ON THE EVALUATION OF NUMERICAL DISSIPATION RATE AND VISCOSITY IN A COMMERCIAL CFD CODE

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1 June 30 - July 3, 2015 Melbourne, Ausralia 9 7B-1 ON THE EVALUATION OF NUMERICAL DISSIPATION RATE AND VISCOSITY IN A COMMERCIAL CFD CODE G. Casiglioni and J. A. Domaradzki Deparmen of Aerospace and Mechanical Engineering Universiy of Souhern California Los Angeles, California casigli@usc.edu, jad@usc.edu ABSTRACT Recenly i has become increasingly clear ha he role of a numerical dissipaion, originaing from he discreizaion of governing equaions of fluid dynamics, rarely can be ignored while using explici or implici Large Eddy Simulaions (LES). The numerical dissipaion inhibis he predicive capabiliies of LES whenever i is of he same order of magniude or larger han he subgrid-scale (SGS) dissipaion. The need o esimae he numerical dissipaion is mos pressing for low order mehods employed by commercial CFD codes. Following he recen work of Schranner e al. (2015) he equaions and procedure for esimaing he numerical dissipaion rae and he numerical viscosiy in a commercial code will be presened. The mehod allows o compue he numerical dissipaion rae and numerical viscosiy in he physical space for arbirary sub-domains in a self-consisen way, using only informaion provided by he code in quesion. I is he firs ime his analysis has been applied o low-order incompressible and compressible solvers. The procedure is esed for a hree-dimensional Taylor-Green vorex flow and compared wih benchmark resuls obained using an accurae, incompressible specral solver. INTRODUCTION Direc Numerical Simulaions (DNS) of urbulen flows are excessively compuaionally expensive for complex geomeries and/or high Reynolds number flows due o he wide separaion of physical scales ha need o be resolved. A relaively successful way o reproduce he dynamics of Navier-Sokes (N-S) equaions while reducing he number of degrees of freedom is he Large Eddy Simulaions (LES) approach. In LES he number of degrees of freedom is reduced by means of a spaial filer ha suppresses he effecs of small scales a he cos of inroducing subgrid scale (SGS) unknowns (i.e. for he incompressible N-S he SGS sress ensor) which mus be explicily modeled (Pope, 2000; Sagau, 2006; Garnier e al., 2009). An alernaive approach is o use he numerical dissipaion coming from he discreizaion of he N-S equaions as an implici LES (ILES) model. The sraegy of using he runcaion error as implici model dubbed monoonically inegraed LES (MILES) originaed wih he idea of Boris e al. (1992) and is reviewed by Grinsein e al. (2007). MILES approach has been conroversial and as such i has been subjec of rigorous invesigaions (Garnier e al., 1999; Domaradzki e al., 2003). These sudies have no been paricularly encouraging. Even when MILES appears o reproduce qualiaively he dynamics of N-S equaions a more in deph, quaniaive invesigaion has shown ha his is no he case. Broadly speaking in hese sudies have been presened wo scenarios. The firs one (Garnier e al., 1999) is ha he numerical dissipaion is excessive wih respec o he correc SGS dissipaion leading o poor resuls boh in ILES and explici LES (ELES) configuraion. The oher opion is ha he scheme is under-dissipaive (wih respec o he correc SGS dissipaion) leading o good resuls for shor ime inegraions and poor resuls for long ime inegraions due o accumulaion of energy in he high wave numbers (Domaradzki e al., 2003). The laer case can poenially be adjused eiher by filering or wih he addiion of an explici SGS model. Boh siuaions shows a fundamenal deficiency of he MILES approach: here is no mechanisms embedded in he numerics o ensure correc amoun of SGS dissipaion. This observaion, ha can appear crudely obvious, suggesed ha raher han calling his approach ILES i should be called under-resolved DNS (UDNS) furhermore he erm ILES should be reserved for schemes ha are designed in such a way ha he discreizaion error provides he correc amoun of SGS dissipaion. An example of such schemes is he adapive local deconvoluion mehod (ALDM) of Hickel e al. (2006). In he ALDM he discreizaion is based on a soluion-adapive deconvoluion operaor which allows o conrol he runcaion error so ha he numerical viscosiy can mach he heoreical values for isoropic urbulence. Early ELES sudies poined ou ha low-order models are no suiable in he ELES framework as he ineracion beween numerical dissipaion and he SGS dissipaion (Kravchenko & Moin, 1997) would negaively affec he resuls, while for highorder/specral mehods he leading source of error is aliasing of he non-linear erm. Recenly i has been shown ha a very coarse resoluions even formally high-order mehods can suffer from he ineracion beween numerical dissipaion and SGS dissipaion making he addiion of an ELES model derimenal o he performance of he code (Cadieux e al., 2014). On he oher hand UDNS simulaions (ofen improperly called ILES) are becoming more and more popular. The reason for his rend and he araciveness of he approach are due o several reasons: is simpliciy, he lack of a universal explici SGS model and a qualiaive behavior ha mimics he dynamics of N-S equaions. Furhermore UDNS simulaion ofen are validaed wih experi- 1

2 menal resuls which suffer from high degree of uncerainy. Recenly he need of quanifying he numerical dissipaion has been addressed by Schranner e al. (2015) who developed a new mehodology ha allows he quanificaion of numerical dissipaion in a self consisen way, using only informaion provided by he code analyzed. Through he quanificaion of numerical dissipaion, he ool proposed by Schranner e al. (2015) is a rigorous mehod o judge a poseriori he qualiy of a given simulaion allowing for an imparial assessmen of he impac of he numerical dissipaion. Here we apply his mehodology for he firs ime o a low-order incompressible and compressible solver, validaing he mehod wih he goal o exend i o more realisic configuraions. EQUATIONS AND METHOD Analyical Form Transpor energy equaion for compressible Navier- Sokes (N-S) is ρe + [ ] u i τ i j (ρe + p)uj = q j, (1) where u i are he componens of he velociy vecor, p he pressure, ρ he densiy and e he oal energy per uni mass. The consiuive relaion beween sress and srain rae for a Newonian fluid is [( ui τ i j = µ + u ) j 2 ] u k δ i j, (2) x i 3 x k he hea flux q i is defined as q i = k T x i, (3) where µ is he dynamic viscosiy, k is he hermal conduciviy, and T is he emperaure. The definiion of oal energy is Afer some manipulaions he inegral form of Eq. (6) can be wrien as V + ρe kin dv + A ( p u j u i + τ i j (ρe kin u j + u j p u i τ i j ) n j da ) dv = V Ekin + F kin + F vis W p + ε vis = 0, (7) where F kin,, F vis are he kineic energy, acousic and viscous fluxes, n j he ouward uni vecor normal o he surface A, W p he work due o pressure and ε vis he viscous dissipaion. Noice ha in he incompressible limi W p = 0 as u j / = 0. We can define a dissipaion funcion ε as ε = V 1 ν τ u i i j dv, (8) herefore εν = ε vis, where ν is he kinemaic viscosiy ( ν = µ/ρ ). Noe ha he above equaion is exac only if ν = cons in V. Discreized Form If we assume a FV spaial discreizaion and generic discreizaion in ime hen he Eq. (7) is conaminaed by he runcaion and aliasing errors so we can define a local residual ε(m) n = Ekin (m) + F kin (m) + Fac (m) Fvis (m) W p (m) + εvis (m), (9) where he subscrip [ ] (m) refers o he mh conrol volume. We call he residual numerical dissipaion rae because i has been shown ha if inegraed over a sufficienly large conrol volume i has a predominanly dissipaive characer (Schranner e al., 2015; Domaradzki & Radhakrishnan, 2005). Following his definiion we can recover he numerical kinemaic viscosiy as e = e in + e kin, (4) where e in is he specific energy per uni mass and e kin = 1 2 u iu i he kineic energy. Following he procedure of Schranner e al. (2015) he ranspor equaion, Eq. (1), can be separaed ino he conribuion of inernal energy ρe in and kineic energy ρe kin + ρein u j + ρekin u j = p u j + τ i j u i q j, (5) = u j p + u i τ i j. (6) ν n (m) = εn (m) ε (m). (10) We can exend he above definiions o a sub-domain or o he enire compuaional domain sub = M m (m) ; ν n sub = εn sub ε sub, (11) where M is oal number of adjacen cells of a given subdomain. PROCEDURE This is a poseriori procedure, all quaniies come from he compued flow field. For he ime discreizaion a 2

3 second-order hree-poins finie difference formula is used. Kineic fluxes are calculaed as follow ( ) F(m) kin = (ρe kin u j n j ) (r) A (r), (12) r (m) where subscrip [ ] (r) represens he sum over he rh face of he mh conrol volume and A (r) is he area of he rh face. The oher fluxes are compued in similar fashion. Volume erms are calculaed as (for example he kineic energy) ( ) E(m) kin 1 = 2 ρu iu i V (m), (13) (m) where V (m) is he volume of he mh conrol volume. In he infinie Re number limi he viscous erms in Eq. (9) are dropped. In he incompressible limi (Ma < 0.3) he work due o pressure should be zero, his is no he case if he flow field is no divergence free (as i can happen wih loworder incompressible schemes). Also noe ha if we ake a periodic box he conribuion of flux erms cancels ou. NUMERICAL METHOD Sar-CCM+ solves eiher he incompressible or compressible hree-dimensional inegral Navier-Sokes equaions in conservaive form. The equaions are solved in a precondiioned dimensional form on an unsrucured grid (CD-Adapco, 2013). The ime is advanced hrough a dual ime-sepping implici scheme. The incompressible solver uses a Rhie-Chow pressure-velociy coupling and a SIM- PLE algorihm, while he for he compressible solver he inviscid fluxes are evaluaed by using he Weiss-Smih precondiioned Roe s flux-difference spliing scheme. Boh schemes are formally a bes second order accurae. The viscous fluxes are evaluaed by a sandard cenral difference scheme. VALIDATION This is he same case which was analyzed previously o validae he procedure applied o a research code ha implemens he ALDM mehod as an implici LES model (Schranner e al., 2015; Hickel e al., 2006). The Taylor Green Vorex (TGV) problem is originally described in Taylor & Green (1937). The iniial condiions, following (Shu e al., 2005), are u = u 0 [sin(x)cos(y)cos(z)], (14) v = u 0 [cos(x)sin(y)cos(z)], (15) w = 0, (16) ρ = ρ 0, (17) p = p 0 + ρ 0 16 [(cos(2x) + cos(2y))(cos(2z) + 2) 2], (18) where subscrip [ ] 0 indicaes reference quaniies. The iniial flow field is le o evolve for 10 non-dimensional ime unis. The ime uni is T = l 0 /u 0 where l 0 = 1 is he reference lengh. The ime sep is kep consan as = T /50. Sar-CCM+ solves he dimensional form of Figure 1. Time-evoluion of numerical dissipaion rae εo, n Re =. Sar-CCM+ wih 64 3 cells (blue line), Sar- CCM+ wih cells (red dashed line) and from Schranner e al. (2015) (black circles). he N-S. Under ideal gas assumpion, o obain he same iniial condiion as in Schranner e al. (2015) and Hickel e al. (2006) we need o pick he following reference values: u 0 = Ma γ p 0 /ρ 0 and ρ 0 = p 0 /(R T 0 ). The reference pressure is p 0 = Pa, he Mach number Ma = 845, he hea capaciy raio γ = 1.4, he reference emperaure T 0 = 300 K and he specific gas consan R = J/(kg K). Even hough he code solves he dimensional N-S equaions, all resuls will be presened in non-dimensional form. Unless saed oherwise he resoluion for all simulaions is of 64 cells per spaial direcion for a cube wih sides ha are 2π long and ha has periodic boundary condiions. Iniially all simulaions are run wih he incompressible solver for a direc comparison wih he work of Schranner e al. (2015), aferwards seleced cases are run wih he compressible solver. Full periodic domain For he firs es case we use as a conrol volume he whole domain so ha we can calculae he oal numerical dissipaion rae wihou flux erms. For Re = we show o ploed agains ime in Fig. 1. The numerical dissipaion rae is quie high, i is around hree imes as much as in Schranner e al. (2015) hroughou he emporal domain. To make sure ha he o is calculaed properly his case has been repeaed wih 128 cells per spaial direcion. This case can also be seen in Fig. 1. We can observe ha for he laminar par of he simulaion ( < 3.5 ) here is indeed fairly good agreemen wih he reference resuls. Ensrophy (Ω = 1/2 V ω iω i dv, where ω i are he componen of he voriciy vecor) is ofen used in TGV simulaions as a meric o judge he degree of resoluion. I is hypohesized ha for he inviscid TGV problem ensrophy should increase o infiniy a he non-dimensional ime 5 (Shu e al., 2005). The resoluion of his singulariy is a common meric o evaluae a given scheme/resoluion for a TGV simulaion. In Fig. 2 we can see ha for our curren simulaion we fall in beween he behavior of a specral scheme, which has very lile numerical dissipaion and a fifh-order WENO scheme which has subsanial numerical dissipaion, wih Sar-CCM+ closer o he laer. This is expeced since he curren scheme is formally a bes second order, herefore quie dissipaive. The resul for he case is also included. While his is no a fair comparison neiher wih he specral nor he WENO schemes used wih he 64 3 resolu- 3

4 Ω/Ω(0) WENO specral Figure 2. Time-evoluion of normalized ensrophy Ω()/Ω(0), Re =. Sar-CCM+ wih 64 3 cells (blue line), wih cells (red dashed line), WENO (black squares) and specral (black circles) from Shu e al. (2005). curren Figure 4. Time-evoluion of numerical dissipaion rae εo, n Re = 100. Sar-CCM+ (blue line) and from Schranner e al. (2015) (black circles). 3.5 curren Figure 3. Time-evoluion of numerical dissipaion rae εo, n Re = Sar-CCM+ (blue line) and from Schranner e al. (2015) (black circles). ion, i is sill a useful way o show ha he numerical dissipaion of he curren scheme approaches he specral numerical dissipaion when he resoluion increases. For he same configuraion, as in he reference paper, a sweep of Reynolds number has been ran (Re =, 3000, 1600, 800, 400, 200, and 100). We can see o ploed agains ime in Figures 3 4 for wo of hose cases. The general rend is he same as in he infinie Reynolds number case, and he peak is abou 2.5 imes he peak in he reference sudy. In Fig. 5 we show a full sweep of Reynolds numbers o beer appreciae he behavior of he numerical dissipaion rae as a funcion of he Reynolds number. Sub-domains As es cases we choose some of he sub-domains used in Schranner e al. (2015). The firs cubic sub-domain used in he reference was an ocan of he domain. Due o symmeries of he TGV problem his case is only useful o check ha all fluxes cancel ou. The second es case simulaed is he cubic sub-domain one (CSD1) which consiss of cells 10 hrough 56 in all spaial direcions. The resuls for he infinie Reynolds number case are shown in Fig. 6. Here and in he following Figures erms of he kineic energy balance are ploed wih a negaive sign in order o sum up o. All he curren resuls are ploed agains he resuls from he reference paper and also ploed are he various erms Figure 5. Time-evoluion of numerical dissipaion rae εo. n Re = (blue line), Re = 3000 (red dashed line), Re = 1600 (black dash-doed line), Re = 800 (magena doed line), Re = 400 (solid blue line wih markers), Re = 200 (red dashed line wih markers), and Re = 100 (black dashdoed line wih markers). ha conribue o he numerical dissipaion rae. We can see ha we recover he same rend observed in he infinie Reynolds number case wih he full periodic box, i.e. for he laminar par of he simulaion he resuls agree wih he specral reference while he peak in he urbulen par of he simulaion is abou 3 imes as high. In Fig. 6 we can see a noisy oscillaory behavior in he acousic flux erm and he work due o pressure. These oscillaions are ou of phase and cancel each oher ou. We speculae ha his is an unphysical energy exchange beween he wo erms due o he fac ha incompressibiliy is no sricly enforced. For he finie Re cases we see he same rend observed for he periodic box cases. A sweep of Re has been performed, here we repor only wo cases: Re = 3000, 100 (see Figures 7 8). To beer visualize he addiional dissipaion provided by he scheme wih respec o he physical dissipaion we have ploed he molecular viscosiy ν ogeher wih he oal effecive viscosiy ν + ν n, i.e. a sum of he numerical and he physical viscosiy, for he Re = 3000 case (see Fig. 9). The only non-cubic sub-domain considered here is he NCSD2 from reference Schranner e al. (2015), i.e. for he x-direcion cells 8 o 23 and for he y and z-direcion cells 1 o 31. This sub-domain is no chosen randomly, i was seleced because his is he region wihin he ocan wih 4

5 ǫ n E kin / kin W p E kin / F kin F vis ε vis W p Figure 6. Time-evoluion of numerical dissipaion rae εsub n, Re =, sub-volume CSD1. Numerical dissipaion rae εsub n (blue line), rae of change of kineic energy Ekin sub / (black do-dashed line), kineic energy flux Fsub kin (blue dodashed line wih markers), acousic flux Fsub ac (magena doed line) and from Schranner e al. (2015) (black circles). E kin / kin F vis ε vis W p Figure 8. Time-evoluion of numerical dissipaion rae εsub n, Re = 100, sub-volume CSD1. Numerical dissipaion rae εsub n (blue line), rae of change of kineic energy Esub kin / (black do-dashed line), kineic energy flux Fsub kin (blue do-dashed line wih markers), acousic flux Fsub ac (magena solid line wih markers), viscous flux Fvis sub (black dashed line wih markers) viscous dissipaion rae εsub vis (red dashed line) and from Schranner e al. (2015) (black circles) ν ν n Figure 7. Time-evoluion of numerical dissipaion rae εsub n, Re = 3000, sub-volume CSD1. Numerical dissipaion rae εsub n (blue line), rae of change of kineic energy Esub kin / (black do-dashed line), kineic energy flux Fsub kin (blue do-dashed line wih markers), acousic flux Fsub ac (magena solid line wih markers), viscous flux Fvis sub (black dashed line wih markers) viscous dissipaion rae εsub vis (red dashed line) and from Schranner e al. (2015) (black circles). non-negligible numerical dissipaion wihin an ocan. The resuls for he subdomain NCSD2 are shown in Fig. 10. Surface versus Volume Inegrals For cerain incompressible solvers pressure can be defined up o a consan wihou changing he dynamics of he flow field since only he gradien of pressure appears direcly in he momenum equaions. For his reason simply adding W p sub o he energy balance can poenially lead o wrong esimaes of he numerical dissipaion rae. One way o check for consisency of he resuls is o recas he pressure erms as F(m) ac W p (m) = p (m), where p (m) is he numerical ( form of he pressure ranspor erm defined as p p (m) = u j )(m) V (m). As can be seen comparing Figures 6 and 11 he wo mehods are equivalen. Oher surface ν Figure 9. Time-evoluion of physical and numerical viscosiy, Re = 3000, sub-volume CSD1. Toal viscosiy ν + ν n (blue line), physical viscosiy ν (red dashed line) E kin / F kin W p Figure 10. Time-evoluion of numerical dissipaion rae εsub n, Re =, sub-volume NCSD2, including work due o pressure. Numerical dissipaion rae εsub n (blue line), rae of change of kineic energy Esub kin / (black dodashed line), kineic energy flux Fsub kin (blue do-dashed line wih markers), acousic flux Fsub ac (magena solid line wih markers), W p sub (red dashed line wih markers) and from Schranner e al. (2015) (black circles). 5

6 ǫ n E kin / F kin p Figure 11. Time-evoluion of numerical dissipaion rae εsub n, Re =, sub-volume CSD1 wih pressure gradien ranspor erm. Numerical dissipaion rae εsub n (blue line), rae of change of kineic energy Esub kin / (black dodashed line), kineic energy flux Fsub kin (red dashed line), acousic flux Fsub ac (magena doed line), p sub (magena solid line wih markers) and from Schranner e al. (2015) (black circles). ε vis Figure 12. Time-evoluion of numerical dissipaion rae εsub n for a compressible solver, Ma = 845, Re = 3000, sub-volume CSD1. Numerical dissipaion rae εsub n (blue line), viscous dissipaion rae εsub vis (red dashed line) and from Schranner e al. (2015) (black circles). erms have been calculaed as volume erms wih similar conclusions; here seem o be negligible differences in calculaing he quaniies in he kineic energy balance as eiher volume or surface inegrals. Effecs of Compressibiliy All simulaions so far have been performed wih an incompressible solver o have a one o one comparison wih he resuls available in lieraure. Due o convergence issues for he compressible case he ime sep has been reduced o = T /500. For he CSD1 sub-domain all Reynolds number cases have been run for a complee comparison wih he incompressible simulaions. Here we repor only he case for Re = 3000, see Fig. 12 where he physical and he numerical dissipaion raes are ploed. We find a good agreemen wih he incompressible solver a all Reynolds numbers. DISCUSSION AND CONCLUSIONS Even lacking a direc validaion hrough an exernal specral code as in he reference paper of Schranner e al. (2015) he mehods applied o a low-order incompressible and compressible schemes compares well o he available reference daa and i is self consisen. I should be noed ha for low-order schemes he work due o pressure, even hough should be zero, migh no be negligible. The exension o a compressible solver, a leas a low Mach numbers, is sraighforward. From our resuls i seems o no maer if erms in he discree kineic energy balance, Eq. (9), are calculaed as fluxes or as volume erms. The mehod seems self consisen and robus as all our resuls are in qualiaive agreemen wih he resuls of Schranner e al. (2015). Therefore we conclude ha he mehod works well for loworder schemes and ha i is ready o be applied o more realisic flow geomeries. REFERENCES Boris, J.P., Grinsein, F.F., Oran, E.S. & Kolbe, R.L New insighs ino large eddy simulaion. Fluid Dyn. Res. 10, Cadieux, F., Domaradzki, J. A., Sayadi, T. & Bose, T DNS and LES of laminar separaion bubbles a moderae Reynolds numbers. ASME J. Fluids Eng. 136, CD-Adapco 2013 Sar-CCM+ manual. Version Domaradzki, J.A. & Radhakrishnan, S Eddy viscosiies in implici large eddy simulaions of decaying high Reynolds number urbulence wih and wihou roaion. Fluid Dyn. Res. 36, Domaradzki, J. A., Xiao, Z. & Smolarkiewicz, P. K Effecive eddy viscosiies in implici large eddy simulaions of urbulen flows. Phys. Fluids 15 (12), Garnier, E., Adams, N. & Sagau, P Large eddy simulaion for compressible flows. Springer. Garnier, E., Mossi, M., Sagau, P., Come, P. & Deville, M On he use of shock-capuring schemes for largeeddy simulaion. J. Comp. Phys. 153 (2), Grinsein, F., Margolin, L. & Rider, W Implici Large Eddy Simulaion: Compuing Turbulen Flow. Cambridge Universiy Press. Hickel, S., Adams, N.A. & Domaradzki, J.A An adapive local deconvoluion mehod for implici LES. J. Comp. Phys. 213, Kravchenko, A.G. & Moin, P On he effec of numerical errors in large eddy simulaions of urbulen flows. J. Comp. Phys. 131 (2), Pope, S. B Turbulen flows. Cambridge Universiy Press. Sagau, P Large eddy simulaion for incompressible flows: an inroducion, 3rd edn. Springer. Schranner, F.S., Domaradzki, J.A., Hickel, S. & Adams, N.A Assessing he numerical dissipaion rae and viscosiy in numerical simulaions of fluid flows. Comp. & Fluids 114, Shu, C.-W., Don, W.-S., Golieb, D., Schilling, O. & Jameson, L Numerical convergence sudy of nearly incompressible, inviscid TaylorGreen vorex flow. J. Sci. Comp. 24 (1). Taylor, G.I. & Green, A.E Mechanism of he producion of small eddies from large ones. Proc. R. Soc. Lond. A 158 (895),