A Pressure-Based Algorithm for Multi-Phase Flow at all Speeds

Transcription

1 A ressure-based Algorithm for Multi-hase Flow at all Seeds F. Moukalled* American University of Beirut, Faculty of Engineering & Architecture, Mechanical Engineering Deartment,.O.Box Riad El Solh, Beirut Lebanon M. Darwish American University of Beirut, Faculty of Engineering & Architecture, Mechanical Engineering Deartment,.O.Box Riad El Solh, Beirut Lebanon B. Sekar Air Force Research Laboratory AFRL/RTC Wright-atterson AFB OH ,USA Keywords: Multi-hase flow, ressure-based algorithm, all seed flows, finite volume method. roosed running head: An algorithm for Multi-hase Flow at All Seeds * Corresonding Author

2 An algorithm for Multi-hase Flow at All Seeds 2 Abstract A new finite volume-based numerical algorithm for redicting incomressible and comressible multi-hase flow henomena is resented. The technique is equally alicable in the subsonic, transonic, and suersonic regimes. The method is formulated on a nonorthogonal coordinate system in collocated rimitive variables. ressure is selected as a deendent variable in reference to density because changes in ressure are significant at all seeds as oosed to variations in density, which become very small at low Mach numbers. The ressure equation is derived from overall mass conservation. The erformance of the new method is assessed by solving the following two-dimensional two-hase flow roblems: (i) incomressible turbulent bubbly flow in a ie, (ii) incomressible turbulent air-article flow in a ie, (iii) comressible dilute gas-solid flow over a flat late, and (iv) comressible dusty flow in a converging diverging nozzle. redictions are shown to be in excellent agreement with ublished numerical and/or exerimental data.

3 An algorithm for Multi-hase Flow at All Seeds 3 Nomenclature A (k ),.. coefficients in the discretized equation for φ (k ). B (k ) source term in the discretized equation for φ (k ). B (k ) body force er unit volume of fluid/hase k. C ρ coefficient equals to 1/ R T (k ). (k D ) [φ (k ) ] matrix oerator defined in Eq. (14). H [ φ ] the H oerator. H [ ] the vector form of the oerator. I (k ) u ( k) D f H inter-hase momentum transfer. J diffusion flux of φ across cell face f. J f (k )C convection flux of φ (k ) across cell face f. (k ) M & mass source er unit volume. ressure. r, rt laminar and turbulent randtl number for fluid/hase k. q& heat generated er unit volume of fluid/hase k. Q (k ) general source term of fluid/hase k. r (k ) volume fraction of fluid/hase k. S f surface vector. t time. T (k ) temerature of fluid/hase k. U f interface flux velocity ( S ) u velocity vector of fluid/hase k. v of fluid/hase k. u,v,.. velocity comonents of fluid/hase k. x, y Cartesian coordinates. a,b the maximum of a and b. Greek Symbols ρ (k ) density of fluid/hase k. Γ (k ) diffusion coefficient of fluid/hase k. f f

4 An algorithm for Multi-hase Flow at All Seeds 4 Φ (k ) dissiation term in energy equation of fluid/hase k. φ (k ) general scalar quantity associated with fluid/hase k. [ φ ] µ ( k ), µ ( k ) t the oerator. laminar and turbulent viscosity of fluid/hase k. Ω δt cell volume. time ste. Subscrits f refers to control volume face f. Suerscrits C D refers to the grid oint. refers to convection contribution. refers to diffusion contribution. refers to fluid/hase k. * refers to udated value at the current iteration. (k ) o refers to values of fluid/hase k from the revious iteration. (k ) refers to correction field of hase/fluid k. m refers to fluid/hase m. old refers to values from the revious time ste.

5 Introduction The last two decades have witnessed a substantial transformation in the Comutational Fluid Dynamics (CFD) industry; from a research means confined to research laboratories, CFD has emerged as an every day engineering tool for a wide range of industries (Aeronautics, Automobile, Chemical rocessing, etc ). This increasing deendence on CFD is due to a multitude of factors that have rendered ractical the simulation of large comlex industrialtye roblems. Some of these factors are directly related to the maturity of several numerical asects at the core of CFD. These include: multi-grid acceleration techniques [1-4] with enhanced equation solvers [5,6] that have decreased the comutational cost of tackling large roblems, better discretization techniques with bounded high resolution schemes [13-18] yielding more accurate results, unstructured grid methods [7-12] that simlify the descrition of comlex geometry, as well as imroved ressure-velocity (and density) couling algorithms for fluid flow at all seeds [19-27] resulting in better convergence behavior. The exonential increase in microrocessors ower and the associated decrease in unit cost have also benefited the CFD industry. Multirocessor systems with large memory, set u at a fraction of the cost of the suer comuters of a decade ago, have ushed the limits on the size of the roblems that can be tackled. Challenges still abound in a number of fields: increasing the robustness of the emloyed numerical schemes, imroving the accuracy of the used models, extending the many advances in the simulation of single fluid flows [28-34] to multi-hase flows [35], are just a few areas of current research interest. For multi-hase flow simulation, the basic difficulty [36] stems from the increased algorithmic comlexity that need to be addressed when dealing with multile sets of continuity and momentum equations that are inter-couled (interchange momentum by inter-hase mass and momentum transfer, etc.) both satially and across fluids. That is, on to of the velocity-ressure couling for each hasic continuity-

6 An algorithm for Multi-hase Flow at All Seeds 6 momentum set, there exist a number of inter-fluid couling relations. This is further comlicated in the simulation of suersonic multi-hase flows, or in general when develoing an all seed flow multi-hase algorithm. Desite these comlexities, successful segregated incomressible ressure-based solution algorithms have been devised, such as the ISA variants develoed by the Salding Grou at Imerial College [37-39] and the set of algorithms develoed by the Los Alamos Scientific Laboratory (LASL) grou [40-42]. For comressible flow simulations at high Mach number, secial treatment is needed to resolve the density-velocity and density-ressure coulings. Algorithms for all-seed multi-hase flow simulation were recently resented and new ones derived in [36] following a ressurebased aroach but none was imlemented or tested. Actually to the authors knowledge, no work dealing with a ressure-based method caable of redicting multi-hase flow henomena at all seeds has been reorted in the literature. It is the objective of this work to test a newly develoed multi-hase ressure-based solution rocedure that is equally valid at all Reynolds and Mach number values. In what follows the governing equations for comressible multi-hase flows are first resented and their discretization outlined so as to lay the ground for the derivation of the ressure-correction equation, which is obtained from overall mass conservation. This class of algorithms is denoted as the Mass Conservation Based Algorithms (MCBA) [36]. Then, a brief descrition of the solution rocedure is given and the caability of the newly develoed algorithm to redict multi-hase flow at all seeds demonstrated by resenting solutions to four test roblems sanning the entire subsonic to suersonic sectrum over a wide range of hysical conditions (from turbulent incomressible bubbly flows to suersonic air-article flows). The roblems solved are: (i) turbulent incomressible bubbly flow in a ie, (ii) turbulent incomressible air-article flow in a ie, (iii) comressible dilute air-article flow over a flat late, and (iv) inviscid transonic dusty flow in a converging-diverging nozzle.

7 An algorithm for Multi-hase Flow at All Seeds 7 Furthermore, the accuracy of the method is demonstrated by comaring results against ublished exerimental and/or numerical data. The Governing Equations In multi-hase flow the various fluids/hases coexist with different concentrations at different locations in the flow domain and move with unequal velocities. Thus, the equations governing multi-hase flows are the conservation laws of mass, momentum, and energy for each individual fluid. For turbulent multi-hase flow situations, an additional set of equations may be needed deending on the turbulence model used. These equations should be sulemented by a set of auxiliary relations. The various conservation equations are: ( r ρ ) ( k ( ) ) t + r ρ u = r M& (1) k ( r u ( ) ρ ) ( k ) ( k) ( k) ( k + ( r ρ u u ) = [ r ( µ + µ ) u ] + r ( + B ) ) + I t ( k ) ( r ρ T ) ( k ) ( k) + ( r ρ u T ) t r + c β T + t = r µ µ t + r rt t T I E ( u ) ( u ) + Φ + q& + where the meanings of the various terms are as given in the nomenclature. In addition to the above mass, momentum, and energy conservation equations (Eqs. (1)-(3)), a geometric conservation equation is needed for multi-hase flow. hysically, this equation is a statement indicating that the sum of volumes occuied by the different fluids, r, within a cell is equal to the volume of the cell containing the fluids, and is given as: c M (2) (3) r (k ) = 1 (4) k Because a static mesh is used, Eq. (4) does not include a transient term. The effect of turbulence on interfacial mass, momentum, and energy transfer is difficult to model and is still an active area of research. Similar to single-fluid flow, researchers have

8 An algorithm for Multi-hase Flow at All Seeds 8 advertised several flow-deendent models to describe turbulence. These models vary in comlexity from simle algebraic [43] models to state-of-the-art Reynolds-stress [44] models. In this work, the widely used two-equation k-ε turbulence model [45] with multihase secific modifications is adoted. The hasic conservation equations governing the turbulence kinetic energy and turbulence dissiation rate (ε) for the k th fluid are given by: ( k) ( r ρ k ) ( k) ( k) µ ( k) ( ) ( ) t + ( k ) ( r ρ ε ) ( k) + ( r ρ u ε ) t r ρ r ε k ρ t u k = r k + r ρ G ε + I k (5) σ k ( k) ( c G c ε ) + I 1ε = r 2ε ε µ σ t ε where and reresent the interfacial turbulence terms. The turbulent viscosity is I k calculated as: I ε 2 ( k )[ k ] µ t = Cµ ρ (7) ε For two-hase flows, several extensions of the k-ε model that are based on calculating the turbulent viscosity by solving the k and ε equations for the carrier or continuous hase only have been roosed in the literature [46-51]. In a recent article, Cokljat and Ivanov [45] resented a hase couled k-ε turbulence model, intended for the cases where a non-dilute secondary hase is resent, in which the k-ε transort equations for all hases are solved. Since the method is still not well develoed, the first aroach in which only the k and ε equations for the carrier hase are solved is adoted in this work. If a tyical reresentative variable associated with hase is denoted by φ, the above conservation equations can be resented via the following general hasic equation: ε + ( r ρ φ ) ( k ( ) ( ) ) t + r ρ u φ = r Γ φ + r Q (8) where the exression for Γ and Q can be deduced from the arent equations. (6)

9 An algorithm for Multi-hase Flow at All Seeds 9 The resented set of differential equations has to be solved in conjunction with constraints on certain variables reresented by algebraic relations. These auxiliary relations include the equations of state and the interfacial mass, momentum, energy, and turbulence energy transfers. For a comressible multi-hase flow, auxiliary equations of state relating density to ressure and temerature are needed. For the k th hase, such an equation can be written as: ρ =ρ (,T ) (9) Several models have been develoed for comuting the interfacial mass, momentum, energy, and turbulence energy transfers terms. Details regarding the closures used here are given in the results section. In order to resent a comlete mathematical roblem, thermodynamic relations may be needed and initial and boundary conditions should sulement the above equations. Discretization rocedure Integrating the general conservation equation (8) over a finite volume (Fig. 1) yields: Ω ( r ρ φ ) dω + ( r ρ u φ ) dω t Ω = ( r Γ φ ) dω + Ω Where Ω is the volume of the control cell (Fig. 1). Using the divergence theorem to transform the volume integral into a surface integral and then relacing the surface integral by a summation of the fluxes over the sides of the control volume, equation (10) is transformed to: ( ) t r ρ φ Ω ( ) Ω r Q dω (10) + J D C nb + J nb = r Q Ω (11) nb =e,w,n,s,t,b ( k ) D ( k ) C where J and J are the diffusive and convective fluxes, resectively. The nb nb discretization of the diffusion term is second order accurate and follows the derivations

10 An algorithm for Multi-hase Flow at All Seeds 10 resented in [35]. For the convective terms and for the calculation of interface densities, the third-order SMART [13] scheme is emloyed and imlemented within the context of the Normalized Variables and Sace Formulation (NVSF) methodology [15]. Moreover, the integral value of the source term over the control volume is obtained by assuming the estimate of the source at the control volume center to reresent the mean value over the whole control volume. Furthermore, the additional terms aearing in the momentum and energy equations, not featured in equation (10), are treated exlicitly and their discretization is analogous to that of the ordinary diffusion flux. Substituting the face values by the functional relationshis relating them to the neighboring node values, Eq. (11) is transformed after some algebraic maniulations into the following discretized equation: A φ = A φ + B (12) NB NB NB (k where the coefficients A and A deend on the selected scheme and is the source ) NB B term of the discretized equation. In comact form, the above equation can be written as φ = H A NBφ NB NB [ φ ] = A + B The discretization rocedure for the momentum equation yields an algebraic equation of the form: Ω 0 u ( k) ( k) ( k) ( k) A u = H [ u ] r D ( ) where D = (14) Ω 0 v A On the other hand, the hasic mass-conservation equation (Eq. (1)) can be either viewed as a hasic volume fraction equation: (13) r = H [ r ] (15) or as a hasic continuity equation to be used in deriving the ressure correction equation:

11 An algorithm for Multi-hase Flow at All Seeds 11 Old ( r ρ ) ( r ρ ) k [ ] Ω + r ρ u ( ) S = r M& δt (16) where the oerator reresents the following oeration: [ Θ]= Θ f (17) f= NB() Solution rocedure The number of equations describing an n-fluid flow situation are: n hasic momentum equations, n hasic volume fraction (or mass conservation) equations, a geometric conservation equation, and for the case of a comressible flow an additional n auxiliary ressure-density relations. Moreover, the variables involved are the n hasic velocity vectors, the n hasic volume fractions, the ressure field, and for a comressible flow an additional n unknown hasic density fields. In the current work, the n momentum equations are used to calculate the n velocity fields, n-1 volume fraction (mass conservation) equations are used to calculate n-1 volume fraction fields, and the last volume fraction field calculated using the geometric conservation equation (n) r = 1 r (18) k n The remaining volume fraction equation can be used to calculate the ressure field that is shared by all hases. However, instead of using this last volume fraction equation, in the class of Mass Conservation Based Algorithms (MCBA) the global conservation equation is emloyed, i.e. the sum of the individual mass conservation equations, to derive a ressure correction equation as outlined next. The ressure Correction Equation To derive the ressure-correction equation, the mass conservation equations of the various hases are added to yield the global mass conservation equation given by: ( r ρ ) ( r ρ ) Old k Ω + ( r ρ u ( ) S) = r M = 0 k t & k δ (19)

12 An algorithm for Multi-hase Flow at All Seeds 12 In the redictor stage a guessed or an estimated ressure field from the revious iteration, denoted by o, is substituted into the momentum equations. The resulting velocity fields denoted by u (k )* which now satisfy the momentum equations will not, in general, satisfy the mass conservation equations. Thus, corrections are needed in order to yield velocity and ressure fields that satisfy both equations. Denoting the corrections for ressure, velocity, and density by, u (k ), and ρ ( k ) resectively, the corrected fields are written as: = ο +, * ο u = u + u, ρ = ρ + ρ (20) where the suerscrit o refers to values from the revious iterations. Hence the equations solved in the redictor stage are: * * ο ο u = H [ u ] r D (21) While the final solutions satisfy u = H [ u ] r D (22) Subtracting the two equation sets ((22) and (21)) from each other yields the following equation involving the correction terms: ο u = H [ u ] r D Moreover, the new density and velocity fields, ρ and u, will satisfy the overall mass conservation equation if: k ο old ( r ρ ) ( r ρ ) k [ r ο u ( ) S = 0 δt Ω + Exanding the (ρ u ) term, one gets ( ρ * +ρ )u ( * + u )=ρ * u * +ρ * u +ρ u * +ρ (k ρ (23) ] (24) ) u (k ) (25) Substituting equations (25) and (23) into equation (24), rearranging, and relacing density correction by ressure correction, the final form of the ressure-correction equation is written as:

13 An algorithm for Multi-hase Flow at All Seeds 13 ο ο ο * * [ r U C ] [ ρ ( ρ r r D ) S] Ω ο r Cρ + k δt old (26) ο * r ρ ( ρ ) r ο * * = Ω + [ r ρ U ] k δt The corrections are then alied to the velocity, ressure, and density fields using the following equations: (k u )* = u o r ( k)o ( D k), * = o +, ρ * = ρ o + C ρ (27) Numerical exeriments using the above aroach to simulate air-water flows have shown oor conservation of the lighter fluid. This roblem can be considerably alleviated by normalizing the individual continuity equations, and hence the global mass conservation equation, by means of a weighting factor such as a reference density ρ (which is fluid deendent). This aroach has been adoted in solving all roblems resented in this work (see [36] for details). The MCBA-SIMLE Algorithm The overall solution rocedure is an extension of the single-hase SIMLE algorithm into multi-hase flows. Since the ressure correction equation is derived from overall mass conservation, it is denoted by MCBA-SIMLE [36]. The sequence of events in the MCBA- SIMLE is as follows: 1. Solve the hasic momentum equations for velocities. 2. Solve the ressure correction equation based on global mass conservation. 3. Correct velocities, densities, and ressure. 4. Solve the hasic mass conservation equations for volume fractions. 5. Solve the hasic scalar equations (k, ε, T, etc ). 6. Return to the first ste and reeat until convergence.

14 An algorithm for Multi-hase Flow at All Seeds 14 Results and Discussion The erformance of the above-described solution rocedure is assessed in this section by resenting solutions to four two-dimensional two-hase flow roblems sanning the entire subsonic to suersonic sectrum. The first two roblems deal with incomressible turbulent flows while the last two roblems are concerned with comressible flows. Comuted results are comared against available exerimental data and/or numerical/theoretical values. In all roblems, the first hase reresents the continuous hase (denoted by a suerscrit ), which must be fluid, and the second hase is the diserse hase (denoted by a suerscrit (d)), which may be solid articles or fluid. Unless otherwise secified the third order SMART scheme is used in all comutations reorted in this study. roblem 1:Turbulent uward bubbly flow in a ie The roblem considered involves the rediction of radial hase distribution in turbulent uward air-water flow in a ie. Many exerimental and numerical studies addressing this roblem have aeared in the literature [52-60]. Most of these studies have indicated that the lateral forces that most strongly affect the void distribution are the turbulent stresses and the lateral lift force. As such, in addition to the usual drag force, the lift force is considered as art of the interfacial force terms in the momentum equations. In the resent work, the interfacial drag forces er unit volume are given by: x (c ) x (d) C ( I M ) = IM D ( ) D = D ρ r (d) r V sli ( u (d) u ) (28) r y (c ) y (d) C ( I M ) D = I ( M ) D = D ρ r (d) r V sli ( v (d) v ) (29) r where r is the bubble radius. The drag coefficient C D varies as a function of the bubble Reynolds and Weber numbers defined as: Re r r 2 = 2 V sli We = 4ρ Vsli (30) ν l σ

15 An algorithm for Multi-hase Flow at All Seeds 15 where σ, the surface tension, is given a value of N/m for air-water systems. The following correlations, which take the shae of the bubble into consideration, are utilized [61,62]: C D = 16 for Re Re < C D = Re for 0.49 < Re < 100 C D = for Re Re >> 100 C D = 8 for Re 3 >> 100 and We > 8 C D = We for Re >> 100 and Re > / We Many investigators have considered the modeling of lift forces [61-65]. Based on their work, the following exressions are emloyed for the calculation of the interfacial lift forces er unit volume: (d) (d) (d) ( I ) ( I ) = C ρ r ( u u ) ( u ) M L = (32) M L 1 where C 1 is the interfacial lift coefficient calculated from: C 1 = C 1a ( , r (d) ) (33) where a, b denotes the minimum of a and b and C 1a is an emirical constant. Besides the drag and lift interfacial forces, the effect of bubbles on the turbulent field is very imortant, as the distribution of bubbles affects the turbulence field in the liquid hase and at the same time the liquid hase s turbulence is influenced by the bubbles. In this work, turbulence is assumed to be a roerty of the continuous liquid hase and the turbulent kinematics viscosity of the disersed air hase (d) is assumed to be a function of that of the continuous hase. The turbulent viscosity of the continuous hase is comuted by solving the following modified transort equations for the turbulent kinetic energy k and its dissiation rate ε that take into account the interaction between the hases: (31)

16 An algorithm for Multi-hase Flow at All Seeds 16 ( r ρ k ) + ( r ρ u k ) t r ρ = r ν ( G ε ) + ρ t k r + r b σ ρ r ν l ν + σ t k k + (34) ( r ρ ε ) + ( r ρ u ε ) t r ρ ε k = r ( c G c ε ) 1ε 2ε ρ ν + ρ l ν + σ ν t σ r t ε ε ε r + r c 1ε b ε k + (35) where G is the well known volumetric roduction rate of k by shear forces, σ r the turbulent Schmidt number for volume fractions, and b is the roduction rate of k by drag due to the motion of the bubbles through the liquid and is given by: b (d) C bc Dρ r r Vsli = (36) r In Eq. (36) C b is an emirical constant reresenting the fraction of turbulence induced by bubbles that goes into large-scale turbulence of the liquid hase. Moreover, as suggested in [60], the flux reresenting the interaction between the fluctuating velocity and volume fraction is modeled via a gradient diffusion aroximation and added as a source term in the continuity ( ( ρ D r ) and momentum ( ( ρ D r ) diffusion coefficient D given by: u equations with the D = ν t (37) σ r The turbulent kinematics eddy viscosity of the disersed and continuous hases are related through: ν (d) t = ν t (38) σ f where σ f is the turbulent Schmidt number for the interaction between the two hases. The above described turbulence model is a modified version of the one described in [60] in which the turbulent kinematics viscosities of both hases are allowed to be different in contrast to

17 An algorithm for Multi-hase Flow at All Seeds 17 what is done in [60]. This is accomlished through the introduction of the σ arameter. As such, different diffusion coefficients (D ) are used for the different hases. Results are comared to exerimental data from [52,63]. Two exeriments were simulated using the above described treatment and results comared to exerimental data. The two exeriments differ in the Reynolds number, the bubbles diameter and the inlet conditions. In the exeriment of Seriwaza et al [52] the Reynolds number based on suerficial liquid velocity and ie diameter is 8x10 4, the inlet suerficial gas and liquid velocities are and 1.36 m/s, resectively, and the inlet void fraction is 5.36x10-2 with no sli between the incoming hases. Moreover, the bubble diameter is taken as 3 mm [60], while the fluid roerties are taken as ρ =1000 Kg/m 3, ρ (d) =1.23 Kg/m 3, and ν l = 10-6 m 2 /s. In the exeriment of Lahey et al. [63] the Reynolds number is based on suerficial liquid velocity and ie diameter of 5x10 4, the inlet suerficial gas and liquid velocities are 0.1 and 1.08 m/s, resectively. Both roblems are solved using the same values for all constants in the model with C 1a =0.075, σ f =0.5, σ r =0.7, and C b =0.05. redicted radial rofiles of the vertical liquid velocity and void fraction resented in Figs. 2(a) and 2(b) using a grid of size 96x32 control volumes concur very well with measurements and comare favorably with numerical rofiles reorted by Boisson and Malin [60] (Fig. 2(a)) and HOENICS [65] (Fig. 2(b)). As shown, the void fraction rofile indicates that gas is taken away from the ie center. This is caused by the lift force, which drives the bubbles towards the wall. roblem 2:Turbulent air-article flow in a vertical ie In roblem 2, the uward flow of a dilute gas-solid mixture in a vertical ie is simulated. As in the revious roblem, the axi-symmetric form of the gas and articulate transort equations are emloyed. The effects of interfacial virtual mass and lift forces are small and may be neglected, as reorted in several studies [66-68], and the controlling interfacial force f

18 An algorithm for Multi-hase Flow at All Seeds 18 is drag (see Harlow and Amsden [69]). Denoting the continuous and disersed hases by suerscrits and (d), resectively, the drag in the x- and y-momentum equations are given by: x x (d) 3 CD (d) (d) ( I ) ( I ) = ρ r V ( u u M D = M D sli ) (39) 8 r y y (d) 3 CD (d) (d) ( I ) ( I ) = ρ r V ( v v M D = M D sli ) (40) 8 r where r reresents the article s radius, C D the drag coefficient comuted from: C D = 24 for Re Re < 1 C D = Re Re C D = 0.44 for Re > 1000 ( ) for 1 < Re < 1000 and Re the Reynolds number based on the article size as defined in Eq.(30). (41) As before, turbulence is assumed to be a roerty of the continuous gas hase and the turbulent kinematics viscosity of the disersed article hase (d) is assumed to be a function of that of the continuous hase. Again, turbulence modulation due to the resence of articles is redicted using a two-hase k-ε model. Several extensions of the k-ε model for carrierhase turbulence modulation have been roosed in the literature [46-51] and the modification of Chen and Wood [48], which introduces additional source terms into the turbulence transort equations, is adoted here. These source terms are always negative and act to reduce k and ε. However, deending on the relative extent of reductions in k and ε, the turbulent viscosity may be either reduced or increased by the resence of articles. Thus, the turbulent viscosity is comuted by solving the turbulence transort equations (Eqs. (5) and (6)) for the continuous hase with I and I evaluated using the following relations suggested by Chen and Wood [48]: k ε

19 An algorithm for Multi-hase Flow at All Seeds 19 τ (d) (d) k τ I = 2ρ r r 1 e e k τ (42) I ε = 2ρ (d) r r (d) ε (43) τ where τ and τ e are timescales characterizing the article resonse and large-scale turbulent motion, resectively, and are comuted from: τ = ρ(d) r (d) F D τ e = k V sli ε where F D is the magnitude of the inter-hase drag force er unit volume. The turbulent kinematics eddy viscosity of the disersed hase is found using Eq. (38). The model is validated against the exerimental results of Tsuji et al [66]. In their exeriments, the vertical ie has an internal diameter of 30.5 mm. Results are relicated here for the case of an air Reynolds number, based on the ie diameter, of 3.3x10 4 and a mean air inlet velocity of 15.6 m/s using articles of diameter 200 µm and density 1020 Kg/m 3. In the comutations, the mass-loading ratio at inlet is considered to be 1 with no sli between the hases, and σ f and σ r are set to 5 and (i.e. the interaction terms included for bubbly flows are neglected here), resectively. Figure 3 shows the fully develoed gas and articles mean axial velocity rofiles generated using a grid of size 96x40 C.V. It is evident that there is generally a very good agreement between the redicted and exerimental data with the gas velocity being slightly over redicted and the articles velocity slightly under redicted. Moreover, close to the wall, the model redictions indicate that the articles have higher velocities than the gas, which is in accord with the exerimental results of Tsuji et al. [66]. (44)

20 An algorithm for Multi-hase Flow at All Seeds 20 roblem 3:Comressible dilute air-article subsonic flow over a flat late This is a well-studied roblem [66-76] suitable as a benchmark test. It is known that twohase flow greatly changes the main features of the boundary layer over a flat late. Tyically, three regions are defined in the two-hase boundary layer (Fig. 4), based on the imortance of the sli velocity between the two hases: a large-sli region close to the leading edge, a moderate-sli region further down, and a small-sli one far downstream. The characteristic scale in this two-hase roblem is the relaxation length λ e [73], defined as: λ e = 2 9 ρ (d) r 2 u µ (45) where ρ (d) and r are resectively the density and radius of the articles, µ the viscosity of the fluid, and u the free stream velocity. The three regions are defined according to the order of magnitude of the sli arameter x* = x/ λ e. In the simulation, the viscosity of the fluid is considered to be a function of temerature and varies according to [73]: µ T =µ ref T ref 0.6 where the reference viscosity and temerature are µ ref =1.86x10-5 N.s/m 2 and T ref =303 K, resectively. Even though variations in gas density are small under the conditions considered, these variations are not neglected and the flow is treated as comressible for the continuous hase and as incomressible for the disersed hase. Moreover, drag is the only interfacial force retained due to its dominance over other interfacial forces. Denoting the continuous and disersed hases by suerscrits and (d), resectively, this force is comuted as [73]: x x (d) 9 C D (d) (d) ( I ) ( I ) = r µ ( u u M D = M ) (47) D 2 r y y (d) 9 C D (d) (d) ( I ) ( I ) = r µ ( v v M D 2 = M ) (48) D 2 r where the drag coefficient is given by: 2 (46)

21 An algorithm for Multi-hase Flow at All Seeds C D = Re + Re (49) 50 6 In the energy equation, heat transfer due to radiation is neglected and only convective heat transfer around an isolated article is considered. Moreover, the articles have no individual random motion, mutual collisions, and other interactions among them. Therefore, only the rocess of drag and heat transfer coule the articles with the gas. Under such conditions, the interfacial terms in the gas (continuous hase) and articles (disersed hase) energy equations reduce to [73]: I E = Q g + F g.u (d) (50) I E (d) = Q g (51) where: F g = I M x ( ) D i + I y ( M ) D j (52) Nu = Re Q g = 3 2 r (d) λ Nu r r ( ) 1 3 (53) ( T (d) T ) (54) In the above equations, Nu is the Nusselt number, r the gas randtl number, λ the gas thermal conductivity, T the temerature, and other arameters are as defined earlier. In the simulation, the article diameter is chosen to be 10 µm, the article Reynolds number is assumed to be equal to 10, the material density is 1766 kg/m 3, and the randtl number is set to The south boundary (wall) is treated as a no-sli wall boundary for the gas hase (i.e. both comonents of the gas velocity are set to zero), while the article hase encounters sli wall conditions (i.e. the normal fluxes are set to zero). The gas and the articles enter the comutational domain under thermal and dynamical equilibrium conditions. A mass load ratio of 1 between the articles hase and the gas hase is used. Results are dislayed using the following dimensionless variables in order to bring all quantities to the same order of magnitudes:

22 An algorithm for Multi-hase Flow at All Seeds 22 x* = x λ e,y*= y λ e Re,u*= u u,v*= v u Re Re = ρuλ e µ Figure 5 shows the results for the steady flow obtained on a rectangular domain with a mesh of density 104x48 C.V. stretched in the y-direction. The figure rovides the develoment of gas and articles velocity rofiles within the three regions mentioned earlier. In the near leading edge area (x*=0.1), the gas velocity is adjusted at the wall to obtain the no-sli condition as for the case of a ure gas boundary layer. The articles have no time to adjust to the local gas motion and there is a large velocity sli between the hases. In the transition region (x*=1), significant changes in the flow roerties take lace. The interaction between the hases causes the articles to slow down and the gas to accelerate as aarent in the lots. In the far downstream region (x*=5), the articles have enough time to adjust to the state of the gas motion. The sli is very small and the solution tends to equilibrium. These results are in excellent agreement with the numerical solutions reorted by Thevand et al. [76] lotted in Fig. (5), validating the roosed methodology. roblem 4:Inviscid transonic dusty flow in a converging-diverging nozzle As a final test for the newly suggested numerical rocedure, dilute two-hase transonic flow in an axi-symmetric converging-diverging rocket nozzle is considered. Several researchers have analyzed the roblem and data is available for comarison [77-86]. In most of the reorted studies, a shorter diverging section, in comarison with the one considered here, has been used when redicting the two-hase flow. Two-hase results for the long configuration have only been reorted by Chang et al. [81]. The flow is assumed to be inviscid and the single-hase results are used as an initial guess for solving the two-hase roblem. The hysical configuration (Fig. 6) is the one described in [81]. The viscosity of the fluid, which is solely used in the calculation of the interfacial drag force, varies with the temerature according to Sutherland s law for air: (55)

23 An algorithm for Multi-hase Flow at All Seeds 23 T T µ = 1.458x10 6 (56) T The couling between gas and article hases is through the interfacial momentum and energy terms. The force exerted on a single article moving through a gas is given as [82] f x = 6πr f D µ u (d) u f y = 6πr f D µ ( ) (57) ( v (d) v ) (58) so that for N articles in a unit volume the effective drag force is (d) x x (d) 9 r (d) ( I ) ( I ) = f µ ( u u M D = M D D ) (59) 2 r (d) y y (d) 9 r (d) ( I ) ( I ) = f µ ( v v M D 2 = M D D ) (60) 2 r 2 where f D is the ratio of the drag coefficient C D to the stokes drag C D0 =24/Re and is given by [81] Re f D = Re x Re Re < 3x10 5 (61) The heat transferred from gas to article hase er unit volume is given as [82] Q g = 3 2 r (d) ( ) (62) r λ Nu T (d) T (c ) Where λ is the thermal conductivity of the gas and Nu, the Nusselt number, is written as [82] Nu = Re 0.55 r c 0.33 The gas-article inter-hase energy term is given by (63) I E 9 r (d) (d) (d) 9 r (d) 3 r (d) ( u u ) u + f µ ( v v ) v + λ Nu( T T ) (d) = f Dµ 2 d 2 D d (64) 2 r 2 r 2 r I E (d) = 3 2 r (d) r λ Nu( T T (d) ) (65) where the first two terms on the right-hand side of equation (64) reresent the energy exchange due to momentum transfer.

24 An algorithm for Multi-hase Flow at All Seeds 24 The hysical quantities emloyed are similar to those used in [81]. The gas stagnation temerature and ressure at inlet to the nozzle are 555 ºK and 10.34x10 5 N/m 2, resectively. The secific heat for the gas and articles are 1.07x10 3 J/KgºK and 1.38x10 3 J/KgºK, resectively, and the article density is kg/m 3. With a zero inflow velocity angle, the fluid is accelerated from subsonic to suersonic seed in the nozzle. The inlet velocity and temerature of the articles are resumed to be the same as those of the gas hase. Results for two article sizes of radii 1 and 10 µm with the same mass fraction φ=0.3 are resented using a grid of size 188x80 C.V. Figures 7(a) and 7(b) show the article volume fraction contours while Figures 7 and 7(d) dislay the velocity distribution. For the flow with articles of radius 1 µm, a shar change in article density is obtained near the uer wall downstream of the throat, and the article density decreases to a small value. With the large article flow (10 µm), however, a much larger article-free zone aears due to the inability of the heavier articles to turn around the throat corner. These findings are in excellent agreement with ublished results reorted in [81] and others using different methodologies. In addition the contour lines are similar to those reorted by Chang et al. [81]. A quantitative comarison of current redictions with ublished exerimental and numerical data is resented in Fig. 8 through gas Mach number distributions along the wall (Fig. 8(a)) and centerline (Fig. 8(b)) of the nozzle for the one-hase and two-hase flow situations with articles of radii 10 µm. As can be seen, the one-hase redictions fall on to of exerimental data reorted in [84-86]. Along the centerline of the nozzle, current redictions are of quality better than those obtained by Chang et al. [81]. Since the nozzle contour has a raid contraction followed by a throat with a small radius of curvature, the flow near the throat wall is overturned and inclined to the downstream wall. A weak shock is thus formed to turn the flow arallel to the wall. This results in a sudden dro in the Mach number value and as deicted in Fig. 8(b),

25 An algorithm for Multi-hase Flow at All Seeds 25 this sudden dro is correctly envisaged by the solution algorithm with the value after the shock being slightly over redicted. Due to the unavailability of two-hase flow data, redictions are comared against the numerical results reorted in [81]. As dislayed in Figs. 8(a) and 8(b), both solutions are in good agreement with each other indicating once more the correctness of the calculation rocedures. The lower gas Mach number in the two-hase flow is caused by the heavier articles (ρ (d) >>ρ ), which reduce the gas velocity. Moreover, owing to the article-free zone, the Mach number difference between the one- and two-hase flows along the wall is smaller than that at the centerline. Closing Remarks A new finite volume-based numerical rocedure for the calculation of multi-hase flows at all seeds was resented. The virtues of the method were demonstrated by solving four twohase flow roblems sanning the entire subsonic to suersonic sectrum over a wide range of hysical conditions: turbulent bubbly flow in a ie, turbulent air-article flow in a ie, subsonic comressible air-article flow over a flat late, and transonic dusty flow in a converging diverging nozzle. Results generated were comared against exerimental and/or numerical simulation data where available. The accuracy of the redicted quantities, which was shown to be similar or better than that obtained with secial urose methods, was a clear demonstration of the effectiveness of the new method as a tool for modeling multihase flows at all seeds. Acknowledgments The financial suort rovided by the Euroean Office of Aerosace Research and Develoment (EOARD) (SC ) is gratefully acknowledged.

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