ON THE PERFORMANCE OF MASS CONSERVATION BASED ALGORITHMS FOR MULTI-PHASE FLOWS

Transcription

1 ON THE ERFORMANCE OF MASS CONSERVATION BASED ALGORITHMS FOR MULTI-HASE FLOWS F. Moualled and M. Dawish Ameican Univesity of Beiut, Faculty of Engineeing & Achitectue, Mechanical Engineeing Depatment,.O.Box Riad El Solh, Beiut Lebanon ABSTRACT This wo is concened with the implementation and testing, of fou incompessible-segegated multi-phase flow algoithms that belong to the Mass Consevation Based Algoithms (MCBA) goup in which the pessue coection equation is deived fom oveall mass consevation. The pessue coection schemes in these algoithms ae based on SIMLE, SIMLEX, ISO, and RIME. Solving two one-dimensional two-phase flow poblems spanning the spectum fom bubbly to gas-solid flows assesses the pefomance and accuacy of the multiphase algoithms. The main outcome of this study is a clea demonstation of the capability of all MCBA algoithms to deal with multi-phase flow situations. Moeove, esults displayed in tems of convegence histoy plots and CU-times, indicate that the pefomances of the MCBA vesions of SIMLE and SIMLEX ae vey close. As expected, the RIME algoithm is found to be the most expensive due to its explicit teatment of the phasic momentum equations. The ISO algoithm is geneally moe expensive than SIMLE and its pefomance depends on the type of flow and solution method used. KEY WORDS Multi-phase algoithms, bubbly flows, ai-paticle flows. 1. INTRODUCTION The extensive developments that have taen place in Computational Fluid Dynamics (CFD) ove the last thee decades have established this still evolving technology as a eliable and essential tool fo the simulation and optimization of a wide vaiety of engineeing fluid flow pocesses (mixing, solidification, tubulence, ). Seveal issues that wee hindeing its pogess have been addessed and emedies suggested. Concens elated to accuacy wee assuaged though the development of High Resolution (HR) schemes [1-3]. Moeove, bette solution algoithms [4-7], solves, and multi-gid techniques have geatly educed the computational cost and made it feasible to solve eal life poblems. While high-esolution schemes, solves, multi-gid techniques, etc can be applied to simulate both singleand multi-phase flows, nealy all developments in solution algoithms have been diected towads the simulation of single-fluid flow. In paticula, many segegated single-fluid solution algoithms wee developed such as the well-nown SIMLE [4], ISO [5], RIME [8], and SIMLEX [6] algoithms, to site a few. On the othe hand, developments in solution algoithms fo simulating multi-phase flow phenomena have lagged behind that of single-phase flow algoithms due to the much highe computational cost involved, the numeical difficulties that had to be fist addessed in the simulation of single-phase flow, and the incease in algoithmic complexity. While the majo difficulties in the simulation of single-phase flow stems fom the coupling between the momentum and continuity equations, in the simulation of multi-phase flow phenomena, this poblem is futhe complicated by the fact that thee ae as many sets of continuity and momentum equations as thee ae fluids, that they ae all coupled togethe in vaious ways and that the fluids shae space. Despite these complexities, successful segegated pessue-based solution algoithms have been devised. The ISA vaiants devised by the Spalding goup at Impeial College [9] and the set of algoithms devised by the Los Alamos Scientific Laboatoy (LASL) goup [10] ae examples of multi-phase algoithms. Recently, Dawish et al. [11] extended the lage numbe of segegated single-fluid flow algoithms eviewed in [7] to pedict multi-phase flow phenomena and showed that the pessue coection equation can be deived eithe by using the geometic consevation equation o the oveall mass consevation equation. Depending on the chosen equation, the segegated pessue-based multi-phase flow algoithms wee classified as eithe the Geometic Consevation Based family of Algoithms (GCBA) o the Mass Consevation Based family of Algoithms (MCBA). Many of these algoithms have neithe been tested no implemented in CFD codes.

2 The objective of the pesent wo is to implement and test fou multi-phase algoithms fom the MCBA goup and to assess thei elative pefomance by solving two one-dimensional incompessible two-phase flow poblems encompassing gas-solid flows in addition to bubbly flows on seveal gid sizes. 2. THE GOVERNING EQUATIONS In incompessible multi-phase flow the vaious fluids/phases coexist with diffeent concentations at diffeent locations in the flow domain and move with unequal velocities. Thus, the equations govening multiphase flows ae the consevation laws of mass and momentum fo each individual fluid. If a typical epesentative vaiable associated with phase () is denoted by φ ), the consevation equations can be witten using a geneal phasic equation as: ( ) φ +. u φ = t (1). Γ φ + Q whee () efes to the volume faction ( () /), () the phasic density, and u () the velocity vecto. Moeove, the diffusion coefficient Γ () and souce tem Q () ae specific fo a paticula meaning of φ. Fo incompessible lamina multi-phase flow, auxiliay elations ae needed to close the system of equations including the geometic consevation ( ) equation = 1 and the intefacial mass and momentum tansfes. In this wo, only intefacial momentum tansfe is of inteest and its closue will be detailed late. 2. DISCRETIZATION ROCEDURE The geneal consevation equation (1) is integated ove a finite volumeto yield: ( ) ( φ ) ( ) d +.( u φ ) t.( Γ φ ) d + Q d = d Whee is the volume of the contol cell. Using the divegence theoem to tansfom the volume integal into a suface integal, eplacing the suface integals by a summation of the fluxes ove the sides of the contol volume, and then discetizing these fluxes using suitable intepolation pofiles (the High Resolution SMART [1] scheme is employed and applied within the context of the NVSF methodology [3]) the following algebaic equation esults: ( ) ( ) ( ) ( ) = ANBφ NB B NB (2) A φ + (3) as In compact fom, the above equation can be witten ANB NB ( ) NB [ ] φ φ = + B φ = H (4) A An equation simila to equation (4) is obtained at each gid point in the domain and the collection of these equations foms a system that is solved iteatively. The discetization pocedue fo the momentum equation yields an algebaic equation of the fom: u ( ) = H ( ) ( ) [ u ] D ( ) Futhemoe, the phasic mass-consevation equation can be viewed as a phasic volume faction equation, which can be witten as: ( ) [ ] H = (6) o as a phasic continuity equation to be used in deiving the pessue coection equation: ( ) ( ) δt Old + [ u ] ( ).S = M& whee the opeato epesents the following opeation: Θ = Θ f (8) [ ] f = nb() (5) (7) 3. RESSURE CORRECTION EQUATION To deive the pessue-coection equation, the mass consevation equations of the vaious fluids ae added to yield the global mass consevation equation given by: Old ( ) ( ) + δt = 0 (9) ( u.s) ( ) Denoting the coections fo pessue and velocity by and u ( ), espectively, the coected fields ae witten as: ( ) * ( ) = +, u = u + u (10) Combining equations (5), (9), and (10), the final fom of the pessue-coection equation is obtained as: * { [ ( D ).S]} ( + + )* ( ) δt Old * * [ U ] * () ( H[ u ]) [.S] ) If the [ u ] ( = (11) H tem in the above equation is etained, thee will esult a pessue coection equation elating the pessue coection value at a point to all values in the domain. To facilitate implementation and educe cost, simplifying assumptions elated to this tem have been intoduced. Depending on these assumptions,

3 diffeent algoithms ae obtained. These algoithms wee accoded a full-length pape [11] of discussion to which inteested eades ae efeed. The coections ae then applied to the velocity and pessue fields using the following equations: * * u = u D, = + (12) 4. THE MCBA SOLUTION ROCEDURE The sequence of events in the MCBA is as follows: Solve the phasic momentum equations fo velocities. Solve the pessue coection equation based on global mass consevation. Coect velocities and pessue. Solve the phasic mass consevation equations fo volume factions. Retun to the fist step and epeat until convegence. 5. RESULTS AND DISCUSSION Due to the lage numbe of paametes affecting the pefomance of the vaious multi-phase Mass Consevation Based Algoithms and to allow a thoough testing of these algoithms, one-dimensional two-phase flow poblems ae consideed. A total of eight poblems have been thooughly investigated anging fom dilute bubbly flows to dense gas-solid flows. Due to space limitation only two epesentative poblems ae discussed hee. The fist one deals with a gas-solid situation wheeas the second poblem is concened with a bubbly flow. Results ae pesented in tems of the convegence histoy and the CU-time needed to convege the solution to a set level. edictions ae compaed against available numeical/theoetical values. Computations ae teminated when the maximum nomalized esidual of all vaiables dops below a vey small numbe ε s. In geneal, it is found that equiing the oveall mass esiduals to be satisfied to within ε s is a vey stingent and sufficient equiement. The effects of gid efinement on accuacy and convegence ae studied by solving the poblems on fou gid systems of sizes 20, 40, 80, and 160 contol volumes with ε s assigned the value of. To allow a compaative assessment of pefomance, the CU times ae epoted in the fom of chats. Moeove, all CU times ae nomalized by the time needed by MCBA-SIMLE to each the set esiduals on the coasest gid. (c) (d) (c) (c) inlet (d) (d) inlet Fig. 1 hysical domain. Fee slip walls o Symmety planes L g=0 o 10 m/s 2 =0 oblem 1: Gas-solid flow The physical situation is depicted in Fig. 1. Depending on the set densities, it epesents eithe the steady flow of solid paticles suspended in a fee steam of ai o the steady flow of ai bubbles in a steam of wate. The slip between the phases detemines the dag, which is the sole diving foce fo the paticle-bubble/aiwate motion (g=0). In the suspension, the intepaticle/bubble foces ae neglected. Diffusion within both phases is set to zeo while the inte-phase dag foce is calculated as: ( c) ( d ) 3 C D ( d ) ( c) ( d ) ( c) I M = I M = Vslip ( u u ) (13) 8 Vslip ( d ) ( c) p = u u (14) The dag coefficient, C D, is set to The tas is to calculate the paticle/bubble-velocity distibution as a function of position. If the flow field is extended fa enough (hee computations ae pefomed ove a length of L=2m), the paticle/bubble and fluid phases ae expected to appoach an equilibium velocity given by: U equilibium = (c) (c) inlet + (d) (d) inlet (15) At inlet, the ai and paticle velocities ae 5 m/s and 1 m/s, espectively. The physical popeties of the two phases ae: ( d ) ( c) ( d ) 5 / = 2000, p = 1mm, inlet = 10 Due to the dilute concentation of the paticles, the fee steam velocity is moe o less unaffected by thei pesence and the equilibium velocity is nealy equal to the inlet fee steam velocity. Based on this obsevation, Mosi and Alexande [12] obtained an analytical solution fo the paticle velocity u (d) as a function of the position x. As shown in Fig. 2(a) the pedicted paticle velocity distibution falls on top of the analytical solution [12], which is an indication of the accuacy of the numeical pocedue. The convegence histoies of the vaious MCBA ove the fou gid netwos used ae displayed in Figs. 2(b)-2(e). Fo all algoithms, the equied numbe of iteations inceases as the gid size inceases, with ISO (Fig. 2(b)) equiing the minimum and RIME (Fig. 2(d)) the maximum numbe of iteations on all gids. The convegence histoies of SIMLE and SIMLEX (Figs. 2(c) and 2(e), espectively) ae vey simila equiing nealy the same numbe of iteations on all gids. oblem 2: Bubbly flow Fo the same configuation displayed in Fig. 1, the continuous phase is consideed to be wate and the dispese phase to be ai. The esulting flow is denoted in the liteatue by bubbly flow. With the exception of ( d ) ( c) 3 ( d ) inlet / = 10 and at inlet = 0. 1, othe physical popeties and inlet conditions ae the same as those consideed ealie. The coect physical solution is that the bubble and continuous phase velocities both each the equilibium velocity of 4.6 m/s (Eq. (15)) in a distance too small to be coectly esolved by any of the gid netwos used. Results fo this case ae pesented in Fig. 3. Axial.

4 velocity distibution fo both wate and ai ae displayed in Fig. 3(a). As expected, both phases each the equilibium velocity of 4.6 m/s ove a vey shot distance fom the inlet section and emain constant aftewad. The elative convegence chaacteistics of the vaious algoithms emain the same. Howeve, all algoithms equie lage numbe of iteations as compaed to the gas solid flow case due to the stonge coupling between the phases. Consistently, the ISO (Fig. 3(b)) and RIME (Fig. 3(d)) algoithms need the lowest and highest numbe of iteations, espectively. As in the pevious two cases, the convegence attibutes of SIMLE (Fig. 3(c)) and SIMLEX (Fig. 3(e)) ae vey simila SIMLEST-RIME Iteation (d) aticle Velocity Exact 1-D analytical solution Cuent edictions SIMLEX Axial distance ISO (a) Iteation (b) SIMLE Iteation (c) Iteation (e) Fig. 2 (a) Compaison between the analytical and numeical paticle velocity distibutions, (b)-(e) convegence histoies on the diffeent gid systems. CU time The nomalized CU effots equied by the vaious algoithms ove all gids ae depicted in Fig. 4. The chats clealy show that the CU time inceases with inceasing gid density. Fo the gas-solid poblem (Fig. 4(a)), it is had to see any noticeable diffeence in the CU times fo SIMLE, SIMLEX, and ISO. The wost pefomance is fo RIME, which uses a fully explicit solution scheme. The nomalized CU time of RIME fo the bubbly flow poblem (Figs. 4(b)) is lowe than in the pevious poblem due to a highe ate of incease in the time needed by othe algoithms (the computational time of all algoithms has inceased). The elative pefomance of the vaious algoithms is nealy as descibed ealie with the time equied by of ISO, SIMLE, and SIMLEX being on aveage the same. The RIME algoithm howeve, equies nealy thee folds the time needed by SIMLE, which epesents a noticeable impovement. By compaing the behavio of the vaious algoithms in both poblems, it is clea that the pefomance of SIMLE and SIMLEX is consistent and equie, on aveage, the least computational effot. The RIME algoithm is the most expensive to use on all gids and fo all physical situations pesented hee. Most impotantly

5 howeve, is the fact that all these algoithms can be used to pedict multi-phase (in this case two-phase) flows. SIMLEX 5 Velocity Liquid Gas Axial Distance (a) Iteation (e) Fig. 3 (a) Liquid and gas velocity distibutions, (b)-(e) convegence histoies on the diffeent gid systems. ISO Iteation (b) SIMLE simple simplex piso pime 25 (a) Iteation 15 (c) 10 SIMLEST-RIME 5 0 simple simplex piso pime (b) Fig. 4 Nomalized CU-times fo the hoizontal (a) gassolid, and (b) bubbly flow poblem Iteation (d)

6 5. CONCLUSION Fou MCBA algoithms fo the simulation of incompessible multi-phase flows wee implemented, tested, and thei elative pefomance assessed by solving a vaiety of one-dimensional two-phase flow poblems. Results obtained demonstated that all MCBA multiphase algoithms ae capable of dealing with a wide vaiety of incompessible multi-phase flow poblems. The convegence histoy plots and CU-times pesented, indicated simila pefomances fo SIMLE and SIMLEX. The ISO algoithm was in geneal moe expensive than SIMLE. Moeove, the RIME algoithm was the most expensive to use. 6. ACKNOWLEDGEMENT The financial suppot povided by the Univesity Reseach Boad of the Ameican Univesity of Beiut though Gant No is gatefully acnowledged. REFERENCES [1] Gasell,.H. and Lau, A.K.C., Cuvatue Compensated Convective Tanspot: SMART, A New Boundedness eseving Tanspot Algoithm, Int. J. Num. Meth. Fluids, 8, 1988, [2] Leonad, B.., Locally Modified Quic Scheme fo Highly Convective 2-D and 3-D Flows, in Taylo, C. and Mogan, K. (Eds.) Numeical Methods in Lamina and Tubulent Flows, 15 (Swansea, U.K: ineidge ess, 1987) [3] Dawish, M.S. and Moualled, F., Nomalized Vaiable and Space Fomulation Methodology Fo High- Resolution Schemes, Numeical Heat Tansfe, at B, 26, 1994, [4] atana, S.V. and Spalding, D.B., A Calculation ocedue fo Heat, Mass and Momentum Tansfe in Thee Dimensional aabolic Flows, Int. J. Heat & Mass Tans., 15, 1972, [5] Issa, R.I., Solution of the Implicit Discetized Fluid Flow Equations by Opeato Splitting, Mechanical Engineeing Repot, FS/82/15, (Impeial College, London, 1982). [6] Van Doomaal, J.. and Raithby, G. D., An Evaluation of the Segegated Appoach fo edicting Incompessible Fluid Flows, ASME ape 85-HT-9, oc. National Heat Tansfe Confeence, Denve, Coloado, August 4-7, [7] Moualled, F. and Dawish, M., A Unified Fomulation of the Segegated Class of Algoithms fo Fluid Flow at All Speeds, Numeical Heat Tansfe, at B, 40, 2001, [8] Malisa, C.R. and Raithby, G.D., Calculating 3-D fluid Flows Using non-othogonal Gid, oc. Thid Int. Conf. on Numeical Methods in Lamina and Tubulent Flows, Seattle, 1983, [9] Spalding, D.B., Numeical Computation of Multi- hase Fluid Flow and Heat Tansfe, in Taylo C. and Mogan K. (Eds.), Recent Advances in Numeical Methods in Fluid, 1, 1980, [10] Amsden, A.A., Halow F.H., KACHINA: An Euleian Compute ogam fo Multifield Flows, Repot LA-NUREG-5680, [11] Dawish, M., Moualled, F., and., Sea, B., A Unified Fomulation of the Segegated Class of Algoithms fo Multi-phase Flow at All Speeds, Numeical Heat Tansfe, at B, 40(2), 2001, [12] Mosi, S.A. and Alexande, A.J., An investigation of aticle Tajectoies in Two-hase Flow System, Jounal of Fluid Mechanics, 55(2),1972, [13] Baghdadi, A.H.A. Numeical Modelling of Two- hase Flow With Inte-hase Slip, h.d. Thesis, Impeial College, Univesity of London, 1979.