Grid-independent large-eddy simulation of compressible turbulent flows using explicit filtering

Transcription

1 Center for Turbulence Researc Proceedings of te Summer Program Grid-independent large-edd simulation of compressible turbulent flows using explicit filtering B D. You, S. T. Bose AND P. Moin Te governing equations for large-edd simulation (LES) are derived from te application of a low-pass filter to te Navier-Stokes equations. It is often assumed tat discrete operations performed on a particular grid act as an implicit filter, causing simulation results to be sensitive to te mes resolution. Alternativel, explicit filtering separates te filtering operations from discretization operations, and ence alleviates te grid sensitivities. Obtaining a grid-independent LES solution in incompressible flow as been successfull demonstrated b Bose, Moin & You [Ps. Fluids, 22, , 2010]. In te present stud, we investigate te use of explicit filtering in obtaining a grid-independent LES solution for compressible turbulent flow. Te convergence of simulations using a fixed filter widt wit varing mes resolutions to a true large-edd simulation solution is analzed for compressible turbulent flow troug a cannel wit periodic constrictions at a bulk Renolds number of 10 5 and Mac number of 0.6. Te mean and turbulent statistics obtained b te conventional implicitl filtered LES and b te present explicitl filtered LES are compared at two different mes resolutions. Altoug grid independence is not full acieved, tis is te first application of explicitl filtered LES to te compressible governing equations and to a quasi-complex geometr. 1. Introduction In LES, te separation of large- and small-scale turbulent fluid motions is performed b appling a low-pass filter to te Navier-Stokes equations. In conventional LES, te filtering operation is implicitl applied to te Navier-Stokes equations. One of te most notable drawbacks associated wit te conventional implicit-filter LES is tat te simulation result is igl dependent on te numerical grid emploed because of te inerent dependence of te filtering operation on te numerical discretization. As a consequence of te grid-dependenc, te implicit-filter LES is sensitive to numerical errors. A priori analses of direct numerical simulation data of canonical turbulent flows sow tat te truncation error and aliasing error (due to discretization of nonlinear terms) can dominate te contribution of te subgrid-scale (SGS) turbulence model tat is necessar to close te filtered nonlinear terms in te governing equations (Cow & Moin 2003). Numerical errors not onl unfavorabl affect te SGS model but also introduce numerical instabilit. As remedies for te numerical instabilit, researcers ave added numerical (artificial) dissipation to te molecular (psical) dissipation or ave performed a posteriori low-pass filtering to attenuate te unresolved ig-wavenumber motions causing numerical instabilit. However, te artificial dissipation and ad-oc low-pass filtering also Department of Mecanical Engineering, Carnegie Mellon Universit, Pittsburg, PA 15213

2 204 You, Bose & Moin damp out dnamicall important large-scale fluid motions and, terefore, significantl degrade te simulation result. Te detrimental effects of numerical errors on te numerical stabilit and SGS modeling can be effectivel eliminated b an explicit filtering tecnique wic separates discretization and filtering operations, tereb allowing a grid-independent simulation result. In contrast to a posteriori low-pass filtering applied directl to a simulation result, te explicit filtering tecnique applies a low-pass filter onl to te nonlinear terms. Tis prevents te production of contaminated ig-wavenumber motions, and elps to maintain numerical stabilit and accurac. Bose, Moin & You (2010) recentl developed a metodolog for explicitl filtered LES of incompressible turbulent flows, and successfull demonstrated tat explicitl filtered LES can predict grid-independent stable solutions of incompressible turbulent flows. Te objective of te proposed stud is to exploit te explicitl filtered LES concept for accurate and stable predictions of compressible turbulent flows. For compressible LES, explicit filtering is necessar for nonlinear terms in bot te momentum and energ equations, wereas te energ equation is not considered in te incompressible flow case. Altoug te concept of explicitl filtered LES as been known for some time (Lund & Kaltenbac 1995; Gullbrand 2003), verification and validation of te concept for obtaining a grid-independent LES solution was onl recentl fulfilled (Bose, Moin & You 2010). To date, te use of explicitl filtered LES for compressible turbulent flows is not found in te literature. In te present researc, an effective metodolog for grid-independent LES of compressible turbulent flows using explicit filtering, is developed and validated. 2. Computational metodolog Te explicitl filtered governing equations for compressible large-scale fluid motions are as follows: ρ t + ρṽ k x k = 0, (2.1) ρṽ k t + ρṽ kṽ l x l = p + σ kl τ kl + f k, (2.2) x k x l x l were C v ρ T t + C v ρ T ṽ k x k = p ṽ k ṽ k + σ ik + ( α T ) q k, (2.3) x k x i x k x k x k σ kl = 2 3 µ v ( j vk δ kl + µ + v ) l, x j x l x k t is te time. x k, v k, and f k are te spatial coordinate, velocit, and bod force in te k-direction, respectivel. ρ is te densit, p is te pressure, T is te temperature, C v is te specific eat at constant volume, α is te termal conductivit, and µ is te molecular viscosit. τ kl and q k are te subfilter-scale model terms for te momentum and energ equations, and defined as τ kl = ρv k v l ρṽ k ṽ l and q k = C v ρt v k C v ρ T ṽ k, respectivel. Te global coefficient model b Vreman (2004) is extended to model te subfilter-scale stress τ kl and flux q k (You & Moin 2008). Te current simulations utilize a constant global coefficient suggest b Vreman, altoug future simulations will be performed using a dnamicall

3 Grid-independent LES of compressible turbulent flows 205 determined model coefficient (You & Moin 2007a,b). In equations (2) and (3), additional filtering ( ), i.e., explicit filtering, is applied to nonlinear products ρṽ k ṽ l and C v ρ T ṽ k. Tis particular coice of te convective terms arises from te inclusion of te resolvable portions of te traditional subgrid stress tensor; for instance, including te ρṽ k ṽ l ρṽ k ṽ l component from te ρv k v l ρṽ k ṽ l subgrid stress tensor te momentum equation. Tis is inclusion of te Leonard stress term in te momentum and energ equations. Te explicit filter widt is not necessaril identical to te grid spacing (implicit filter widt), and terefore, filtering and discretization operations can be formall separated. Finite difference/volume operators produce numerical errors tat are significant at ig wavenumbers, and tus te filter widt needs to be cosen to damp out ig-wavenumber quantities contaminated b te numerical error. In general, te explicit filter widt is defined to be sufficientl larger tan te grid spacing so tat onl te numericall wellresolved fluid motions are preserved. Ten, a series of simulations is performed until a grid-independent solution is obtained b refining te grid spacing wile keeping te explicit filter widt and effective resolution for te flow field. An unstructured-grid finite-volume LES code for compressible flows, CarLES, developed at te Center for Turbulence Researc, is modified to ave an explicit filtering capabilit. For grid-independent LES, te sape and widt of te spatial filter sould be consistent regardless of te grid resolution. In te present stud, te differential filters suggested b Germano (1986), wic commute wit second-order accurac wit differentiation, are emploed for te explicit filtering operations. 3. Explicit filtering Differential filters (Germano 1986) in te following form are used to perform te explicit filtering: φ (γ φ ) = φ, (3.1) were φ corresponds to a filtered function, φ is an unfiltered function, and te spatiall varing function γ is associated wit te filter widt of te kernel. It is required tat γ 0 at te wall boundaries wic is equivalent to requiring a vanising filter widt at te wall. Te filter widt is determined b matcing te second moment of te differential filter kernel wit te second moment of a sperical top at function wit radius, f (Marsden et al. 2002). However, te second moment of te differential filter kernel is not known a priori because it requires knowledge of te Green s function at eac spatial location as a function of γ. Instead, in te present stud, te second moment of te infinite space Green s function is computed assuming tat γ is constant, wic ields te relation γ = 2 f 10. Te filter widt (γ) is polnomiall relaxed suc tat γ vanises at te walls in tis case. Te filter widt reported in Table 1 is te maximum filter widt in te domain normalized b te eigt of te ill. On te coarsest mes, tis corresponds to a filter tat is nominall 1.65 c, were c is a nominal grid spacing near te inlet centerline.

4 206 You, Bose & Moin Case implicitl filtered LES explicitl filtered LES explicitl filtered LES Mes (Nx N Nz ) f / (max) Table 1. Numerical parameters for cannel flow simulations. Nx, N, and Nz denote te number of grid points used in te streamwise, wall-normal, and spanwise directions, respectivel. f / is te maximum filter widt x/ Figure 1. Computational grid ( ) for large-edd simulation of compressible flow troug a cannel wit periodic constrictions. 4. Flow configuration One of te computational grids considered in te present simulations is sown in figure 1. Te flow configuration was first introduced b Mellen, Fr olic & Rodi (2000) and modified later b Fr olic et al. (2005). Altoug te flow configuration as been extensivel studied in te incompressible flow regime, no experimental and computational studies in te compressible flow regime are found in te literature. Te domain extents in te streamwise, wall-normal, and spanwise directions are Lx = 9, L = 35, and Lz = 4.5, respectivel, were is te eigt of periodic ills. Te Renolds number, based on te ill eigt and te bulk velocit of 105, and te bulk Mac number of 0.6 are considered in te present simulations. Periodic boundar conditions are applied in te streamwise and spanwise directions wereas no-slip conditions are applied on te top and bottom walls were an isotermal condition is also imposed. Te isotermal wall is specified to be T = 1400K; altoug tis temperature is artificiall selected for tis test problem, it ma be indicative of wall temperatures in combustion applications. 5. Results and discussion Te flow configuration is relativel simple and allows ig level control of mes distribution and qualit wile involves a variet of flow penomena suc as flow separation, recirculation, and reattacment. Cannel flow simulations are performed wit a fixed mass

5 Grid-independent LES of compressible turbulent flows 207 (a) (b) (c) Figure 2. Contours of te streamwise velocit in an x plane. (a) Explicitl filtered LES on te mes; (b) explicitl filtered LES on te mes; and (c) implicitl filtered LES on te mes. flow rate. Explicitl filtered LESs are conducted on two different meses: , and , wile te filter widt remains constant. For a comparison, an implicitl filtered LES is performed on te mes. Te number of grid points and te filter widt considered in te present simulations are summarized in table 1. Simulations are started from uniform initial flow perturbed wit random numbers wit amplitudes approximatel 5% of te streamwise velocit. Te initial field, ten, is allowed to evolve for about 10 domain-troug flow times so tat turbulent structures are full developed. Te mean turbulence statistics are collected for a time period of at least 15 domain-troug flow times, tereafter. Te streamwise velocit contours in an x plane are sown in figures 2(a) and (b) for te two explicitl filtered LESs. Te same contours for an implicitl filtered LES calculation on te mes is sown in figure 2(c). Some qualitative agreement in te size of te flow structures is found between te two explicitl filtered LESs. Te implicitl filtered LES sows te presence of smaller-scale corrugated structures, wic are not present in te explicitl filtered LESs.

6 208 You, Bose & Moin x/ = U/U bulk Figure 3. Profiles of te mean streamwise velocit along te wall-normal direction at four streamwise locations. Solid lines: explicitl filtered LES on te mes; dased lines: explicitl filtered LES on te mes; das-dotted lines: implicitl filtered LES on te mes. x/ = x/ = u u /U 2 bulk v v /U 2 bulk x/ = x/ = w w /U 2 bulk v w /U 2 bulk Figure 4. Profiles of te mean Renolds stresses along te wall-normal direction at four streamwise locations. Solid lines: explicitl filtered LES on te mes; dased lines: explicitl filtered LES on te mes; das-dotted lines: implicitl filtered LES on te mes. Te mean and turbulence statistics are averaged in time and over te spanwise direction. Figure 3 sows te mean streamwise velocit profiles at four different streamwise locations. Te size of te mean separation bubble is found to be smaller in te explicitl filtered LES on te fine mes ( mes) tan tose predicted b te explicitl filtered and implicitl filtered LESs on te coarse mes ( mes). In general, te mean velocit profiles predicted b te explicitl filtered LES on te coarse mes (dased lines) are found to be closer to te profiles predicted b te explicitl filtered LES on te fine mes (solid lines) tan tose predicted b te implicitl filtered LES on te same coarse mes (cain-dased lines).

7 Grid-independent LES of compressible turbulent flows 209 Figure 4 sows te profiles of te Renolds normal and sear stresses for te tree LESs. Altoug grid-convergence of te explicitl filtered LES solutions are not obtained at te present two different grid resolution, te agreement between te two explicitl filtered LES solutions is found to be consistentl better tan te agreement between te implicitl filtered LES and te fine-mes explicitl filtered LES solutions. 6. Concluding remarks A computational metodolog for explicitl filtered large-edd simulation of compressible turbulent flows as been developed. Te explicitl filtered large-edd simulation metodolog as been utilized to obtain grid-independent mean and turbulent statistics of compressible turbulent flow troug a cannel wit periodic constrictions. Te application of an explicit filter to te large-edd simulation equations as enabled an unambiguous separation of te scales tat are resolved in te simulation and tose tat must be modeled. Te explicit filtering is expected to elp us overcome te difficult of te conventional implicitl filtered LES in assessing te predictive capabilit of subfilterscale turbulence models because of te sensitivit of te subfilter models to numerical errors. An explicitl filtered LES and a direct numerical simulation of te same flow on a finer mes are currentl in progress. Detailed comparison wit te anticipated data will be performed to assess te grid-convergence of te explicitl filtered LES solutions and predictive capabilit of te emploed subfilter model. REFERENCES Bose, S. T., Moin, P. & You, D Grid-independent large-edd simulation using explicit filtering. Psics of Fluids 22 (10), Cow, F. & Moin, P A furter stud of numerical errors in large edd simulation. Journal of Computational Psics 184, Frölic, J., Mellen, C. P., Rodi, W., Temmerman, L. & Lescziner, M. A Higl resolved large-edd simulation of separated flow in a cannel wit streamwise periodic constrictions. Journal of Fluid Mecanics 526, Germano, M Differential filters of elliptic tpe. Psics of Fluids 29, Gullbrand, J Grid-independent large-edd simulation in turbulent cannel flow using tree-dimensional explicit filtering. Annual Researc Briefs Center for Turbulence Researc, Stanford Universit/NASA, California. Lund, T. S. & Kaltenbac, H.-J Experiments wit explicit filtering for LES using a finite-difference metod. Annual Researc Briefs Center for Turbulence Researc, Stanford Universit/NASA, California. Marsden, A. L., Vasilev, O. & Moin, P Construction of commutative filters for LES on unstructured meses. Journal of Computational Psics 175 (2), Mellen, C. P., Frölic, J. & Rodi, W Large-edd simulation of te flow over periodic ills. In Proceedings of IMACS World Congress (ed. M. Deville & R. Owens). Lausanne. Vreman, A. W An edd-viscosit subgrid-scale model for turbulent sear flow: algebraic teor and applications. Psics of Fluids 16 (10), You, D. & Moin, P. 2007a A dnamic global-coefficient subgrid-scale edd-viscosit

8 210 You, Bose & Moin model for large-edd simulation in complex geometries. Psics of Fluids 19 (6), You, D. & Moin, P. 2007b A dnamic global-coefficient subgrid-scale model for largeedd simulation of turbulent scalar transport in complex geometries. Annual Researc Briefs, Center for Turbulence Researc, Stanford/NASA, California. You, D. & Moin, P A dnamic global-coefficient subgrid-scale model for compressible turbulence in complex geometries. Annual Researc Briefs Center for Turbulence Researc, Stanford/NASA, California.