west virginia university part -ii: advanced turbulence...

200

Overview of Turbulence Modelsfor Industrial Applications:

Professor Ismail B. CelikWest Virginia University

[email protected] ; (304) 293 3111

Part -II: Advanced Turbulence Modeling

201

Outline: Part-II•Introduction: laminar versus turbulent flow

•Governing equations

•Averaging techniques

•Two equation models

•Compressibility effects

•Reynolds-stress-transport models (RSTM)

•Algebraic stress models (ASM)

•Large eddy simulations (LES)

•Summary and conclusions

202

Introduction: Flow Regimes

Steady and Unsteady Laminar and Turbulent Flow

203

Introduction: What is Turbulence?

• What is turbulence?– Fluid flow occurs primarily in two regimes: laminar and

turbulent flow regimes.– Laminar flow:

• smooth, orderly flow restricted (usually) to low values ofkey parameters- Reynolds number, Grashof number,Taylor number, Richardson number.

– Turbulent flow:• fluctuating, disorderly (random) motion of fluids

204

• Turbulent fluid motion is an irregular condition of flow inwhich various quantities show a random variation with timeand space coordinates, so that statistically distinct averagevalues can be discerned. (Hinze, 1975)

• Beyond the critical values of some dimensionlessparameters (e.g. Reynolds number) the laminar flowbecomes unstable and transitions itself into a more stablebut chaotic mode called turbulence characterized byunsteady, and spatially varying (three-dimensional) randomfluctuations which enhance mixing, diffusion, entrainment,and dissipation.

Introduction: What is Turbulence?

205

Laminar Flow Examples

(Woods et al., 1988) (Van dyke, 1982)

206

Turbulent Flow Examples

(Van Dyke, 1982) (Van Dyke, 1982)

207

Turbulence Scales

• Velocity (fluctuations): u• Length (eddy size): • Time, τ = /u• Turbulence Reynolds

number– Ret = u /ν

• Turbulent kinetic energy: k~ 3u2/2

• Dissipation rate: ε ~ u3/ • Kolmogorov scales:

– τK = (ν/ε)1/2

– K = (ν3/ε)1/4

– uK = (νε)1/4

Large eddies in a turbulentboundary layer (Tennekesand Lumley, 1992):

~ Lt = boundary layer thickness

208

Governing Equations

•Conservation of Mass:

•Conservation of Momentum (Navier-Stokes Equations):

•Conservation of Energy:

0)U(xt i

i

=ρ∂∂+

∂ρ∂

j

ij

iij

ii xx

P)UU(x

)U(t ∂

τ∂+

∂∂=ρ

∂∂+ρ

∂∂

direction)-j influx (heat xtkq

DtDP

xq

x)hU(

)h(t

jj

j

j

j

j

∂∂−=

++∂∂

−=∂ρ∂

+ρ∂∂ ΦΦΦΦ

e,temperaturT enthalpy,h ==

209

Governing Equations (Continued)

•Stress-strain relation

•Viscous dissipation

•Equation of state

; Kronecker’s delta

j

iij x

U∂∂τ=ΦΦΦΦ

ij~~j

i

i

jij U3

2xU

xU

δ⋅−

∂∂+

∂∂

µ=τ ∇

)T,(funcP ρ=

ji if 1ji if 0ij

=≠=δ

210

Stationary Turbulence

Averaging Techniques: Reynolds Averaging

Unstationary Turbulence

211

• U = + u; Notation u = u´ = fluctuating component of U(x,t)• Time average:

• Ensemble average:

• Phase Averaging:

– t = window width

∞→>==<+

tasLimit;dt)t(Ut

1UUtt

t

0

0

∆∆∆∆∆∆∆∆

∆∆∆∆

( ) LargeN;t,xUN1U

N

1ii →>=<

=

( ) +>=<−

2t

2t

d)t,x(Ut

1t,xU∆∆∆∆

∆∆∆∆∆∆∆∆ττ

Averaging Techniques: Reynolds Averaging

212

• For flows with significant variations in fluid properties,fluctuations in density and viscosity etc., can not be neglected.In such flows a density weighted average (Favre average) ismore appropriate.

– U = U + u; decomposition of U(x,t)

• Definition U = /< >, = Favre average– note: = -/< > = - /< > 0,– but = 0.

• uv = <uv> + < uv>/< > - /< >2

Averaging Techniques: Favre Averaging

213

• = + <V>; < > = ; <V>> = <V>

• <dU/dt> = d()/dt; <d(UV)/dx> = d (<UV>)/dx– average of a derivative = derivative of the average

• = 0; average of the fluctuations is zero , (not for Favreaveraging)

• <UV> = <V> + <uv> ; <uv> 0. (non linear terms!)

• Comment: Average of linear terms is the same with the averagedquantities substituted, Non-linear terms, e.g. d(UV)/dx, lead to extraterms that need to be calculated separately.

Averaging Rules

214

Two- Equation Models: Exact k- Equations

• Exact equation can be derived for turbulent kinetic energy, k,and its dissipation rate, , from Naiver-Stokes Equation.

k-Equation

;PDiffDtDk

kk ρερ −+=

; diffusiondiffusion TurbulentLaminar

; u'puuu21

xkq jjii

jk j

−−∂∂= µ

j

i

j

i

j

iji

j

itijk

xu

xu

xUuu

xUP

∂∂

∂∂ν=ε

∂∂ρ−=

∂∂τ=

j

kk x

qDiff∂∂+=

Production:

Dissipation:

215

Two- Equation Models: k- Equations

Similarly exact equations can be obtained for any quantity

nch

2/mnch

mch kuZ ll ==

(See Rodi, 1980; Menter and Scheueres, 1998, also the work book)

e.g.: : dissipation per unit turbulence kinetic energy (Specific dissipation rate);tch=()-1 or rms fluctuation voticity (enstropy)

: Const u/l = c k1/2/l=cε/k; -Equation(see Wilcox, 1993)

DissPDiff

Dt

D−+=ρ

εεεερ DissPDiff

DtD −+=

-equation:

(See workbook for details)

216

Two- Equation Models: Modeling assumptions

• Turbulent diffusion is proportional to gradient of the mean flowproperties (analogous to heat conduction qhj=-khdT/dxj)

• The principal axes of turbulent stresses and mean-strain rate Sij arealigned (Not valid for many flows)

•Small turbulent eddies are isotropic (Valid at high turbulence Reynoldsnumber)

•Turbulence phenomenon is consistent in symmetry, invariance (e.g.coordinate invariance), permutation, and physical observations(consistency and reliability)

•Turbulence phenomenon can be characterized by one velocity scaleuchk1/2, and one length scale, lch .

tijτ

ε2

3k

217

• Turbulent eddy viscosity; (Dimensional analysis)

• Diffusion fluxes

• (See e.g. Speziale 1995, Hanjalic &Launder, 1972; p. 168 Shyy et al., 1997) <add notes from Shyy et al and Ching and Shenq p.38 to work book or lecture notes)

• Valid for equilibrium, Pk ε (locally) flows at high Reynoldsnumber, see e.g. Hanjalic and Launder, 1972)

Two- Equation Models: Modeling of k- equations

εε

ενν µµ

ku

t ;k

;ku ;kc ;uc

chch

2/3

ch

2/1ch

2

tchcht

===

===

ll

l

( )j

effkkjxkq

∂∂= ΓΓΓΓ

ch2 t/CDiss εεε =

ch

k1 t

PCP εε =

218

Turbulent Kinetic Energy budget: Wake flow(After Menter & Scheuerer, 1998)

219

Two- Equation Models: Variants-1

1.150.89271.901.15 + 0.25P/ε0.09Nonequilibrium

0.71790.71791.681.42 -0.085RNG

1.31.01.921.440.09Standard

σεσkCε2Cε1CµModel

30

1

1

βηη

ηη

+

−

Table 2.0: Comparison of coefficients adopted by different models (Shyy et al., 1997)

( ) 0.015 ;38.4 ;SS2S ;S 021

ijij ==== βηε

η ΚΚΚΚ

Low-Reynolds number & near wall effects222111 CC ;C ; ffCfCC εεεεµµµ

Additional source term, S = E; also ε ε* = ε - ε0

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Example: Launder-Sharma Model (see Wilcox, 1993, and Bardina et al; 1997 for other models)

3.1 ,0.1 ,09.0C ,92.1C ,44.1C

yU2 ;

yk2

k Re;e3.01f ;1f ;ef

k21

2

2

2

t

2

0

t*2

tRe

2150

Re1/4.3 2t

2t

=====

∂∂=

∂∂=

≅=−=== −

+−

εµεε

µ

σσ

νννε

νν

νε

ΕΕΕΕ

Note: This model is asymptotically consistent near the wall.3

xy22 y~ ,y

2~k ,y~k ,oy As τνε→

221

Two- Equation Models: Variants-2• Shear-Stress-Transport (SST) effects (see Menter, 1994)

– Use standard two-equation (e.g. k-ω) model outside of the boundary layer (orshear layer), but inside use

• Ω = (dU/dy), F2 = [0,1] blending function

• Dominant Strain-rate Effects– Near stagnation regions dU/dx is large

• Pk ~ (u2 – v2)(∂U/∂x)– Kato-Launder modification:

• Curvature, Buoyancy and Rotation effects- Zero Equation Models– lmix = lm0 [ 1 + βRi ]-1; Ri = Richardson number (see later)

( ) 31.0 ;,max ; 121

11

21

≅Ω== aFa

kakaPt ωνερτ κ

[ ]i

ilmijtk x

UkSP∂∂−Ω=

322 2

122ν

222

Two- Equation Models: Variants-3• Buoyancy effects: production/destruction of k & ε is affected by buoyancy forces

– Use Boussinesq approximation: ρ => ρ + ∆ρ only in the body force term; g∆ρ = gβT

• Additional source terms in the k-ε equations

• Similarly

• Ri = Rig = -Gk/Pk ; in 2D Rig = ( g/ρ)(∂ρ/∂x)/(∂u/∂y)2

– Cε3 = 1.0 (horizontal layers); Cε3 = 0 (vertical layers)

• Ri =Rif = - (1/2) Gv2/(Pk+Gk); Flux Richardson Number, (Rodi, 1980)

– Rif = -Gk/(Pk+Gk) Horizontal layers, Rif = 0 (vertical layers); Cε3 = 0.8

– May also use Gε = Cε1Cε3max(Gk, 0)

• If Ri > 0 (∂ρ/∂y>0) stable; otherwise unstable flow

;y

g''vgG tk ∂

∂

=−= ρσ

νρφ

( )( ) ( )εεεεkGPRC1CG kki31 ++=

yg''vgG t

k ∂∂

=−= ΦΦΦΦφσ

νβφβ

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• Streamline curvature effects: modify production and destruction of k and ε byanalogy to buoyancy

– The Curvature Richardson number can be defined (for various definitionssee notes, also see Sloan et al., 1986) as:

• U= velocity tangent to the curved surface, R = radius of curvature• n = direction normal to the curved surface

– Rotating Flows: similar to curvature effects use

• W= swirl velocity, U=axial velocity, r=radial distance

( ) ( )

∂∂

∂∂=

2

2g nU

nUR

RU2Ri

( )[ ] ( )22g rU

rW2r

U rW2Ri ∂

∂+∂∂=

224

Two- Equation Models: Variants-5Two-Layer Models [2L] • Use two-eq. model away from the wall • Near the wall (Say for y/ < 0.1) use one-eq. Model Example (Chen and Patel, 1987):

08.5A ;70A ;ykRe ;cc

AReexp1yc ;k

AReexp1yc ;kc

21

t4

3

1

t1

23

t1

21

t

====

−−==

−−==

−εµµ

εε

ε

µµµµ

νκ

ε

ν

ll

ll

Comments: - Saves computer storage and time, increases robustness - Avoids solving the troublesome and weakest modeled ε-equation in the critical near wall regions.

225

Two- Equation Models: Examples

0.086 – 0.0950.2200.1750.122Round Jet0.100 – 0.1100.1430.1420.102Plane Jet0.1150.1090.1000.100Mix. Layer0.3650.3390.3080.256Far WakeExperimentalSARNGK-εεεε (SST)Flow

(After Menter and Scheurer, 1998)

Spreading rates for free shear flows: )0.5U (U widthhalf ;dxd

RateSpread2

12

1

∞=== δδ

(After Chen, et al., 1998)Table 4: Comparison of predicted spread rate for free shear layers

Table 3: Performance of two-equation models for free shear layers

226


(After Menter and Scheurer, 1998) (After Menter and Scheurer, 1998)

(After Menter and Scheurer, 1998) (After Menter and Scheurer, 1998)

227


(After Bardina et al., 1997)(After Bardina et al., 1997)

(After Bardina et al., 1997)

228

Axial Dis tance from He ad (mm)

Rad

ialD

ista

nce

from

Axi

s(m

m)

0 10 20 30 40 500

10

20

30

k-εRNG k-ε

2 VP

0 1 0 2 0 3 0 4 0 5 0A x ia l D is ta n c e fro m H e a d ( m m )

0

1 0

2 0

3 0

Rad

ialD

ista

nce

from

Axi

s(m

m)

k - εR N G k - εL o w - R e k - ε (L B )M o rs e

2 0 V P

C A = 9 0 o

Streamlines of the intake case at 90o CA Streamlines of the engine case at 180o CA

Profiles of axial velocity at 90° CA for various RANS turbulencemodels (Yavuz and Celik, 1999)

Intake Engine (Piston-Bowl)

k-k-

RNG k-RNG k-

229

Effects of Compressibility• In compressible flows significant density changes occur even if the

pressure changes are small; Dρ/dt ≠ 0; ∇ .u ≠ 0

• For shock free, non-supersonic flows the Markovin hypothesis can beused, i.e. the effect of density fluctuations on turbulence is small if

• Favre averaged equations should be used with proper account ofdilation, ∇ .u , and the second coefficient of viscosity (see e.g.Vandromme, 1995)

• The k -ε equations should be modified to account for the dilatationdissipation as a function of the Mach number (see e.g. Zeman, 1990)

• Wall functions should be modified (see e.g. Wilcox, 1993) to includedensity changes near the wall, and the Mach number effects on the log-law coefficients.

1<<′

ρρ

230

Compressibility Effects: Examples

(After Bardina et al., 1997)

Fig 5.9: Comparison of computed and measured surfacepressure and heat transfer for Mach 9.2 flow past a 40cylinder flare. (After Wilcox, 1993)

231

Compressibility Effects: Examples

(After Menter and Scheuerer, 1998)

232

Two- Equation Models: Assessment-1

• Simple, robust, and easy to apply to complex industrial flows. Norestriction other than performance and accuracy concerns.

• Eddy viscosity usually improves stability and convergence , but -equation, especially when used with low Re-corrections can causeconvergence problems.

• In general, the results can be rated as good to fair except for somecertain cases for which the model variants are calibrated.

• Transport effects are partially taken into account via k- type equations,but the history effects on Reynolds stresses are not.

233

Two- Equation Models: Assessment-2

• Deficiencies of Boussinesq Approximation (i.e. eddy viscosity models)– Principal axias of Reynolds stresses are aligned with those of mean stain

rate; not necessarily so in reality (dU/dy = 0 does not always imply

– Normal stresses are usually not well predicted; local isotropy assumptionwhich is implicitly inherent to these models is not always valid

– In general these models are not good for flows with extra rate of strains(rotation, curvature, buoyancy, secondary motion, sudden acceleration etc.)

• Remedy: Reynolds Stress Transport, Models (RSTM)– Most of these short falls can be rectified by solving for the Reynolds stresses

explicitly using appropriate transport equations

– This is also known as Second Moment Closure Models (SMCM)

0uvxy =−= ρτ

234

Turbulent scalar fluxes and variance:

• For problems involving buoyancy effects or density fluctuations (e.g.combustion, mixture fraction) turbulent fluxes , and variance (orrms fluctuations) appear in the equations. (Rodi, 1980)

( )

isotropy) local in 0 ( destruction viscous

ll

j

correlationgradient scalar-pressure

iproductionbouyancy

2i

production field-mean

j

ij

jji

transportdiffusive

illil

transportconvective

l

il

of change time rate

i

xxu

- x

p1g-

xU

ux

uu1iuxx

uU

tu

i=

=

∂∂

∂∂

+∂∂+

∂∂

−∂∂−

+∂∂−=

∂∂

+∂

∂

ϕνϕ

ρϕβ

ϕΦρϕδρ

ϕϕϕ

ϕπ

( )

jj

field mean by theproduction P

jj

transportdiffusive

2j

j

convectivetransport

j

2

j

changeof rate

2

xx2-

xu2 u

xxU

t==

∂∂

∂∂

∂∂−

∂∂−=

∂∂+

∂∂

ϕϕ ε

ϕϕΓφ

Φϕϕϕϕ

Destruction of 2

ϕju−2ϕ

235

Reynolds Stress Transport Models:Exact Equations

• Diff. ij =Diffusion (molecular + turbulent transport)

• Pij = Production

• Πij = redistribution or pressure-strain term

• εij = Dissipation (relation to dissipation rate of k , (3/2) ε δij = εij )

Pij= ; Production rate by the mean flow

k

j

k

iij x

uxu2=

∂∂

∂∂µε ; Dissipation Rate

t)( t

ij

∂τ∂ =+

k

t

k x)(

U ij

∂τ∂

Diffij + Pij +ij - ij

+ ijk

k

t

k

Cx

)(x

ij

∂τ∂

µ∂∂

jkiikjjikijk pupu uuuC δδρ +⋅+=

k

jt

k

it

xU

xU

ikjk ∂∂

τ∂∂τ +

iji

j

j

iij ps2

xu

xup =

∂∂

+∂∂=ΠΠΠΠ

Diff = ;

236

Reynolds Stress Transport Models:Assumptions

• The modeling of various terms in the RST equations is quitecomplicated and involves a deeper understanding of physics ofturbulent flows.

• A brief discussion is presented in the work book. Interested readers canfind excellent reviews in the following references: Rodi (1980), Shih(1996), Leschziner (1998), and Speziale (1998)

• Here a commonly used RSTM by Launder, Reece and Rodi (1975)namely LRR model, is given as an example. Most other models arebased on LRR model, they differ primarily in modeling of the pressurestrain term ij.

237

Reynolds Stress Transport Models:Modeled Eqs.

The LRR model (Launder-Reece-Rodi, 1975)

“In their original paper, Launder, Reece and Rodi recommend C1 = 1.5, C2 = 0.4, Cs = 0.11, C = 0.15,C1 = 1.44 and C2 = 1.90.” (Wilcox, 1993, pp. 232)

(Note here “n” is the distance normal to the surface, but P is not the pressure !)

238

Reynolds Stress Transport Models:Examples.

(Hogg, et. al., 1989)

239

Reynolds Stress Models: Assessment.

• Most rigorous of all models• Have great potential for remedying the short comings of Boussinesq

approximation without ad hoc corrections• Physically realistic predictions for flows with curved streamlines, system

rotation, stratification, sudden changes in mean strain rate, secondarymotion and anisotropy.

• The most problematic equation is still the -equation• These models are mathematically complex, numerically challenging

and computationally expensive.• Wall functions and viscous damping functions are still necessary for

wall bounded and free surface flows.• Possible remedy(a compromise): non-linear eddy viscosity models

(NLEVM) and algebraic stress models (ASM)

240

Non-linear Eddy Viscosity Models - NLEVM• Assume that the Boussinesq approximation is the first term in a series expansion of

functionals (see Wilcox, 1993; Leschziner, 1997; Speziale, 1998). Here we give as anexample the Shih et al. (1993) model. See also the work book.

( )

( ) ijklkl3

3

7ijklkl3

3

6ijmklmkllikljkljklik3

3

5kliljkjlik3

3

4

ijklkljkik2

2

3ikjkjkik2

2

2ijklklijik2

2

1ijij

S~kCSSS~

kCS32SS~

kCSSS~kC

31

~kC SS~

kCSS31SS~

kCS~kC2a

ΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩ

ΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩΩ

εεδ

εε

δεε

δεεµ

++

−++++

−+++

−+−=

ijji

ij k32

kuu

a δ−≡ ΩΩΩΩ9.0S25.132C++

=µ ( ) ( )2

21

~

3321 yk2 ;

S1000

119,15,3C,C,C

∂∂−=

+= νεε

ijijSS2kSε

=ijij2k ΩΩΩΩΩΩΩΩΩΩΩΩ

ε= ( ) ( ) 3

7654 C16,16,0,80C,C,C,C µ−=

241

Algebraic Stress Models - ASM• The traditional ASM’s can be viewed as implicit NLEVM. The most

commonly used ASM was proposed by Rodi (1976, 1980)

• Assumption: Transport of τtij is proportional to transport of k

– D(τtij)/Dt - Diff( τt

ij ) = τtij /k [ Dk/Dt - Diff ( k ) ] = τt

ij /k [Pk + G - ε ]

• Result:– aij = Fa [ Pij /ε - (2/3)P/ε δij ] + (1-c3)[ Gij/ε - (2/3) G/ε ]– Fa = (1- γ)/[c1 - 1 + (P+G)/ε]– Pij = (τt

il Uj/ xl + τtjl Ui/ xl )/ρ ; Gij = - β [gi <uj ϕ> +gj<ui ϕ>]

– Pk = Pii/2 ; Gk = Gii/2

• Typical values for the constants are(Rodi, 1980):– γ = 0.6, c1 = 1.8, c3 = 0.5– Since τt

ij appears in Pij and aij this equation needs to be solved iteratively.

242

ASM & k- model: Examples

Swirling flow; (After Sloan et al., 1986) Swirling flow; (After Sloan et al., 1986)

Fig. 1. Qualitative representation of a combined vortex.Fig. 2. Qualitative spatial distribution of the stream function as induced by a strongly swirling flow.

243


(After Sloan et al., 1986)

Fig. 14. Comparison of predicted and measured velocityprofiles for Case 4 (data from Yoon71; legend suppliedby table 18)

Fig. 15. Case 4 Fig. 16. Comparison of predicted and measured tangential velocity profiles for case 4 (data from Yoon71; legendsupplied by table 18).

Fig. 3: Cases 3-5Fig. 3. Case 6

244


(After Wilcox, 1993)

(After Chen et al., 1998)

245

NLEUM & k- model: Examples

(After Apsley et al., 1998)(After Apsley et al., 1998)

Fig. 1. Plane Channel flow: comparison of solutions withdifferent models against DNS data ok Kim et al., (1987);(a) u2; (b) v2; (c) -uv

Fig. 4. High-lift aerofoil: mean-velocity and Reynolds-stress profilesat 82.5% chord; (a) U; (b) -uv; (c) u2; (d) v2.

246

NLEUM & RSTM: Examples

(After 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; © u2; (d) v2

247

NLEUM & RSTM: Examples

(Ref., Apsley et al., 1998)

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

248

Algebraic Stress Models: Assessment

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

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

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

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

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

249

Influence of Inlet Conditions

(After Sloan, et al., 1986)

(After 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).

250

Initial and Boundary ConditionsInlet: Prescribe all unknowns from experiments

example: U, V, W, k, etc.

If k, are not available from experiments:

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

23k

inlet

rms2inlet ==

1.0C diameter; Hydraulic D;DC ;Uhh

3rms ≅==≅ε εεll

or Let ( ) model) -(k kC 10 - 10

t

232t εν=ενν≅λ 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.

251

Numerical Issues: Iteration Convergence

• The CFD solution methodology is usually iterative;

• φn+1 = [A]φn +S ; n= number of iterations– Erorr = abs(φφφφexact -φφφφn) ≈≈≈≈ abs[(φφφφn+1- φφφφn )/(1 -λλλλmax)]<====εεεε ;

– 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 tomonitor overall convergence of profiles over many iterations

• The solution must be fully converged befor any assessment is made– (see Ferziger, 1989 for details)

252

Numerical Issues: Grid Convergence

• Numerical solutions use finite elements or volumes (cells), called girdor mesh to discretize the continuum equations (PDE’s), to obtaindifference 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)

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

implemented.

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

• Make sure that grid convergence and iterative convergence areachieved or characterized. Note that convergence of turbulencequantities are much more difficult.

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

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

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Large Eddy Simulation: Introduction

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Large Eddy Simulation

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Large Eddy Simulation: Filtered Equations

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LES Examples (channel Flow)

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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)

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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 capturedand statistics can be obtained via post-processing

• Require very accurate numerical schemes, at least 4th order in timeand 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 arenecessary. The higher the Re the smaller is the scales, hence thelarger the computational cost and time.

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

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LES and DNS Examples

Fig. 3: Longitudinal hairpin vortices strained behind a backward-facing step; 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)


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

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LES and DNS Examples

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


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

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Summary

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

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

• The boundary conditions, consistency checks and possible pitfallsparticularly w.r.t numerical issues are presented as guidance to modelimplementation.

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

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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 methodis the one that is validated for a similar problem being solved!

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

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

• Best use for CFD is trend analysis and hence reduction in prototpelaboratory testing in design improvements and in new design concepts.

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

west virginia university part -ii: advanced turbulence...

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