—for a node i connected to n nodes j, the governing equation is given by (4) The mass flow rate along a link connecting nodes i and j is determined by Equation (5) where the function f is used to model the three-dimensional flow as one-dimensional. i,j Kutz and Speer [14] expanded from solving for continuity to account for the heat transfer along each link Q by making use of the energy equation, adding another governing i,j Equation (6) at each node. This network approach was applied to a full secondary air system, and results for the rear sealing chamber air pressure were in good agreement with engine test results. m = 0 (4) i,j m = f P , P , T (5) i,j i,j i j i m h + Q = 0 (6) å i,j i,j i,j Using a similar methodology, Ebenhoch and Speer [15] produced a network solver that was applied to three test cases; isothermal flow in a multipass blade cooling system, coolant flow undergoing high heat transfer in a hypersonic vehicle nozzle, and coolant flow in a rotating HP blade. This flexible network solver was deemed to have “sufficient accuracy in predicting coolant mass flow, regional coolant side heat transfer coefficients, and local coolant temperatures”. Gouws et al. [16] created a network solver for a modern combustor that, once calibrated with experimental data, predicted the mass flow splits through cooling holes with high accuracy. Heat transfer elements within the network allowed a prediction of the outlet temperature distribution. It was found that in a cooling system with large numbers of cooling holes, discharge coefficients had a high influence on one-dimensional flow and pressure drop predictions. Jin et al. [17] used a compressible flow network analysis to find the mass flow distribu- tion for the various positions around the trailing edge of an HP turbine blade, as part of an investigation into blade oxidation. This investigation revealed that an inlet metering plate at the blade root reduced the coolant internal pressure, raising the risk of hot gas ingestion, particularly in “off-design or transient conditions”. The network allowed testing and recommendation of changes to reduce this risk. 3. Modelling Methodology 3.1. Flow Network Construction The LOM presented focused only on mass flow continuity and was based on test cases without heat transfer and in incompressible conditions. This allowed the temperature of each node to be neglected, with the model solving only for the pressures at each node. Newton’s method (see [18]) was used to iteratively solve for the internal pressure distri- bution. Static pressures at each node served as the list of unknowns, which could then be used to calculate mass flows through the network. In these conditions, the mass flow m i,j along a link between nodes i and j becomes a function only of the link’s compliance, C , i,j and the pressure difference between the nodes, such that: m = C  f P P (7) i,j i,j i,j i j Int. J. Turbomach. Propuls. Power 2022, 7, 5 5 of 19 Int. J. Turbomach. Propuls. Power 2022, 7, 5 5 of 18 𝑚 = 𝐶 ∙ 𝑓 (𝑃 −𝑃 ) (7) , , , The geometries discussed in this paper used a staggered arrangement of film holes The geometries discussed in this paper used a staggered arrangement of film holes and impingement holes, such that no holes of the same type were located next to one and impingement holes, such that no holes of the same type were located next to one another—this layout is the same as shown in Figure 1. The standard flow network layout another—this layout is the same as shown in Figure 1. The standard flow network layout is shown for a single row of holes and pedestals in Figure 2—in the case of there being is shown for a single row of holes and pedestals in Figure 2—in the case of there being multiple rows, further nodes would be located into the page, with impingement and film multiple rows, further nodes would be located into the page, with impingement and film hol holes es a alternating lternating le left/right ft/right posi positions. tions. The posi The position tion of ea ofch node ha each nodes has three components: po- three components: sition position in the in the streamwise streamwise didir rection, ection, level in level in the cool the cooling ing system system(1 (1 at the bottom, 5 a at the bottom, 5 at t the the top), and spanwise row number. Other than for nodes on the pedestal layer (Level 4), all top), and spanwise row number. Other than for nodes on the pedestal layer (Level 4), all nodes nodes on onthe same layer ar the same layer ar e e of of the same the same type. type. LinLinks ks are cre areacr ted base eated based d on the physical flow on the physical flow paths available from each node’s position, so flow is allowed to transfer to nodes on paths available from each node’s position, so flow is allowed to transfer to nodes on the the same level only on Level 4—the other levels only have links to nodes directly above same level only on Level 4—the other levels only have links to nodes directly above or or below. below. Figure 2. Flow network diagram for a single row. Figure 2. Flow network diagram for a single row. The number label for each node determines its type: The number label for each node determines its type: • 10: Impingement Hole Entry/Inlet Nodes. Boundary nodes with fixed pressure. 10: Impingement Hole Entry/Inlet Nodes. Boundary nodes with fixed pressure. • 20: Impingement Hole Exit Nodes. Node pressure is representative of the flow as it 20: Impingement Hole Exit Nodes. Node pressure is representative of the flow as it leave leaves s t the he ho hole. le. • 30: Impin 30: Impingement gement Point Point No Nodes. des. Nod Nodes es re repr present the p esent the point oint at at which which flow flow stagnate stagnates s on on the inner surface of the outer wall as the jet flow from the hole impinges. the inner surface of the outer wall as the jet flow from the hole impinges. 40: Expansion from Impingement Nodes. Represents flow as it accelerates away from • 40: Expansion from Impingement Nodes. Represents flow as it accelerates away from the i the impingement mpingement poi points, nts, but bef but befor ore e iit t intera interacts cts wi with th a any ny pedestals. In the di pedestals. In the diagram, agram, this point is represented as being above the impingement point for ease of display. this point is represented as being above the impingement point for ease of display. 41: Post Pedestal Nodes. Frictional pressure losses due to pedestals are modelled in • 41: Post Pedestal Nodes. Frictional pressure losses due to pedestals are modelled in the link between nodes 40 and 41. the link between nodes 40 and 41. 42: Film Entry Hole Nodes. As noted in Figure 2, these nodes are only used in the • 42: Film Entry Hole Nodes. As noted in Figure 2, these nodes are only used in the case when the film hole inclination is not 90 —when film holes are perpendicular, the case when the film hole inclination is not 90°—when film holes are perpendicular, 42 nodes are merged with linked 41 nodes. When holes are inclined, flow from one the 42 nodes are merged with linked 41 nodes. When holes are inclined, flow from direction turns by a different angle than flow from the opposite direction, requiring one direction turns by a different angle than flow from the opposite direction, requir- the use of a different dynamic head loss coefficient. ing the use of a different dynamic head loss coefficient. 50: Film Hole Exit Nodes. Boundary nodes with fixed pressure. • 50: Film Hole Exit Nodes. Boundary nodes with fixed pressure. The mass flow functions f P P and compliances C are dependent on the type i,j i j i,j The mass flow functions 𝑓 (𝑃 −𝑃 ) and compliances 𝐶 are dependent on the , , of link. Flow through a hole, occurring for impingement holes between Levels 1 and 2, and type of link. Flow through a hole, occurring for impingement holes between Levels 1 and occurring for film holes between Levels 4 and 5, allows the discharge Equation (8) to be 2, and occurring for film holes between Levels 4 and 5, allows the discharge Equation (8) used—all holes of both types are cylindrical in this geometry. In cases where high coolant to be used—all holes of both types are cylindrical in this geometry. In cases where high migration and thus an uneven outlet flow distribution is expected, it is unreasonable to coolant migration and thus an uneven outlet flow distribution is expected, it is unreason- assume that all film holes will have similar discharge coefficients. To account for this, the able to assume that all film holes will have similar discharge coefficients. To account for discharge coefficient for each hole was updated in each iteration. The two pressure values from the previous iteration were used to calculate the area-averaged velocity through the Int. J. Turbomach. Propuls. Power 2022, 7, 5 6 of 18 hole. This velocity, the pressure ratio across the hole and geometric properties were used to calculate an updated discharge coefficient using a correlation developed by Mazzei et al. [19]. This was applied to both impingement and film holes. The area ratios b and b use a square inlet area with a side length of half the hole pitch, and the expansibility factor # is taken as unity as the air is assumed to be incompressible. 2 p pd d,i i m = # 2r P P , 10 20 1b 2 (8) p p . C pd d, f f m = # 2r P P f 42 1b The connection between Levels 2 and 3 represents the sudden expansion of flow area, when flow leaves the impingement hole and enters the inter-wall cavity. At Level 3, flow is expected to stagnate as it impinges on the outer wall, allowing the form of the Bernoulli equation shown in (9). to be used—this features a loss coefficient k to account for sudden l,ex expansion pressure loss. 1 1 . r A 2 2 P + rU k rU = P ) m = P P (9) 20 20 l,ex 20 30 30 20 2 2 (1 k ) l,ex To account for the total pressure loss produced by a jet impinging on a wall, a similar loss equation is used for the link between Levels 3 and 4. A single node is used in this position at Level 4, as it assumed that there is no circumferential variation in total pressure loss due to impingement. 1 1 . r A 2 2 P = P + rU + k rU ) m = P P (10) 30 40 40 40 30 40 l,i 2 2 (1 + k ) l,i Equation (11) accounts for flow around pedestals on Level 4 between nodes 40 and 41. The friction factor f for flow around pedestals is determined by the correlation developed by Wang [20]. p 2 . r A D h 0.318 m = P P , f = 1.76Re (11) 40 41 max 2 f L When turning flows are employed, the mass flow between nodes 41 and 42 is given by (12). For turning flows, loss coefficients were selected from studies by Miller [21]. In cases with close packed pedestals, (11). is replaced by a flow coefficient in the same form as (12), as pressure losses become functions of impingement onto pedestals and turning between them, rather than by friction. This equation uses the assumption that A is larger than A . 41 42 1 2 1 2 1 2 P + rU = P + rU + k rU 41 41 42 42 l,t 42 2 2 2 (12) . 2 2 2r A A 41 42 ) m = P P 41 42 2 2 A 1+k A ( ) 41 l,t 42 3.2. Computational Implementation The LOM was produced in MATLAB (MathWorks—Natick, MA, USA), using sum- mations for the mass flows out of nodes described in the equations above, as in (4). The initial guess of pressures through the network was selected to ensure that pressure drops and rises occurred along the links that were expected. To enhance the stability of the model, changes between iterations were damped. The value of the damping factor z used was 0.5 in simpler models, such as the geometry featured in Figure 3, but more complex geometries or high mainstream pressure gradients often required z to be reduced further. This slowed convergence of the model, though the solving time remained on the order of seconds, a huge reduction from CFD cases. Convergence of the LOM was attained when the maximum absolute value of mass flow imbalance at any node was less than 10 kg/s. Int. J. Turbomach. Propuls. Power 2022, 7, 5 7 of 19 3.2. Computational Implementation The LOM was produced in MATLAB (MathWorks—Natick, MA, USA), using sum- mations for the mass flows out of nodes described in the equations above, as in (4). The initial guess of pressures through the network was selected to ensure that pressure drops and rises occurred along the links that were expected. To enhance the stability of the model, changes between iterations were damped. The value of the damping factor 𝜁 used was 0.5 in simpler models, such as the geometry featured in Figure 3, but more com- plex geometries or high mainstream pressure gradients often required 𝜁 to be reduced further. This slowed convergence of the model, though the solving time remained on the order of seconds, a huge reduction from CFD cases. Convergence of the LOM was attained −12 when the maximum absolute value of mass flow imbalance at any node was less than 10 kg/s. An additional mass flow check was performed by comparing the inlet and outlet total mass flows. 3.3. Example Results Figure 3 shows a set of example results for the LOM. Flow paths are contoured by their mass flow rate. In this test case, all film holes are ejecting to the same outlet pressure, and all impingement holes have the same inlet pressure. Despite this, the film holes in the central row (2nd spanwise position) clearly have larger outlet flows than those in the outer rows. The position of the impingement holes leads to film holes at the right end (highest streamwise distance) of the array each having ~75% of the coolant flow that each film hole in the centre receives. The unevenness in the outlet flow distribution is not matched at the inlet—the two impingement holes at the left end of the array (lowest streamwise distance) Int. J. Turbomach. Propuls. Power 2022, 7, 5 7 of 18 are the most poorly fed, but receive only 5% less flow than those in the centre of the array. The size of this array and the inclination angle of the film holes are flexible—this particular test case size is ‘6 × 3’, with six hole positions in streamwise direction and three in the An additional mass flow check was performed by comparing the inlet and outlet total spanwise direction. The film hole angle of inclination is 30°. mass flows. Figure 3. Example Figure 3. resultsEx fram omple resu the LOM. lts from the LOM. 3.3. Example Results Figure 3 shows a set of example results for the LOM. Flow paths are contoured by their mass flow rate. In this test case, all film holes are ejecting to the same outlet pressure, and all impingement holes have the same inlet pressure. Despite this, the film holes in the central row (2nd spanwise position) clearly have larger outlet flows than those in the outer rows. The position of the impingement holes leads to film holes at the right end (highest streamwise distance) of the array each having ~75% of the coolant flow that each film hole in the centre receives. The unevenness in the outlet flow distribution is not matched at the inlet—the two impingement holes at the left end of the array (lowest streamwise distance) are the most poorly fed, but receive only 5% less flow than those in the centre of the array. The size of this array and the inclination angle of the film holes are flexible—this particular test case size is ‘6  3’, with six hole positions in streamwise direction and three in the spanwise direction. The film hole angle of inclination is 30 . 4. Computational Methodology CFD simulations in ANSYS Fluent (Ansys—Canonsburg, PA, USA) were used along- side the development of the model for early tests of performance, and to produce some loss correlations where the literature was unable to provide. These tests were run largely in a ‘2  2’ test case size, featuring only two impingement holes, two film holes (inclined at 90 ), and the pedestals that would feature in the flow paths between these positions. An example fluid domain for use in CFD is shown in Figure 4. Based on previous studies, the realizable k # model was chosen for turbulence mod- elling. Fluid density was made to vary according to the ideal gas law, thermal conductivity by kinetic theory, and viscosity by the Sutherland model. Simulations were run such that all flow velocities were well below a Mach Number of 0.3, allowing results from the incompressible LOM solver to be compared. To ensure a y of close to unity on all surfaces, inflation prism layers were used along all walls. Each inflation had 15 layers and a growth rate of 1.2, with a maximum total thickness of 10% of the hole diameter used. Int. J. Turbomach. Propuls. Power 2022, 7, 5 8 of 19 4. Computational Methodology CFD simulations in ANSYS Fluent (Ansys - Canonsburg, Pennsylvania, USA) were used alongside the development of the model for early tests of performance, and to pro- duce some loss correlations where the literature was unable to provide. These tests were run largely in a ‘2 × 2’ test case size, featuring only two impingement holes, two film holes Int. J. Turbomach. Propuls. Power 2022, 7, 5 8 of 18 (inclined at 90°), and the pedestals that would feature in the flow paths between these positions. An example fluid domain for use in CFD is shown in Figure 4. Figure 4. Fluid domain for CFD fluid domain. Figure 4. Fluid domain for CFD fluid domain. Based on previous studies, the realizable 𝑘− 𝜀 model was chosen for turbulence To ensure sufficient mesh refinement, a mesh independence study was performed. modelling. Fluid density was made to vary according to the ideal gas law, thermal con- The two quantities used to judge mesh independence were the total mass flow through the ductivity by kinetic theory, and viscosity by the Sutherland model. Simulations were run system and the coolant migration factor (13), the derivation of which is discussed further in such that all flow velocities were well below a Mach Number of 0.3, allowing results from Section 6.2. For the study conducted, the inlet pressure was set at 1.03 bar, the HP Outlet at the incompre 1.0 ssib bar le LOM , and the solver LP Outlet to be c atompared. To 0.99 bar. The ensure inlet te a y mperatur of close to e was un 300ity on K, and all all walls were surfaces, infla set tion prism as adiabatic. layers were Resultsus of ed the alon mesh g all independence walls. Each infl study ation had are shown 15 laye in rs Figur and e 5 giving a a growth rate of mesh 1.2 size , wiof th appr a maoximately ximum tota 3.5 l thic million kness elements. of 10% of the hole diameter used. Int. J. Turbomach. Propuls. Power 2022, 7, 5 9 of 19 To ensure sufficient mesh refinement, a mesh independence study was performed. Dm The two quantities used to judge mesh independence were the total mass flow through C MF = (13) 2m the system and the coolant migration factor (13), the derivation of which is discussed fur- in ther in Section 6.2. For the study conducted, the inlet pressure was set at 1.03 bar, the HP Outlet at 1.0 bar, and the LP Outlet at 0.99 bar. The inlet temperature was 300 K, and all walls were set as adiabatic. Results of the mesh independence study are shown in Figure 5 giving a mesh size of approximately 3.5 million elements. ∆𝑚 𝐶𝑀𝐹 = (13) 2𝑚 Figure 5. Mesh independence study results. Figure 5. Mesh independence study results. 5. Experimental Methodology 5. Experimental Methodology 5.1. Experimental Set-Up 5.1. Experimental Set-Up To validate the results of the LOM, a simple experimental rig consisting of a Bosch ALS To validate the results of the LOM, a simple experimental rig consisting of a Bosch 25 Blower/Vacuum [22], test piece, and six outlet pipes with orifice plates was constructed— ALS 25 Blower/Vacuum [22], test piece, and six outlet pipes with orifice plates was con- this rig was named the ‘Blower Rig’. For these experiments, two test pieces were used, D3 structed—this rig was named the ‘Blower Rig’. For these experiments, two test pieces were and D6. A cut-out diagram of these is shown in Figure 6b. Figure 6a shows the diagram used, D3 and D6. A cut-out diagram of these is shown in Figure 6b. Figure 6a shows the of a unit block of both test pieces—many of these combined to give a ‘wall block’ with six diagram of a unit block of both test pieces—many of these combined to give a ‘wall block’ with six rows of holes, each with four impingement holes and four effusion holes in a staggered formation (a size of ‘8 × 6’). Total lengths were 248 mm in the streamwise direc- tion and 178 mm in the spanwise direction (both including edge walls). A CAD diagram of the test section is shown in Figure 7. Flow from the blower enters the inlet plenum, contained behind the test piece, and is then forced through the test piece to one of six outlet channels, which are connected by hoses to pipes with orifice plates. Each outlet channel corresponds to one spanwise hole pitch and covers the whole streamwise extent of the test piece. At the entrance to said pipes, valves are used to restrict the flow, creating a spanwise pressure gradient. Mass flow rates from each channel are calculated using pressure measurements taken at either side of the orifice plates. Additional pressure tap- pings are located at the inlet and in the plenum, and at the end of each outlet channel. For this validation, the LOM was expanded to add links for flow leaving the film holes to collect at the channel exit—the point at the end of the channel serves as the new pressure boundary condition for the LOM. Int. J. Turbomach. Propuls. Power 2022, 7, 5 9 of 18 rows of holes, each with four impingement holes and four effusion holes in a staggered formation (a size of ‘8  6’). Total lengths were 248 mm in the streamwise direction and 178 mm in the spanwise direction (both including edge walls). A CAD diagram of the test section is shown in Figure 7. Flow from the blower enters the inlet plenum, contained behind the test piece, and is then forced through the test piece to one of six outlet channels, which are connected by hoses to pipes with orifice plates. Each outlet channel corresponds to one spanwise hole pitch and covers the whole streamwise extent of the test piece. At the entrance to said pipes, valves are used to restrict the flow, creating a spanwise pressure gradient. Mass flow rates from each channel are calculated using pressure measurements taken at either side of the orifice plates. Additional pressure tappings are located at the inlet and in the plenum, and at the end of each outlet channel. For this validation, the LOM Int. J. Turbomach. Propuls. Power 2022, 7, 5 10 of 19 Int. J. Turbomach. Propuls. Power 2022, 7, 5 10 of 19 was expanded to add links for flow leaving the film holes to collect at the channel exit—the point at the end of the channel serves as the new pressure boundary condition for the LOM. (a) (b) (a) (b) Figure 6. Test piece diagrams from Murray et al. [11]: (a) Unit block layout with dimensions. (b) Figure 6. Test piece diagrams from Murray et al. [11]: (a) Unit block layout with dimensions. (b) Figure 6. Test piece diagrams from Murray et al. [11]: (a) Unit block layout with dimensions. Sectioned 3D model. Used with Permission. Sectioned 3D model. Used with Permission. (b) Sectioned 3D model. Used with Permission. Figure 7. CAD model (Section Diagram) of the blower rig test section. Figure 7. CAD model (Section Diagram) of the blower rig test section. Figure 7. CAD model (Section Diagram) of the blower rig test section. 5.5. 2. Ex 2. Ex peri peri me me nn tal tal Unc Unc ertai ertai nn ty Anal ty Anal ysi ysi s s 5.2. Experimental Uncertainty Analysis Pressure measurements were made using First Sensor HCE Series pressure transduc- Pressure measurements were made using First Sensor HCE Series pressure transduc- Pressure measurements were made using First Sensor HCE Series pressure trans- ers [23] and recorded using a PicoLog ADC-24 High Resolution Data Logger [24]. Meas- ers [23] and recorded using a PicoLog ADC-24 High Resolution Data Logger [24]. Meas- ducers [23] and recorded using a PicoLog ADC-24 High Resolution Data Logger [24]. urements were recorded at 5 Hz for 30 s. In the tests conducted, uncertainties for each urements were recorded at 5 Hz for 30 s. In the tests conducted, uncertainties for each Measurements were recorded at 5 Hz for 30 s. In the tests conducted, uncertainties for pressure reading peaked at approximately ±250 Pa. Mass flow values across each orifice pressure reading peaked at approximately ±250 Pa. Mass flow values across each orifice each pressure reading peaked at approximately 250 Pa. Mass flow values across each plate, computed using the pressure drops across the orifice plates, had a maximum un- orifice plate, computed using plate, computed the using pressure the pressur drop es ac drops ross the acrossorifice p the orifice lateplates, s, had had a ma aximum un maximum- certainty of ±0.48 g/s and a minimum of ±0.45 g/s, giving an uncertainty on the total mass certainty of ±0.48 g/s and a minimum of ±0.45 g/s, giving an uncertainty on the total mass uncertainty of 0.48 g/s and a minimum of 0.45 g/s, giving an uncertainty on the total flow rate of up to ±1.2 g/s. Test cases where the relative uncertainty of any individual mass flow rate of up to ±1.2 g/s. Test cases where the relative uncertainty of any individual mass flow was over 25% were not used for LOM comparison. flow was over 25% were not used for LOM comparison. 6. Results and Discussion 6. Results and Discussion 6.6. 1. Ex 1. Ex peri peri me me nn tal tal Val Val id ia dt aitoin on The blower r The blower r ig was r ig was r uu n for n for many many separate tests, between which th separate tests, between which th e valve e valve posit posit ions ions for each channel were varied to produce a new outlet pressure distribution. The pressure for each channel were varied to produce a new outlet pressure distribution. The pressure readings recorded from each test were averaged and used as the input boundary condi- readings recorded from each test were averaged and used as the input boundary condi- tions to the LOM. Figure 8 compares the total inlet mass flow for each test case, for both tions to the LOM. Figure 8 compares the total inlet mass flow for each test case, for both experiments and the LOM. Within the range of cases investigated, the results were in good experiments and the LOM. Within the range of cases investigated, the results were in good agreement, with a maximum deviation of 10.9% from the experiment results. Agreement agreement, with a maximum deviation of 10.9% from the experiment results. Agreement for D6 results was hampered by some manufacturing damage that affected film hole size for D6 results was hampered by some manufacturing damage that affected film hole size and quality. and quality. Int. J. Turbomach. Propuls. Power 2022, 7, 5 10 of 18 mass flow rate of up to1.2 g/s. Test cases where the relative uncertainty of any individual mass flow was over 25% were not used for LOM comparison. 6. Results and Discussion 6.1. Experimental Validation The blower rig was run for many separate tests, between which the valve positions for each channel were varied to produce a new outlet pressure distribution. The pressure readings recorded from each test were averaged and used as the input boundary condi- tions to the LOM. Figure 8 compares the total inlet mass flow for each test case, for both experiments and the LOM. Within the range of cases investigated, the results were in good Int. J. Turbomach. Propuls. Power 2022, 7, 5 11 of 19 agreement, with a maximum deviation of 10.9% from the experiment results. Agreement for D6 results was hampered by some manufacturing damage that affected film hole size and quality. Figure 8. Test case mass flows, experiment result vs. LOM results. Figure 8. Test case mass flows, experiment result vs. LOM results. Figure 9 shows how experimental results from two individual D3 test cases com- Figure 9 shows how experimental results from two individual D3 test cases compared pared with those produced by the LOM. The pressure distribution is assessed using the with those produced by the LOM. The pressure distribution is assessed using the Inlet- Inlet-Outlet Channel Pressure Loss Coefficient (14) —the pressure loss from inlet to chan- Outlet Channel Pressure Loss Coefficient (14) —the pressure loss from inlet to channel nel outlet, normalized by the dynamic head of the total experimental mass flow passing outlet, normalized by the dynamic head of the total experimental mass flow passing through the total impingement hole area. The outlet mass flow in each case is assessed as through the total impingement hole area. The outlet mass flow in each case is assessed as a proportion of that method’s total mass flow. Across the range of test cases conducted, a proportion of that method’s total mass flow. Across the range of test cases conducted, individual channel outlet flow shares of the LOM generally fell within 20% of experi- individual channel outlet flow shares of the LOM generally fell within 20% of experimental r mental re esults, rising sults, risin only in g only extreme in extreme cas cases. The e high s. The hig spanwise h spapr nwi essur se press e gradient ure gradi imposed ent imin pos the ed test in the test ca case of Figur se of e 9 Fi bgure clearly 9b cl leads early lea to a mor ds to e a more uneven outl uneven outlet m et mass flow distribution, ass flow distr as ibution, row 1 receives ~10% of the total inlet mass flow, whilst rows 5 and 6 receive close to 20% each—in as row 1 receives ~10% of the total inlet mass flow, whilst rows 5 and 6 receive close to the near-uniform outlet pressure case of Figure 9a the flow distribution is more even. 20% each—in the near-uniform outlet pressure case of Figure 9a the flow distribution is more even. P P in out C = (14) P 𝑃 . −𝑃 1 in 𝐶 = 2 r A (14) 1 𝑚 i 2 𝜌 𝐴 Int. Int. J. Turb J. Turbomach. omach. Prop Propuls. uls. PoPower wer 2022 2022 , 7,, 5 7, 5 12 of 11 19 of 18 (a) (b) Figure 9. Experiment vs. LOM flow distribution for: (a) near-uniform spanwise pressure and (b) Figure 9. Experiment vs. LOM flow distribution for: (a) near-uniform spanwise pressure and (b) high high spanwise pressure gradient in 2 D3 blower rig tests. spanwise pressure gradient in 2 D3 blower rig tests. 6.2. Effects 6.2. Effects of Geome of Geometric tric PaParameters rameters Following validation of the LOM using the blower rig, it was used to analyse the Following validation of the LOM using the blower rig, it was used to analyse the efef fefect ct ofof seselected lected ge geometric ometric pa parameters rameters on on co coolant olant mmigration. igration. To T s oim simplify plify an analysis, alysis, ththe e investigated domain was reduced to a ‘2 2’ size, as used for CFD simulations in Section 4. investigated domain was reduced to a ‘2 × 2’ size, as used for CFD simulations in Section . . . In such a set-up, the proportion of total coolant that migrates, Dm/2m , (Dm being the 4. In such a set-up, the proportion of total coolant that migrates, ∆𝑚 /2𝑚 , (∆𝑚 being the in difference between the two outlet mass flows) is a function only of the pressure drops from difference between the two outlet mass flows) is a function only of the pressure drops the coolant supply (inlet pressure P ) to the two outlets with exit pressures P and P , the from the coolant supply (inlet pressure 0c 𝑃 ) to the two outlets with exit pressures 1 𝑃 an 2 d density, the viscosity, and the length scales s, d , d , L , L and L . In non-dimensional 𝑃 , the density, the viscosity, and the length scales 𝑠 , 𝑑 , 𝑑 , 𝐿 , 𝐿 and 𝐿 . In non-di- i f i f pd terms, the Coolant Migration Factor C MF is a function of the Pressure Drop Ratio PDR, mensional terms, the Coolant Migration Factor 𝐶𝑀𝐹 is a function of the Pressure Drop the system Reynolds number Re (15), and the ratio of the impingement hole diameter to sys Ratio 𝑃𝐷𝑅 , the system Reynolds number (15), and the ratio of the impingement the other geometric parameters (d /d , L /d , L /d and L /d ). f i i i f i pd i hole diameter to the other geometric parameters (𝑑 /𝑑 , 𝐿 /𝑑 , 𝐿 /𝑑 and 𝐿 /𝑑 ). . p d 𝑑 P 𝑃 P−𝑃 r 𝜌 ∆𝑚 Dm 𝑃 P −𝑃 P i 0c 1 0c 2 (15) 𝐶𝑀𝐹 = C MF = ,𝑃 , P𝐷𝑅= D R = ,𝑅 , Re𝑒 = = (15) sys 2𝑚 𝑃 P −𝑃 P m 𝜇 2m 0c 1 in To give context to these parameters, the 𝐶𝑀𝐹 serves as a measure of how much flow To give context to these parameters, the C MF serves as a measure of how much is being ‘stolen’ from one film hole by the other. A value of 0 means that the outlet flow flow is being ‘stolen’ from one film hole by the other. A value of 0 means that the outlet distribution is uniform. Any value between 0 and 0.5 means that some migration is occur- flow distribution is uniform. Any value between 0 and 0.5 means that some migration is ring, with a value of 0.5 indicating that the low pressure film hole is the outlet for all cool- occurring, with a value of 0.5 indicating that the low pressure film hole is the outlet for all ant in the system. If the 𝐶𝑀𝐹 > 0.5, the High Pressure film hole is ingesting mainstream coolant in the system. If the C MF > 0.5, the High Pressure film hole is ingesting mainstream flow. The 𝑃𝐷𝑅 quantifies the external pressure gradient relative to the coolant supply— flow. The PDR quantifies the external pressure gradient relative to the coolant supply—a a 𝑃𝐷𝑅 of unity means the external pressure gradient is flat. PDR of unity means the external pressure gradient is flat. Multiple series of simulations with the model were run to determine each geometric Multiple series of simulations with the model were run to determine each geometric factor’s effect on the 𝐶𝑀𝐹— 𝑅𝑃𝐷 relationship, which sets the level of coolant migration factor ’s effect on the C MFPDR relationship, which sets the level of coolant migration across across a range of operating conditions. In a 2 × 2 set-up where all features maintain con- a range of operating conditions. In a 2  2 set-up where all features maintain constant stant size (i.e., film holes are of the same diameter, etc.), a pressure drop ratio of 1 causes size (i.e., film holes are of the same diameter, etc.), a pressure drop ratio of 1 causes no no cool coolant ant migration. migrationAs . As the the PD 𝑃𝐷𝑅 R gr grow ows, s, coolant coolant migration migratioincr n incr eases, eases as , as sh shown own in in Figur Figure e 10a. 10This a. Thwas is wa found s found t to be o b re epeatable repeatabl if e i all f ageometric ll geometric ratios ratios were were kept kepconstant, t constantas , as the the sys- system tem R Reynolds eynolds number wa number was found s found to ha to have avminimal e a minim efa fect—this l effect—this is is shown shown in Figure 10b, in Figure 10b, where where the proportion of m the proportion of migrating igrating co coolantolan hardly t hard changes ly changes for wide for w ranges ide rang of R es o e f . . sys 𝑅𝑒 𝑅𝑒 Int. J. Turbomach. Propuls. Power 2022, 7, 5 13 of 19 Int. J. Turbomach. Propuls. Power 2022, 7, 5 12 of 18 (a) (b) Figure 10. Non-Dimensional Relations for 𝐶𝑀𝐹 : (a) 𝐶𝑀𝐹 vs. 𝑃𝐷𝑅 for 2 × 2 Model, (b) 𝐶𝑀𝐹 vs. 𝑃𝐷𝑅 Figure 10. Non-Dimensional Relations for C MF: (a) C MF vs. PDR for 2  2 Model, (b) C MF vs. for 2 × 2 Model at multiple PDRs. PDR for 2  2 Model at multiple PDRs. Figure 11a shows the effects of scaling the impingement hole and film hole diameters Figure 11a shows the effects of scaling the impingement hole and film hole diameters independently. Scaling each hole diameter has opposite effects—increasing the impinge- independently. Scaling each hole diameter has opposite effects—increasing the impinge- ment hole diameter reduces the proportion of coolant migrating, whereas increasing the ment hole diameter reduces the proportion of coolant migrating, whereas increasing the film hole diameter leads to greater migration. These effects are a product of how the pres- film hole diameter leads to greater migration. These effects are a product of how the sure pressur losse e s losses through thr ou each skin affect gh each skin af the aver fect the age average pressure in pressur the internal e in the internal cavity. As cavity noted . As in the re noted in lated the work [10], the imping related work [10], theeme impingement nt process do process minates pr dominates essure pr losse essur s in e losses doublein - wall e double-wall ffusion cooling effusion sy cooling stems, so systems, even in cas so even es win ith cases high extern with high al pressu external re gr pr ad essur ients, the e gra- impingement dients, the impingement hole mass flhole ows r mass emain c flows lose remain to equa close l and to tequal he flow and expan the flow dingexpanding from im- pingement po from impingement ints is large points ly at the is largely sameat prthe essure. Only same pressur aftere. the Only impingement proc after the impingement ess does process does the external pressure gradient influence the flow distribution, unless the the external pressure gradient influence the flow distribution, unless the array features a array features a variation in pedestal row constrictions, such as in Figure 3. Increasing the variation in pedestal row constrictions, such as in Figure 3. Increasing the impingement impingement hole diameter reduces the pressure drop across the inner skin, creating a hole diameter reduces the pressure drop across the inner skin, creating a larger pressure larger pressure margin across all film hole positions. The pressure drop ratio across the margin across all film hole positions. The pressure drop ratio across the outer skin is thus outer skin is thus reduced, limiting coolant migration. In contrast, increasing the film reduced, limiting coolant migration. In contrast, increasing the film hole diameter reduces hole diameter reduces pressure drops across the outer skin and shifts more pressure loss pressure drops across the outer skin and shifts more pressure loss to the impingement to the impingement process, increasing the pressure drop ratio across the outer skin and process, increasing the pressure drop ratio across the outer skin and increasing coolant increasing coolant migration. This implies that a low porosity outer skin is much less migration. This implies that a low porosity outer skin is much less vulnerable to coolant vulnerable to coolant migration than a high porosity one, though coolant migration effects migration than a high porosity one, though coolant migration effects must be considered must be considered alongside the likely reduction of film cooling effectiveness caused alongside the likely reduction of film cooling effectiveness caused by reduced outer skin by reduced outer skin porosity. If the external pressure distribution is known, it can be porosity. If the external pressure distribution is known, it can be beneficial to reduce the beneficial to reduce the diameter of film holes in low pressure regions whilst maintaining diameter of film holes in low pressure regions whilst maintaining high hole diameters in high hole diameters in high pressure ones—holes at the SS would have a greater pressure high pressure ones—holes at the SS would have a greater pressure drop per unit mass, drop per unit mass, compensating for the imbalance in outer skin pressure drop ratio and compensating for the imbalance in outer skin pressure drop ratio and limiting coolant limiting coolant migration. migration. The effect of scaling the hole lengths is shown in Figure 11b. In the range investi- The effect of scaling the hole lengths is shown in Figure 11b. In the range investigated, gated, increasing the length of the impingement hole reduced coolant migration, whereas increasing the length of the impingement hole reduced coolant migration, whereas in- increasing the length of film holes increased it. Changing the length of each hole alters creasing the length of film holes increased it. Changing the length of each hole alters its its discharge coefficient—for a length/diameter ratio below 2, C increases with the hole discharge coefficient—for a length/diameter ratio below 2, 𝐶 increases with the hole length as a longer hole length allows secondary flows to be suppressed. C begins to length as a longer hole length allows secondary flows to be suppressed. 𝐶 begins to de- decrease as the length to diameter ratio exceeds 2.5 due to excess friction. Therefore, in the crease as the length to diameter ratio exceeds 2.5 due to excess friction. Therefore, in the range investigated, discharge coefficients are increasing, leading to similar trends as those range investigated, discharge coefficients are increasing, leading to similar trends as those seen for scaling the hole diameters. A high C reduces pressure losses due to impingement d,i seen for scaling the hole diameters. A high 𝐶 reduces pressure losses due to impinge- and reduces the pressure drop ratio across the outer skin, and a high C has the opposite d, f ment and reduces the pressure drop ratio across the outer skin, and a high 𝐶 has the effect. As significant heat transfer takes place on the internal surfaces of the holes, cooling opposite effect. As significant heat transfer takes place on the internal surfaces of the holes, performance would need to be considered alongside migration in optimising the wall cooling performance would need to be considered alongside migration in optimising the thicknesses for overall cooling performance. wall thicknesses for overall cooling performance. Int. J. Turbomach. Propuls. Power 2022, 7, 5 14 of 19 Int. J. Turbomach. Propuls. Power 2022, 7, 5 13 of 18 Int. J. Turbomach. Propuls. Power 2022, 7, 5 14 of 19 (a) (b) (a) (b) Figure 11. Effect of scaling factors on 𝐶𝑀𝐹 : (a) 𝐶𝑀𝐹 vs. scaling factors for hole diameters at 1.5 𝑃𝐷𝑅 ; (b) 𝐶𝑀𝐹 vs. hole length/diameter ratios at 1.5 𝑃𝐷𝑅 . Figure 11. Effect of scaling factors on 𝐶𝑀𝐹 : (a) 𝐶𝑀𝐹 vs. scaling factors for hole diameters at 1.5 Figure 11. Effect of scaling factors on C MF: (a) C MF vs. scaling factors for hole diameters at 1.5 PDR; 𝑃𝐷𝑅 ; (b) 𝐶𝑀𝐹 vs. hole length/diameter ratios at 1.5 𝑃𝐷𝑅 . (b) C MF vs. hole length/diameter ratios at 1.5 PDR. In the context of an engine vane, multiple 𝑃𝐷𝑅 s would need to be considered when In the context of an engine vane, multiple 𝑃𝐷𝑅 s would need to be considered when investigating any given hole position. Positions partway along the chord would both gain In the context of an engine vane, multiple PDRs would need to be considered when investigating any given hole position. Positions partway along the chord would both gain coolant from higher pressure positions and lose coolant to lower pressure ones. investigating any given hole position. Positions partway along the chord would both gain coolant from higher Figure 12 show pressure s the effect posit of ch ions an angi d lose coolant to lower pressure ones. ng the pedestal width on the coolant migration. coolant from higher pressure positions and lose coolant to lower pressure ones. As the pe Figuredest 12 sho al width to ws the effect impingement of changihole d ng the iam pede eter ra stal ti width on o goes to the two, the a coolantv migr ailabl ation. e flow Figure 12 shows the effect of changing the pedestal width on the coolant migration. As the pe a As rea the goes to pedestal desta zero. In this regi l width to width to impingement impingement on, the flow ve hole d holelo idiameter am city through the pedest eter raratio tio goes to goes to two, the a two, al ar the ray i v available as ila hbl igh, e flle ow flow ad- area goes to zero. In this region, the flow velocity through the pedestal array is high, lead- ing t areaogoes high pr to zer essu o.re lo In sses this ir n egion, the region the flow —this ha velocity s a simil thra ough r effect the topedestal reducingarray the film ho is high, le ing to high pressure losses in the region—this has a similar effect to reducing the film hole diameter leading to , as less of the syst high pressure losses em pressu in the r re losse egion—this s occu has r ina th similar e impinge effect ment to r process, educing lea thed film ing diameter to a higher p hole diameter , as less of the syst ressure m , as less of arg the em pressu in across the system re losse pressur oute s occ r s e k losses in ur an in t d le occur he impinge ss coo in the lant m impingement m ent igr process, ation. When t lea pr d ocess, ing h e to a higher p pedestal w leading to a idt rhigher essure m h is lepr ss than essur argin across the e 1.5 t mar imes the gin acroute oss impi the r s ngement k outer in anskin d le hole ss coo and diameter less lant coolant m , the igrat effect migration. ion. When t of the ped- When he the pedestal width is less than 1.5 times the impingement hole diameter, the effect of the pedestal w estal width idt oh is le n coolant mig ss than 1.5 t ration imes the is muc impi h weaker, ngement as pressu hole diameter re losses are dist , the effect inct of ly les the ped- s than estal width pedestal width on coolant mig on coolant ration migration is mucis h weaker, much weaker as pressu , as pr re loss essur es are dist e losses ar inct e distinctly ly less than less those occurring through either set of holes. than those occurring through either set of holes. those occurring through either set of holes. Figure 12. 𝐶𝑀𝐹 vs. pedestal width/impingement hole diameter at 1.5 𝑃𝐷𝑅 . Figure 12. C MF vs. pedestal width/impingement hole diameter at 1.5 PDR. Figure 12. 𝐶𝑀𝐹 vs. pedestal width/impingement hole diameter at 1.5 𝑃𝐷𝑅 . 6.3. Full-Vane Analysis 6.3. Full-Vane Analysis 6.3. Full-Vane Analysis In order to demonstrate the issue of coolant migration within a double-wall effusion In order to demonstrate the issue of coolant migration within a double-wall effusion cooled nozzle guide vane, the LOM was adapted to fit a modern engine NGV, using flow cooled In order to demonstra nozzle guide vane, the LOM te the issue of wa coola s adapted to nt migra fit t a modern ion within en a doubl gine NG e-w V, using all effusi flo onw properties found experimentally by Holgate et al. [4]. A network diagram of this application cooled propert nozzle ies fo guide und exp vae ne, the LOM rimentally b w y Ho as ad lgat apted to e et al. [4 fit ] a modern . A network end gine iagram NG o V, using f this apflo plic w a- is shown in Figure 13. Due to the very high pressure gradient around the early Suction properties found experimentally by Holgate et al. [4]. A network diagram of this applica- tion is shown in Figure 13. Due to the very high pressure gradient around the early Suction Surface, the LOM was unable to resolve the internal flow-field for a continuous cavity, so tion is shown in Figure 13. Due to the very high pressure gradient around the early Suction Surface, the LOM was unable to resolve the internal flow-field for a continuous cavity, so results shown are for a test case where the LE’s internal cavity has been separated from Sur resu faclts sh e, the ow LO n are M wa fo s r una a te bst cas le to res e wher olve et the LE’ he interna s inte l flow-f rnal cav ield for ity h aa cont s been inuo separated from us cavity, so both the PS and SS. This shows a series of zero mass flows through the pedestal array. In resu both the PS and SS. Th lts shown are for a te is shows a ser st case wherie es of the LE’ zero m s inte ass rnal cav flows thro ity h ug as been h the pedestal arr separated from ay. In this configuration, the maximum film hole coolant flows are observed at the early SS, whilst both the PS and SS. Th this configuration, this shows a ser e maximum fi ies of lm hole zero m cooalant ss flows thro flows are ug obs h the pedestal arr erved at the ear ay ly SS . In , film holes around the LE produce little flow due to the much lower overall pressure margin this configuration, the maximum film hole coolant flows are observed at the early SS, Int. J. Turbomach. Propuls. Power 2022, 7, 5 15 of 19 Int. J. Turbomach. Propuls. Power 2022, 7, 5 14 of 18 whilst film holes around the LE produce little flow due to the much lower overall pressure margin for that section. In order to achieve a more even outlet coolant flow around the for that section. In order to achieve a more even outlet coolant flow around the vane, it vane, it would be necessary to increase hole sizes at the LE to allow more flow through, would be necessary to increase hole sizes at the LE to allow more flow through, or to reduce or to reduce hole sizes around the SS to restrict coolant flow. This presents a significant hole sizes around the SS to restrict coolant flow. This presents a significant advantage of advantage of the LOM, which can be used to rapidly assess the effects of many minor the LOM, which can be used to rapidly assess the effects of many minor design changes in design changes in an attempt to optimize the outlet flow distribution. an attempt to optimize the outlet flow distribution. Figure 13. Flow network for a double-wall effusion cooled NGV. Figure 13. Flow network for a double-wall effusion cooled NGV. CFD was used to review the effect of walls across the span on the outlet mass flow CFD was used to review the effect of walls across the span on the outlet mass flow distribution, the results of which are shown in Figure 14. This figure compares the average distribution, the results of which are shown in Figure 14. This figure compares the average outlet mass flow per row of holes, normalized by the value seen at the stagnation line, for outlet mass flow per row of holes, normalized by the value seen at the stagnation line, for three different set-ups; a continuous cavity (no walls), separated sections (LE, PS and SS three different set-ups; a continuous cavity (no walls), separated sections (LE, PS and SS separate, as in Figure 13 and completely separated (walls between every row of holes). It is separate, as in Figure 13 and completely separated (walls between every row of holes). It immediately clear from these results that without walls through the cavity, there is massive is immediately clear from these results that without walls through the cavity, there is mas- migration of coolant to the early suction surface, resulting in a significant imbalance in sive migration of coolant to the early suction surface, resulting in a significant imbalance the outlet coolant distribution. In this particular case, the use of walls across the span is in the outlet coolant distribution. In this particular case, the use of walls across the span is imperative to maintaining any sort of adequate film cooling at the LE. imperative to maintaining any sort of adequate film cooling at the LE. Even in the cases where the LE, PS and SS were separated from one another, the average per-hole coolant outlet mass flows were significantly greater for the SS than for other sections. To demonstrate potential methods of evening the outlet flow distribution, the LOM was used to scale up the diameters of film holes around the PS and LE, whilst scaling down the diameters of all film holes on the SS by the same proportion. Results of these tests are shown in Figure 15, which indicates that the hole diameters would need to be scaled by around 80% if the designer were pursuing a more uniform outlet flow. This scaling gives a maximum deviation in the per-hole average coolant mass flow of 19% from the stagnation row value, compared to 118% when no scaling is used. In an engine case, Int. J. Turbomach. Propuls. Power 2022, 7, 5 16 of 19 Int. J. Turbomach. Propuls. Power 2022, 7, 5 15 of 18 Int. J. Turbomach. Propuls. Power 2022, 7, 5 16 of 19 greater heat loads around the SS would lead to a greater demand for film cooling there, but maintaining adequate film cooling to the LE presents more of a challenge. Figure 14. CFD results for outlet mass flow distribution for different internal cavities. Even in the cases where the LE, PS and SS were separated from one another, the av- erage per-hole coolant outlet mass flows were significantly greater for the SS than for other sections. To demonstrate potential methods of evening the outlet flow distribution, the LOM was used to scale up the diameters of film holes around the PS and LE, whilst scaling down the diameters of all film holes on the SS by the same proportion. Results of these tests are shown in Figure 15, which indicates that the hole diameters would need to be scaled by around 80% if the designer were pursuing a more uniform outlet flow. This scaling gives a maximum deviation in the per-hole average coolant mass flow of 19% from the stagnation row value, compared to 118% when no scaling is used. In an engine case, greater heat loads around the SS would lead to a greater demand for film cooling there, but maintaining adequate film cooling to the LE presents more of a challenge. Figure 14. CFD results for outlet mass flow distribution for different internal cavities. Figure 14. CFD results for outlet mass flow distribution for different internal cavities. Even in the cases where the LE, PS and SS were separated from one another, the av- erage per-hole coolant outlet mass flows were significantly greater for the SS than for other sections. To demonstrate potential methods of evening the outlet flow distribution, the LOM was used to scale up the diameters of film holes around the PS and LE, whilst scaling down the diameters of all film holes on the SS by the same proportion. Results of these tests are shown in Figure 15, which indicates that the hole diameters would need to be scaled by around 80% if the designer were pursuing a more uniform outlet flow. This scaling gives a maximum deviation in the per-hole average coolant mass flow of 19% from the stagnation row value, compared to 118% when no scaling is used. In an engine case, greater heat loads around the SS would lead to a greater demand for film cooling there, but maintaining adequate film cooling to the LE presents more of a challenge. Figure 15. LOM results for outlet mass flow distributions in an NGV for scaling hole diameters. Figure 15. LOM results for outlet mass flow distributions in an NGV for scaling hole diameters. 6.4. Future Work Future work will address the effect of heat transfer on coolant migration, combining the Flow Network Model with a Low Order Thermal Network through the solid based on that developed by Murray et al. [25]. Given the high rise in coolant temperature as it passes through an engine vane’s double-wall cooling system, it is expected that coolant migration would be affected by heat transfer. Significant increases in the coolant temperature in the cavity would lead a reduction in the density of the coolant. A lower density would produce smaller outlet mass flows for the same pressure drops, leading to a reduction in the coolant migration factor. As noted previously, geometric changes made to combat coolant Figure 15. LOM results for outlet mass flow distributions in an NGV for scaling hole diameters. Int. J. Turbomach. Propuls. Power 2022, 7, 5 16 of 18 migration will have knock-on effects on the cooling performance of the geometry—it is important for these concerns to be considered alongside one another, as either could lead to catastrophic failure of the engine component. 7. Conclusions A low order flow network model was created to assess mass flow and pressure distributions through double-wall effusion cooling systems. The LOM has been developed to use empirical loss coefficients and discharge coefficient correlations to solve mass flows in and out of nodes in a network, solving for the pressures at internal nodes within the cooling system. This allows assessment of the vulnerability of a cooling scheme’s geometry to coolant migration under the influence of an external static pressure gradient. Reducing the coolant migration from high to low pressure regions is key to preventing localized areas of poor external film cooling effectiveness. The results of the LOM were validated using the Blower Rig, whereby a blower forced flow through a Double-Wall Effusion Cooling test piece, through which the flow was divided into six outlet channels and mass flows were measured independently. This allowed evaluation of the effect of an imposed spanwise pressure gradient on the outlet flow distribution. Results compared favourably with those of the LOM. The LOM performed a series of scaling simulations to show the effect of different geometric parameters on coolant migration. These tests showed that the effective method in reducing the Coolant Migration Factor was to minimise the Pressure Drop Ratio across the outer wall. This was done by reducing the share of the overall pressure losses caused by impingement—the geometric changes that caused this included increasing the diameter and length of the impingement holes, and reducing the diameter and length of the film holes. Large pedestal diameters, and thus low-inter pedestal spacing, produced the same results as small film hole diameters, but only at very high values. It has been suggested that a non-uniform distribution of hole diameters could be used to compensate for the effects of high external static pressure gradients on coolant migration. This was tested for a full vane example, where increasing the diameters of holes at the LE and SS whilst reducing those on the SS gave a far more uniform outlet mass flow distribution. Author Contributions: M.v.d.N. and P.I. established the goals of the work and the design of exper- iments. M.v.d.N. developed the Low Order Flow Network Model and performed CFD work and experiments under the supervision of P.I. All authors have read and agreed to the published version of the manuscript. Funding: This work is funded by Rolls Royce plc. and the EPSRC, under the CDT in Gas Turbine Aerodynamics (No. EP/L015943/1) and the Transpiration Cooling Systems for Jet Engine Turbines and Hypersonic Flight Programme Grant (No. EP/P000878/1). Data Availability Statement: Not Applicable. Acknowledgments: The authors wish to thank of Alexander Murray and the technicians at the Oxford Thermofluids Institute for their contributions to setting up experiments, and Luca Di Mare for help with numerical work. Additional thanks go to Ignacio Mayo, formerly of Rolls Royce plc., for his oversight of this study. Conflicts of Interest: The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Int. J. Turbomach. Propuls. Power 2022, 7, 5 17 of 18 Nomenclature A Area m C Discharge Coefficient C Pressure Coefficient C MF Coolant Migration Factor d, D Diameter (m) f Friction Factor h Specific Enthalpy (J/kg) H TC Heat Transfer Coefficient W/m K) k Loss Coefficient L Length (m) LE Leading Edge m Mass Flow Rate (kg/s) N GV Nozzle Guide Vane P Pressure (Pa) PDR Pressure Difference Ratio PS Pressure Surface Re Reynolds Number s Spacing (m) SFC Specific Fuel Consumption (kg/N s) T Temperature (K) T ET Turbine Entry Temperature (K) u, U Velocity (m/s) b Area Ratio # Expansibility Factor m Viscosity (Pa s) r Density kg/m Subscripts 0 Total c Coolant ext Exterior f Film/Effusion Hole h Hydraulic i Impingement Hole in Inlet out Outlet pd Pedestal sys System t Turning throat Throat w Wall ¥ Mainstream References 1. Van de Noort, M.; Ireland, P.T. A Low Order Flow Network Model for Double-Wall Effusion Cooling Systems. In Proceedings of the European Turbomachinery Conference ETC14 2021, Paper No. 666, Gdansk, Poland, 12–16 April 2021. 2. Bunker, R.S. Gas Turbine Heat Transfer: 10 Remaining Hot Gas Path Challenges. In Proceedings of the ASME Turbo Expo 2006: Power for Land, Sea, and Air, Barcelona, Spain, 8–11 May 2006. 3. Murray, A.V.; Ireland, P.T.; Rawlinson, A.J. An Integrated Conjugate Computational Approach for Evaluating the Aerothermal and Thermomechanical Performance of Double-Wall Effusion Cooled Systems. In Proceedings of the ASME Turbo Expo 2017: Turbomachinery Technical Conference and Exposition, Charlotte, NC, USA, 26–30 June 2017. 4. Holgate, N.; Cresci, I.; Ireland, P.; Rawlinson, A. Prediction and augmentation of nozzle guide vane film cooling hole pressure margin. In Proceedings of the 12th European Conference on Turbomachinery Fluid dynamics & Thermodynamics, Stockholm, Sweden, 3–7 April 2017. 5. Campbell, C.X.; Morrison, J.A. Turbine Airfoil With a Compliant Outer Wall. U.S. Patent 8,147,196 B2, 3 April 2012. 6. Devore, M.A.; Paauwe, C.S. Turbine Airfoil with Improved Cooling. U.S. Patent 7,600,966 B2, 13 October 2009. Int. J. Turbomach. Propuls. Power 2022, 7, 5 18 of 18 7. Chyu, M.K.; Hsing, Y.C.; Natarajan, V. Convective Heat Transfer of Cubic Fin Arrays in a Narrow Channel. J. Turbomach. 1998, 120, 363–367. [CrossRef] 8. Siw, S.C.; Fradeneck, A.D.; Chyu, M.K.; Alvin, M.A. The Effects of Different Pin-Fin Arrays on Heat Transfer and Pressure Loss in a Narrow Channel. In Proceedings of the ASME Turbo Expo 2015: Turbine Technical Conference and Exposition, Montreal, QC, Canada, 15–19 June 2015. 9. Bamba, T.; Kumagai, T.; Mimura, F.; Yamane, T.; Fukuyama, Y.; Usui, T.; Yoshida, T. Leading Edge Cooling Performance of an Integrated Cooling Configuration. In Proceedings of the ASME Turbo Expo 2008: Power for Land, Sea, and Air, Berlin, Germany, 9–13 June 2008. 10. Andrews, G.E.; Asere, A.A.; Husain, C.I.; Mkpadi, M.C. Full Coverage Impingement Heat Transfer: The Variation in Pitch to Diameter Ratio at a Constant Gap. In AGARD Heat Transfer and Cooling in Gas Turbines; AGARD: Bergen, Norway, 1985. 11. Murray, A.V.; Ireland, P.T.; Romero, E. Experimental and Computational Methods for the Evaluation of Double-Wall, Effusion Cooling Systems. In Proceedings of the ASME Turbo Expo 2019: Turbomachinery Technical Conference and Exposition, Phoenix, AZ, USA, 17–21 June 2019. 12. Wambersie, A.; Wong, H.; Ireland, P.; Mayo, I. Experiments of Transpiration Cooling Inspired Panel Cooling on a Turbine Blade Yielding Film Effectiveness Levels over 95%. Int. J. Turbomach. Propuls. Power 2021, 6, 16. [CrossRef] 13. Rose, J.R. NASA TM-73774: FLOWNET: A Computer Program for Calculating Secondary Flow Conditions in a Network of Turbomachinery; NASA: Cleveland, OH, USA, 1978. 14. Kutz, K.J.; Speer, T.M. Simulation of the Secondary Air System of Aero Engines. In Proceedings of the ASME 1992 International Gas Turbine and Aeroengine Congress and Exposition, Cologne, Germany, 1–4 June 1992. 15. Ebenhoch, G.; Speer, T.M. Simulation of Cooling Systems in Gas Turbines. In Proceedings of the ASME 1994 International Gas Turbine and Aeroengine Congress and Exposition, The Hague, The Netherlands, 13–16 June 1994. 16. Gouws, J.; Morris, R.; Visser, J. Modelling of a gas turbine combustor using a network solver. S. Afr. J. Sci. 2006, 102, 533–536. 17. Jin, H.; Riddle, A.; Cooke, L. A Compressible Flow Network Analysis for Design Upgrade of Industrial Aeroderivative High Pressure Turbine Blades. In Proceedings of the ASME Turbo Expo 2008: Power for Land, Sea, and Air, Berlin, Germany, 9–13 June 2008. 18. Remani, C. Numerical Methods for Solving Systems of Non-Linear Equations; Lakehead University: Ontario, QC, Canada, 2013. 19. Mazzei, L.; Winchler, L.; Andreini, A. Development of a numerical correlation for the discharge coefficient of round inclined holes with low crossflow. Comput. Fluids 2017, 152, 182–192. [CrossRef] 20. Wang, Z. The Application of Thermochromic Liquid Crystals to Detailed Turbine Blade Cooling Measurements. Ph.D. Thesis, University of Oxford, Oxford, UK, 1991. 21. Miller, D.S. Internal Flow Systems, 2nd ed.; Miller Innovations: Bedford, UK, 2009. 22. Robert Bosch Power Tools GmbH. ALS 2400|25|2500|28|30; Robert Bosch Power Tools GmbH: Stuttgart, Germany, 2017. 23. First Sensor. HCE Series: Miniature Amplified Pressure Sensors Datasheet; First Sensor: Berlin, Germany, 2020. 24. Pico Technology. ADC-20 and ADC-24: High Resolution Data Logger Datasheet; Pico Technology: St Neots, UK, 2019. 25. Murray, A.V.; Ireland, P.T.; Romero, E. An Experimentally Validated Low Order Model of the Thermal Response of Double-Wall Effusion Cooling Systems for HP Turbine Blades. In Proceedings of the ASME Turbo Expo 2020: Turbomachinery Technical Conference and Exposition, Online, 21–25 September 2020. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Turbomachinery, Propulsion and Power Multidisciplinary Digital Publishing Institute

A Low Order Flow Network Model for Double-Wall Effusion Cooling Systems

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International Journal of Turbomachinery Propulsion and Power Article A Low Order Flow Network Model for Double-Wall Effusion Cooling Systems Michael van de Noort * and Peter Ireland Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, UK; peter.ireland@eng.ox.ac.uk * Correspondence: michael.vandenoort@eng.ox.ac.uk † This paper is an extended version of our paper published in Proceedings of the European Turbomachinery Conference ETC14 2021, Paper No. 666, Gdansk, Poland, 12–16 April 2021. Abstract: The high pressure turbine nozzle guide vane of a modern aeroengine experiences large heat loads and thus requires both highly effective internal and external cooling. This can be accomplished with double-wall effusion cooling, which combines impingement, pin-fin and effusion cooling. The combination of three cooling mechanisms causes high pressure losses, increasing potential for the migration of coolant towards low pressure regions, subsequently starving effusion holes on the leading edge of coolant supply. This paper presents a low order flow network model to rapidly assess the pressure and mass flow distributions through such cooling schemes for a flexible set of geometric and flow conditions. The model is subsequently validated by a series of experiments with varying mainstream pressure gradients. Results from the model are used to indicate design parameters to reduce the effect of coolant migration, and to minimise the risk of destructive hot gas ingestion. Keywords: coolant migration; flow networks; double-wall effusion cooling Citation: van de Noort, M.; Ireland, P. 1. Introduction A Low Order Flow Network Model Studies into the thermal efficiency of gas turbines have come to the consensus that the for Double-Wall Effusion Cooling specific fuel consumption can be improved by increasing the turbine entry temperature. To Systems. Int. J. Turbomach. Propuls. maximise this benefit, modern aeroengines use TETs well in excess of the turbine compo- Power 2022, 7, 5. https://doi.org/ nents’ melting temperatures. Cooling systems are required to prevent component failure, 10.3390/ijtpp7010005 but these reduce power output by reducing the air available for combustion and causing Academic Editor: Francesco mixing losses in the turbine, which reduce aerodynamic efficiency. Double-wall effusion Martelli cooling is viewed as a potential solution to these challenges—the high porosity make-up Received: 9 December 2021 of the system brings it close to the goal of accomplishing micro-cooling. Bunker [1,2] Accepted: 31 January 2022 noted that mirco-cooling has two goals—“spread[ing] out the cooling network in a series Published: 2 February 2022 of smaller and highly distributed channels” and “bringing the cooling fluid closer to the outer surface of the airfoil”. This is accomplished by combining three commonly used Publisher’s Note: MDPI stays neutral cooling techniques—impingement cooling, pin-fin cooling and effusion (or film) cooling, with regard to jurisdictional claims in with features densely packed to spread out the coolant, and double-walled to allow coolant published maps and institutional affil- to sit close to the outer surface. iations. Figure 1 shows an example of double-wall effusion cooling being employed in a modern commercial engine turbine blade. This combination of cooling techniques leads to high convection cooling efficiency (1) and overall cooling effectiveness (2), but also Copyright: © 2022 by the authors. produces high pressure losses. T T Licensee MDPI, Basel, Switzerland. c,e c,i h = (1) conv This article is an open access article T T m c,i distributed under the terms and T T ¥ m conditions of the Creative Commons # = (2) T T c,i Attribution (CC BY-NC-ND) license (https://creativecommons.org/ licenses/by-nc-nd/4.0/). Int. J. Turbomach. Propuls. Power 2022, 7, 5. https://doi.org/10.3390/ijtpp7010005 https://www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2022, 7, 5 2 of 19 𝑇 −𝑇 Int. J. Turbomach. Propuls. Power 2022, 7, 5 2 of 18 𝜀 = (2) 𝑇 −𝑇 Figure 1. Turbine blade using double-wall effusion cooling scheme—From Murray et al., 2017 [3]. Figure 1. Turbine blade using double-wall effusion cooling scheme—From Murray et al., 2017 [3]. Used with Permission. Used with Permission. The high The high pressure pressure losses caused losses caused b by comb y combining ining thr three co ee cooling oling schemes schemecan s can make make d double- ou- ble-wall effusion cooling systems vulnerable to coolant migration—the movement of cool- wall effusion cooling systems vulnerable to coolant migration—the movement of coolant in ant in the pedestal cavity toward low pressure regions under the influence of an external the pedestal cavity toward low pressure regions under the influence of an external pressure p gradient. ressure grCoolant adient. Cmigration oolant migr can ation can be particularly be particularly concerning concerning in components in components where where the the cool coolant a supply nt suppl toy to mainstrea mainstream stag m st nation agnati pr on pressure ra essure ratio is tio low—the is low—the ma main example in examof ple of this being the Turbine’s High Pressure NGV, where the mainstream stagnation pressure is only this being the Turbine’s High Pressure NGV, where the mainstream stagnation pressure ~3% less than that of the coolant supplied from the HP Compressor. In a configuration such is only ~3% less than that of the coolant supplied from the HP Compressor. In a configu- as that shown in Figure 1, where the cavity between the walls is continuous along both ration such as that shown in Figure 1, where the cavity between the walls is continuous surfaces, the streamwise pressure gradient will lead to coolant in the cavity migrating away along both surfaces, the streamwise pressure gradient will lead to coolant in the cavity from the LE sections and toward film holes further along the PS and SS. Spanwise pressure migrating away from the LE sections and toward film holes further along the PS and SS. gradients can have similar effects, causing cooling coolant to move from the hub to the Spanwise pressure gradients can have similar effects, causing cooling coolant to move tip. Results produced by Holgate et al. [4] showed that for an engine representative NGV, from the hub to the tip. Results produced by Holgate et al. [4] showed that for an engine the external static pressure coefficient C (3) could fall from 0.2 to 1.2 along the early representative NGV, the external static pressure coeff p icient 𝐶 (3) could fall from −0.2 to suction surface alone—given the small pressure margin at the leading edge, this provides a −1.2 along the early suction surface alone—given the small pressure margin at the leading major driving force for coolant migration. edge, this provides a major driving force for coolant migration. 𝑃 −𝑃 P P ext ext, max C 𝐶 == (3) 1 (3) r U 𝜌 𝑈 ¥ ¥ throat To combat coolant migration, recent patented designs using this cooling scheme (e.g., To combat coolant migration, recent patented designs using this cooling scheme [5,6]) use walls across the span of the cavity to create separate cooling zones for the PS, LE (e.g., [5,6]) use walls across the span of the cavity to create separate cooling zones for the PS, and SS, but this reduces the effectiveness of pin-fin cooling due to reduced throughflow LE and SS, but this reduces the effectiveness of pin-fin cooling due to reduced throughflow and turbulence generation. and turbulence generation. Coolant Coolant migr migration ation can h can have ave severel severely y de detrimental trimental impact impacts s on t on the he ffilm ilm cool cooling ing e ef ffec fec- - tiveness in both the region the coolant migrates away from and the region it migrates to. tiveness in both the region the coolant migrates away from and the region it migrates to. If coolant m If coolant migration igration is h is high igh enough, th enough, the e region region los losing ing cool coolant ant c can an ess essentially entially b be e st starved, arved, lea leading ding t to o de destr struct uctive ive hot hot g gas as ingestion, ingestion,which which w willileventually l eventually cause causcomponent e componen failur t faile. ure. In the region that the coolant migrates to, a high pressure margin across the outer skin causes In the region that the coolant migrates to, a high pressure margin across the outer skin film hole flows to be ejected into the mainstream with high velocity, potentially leading to causes film hole flows to be ejected into the mainstream with high velocity, potentially jet-lift off. In both regions, film cooling effectiveness would be significantly reduced, so the component would experience much greater heat loads and would likely have a much Int. J. Turbomach. Propuls. Power 2022, 7, 5 3 of 18 shorter operating life. Predicting and controlling coolant migration is key to ensuring the proper development of a coolant film for external cooling. This paper summarizes work done to model coolant migration within double-wall effusion cooling systems under the influence of mainstream pressure gradients, done through the use of low order flow network models (LOMs). The LOM developed is a high speed, iterative solver that produces mass flow and pressure distributions throughout cooling arrays for given boundary conditions. This allowed assessment of a high number of design possibilities that would’ve been too computationally expensive to do in a time- efficient manner using CFD. 2. Related Work 2.1. Double-Wall Effusion Cooling Systems Research has been conducted into the application of double-wall effusion cooling systems since the 1970s, but their application in modern engines is limited due to issues of manufacturing difficulty and high thermomechanical stresses. As transpiration cooling research is becoming more prevalent, double-wall effusion cooling has become a topic of increased interest—the high manufactured porosity, complex internal flow field, and low film hole blowing ratio lead to such schemes being dubbed as “quasi-transpiration” cooling by Murray et al. [3], who conducted a series of computational tests to find the effects of various geometric parameters on the thermomechanical stresses developed. Increases of the passage height, pedestal diameter and hole diameters were shown to reduce the average thermomechanical stress, as did reducing the pedestal spacing. Flow travelling through arrays of pedestals can undergo high pressure loss due to the wakes generated behind the pedestals. Research by Chyu et al. [7] and Siw et al. [8] showed that staggered arrays of pedestals caused much greater pressure drops than in-line arrays. The pedestal shape can also have a large influence—cylindrical pedestals produce less pressure loss than those of square or diamond (square rotated by 45 ) shapes [7], but also produce lower HTCs. Bamba et al. [9] found experimentally that in regions where pin-fins did not signif- icantly obstruct the passage of flow from impingement to film hole entry, such as at the LE, the contribution of the pin-fins to the pressure loss was negligible. In these regions, pedestals enhance cooling only by increasing the surface area over which heat transfer can take place. The dominant form of pressure loss in these systems, however, is from impingement cooling. Andrews et al. [10] stated that “for impingement/effusion applications it is . . . valid to assume that all the pressure loss would occur at the impingement plate”. Greater pressure losses generally equate to increased cooling performance, as is the case for pin-fin arrays. Murray et al. [11] conducted a comprehensive study of the effect of the cooling scheme’s porosity on its performance. Higher porosity geometries (particularly those with high porosity in the walls) required less coolant mass flow for a given pressure drop and exhibited “relatively high cooling effectiveness” at low coolant flow rates. The au- thors suggested cooling around the blade could be optimised by varying the porosity as required—high pressure drop regions (i.e., the SS) should use low porosity cooling geometries to minimise aerodynamic losses, whilst areas such as the PS should use higher porosity geometries for increased coolant flow and better effusion cooling performance. Wambersie et al. [12] conducted experimental tests using high-porosity panels, in- spired by transpiration-cooling designs. Whilst these panels gave exceptionally high film effectiveness levels at standard blowing rates, at low flow rates the low pressure margin made the panels susceptible to mainstream flow ingestion, localised to certain regions as a result of high streamwise and spanwise pressure gradients. Int. J. Turbomach. Propuls. Power 2022, 7, 5 4 of 18 2.2. Flow Networks The models developed for this paper were based on flow network solvers developed by Rose [13], and later by Kutz and Speer [14]. In these systems, flow networks were produced by creating lists of nodes, each representing a specific point in the cooling system. Nodes are connected by links, along which fluid is allowed to flow, forming a network of connected nodes. Boundary nodes form the inlet and outlet of flow within a network. In order to solve for the flow through a network, the law of mass conservation is applied at each internal node—for a node i connected to n nodes j, the governing equation is given by (4) The mass flow rate along a link connecting nodes i and j is determined by Equation (5) where the function f is used to model the three-dimensional flow as one-dimensional. i,j Kutz and Speer [14] expanded from solving for continuity to account for the heat transfer along each link Q by making use of the energy equation, adding another governing i,j Equation (6) at each node. This network approach was applied to a full secondary air system, and results for the rear sealing chamber air pressure were in good agreement with engine test results. m = 0 (4) i,j m = f P , P , T (5) i,j i,j i j i m h + Q = 0 (6) å i,j i,j i,j Using a similar methodology, Ebenhoch and Speer [15] produced a network solver that was applied to three test cases; isothermal flow in a multipass blade cooling system, coolant flow undergoing high heat transfer in a hypersonic vehicle nozzle, and coolant flow in a rotating HP blade. This flexible network solver was deemed to have “sufficient accuracy in predicting coolant mass flow, regional coolant side heat transfer coefficients, and local coolant temperatures”. Gouws et al. [16] created a network solver for a modern combustor that, once calibrated with experimental data, predicted the mass flow splits through cooling holes with high accuracy. Heat transfer elements within the network allowed a prediction of the outlet temperature distribution. It was found that in a cooling system with large numbers of cooling holes, discharge coefficients had a high influence on one-dimensional flow and pressure drop predictions. Jin et al. [17] used a compressible flow network analysis to find the mass flow distribu- tion for the various positions around the trailing edge of an HP turbine blade, as part of an investigation into blade oxidation. This investigation revealed that an inlet metering plate at the blade root reduced the coolant internal pressure, raising the risk of hot gas ingestion, particularly in “off-design or transient conditions”. The network allowed testing and recommendation of changes to reduce this risk. 3. Modelling Methodology 3.1. Flow Network Construction The LOM presented focused only on mass flow continuity and was based on test cases without heat transfer and in incompressible conditions. This allowed the temperature of each node to be neglected, with the model solving only for the pressures at each node. Newton’s method (see [18]) was used to iteratively solve for the internal pressure distri- bution. Static pressures at each node served as the list of unknowns, which could then be used to calculate mass flows through the network. In these conditions, the mass flow m i,j along a link between nodes i and j becomes a function only of the link’s compliance, C , i,j and the pressure difference between the nodes, such that: m = C  f P P (7) i,j i,j i,j i j Int. J. Turbomach. Propuls. Power 2022, 7, 5 5 of 19 Int. J. Turbomach. Propuls. Power 2022, 7, 5 5 of 18 𝑚 = 𝐶 ∙ 𝑓 (𝑃 −𝑃 ) (7) , , , The geometries discussed in this paper used a staggered arrangement of film holes The geometries discussed in this paper used a staggered arrangement of film holes and impingement holes, such that no holes of the same type were located next to one and impingement holes, such that no holes of the same type were located next to one another—this layout is the same as shown in Figure 1. The standard flow network layout another—this layout is the same as shown in Figure 1. The standard flow network layout is shown for a single row of holes and pedestals in Figure 2—in the case of there being is shown for a single row of holes and pedestals in Figure 2—in the case of there being multiple rows, further nodes would be located into the page, with impingement and film multiple rows, further nodes would be located into the page, with impingement and film hol holes es a alternating lternating le left/right ft/right posi positions. tions. The posi The position tion of ea ofch node ha each nodes has three components: po- three components: sition position in the in the streamwise streamwise didir rection, ection, level in level in the cool the cooling ing system system(1 (1 at the bottom, 5 a at the bottom, 5 at t the the top), and spanwise row number. Other than for nodes on the pedestal layer (Level 4), all top), and spanwise row number. Other than for nodes on the pedestal layer (Level 4), all nodes nodes on onthe same layer ar the same layer ar e e of of the same the same type. type. LinLinks ks are cre areacr ted base eated based d on the physical flow on the physical flow paths available from each node’s position, so flow is allowed to transfer to nodes on paths available from each node’s position, so flow is allowed to transfer to nodes on the the same level only on Level 4—the other levels only have links to nodes directly above same level only on Level 4—the other levels only have links to nodes directly above or or below. below. Figure 2. Flow network diagram for a single row. Figure 2. Flow network diagram for a single row. The number label for each node determines its type: The number label for each node determines its type: • 10: Impingement Hole Entry/Inlet Nodes. Boundary nodes with fixed pressure. 10: Impingement Hole Entry/Inlet Nodes. Boundary nodes with fixed pressure. • 20: Impingement Hole Exit Nodes. Node pressure is representative of the flow as it 20: Impingement Hole Exit Nodes. Node pressure is representative of the flow as it leave leaves s t the he ho hole. le. • 30: Impin 30: Impingement gement Point Point No Nodes. des. Nod Nodes es re repr present the p esent the point oint at at which which flow flow stagnate stagnates s on on the inner surface of the outer wall as the jet flow from the hole impinges. the inner surface of the outer wall as the jet flow from the hole impinges. 40: Expansion from Impingement Nodes. Represents flow as it accelerates away from • 40: Expansion from Impingement Nodes. Represents flow as it accelerates away from the i the impingement mpingement poi points, nts, but bef but befor ore e iit t intera interacts cts wi with th a any ny pedestals. In the di pedestals. In the diagram, agram, this point is represented as being above the impingement point for ease of display. this point is represented as being above the impingement point for ease of display. 41: Post Pedestal Nodes. Frictional pressure losses due to pedestals are modelled in • 41: Post Pedestal Nodes. Frictional pressure losses due to pedestals are modelled in the link between nodes 40 and 41. the link between nodes 40 and 41. 42: Film Entry Hole Nodes. As noted in Figure 2, these nodes are only used in the • 42: Film Entry Hole Nodes. As noted in Figure 2, these nodes are only used in the case when the film hole inclination is not 90 —when film holes are perpendicular, the case when the film hole inclination is not 90°—when film holes are perpendicular, 42 nodes are merged with linked 41 nodes. When holes are inclined, flow from one the 42 nodes are merged with linked 41 nodes. When holes are inclined, flow from direction turns by a different angle than flow from the opposite direction, requiring one direction turns by a different angle than flow from the opposite direction, requir- the use of a different dynamic head loss coefficient. ing the use of a different dynamic head loss coefficient. 50: Film Hole Exit Nodes. Boundary nodes with fixed pressure. • 50: Film Hole Exit Nodes. Boundary nodes with fixed pressure. The mass flow functions f P P and compliances C are dependent on the type i,j i j i,j The mass flow functions 𝑓 (𝑃 −𝑃 ) and compliances 𝐶 are dependent on the , , of link. Flow through a hole, occurring for impingement holes between Levels 1 and 2, and type of link. Flow through a hole, occurring for impingement holes between Levels 1 and occurring for film holes between Levels 4 and 5, allows the discharge Equation (8) to be 2, and occurring for film holes between Levels 4 and 5, allows the discharge Equation (8) used—all holes of both types are cylindrical in this geometry. In cases where high coolant to be used—all holes of both types are cylindrical in this geometry. In cases where high migration and thus an uneven outlet flow distribution is expected, it is unreasonable to coolant migration and thus an uneven outlet flow distribution is expected, it is unreason- assume that all film holes will have similar discharge coefficients. To account for this, the able to assume that all film holes will have similar discharge coefficients. To account for discharge coefficient for each hole was updated in each iteration. The two pressure values from the previous iteration were used to calculate the area-averaged velocity through the Int. J. Turbomach. Propuls. Power 2022, 7, 5 6 of 18 hole. This velocity, the pressure ratio across the hole and geometric properties were used to calculate an updated discharge coefficient using a correlation developed by Mazzei et al. [19]. This was applied to both impingement and film holes. The area ratios b and b use a square inlet area with a side length of half the hole pitch, and the expansibility factor # is taken as unity as the air is assumed to be incompressible. 2 p pd d,i i m = # 2r P P , 10 20 1b 2 (8) p p . C pd d, f f m = # 2r P P f 42 1b The connection between Levels 2 and 3 represents the sudden expansion of flow area, when flow leaves the impingement hole and enters the inter-wall cavity. At Level 3, flow is expected to stagnate as it impinges on the outer wall, allowing the form of the Bernoulli equation shown in (9). to be used—this features a loss coefficient k to account for sudden l,ex expansion pressure loss. 1 1 . r A 2 2 P + rU k rU = P ) m = P P (9) 20 20 l,ex 20 30 30 20 2 2 (1 k ) l,ex To account for the total pressure loss produced by a jet impinging on a wall, a similar loss equation is used for the link between Levels 3 and 4. A single node is used in this position at Level 4, as it assumed that there is no circumferential variation in total pressure loss due to impingement. 1 1 . r A 2 2 P = P + rU + k rU ) m = P P (10) 30 40 40 40 30 40 l,i 2 2 (1 + k ) l,i Equation (11) accounts for flow around pedestals on Level 4 between nodes 40 and 41. The friction factor f for flow around pedestals is determined by the correlation developed by Wang [20]. p 2 . r A D h 0.318 m = P P , f = 1.76Re (11) 40 41 max 2 f L When turning flows are employed, the mass flow between nodes 41 and 42 is given by (12). For turning flows, loss coefficients were selected from studies by Miller [21]. In cases with close packed pedestals, (11). is replaced by a flow coefficient in the same form as (12), as pressure losses become functions of impingement onto pedestals and turning between them, rather than by friction. This equation uses the assumption that A is larger than A . 41 42 1 2 1 2 1 2 P + rU = P + rU + k rU 41 41 42 42 l,t 42 2 2 2 (12) . 2 2 2r A A 41 42 ) m = P P 41 42 2 2 A 1+k A ( ) 41 l,t 42 3.2. Computational Implementation The LOM was produced in MATLAB (MathWorks—Natick, MA, USA), using sum- mations for the mass flows out of nodes described in the equations above, as in (4). The initial guess of pressures through the network was selected to ensure that pressure drops and rises occurred along the links that were expected. To enhance the stability of the model, changes between iterations were damped. The value of the damping factor z used was 0.5 in simpler models, such as the geometry featured in Figure 3, but more complex geometries or high mainstream pressure gradients often required z to be reduced further. This slowed convergence of the model, though the solving time remained on the order of seconds, a huge reduction from CFD cases. Convergence of the LOM was attained when the maximum absolute value of mass flow imbalance at any node was less than 10 kg/s. Int. J. Turbomach. Propuls. Power 2022, 7, 5 7 of 19 3.2. Computational Implementation The LOM was produced in MATLAB (MathWorks—Natick, MA, USA), using sum- mations for the mass flows out of nodes described in the equations above, as in (4). The initial guess of pressures through the network was selected to ensure that pressure drops and rises occurred along the links that were expected. To enhance the stability of the model, changes between iterations were damped. The value of the damping factor 𝜁 used was 0.5 in simpler models, such as the geometry featured in Figure 3, but more com- plex geometries or high mainstream pressure gradients often required 𝜁 to be reduced further. This slowed convergence of the model, though the solving time remained on the order of seconds, a huge reduction from CFD cases. Convergence of the LOM was attained −12 when the maximum absolute value of mass flow imbalance at any node was less than 10 kg/s. An additional mass flow check was performed by comparing the inlet and outlet total mass flows. 3.3. Example Results Figure 3 shows a set of example results for the LOM. Flow paths are contoured by their mass flow rate. In this test case, all film holes are ejecting to the same outlet pressure, and all impingement holes have the same inlet pressure. Despite this, the film holes in the central row (2nd spanwise position) clearly have larger outlet flows than those in the outer rows. The position of the impingement holes leads to film holes at the right end (highest streamwise distance) of the array each having ~75% of the coolant flow that each film hole in the centre receives. The unevenness in the outlet flow distribution is not matched at the inlet—the two impingement holes at the left end of the array (lowest streamwise distance) Int. J. Turbomach. Propuls. Power 2022, 7, 5 7 of 18 are the most poorly fed, but receive only 5% less flow than those in the centre of the array. The size of this array and the inclination angle of the film holes are flexible—this particular test case size is ‘6 × 3’, with six hole positions in streamwise direction and three in the An additional mass flow check was performed by comparing the inlet and outlet total spanwise direction. The film hole angle of inclination is 30°. mass flows. Figure 3. Example Figure 3. resultsEx fram omple resu the LOM. lts from the LOM. 3.3. Example Results Figure 3 shows a set of example results for the LOM. Flow paths are contoured by their mass flow rate. In this test case, all film holes are ejecting to the same outlet pressure, and all impingement holes have the same inlet pressure. Despite this, the film holes in the central row (2nd spanwise position) clearly have larger outlet flows than those in the outer rows. The position of the impingement holes leads to film holes at the right end (highest streamwise distance) of the array each having ~75% of the coolant flow that each film hole in the centre receives. The unevenness in the outlet flow distribution is not matched at the inlet—the two impingement holes at the left end of the array (lowest streamwise distance) are the most poorly fed, but receive only 5% less flow than those in the centre of the array. The size of this array and the inclination angle of the film holes are flexible—this particular test case size is ‘6  3’, with six hole positions in streamwise direction and three in the spanwise direction. The film hole angle of inclination is 30 . 4. Computational Methodology CFD simulations in ANSYS Fluent (Ansys—Canonsburg, PA, USA) were used along- side the development of the model for early tests of performance, and to produce some loss correlations where the literature was unable to provide. These tests were run largely in a ‘2  2’ test case size, featuring only two impingement holes, two film holes (inclined at 90 ), and the pedestals that would feature in the flow paths between these positions. An example fluid domain for use in CFD is shown in Figure 4. Based on previous studies, the realizable k # model was chosen for turbulence mod- elling. Fluid density was made to vary according to the ideal gas law, thermal conductivity by kinetic theory, and viscosity by the Sutherland model. Simulations were run such that all flow velocities were well below a Mach Number of 0.3, allowing results from the incompressible LOM solver to be compared. To ensure a y of close to unity on all surfaces, inflation prism layers were used along all walls. Each inflation had 15 layers and a growth rate of 1.2, with a maximum total thickness of 10% of the hole diameter used. Int. J. Turbomach. Propuls. Power 2022, 7, 5 8 of 19 4. Computational Methodology CFD simulations in ANSYS Fluent (Ansys - Canonsburg, Pennsylvania, USA) were used alongside the development of the model for early tests of performance, and to pro- duce some loss correlations where the literature was unable to provide. These tests were run largely in a ‘2 × 2’ test case size, featuring only two impingement holes, two film holes Int. J. Turbomach. Propuls. Power 2022, 7, 5 8 of 18 (inclined at 90°), and the pedestals that would feature in the flow paths between these positions. An example fluid domain for use in CFD is shown in Figure 4. Figure 4. Fluid domain for CFD fluid domain. Figure 4. Fluid domain for CFD fluid domain. Based on previous studies, the realizable 𝑘− 𝜀 model was chosen for turbulence To ensure sufficient mesh refinement, a mesh independence study was performed. modelling. Fluid density was made to vary according to the ideal gas law, thermal con- The two quantities used to judge mesh independence were the total mass flow through the ductivity by kinetic theory, and viscosity by the Sutherland model. Simulations were run system and the coolant migration factor (13), the derivation of which is discussed further in such that all flow velocities were well below a Mach Number of 0.3, allowing results from Section 6.2. For the study conducted, the inlet pressure was set at 1.03 bar, the HP Outlet at the incompre 1.0 ssib bar le LOM , and the solver LP Outlet to be c atompared. To 0.99 bar. The ensure inlet te a y mperatur of close to e was un 300ity on K, and all all walls were surfaces, infla set tion prism as adiabatic. layers were Resultsus of ed the alon mesh g all independence walls. Each infl study ation had are shown 15 laye in rs Figur and e 5 giving a a growth rate of mesh 1.2 size , wiof th appr a maoximately ximum tota 3.5 l thic million kness elements. of 10% of the hole diameter used. Int. J. Turbomach. Propuls. Power 2022, 7, 5 9 of 19 To ensure sufficient mesh refinement, a mesh independence study was performed. Dm The two quantities used to judge mesh independence were the total mass flow through C MF = (13) 2m the system and the coolant migration factor (13), the derivation of which is discussed fur- in ther in Section 6.2. For the study conducted, the inlet pressure was set at 1.03 bar, the HP Outlet at 1.0 bar, and the LP Outlet at 0.99 bar. The inlet temperature was 300 K, and all walls were set as adiabatic. Results of the mesh independence study are shown in Figure 5 giving a mesh size of approximately 3.5 million elements. ∆𝑚 𝐶𝑀𝐹 = (13) 2𝑚 Figure 5. Mesh independence study results. Figure 5. Mesh independence study results. 5. Experimental Methodology 5. Experimental Methodology 5.1. Experimental Set-Up 5.1. Experimental Set-Up To validate the results of the LOM, a simple experimental rig consisting of a Bosch ALS To validate the results of the LOM, a simple experimental rig consisting of a Bosch 25 Blower/Vacuum [22], test piece, and six outlet pipes with orifice plates was constructed— ALS 25 Blower/Vacuum [22], test piece, and six outlet pipes with orifice plates was con- this rig was named the ‘Blower Rig’. For these experiments, two test pieces were used, D3 structed—this rig was named the ‘Blower Rig’. For these experiments, two test pieces were and D6. A cut-out diagram of these is shown in Figure 6b. Figure 6a shows the diagram used, D3 and D6. A cut-out diagram of these is shown in Figure 6b. Figure 6a shows the of a unit block of both test pieces—many of these combined to give a ‘wall block’ with six diagram of a unit block of both test pieces—many of these combined to give a ‘wall block’ with six rows of holes, each with four impingement holes and four effusion holes in a staggered formation (a size of ‘8 × 6’). Total lengths were 248 mm in the streamwise direc- tion and 178 mm in the spanwise direction (both including edge walls). A CAD diagram of the test section is shown in Figure 7. Flow from the blower enters the inlet plenum, contained behind the test piece, and is then forced through the test piece to one of six outlet channels, which are connected by hoses to pipes with orifice plates. Each outlet channel corresponds to one spanwise hole pitch and covers the whole streamwise extent of the test piece. At the entrance to said pipes, valves are used to restrict the flow, creating a spanwise pressure gradient. Mass flow rates from each channel are calculated using pressure measurements taken at either side of the orifice plates. Additional pressure tap- pings are located at the inlet and in the plenum, and at the end of each outlet channel. For this validation, the LOM was expanded to add links for flow leaving the film holes to collect at the channel exit—the point at the end of the channel serves as the new pressure boundary condition for the LOM. Int. J. Turbomach. Propuls. Power 2022, 7, 5 9 of 18 rows of holes, each with four impingement holes and four effusion holes in a staggered formation (a size of ‘8  6’). Total lengths were 248 mm in the streamwise direction and 178 mm in the spanwise direction (both including edge walls). A CAD diagram of the test section is shown in Figure 7. Flow from the blower enters the inlet plenum, contained behind the test piece, and is then forced through the test piece to one of six outlet channels, which are connected by hoses to pipes with orifice plates. Each outlet channel corresponds to one spanwise hole pitch and covers the whole streamwise extent of the test piece. At the entrance to said pipes, valves are used to restrict the flow, creating a spanwise pressure gradient. Mass flow rates from each channel are calculated using pressure measurements taken at either side of the orifice plates. Additional pressure tappings are located at the inlet and in the plenum, and at the end of each outlet channel. For this validation, the LOM Int. J. Turbomach. Propuls. Power 2022, 7, 5 10 of 19 Int. J. Turbomach. Propuls. Power 2022, 7, 5 10 of 19 was expanded to add links for flow leaving the film holes to collect at the channel exit—the point at the end of the channel serves as the new pressure boundary condition for the LOM. (a) (b) (a) (b) Figure 6. Test piece diagrams from Murray et al. [11]: (a) Unit block layout with dimensions. (b) Figure 6. Test piece diagrams from Murray et al. [11]: (a) Unit block layout with dimensions. (b) Figure 6. Test piece diagrams from Murray et al. [11]: (a) Unit block layout with dimensions. Sectioned 3D model. Used with Permission. Sectioned 3D model. Used with Permission. (b) Sectioned 3D model. Used with Permission. Figure 7. CAD model (Section Diagram) of the blower rig test section. Figure 7. CAD model (Section Diagram) of the blower rig test section. Figure 7. CAD model (Section Diagram) of the blower rig test section. 5.5. 2. Ex 2. Ex peri peri me me nn tal tal Unc Unc ertai ertai nn ty Anal ty Anal ysi ysi s s 5.2. Experimental Uncertainty Analysis Pressure measurements were made using First Sensor HCE Series pressure transduc- Pressure measurements were made using First Sensor HCE Series pressure transduc- Pressure measurements were made using First Sensor HCE Series pressure trans- ers [23] and recorded using a PicoLog ADC-24 High Resolution Data Logger [24]. Meas- ers [23] and recorded using a PicoLog ADC-24 High Resolution Data Logger [24]. Meas- ducers [23] and recorded using a PicoLog ADC-24 High Resolution Data Logger [24]. urements were recorded at 5 Hz for 30 s. In the tests conducted, uncertainties for each urements were recorded at 5 Hz for 30 s. In the tests conducted, uncertainties for each Measurements were recorded at 5 Hz for 30 s. In the tests conducted, uncertainties for pressure reading peaked at approximately ±250 Pa. Mass flow values across each orifice pressure reading peaked at approximately ±250 Pa. Mass flow values across each orifice each pressure reading peaked at approximately 250 Pa. Mass flow values across each plate, computed using the pressure drops across the orifice plates, had a maximum un- orifice plate, computed using plate, computed the using pressure the pressur drop es ac drops ross the acrossorifice p the orifice lateplates, s, had had a ma aximum un maximum- certainty of ±0.48 g/s and a minimum of ±0.45 g/s, giving an uncertainty on the total mass certainty of ±0.48 g/s and a minimum of ±0.45 g/s, giving an uncertainty on the total mass uncertainty of 0.48 g/s and a minimum of 0.45 g/s, giving an uncertainty on the total flow rate of up to ±1.2 g/s. Test cases where the relative uncertainty of any individual mass flow rate of up to ±1.2 g/s. Test cases where the relative uncertainty of any individual mass flow was over 25% were not used for LOM comparison. flow was over 25% were not used for LOM comparison. 6. Results and Discussion 6. Results and Discussion 6.6. 1. Ex 1. Ex peri peri me me nn tal tal Val Val id ia dt aitoin on The blower r The blower r ig was r ig was r uu n for n for many many separate tests, between which th separate tests, between which th e valve e valve posit posit ions ions for each channel were varied to produce a new outlet pressure distribution. The pressure for each channel were varied to produce a new outlet pressure distribution. The pressure readings recorded from each test were averaged and used as the input boundary condi- readings recorded from each test were averaged and used as the input boundary condi- tions to the LOM. Figure 8 compares the total inlet mass flow for each test case, for both tions to the LOM. Figure 8 compares the total inlet mass flow for each test case, for both experiments and the LOM. Within the range of cases investigated, the results were in good experiments and the LOM. Within the range of cases investigated, the results were in good agreement, with a maximum deviation of 10.9% from the experiment results. Agreement agreement, with a maximum deviation of 10.9% from the experiment results. Agreement for D6 results was hampered by some manufacturing damage that affected film hole size for D6 results was hampered by some manufacturing damage that affected film hole size and quality. and quality. Int. J. Turbomach. Propuls. Power 2022, 7, 5 10 of 18 mass flow rate of up to1.2 g/s. Test cases where the relative uncertainty of any individual mass flow was over 25% were not used for LOM comparison. 6. Results and Discussion 6.1. Experimental Validation The blower rig was run for many separate tests, between which the valve positions for each channel were varied to produce a new outlet pressure distribution. The pressure readings recorded from each test were averaged and used as the input boundary condi- tions to the LOM. Figure 8 compares the total inlet mass flow for each test case, for both experiments and the LOM. Within the range of cases investigated, the results were in good Int. J. Turbomach. Propuls. Power 2022, 7, 5 11 of 19 agreement, with a maximum deviation of 10.9% from the experiment results. Agreement for D6 results was hampered by some manufacturing damage that affected film hole size and quality. Figure 8. Test case mass flows, experiment result vs. LOM results. Figure 8. Test case mass flows, experiment result vs. LOM results. Figure 9 shows how experimental results from two individual D3 test cases com- Figure 9 shows how experimental results from two individual D3 test cases compared pared with those produced by the LOM. The pressure distribution is assessed using the with those produced by the LOM. The pressure distribution is assessed using the Inlet- Inlet-Outlet Channel Pressure Loss Coefficient (14) —the pressure loss from inlet to chan- Outlet Channel Pressure Loss Coefficient (14) —the pressure loss from inlet to channel nel outlet, normalized by the dynamic head of the total experimental mass flow passing outlet, normalized by the dynamic head of the total experimental mass flow passing through the total impingement hole area. The outlet mass flow in each case is assessed as through the total impingement hole area. The outlet mass flow in each case is assessed as a proportion of that method’s total mass flow. Across the range of test cases conducted, a proportion of that method’s total mass flow. Across the range of test cases conducted, individual channel outlet flow shares of the LOM generally fell within 20% of experi- individual channel outlet flow shares of the LOM generally fell within 20% of experimental r mental re esults, rising sults, risin only in g only extreme in extreme cas cases. The e high s. The hig spanwise h spapr nwi essur se press e gradient ure gradi imposed ent imin pos the ed test in the test ca case of Figur se of e 9 Fi bgure clearly 9b cl leads early lea to a mor ds to e a more uneven outl uneven outlet m et mass flow distribution, ass flow distr as ibution, row 1 receives ~10% of the total inlet mass flow, whilst rows 5 and 6 receive close to 20% each—in as row 1 receives ~10% of the total inlet mass flow, whilst rows 5 and 6 receive close to the near-uniform outlet pressure case of Figure 9a the flow distribution is more even. 20% each—in the near-uniform outlet pressure case of Figure 9a the flow distribution is more even. P P in out C = (14) P 𝑃 . −𝑃 1 in 𝐶 = 2 r A (14) 1 𝑚 i 2 𝜌 𝐴 Int. Int. J. Turb J. Turbomach. omach. Prop Propuls. uls. PoPower wer 2022 2022 , 7,, 5 7, 5 12 of 11 19 of 18 (a) (b) Figure 9. Experiment vs. LOM flow distribution for: (a) near-uniform spanwise pressure and (b) Figure 9. Experiment vs. LOM flow distribution for: (a) near-uniform spanwise pressure and (b) high high spanwise pressure gradient in 2 D3 blower rig tests. spanwise pressure gradient in 2 D3 blower rig tests. 6.2. Effects 6.2. Effects of Geome of Geometric tric PaParameters rameters Following validation of the LOM using the blower rig, it was used to analyse the Following validation of the LOM using the blower rig, it was used to analyse the efef fefect ct ofof seselected lected ge geometric ometric pa parameters rameters on on co coolant olant mmigration. igration. To T s oim simplify plify an analysis, alysis, ththe e investigated domain was reduced to a ‘2 2’ size, as used for CFD simulations in Section 4. investigated domain was reduced to a ‘2 × 2’ size, as used for CFD simulations in Section . . . In such a set-up, the proportion of total coolant that migrates, Dm/2m , (Dm being the 4. In such a set-up, the proportion of total coolant that migrates, ∆𝑚 /2𝑚 , (∆𝑚 being the in difference between the two outlet mass flows) is a function only of the pressure drops from difference between the two outlet mass flows) is a function only of the pressure drops the coolant supply (inlet pressure P ) to the two outlets with exit pressures P and P , the from the coolant supply (inlet pressure 0c 𝑃 ) to the two outlets with exit pressures 1 𝑃 an 2 d density, the viscosity, and the length scales s, d , d , L , L and L . In non-dimensional 𝑃 , the density, the viscosity, and the length scales 𝑠 , 𝑑 , 𝑑 , 𝐿 , 𝐿 and 𝐿 . In non-di- i f i f pd terms, the Coolant Migration Factor C MF is a function of the Pressure Drop Ratio PDR, mensional terms, the Coolant Migration Factor 𝐶𝑀𝐹 is a function of the Pressure Drop the system Reynolds number Re (15), and the ratio of the impingement hole diameter to sys Ratio 𝑃𝐷𝑅 , the system Reynolds number (15), and the ratio of the impingement the other geometric parameters (d /d , L /d , L /d and L /d ). f i i i f i pd i hole diameter to the other geometric parameters (𝑑 /𝑑 , 𝐿 /𝑑 , 𝐿 /𝑑 and 𝐿 /𝑑 ). . p d 𝑑 P 𝑃 P−𝑃 r 𝜌 ∆𝑚 Dm 𝑃 P −𝑃 P i 0c 1 0c 2 (15) 𝐶𝑀𝐹 = C MF = ,𝑃 , P𝐷𝑅= D R = ,𝑅 , Re𝑒 = = (15) sys 2𝑚 𝑃 P −𝑃 P m 𝜇 2m 0c 1 in To give context to these parameters, the 𝐶𝑀𝐹 serves as a measure of how much flow To give context to these parameters, the C MF serves as a measure of how much is being ‘stolen’ from one film hole by the other. A value of 0 means that the outlet flow flow is being ‘stolen’ from one film hole by the other. A value of 0 means that the outlet distribution is uniform. Any value between 0 and 0.5 means that some migration is occur- flow distribution is uniform. Any value between 0 and 0.5 means that some migration is ring, with a value of 0.5 indicating that the low pressure film hole is the outlet for all cool- occurring, with a value of 0.5 indicating that the low pressure film hole is the outlet for all ant in the system. If the 𝐶𝑀𝐹 > 0.5, the High Pressure film hole is ingesting mainstream coolant in the system. If the C MF > 0.5, the High Pressure film hole is ingesting mainstream flow. The 𝑃𝐷𝑅 quantifies the external pressure gradient relative to the coolant supply— flow. The PDR quantifies the external pressure gradient relative to the coolant supply—a a 𝑃𝐷𝑅 of unity means the external pressure gradient is flat. PDR of unity means the external pressure gradient is flat. Multiple series of simulations with the model were run to determine each geometric Multiple series of simulations with the model were run to determine each geometric factor’s effect on the 𝐶𝑀𝐹— 𝑅𝑃𝐷 relationship, which sets the level of coolant migration factor ’s effect on the C MFPDR relationship, which sets the level of coolant migration across across a range of operating conditions. In a 2 × 2 set-up where all features maintain con- a range of operating conditions. In a 2  2 set-up where all features maintain constant stant size (i.e., film holes are of the same diameter, etc.), a pressure drop ratio of 1 causes size (i.e., film holes are of the same diameter, etc.), a pressure drop ratio of 1 causes no no cool coolant ant migration. migrationAs . As the the PD 𝑃𝐷𝑅 R gr grow ows, s, coolant coolant migration migratioincr n incr eases, eases as , as sh shown own in in Figur Figure e 10a. 10This a. Thwas is wa found s found t to be o b re epeatable repeatabl if e i all f ageometric ll geometric ratios ratios were were kept kepconstant, t constantas , as the the sys- system tem R Reynolds eynolds number wa number was found s found to ha to have avminimal e a minim efa fect—this l effect—this is is shown shown in Figure 10b, in Figure 10b, where where the proportion of m the proportion of migrating igrating co coolantolan hardly t hard changes ly changes for wide for w ranges ide rang of R es o e f . . sys 𝑅𝑒 𝑅𝑒 Int. J. Turbomach. Propuls. Power 2022, 7, 5 13 of 19 Int. J. Turbomach. Propuls. Power 2022, 7, 5 12 of 18 (a) (b) Figure 10. Non-Dimensional Relations for 𝐶𝑀𝐹 : (a) 𝐶𝑀𝐹 vs. 𝑃𝐷𝑅 for 2 × 2 Model, (b) 𝐶𝑀𝐹 vs. 𝑃𝐷𝑅 Figure 10. Non-Dimensional Relations for C MF: (a) C MF vs. PDR for 2  2 Model, (b) C MF vs. for 2 × 2 Model at multiple PDRs. PDR for 2  2 Model at multiple PDRs. Figure 11a shows the effects of scaling the impingement hole and film hole diameters Figure 11a shows the effects of scaling the impingement hole and film hole diameters independently. Scaling each hole diameter has opposite effects—increasing the impinge- independently. Scaling each hole diameter has opposite effects—increasing the impinge- ment hole diameter reduces the proportion of coolant migrating, whereas increasing the ment hole diameter reduces the proportion of coolant migrating, whereas increasing the film hole diameter leads to greater migration. These effects are a product of how the pres- film hole diameter leads to greater migration. These effects are a product of how the sure pressur losse e s losses through thr ou each skin affect gh each skin af the aver fect the age average pressure in pressur the internal e in the internal cavity. As cavity noted . As in the re noted in lated the work [10], the imping related work [10], theeme impingement nt process do process minates pr dominates essure pr losse essur s in e losses doublein - wall e double-wall ffusion cooling effusion sy cooling stems, so systems, even in cas so even es win ith cases high extern with high al pressu external re gr pr ad essur ients, the e gra- impingement dients, the impingement hole mass flhole ows r mass emain c flows lose remain to equa close l and to tequal he flow and expan the flow dingexpanding from im- pingement po from impingement ints is large points ly at the is largely sameat prthe essure. Only same pressur aftere. the Only impingement proc after the impingement ess does process does the external pressure gradient influence the flow distribution, unless the the external pressure gradient influence the flow distribution, unless the array features a array features a variation in pedestal row constrictions, such as in Figure 3. Increasing the variation in pedestal row constrictions, such as in Figure 3. Increasing the impingement impingement hole diameter reduces the pressure drop across the inner skin, creating a hole diameter reduces the pressure drop across the inner skin, creating a larger pressure larger pressure margin across all film hole positions. The pressure drop ratio across the margin across all film hole positions. The pressure drop ratio across the outer skin is thus outer skin is thus reduced, limiting coolant migration. In contrast, increasing the film reduced, limiting coolant migration. In contrast, increasing the film hole diameter reduces hole diameter reduces pressure drops across the outer skin and shifts more pressure loss pressure drops across the outer skin and shifts more pressure loss to the impingement to the impingement process, increasing the pressure drop ratio across the outer skin and process, increasing the pressure drop ratio across the outer skin and increasing coolant increasing coolant migration. This implies that a low porosity outer skin is much less migration. This implies that a low porosity outer skin is much less vulnerable to coolant vulnerable to coolant migration than a high porosity one, though coolant migration effects migration than a high porosity one, though coolant migration effects must be considered must be considered alongside the likely reduction of film cooling effectiveness caused alongside the likely reduction of film cooling effectiveness caused by reduced outer skin by reduced outer skin porosity. If the external pressure distribution is known, it can be porosity. If the external pressure distribution is known, it can be beneficial to reduce the beneficial to reduce the diameter of film holes in low pressure regions whilst maintaining diameter of film holes in low pressure regions whilst maintaining high hole diameters in high hole diameters in high pressure ones—holes at the SS would have a greater pressure high pressure ones—holes at the SS would have a greater pressure drop per unit mass, drop per unit mass, compensating for the imbalance in outer skin pressure drop ratio and compensating for the imbalance in outer skin pressure drop ratio and limiting coolant limiting coolant migration. migration. The effect of scaling the hole lengths is shown in Figure 11b. In the range investi- The effect of scaling the hole lengths is shown in Figure 11b. In the range investigated, gated, increasing the length of the impingement hole reduced coolant migration, whereas increasing the length of the impingement hole reduced coolant migration, whereas in- increasing the length of film holes increased it. Changing the length of each hole alters creasing the length of film holes increased it. Changing the length of each hole alters its its discharge coefficient—for a length/diameter ratio below 2, C increases with the hole discharge coefficient—for a length/diameter ratio below 2, 𝐶 increases with the hole length as a longer hole length allows secondary flows to be suppressed. C begins to length as a longer hole length allows secondary flows to be suppressed. 𝐶 begins to de- decrease as the length to diameter ratio exceeds 2.5 due to excess friction. Therefore, in the crease as the length to diameter ratio exceeds 2.5 due to excess friction. Therefore, in the range investigated, discharge coefficients are increasing, leading to similar trends as those range investigated, discharge coefficients are increasing, leading to similar trends as those seen for scaling the hole diameters. A high C reduces pressure losses due to impingement d,i seen for scaling the hole diameters. A high 𝐶 reduces pressure losses due to impinge- and reduces the pressure drop ratio across the outer skin, and a high C has the opposite d, f ment and reduces the pressure drop ratio across the outer skin, and a high 𝐶 has the effect. As significant heat transfer takes place on the internal surfaces of the holes, cooling opposite effect. As significant heat transfer takes place on the internal surfaces of the holes, performance would need to be considered alongside migration in optimising the wall cooling performance would need to be considered alongside migration in optimising the thicknesses for overall cooling performance. wall thicknesses for overall cooling performance. Int. J. Turbomach. Propuls. Power 2022, 7, 5 14 of 19 Int. J. Turbomach. Propuls. Power 2022, 7, 5 13 of 18 Int. J. Turbomach. Propuls. Power 2022, 7, 5 14 of 19 (a) (b) (a) (b) Figure 11. Effect of scaling factors on 𝐶𝑀𝐹 : (a) 𝐶𝑀𝐹 vs. scaling factors for hole diameters at 1.5 𝑃𝐷𝑅 ; (b) 𝐶𝑀𝐹 vs. hole length/diameter ratios at 1.5 𝑃𝐷𝑅 . Figure 11. Effect of scaling factors on 𝐶𝑀𝐹 : (a) 𝐶𝑀𝐹 vs. scaling factors for hole diameters at 1.5 Figure 11. Effect of scaling factors on C MF: (a) C MF vs. scaling factors for hole diameters at 1.5 PDR; 𝑃𝐷𝑅 ; (b) 𝐶𝑀𝐹 vs. hole length/diameter ratios at 1.5 𝑃𝐷𝑅 . (b) C MF vs. hole length/diameter ratios at 1.5 PDR. In the context of an engine vane, multiple 𝑃𝐷𝑅 s would need to be considered when In the context of an engine vane, multiple 𝑃𝐷𝑅 s would need to be considered when investigating any given hole position. Positions partway along the chord would both gain In the context of an engine vane, multiple PDRs would need to be considered when investigating any given hole position. Positions partway along the chord would both gain coolant from higher pressure positions and lose coolant to lower pressure ones. investigating any given hole position. Positions partway along the chord would both gain coolant from higher Figure 12 show pressure s the effect posit of ch ions an angi d lose coolant to lower pressure ones. ng the pedestal width on the coolant migration. coolant from higher pressure positions and lose coolant to lower pressure ones. As the pe Figuredest 12 sho al width to ws the effect impingement of changihole d ng the iam pede eter ra stal ti width on o goes to the two, the a coolantv migr ailabl ation. e flow Figure 12 shows the effect of changing the pedestal width on the coolant migration. As the pe a As rea the goes to pedestal desta zero. In this regi l width to width to impingement impingement on, the flow ve hole d holelo idiameter am city through the pedest eter raratio tio goes to goes to two, the a two, al ar the ray i v available as ila hbl igh, e flle ow flow ad- area goes to zero. In this region, the flow velocity through the pedestal array is high, lead- ing t areaogoes high pr to zer essu o.re lo In sses this ir n egion, the region the flow —this ha velocity s a simil thra ough r effect the topedestal reducingarray the film ho is high, le ing to high pressure losses in the region—this has a similar effect to reducing the film hole diameter leading to , as less of the syst high pressure losses em pressu in the r re losse egion—this s occu has r ina th similar e impinge effect ment to r process, educing lea thed film ing diameter to a higher p hole diameter , as less of the syst ressure m , as less of arg the em pressu in across the system re losse pressur oute s occ r s e k losses in ur an in t d le occur he impinge ss coo in the lant m impingement m ent igr process, ation. When t lea pr d ocess, ing h e to a higher p pedestal w leading to a idt rhigher essure m h is lepr ss than essur argin across the e 1.5 t mar imes the gin acroute oss impi the r s ngement k outer in anskin d le hole ss coo and diameter less lant coolant m , the igrat effect migration. ion. When t of the ped- When he the pedestal width is less than 1.5 times the impingement hole diameter, the effect of the pedestal w estal width idt oh is le n coolant mig ss than 1.5 t ration imes the is muc impi h weaker, ngement as pressu hole diameter re losses are dist , the effect inct of ly les the ped- s than estal width pedestal width on coolant mig on coolant ration migration is mucis h weaker, much weaker as pressu , as pr re loss essur es are dist e losses ar inct e distinctly ly less than less those occurring through either set of holes. than those occurring through either set of holes. those occurring through either set of holes. Figure 12. 𝐶𝑀𝐹 vs. pedestal width/impingement hole diameter at 1.5 𝑃𝐷𝑅 . Figure 12. C MF vs. pedestal width/impingement hole diameter at 1.5 PDR. Figure 12. 𝐶𝑀𝐹 vs. pedestal width/impingement hole diameter at 1.5 𝑃𝐷𝑅 . 6.3. Full-Vane Analysis 6.3. Full-Vane Analysis 6.3. Full-Vane Analysis In order to demonstrate the issue of coolant migration within a double-wall effusion In order to demonstrate the issue of coolant migration within a double-wall effusion cooled nozzle guide vane, the LOM was adapted to fit a modern engine NGV, using flow cooled In order to demonstra nozzle guide vane, the LOM te the issue of wa coola s adapted to nt migra fit t a modern ion within en a doubl gine NG e-w V, using all effusi flo onw properties found experimentally by Holgate et al. [4]. A network diagram of this application cooled propert nozzle ies fo guide und exp vae ne, the LOM rimentally b w y Ho as ad lgat apted to e et al. [4 fit ] a modern . A network end gine iagram NG o V, using f this apflo plic w a- is shown in Figure 13. Due to the very high pressure gradient around the early Suction properties found experimentally by Holgate et al. [4]. A network diagram of this applica- tion is shown in Figure 13. Due to the very high pressure gradient around the early Suction Surface, the LOM was unable to resolve the internal flow-field for a continuous cavity, so tion is shown in Figure 13. Due to the very high pressure gradient around the early Suction Surface, the LOM was unable to resolve the internal flow-field for a continuous cavity, so results shown are for a test case where the LE’s internal cavity has been separated from Sur resu faclts sh e, the ow LO n are M wa fo s r una a te bst cas le to res e wher olve et the LE’ he interna s inte l flow-f rnal cav ield for ity h aa cont s been inuo separated from us cavity, so both the PS and SS. This shows a series of zero mass flows through the pedestal array. In resu both the PS and SS. Th lts shown are for a te is shows a ser st case wherie es of the LE’ zero m s inte ass rnal cav flows thro ity h ug as been h the pedestal arr separated from ay. In this configuration, the maximum film hole coolant flows are observed at the early SS, whilst both the PS and SS. Th this configuration, this shows a ser e maximum fi ies of lm hole zero m cooalant ss flows thro flows are ug obs h the pedestal arr erved at the ear ay ly SS . In , film holes around the LE produce little flow due to the much lower overall pressure margin this configuration, the maximum film hole coolant flows are observed at the early SS, Int. J. Turbomach. Propuls. Power 2022, 7, 5 15 of 19 Int. J. Turbomach. Propuls. Power 2022, 7, 5 14 of 18 whilst film holes around the LE produce little flow due to the much lower overall pressure margin for that section. In order to achieve a more even outlet coolant flow around the for that section. In order to achieve a more even outlet coolant flow around the vane, it vane, it would be necessary to increase hole sizes at the LE to allow more flow through, would be necessary to increase hole sizes at the LE to allow more flow through, or to reduce or to reduce hole sizes around the SS to restrict coolant flow. This presents a significant hole sizes around the SS to restrict coolant flow. This presents a significant advantage of advantage of the LOM, which can be used to rapidly assess the effects of many minor the LOM, which can be used to rapidly assess the effects of many minor design changes in design changes in an attempt to optimize the outlet flow distribution. an attempt to optimize the outlet flow distribution. Figure 13. Flow network for a double-wall effusion cooled NGV. Figure 13. Flow network for a double-wall effusion cooled NGV. CFD was used to review the effect of walls across the span on the outlet mass flow CFD was used to review the effect of walls across the span on the outlet mass flow distribution, the results of which are shown in Figure 14. This figure compares the average distribution, the results of which are shown in Figure 14. This figure compares the average outlet mass flow per row of holes, normalized by the value seen at the stagnation line, for outlet mass flow per row of holes, normalized by the value seen at the stagnation line, for three different set-ups; a continuous cavity (no walls), separated sections (LE, PS and SS three different set-ups; a continuous cavity (no walls), separated sections (LE, PS and SS separate, as in Figure 13 and completely separated (walls between every row of holes). It is separate, as in Figure 13 and completely separated (walls between every row of holes). It immediately clear from these results that without walls through the cavity, there is massive is immediately clear from these results that without walls through the cavity, there is mas- migration of coolant to the early suction surface, resulting in a significant imbalance in sive migration of coolant to the early suction surface, resulting in a significant imbalance the outlet coolant distribution. In this particular case, the use of walls across the span is in the outlet coolant distribution. In this particular case, the use of walls across the span is imperative to maintaining any sort of adequate film cooling at the LE. imperative to maintaining any sort of adequate film cooling at the LE. Even in the cases where the LE, PS and SS were separated from one another, the average per-hole coolant outlet mass flows were significantly greater for the SS than for other sections. To demonstrate potential methods of evening the outlet flow distribution, the LOM was used to scale up the diameters of film holes around the PS and LE, whilst scaling down the diameters of all film holes on the SS by the same proportion. Results of these tests are shown in Figure 15, which indicates that the hole diameters would need to be scaled by around 80% if the designer were pursuing a more uniform outlet flow. This scaling gives a maximum deviation in the per-hole average coolant mass flow of 19% from the stagnation row value, compared to 118% when no scaling is used. In an engine case, Int. J. Turbomach. Propuls. Power 2022, 7, 5 16 of 19 Int. J. Turbomach. Propuls. Power 2022, 7, 5 15 of 18 Int. J. Turbomach. Propuls. Power 2022, 7, 5 16 of 19 greater heat loads around the SS would lead to a greater demand for film cooling there, but maintaining adequate film cooling to the LE presents more of a challenge. Figure 14. CFD results for outlet mass flow distribution for different internal cavities. Even in the cases where the LE, PS and SS were separated from one another, the av- erage per-hole coolant outlet mass flows were significantly greater for the SS than for other sections. To demonstrate potential methods of evening the outlet flow distribution, the LOM was used to scale up the diameters of film holes around the PS and LE, whilst scaling down the diameters of all film holes on the SS by the same proportion. Results of these tests are shown in Figure 15, which indicates that the hole diameters would need to be scaled by around 80% if the designer were pursuing a more uniform outlet flow. This scaling gives a maximum deviation in the per-hole average coolant mass flow of 19% from the stagnation row value, compared to 118% when no scaling is used. In an engine case, greater heat loads around the SS would lead to a greater demand for film cooling there, but maintaining adequate film cooling to the LE presents more of a challenge. Figure 14. CFD results for outlet mass flow distribution for different internal cavities. Figure 14. CFD results for outlet mass flow distribution for different internal cavities. Even in the cases where the LE, PS and SS were separated from one another, the av- erage per-hole coolant outlet mass flows were significantly greater for the SS than for other sections. To demonstrate potential methods of evening the outlet flow distribution, the LOM was used to scale up the diameters of film holes around the PS and LE, whilst scaling down the diameters of all film holes on the SS by the same proportion. Results of these tests are shown in Figure 15, which indicates that the hole diameters would need to be scaled by around 80% if the designer were pursuing a more uniform outlet flow. This scaling gives a maximum deviation in the per-hole average coolant mass flow of 19% from the stagnation row value, compared to 118% when no scaling is used. In an engine case, greater heat loads around the SS would lead to a greater demand for film cooling there, but maintaining adequate film cooling to the LE presents more of a challenge. Figure 15. LOM results for outlet mass flow distributions in an NGV for scaling hole diameters. Figure 15. LOM results for outlet mass flow distributions in an NGV for scaling hole diameters. 6.4. Future Work Future work will address the effect of heat transfer on coolant migration, combining the Flow Network Model with a Low Order Thermal Network through the solid based on that developed by Murray et al. [25]. Given the high rise in coolant temperature as it passes through an engine vane’s double-wall cooling system, it is expected that coolant migration would be affected by heat transfer. Significant increases in the coolant temperature in the cavity would lead a reduction in the density of the coolant. A lower density would produce smaller outlet mass flows for the same pressure drops, leading to a reduction in the coolant migration factor. As noted previously, geometric changes made to combat coolant Figure 15. LOM results for outlet mass flow distributions in an NGV for scaling hole diameters. Int. J. Turbomach. Propuls. Power 2022, 7, 5 16 of 18 migration will have knock-on effects on the cooling performance of the geometry—it is important for these concerns to be considered alongside one another, as either could lead to catastrophic failure of the engine component. 7. Conclusions A low order flow network model was created to assess mass flow and pressure distributions through double-wall effusion cooling systems. The LOM has been developed to use empirical loss coefficients and discharge coefficient correlations to solve mass flows in and out of nodes in a network, solving for the pressures at internal nodes within the cooling system. This allows assessment of the vulnerability of a cooling scheme’s geometry to coolant migration under the influence of an external static pressure gradient. Reducing the coolant migration from high to low pressure regions is key to preventing localized areas of poor external film cooling effectiveness. The results of the LOM were validated using the Blower Rig, whereby a blower forced flow through a Double-Wall Effusion Cooling test piece, through which the flow was divided into six outlet channels and mass flows were measured independently. This allowed evaluation of the effect of an imposed spanwise pressure gradient on the outlet flow distribution. Results compared favourably with those of the LOM. The LOM performed a series of scaling simulations to show the effect of different geometric parameters on coolant migration. These tests showed that the effective method in reducing the Coolant Migration Factor was to minimise the Pressure Drop Ratio across the outer wall. This was done by reducing the share of the overall pressure losses caused by impingement—the geometric changes that caused this included increasing the diameter and length of the impingement holes, and reducing the diameter and length of the film holes. Large pedestal diameters, and thus low-inter pedestal spacing, produced the same results as small film hole diameters, but only at very high values. It has been suggested that a non-uniform distribution of hole diameters could be used to compensate for the effects of high external static pressure gradients on coolant migration. This was tested for a full vane example, where increasing the diameters of holes at the LE and SS whilst reducing those on the SS gave a far more uniform outlet mass flow distribution. Author Contributions: M.v.d.N. and P.I. established the goals of the work and the design of exper- iments. M.v.d.N. developed the Low Order Flow Network Model and performed CFD work and experiments under the supervision of P.I. All authors have read and agreed to the published version of the manuscript. Funding: This work is funded by Rolls Royce plc. and the EPSRC, under the CDT in Gas Turbine Aerodynamics (No. EP/L015943/1) and the Transpiration Cooling Systems for Jet Engine Turbines and Hypersonic Flight Programme Grant (No. EP/P000878/1). Data Availability Statement: Not Applicable. Acknowledgments: The authors wish to thank of Alexander Murray and the technicians at the Oxford Thermofluids Institute for their contributions to setting up experiments, and Luca Di Mare for help with numerical work. Additional thanks go to Ignacio Mayo, formerly of Rolls Royce plc., for his oversight of this study. Conflicts of Interest: The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Int. J. Turbomach. Propuls. Power 2022, 7, 5 17 of 18 Nomenclature A Area m C Discharge Coefficient C Pressure Coefficient C MF Coolant Migration Factor d, D Diameter (m) f Friction Factor h Specific Enthalpy (J/kg) H TC Heat Transfer Coefficient W/m K) k Loss Coefficient L Length (m) LE Leading Edge m Mass Flow Rate (kg/s) N GV Nozzle Guide Vane P Pressure (Pa) PDR Pressure Difference Ratio PS Pressure Surface Re Reynolds Number s Spacing (m) SFC Specific Fuel Consumption (kg/N s) T Temperature (K) T ET Turbine Entry Temperature (K) u, U Velocity (m/s) b Area Ratio # Expansibility Factor m Viscosity (Pa s) r Density kg/m Subscripts 0 Total c Coolant ext Exterior f Film/Effusion Hole h Hydraulic i Impingement Hole in Inlet out Outlet pd Pedestal sys System t Turning throat Throat w Wall ¥ Mainstream References 1. Van de Noort, M.; Ireland, P.T. A Low Order Flow Network Model for Double-Wall Effusion Cooling Systems. 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Journal

International Journal of Turbomachinery, Propulsion and PowerMultidisciplinary Digital Publishing Institute

Published: Feb 2, 2022

Keywords: coolant migration; flow networks; double-wall effusion cooling

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