Glazunov A.V. Institute of Numerical Mathematics RAS glazunov@inm.ras.ru Large-eddy simulation of shear-driven turbulence using dynamic SFS closure and high-order numerical scheme
Synoptical variations Boundary-Layer flows DNS requirement Now: Energy production range Modern computers are not able Inertial range to do such calculations ! In the future: Principal impossibility to specify Dissipation range practically applicable boundary conditions. In DNS model one needs to prescribe each leaf. LES is the most perspective DNS (Direct Numerical Simulation) approach for practical use LES (Large Eddy Simulation) Three approaches for RANS (Reynolds Averaged Navier-Stokes) turbulence simulation
Mathematical formulation Navier-Stokes equations. Approximate form of the momentum and mass conservation laws in viscous incompressible fluid. Spatial filtering It’s usually assumed that filter commutes with operator of differentiation. It’s not always the case near the boundary and/or after discretization Re-independent statistics of LES large-scale motions in turbulent flows ( observation and high-Re DNS data ) gives us hope of possibility: 1. To neglect the viscous term. 2. To find closure: τ ≈ Τ ( u , u , u ) ij ij k l m Central problem of LES modeling. Universal approach isn’t known. The most difficult in anisotropic wall-bounded flows (if the energy production range isn’t strongly separated from dissipation range and/or inertial range can’t be resolved by the numerical model)
One more application (turbulent flow in domain of complex geometry ): The example of the particles transport by the turbulent flow among buildings Particles concentration Wind fraud CFD (computational fluid dynamics) Particles are ejecting near the surface (along the dotted line). The maximal number of particles is about 10 000 000.
1. Viscous third-order finite difference approximation of nonlinear terms in momentum balance equation. 2. Numerically stable but dissipative predictor-corrector time- marching scheme. 3. Very simple and numerically effective K- ε closure gives results, which are very similar to standard Smagorinsky model (far away from real turbulent stress in anisotropic turbulent flow) . 4. Iterative (SOR with red-black ordering) Poisson equation solver doesn’t provide exact mass conservation. All these make the model too dissipative and incapable to reproduce properly not only the variances but even the mean values of wind velocity in complex wall-bounded turbulent flow Don’t trust such models, they don’t exist yet! Can we return to presented simulation anytime? Let us begin with the simplest, but well-studied, anisotropic turbulent wall-bounded flow - channel flow
Snapshot of wall-normal velocity isosurfaces ( + 1.5 U * ) in rough wall bounded channel flow. LES instantaneous data (reconstructed); fragment of domain. Stream direction t n a t s n o c e r u s s e r p t n e i d a r g Averaged streamwise Two rough infinitely velocity profile wide parallel plates
Snapshot of wall-parallel velocity fluctuations at the centerline of rough wall bounded channel flow. LES instantaneous data (reconstructed); fragment of domain.
Some typical characteristics of channel flow Laminar flow Turbulent flow Z 0 for rough- wall-bounded flow Planar averaged steamwise velocity Normalized averaged steamwise velocity with the same viscosity and pressure gradient (in logarithmic coordinates) (up to channel centerline) Streamwise Spanwise Wallnormal Turbulent momentum flux Normalized velocity rms and correlation of streamwise (up to channel centerline) and wall-normal velocity
Some typical characteristics of channel flow Normalized spectra of streamwise and wall-normal velocities at the invariance different heights above surface. anisotropy
The main requirements 1. Model should be able to reproduce mean values and variability of velocity in high-Re rough wall bounded flow. 2. Model should be applicable to the domains of complex geometry (finite-difference or finite-elements schemes) 3. Model should be suitable for parallelizing (explicit algorithms are preferable) and restrictions 1. We should avoid similarity (Monin-Obukhov) arguments while the constriction of near- the-wall model. 2. The assumption of statistical homogeneity in some spatial direction isn’t applicable (no averaging) 3. Model shouldn’t have the preferable spatial directions (3-d filtering). The main difficulty: Numerical errors of approximation of non-liner terms in momentum balance equation (truncation and aliasing) are of the same order or larger (for low-order schemes) than turbulent stress terms! (Ghosal, 1995, Kravchenko and Moin 1996, Park et al. 2004, Brandt 2005 …) The dissipative up-flux schemes must be avoided due to systematic impact of numerical dissipation.
Turbulent closure Rate-of-strain tensor 1) Eddy viscosity models (wide class of closures, includes for example: K-l, K- ε …) Are written by analogy with molecular viscosity, but in this tensor form provide zero dissipation with solid-type rotation and not-constant K T The main advantage –provide energy cascade down to subgrid/subfilter scales The main disadvantage – don’t provide big positive correlation between real and modeled subfilter turbulent stress Very often more complicated eddy-viscosity models don’t give any improvement with respect to simplest Smagorinsky model (Neumann and Richtmyer, 1950; Smagorinsky 1963): Local balance of production and dissipation of subgrid energy Filter width scale Mathematically justified (Ladyzhenskaya, 1967) For the inertial range the theoretical estimation C ≈ 0.18 exist (Lilly,1967) In standard form Smagorinsky model isn’t applicable for the modeling of wall-bounded flow (mainly due to uncertainty of coefficient C in energy production range) . Modified Smagorinsky (wall damping functions (Masson,1992) (Masson and Thomson,1992)) is slighly better
Turbulent closure 2) SSM - scale-similarity model (Bardina , et al., 1983), (Sagaut, 1998), (Layton and Lewandowski 2002): The simplest case of ADM (approximately deconvolution model): ( Stolts et.al. , 2001 ), (Gullbrand et.al.2002), (Layton, 2004) τ = − * * u u u u ij i j i j The main advantage – in the case of smooth filter provides high correlation of subfilter- scale (but not subgrid !) modeled stress with real stress. The main disadvantage – don’t produce enough dissipation. 3) MM - mixed model (Zang , et al., 1993): Combines advantages of SSM and Smagorinsky model. But the problem of uncertainty of coefficient C still exists.
Turbulent closure 4) Dynamic approach (Germano, 1991): - test filter Germano identity: Dynamic approach is based on the main assumption of the LES modelling – the invariance of turbulent closure inside some (may be quit narrow) spectral range. We suppose that one can use the same turbulent model with base model filter and with additional test filter. Invariance of the constants isn’t required in common case (scale-dependent dynamic models (Porte-Agel et al.,2000)) overspecified problem with respect to C For the mixed model Germano identity gives: Ratio of test filter scale to basic filter scale It’s slightly simplified formulation of dynamic model: 1. The numerical errors of discrete model may play the role of the additional implicit filter 2. The definition of the scale of smooth filters is not formulated mathematically 3. The spectral cut-off filter isn’t applicable in wall-normal direction
Turbulent closure Local in space application of Germano identity and minimizing of errors ε with least squire method gives: (+) we don’t deviate from the locality of closure. (-) ill-conditioned problem gives large amount of negative and big positive values of C. Didn’t work without additional restrictions, which are not justified physically. (-) It’s assumed that C is constant in space over entire filter width. At the case of smooth filters and/or anisotropy this assumption is questionable. Additional minimization over: 1) Homogeneous space directions (Ghosal et al. 1995) 2) Fluid pathlines (Meneveau, et al. 1996). (+) usually doesn’t give negative values of C (-) turbulent closure isn’t local ! (-) may be: gives reasonable values of C only in apriority tests and measurements, but not after including into the dynamic procedure of numerical model with very high Re.
Turbulent closure decision variable We propose: ε 0,58 0,56 τ v 0,54 0,52 0,50 generalized solution of 0,48 over-specified problem 0,46 Conjugate gradients method 0,44 at each time step of the model 0,42 0,40 (+) locality of closure. 0 2 4 6 8 10 12 14 16 18 20 (+) doesn’t give large amount of negative and big positive values of C iteration (+) reduce error ε (-) numerically expensive (consumes ~4-5 times more computational resources in comparison with ordinary dynamic model) (+) gives more smooth fields of viscosity coefficient, so permits to increase model time step
The areas with restricted values are not colored − ( L H ) M 1 = − ∆ ij ij ij ∆ = − 2 T 2 2 T C A A (2 C ) A ( L H ) s 2 s 2 M M ij ij Local least-squire method Conjugate gradients method
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