Simulations of the turbulent channel flow at Re τ = 180 with projection-based finite element variational multiscale methods

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1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Met. Fluids 7; 55:47 49 Publised online 4 Marc 7 in Wiley InterScience ( DOI:./fld.46 Simulations of te turbulent cannel flow at Re τ = 8 wit projection-based finite element variational multiscale metods Volker Jon, and Micael Roland FR 6. Matematik, Universität des Saarlandes, Postfac 5 5, Saarbrücken 664, Germany SUMMARY Projection-based variational multiscale (VMS) metods, witin te framework of an inf sup stable second order finite element metod for te Navier Stokes equations, are studied in simulations of te turbulent cannel flow problem at Re τ = 8. For comparison, te Smagorinsky large eddy simulation (LES) model wit van Driest damping is included into te study. Te simulations are performed on very coarse grids. Te VMS metods give often considerably better results. For second order statistics, owever, te differences to te reference values are sometimes rater large. Te dependency of te results on parameters in te eddy viscosity model is muc weaker for te VMS metods tan for te Smagorinsky LES model wit van Driest damping. It is sown tat one uniform refinement of te coarse grids allows an underresolved direct numerical simulations (DNS). Copyrigt q 7 Jon Wiley & Sons, Ltd. Received July 6; Revised 9 January 7; Accepted January 7 KEY WORDS: variational multiscale metods; inf sup stable second order finite element metods; turbulent cannel flow; Smagorinsky LES model wit van Driest damping. INTRODUCTION Turbulent incompressible flows occur in many processes in nature and industry. Te accurate simulation of suc flows is, owever, still a major callenge. Due to te limited resolution of discretizations for te underlying incompressible Navier Stokes equations, it is not possible to resolve (and tus to simulate) all scales of a turbulent flow []. Te unresolved scales are important for te turbulent caracter of te flow and teir influence onto te resolved scales as to be taken into account by means of a turbulence model. Tere are many approaces for turbulence modelling, for instance, k ε models, classical large eddy simulation (LES) models, Navier Stokes-α-models, Correspondence to: Volker Jon, FR 6. Matematik, Universität des Saarlandes, Postfac 5 5, Saarbrücken 664, Germany. jon@mat.uni-sb.de Copyrigt q 7 Jon Wiley & Sons, Ltd.

2 48 V. JOHN AND M. ROLAND approximate deconvolution models or variational multiscale (VMS) models. We will study in tis paper a class of VMS models at te bencmark problem of te turbulent cannel flow at Re τ = 8 and compare te results wit a classical LES model. Classical LES metods are currently one of te most popular approaces for te simulation of incompressible turbulent flows. A classical LES metod starts by decomposing te flow field into large (resolved) scales and small (unresolved) scales. A caracteristic feature of tese metods is te definition of te large scales by an average in space, using a convolution wit an appropriate filter function or a so-called differential filter (wic is an approximation of a convolution operator). Te aim of classical LES metods consists in simulating accurately only te large scales of te flow field, tereby modelling te influence of te small scales onto te large ones wit te elp of a turbulence model, see te monograps [ 4] for details. Widely used classical LES models are Smagorinsky-type models [5], for instance, te dynamic Smagorinsky model by Germano et al. [6] and Lilly [7]. A more recently proposed alternative approac for incompressible turbulent flow simulations are VMS metods. Teir development is based on general ideas for te simulation of multiscale penomena from [8, 9]. Te first presentation of a VMS approac for turbulent flows can be found in []. Te basis of a VMS metod for incompressible flows is a decomposition of te flow field into tree scales; resolved large scales; resolved small scales; and unresolved small scales []. Similar to classical LES models, te goal of a VMS metod consists in simulating all resolved scales, or at least te large scales, accurately. However, an essential difference to classical LES metods consists in te way of defining te scales. In VMS metods, te scales are defined by projections into appropriate subspaces of te space in wic te variational formulation of te problem is given. Anoter essential difference consists in te way of applying te turbulence model wic models te influence of te unresolved scales. Wereas in te classical LES approac, te turbulence model is applied directly to all resolved scales, a VMS metod applies it directly only to te resolved small scales. By te coupling of scales, tere is an indirect influence of te turbulence model to te resolved large scales as well. A VMS metod tries to restrict in tis way te direct application of te turbulence model to te scales were it is needed. Tis is similar to te goal of te dynamic Smagorinsky LES model by Germano and Lilly, wic dynamically adjusts a factor in te Smagorinsky model to control te influence of tis model in te simulations. For an introduction to VMS models and teir relations and differences to classical LES models, we refer to te survey papers [, 3]. Since presenting te idea of using te VMS approac for turbulent flow simulations in [], a number of numerical studies ave been publised applying metods of tis kind. Among te first ones are [4, 5] using Fourier spectral metods for te simulation of omogeneous isotropic turbulence and turbulent cannel flows (at Re τ {8, 395}), respectively. Te study [5] was later complemented by [6] for Re τ = 59 and by [7], wic studies te dependency of te results on te separation of scales in terms of wave numbers. In [8], te so-called planar VMS metod was applied to turbulent cannel flow simulations at Re τ {8, 59}. In te planar VMS metod, a Fourier Galerkin metod was used in streamwise and spanwise direction, like in te Fourier spectral metod, wereas a finite volume discretization was applied in wall normal direction. Te scale separation of te VMS approac as been performed only in planes ortogonal to te wall normal direction. Te same autors also developed te local VMS metod wic is based on a discontinuous Galerkin discretization. Turbulent cannel flow simulations wit tis metod can be found, for instance, in [9] (Re τ = ) and [] (Re τ {, 395}). In [, ], a VMS metod for a second order, energy conserving finite volume metod is Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld

3 PROJECTION-BASED FINITE ELEMENT VARIATIONAL MULTISCALE METHODS 49 presented. Tis metod defines te large scales by restricting te discrete functions to te next coarser grid and prolongating te coarse functions appropriately back to te fine grid. Turbulent cannel flow computations are presented for Re τ {8, 59}. For furter applications of te VMS idea in te simulation of turbulent flows, we refer to [3, 4]. Numerical studies of VMS metods wic include a comparison wit te classical dynamic Smagorinsky LES model, for instance, in [4, 5, 8,, 4], sow tat te VMS metods give in general better results. Te parameters of a VMS metod in te context of finite element metods (FEMs) are finite element spaces wic define a scale separation and te turbulence model acting directly on te resolved small scales. Concerning te turbulence model, almost all simulations wic can be found in te literature so far use Smagorinsky-type models, often te standard Smagorinsky model [5], see Section 3 for details. Wit respect to te spaces for te scale separation, tere are principally different realizations. Te first one utilizes separate finite element spaces for te large scales and for te resolved small scales. It requires te solution of equations for te resolved small scales. For te large scales, standard finite element spaces are used. On te one and, te finite element spaces for te resolved small scales ave to ave in some sense a better resolution tan te finite element spaces for te large scales. On te oter and, te solution of te equations for te resolved small scales sould be not too expensive. For tese reasons, it is proposed to use bubble functions for te resolved small scales, wic leads to a localization of te small-scale problems. Tis so-called bubble VMS metod as been studied, for instance, in [5, 6]. A second realization of a VMS metod uses a standard finite element space for all resolved scales and an additional finite element space for te large space, see [7] and Section 3 for detailed descriptions. Tis so-called projection-based VMS metod will be used in te numerical simulations presented in tis paper. Te VMS metod of [, ], wic is based on a finite volume metod, as a similar spirit like te two-level version of te projection-based finite element VMS metod, wic was sortly mentioned in [7] and presented in detail (for convection diffusion equations) in [8]. For a detailed description of bubble and projection-based VMS metods, we refer to [9]. Atirdway of acieving a scale separation in finite element VMS metods was used in [3]. Tis metod exploits te ierarcy of ierarcical basis functions for velocity and pressure: ansatz functions up to a certain polynomial degree represent te large scales and te ansatz functions of iger degree te resolved small scales. A natural framework of VMS metods are finite element metods since tey are based on a variational formulation of te underlying equation. Finite element metods ave so far been widely used in te simulation of laminar flows. It as been sown tat in particular iger order finite element metods (at least second order velocity and first order pressure) lead to quite accurate results, for instance, in [3, 3] for flows around a cylinder. Tus, it seems naturally to apply finite element discretizations also in te turbulent regime. However, finite element metods ave been used far less tan Fourier spectral metods, finite difference metods (FDMs) or finite volume metods (FVMs) for te simulation of turbulent flows. Tis migt ave several reasons. Te oter metods are traditionally popular in te engineering community and most of te simulations ave been performed by scientist aving an engineering background. Anoter reason migt be te iger complexity of implementing finite element metods. Tere are only very few examples for te application of iger order finite element metods in turbulent flow simulations, like [9,, 7]. In fact, to our best knowledge, te current paper presents te first study of a turbulent cannel flow wic uses inf sup stable second order finite elements for te velocity and first order finite elements for te pressure. It will be sown tat for te turbulent cannel flow wit Re τ = 8 good Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld

4 4 V. JOHN AND M. ROLAND results for te mean velocity profile are obtained even on very coarse grids wit rougly 7 and 9 degrees of freedom (d.o.f.). Te results for second order statistics are less accurate, owever, in view of te coarseness of te grids often still satisfactory. For te coarse grids, te application of a turbulence model becomes necessary. A question of interest to explore is up to wic fineness of te meses turbulence models are needed. It will be sown tat refining tese coarse grids once and obtaining tus 6 and 3 d.o.f., respectively, te application of te Galerkin finite element discretization is already possible, wic can be interpreted as an underresolved direct numerical simulation (DNS) or as a Monotone integrated LES (MILES) approac. Te paper is organized as follows. In Section, te turbulent cannel flow problem, te discretizations, te grids and te computation of te statistics of interest are explained in detail. Te computations on te coarse grids are presented in Section 3. Tis section contains also a detailed description of te used turbulence models. Section 4 briefly presents te results for te Galerkin finite element metod on te finer grids. Te results of te computational studies are summarized in Section 5.. SET-UP OF THE NUMERICAL SIMULATIONS.. Te turbulent cannel flow at Re τ = 8 Te turbulent cannel flow is governed by te (non-dimensionalized) incompressible Navier Stokes equations u t (Re τ D(u)) + (u )u + p = f in (, T ] Ω u = in [, T ] Ω () were Ω = ( π, π) (, H) ( 3 π, 3 π) D(u) = ( u + u T )/ being te velocity deformation tensor, H = being te cannel alf widt and Re τ = 8 being te Reynolds number based on te cannel alf widt, te kinematic viscosity ν of te fluid and te sear or friction velocity u τ,see[] for te definition of u τ. Te dimensions of te cannel are standard ones for tis Reynolds number [, 33]. Tere are periodic conditions in te streamwise x- and te spanwise z-direction for te velocity u on te boundary and no-slip conditions for te solid walls at y = and. Te definition of an initial condition in our simulations is based on te discrete mean velocity profile Umean DNS (y) from te data file can8.means provided in [33]. Te discrete mean velocity profile is interpolated linearly, giving Umean DNS,lin (y), and noise is added in te same form as in [] u (; x, y, z) = Umean DNS,lin (y) +. U bulk ψ u (; x, y, z) =. U bulk ψ () u 3 (; x, y, z) =. U bulk ψ Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld

5 PROJECTION-BASED FINITE ELEMENT VARIATIONAL MULTISCALE METHODS 4 Te bulk velocity is computed by U bulk = H H U DNS,spline mean (y) dy = (3) were Umean DNS,spline (y) is a cubic spline interpolation of Umean DNS (y). Te noise is given by a random function ψ wit values in [, ]. Tis function, in C ++-notation rand() ψ = [, ] RAND MAX() is called for eac degree of freedom and eac component of te velocity. Altogeter, te initial velocity field () is obtained by disturbing te linearly interpolated mean velocity profile Umean DNS,lin (y) by a random velocity fluctuation of up to % of te bulk velocity U bulk eiter in negative or positive direction. Since te flow is incompressible, te bulk velocity sould be constant during te simulations. However, finite element functions are in general only discretely divergence free. Tus, a finite element discretization will not lead automatically to a conservation of te bulk velocity. We account for te difference of te computed bulk velocity and (3) by a dynamic adjustment of te rigt-and side of te Navier Stokes equations. Te flow is driven by a pressure gradient. Let U bulk,sim (t n ) be te bulk velocity of te computed solution at time t n. Ten, we define te rigt-and side of te Navier Stokes equations () at t n+ by U bulk U bulk,sim (t n ) f = + Δt n (4) were Δt n is te lengt of te time step. Tat means, if U bulk,sim (t n )<U bulk, te flow will be accelerated wic leads to an increase in te bulk velocity of te computed solution. In te case U bulk,sim (t n )>U bulk, te mean speed of te flow will be slowed down below and U bulk,sim becomes smaller. If te dynamic adjustment of te driving force (4) were not applied, we could observe increase as well as decrease in te bulk velocity of te computed solution, depending on te turbulence model. Using (4), te bulk velocity still sowed some oscillations but it stayed always close (differences in general far less tan %) to te value given in (3)... Te discretization Here, we will restrict ourselves to te description of te discretization of te Navier Stokes equations (). Te treatment of additional terms in te turbulence models is presented in Section 3. Standard notations for Lebesgue and Sobolev spaces are used. Let (, ) denote te inner product in (L (Ω)) d, d. Te space V is defined by V ={v (H (Ω)) 3 : v = on y = andy = } and te space Q = L (Ω), te space of all functions from L (Ω) wit integral mean value zero. Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld

6 4 V. JOHN AND M. ROLAND Our approac of discretizing () is as follows:. Discretize () by te Crank Nicolson sceme in time. Tis gives for te discrete time t n te system u n + Δt n[ (Re τ D(u n )) + (u n )u n ]+Δt n p n = u n + Δt nf n + Δt nf n Δt n[ (Re τ D(u n )) + (u n )u n ] (5) u n = were Δt n is te lengt of te time step from t n to t n and u n = u(t n ), etc. Note tat tere is an inconsistency in te temporal discretization of te pressure, see [3] for a discussion of tis issue.. Transform (5) into a variational form: find (u n, p n ) V Q suc tat (u n, v) + Δt n[(re τ D(u n ), D(v)) + ((u n )u n, v)] Δt n (p n, v) = (u n, v) + Δt n(f n, v) + Δt n(f n, v) Δt n[(re τ D(u n ), D(v)) +((u n )u n, v)] (6) ( u n, q) = for all (v, q) V Q. 3. Solve (6) by a fixed point iteration: given u () n find (u (k) n, p n (k) ) V Q (u (k) n, v) + Δt n[(re τ D(u (k) n ), D(v)) + ((u(k ) = u n, solve te linear system (Oseen system): n )u (k) n, v)] Δt n(p (k), v) = (u n, v) + Δt n(f n, v) + Δt n(f n, v) Δt n[(re τ D(u n ), D(v)) + ((u n )u n, v)] ( u n, q) = for all (v, q) V Q, k =,, Discretize (7) by te Q /P disc finite element metod, i.e. te velocity is approximated by a piecewise triquadratic continuous function and te pressure by a piecewise linear discontinuous function. We would like to present some motivations for coosing tis way of discretizing (). Te Crank Nicolson sceme is well known to be an accurate and efficient temporal discretization of te incompressible Navier Stokes equations, see [34 36]. Likewise, te Q /P disc finite element discretization is known to be among te best performing finite elements for incompressible flows, see in particular [37] and our own experiences in [3, 3, 35, 38]. Note tat finite element metods for te pysically correct deformation tensor formulation are considerably more expensive tan for te gradient formulation (Re τ u, v), cf. te discussion of tis topic in [3]. Te linearization by a fixed point iteration was sown to be more efficient tan using a Newton metod in [35]. Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld n (7)

7 PROJECTION-BASED FINITE ELEMENT VARIATIONAL MULTISCALE METHODS y Figure. Distributions of te degrees of freedom in wall normal direction, level, left: y [, ], rigt: zoom near te wall; from top to bottom: Grid, l = ; Grid, l = ; Grid, l = 4; Grid, l = 4. Te Crank Nicolson sceme was applied wit an equidistant time step of Δt =. (Δt + := u τ Re τ Δt =.36, were te value u τ = of te statistically steady state as been used). Tis is considerably smaller tan te Kolmogorov time scale and it fits into te range of te time step proposed in [39]. Simulations on two different grids were performed in our numerical studies. Te grids were obtained by uniform refinement from a coarsest grid (level ) using a subdivision of te exaedral mes cells into eigt smaller mes cells. In te periodic directions (x and z), te grid spacings are uniform. Te coarsest grid in streamwise and spanwise direction is a uniform grid. In te wall normal direction, one as to use non-uniform grids wic become finer towards te walls. We will study two different grids, wic describe te distribution of te grid points in wall normal direction as follows: Grid : y i = + tan(γ(i/n y )), i =,...,N y tan(γ) wit γ =.75, see [39]; also[9, 4] for turbulent cannel flows wit different Reynolds numbers were also different stretcing factors γ ave been used; Grid : ( ) iπ y i = cos, i =,...,N y N y see [, ]. Here, N y is te number of mes cell layers in wall normal direction. Consequently, te number of grid points in y-direction, including te points on te boundaries, is N. Concerning te coarsest grid, we will study grids wit two layers (denoted by l = ) and wit four layers (l = 4). After eac uniform refinement step, te points of te new grid are translated into wall normal direction suc tat te prescribed distribution is attained. Since te velocity is approximated wit te Q finite element, tere are N layers of d.o.f. in y direction, including te boundary. Note tat due to te definition of te Q finite element, te layers of d.o.f. between te grid points do not obey te prescribed distribution but tey are located alf way between te grid points, see Figure. Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld

8 44 V. JOHN AND M. ROLAND Table I. Information on te grids used in te computations. Level l Cells N y Vel. d.o.f. Press. d.o.f. Grid, min Grid, min Let = Re τ y = 8y, y [, ], be te distance from te wall measured in wall units (or viscous lengts). Te distributions of te d.o.f. in wall normal direction for our numerical simulations on level are presented in Figure. It can be seen tat te yperbolic tangent function (Grid ) places te d.o.f. wit a iger density closer to te wall tan te cosine function (Grid ). Information concerning te grids are given in Table I: te number of mes cells, te number of d.o.f., te distance of te d.o.f. next to te walls min and te number N y of mes cell layers are given. It is ard to compare te fineness of te grids to resolutions of completely different discretizations like Fourier Galerkin metods. Information on te number of d.o.f. are given for te discontinuous Galerkin metods used in [9]. Level of our grids as in bot cases fewer d.o.f. tan te grids in [9], wereas te number of d.o.f. on level 3 as te same order of magnitude like in [9]..3. Statistics of interest We will use te reference data for te turbulent cannel flow at Re τ = 8 wic can be found in te data files belonging to te DNS simulations from [33]. Let s denote te spatial averaging over te directions of omogeneity. Since uniform grids are used in streamwise and spanwise direction, te spatial averaging can be performed by te aritmetic mean. Let u (t, x, y, z) be te computed flow field. Ten, te spatial mean velocity at time t n in te plane y = const. is computed by U (t n, y) := u (t n, x, y, z) s = N x N z N x N z i= j= u (t n, x i, y, z j ) were N x (N z ) is te number of d.o.f. in te streamwise (spanwise) direction in te plane y = const. Tis is done for all planes y = const. wic contain velocity d.o.f. in wall normal direction, see Figure. Te average in time will be denoted by t. Since equidistant time steps will be used in te computations, te aritmetic mean is again applied. Tus, te mean velocity profile is given by U mean (y) := u (t n, x, y, z) s t = N t + N t n= U (t n, y) We will present results for te first component Umean (y) of U mean (y). Te simulated friction velocity uτ is defined as te average of te computed friction velocities at bot walls, were te friction velocity at eac wall is approximated by a one-sided difference ( uτ := U mean ( min ) U mean ( y+ min ) ) min min Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld

9 PROJECTION-BASED FINITE ELEMENT VARIATIONAL MULTISCALE METHODS 45 Second order statistics of interest in turbulent cannel flows are te off-diagonal Reynolds stresses and te root mean square (rms) turbulence intensities. Unfortunately, te definition of te averaged Reynolds stresses does not seem to be unique in te literature. Denoting by u te streamwise velocity component and by v te wall normal component, te Reynolds stress R = R uv = R xy can be defined by or alternatively by R := uv s t u s t v s t (8) R := uv s t u s v s t = uv s u s v s t (9) Te form (8) can be found, for instance, in [4] for teoretical considerations and in [,, 4] for computations. Formula (9) was used, for instance, in [5], were in particular (u u s ) s / t = u u u s u + u s u s s / t = u u s u s u s + u s u s / t = ( R )/ was considered. Te definitions (8) and (9) are in general not identical. Spatial and temporal averaging can be intercanged in (8) wereas tis is not possible in (9). It is not clear wic form, (8) or (9), was used to compute te reference data in [33]. Te computation of te statistics in te numerical studies presented below follows [4]. Te off-diagonal Reynolds stresses of te DNS from [33] can be approximated by R DNS ij R ij + A ij s t, i, j =,, 3, i = j were te approac (8) is used to compute Rij and Aij stands for te modelled subgrid scale stresses. A normalization wit (uτ ) was used to compute te off-diagonal Reynolds stresses presented below R, ij := R ij + A ij s t (uτ, i, j =,, 3, i = j () ) Concerning te diagonal stresses, teir deviation from isotropy can be approximated by [4] R DNS ii 3 3 R DNS j= jj R ii + A ii s t 3 3 (R jj + A jj s t ), i =,, 3 Ten, te rms turbulence intensities given for te simulations below are computed wit a normalization wit uτ, for instance u, rms := j= R + A s t 3 3j= (R jj + A jj s t ) Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld u τ / ()

10 46 V. JOHN AND M. ROLAND Te form of te subgrid scale model A for te different models used in te computations, togeter wit remarks for evaluating A, is given in Section 3. Let Rii MKM denote te reference data of [33]. We compare te rms turbulence intensities, for instance, u, rms, to R MKM 3 (RMKM + R MKM + R MKM 33 ) / Tis approac follows [4]. Note tat oter comparisons can be found in te literature as well. For instance, in [5], te term (u u s ) s t / (plus influence of te turbulence model) is compared to (R MKM ) /. Since te turbulent cannel flow is statistically symmetric at y =, only te averaged values on alf te cannel are given below. Tese are computed by averaging te values obtained on bot alves of te cannel. Starting wit te initial condition (), te turbulent cannel flows were simulated wit eac metod in te time interval [, ] s (t + [, 36]) to obtain a fully developed flow field wic can be considered to be independent of te initial condition. We cecked wit oter initial conditions (perturbed laminar flow profile) tat te initial simulation time of s was sufficient for acieving tis goal. Te mean velocity profiles were almost indistinguisable. Te differences in te second order statistics were somewat larger, owever, neiter te form of te curves, nor te magnitude of te values canged considerably. Wit te results at t = s, te simulations were started anew for computing te time averages. Te lengt of te time interval for computing te time averages was also s. Te simulations were performed wit te code MooNMD [43]. 3. SIMULATIONS ON COARSE GRIDS WITH THE APPLICATION OF TURBULENCE MODELS 3.. Te turbulence models In tis section, te simulation of te turbulent cannel flow at Re τ = 8 is considered on grids wic are too coarse to allow te application of te Galerkin finite element metod. Te coarse grids described in Section are refined twice to obtain tese grids (level ). Te corresponding number of d.o.f. are given in Table I. Te Galerkin finite element metod blows up in final times on tese grids. Tus, te application of a turbulence model becomes necessary. Te following turbulence models are studied: te Smagorinsky LES model [5] wit van Driest damping [44] (SvD), see also []; te fully implicit projection-based VMS metod wit piecewise constant large-scale tensors (VMS P), see [7]; te fully implicit projection-based VMS metod wit discontinuous piecewise linear tensors (VMS P), see [7]. It was sown in [, ] tat on grids wic allow te application of a Galerkin metod (underresolved or coarse DNS), tis metod outperforms in general te metods wit turbulence models. Tus, it is important to use grids wic are sufficiently coarse to make te use of turbulence models meaningful. Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld

11 PROJECTION-BASED FINITE ELEMENT VARIATIONAL MULTISCALE METHODS 47 Te traditional Smagorinsky LES model introduces te additional term (ν T D(u ), D(v )), ν T = C s δ K D(u ) F () into te Galerkin finite element metod of te Navier Stokes equations. In (), C s is a user-cosen constant, K denotes a mes cell, δ K is a parameter explained below and F is te Frobenius norm of a tensor. Usually, δ K = c K is cosen were c [, ] depends on te discretization (order of te finite element metod, for Q,weusec = ) and K is a measure of te cell widt. Te grids we are using consist of anisotropic mes cells near te wall. Te possible measures range from te sortest edge to te diameter of te mes cells. Wic is an appropriate one will be studied below, see Section 3.. Anoter caracteristic feature of te Smagorinsky LES model is te production of too muc energy dissipation near te walls. Tis issue will be addressed by te so-called van Driest damping, i.e. te introduction of a damping factor in te viscous sublayer region ( <5). Instead of te turbulent viscosity () ( y ν T = C s δ + )) K D(u ) F ( exp, <5 (3) A is used wit A = 6 and te same constant C s as in (). Te terms of te Smagorinsky model wit van Driest damping (), (3) were treated implicitly in our computations. Te model of te subgrid-scale stresses, wic is needed to compute te Reynolds stresses () and te rms turbulence intensities (), as te form A = ν T D(u ). Te projection-based VMS metod as te following form (continuous in time): find u :[, T ] V, p :(, T ] Q and G H :[, T ] L H satisfying ( u ) t, v + (Re τ D(u ), D(v )) + ((u )u, v ) (p, v ) +(ν T (D(u ) G H ), D(v )) = (f, v ) for all v V (4) (q, u ) = for all q Q (D(u ) G H, L H ) = for all L H L H Here, V Q is a standard, inf sup stable pair of finite element spaces, in our simulations V = Q, Q = P disc. Te tensor-valued space L H represents te large scales of te velocity deformation tensor. Tey are defined in te tird equation of (4) by an L -projection. Te coice of L H controls te scale separation: te larger L H becomes, te smaller becomes te part of te resolved small scales among all resolved scales. Since iger order finite elements were used for velocity and pressure, L H can be defined on te same grid as V Q wit low order polynomials. We used in te simulations L H = P (piecewise constant tensors) and L H = P disc (piecewise linear but discontinuous tensors). Te resulting metods will be called VMS P and VMS P, respectively. In comparison to te variational form of te Navier Stokes equations, tere is one additional term in te momentum equation of (4), te most rigt one on te left-and side. Te difference D(u ) G H represents small scales since G H are large scales of D(u ). Tus, tis term adds te additional turbulent viscosity ν T to te small scales of te flow field. Tis is exactly one of Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld

12 48 V. JOHN AND M. ROLAND te basic ideas of VMS metods. For te turbulent viscosity ν T, we use te Smagorinsky model given in (). Of course, te van Driest damping could ave been used easily for te VMS metods. However, in our opinion, a potential advantage of te VMS approac is tat simple models for te influence of te unresolved scales can be applied since tese models act directly only on a part of te resolved scales and te importance of te model is reduced in tis way. A constant Smagorinsky model was used in te first numerical simulations of turbulent flows wit VMS metods [4, 5] and a dynamic model, for instance, in [45] (witout comparison to te constant Smagorinsky model) and in [7]. In[7], it was sown tat te constant Smagorinsky model witin te VMS metods led to igly accurate results for appropriate scale separations (in terms of wave numbers of a Fourier spectral metod) but te dynamic model was less sensitive to te cosen scale partition. Bot models witin a two-level VMS metod based on a second order finite volume metod were compared in turbulent cannel flow problems in []. It turned out tat te use of te constant coefficient Smagorinsky model (witout van Driest damping) led to better results tan te dynamic model wit even less computational effort. Similar observations are reported for turbulent flow simulations in a diffuser [46]. Moreover, for a VMS metod based on a finite volume discretization applied to compressible flows, [4], it was found tat te dynamic computation of te Smagorinsky parameter does not improve te results obtained by considering a constant parameter. In view of tese experiences, te use of a dynamic model in te VMS metods seems not to be promising at te moment, at least not for second order discretizations. Te additional term in te momentum equation in (4) is treated implicitly. For details of te algoritm, we refer to [7]. For computing te Reynolds stresses () and te rms turbulence intensities (), te model of te subgrid-scale stresses as te form A = ν T (D(u ) G H ) for te projection-based VMS metod. Te coice of te constant C s is te main issue in Smagorinsky-type models. Te traditional value for turbulent cannel flows is C s =. [47]. However, it is known, [48], tat a good value depends on te actual type of te mes, te refinement level of te mes and most probably also on te underlying discretization. Te numerical studies in [48] sow tat even te dynamic Smagorinsky model often does not give good values for C s (wic is for tis model a function in space and time). To study te influence of te value of C s for te different models, besides te traditional value, we will present also results for C s {.,.5}. Te quantities () and () were computed in all positions were te velocity possesses d.o.f. Tese d.o.f. are located in particular at faces of te mes cells. However, D(u ) and G H are discontinuous finite element functions. For computing A, tese functions are mapped to continuous functions by computing a local weigted average. Let (x, y, z) be te position of a velocity degree of freedom and ω be te union of all mes cells were tis degree of freedom belongs to. Ten D(u )(x, y, z) := K ω K D(u ) K (x, y, z) were K is te volume of te mes cell K and K denotes te restriction of te finite element function to K. An analogous formula is used for G H. In tis way, one obtains te continuous Q finite element functions D(u ) and G H and tese functions are used to compute A in te SvD and in te VMS metods. Note tat te issue of smooting te derivatives of te velocity arises from te low regularity of standard finite element functions. Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld

13 PROJECTION-BASED FINITE ELEMENT VARIATIONAL MULTISCALE METHODS U mean 5 SvD, C s =.5, grid, diameter SvD, C s =.5, grid, geom. mean SvD, C s =.5, grid, sortest edge VMS P, C s =.5, grid, diameter VMS P, C s =.5, grid, geom. mean VMS P, C s =.5, grid, sortest edge Figure. Mean velocity profile obtained wit K being te diameter of te mes cells, K = 3 x y z and K being te sortest edge of te mes cells. 3.. Simulations wit different measures for te widt of a mes cell Te parameter δ K of te Smagorinsky LES model wit van Driest damping () (3) as well as of te projection-based VMS metods (4) involves a measure K of te size of te mes cells. Figure presents some representative results wic are obtained wit: K being te diameter of te mes cells (longest distance between two points of te mes cells); K = 3 x y z, were x, y, z are te sizes of te edges of te mes cells in te coordinate directions (geometric mean); K being te sortest edge. Te use of te geometric mean can be found in te literature [8,,, 4]. Figure presents mean velocity profiles Umean for te SvD on Grid, level, l = 4, C s =.5 and te projectionbased VMS P on Grid, level, l = 4, C s =.5. Te results were K is cosen to be te sortest edge are te best ones for bot models. Te curves for te VMS metod wit K being te geometric mean and K being te sortest edge are almost indistinguisable. For te SvD, te result becomes considerably worse for K being te geometric mean. If K is cosen to be te diameter, very bad results are obtained for bot metods. In te remainder of te paper, we will only present results wic are obtained wit K cosen to be te sortest edge of te mes cells Simulations on grids wit an initial subdivision into two layers Te results of te simulations on te grids were te initial grid consists of two mes cell layers, l =, and wic were refined twice are presented in Figures 3 7. Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld

14 4 V. JOHN AND M. ROLAND U mean SvD, C s =. SvD, C s =.5 VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s =.5 U mean SvD, C s =. SvD, C s =.5 VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s = Figure 3. Simulations on te grids wit an initial subdivision in y-direction into two layers, level, computed mean velocity profiles, Grid (left) and Grid (rigt) U mean :difference to reference 3 SvD, C s =. SvD, C s =.5 VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s =.5 U mean :difference to reference SvD, C s =. SvD, C s =.5 VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s = Figure 4. Simulations on te grids wit an initial subdivision in y-direction into two layers, level, differences to te reference mean velocity profile, Grid (left) and Grid (rigt). Te results for te streamwise mean velocity, Figures 3 and 4, are muc better for te VMS metods, in particular for VMS P, tan for te SvD metod. Te best results ave been obtained wit VMS P and C s =.5. Even on tese very coarse grids, te mean velocity profile is rater close to te reference one. Among te VMS metods, only te results for VMS P wit C s =. are unsatisfactory. Te mean velocity profiles computed wit te SvD metod tend towards te profile for a laminar flow. Te computed rms turbulent intensities urms, are sown in Figure 5. We present only results for urms, since te evaluation of te results for vrms, and wrms, would be very similar. Te curves obtained wit te VMS metods (save VMS P, C s =., Grid ) ave in principle te correct form but te values are overpredicted. Tis overprediction is somewat smaller on Grid. Te curves computed wit te SvD metod do not possess te correct form. In addition, tere is a eavy underprediction of te values at te wall for C s =.. Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld

15 PROJECTION-BASED FINITE ELEMENT VARIATIONAL MULTISCALE METHODS 4 u rms SvD, C s =. SvD, C s =.5 VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s =.5 u rms SvD, C s =. SvD, C s =.5 VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s = Figure 5. Simulations on te grids wit an initial subdivision in y-direction into two layers, level, u, rms, Grid (left) and Grid (rigt)..5 R.5.5 SvD, C s =. SvD, C s =.5 VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s =.5 R SvD, C s =. SvD, C s =.5 VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s = Figure 6. Simulations on te grids wit an initial subdivision in y-direction into two layers, level, R,, Grid (left) and Grid (rigt). Te computed off-diagonal Reynolds stresses R, are presented in Figure 6. In particular for Grid, te influence of te model A in computing te Reynolds stress at te wall for te VMS metods can be seen. Again, te VMS metods overpredict te absolute values quite a lot. A correct form of te curve is obtained only on Grid for all metods wit C s =.5 (also te SvD metod). Te coarse near wall resolution of Grid leads to peaks wic are too far away from te wall. Note tat in comparable simulations (second order FVMs) from [, ] on a muc finer grid (3 3 = grid cells), te second order statistics (turbulent kinetic energy) also sow rater large differences to te reference curve. Tus, te use of second order spatial discretizations, wic ave muc lower order tan, for instance, spectral metods, migt be an important reason for te considerable differences. Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld

16 4 V. JOHN AND M. ROLAND U : difference to reference VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s =. VMS P, C s =.5 VMS P, C s = U VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s =. VMS P, C s =.5 VMS P, C s = Figure 7. Simulations on te grids wit an initial subdivision in y-direction into two layers, level, parameter study for te VMS metods, Grid, difference to te mean velocity profile (left), u, rms (rigt). A parameter study for C s in te VMS metods reveals tat te results are best for C s =.5, Figure 7. Te coice of C s as a greater impact on te results of VMS P tan on te results of VMS P. Te great impact of C s in te SvD metod can be observed already in Figures 3 6. Concerning te two distributions of te grid points in y-direction, te results on Grid are in general better. Tis can be observed in particular for te second order statistics in Figures 5 and 6. In summary, muc better results are obtained on tese very coarse grids wit te VMS metods (VMS P, C s =.5; VMS P, C s {.5,.}) tan wit te SvD metod. Te mean streamwise velocities are predicted quite well and te curves for te second order statistics are predicted qualitatively correctly (te off-diagonal Reynolds stress only Grid ). However, te (absolute) values of te second order statistics are considerably overpredicted Simulations on grids wit an initial subdivision into four layers Te results on te finer grids, wic possess four mes cell layers in y-direction on level, l = 4, and wic are refined twice, are presented in Figures 8. Te SvD metod wit C s =. gives poor results. Even te streamwise mean velocity profile is very badly captured. Concerning te VMS metods, te mean velocity profiles wit C s =. are somewat better on bot grids tan wit C s =.5, see also Figure. Te differences to te reference profile on Grid are sligtly smaller tan on Grid. Te values for te rms turbulent intensities u, rms are too large for te VMS metods, Figure 9. Tese metods give better results wit C s =. tan wit C s =.5. Te SvD metod wit C s =.5 computes similar results to te VMS metods. Te evaluation of te results for v, rms and wrms, (not sown ere) is quite similar. Similar observations as for u, rms can be made for R,, Figure. Te computed (absolute) values of te VMS metods are too large. Again, better results are obtained wit C s =.. Te SvD metod wit C s =.5 beaves similarly to VMS P wit C s =.. A parameter study for te turbulence models, Figures and, sows tat coosing even smaller values tan.5 worsens te results. However, te qualitative form of te curves for te VMS metods is correct for all parameters. Tis is in contrast to te SvD metod wit Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld

17 PROJECTION-BASED FINITE ELEMENT VARIATIONAL MULTISCALE METHODS 43 U mean :difference to reference 3 4 SvD, C s =. SvD, C s =.5 VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s = U mean :difference to reference SvD, C s =. SvD, C s =.5 VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s = Figure 8. Simulations on te grids wit an initial subdivision in y-direction into four layers, level, differences to te reference mean velocity profile, Grid (left) and Grid (rigt). u rms SvD, C s =. SvD, C s =.5 VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s =.5 u rms SvD, C s =. SvD, C s =.5 VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s = Figure 9. Simulations on te grids wit an initial subdivision in y-direction into four layers, level, u, rms, Grid (left) and Grid (rigt). C s =.. It can be clearly observed tat te coice of te parameter C s as a muc smaller influence in te VMS metods tan in te SvD metod. Tis reflects te pilosopy of VMS metods to apply a turbulence model only to scales were it is necessary and not to all scales. Tus, a variation of te parameters will affect muc fewer scales directly and ence cange te solution less. Altogeter, VMS P and VMS P wit C s =. and SvD wit C s =.5 give te best results in our numerical studies wit an initial subdivision into four layers. Te results of VMS P and VMS P wit C s =.5 are sligtly worse wereas SvD wit C s =. gives completely wrong results. Te results on Grid are sligtly better tan on Grid Computational costs Te computing times of a number of simulations are given in Table II. A preconditioned flexible GMRES metod wit a multiple discretization multi-level preconditioner was used as solver, see Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld

18 44 V. JOHN AND M. ROLAND R SvD, C s =. SvD, C s =.5 VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s = R.5.5 SvD, C s =. SvD, C s =.5 VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s = Figure. Simulations on te grids wit an initial subdivision in y-direction into four layers, level,, Grid (left) and Grid (rigt). R, 4 U mean :difference to reference SvD, C s =. SvD, C s =.5 SvD, C s =. U mean :difference to reference VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s =. VMS P, C s =.5 VMS P, C s = Figure. Simulations on te grids wit an initial subdivision in y-direction into four layers, level, Grid, difference to te mean velocity profile, SvD (left), VMS metods (rigt). [3, 35] for details. Rougly two-tird of te computing time was spend for solving te non-linear systems and approximately one-tird for assembling te matrices. Te absolute computing times depend on te computer and te stopping criteria for te iterations. We would like to concentrate te evaluation on te comparison of te different metods and grids. It can be seen tat SvD was always te fastest metod. Te computations wit VMS P were somewat slower and te computations wit VMS P took again somewat more time. Tis corresponds to te observations in [7]. Te computing times of VMS P were 7 33% longer tan of SvD, owever, te results are sometimes considerably better. We do not report computing times wit l = 4, C s =., since te results obtained wit SvD were so different to te oter results tat a comparison of te computing times is meaningless. Concerning te two grids, te computations on Grid were in general faster. Te reason is tat te mes cells close to te boundary possess a muc iger aspect ratio on Grid and our solver Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld

19 PROJECTION-BASED FINITE ELEMENT VARIATIONAL MULTISCALE METHODS 45 u rms SvD, C s =. SvD, C s =.5 SvD, C s =. u rms VMS P, C s =. VMS P, C s =.5 VMS P, C s =. VMS P, C s =. VMS P, C s =.5 VMS P, C s = Figure. Simulations on te grids wit an initial subdivision in y-direction into four layers, level, Grid, u, rms, SvD (left), VMS metods (rigt). Table II. Computing times in seconds, in parenteses: percentage to computing time wit SvD. Parameters SvD VMS P VMS P l =, C s =.5, Grid () 995 (33) 3 68 (38) l =, C s =., Grid 8 () () () l = 4, C s =.5, Grid () (7) (6) l = 4, C s =.5, Grid () (7) (5) looses some efficiency on grids wit ig aspect ratios. Doubling te initial subdivision of te domain from l = tol = 4, wic rougly doubles te number of d.o.f., leads to an increase in te computing times wit a factor 3 4. However, also ere as to be noted tat te results for l = and 4 are rater different since on te grids obtained wit l = 4 more details of te flow field are simulated. Tis of course requires additional computational efforts. Altogeter, te better results of te VMS metods need somewat iger computational costs. Te reduction of tese costs is an important topic for future researc. 4. GALERKIN FINITE ELEMENT METHOD ON FINER GRIDS Tis section studies te question of weter te use of turbulence models is still necessary for grids finer tan tose used in Section 3. It will be sown tat refining tose grids once, it is possible to apply even te Galerkin finite element metod for te simulation of te turbulent cannel flow at Re τ = 8. Tat means, no additional stabilization or modelling terms are used. Note tat tis property is studied in te present paper for te Q /P disc finite element. Oter finite element metods migt need different refinement levels for te Galerkin metod to be applicable. Te usage of te Galerkin finite element metod on a rater coarse mes can be interpreted as an underresolved DNS or as a so-called MILES metod. In a MILES metod, te numerical diffusion takes te role of a turbulence model. Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld

20 46 V. JOHN AND M. ROLAND U mean :difference to reference.5.5 Galerkin, l=, grid Galerkin, l=, grid Galerkin, l=4, grid Galerkin, l=4, grid R.5.5 Galerkin, l=, grid Galerkin, l=, grid Galerkin, l=4, grid Galerkin, l=4, grid Figure 3. Galerkin finite element metod on level 3, difference to te reference mean velocity profile (left) and R, (rigt)..5 Galerkin, l=, grid Galerkin, l=, grid Galerkin, l=4, grid Galerkin, l=4, grid.5 Galerkin, l=, grid Galerkin, l=, grid Galerkin, l=4, grid Galerkin, l=4, grid u rms.5 u rms Figure 4. Galerkin finite element metod on level 3, u, rms (left) and v, rms (rigt). Te results on te different grids are presented in Figures 3 and 4. Te corresponding numbers of d.o.f. on level 3 are given in Table I. Concerning te mean velocity profile, te curves lie almost on top of eac oter suc tat we present only te differences to te reference profile. Tese differences are somewat smaller tan for te simulations wit turbulence models on level. Using an initial subdivision into two layers, l =, leads close to te boundary, 5andin [, 4] to sligtly smaller differences tan using an initial subdivision into four layers, l = 4. Te results for te Reynolds stress R, are still quite far away from te reference curve. On te coarser grids, l =, te results are somewat oscillatory. Te rms turbulence intensities urms, and vrms, are presented in Figure 4. Te curves are closer to te reference curves tan for te simulations wit te turbulence models on level. Some oscillations can be observed towards te centre of te cannel, in particular for te computations on Grid. Te results obtained on Grid are altogeter sligtly better. In summary, te results for te mean velocity profile are satisfactory. Te results for te second order statistics are not so good. However, one sould keep in mind tat on te one and te grids are Copyrigt q 7 Jon Wiley & Sons, Ltd. Int. J. Numer. Met. Fluids 7; 55:47 49 DOI:./fld