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Linear stability analysis reveals exclusion zone for sliding bed transport

Linear stability analysis reveals exclusion zone for sliding bed transport A bend or any another pipe component disturbs solids transport in pipes. Longitudinal pressure profiles downstream of such a component may show a stationary transient harmonic wave, as revealed by a recent settling slurry laboratory experiment. Therefore the fundamental transient response of the two-layer model for fully stratified flow is investigated as a first approach. A linear stability analysis of the sliding bed configuration is conducted. No stationary transient harmonic waves are found in this analysis, but adaptation lengths for exponential recovery are quantified. An example calculation is given for a 0.1 m diameter pipeline. Also consequences for long stretches of pipe line emerged. A so far undiscovered exclusion zone is found in the I-V diagram. This exclusion zone is situated adjacent to the deposit limit velocity locus curve. This simplified physical system reveals that flow velocities should be taken about 10% greater than the calculated maximum deposit limit velocity for stable converging flow. Keywords: Stratification; Hydrotransport; Transients; Bed layer. RODUCTION It is good practice in hydraulic laboratories to situate test sections at a safe distance from upstream bends and inlets. For hydraulic transport of solids there are however virtually no guidelines on appropriate distance. Colwell and Shook (1988) mention a distance of 50D based on measurements in the pseudo-homogeneous hydrotransport regime. In the literature, on many occasions the distance to the inlet is not mentioned and authors assume that equilibrium conditions are being measured, i.e. the results are assumed independent of downstream position at their measuring section. A bend or any other component disturbs the flow, and the flow has to revert again to an asymptotic equilibrium situation via transient development. This may lead to problems in the erpretation of existing data, the risk being that the flow is not fully developed, or worse, that it is being measured in the trough or at the crest of stationary harmonic waves. Pressure gradient measurements over adjacent ervals at a 0.1 meter diameter pipe, Matousek and Krupicka (2013), po at a possibility that stationary harmonic waves could be present. Motivated by the data presented in Matousek and Krupicka (2013), longitudinal pressure profile measurements commenced in a horizontal 0.04 m diameter pipe at Delft University of Technology, Figure 1. A straight pipe section situated downstream of a 180 degree r/D = 5 bend is equipped with a series of 9 pressure taps distributed over a distance of 3 m. The measured pressure profile and associated local pressure gradient profile are shown in Figure 2. These are 300 s averages, equivalent to about 20 mixture circulations. It shows that the pressure gradient varies harmonically with distance, and that within a distance of 70D no asymptotic equilibrium condition is reached. Silica sand is employed. The spatial volumetric solids concentration in the flow loop is 20%, the median grain size is 0.325 mm. The flow velocity is 1.5 m/s, whereas the deposit limit velocity is at about 1 m/s. For this condition the delivered concentration is 18% by volume. Talmon (1999) drew attention to the availability of linear stability and normal mode analysis to investigate the dynamics of stratified slurry flow in pipes. In this case it is no longer assumed that the flow is independent of streamwise distance. In the current paper the same technique is applied, but now transient stationary behaviour is analysed. A macroscopic system approach is followed. A simplified schematisation of the physics is applied and it is not ventured o detailed descriptions of physical processes at the erface or in the suspension, such as in for example Matousek (2009), Krupicka and Matousek (2010), Matousek and Krupicka (2014), Matousek et al. (2014), Kaushal et al. (2005) and Kaushal and Tomita (2013). In fact the analysis pertains to the very basic behaviour of the two-layer structure and is a first step in a process to develop theory for longitudinal development of stratified sand-water flows in pipes. The methodology is to first quantify the asymptotic equilibrium condition and next to quantify transient behaviour towards this equilibrium. Externally imposed are the flow rate, equilibrium bed level and friction coefficients. In this case the velocities of bed and suspension are calculated explicitly via the solution of a quadratic equation in the flow velocity of the suspension Us, a method similar to that described by Jones (2011). Transient development is described by the same equations, but now including advection and mass acceleration terms. Fig. 1. Line drawing essential components of test set-up. Fig. 2. Measured fluid pressures and associated pressure gradient of slurry as a function of distance from the upstream bend, flow velocity 1.5 m/s. Whiskers denote standard deviation of time series and are plotted at locations of upstream pressure taps. sliding sand bed Fig. 3. Definition sketch of a sliding bed layer in a two-layer schematisation, showing bed layer, suspension, flow velocities, shear stresses and fluid pressure profile. The present analysis, which reveals an exclusion zone but not the sought harmonic variation, forms a basis for ongoing analysis considering more physical processes; suspension transport and mass-exchange between the bed layer and the suspension. Governing equations Assumptions in the present two-layer model (2-LM) for horizontal stratified solids transport are: - sliding bed regime, solids transport by sliding bed only, - uniform velocity distribution per layer, - vertical accelerations are neglected (1-D equations), - constant friction coefficients, as for fully developed uni-directional flow, - en-bloc sliding of the bed (no shear in bed layer), loose porous bed layer (constant porosity), - timeindependent. The cross section of the 2-LM configuration is sketched in Figure 3. The overriding fluid flow is referred to as suspension to comply with past and future nomenclature when suspended load transport is considered. The total cross-sectional area (A) and the flow rate (Q) are constant with distance x: A = Ab + As , Q = UA = Abub + As us (1), (2) Flow rates of bed and suspension do not vary along the length of the pipeline in the present analysis because there is no mass exchange through the bed-suspension erface: Abub = constant , As us = constant (3), (4) The bed layer is modelled as a granular layer subject to Coulomb grain friction, pore fluid friction, longitudinal hydraulic gradient, shear stresses exerted by the overlain fluid flow, mass acceleration and effects from a longitudinal bed level slope. The momentum balance of the bed layer is: ub P = - Ab + W - bot Pbot x x F z - s n Ab ( b - w ) g + Ab ( s - w ) g b Fw x b Ab ub (5) The last term on the right hand side accounts for a driving force which originates from density differences between the suspended load layer and the pore water in the bed layer when the erface is non-horizontal. In this term the granular character of the bed layer is preserved: the grain skeleton transfers forces from bed level variations to the pipe wall. The density of the bed layer is: b = n w + (1 ­ n) sand. The momentum balance of the suspension is: of an exponential function and a cosine function: us ' ub ' ikx U , U = us , ub e + complex conjugate (13) This is a short-hand notation for: u P s As us s = - As - W - top Ptop x x (6) us ' ub ' U , U = ­k x us , ub e i cos(kr x) (14) Elimination of the fluid pressure gradient between both momentum equations gives: where k = k r + ik i (15) s us - bot u s u W W ) + top - b ub b + ( + As Ab As x x Ptop (7) Pbot F z - s n ( b - w ) g + ( s - w ) g b = 0 Ab Fw x The real part kr of the wave number k represents the wave number of harmonic variations. The imaginary part ki of k indicates whether the perturbations amplify with downstream distance (ki < 0), or are damped (ki > 0). Differences in amplitude and phase of the harmonic waves are accommodated in the complex amplitudes us , ub . Fundamental relations for differentiation are: The shear stresses are modelled by: top = top 2 s us , = u ' us '/ U = -i k s , x U (8), (9) (10) u ' ub '/ U = -ik b , x U (16), (17) s (u s - ub ) 2 2 bot = bot b ub Linearization Harmonic variations of the variables are substituted in the continuity equation (Eq. 2) and the combined momentum balance of both layers, Eq. (7): The characteristic adaptation length of perturbations is x0 = 1/ki. Four typical responses are shown in Figure 4. Linearization of the equations and mathematical relations for calculating the wave number k are given in Section B and C of the Supplementary material. CALCULATIONS Linear stability analysis (normal mode analysis) A series of calculations is conducted where the bed height is fixed and the superficial velocity is varied. For each superficial velocity, the velocity of the bed layer and suspension layer for the equilibrium of the two-layer configuration are calculated first by the 2-LM model (Section A of the Supplementary material). Next the transient response according to the normal mode analysis is calculated. Parameter values of a typical calculation are listed in Table 1. The listed values for wall friction coefficients (top, bot and µs) are more or less established values. ub = U b + ub ', us = U s + us ' (11), (12) where Us and Ub, are the asymptotic mean velocities. The asymptotic mean velocities Us and Ub, are determined first. Bed level height zb is inputted and Us and Ub are calculated by a 2LM model, see Section A of the Supplementary material. The harmonic variations are modelled mathematically by the product kr = +, ki = - ; spatially grow ing kr = +, ki = 0 ; resonant kr = +, ki = + ; underdamped kr = 0, ki = + ; overdamped zb -1 0 Fig. 4. Schematisation of typical responses to a perturbation: course of bed level height with distance. Linear stability analysis reveals exclusion zone for sliding bed transport Table 1. Parameters in linear stability analysis. Model parameters diameter pipe, D [m] density sand [kg/m ] density suspension [kg/m3] density water [kg/m ] porosity bed, n [­] friction factor Darcy-Weisbach pipe wall, top [­] friction factor Darcy-Weisbach bed shear stress, [­] Coulomb friction coefficient, s [­] friction pore fluid against pipe wall, bot [­] Value 0.1 2650 1000 1000 0.45 0.015 0.07 0.4 0.004 gradient I is calculated by considering the momentum balance of the upper layer: I= 2 2 A 4 s U 2 Ptop top U s W U s U b + - As w gD D 8 U D 8 U U (18) The linear stability analysis shows that there exists a region of amplifying diverging solutions, see Figure 6. This region is bounded by a marginal stability curve (ki = 0) and the deposit limit curve. Curves of iso-decay rates are inscribed in this I-V diagram. Indicated are conditions with characteristic lengths quantifying 63% decay at x0 = 20D and 50D (kiD = 0.05, 0.02). A graphical representation of these transient responses is shown in Figure 7. The type of analysis earlier depicted in Figure 5 is along one curve of z/D = constant. From Figure 6 it is concluded that for example at U = 2.25 m/s and z/D = 0.2, 63% adaptation of the two-layer configuration is reached at a distance of 20 pipe diameters downstream of the bend (20D). An adaptation of 95% is than reached at x = 40D. Multi-valued delivered concentration near deposit limit velocity A common approach is to quantify the hydraulic gradient as a function of flow velocity and delivered concentration. Figure 8 shows that close to the deposit limit velocity, multiple bed heights are possible at a given delivered solids concentration. The ones that are situated in the amplification region (ki < 0) are unattainable, and the solution will diverge with respect to the calculated asymptotic equilibrium. Without resorting to linear stability analysis, the presence of the exclusion zone can also be inferred in hindsight from a series of equilibrium calculations with a standard 2-LM model. The boundary condition is the delivered concentration entering the pipe. If the sliding bed conditions are outside the exclusion zone, an increase in bed level leads to an increase in solids transport rate (greater C vd , see Figure 8). In that case more solids are discharged from the pipe than are entering, and as a consequence the bed level will drop again. This is a self-stabilising The selected value for erface friction coefficient yields realistic deposit limit velocities in the 0.1 m diameter pipe. The calculated complex component of the wavenumber k is graphically displayed as a function of flow velocity in Figure 5. For the real part it is calculated kr = 0 for all circumstances, indicating no harmonic sinusoidal wave development. The equilibrium calculation, also shown in Figure 5, gives a bed velocity Ub which increases approximately linearly with superficial flow velocity. The calculated deposit limit velocity is 1.9 m/s (at z/D = 0.2). The calculations show that the imaginary part of the complex wave number kiD changes sign at about 2.1 m/s. For lower velocities, downstream amplifying perturbations are found (divergence). For higher velocities, downstream decaying perturbations are found (convergence). I-V diagram Wilson (1979) created an I-V diagram showing a deposit limit velocity locus curve (= a curve describing the edge of the stationary bed layer regime: Ub = 0). Hydraulic gradients are again calculated with the 2-LM equilibrium calculation method given in Section A of the Supplementary material, and the deposit limit velocity is reached when a bed velocity of Ub = 0 is calculated. Figure 6 shows the I-V diagram (bed layer only configuration, parameters according to Table 1). The hydraulic 0.2 0.15 0.1 ki D [-] 0.05 0 -0.05 -0.1 -0.15 -0.2 1.6 1.8 2 U [m /s] 2.2 2.4 ki D @z/D=0.2 Ub [m/s] @z/D=0.2 0.8 0.7 0.6 Ub [m/s] 0.5 0.4 0.3 0.2 0.1 0 2.6 Fig. 5. Outcome of stability calculations at a relative bed level of z/D = 0.2: flow velocity of the bed layer Ub at asymptotic equilibrium and calculated complex wave number ki of perturbations indicating amplification, resonance or damping. 0.25 0.2 0.15 I [-] z/D=0.4 z/D=0.3 z/D=0.2 z/D=0.1 water Vdl kiD=0 (x0=inf) ki D=0.02 (x0=50D) ki D=0.05 (x0=20D) 0.1 0.05 0 0 0.5 1 U [m/s] 1.5 2 2.5 divergence (exclusion zone) Fig. 6. Curves of iso-adaptation lengths inscribed in an I-V diagram quantifying the hydraulic gradient and deposit limit velocity as a function of flow velocity for a pipe line diameter of 0.1 m (for further parameters: Table 1). 1 0.8 0.6 0.4 0.2 0 0 30 60 90 120 150 180 ki D = 0.05 (x0=20D) ki D = 0.02 (x0=50D) 95 % 63 % x/D [-] Fig. 7. Exponential adaptation length profiles of overdamped system. Characteristic 63% and 95% adaptation lengths for kiD = 0.02 and 0.05 are indicated. 0.012 0.01 0.008 Cvd [-] 0.006 0.004 0.002 0 0 0.1 0.2 z/D [-] 0.3 0.4 0.5 U=2.1 [m/s] U=1.9 [m/s] U=1.7 [m/s] U=1.5 [m/s] divergence Fig. 8. Calculated volumetric delivered concentration Cvd for sliding bed layer transport mode. Down sloping branches (divergence) are situated in the exclusion zone of Figure 6. mechanism. If on the other hand, the conditions are in the exclusion zone, an increase in bed level at the exit of the pipe leads to a lowering of the exiting solids transport rate. The amount of solids in the pipe increases, increasing the bed level further. This is an unstable situation. The delivered concentrations shown in Figure 8 are low because bed velocities are small under these conditions. In reality there will be a sheet flow layer at the bed surface providing the majority of solids transport, and the delivered concentration will be significantly higher. As a consequence of the presence of this exclusion zone, it is not unreasonable to design transportation systems at a velocity about 10% higher than the calculated maximum Vdl. This corroborates with a 10% safety margin for deposit velocity which is applied in case studies presented in Matousek (2004). This stems from experience with laboratory tests; the specific energy consumption is low and the flow is stable at velocities slightly above the deposition-limit velocity, Matousek (2014). CONCLUSION In a quest to quantify adaption lengths of two-layer fully stratified flow, an exclusion zone is discovered where no downstream converging 2-LM solution can develop. This zone is situated adjacent to the deposit limit velocity locus curve. For stable conditions outside this exclusion zone, adaptation lengths for the two-layer structure are quantified. Adaptation lengths shorten with increasing superficial flow velocity. As a consequence of the presence of this exclusion zone, it is not unreasonable to design transportation systems at a velocity about 10% higher than the calculated maximum Vdl. Similar analyses including more physical processes are being conducted to capture the mechanism of wave development. Acknowledgement. MSc-students J.K.H. Choi and C.J.A. den Hertog are thanked for pressure profile measurement. REFERENCES Colwell, J.M., Shook, C.A., 1988. The entry length for slurries in horizontal pipeline flows. Canadian J. Chemical Engineering, 66, 714­720. Jones, T.F., 2011. A spreadsheet version of the two-layer model for solid-liquid pipeflow. In: Proc. 15th . Conf. Transport & Sedimentation of Solid Particles, Wroclaw, Poland, pp. 101­114. Kaushal, D.R., Tomita, Y., 2013. Prediction of concentration distribution in pipeline flow of highly concentrated slurry. Particulate Science and Technology, 31, 1, 28­34. Kaushal, D.R., Sato, K., Toyota, T., Funatsu, K., Tomita, Y., 2005. Effect of particle size distribution on pressure drop and concentration profile in pipeline flow of highly concentrated slurry. . J. Multiphase Flow, 31, 7, 809­823. Krupicka, J., Matousek, V., 2010. Modeling of settling-slurry around deposition-limit velocity. In: Proc. Hydrotransport 18, 22­24 September, Rio de Janeiro, Brazil, pp. 143­154. Matousek, V., 2004. Dredge pumps and slurry transport. Lecture notes OE4625, Delft University of Technology. Matousek, V., 2009. Predictive model for frictional pressure drop in settling-slurry pipe with stationary deposit. Powder Technology, 192, 367­374. Matousek, V., 2014. Personal communication. Matousek, V., Krupicka, J., 2013. Different types of unsteady flow of solids generated in laboratory slurry loop. In: Proc. 16th . conf. Transport & Sedimentation of Solid Particles, Rostock, Germany, pp.19­32. Matousek, V., Krupicka, J., 2014. One-dimensional modeling of concentration distribution in pipe flow of combined-load slurry. Powder Technology, 260, 42­51. Matousek, V., Krupicka, J., Penik, V., 2014. Distribution of medium-to-coarse glass beads in slurry pipe flow: evaluation of measured concentration profiles. Particulate Science and Technology, 32, 2, 186­196. Talmon, A.M., 1999. Mathematical analysis of the amplification of density variations in long-distance sand transport pipelines. In: Proc. Hydrotransport 14, BHRG, Cranfield, UK. BHRG Cranfield UK, Maastricht, the Netherlands, pp. 3­20. Wilson, K.C., 1979. Deposit-limit nomograms for particles of various densities in pipeline flow. In: Proc. Hydrotransport 6, BHRG, Cranfield, UK. BHRG Cranfield UK, Canterbury, UK, pp. A1­A12. Nomenclature A = cross-sectional area of pipe, Ab = cross-sectional area of bed layer, As = cross-sectional area of suspension, Cvd = volumetric delivered solids concentration, D = pipe diameter, Frb = densimetric Froude number of bed layer, Fn = normal granular force of bed on pipe wall, Fw = immersed weight of bed layer, g = gravity, I = hydraulic gradient, i = ­1, k = kr+i ki = complex wave number, n = porosity of bed layer, P = pressure at the crest of the pipe line's cross-section, Pbot = circumferential contact length of bed layer and pipe, Ptop = circumferential contact length of suspension and pipe, Q = flow rate, r = bend radius U = superficial flow velocity, Ub = asymptotic velocity of bed layer, Us = asymptotic velocity of the suspension, ub = velocity of bed layer, us = velocity of the suspension, W = width of bed surface, x = length-coordinate, x0 = 63% adaptation length, zb = bed surface level, us', ub' = perturbations of the velocities of suspension and bed layer, us , ub = complex dimensionless amplitudes, bot = friction coefficient of pore fluid with pipe, top = Darcy-Weisbach friction factor of pipe wall - suspension, = Darcy-Weisbach friction factor of bed-layer - suspension, s = Coulomb friction coefficient of solids bed against pipe, b = density of the bed-layer (including pore fluid), s = average density of suspension, sand = sand density, w = density of pore fluid (water), bot = hydrodynamic shear stress at bottom of pipeline, = shear stress at erface between both layers, top = shear stress at top section of pipeline. SUPPLEMENTARY MATERIAL Supplementary nomenclature B. Linearisation of equations and calculation of wave number k The continuity equation, Eq. (2), is written as: aij, bij = linearisation coefficients, = volumetric solids concentration of the suspension, c0 1, 2 = transfer coefficients, 1, 2, 3 = varying shape parameters, = shape coefficient, = varying shape parameter. A. Solution of 2-LM equilibrium model Ab U -U = s A U s - Ub and As U - Ub = A U s - Ub (S12), (S13) Subsequent linearisation gives: Ab' A' u` u' = - s = - 2 b + 1 s A A U U (S14) Asymptotic equilibrium of forces, from Eq. (7), is described by: where Ab' and As' are fluctuations of the cross-sectional area, and: WD WD top + + Ab sU 2 sU 2 As F ( - w ) gD - s n b =0 sU 2 Fw Ptop D bot Pbot D A - U2 A b s s 1 = (S1) U (U - U s ) U (U - U b ) , 2 = 2 (U s - U b ) 2 (U s - U b ) (S15), (S16) and 2 = - 1 Ab As (S17) Relations for shear stresses, Eq. (8), (9) and (10), are substituted. The velocity of the bed layer is next eliminated between Eq. (S1) and Eq. (2). For fixed zb/D the solution is given by: the continuity equation of the bottom layer, Eq. (3), gives: ' ' Ub Ab Ab ub + =0 U A A U The suspension layer flows faster, therefore: 1 = positive and 2 = negative. At small bed velocities: 1 = O(1). Linearisation of U U a s +b s + c = 0 U U with: a = 1 + (S2) (S18) top Ab 3 Ptop 8 bot b As Pbot - A W s A W (S3) b = -2 + 2 8bot b As Pbot s A W (S4) F ( - w ) gD 8 Ab As c = 1 - s n b Fw A WD sU 2 8 A P - bot b s bot s A W (S5) U s -b + b2 - 4ac = U 2a (S6) Linearisation of the combined momentum equation, Eq. (7), is accomplished by multiplication of Eq. (7) with D/(sU2) and substitution of decomposed velocities Eq. (11) and Eq. (12). The x-derivatives of the asymptotic velocities Ub and Us are zero. The derivatives of the fluctuating velocities are given by Eq. (16) and Eq. (17). The ernal two-layer geometry varies and shear stresses vary. For these variables similar decompositions are substituted as for the other variables. This produces contributions consisting only of asymptotic mean values of geometry, shear stress and Coulomb bed friction plus contributions containing fluctuating variables. The former collection of terms defines the asymptotic solution, Eq. (S1), and is by definition zero. These terms are therefore removed from the equation. Products of two or more fluctuating variables are discarded because only linear terms are retained in the linearisation process. The resulting linearised combined momentum equation is: ' U u' U u' 1 c0 A Ab ikD s s - b b b + U U U U Fr 2 1 - n WD A s b ' WD WD top ' Ptop D bot ' )+ + 2 ( + - Ab sU As sU 2 As sU 2 ' ' Geometrical dimensions of the 2-LM configuration are quantified as a function of bed height: W = 2 zb ( D - zb ) , Pbot = arccos(1 - 2 zb )D D (S7), (S8) (S9) (S10) Pbot D Ab Ptop = D - Pbot Ab = Pbot D WD z + (-1 + 2 b ) D 4 4 + - WD WD top Ptop D bot Pbot D + - + Ab sU 2 As sU 2 Ab sU 2 As =0 Frb 2 Fw s Fn Fn =2 Fw W + ( -1 + 2 Pbot zb ) Pbot D z + (-1 + 2 b )W D (S19) (S11) with: Frb = sU ( b - w ) gD (S20) The geometrical dimensions Ab, W, Pbot and Ptop are a function of bed level height zb and pipe diameter D. Analytical expressions for these are given in Section A. The velocities Ub and Us in the stratified asymptotic configuration are calculated with the 2-LM model given in Section A. Solving the linearised equation for complex wave number The linearised equations are represented in a matrix, Eq. (S21). The first row represents the continuity equation of the bed layer, which is a linearisation of Eq. (3). The second row represents the momentum balance of both layers, which is a linearisation of Eq. (7). The aij components represent the mass and momentum equations without variation of frictional forces. The bij components represent the variation of these. ' us 0 a12 U ' = 0 a22 + b22 ub U shape of the two-layer geometry (i.e. variation of combinations of Pbot, Ptop, W, Ab, As and Fn/Fw with variation in bed level zb). Expressions for the i-coefficients in Eq. (S29) are given in Section C. To solve for kD the determinant is set to zero: DET = a11 (a22 + b22 ) - a12 (a21 + b21 ) = 0 (S30) The products aij.aij exclusively lead to a ikD term representing stream wise influences, being: accelerations and variations in bed level. The products aij.bij represent the influence of friction forces. The determinant reads: 2 2 U U -1 b b + 2 s + s U U DET = 0 = ikD 1 2 c0 A U s - U b Fr 2 1 - n WD U U b a11 a + b 21 21 (S21) top U s 2 Ptop D A U U -2 s + b - 1 + 2 2 + s 2 U U A 8 8 U As -1 bot A b U b Pbot D F A 2 - b 3 + s 12 n b 1 A s U Ab Frb Fw A (S31) This equation gives kr = 0 (= no harmonic waves) and ki as representative for the damping length. Considering the sign of 1 and 2, the coefficient of the ikD-term is always negative. The sign of the sum of the last six terms determines whether perturbations amplify or are damped in downstream direction. C. Quantification of geometrical variations in which: a11 = 1 Ub U (S22) a12 = - 2 a21 = Us U (or a12 = - 2 U b Ab ) + U A (S23) Us c A ikD + 12 0 ikD U Frb 1 - n WD (S24) a22 = - b Ub c A ikD - 22 0 ikD s U Frb 1 - n WD (S25) The linearised expression for the friction force terms in the combined momentum equation is, see Eq. (S19): ' ' ' top Ptop D bot Pbot D + ' - + sU 2 sU 2 As sU 2 Ab sU 2 b21 = 2( U s Ub - ) 8 U U top Ptop D U s A + ) + 12 8 As U Frb WD (S26) top Ptop D sU 2 As bot Pbot D s Fn - - 2 2 sU Ab Frb Fw (S32) b22 = -2( +bot U s Ub - ) 8 U U b Pbot D Ub A ) - 22 s Ab U Frb WD (S27) Elimination of between Eq. (S13) and Eq. (S32) and expressing geometric fluctuations o the fluctuation of the bed area Ab gives: ' ' ' top Ptop D bot Pbot D + - sU 2 sU 2 As sU 2 Ab where: = WD + WD As Ab and: (S28) top Ptop D F P D A' + - s2 n 1 + 2 - bot 2 bot 3 b 2 Fw As Ab sU sU Frb A where: (S33) = - s + Frb2 Fn 4 W 1 + Fw D P A 16 W top Ptop A 2 - bot 2 bot 3 2 2 D sU D As sU D Ab (S29) 1 = Fn / Fw 1 1 - Fn / Fw Ab / A Ab / A The parameter follows from the last four terms in Eq. (S19). These terms represent the influence of variations of the z A Pbot Fw A D 3 = - - ( ) 2(1 - 2 b ) Ab W Fn As 4 W D (S34) 2 = Ptop / As 1 1 - Ptop / As Ab / A Ab / A 3 = (S35) Pbot / Ab 1 1 - Ab / A Pbot / Ab Ab / A z A D 3 2D3 =- + - ( ) 2(1 - 2 b ) 2 Ab 4 W D 4 PtopW z 2 D3 A D 3 = - - ( ) 2(1 - 2 b ) 2 4 PbotW As 4 W D (S36) The coefficients 1, 2, 3 basically depend on the relative bed level. Their functional dependency is depicted in Figure S1. 1000 beta1, beta2, - beta3 [-] 100 10 1 0.1 0.01 0 0.2 0.4 z/D [-] 0.6 0.8 1 beta1 beta2 -beta3 Fig. S1. Graphical depiction of the coefficients 1, 2, 3 which quantify the influence of bed-level variations on the momentum balance through geometric variation of the cross-sectional area. Received 17 October 2014 Accepted 26 January 2015 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Hydrology and Hydromechanics de Gruyter

Linear stability analysis reveals exclusion zone for sliding bed transport

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Abstract

A bend or any another pipe component disturbs solids transport in pipes. Longitudinal pressure profiles downstream of such a component may show a stationary transient harmonic wave, as revealed by a recent settling slurry laboratory experiment. Therefore the fundamental transient response of the two-layer model for fully stratified flow is investigated as a first approach. A linear stability analysis of the sliding bed configuration is conducted. No stationary transient harmonic waves are found in this analysis, but adaptation lengths for exponential recovery are quantified. An example calculation is given for a 0.1 m diameter pipeline. Also consequences for long stretches of pipe line emerged. A so far undiscovered exclusion zone is found in the I-V diagram. This exclusion zone is situated adjacent to the deposit limit velocity locus curve. This simplified physical system reveals that flow velocities should be taken about 10% greater than the calculated maximum deposit limit velocity for stable converging flow. Keywords: Stratification; Hydrotransport; Transients; Bed layer. RODUCTION It is good practice in hydraulic laboratories to situate test sections at a safe distance from upstream bends and inlets. For hydraulic transport of solids there are however virtually no guidelines on appropriate distance. Colwell and Shook (1988) mention a distance of 50D based on measurements in the pseudo-homogeneous hydrotransport regime. In the literature, on many occasions the distance to the inlet is not mentioned and authors assume that equilibrium conditions are being measured, i.e. the results are assumed independent of downstream position at their measuring section. A bend or any other component disturbs the flow, and the flow has to revert again to an asymptotic equilibrium situation via transient development. This may lead to problems in the erpretation of existing data, the risk being that the flow is not fully developed, or worse, that it is being measured in the trough or at the crest of stationary harmonic waves. Pressure gradient measurements over adjacent ervals at a 0.1 meter diameter pipe, Matousek and Krupicka (2013), po at a possibility that stationary harmonic waves could be present. Motivated by the data presented in Matousek and Krupicka (2013), longitudinal pressure profile measurements commenced in a horizontal 0.04 m diameter pipe at Delft University of Technology, Figure 1. A straight pipe section situated downstream of a 180 degree r/D = 5 bend is equipped with a series of 9 pressure taps distributed over a distance of 3 m. The measured pressure profile and associated local pressure gradient profile are shown in Figure 2. These are 300 s averages, equivalent to about 20 mixture circulations. It shows that the pressure gradient varies harmonically with distance, and that within a distance of 70D no asymptotic equilibrium condition is reached. Silica sand is employed. The spatial volumetric solids concentration in the flow loop is 20%, the median grain size is 0.325 mm. The flow velocity is 1.5 m/s, whereas the deposit limit velocity is at about 1 m/s. For this condition the delivered concentration is 18% by volume. Talmon (1999) drew attention to the availability of linear stability and normal mode analysis to investigate the dynamics of stratified slurry flow in pipes. In this case it is no longer assumed that the flow is independent of streamwise distance. In the current paper the same technique is applied, but now transient stationary behaviour is analysed. A macroscopic system approach is followed. A simplified schematisation of the physics is applied and it is not ventured o detailed descriptions of physical processes at the erface or in the suspension, such as in for example Matousek (2009), Krupicka and Matousek (2010), Matousek and Krupicka (2014), Matousek et al. (2014), Kaushal et al. (2005) and Kaushal and Tomita (2013). In fact the analysis pertains to the very basic behaviour of the two-layer structure and is a first step in a process to develop theory for longitudinal development of stratified sand-water flows in pipes. The methodology is to first quantify the asymptotic equilibrium condition and next to quantify transient behaviour towards this equilibrium. Externally imposed are the flow rate, equilibrium bed level and friction coefficients. In this case the velocities of bed and suspension are calculated explicitly via the solution of a quadratic equation in the flow velocity of the suspension Us, a method similar to that described by Jones (2011). Transient development is described by the same equations, but now including advection and mass acceleration terms. Fig. 1. Line drawing essential components of test set-up. Fig. 2. Measured fluid pressures and associated pressure gradient of slurry as a function of distance from the upstream bend, flow velocity 1.5 m/s. Whiskers denote standard deviation of time series and are plotted at locations of upstream pressure taps. sliding sand bed Fig. 3. Definition sketch of a sliding bed layer in a two-layer schematisation, showing bed layer, suspension, flow velocities, shear stresses and fluid pressure profile. The present analysis, which reveals an exclusion zone but not the sought harmonic variation, forms a basis for ongoing analysis considering more physical processes; suspension transport and mass-exchange between the bed layer and the suspension. Governing equations Assumptions in the present two-layer model (2-LM) for horizontal stratified solids transport are: - sliding bed regime, solids transport by sliding bed only, - uniform velocity distribution per layer, - vertical accelerations are neglected (1-D equations), - constant friction coefficients, as for fully developed uni-directional flow, - en-bloc sliding of the bed (no shear in bed layer), loose porous bed layer (constant porosity), - timeindependent. The cross section of the 2-LM configuration is sketched in Figure 3. The overriding fluid flow is referred to as suspension to comply with past and future nomenclature when suspended load transport is considered. The total cross-sectional area (A) and the flow rate (Q) are constant with distance x: A = Ab + As , Q = UA = Abub + As us (1), (2) Flow rates of bed and suspension do not vary along the length of the pipeline in the present analysis because there is no mass exchange through the bed-suspension erface: Abub = constant , As us = constant (3), (4) The bed layer is modelled as a granular layer subject to Coulomb grain friction, pore fluid friction, longitudinal hydraulic gradient, shear stresses exerted by the overlain fluid flow, mass acceleration and effects from a longitudinal bed level slope. The momentum balance of the bed layer is: ub P = - Ab + W - bot Pbot x x F z - s n Ab ( b - w ) g + Ab ( s - w ) g b Fw x b Ab ub (5) The last term on the right hand side accounts for a driving force which originates from density differences between the suspended load layer and the pore water in the bed layer when the erface is non-horizontal. In this term the granular character of the bed layer is preserved: the grain skeleton transfers forces from bed level variations to the pipe wall. The density of the bed layer is: b = n w + (1 ­ n) sand. The momentum balance of the suspension is: of an exponential function and a cosine function: us ' ub ' ikx U , U = us , ub e + complex conjugate (13) This is a short-hand notation for: u P s As us s = - As - W - top Ptop x x (6) us ' ub ' U , U = ­k x us , ub e i cos(kr x) (14) Elimination of the fluid pressure gradient between both momentum equations gives: where k = k r + ik i (15) s us - bot u s u W W ) + top - b ub b + ( + As Ab As x x Ptop (7) Pbot F z - s n ( b - w ) g + ( s - w ) g b = 0 Ab Fw x The real part kr of the wave number k represents the wave number of harmonic variations. The imaginary part ki of k indicates whether the perturbations amplify with downstream distance (ki < 0), or are damped (ki > 0). Differences in amplitude and phase of the harmonic waves are accommodated in the complex amplitudes us , ub . Fundamental relations for differentiation are: The shear stresses are modelled by: top = top 2 s us , = u ' us '/ U = -i k s , x U (8), (9) (10) u ' ub '/ U = -ik b , x U (16), (17) s (u s - ub ) 2 2 bot = bot b ub Linearization Harmonic variations of the variables are substituted in the continuity equation (Eq. 2) and the combined momentum balance of both layers, Eq. (7): The characteristic adaptation length of perturbations is x0 = 1/ki. Four typical responses are shown in Figure 4. Linearization of the equations and mathematical relations for calculating the wave number k are given in Section B and C of the Supplementary material. CALCULATIONS Linear stability analysis (normal mode analysis) A series of calculations is conducted where the bed height is fixed and the superficial velocity is varied. For each superficial velocity, the velocity of the bed layer and suspension layer for the equilibrium of the two-layer configuration are calculated first by the 2-LM model (Section A of the Supplementary material). Next the transient response according to the normal mode analysis is calculated. Parameter values of a typical calculation are listed in Table 1. The listed values for wall friction coefficients (top, bot and µs) are more or less established values. ub = U b + ub ', us = U s + us ' (11), (12) where Us and Ub, are the asymptotic mean velocities. The asymptotic mean velocities Us and Ub, are determined first. Bed level height zb is inputted and Us and Ub are calculated by a 2LM model, see Section A of the Supplementary material. The harmonic variations are modelled mathematically by the product kr = +, ki = - ; spatially grow ing kr = +, ki = 0 ; resonant kr = +, ki = + ; underdamped kr = 0, ki = + ; overdamped zb -1 0 Fig. 4. Schematisation of typical responses to a perturbation: course of bed level height with distance. Linear stability analysis reveals exclusion zone for sliding bed transport Table 1. Parameters in linear stability analysis. Model parameters diameter pipe, D [m] density sand [kg/m ] density suspension [kg/m3] density water [kg/m ] porosity bed, n [­] friction factor Darcy-Weisbach pipe wall, top [­] friction factor Darcy-Weisbach bed shear stress, [­] Coulomb friction coefficient, s [­] friction pore fluid against pipe wall, bot [­] Value 0.1 2650 1000 1000 0.45 0.015 0.07 0.4 0.004 gradient I is calculated by considering the momentum balance of the upper layer: I= 2 2 A 4 s U 2 Ptop top U s W U s U b + - As w gD D 8 U D 8 U U (18) The linear stability analysis shows that there exists a region of amplifying diverging solutions, see Figure 6. This region is bounded by a marginal stability curve (ki = 0) and the deposit limit curve. Curves of iso-decay rates are inscribed in this I-V diagram. Indicated are conditions with characteristic lengths quantifying 63% decay at x0 = 20D and 50D (kiD = 0.05, 0.02). A graphical representation of these transient responses is shown in Figure 7. The type of analysis earlier depicted in Figure 5 is along one curve of z/D = constant. From Figure 6 it is concluded that for example at U = 2.25 m/s and z/D = 0.2, 63% adaptation of the two-layer configuration is reached at a distance of 20 pipe diameters downstream of the bend (20D). An adaptation of 95% is than reached at x = 40D. Multi-valued delivered concentration near deposit limit velocity A common approach is to quantify the hydraulic gradient as a function of flow velocity and delivered concentration. Figure 8 shows that close to the deposit limit velocity, multiple bed heights are possible at a given delivered solids concentration. The ones that are situated in the amplification region (ki < 0) are unattainable, and the solution will diverge with respect to the calculated asymptotic equilibrium. Without resorting to linear stability analysis, the presence of the exclusion zone can also be inferred in hindsight from a series of equilibrium calculations with a standard 2-LM model. The boundary condition is the delivered concentration entering the pipe. If the sliding bed conditions are outside the exclusion zone, an increase in bed level leads to an increase in solids transport rate (greater C vd , see Figure 8). In that case more solids are discharged from the pipe than are entering, and as a consequence the bed level will drop again. This is a self-stabilising The selected value for erface friction coefficient yields realistic deposit limit velocities in the 0.1 m diameter pipe. The calculated complex component of the wavenumber k is graphically displayed as a function of flow velocity in Figure 5. For the real part it is calculated kr = 0 for all circumstances, indicating no harmonic sinusoidal wave development. The equilibrium calculation, also shown in Figure 5, gives a bed velocity Ub which increases approximately linearly with superficial flow velocity. The calculated deposit limit velocity is 1.9 m/s (at z/D = 0.2). The calculations show that the imaginary part of the complex wave number kiD changes sign at about 2.1 m/s. For lower velocities, downstream amplifying perturbations are found (divergence). For higher velocities, downstream decaying perturbations are found (convergence). I-V diagram Wilson (1979) created an I-V diagram showing a deposit limit velocity locus curve (= a curve describing the edge of the stationary bed layer regime: Ub = 0). Hydraulic gradients are again calculated with the 2-LM equilibrium calculation method given in Section A of the Supplementary material, and the deposit limit velocity is reached when a bed velocity of Ub = 0 is calculated. Figure 6 shows the I-V diagram (bed layer only configuration, parameters according to Table 1). The hydraulic 0.2 0.15 0.1 ki D [-] 0.05 0 -0.05 -0.1 -0.15 -0.2 1.6 1.8 2 U [m /s] 2.2 2.4 ki D @z/D=0.2 Ub [m/s] @z/D=0.2 0.8 0.7 0.6 Ub [m/s] 0.5 0.4 0.3 0.2 0.1 0 2.6 Fig. 5. Outcome of stability calculations at a relative bed level of z/D = 0.2: flow velocity of the bed layer Ub at asymptotic equilibrium and calculated complex wave number ki of perturbations indicating amplification, resonance or damping. 0.25 0.2 0.15 I [-] z/D=0.4 z/D=0.3 z/D=0.2 z/D=0.1 water Vdl kiD=0 (x0=inf) ki D=0.02 (x0=50D) ki D=0.05 (x0=20D) 0.1 0.05 0 0 0.5 1 U [m/s] 1.5 2 2.5 divergence (exclusion zone) Fig. 6. Curves of iso-adaptation lengths inscribed in an I-V diagram quantifying the hydraulic gradient and deposit limit velocity as a function of flow velocity for a pipe line diameter of 0.1 m (for further parameters: Table 1). 1 0.8 0.6 0.4 0.2 0 0 30 60 90 120 150 180 ki D = 0.05 (x0=20D) ki D = 0.02 (x0=50D) 95 % 63 % x/D [-] Fig. 7. Exponential adaptation length profiles of overdamped system. Characteristic 63% and 95% adaptation lengths for kiD = 0.02 and 0.05 are indicated. 0.012 0.01 0.008 Cvd [-] 0.006 0.004 0.002 0 0 0.1 0.2 z/D [-] 0.3 0.4 0.5 U=2.1 [m/s] U=1.9 [m/s] U=1.7 [m/s] U=1.5 [m/s] divergence Fig. 8. Calculated volumetric delivered concentration Cvd for sliding bed layer transport mode. Down sloping branches (divergence) are situated in the exclusion zone of Figure 6. mechanism. If on the other hand, the conditions are in the exclusion zone, an increase in bed level at the exit of the pipe leads to a lowering of the exiting solids transport rate. The amount of solids in the pipe increases, increasing the bed level further. This is an unstable situation. The delivered concentrations shown in Figure 8 are low because bed velocities are small under these conditions. In reality there will be a sheet flow layer at the bed surface providing the majority of solids transport, and the delivered concentration will be significantly higher. As a consequence of the presence of this exclusion zone, it is not unreasonable to design transportation systems at a velocity about 10% higher than the calculated maximum Vdl. This corroborates with a 10% safety margin for deposit velocity which is applied in case studies presented in Matousek (2004). This stems from experience with laboratory tests; the specific energy consumption is low and the flow is stable at velocities slightly above the deposition-limit velocity, Matousek (2014). CONCLUSION In a quest to quantify adaption lengths of two-layer fully stratified flow, an exclusion zone is discovered where no downstream converging 2-LM solution can develop. This zone is situated adjacent to the deposit limit velocity locus curve. For stable conditions outside this exclusion zone, adaptation lengths for the two-layer structure are quantified. Adaptation lengths shorten with increasing superficial flow velocity. As a consequence of the presence of this exclusion zone, it is not unreasonable to design transportation systems at a velocity about 10% higher than the calculated maximum Vdl. Similar analyses including more physical processes are being conducted to capture the mechanism of wave development. Acknowledgement. MSc-students J.K.H. Choi and C.J.A. den Hertog are thanked for pressure profile measurement. REFERENCES Colwell, J.M., Shook, C.A., 1988. The entry length for slurries in horizontal pipeline flows. Canadian J. Chemical Engineering, 66, 714­720. Jones, T.F., 2011. A spreadsheet version of the two-layer model for solid-liquid pipeflow. In: Proc. 15th . Conf. Transport & Sedimentation of Solid Particles, Wroclaw, Poland, pp. 101­114. Kaushal, D.R., Tomita, Y., 2013. Prediction of concentration distribution in pipeline flow of highly concentrated slurry. Particulate Science and Technology, 31, 1, 28­34. Kaushal, D.R., Sato, K., Toyota, T., Funatsu, K., Tomita, Y., 2005. Effect of particle size distribution on pressure drop and concentration profile in pipeline flow of highly concentrated slurry. . J. Multiphase Flow, 31, 7, 809­823. Krupicka, J., Matousek, V., 2010. Modeling of settling-slurry around deposition-limit velocity. In: Proc. Hydrotransport 18, 22­24 September, Rio de Janeiro, Brazil, pp. 143­154. Matousek, V., 2004. Dredge pumps and slurry transport. Lecture notes OE4625, Delft University of Technology. Matousek, V., 2009. Predictive model for frictional pressure drop in settling-slurry pipe with stationary deposit. Powder Technology, 192, 367­374. Matousek, V., 2014. Personal communication. Matousek, V., Krupicka, J., 2013. Different types of unsteady flow of solids generated in laboratory slurry loop. In: Proc. 16th . conf. Transport & Sedimentation of Solid Particles, Rostock, Germany, pp.19­32. Matousek, V., Krupicka, J., 2014. One-dimensional modeling of concentration distribution in pipe flow of combined-load slurry. Powder Technology, 260, 42­51. Matousek, V., Krupicka, J., Penik, V., 2014. Distribution of medium-to-coarse glass beads in slurry pipe flow: evaluation of measured concentration profiles. Particulate Science and Technology, 32, 2, 186­196. Talmon, A.M., 1999. Mathematical analysis of the amplification of density variations in long-distance sand transport pipelines. In: Proc. Hydrotransport 14, BHRG, Cranfield, UK. BHRG Cranfield UK, Maastricht, the Netherlands, pp. 3­20. Wilson, K.C., 1979. Deposit-limit nomograms for particles of various densities in pipeline flow. In: Proc. Hydrotransport 6, BHRG, Cranfield, UK. BHRG Cranfield UK, Canterbury, UK, pp. A1­A12. Nomenclature A = cross-sectional area of pipe, Ab = cross-sectional area of bed layer, As = cross-sectional area of suspension, Cvd = volumetric delivered solids concentration, D = pipe diameter, Frb = densimetric Froude number of bed layer, Fn = normal granular force of bed on pipe wall, Fw = immersed weight of bed layer, g = gravity, I = hydraulic gradient, i = ­1, k = kr+i ki = complex wave number, n = porosity of bed layer, P = pressure at the crest of the pipe line's cross-section, Pbot = circumferential contact length of bed layer and pipe, Ptop = circumferential contact length of suspension and pipe, Q = flow rate, r = bend radius U = superficial flow velocity, Ub = asymptotic velocity of bed layer, Us = asymptotic velocity of the suspension, ub = velocity of bed layer, us = velocity of the suspension, W = width of bed surface, x = length-coordinate, x0 = 63% adaptation length, zb = bed surface level, us', ub' = perturbations of the velocities of suspension and bed layer, us , ub = complex dimensionless amplitudes, bot = friction coefficient of pore fluid with pipe, top = Darcy-Weisbach friction factor of pipe wall - suspension, = Darcy-Weisbach friction factor of bed-layer - suspension, s = Coulomb friction coefficient of solids bed against pipe, b = density of the bed-layer (including pore fluid), s = average density of suspension, sand = sand density, w = density of pore fluid (water), bot = hydrodynamic shear stress at bottom of pipeline, = shear stress at erface between both layers, top = shear stress at top section of pipeline. SUPPLEMENTARY MATERIAL Supplementary nomenclature B. Linearisation of equations and calculation of wave number k The continuity equation, Eq. (2), is written as: aij, bij = linearisation coefficients, = volumetric solids concentration of the suspension, c0 1, 2 = transfer coefficients, 1, 2, 3 = varying shape parameters, = shape coefficient, = varying shape parameter. A. Solution of 2-LM equilibrium model Ab U -U = s A U s - Ub and As U - Ub = A U s - Ub (S12), (S13) Subsequent linearisation gives: Ab' A' u` u' = - s = - 2 b + 1 s A A U U (S14) Asymptotic equilibrium of forces, from Eq. (7), is described by: where Ab' and As' are fluctuations of the cross-sectional area, and: WD WD top + + Ab sU 2 sU 2 As F ( - w ) gD - s n b =0 sU 2 Fw Ptop D bot Pbot D A - U2 A b s s 1 = (S1) U (U - U s ) U (U - U b ) , 2 = 2 (U s - U b ) 2 (U s - U b ) (S15), (S16) and 2 = - 1 Ab As (S17) Relations for shear stresses, Eq. (8), (9) and (10), are substituted. The velocity of the bed layer is next eliminated between Eq. (S1) and Eq. (2). For fixed zb/D the solution is given by: the continuity equation of the bottom layer, Eq. (3), gives: ' ' Ub Ab Ab ub + =0 U A A U The suspension layer flows faster, therefore: 1 = positive and 2 = negative. At small bed velocities: 1 = O(1). Linearisation of U U a s +b s + c = 0 U U with: a = 1 + (S2) (S18) top Ab 3 Ptop 8 bot b As Pbot - A W s A W (S3) b = -2 + 2 8bot b As Pbot s A W (S4) F ( - w ) gD 8 Ab As c = 1 - s n b Fw A WD sU 2 8 A P - bot b s bot s A W (S5) U s -b + b2 - 4ac = U 2a (S6) Linearisation of the combined momentum equation, Eq. (7), is accomplished by multiplication of Eq. (7) with D/(sU2) and substitution of decomposed velocities Eq. (11) and Eq. (12). The x-derivatives of the asymptotic velocities Ub and Us are zero. The derivatives of the fluctuating velocities are given by Eq. (16) and Eq. (17). The ernal two-layer geometry varies and shear stresses vary. For these variables similar decompositions are substituted as for the other variables. This produces contributions consisting only of asymptotic mean values of geometry, shear stress and Coulomb bed friction plus contributions containing fluctuating variables. The former collection of terms defines the asymptotic solution, Eq. (S1), and is by definition zero. These terms are therefore removed from the equation. Products of two or more fluctuating variables are discarded because only linear terms are retained in the linearisation process. The resulting linearised combined momentum equation is: ' U u' U u' 1 c0 A Ab ikD s s - b b b + U U U U Fr 2 1 - n WD A s b ' WD WD top ' Ptop D bot ' )+ + 2 ( + - Ab sU As sU 2 As sU 2 ' ' Geometrical dimensions of the 2-LM configuration are quantified as a function of bed height: W = 2 zb ( D - zb ) , Pbot = arccos(1 - 2 zb )D D (S7), (S8) (S9) (S10) Pbot D Ab Ptop = D - Pbot Ab = Pbot D WD z + (-1 + 2 b ) D 4 4 + - WD WD top Ptop D bot Pbot D + - + Ab sU 2 As sU 2 Ab sU 2 As =0 Frb 2 Fw s Fn Fn =2 Fw W + ( -1 + 2 Pbot zb ) Pbot D z + (-1 + 2 b )W D (S19) (S11) with: Frb = sU ( b - w ) gD (S20) The geometrical dimensions Ab, W, Pbot and Ptop are a function of bed level height zb and pipe diameter D. Analytical expressions for these are given in Section A. The velocities Ub and Us in the stratified asymptotic configuration are calculated with the 2-LM model given in Section A. Solving the linearised equation for complex wave number The linearised equations are represented in a matrix, Eq. (S21). The first row represents the continuity equation of the bed layer, which is a linearisation of Eq. (3). The second row represents the momentum balance of both layers, which is a linearisation of Eq. (7). The aij components represent the mass and momentum equations without variation of frictional forces. The bij components represent the variation of these. ' us 0 a12 U ' = 0 a22 + b22 ub U shape of the two-layer geometry (i.e. variation of combinations of Pbot, Ptop, W, Ab, As and Fn/Fw with variation in bed level zb). Expressions for the i-coefficients in Eq. (S29) are given in Section C. To solve for kD the determinant is set to zero: DET = a11 (a22 + b22 ) - a12 (a21 + b21 ) = 0 (S30) The products aij.aij exclusively lead to a ikD term representing stream wise influences, being: accelerations and variations in bed level. The products aij.bij represent the influence of friction forces. The determinant reads: 2 2 U U -1 b b + 2 s + s U U DET = 0 = ikD 1 2 c0 A U s - U b Fr 2 1 - n WD U U b a11 a + b 21 21 (S21) top U s 2 Ptop D A U U -2 s + b - 1 + 2 2 + s 2 U U A 8 8 U As -1 bot A b U b Pbot D F A 2 - b 3 + s 12 n b 1 A s U Ab Frb Fw A (S31) This equation gives kr = 0 (= no harmonic waves) and ki as representative for the damping length. Considering the sign of 1 and 2, the coefficient of the ikD-term is always negative. The sign of the sum of the last six terms determines whether perturbations amplify or are damped in downstream direction. C. Quantification of geometrical variations in which: a11 = 1 Ub U (S22) a12 = - 2 a21 = Us U (or a12 = - 2 U b Ab ) + U A (S23) Us c A ikD + 12 0 ikD U Frb 1 - n WD (S24) a22 = - b Ub c A ikD - 22 0 ikD s U Frb 1 - n WD (S25) The linearised expression for the friction force terms in the combined momentum equation is, see Eq. (S19): ' ' ' top Ptop D bot Pbot D + ' - + sU 2 sU 2 As sU 2 Ab sU 2 b21 = 2( U s Ub - ) 8 U U top Ptop D U s A + ) + 12 8 As U Frb WD (S26) top Ptop D sU 2 As bot Pbot D s Fn - - 2 2 sU Ab Frb Fw (S32) b22 = -2( +bot U s Ub - ) 8 U U b Pbot D Ub A ) - 22 s Ab U Frb WD (S27) Elimination of between Eq. (S13) and Eq. (S32) and expressing geometric fluctuations o the fluctuation of the bed area Ab gives: ' ' ' top Ptop D bot Pbot D + - sU 2 sU 2 As sU 2 Ab where: = WD + WD As Ab and: (S28) top Ptop D F P D A' + - s2 n 1 + 2 - bot 2 bot 3 b 2 Fw As Ab sU sU Frb A where: (S33) = - s + Frb2 Fn 4 W 1 + Fw D P A 16 W top Ptop A 2 - bot 2 bot 3 2 2 D sU D As sU D Ab (S29) 1 = Fn / Fw 1 1 - Fn / Fw Ab / A Ab / A The parameter follows from the last four terms in Eq. (S19). These terms represent the influence of variations of the z A Pbot Fw A D 3 = - - ( ) 2(1 - 2 b ) Ab W Fn As 4 W D (S34) 2 = Ptop / As 1 1 - Ptop / As Ab / A Ab / A 3 = (S35) Pbot / Ab 1 1 - Ab / A Pbot / Ab Ab / A z A D 3 2D3 =- + - ( ) 2(1 - 2 b ) 2 Ab 4 W D 4 PtopW z 2 D3 A D 3 = - - ( ) 2(1 - 2 b ) 2 4 PbotW As 4 W D (S36) The coefficients 1, 2, 3 basically depend on the relative bed level. Their functional dependency is depicted in Figure S1. 1000 beta1, beta2, - beta3 [-] 100 10 1 0.1 0.01 0 0.2 0.4 z/D [-] 0.6 0.8 1 beta1 beta2 -beta3 Fig. S1. Graphical depiction of the coefficients 1, 2, 3 which quantify the influence of bed-level variations on the momentum balance through geometric variation of the cross-sectional area. Received 17 October 2014 Accepted 26 January 2015

Journal

Journal of Hydrology and Hydromechanicsde Gruyter

Published: Jun 1, 2015

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