301

NLEUM & RSTM: Examples

(From Apsley et al., 1998)

Fig. 5. High-lift aerofoil: streamwise normal stress in aerofoil wake.

Fig. 7. Plane asymmetric diffuser: mean-velocity and Reynolds-stress profiles in the diffuser section (A U; (b) -uv; (c) u2; (d) v2

302

NLEUM & RSTM: Examples

(Ref., Apsley et al., 1998)

Figure 6: Plane asymmetric diffuser: development of the mean velocityprofile along the diffuser

303

Algebraic Stress Models: Assessment

• Have the potential of including the extra strain effects, as well as anisotropy at some cost less than that of RSTM’s

• Mimic the physical behavior by means of mathematical artifacts and careful calibration (Apsley et al. 1997)

• They need to be modified for low-Re effects and near wall treatment similar to the two-Eq. models

• The advantages seems to be less pronounced in 3D than 2D flows.

• Recommended for problems where anisotropy and certain extra strain rate effects are known to dominate

304

Influence of Inlet Conditions

(Ref: Sloan, et al., 1986)

(Ref: Hogg, et al., 1989)

Fig. 31. Comparison of predicted and measured centerlineaxial velocity profiles for Case 7 based on various inletconditions (data from Vu and Gouldin30; legend suppliedby Table 19).

305

Initial and Boundary ConditionsInlet: Prescribe all unknowns from experiments

example: U, V, W, k, ε etc.

If k, ε are not available from experiments:

( ) assumed)or givenintensity e(turbulenc UuuT ;TuU

23

kinlet

rms2inlet ==

1.0c diameter; HydraulicD ; ; eh

3

≅=+=≅ herms DcU llε

orLet ( ) model) -(k kc 10 - 10

t

232t ενεννλ µ=≅

Outlet: Put outlet boundary away from recirculation regions and set , P = Pambient.0x =∂∂φ

Walls: Use wall functions and/or no-slip condition.

Symmetry Axis: Zero derivatives normal to the axis.

306

Numerical Issues: Iteration Convergence

• The CFD solution methodology is usually iterative;

• φn+1 = [A]φn +S ; n= number of iterations– Erorr = abs(φexact -φn) ≈ abs[(φn+1- φn )/(1 -λmax)]< ε ;

– λmax = Largest eigen value of [A]. The eigen values of [A] must be less than one for convergence

• To monitor only E = abs[(φn+1- φn )] may be misleading. Better to monitor overall convergence of profiles over many iterations

• The solution must be fully converged befor any assessment is made

– (see Ferziger, 1989 for details)

307

Numerical Issues: Grid Convergence

• Numerical solutions use finite elements or volumes (cells), called gird or mesh to discretize the continuum equations (PDE’s), to obtain difference equations (FDE’s).

• Discretization error = (exact sol. to PDE) - (sol.to FDE) = φexact - φnum ;

• let h = (∆x ∆y ∆z)1/3 , a typical cell size

– As h ==> 0, φnum ==> φexact

– 1st order method: Eh ≈ (φh - φ2h)– 2nd order method: Eh ≈ (φh - φ2h)/3 – Eh must be calculated and minimized if possible

(see e.g. Ferziger, 1989; Celik and Zhang, 1995 for details)

308

Consistency Checks• Check if the boundary conditions are reasonable and correctly

implemented.

• Check if 10 < y+ < 300 (wall functions), and y+ < 1 (integration through the sub-layer)

• Make sure that grid convergence and iterative convergence are achieved or characterized. Note that convergence of turbulence quantities are much more difficult.

• For unsteady flow calculations convergence at every time step must be ensured.

• The integral mass, momentum and energy balances must be satisfied

309

Large Eddy Simulation: Introduction

310

Large Eddy Simulation

311

Large Eddy Simulation: Filtered Equations

312

LES Examples (channel Flow)

313

Fluctuating velocity: (a) computed (440000 nodes), (b) Catania and Spessa (1996)

Absolute value of the velocity vectors at 1050 crank angle (440000 nodes).

LES Examples

(Ref: Celik et al, 1999)

314

Direct Numerical Simulation- DNS• Navier-Stokes equations are not limited to laminar flows. If they can be

solved accurately as is (DNS) turbulence fluctuations can be captured and statistics can be obtained via post-processing

• Require very accurate numerical schemes, at least 4th order in time and space, or spectral methods (e.g. Fourier, Chebychev expansions)

• Must resolve all scales of turbulence down to Kolmogorov scales. Hence very large number of grid nodes and very small time steps are necessary. The higher the Re the smaller is the scales, hence the larger the computational cost and time.

• DNS solutions are not suitable to industrial applications but solutions exist for low Re, simple flows which can be used to bench mark turbulence models and even experiments!

315

LES and DNS Examples

Fig. 3: Longitudinal hairpin vortices strained behind a backward-facingstep; simulation without subgrid model.

Fig. 5: Plane-averaged velocity, scaled Smagorinsky model

Fig. 4: Time evolution of the wall shear, scaled Smagorinsky model.

(Zang et al., 1993; Galperin and Orszag, editors)

(Lesieur et al., 1993; Galperin and Orszag, editors)

(Zang et al., 1993; Galperin and Orszag, editors)

……… Scaled Smagorinsky model.--------- RNG model______ Dynamic eddy viscosity model.∆ fine direct simulation

316

LES and DNS Examples

Fig. 7: Plane-averaged shear Reynolds stress Fig. 6: Plane-averaged rms turbulent fluctuations

(Zang et al., 1993; Galperin and Orszag, editors)

……… Scaled Smagorinsky model.--------- RNG model______ Dynamic eddy viscosity model.∆ fine direct simulation

317

Summary

• A overview of turbulence models for industrial application is presented. This included most commonly used models staring from zero-equation models to Reynolds Stress Transport models with an introduction to LES.

• The pros and cons of each model are elucidated to help the CFD users in selection of an appropriate turbulence model for their application. An assessment is made with concrete examples.

• The boundary conditions, consistency checks and possible pitfalls particularly w.r.t numerical issues are presented as guidance to model implementation.

• The users are also provided with an extensive list of references for future reading and as a source of detailed information for numerous models.

318

Concluding Remarks

• CFD is still not a mature area which can be used an ordinary software such as”word processing”!. It is somewhat of an art. The best method is the one that is validated for a similar problem being solved!

• Validation (the process of testing the performance of a model for the intended application) is the responsibility of the user. Iteration convergence, and grid convergence errors must be taken into account before reaching conclusions.

• Verification (the process of ensuring a proper implementation of a turbulence model into a code) is the responsibility of code developers but the users must be aware of it.

• Some minimal background in the area of fluid mechanics, numerical methods for partial differential equations, and turbulence, is essential!

319

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