COMBINED VERTEX-BASED CELL-CENTRED FINITE VOLUME METHOD FOR FLOWS IN COMPLEX GEOMETRIES

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1 Thrd Internatonal Conference on CFD n the Mnerals and Process Industres CSIRO, Melbourne, Australa 1-1 December 3 COMBINED VERTEX-BASED CELL-CENTRED FINITE VOLUME METHOD FOR FLOWS IN COMPLEX GEOMETRIES Dane MCBRIDE, Nck CROFT and Mark CROSS Centre for Numercal Modellng and Process Analyss, Unversty of Greenwch, London SE1 9LS ABSTRACT CFD modellng of real-lfe processes often requres solutons n complex three dmensonal geometres, whch can often result n meshes where aspects of t are badly dstorted. Cell-centred fnte volume methods, typcal of most commercal CFD tools, are computatonally effcen but can lead to convergence problems on meshes whch feature cells wth hghly non-orthogonal shapes. The vertex-based fnte volume method handles dstorted meshes wth relatve ease, but s computatonally expensve. A combned vertex-based cell-centred (VB- CC) technque, detaled n ths paper, allows solutons on dstorted meshes that defeat purely cell-centred (CC) solutons. The method utlses the ablty of the vertexbased technque to resolve the flow feld on a dstorted mesh, enablng well establshed cell-centred physcal models to be employed n the soluton of other transported quanttes. The VB-CC method s valdated wth benchmark solutons for thermally drven flow and turbulent flow. An early applcaton of ths hybrd technque s to three-dmensonal flow over an arcraft wng, although t s planned to use t n a wde varety of processng applcatons n the future. NOMENCLATURE φ transported varable p pressure u Γ dffuson coeffcent ρ densty µ dynamc vscosty c specfc heat K thermal conductvty T temperature S source t tme k turbulent knetc energy ε dsspaton rate v t knematc vscosty β thermal coeffcent of volumetrc expanson g gravty T ref reference temperature INTRODUCTION The accuracy of Computatonal Flud Dynamcs (CFD) analyss not only depends upon the qualty of the dscretsaton approaches wthn the target code to accurately model the physcal process, but also on the ablty to solve on a mesh that matches the true geometry of the physcal doman. The Cell-Centred - Fnte Volume dscretsaton method (CC-FV) s well establshed n CFD analyss for modelng physcal processes nvolvng the soluton of flow and s employed n many CFD codes (e.g. CFX, FLUENT, STAR-CD and PHYSICA). Ths technque s computatonally effcen on a hghly orthogonal mesh, usng smple approxmatons to dscretse the terms n the transport equaton, t has low memory requrements and fast smulaton tmes. However, the method s not robust on a hghly nonorthogonal mesh. Correctons have to be made to the usual dscretsaton process to account for non-orthogonalty n the mesh (Crof 1995 & 1998). These correcton terms can ntroduce errors nto the soluton process and lead to dffcultes wth convergence on hghly dstorted meshes (Crof 1998). Modelng real-lfe processes often requres fttng a mesh to complex geometres. Of course, fttng a hghly orthogonal mesh to real-lfe geometry can be one of the most tme consumng aspects of the CFD modellng process. For example, close couplng between dfferent physcal phenomena, such as flow and stress, may result n the occurrence of mesh dstorton durng the soluton process. In such mult-physcs problems, even f one starts wth a hgh qualty mesh t may degrade durng the soluton process. The Vertex- Based Fnte Volume dscretsaton approach (VB-FV), that utlses element pecewse lnear shape functons, (Prakash and Patankar, 1985), handles such dstorted meshes wth relatve ease but s computatonally expensve and requres consderable software restructurng for exstng CFD software tools. In ths paper we outlne a novel dscretsaton approach whch utlses the VB approach for the flow soluton and employs well establshed physcs models that use cell-centred (CC-FV) technques for other transported propertes. The coupled vertex-based cell-centred (VB-CC) hybrd approach allows solutons on hghly dstorted meshes that defeat purely cell-centred solutons and s relatvely straghtforward to embed wthn generc CC based CFD tools. Copyrght 3 CSIRO Australa 351

2 GOVERNING EQUATIONS The transport equaton (1) for the general transport of a scalar varable s used as a startng pont for FV procedures. ρφ + u ( ρ φ ) ( Γ φ ) + S φ The momentum transport equatons () can be wrtten n the same form as above, wth φ u, v or w and Γ µ. The pressure gradent term that forms the man momentum source term s wrtten separately. ρu ( ρuu ) p + µ + S u (1) +. u () The feld must also satsfy mass conservaton: The values of emprc constants employed n equatons (5) to (9) are: Cµ k σε.9; σ 1.; 1.3; C 1.44; C CONTROL VOLUMES In the VB-CC method equatons () and (3) are dscretsed over a vertex-based control volume. The mesh element s subdvded nto a number of sub-control volumes by connectng the element centrod to the element face centre. The sub-control volumes are assembled around the mesh vertex to form the vertex-based control volume, shown n Fgure 1, for a two dmensonal quadratc mesh. The control volume assocated wth equatons (4), (5) and (6) s smply the mesh element. ρ + u ( ρ ) S m (3) Element dvded nto sub-control volumes The general equaton governng heat transfer can be wrtten n the same form as above: ρct ( ρc T ) ( K T ) + S T + u The popular k-ε model (Launder and Spaldng, 1974) nvolves the soluton of the turbulent knetc energy (k) equaton s gven by, ρk ρν + σ lam tg t k ( ρ ) t k µ + k + ρν ρε (4) u (5) and the dsspaton rate (ε ) equaton s as follows: ρε ρν t ε ε + ( ρu ε) µ lam ε + C1 ρνt G C ρ t + (6) σ ε k k where the rate of generaton of the turbulent knetc energy, G, s gven by: u G x v w y z u v u w w v y x z x y z The turbulent vscosty s related to k and ε by, (7) Fgure 1: Vertex-based control volume INTERPOLATION FUNCTIONS Control Volume assembled around mesh vertex The cell-centred approach uses fnte-dfference type approxmatons to descrbe how φ vares between soluton ponts. However non-orthogonal meshes requre correctons to be made to the usual dscretsaton process (Crof 1995, 1998). The dffusve flux across a face requres the ncluson of a secondary gradent term, ths can lead to face fluxes that are no longer computed n terms of neghbourng values. Most of the terms n the dscretsed transport equaton requre face values of φ, for orthogonal meshes ths s acheved through nterpolaton of adacent cell values, for non-orthogonal terms meshes an extra term s requred based on the gradents of φ. The ncluson of these correcton terms can ntroduce errors nto the soluton procedure leadng to dffcultes wth convergence. These errors are multpled when solvng coupled varables and on arbtrary dstorted meshes and dvergence s often encountered. In the vertex-based approach the local varaton of a varable φ wthn an element s descrbed by smple pecewse polynomal functons. The nterpolaton functons employed here are gven n (Taylor et al, 3) who used them n structural analyss. The varables and co-ordnates are approxmated, n local co-ordnates as, k ρc ε µ µ t (9) 35

3 u ( s, p( s, x ( s, n 1 n 1 n 1 N ( s, u N ( s, p N ( s, x (1) The smulatons were performed on a Athlon 1.39Ghz processor for a unform 35 by 35 Cartesan mesh, mesh 1, and dstorted versons of mesh 1, shown n Fgure. where n s the number of nodes of an element. The use of elemental shape functons allows the drect computaton of fluxes n the requred drecton even on a non-orthogonal mesh, see McBrde (3). COMBINED VB-CC METHOD In the soluton of the Naver Stokes equatons the revsed SIMPLER method of Patankar (198) s employed. Correct pressure and couplng s ensured by the method of Prakash and Patankar (1985). Obtanng a flow feld usng vertex-based technques allows vertex-based veloctes to be employed n the transport of other quanttes usng cell-centred technques. Mass s conserved on the boundary of the vertex-based control volume. Snce the element face centrod s a pont on the boundary of the vertex-based control volume, ndrectly mass s also conserved over the mesh element. As mass conservaton s enforced over the vertex-based control volume, any errors resultng from nterpolatng for element face values also decrease. See the work of McBrde (3) for a detaled descrpton of the computatonal approach. EVALUATION OF THE METHOD Results have been developed for a number of benchmark problems, ncludng the thermally drven flow case dscussed below. The approach was then appled to a three-dmensonal flow over an arcraft wng. The results are shown for a unform Cartesan mesh and dstorted versons of the same mesh. THERMALLY DRIVEN FLOW De Vahl Davs and Jones (1983), suggested that buoyancy-drven flow n a square cavty would be a sutable valdaton test case for CFD codes and publshed a set of benchmark results for a number of dfferent Raylegh numbers. Declnng qualty n solutons s often encountered wth ncreasng Raylegh number. The flud contaned n the cavty s assumed ncompressble and ntally statonary. Thermal gradents across the soluton doman result from opposng walls of dfferng temperatures. These thermal gradents lead to buoyancy forces that create flow. The buoyancy forces are calculated usng the Boussnesq approxmaton. Ths approxmaton results n a source per unt volume of the form, S ρβ g T T ) (1) ( ref Fgure : Dstorted versons of Cartesan mesh 1 Plots of the u- along the central vertcal plane and the v- along the central horzontal plane for each mesh and Raylegh number are shown along wth the benchmark maxmum values. Fgure 3 for Raylegh number of 1 3, Fgure 4 for Raylegh number of 1 4, Fgure 5 for Raylegh number of 1 5 and Fgure 6 for Raylegh number of maxmum Fgure 3: Raylegh number of 1 3, u- v

4 Fgure 4: Raylegh number of 1 4, u- v- Fgure 5: Raylegh number of 1 5, u- v maxmum Fgure 6: Raylegh number of 1 6, u- v- For the unform mesh, Table 1 shows VB-CC U max and V max, the maxmum value of the normalsed component along central planes. The Y max and X max, are the normalsed postons of ths maxmum value. The percentage errors of the smulaton results aganst benchmark solutons are shown n brackets for each Raylegh number (R. VB-CC and CC results, on a 35 by 35 unform mesh, compare well wth benchmark solutons from non-unform optmsed meshes. Ra U max Y max V max X max (.9%).189 (1.%).686 (.7%).178 (.1%) (.1%).178 (.43%) (.4%).1 (.6%) (.14%).144 (.38%) 69.6 (1.47%).67 (.97%) (1.18%).144 (3.71%) (.%).333 (1.1%) Table 1: VB-CC: Percentage error wth benchmark values A measure of the error, due to mesh dstorton, for mesh and mesh 3 s shown n Table for VB-CC solutons, usng mesh 1 as the base result. Dvergence was encountered on mesh 3 for a Raylegh number of 1 6. Ra u- v- u- v x x x1-3.3x x x x1-3.7x x1-1.58x1-4.78x1-1.8x x1-3.94x Table : VB-CC: Error due to mesh dstorton Good agreement wth benchmark solutons was obtaned on the unform mesh and the solutons were only slghtly degraded on the dstorted mesh. The queston here s at what cost? In ths two-dmensonal thermally drven problem, the memory demands per soluton pont are 73 Bytes per element-based varable and 8 Bytes per vertex-based varable. Moreover, the convergence behavour and compute tme for the problem may be summarsed as n Table 3 below. Perf. meas. Ra M1- CC M1- VB M- CC M- VB M3- CC M3- VB Its Tme(s) Its Fal 5 Tme(s) Fal 51 Its Fal 7 Tme(s) Fal 4 Its Fal Fal Tme(s) Fal Fal Table 3: Computatonal tmes: for a mesh consstng of 15 elements and 59 nodes. It can be deduced from these results that the hybrd method s approxmately a factor of 4 more expensve n compute tme, per soluton pon on a good qualty mesh. On the dstorted meshes CC solutons could only be 354

5 acheved wthout the ncluson of the non-orthogonal correcton terms, consequently as the mesh degrades CC solutons deterorate. For example, the error on mesh for a Raylegh number of 1 3 was approxmately 1%, see Fgure 7, compared to.5% for VB-CC solutons. The hybrd scheme contnues to produce reasonably accurate solutons as the mesh degrades wth a further compute cost of about a factor of. Fgure 7: Cell-centred solutons: Raylegh number of 1 3, u- v- FLOW OVER AN AIRCRAFT WING One reasonably challengng case study to assess the potental of the hybrd scheme nvolves turbulent ncompressble flow over an arcraft wng usng the k-ε model. The geometry of the wng was taken from ONERA M6 specfcatons; t s a swep sem-span wng wth no twst. The leadng-edge sweep s 3 degrees, tralng edge sweep 15.8 degrees and the taper rato s.56. The smulaton carred out employs a low speed Mach number of.3, gvng a Reynolds number of about 5 mllon. A wall boundary condton was appled to the wng surface for the flow and turbulence model varables. The smulaton was performed on a unform C-Mesh of approxmately 1, elements and a dstorted verson of the mesh, both shown n Fgure 8. Ths s a relatvely coarse mesh, wth complex flow smulatons of flow over the ONERA wng a mesh of at least three to four hundred thousand elements s normally employed. However, the mesh densty here s suffcent to explore the performance of the VB-CC hybrd method on a real-lfe dstorted mesh M1-CC M-CC M3-CC M1-CC M-CC M3-CC Fgure 8: C-Mesh, Dstorted C-Mesh At the outset t s worth sayng that on the C-mesh, the VB-CC hybrd method and the conventonal CC dscretsaton results are very smlar n most respects. For the VB-CC method on both the C-mesh and the dstorted mesh, plots are shown on two planes, z.558 whch s approxmately half the wng span, and y whch s the symmetry plane. Fgures 9 and 1 show the mach contour plots for the Z-plane and Y-plane respectvely. Although there s some smearng of values on the dstorted mesh, the results have captured the overall trend, dentfyng local mnmum and maxmum values. The u- and w- value range remaned unchanged, beng [ to 17] and [-6.37 to 39] respectvely. The mnmum and maxmum v- values decreased slghtly from [-49.3, 49.3] to [-4.4, 4.4] on the dstorted mesh. The turbulent vscosty contours, Fgure 1, show some smearng of values on the dstorted mesh. The lower values downstream of the wng s tralng edge, on the dstorted mesh, result n hgher turbulent generaton rates n ths regon and hence hgher vscosty values. Ths s caused by the way that the turbulence generaton rate s represented numercally at the tralng edge of the wng shape (McBrde, 3) and ths problem can be elmnated wth a more careful approxmaton. The maxmum turbulent vscosty obtaned on the C-mesh was.3m /s compared to.6m /s on the dstorted mesh. Fgure 9: Mach no. Contour plots on plane z.558 C-mesh, Dstorted C-mesh 355

6 some error n to the flow feld. However, the nonorthogonal errors do not appear to sgnfcantly affect the fnal soluton and local mnmum and maxmum values are dentfed. Obtanng a good flow feld on a dstorted mesh usng a VB method ads the soluton of other transported quanttes usng effcent cell-centred technques. As such, the results obtaned on benchmark problems, usng the VB-CC technque are encouragng. Fgure 9: Mach no. Contour plots on plane y C-mesh Dstorted C-mesh Fgure 1: Vscosty contour plots on plane z.558 C-mesh Dstorted C-mesh Run Tmes and Memory Requrements The smulatons were performed on a Pentum 4 CPU.54GHz. The mesh employed comprsed of 11,41 elements and 18,314 vertces. To acheve the convergence crtera that the Lnorm of the change n the soluton dropped by 5 orders-of-magntude requred 54 teratons on the unform C-mesh and 3 teratons on the dstorted mesh. The tme per teraton/per soluton pont was approxmately 3.3 x 1-5 seconds for each varable beng solved vertex-based and 7. x 1-6 seconds for each varable beng solved cell-centred. Ths yelds, 15.7 seconds per teraton for VB-CC solutons, and 4.6 seconds per teraton for CC solutons. The vertex-based method has consderably more memory requrements than the cell-centred method. The approxmate memory requred per soluton pont s 373 bytes vertex-based compared to 4 bytes cell-centred. CONCLUSION The couplng of the VB-CC hybrd FV dscretsaton method for CFD nvolvng vertex-based flow coupled wth other transported quanttes at the cell-centre (e.g. thermal and turbulent varables) has been presented. The cell-centred dscretsaton of transported quanttes stll ncludes non-orthogonal errors that may n turn ntroduce Although the VB-CC method s approxmately 4 tmes more expensve n compute tme and requres 8 tmes as much memory than the conventonal CC method, ths new hybrd approach does enable solutons on dstorted meshes that defeat purely cell-centred technques whlst enablng well establshed cell-centred physcal models to be retaned. Ths approach s partcularly useful n the generc CFD tool contex because t enables users to explot all the exstng models that they have already developed n CC context wthn the VB-CC framework and to obtan solutons n complex geometry meshes whch have some zones of poor mesh qualty. REFERENCES CROFT, N., PERICLEOUS, K.A. and CROSS, M., (1995), PHYSICA: A mult-physcs envronment for complex flow processes, In C. Taylor and P. Durbetak, edtors, Numercal Methods n Lamnar and Turbulent Flow 95,, CROFT, T.N., (1998), Unstructured Mesh Fnte Volume Algorthms for Swrlng, Turbulen Reactng Flows, Ph.D. thess, The Unversty of Greenwch, London, England. PRAKASH, C. and PATANKAR, S.V., (1985), A Control Volume-Based Fnte-Element Method for solvng the Naver-Stokes equatons usng Equal-order pressure nterpolaton, Numercal Heat Transfer, 8, 59-8 TAYLOR, G.A., BAILEY, C. and CROSS, M., (3), A vertex-based fnte volume method appled to nonlnear materal problems n computatonal sold mechancs, Internatonal ournal for Numercal Methods n Engneerng, 56, PATANKAR, S.V., (198), Numercal Heat Transfer and Flud Flow, Hemsphere Publshng Corporaton. McBRIDE, D., (3), Vertex-Based Dscretsaton Methods for Thermo Flud Flow n a Fnte Volume- Unstructured Mesh Context, Ph.D. thess, The Unversty of Greenwch, London, England. DE VAHL DAVIS, G. and JONES, I.P., (1983), Natural Convecton n a Square Cavty: A Comparson Exercse, Internatonal Journal for Numercal Methods n Fluds, 3, 7-48 LAUNDER, B.E. and SPALDING, D.B., (1974), The Numercal Computaton of Turbulent Flows, Comput. Meth. Appl. Mech. Eng., 3, FLUENT, see CFX, see STAR-CD, see PHYSICA, see 356