west virginia university part -ii: advanced turbulence...
TRANSCRIPT
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> + 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 = <U>/< >, = Favre average– note: <u> = -< u>/< > = - < u>/< > 0,– but < u> = 0.
• uv = <uv> + < uv>/< > - < u
v>/< >2
Averaging Techniques: Favre Averaging
213
• < U + V > = <U> + <V>; < <U> > = <U>; <U><V>> = <U><V>
• <dU/dt> = d(<U>)/dt; <d(UV)/dx> = d (<UV>)/dx– average of a derivative = derivative of the average
• <u> = 0; average of the fluctuations is zero , (not for Favreaveraging)
• <UV> = <U><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
220
Two- Equation Models: Variants-1
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
223
Two- Equation Models: Variants-4
• 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
Two- Equation Models: Examples
(After Menter and Scheurer, 1998) (After Menter and Scheurer, 1998)
(After Menter and Scheurer, 1998) (After Menter and Scheurer, 1998)
227
Two- Equation Models: Examples
(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
ASM & k- model: Examples
(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
ASM & k- model: Examples
(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)
(Zang 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
(Zang et al., 1993; Galperin and Orszag, editors)
……… 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!