100

Overview of Turbulence Modelsfor Industrial Applications:

Professor Ismail B. CelikWest Virginia University

icelik@wvu.edu ; (304) 293 3111

Part-I: Introduction to Turbulence Modeling

101

Outline for Part-I

• Introduction to turbulence; laminar v.s. turbulent flows• Equations of motion and energy• Averaging techniques• The closure problem of turbulence• Mixing length and eddy-viscosity models• One-equation models & k-ε model• Summary• Introduction to Part-II: Advanced Turbulence Models

102

Introduction

• Fluid flow, heat&mass transfer form the life supportsystems for humans and their ecologicalenvironment.

• These systems are governed by principles of mass,momentum, and energy conservation.

• Transport equations are mathematical modelsdescribing the motion &properties of fluids.

103

Introduction (continued)

• 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

104

Introduction (continued)

Steady and Unsteady Laminar and Turbulent Flow

105

Introduction (continued)• Characterization of Turbulence:

– Irregular (disorderly or “random”)– Transient (always unsteady)– Three-dimensional (spatially varying in 3D)– Diffusive: enhances mixing and entrainment– Dissipates kinetic energy into heat– Occur at large Reynolds numbers

106

Introduction ( continued)

• What is turbulence (continued)?– Beyond the critical values of some dimensionless

parameters (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)random fluctuations which enhance mixing, diffusion,entrainment, and dissipation.

107

Introduction (Continued)

• Energy Cascade Concept:• Fluctuations are sustained by vortex stretching under the

action of shear.

• Large eddies are broken into smaller and smaller ones.Smaller ones feed on larger eddies.

• The smallest eddies ( Kolmogorov scale) dissipateenergy to heat by the action of molecular viscosity.

•

108

Laminar Flow Examples

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

Pipe Flow Re = 1.6 x 103 Flow past a circularcylinder Re = 41.0

109

Turbulent Flow Examples

• (After Van Dyke, 1982) (After Van Dyke, 1982)

Turbulent water jetHomogeneous turbulence

behind a grid

110

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

111

Introduction: Why turbulence modeling?

• Memory and timerequirements for DirectNumerical Simulations(DNS) is prohibitive

• Industrial applicationsinvolving complexgeometries, multiphases,and reactions are impossibleto simulate

• Remedy: Solve timeaveraged equations andmodel the turbulencestatistics.

ReH Reτ NDNS Timestep

12,300 360 6.7.106 32,000

30,800 800 4.0.107 47,000

61,600 1,450 1.5.108 63,000

230,000 4,650 2.1.109 114,000

Grid points and time steprequirements for DNS of

channel flow(After Wilcox, 1993)

112

Why turbulence modeling (Continued)?

• Direct Numerical Simulation (DNS) is possible but limited to lowReynolds numbers, and simple geometry&flows.

• The details of unsteady, 3-dimensional effects of turbulence arenot required for design purpose.

• Time averaged or space averaged (filtered) quantities areappropriate and cost effective

• Averaging leads to additional terms, which require semi-empirical models!

113

Transport Equations: Conserved scalar

• Control volume balance: time rate of change = net flux through the surfaces

( ) ( ) ( ) ( ) sin(k)sourceJJJJJJt

zyxbottopsnwe −+−+−+−=

∂∂ ∆∆∆∆∆∆∆∆∆∆∆∆φρ

.)Vol( ∆∆∆∆φρ Je

Jn

Jw

Js

∆ y

∆ x

( ) xJx

JJ wwe ∆∆∆∆∂∂+=

Similarly for Jn, Js, Jtop, Jbot

∂∂−=

xArmJ e

φφ

Convection + Diffusionrateflowmass ;UAm ρ=

( ) ( ) ( )sourcenet

S diffusionconvention

u

Change ofRate Timet φφφρρφ +∇⋅∇=⋅∇+∂

∂

ΓΓΓΓ

114

Governing Equations: Conservation of Mass

• Lagrangian frame of reference: follow a deforming volumecontaining fixed mass.– D(m)/dt = 0 = D( ρ Vol )/dt = 0– VolD ρ/Dt + ρDVol/dt = 0 D()/Dt = material derivative– D( )/dt = d( )/dt + u .∇ ( ); D(Vol)/Dt = ∇ . (u)

• Incompressible fluids: ∇ .(U) =0 <=> D(ρ)/dt =0

Set =1 in the general transport equation to obtainConservation of mass or continuity equation

( ) UDtDUU

tU

t⋅∇ρ+ρ=∇⋅ρ+

ρ∇⋅+∂ρ∂=ρ⋅∇=

∂ρ∂

115

Navier Stokes Equations

• Navier-Stokes equations express the balance of momentumover a control volume

• Momentum is a vector quantity, mv, and hence has threecomponents.

• CV balance: time rate of change = forces (normal and shearforces) acting on the surface + convected momentum

• e.g.

(see work book for a detailed derivation)

( ) ( ) ( ) xUmx

UmUm westeast ∆∆∆∆

∂∂+=

116

Navier Stokes Equations

Cartesian Coordinates, (x,y,z); (U,V,W)x-Momentum Equation

Tensor Notation

( ) ( ) ( ) ( )

IVIIIIIIIIIIzyxx

Pz

WUyVU

xUU

tU

zyx

xzxyxx

∂∂+

∂∂

+∂∂+

∂∂−=

∂∂+

∂∂+

∂∂+

∂∂ τττρρρρ

I=Time rate of Change II=ConvectionIII=Pressure Gradient IV= Viscous diffusion

( ) ( )j

ij

ij

iji

xxP

xUU

tU

∂∂

+∂∂−=

∂∂

+∂

∂ τρρVector Notation( ) ( )

~~~

~ PUUt

Uτρ

ρΘΘΘΘ∇+∇−=⋅∇+

∂

∂

117

Energy Equation

( ) ( )

eTemperatur T ,enthalpy h ,functionndissipatioViscous

);- direction j in fluxt Hea( xTkq;dTCdh

xPU

tP

x)q(

xhU

th

jjp

jj

j

j

j

j

===∂∂−==

∂∂+

∂∂++

∂∂

−=∂

∂+

∂∂

ΦΦΦΦ

ΦΦΦΦρρ

ij~ijij

j

iij

U32S2

xu

δ⋅∇−µ=τ

∂∂τ=ΦΦΦΦ

(does not include pressure)

( ) functiondelta s' Kronecker ji when0 j;i when1ij =≠===δ

118

Stress Tensor• Newtonian Fluids

[ ]

=

zzzyzx

yzyyyx

xzxyxx

ij

τττττττττ

τ : Symmetric 2nd Order Tensor

Normal Stresses Shear Stresses

(ij = ji, e.g. xy = yx)

∇−∂∂+

∂∂µ=τ=τ∇µ−

∂∂µ=τ

∇−∂∂+

∂∂µ=τ=τ∇µ−

∂∂µ=τ

∇−∂∂+

∂∂µ=τ=τ∇µ−

∂∂µ=τ

~~xzzx~zz

~~zyyzx~yy

~~xyyx~xx

U.xW

zU U.

32

xW2

U.yW

zV U.

32

xV2

U.xV

yU U.

32

xU2

119

Equations for Incompressible Fluids/Flows

• Equation of Continuity

• Momentum equation

• Thermal energy equation cp , cv = constant

( ) ( ) Tj

iij

jh

jpj

jp x

UxTk

xTc.U

xTc

tΦΦΦΦ+

∂∂+

∂∂

∂∂=

∂∂+

∂∂ τρρ

∂∂

+∂∂=

i

j

j

iij x

UxU

21s

0U ;0xU

~i

i =⋅∇=∂∂

( ) ( ) ( )i

j

ij

ij

jii gx

S2xP

xUU

tU ρ+

∂µ∂

+∂∂−=

∂ρ∂

+∂ρ∂

120

Stationary Turbulence

Averaging Techniques: Reynolds Averaging

Unstationary Turbulence

121

Averaging Techniques: Reynolds Averaging

• 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

122

Averaging Techniques: Favre averaging

• For flows with significant variations in flow properties,fluctuations in density and viscosity etc. can not be neglected.For these a density weighted average (Favre average) is moreappropriate.– 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

123

Averaging Rules: Averaging

• < 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.

124

Reynolds Averaged Equations:

• incompressible fluids with constant properties

0xu;0

xU

i

i

i

i ==∂∂

∂∂

Note: T=<T>+; Drop < > when not necessary

( ) ( ) ( ) ( )

StressesReynolds;

;uux

Sxx

PUU

xU

t jij

ijji

jij

i−+−=+ ρ∂∂

µ∂∂

∂∂

ρ∂∂

ρ∂∂

( ) ( ) ( )ϕρ∂∂

∂∂τ

∂∂τ

∂∂

∂∂ρ∂ρ

∂∂

jpjj

itij

i

iij

jT

jpj

jp uc

xxU

xU

xTk

xTcU

xTc

t−++

=

∂+

( )

fluxesTurbulent ;uq scaler; conserved

;Suxx

Ux

ii

iii

ii

φ

φρ∂∂

∂∂ρ

∂∂

==

+

−=

ΦΦΦΦ

ΦΦΦΦΓΓΓΓΦΦΦΦ ΦΦΦΦ

jitij uuρτ −=

125

Typical shear flows: Mean flow

Fully developed laminar and turbulentflow in a channel (ref ?)

Experimental turbulent-boundarylayer velocity profiles for variouspressure gradients (ref ?)

126

Typical shear flows: Mean flow

Structure of turbulent flow in a pipe (a) Shear stress (b) Average velocity

127

Typical shear flows: Mean flow

Universal plot of turbulent velocityprofiles in zero pressure gradientAfter Hoffmann and Perry[11]

Universal plot of turbulent temperatureprofiles in zero pressure gradient. AfterHoffmann and Perry [11]

128

Typical shear flows: Mean flow

Comparison of Spalding’s inner-law expressionwith the pipe-flow data of indgren (1965) (ref ?)

Mean-temperature distribution across the layeras a function of molecular Prandtl number

(ref ?)

129

Typical shear flows: Mean flow

Experimental rough-pipe velocity profiles, showing thedownward shift ∆B of the logarithmic overlap layer

Composite plot of the profile-shift parameter ∆B(k+) forvarious roughness geometries, as complied by Clauser (1956)

Boundary layer velocity profiles for rough walls Notation v* = U*=U

5.8)kyln(1U ;60k for )k3.01ln(1B

)k(BByln1U

sss

s

+=>+≅

−+=

+++

+++

κκ

κ

∆∆∆∆

∆∆∆∆ ; U+ = U/U*; y* = yU*/

130

Fluctuating Velocities in a boundary Layer

(After White, 1991)

131

Classical Models: Assumptions

• Turbulent eddies behave like mixing agents and usuallyincrease mixing, hence

– effective diffusivity = molecular (or laminar) + turbulent– (Bousinesq’s eddy viscosity concept)

• Dimensional analysis:t = lchuch ; t = t/ ; not a fluid property

• Analogy to laminar flows: Simple shear flows = (dU/dy) ; -<uv> = t ( dU/dy)

132

Classical Models: Assumptions

• In general:

ijijijlij Pu

32S2 δδµµτ −⋅∇−=

( ) ijijtijtjitij k

32u

32S2uu δρδµµρτ −⋅∇−=−=

+=

i

j

j

iij x

UxU

21S

∂∂

∂∂

; rate of deformation (strain) tensor

=≠

=ji when,1ji when,0

ijδ ; Kronecker’s delta

; Laminar Stresses

; turbulent stresses

( ) energy kinetict turbulen;wvu21uu

21k 222

ii ++==

133

Zero Equation models: Mixing length models

• Prandtl’s Mixing length model:

uch = lmix (dU/dy)

• von Karman uch = lmix (dU/dy) ; lmix = (dU/dy) / (d2U/dy2)]

– lmix = y for y < yo ; (inner region)lmix = for y > yo ; (outer region) ; = shear layer thickness, 0.20

• Combined: lmix = 0.085 tanh ( y / 0.085 )

134

Zero Equation models: Mixing length models

wall free flows: lmix =

Mixing length constants for free shear flows (Wilcox, 1993)

Flow Type: Far Wake Plane Jet Round Jet Plane Mixing Layer

lmix/δ 0.180 0.098 0.080 0.071

135

Zero Eq. Models: Cebeci-Smith Model• Inner layer: y < yo

t = lmix2 (dU/dy)

lmix = fd y ; fd = [1 - exp (-y+ / A+)] van Driest damping function• A+ = 26 func( P/x)

• Outer layer: y > yot = * Ue Fk ;

* = displacement thicknessFk = 0.5 (1 - erf(y / - 0.78) : Klebanoff intermittency factor

– or Fk = [1 + 5.5(y / )6 ]-1

– Compressibility effects: include ( / w), ( / w)– Buoyancy, pressure gradient, Heat and Mass transfer (see notes, Rodi pp.

17-20, see also p. 50 of Wilcox)

Comment: Hard to compute *, can not be generalized to 3D

136

Eddy Viscosity Distribution

*uuZ

2ZZ1ee

2zB

t νκ=κ=

−−−µκ=µ +κ−

016.0C *UC eouter,t ≈δ≈µ ρ

δ+

δρ≈µ 6e

outer,ty5.51

*U016.0

137

Zero Eq. Models: Baldwin-Lowmax model

• Inner layer: y < yot = lmix abs () ; = local vorticity = (x

2 + y2 + z

2)1/2 = (2ij ij)1/2;

lmix = fd y ; fd = [1- exp(-y+ / A+)]

• Outer layer: y > yo t = Ccp Fw Fk (y ; ymax / Ck) ; Ccp = 1.6 , Ck = 0.3Fk = 0.5 [1 - erf(y/ - 0.78) ; Klebanoff intermittency factorFw = min(ymax Fmax ; Cwk ymax Udif /Fmax ] ; Cwk = 1.0F(y) = y abs() Fd

Ccp = 1.6, Ck = 0.3, Cwk = 1.0

Comment: No need to compute *; can be used for 3D flows

−=

i

j

j

iij x

UxU

21

∂∂

∂∂ΩΩΩΩ Vorticity tensor

138

Zero Eq. Models: examples

Comparison of computed and measured channel-flow properties, ReH=13750

_________Baldwin-Lomax; ---------Cebeci-Smith; 0 Mansour et al. (DNS); Halleen-Johnston Correlation

(After Wilcox, 1993)

139

Zero Eq. Models: assessment

Comparison of computed and measured boundary layer velocity profiles and shape factor for nonzero pressuregradient; Cebeci-Smith model (After Wilcox, 1993)

140

Zero Eq. Models: examples

Computed and measured flow properties for Driver’s separated flow;____ Baldwin-Lomax; o Driver (After Wilcox, 1993)

141

Zero Eq. Models: examples

Velocity profiles and pressure distribution Measurements and calculations (Cebeci-Smith model),------basic model; with curvature effects; __________ with low-Rθeffects, ……. Withextrapolated pressure distribution (After Celik and Patel, 1984):

142

Zero Eq. Models: assessment

• Mathematically very simple, numerically robust

• Good for 2D boundary-layer type flows without separation

• Acceptable results even with separation and non-equilibrium flows (e.g. one and1/2 Eq. Model of Johnson-King)

• Calculation of the shear-layer thickness is not trivial

• Extension to 3D flows with complex geometry is difficult.

• Not good for complex flow effects such as curvature, history effects.

• Comment: These models will work well only for the cases for which they havebeen fine tuned (note a different constant for each flow in Table ??).

143

One Eq. models:

• A transport equation is solved for either the length scale, lch,eddy viscosity, vt, the turbulent kinetic energy, k, or a any otherrelated variable

• Generic scalar transport equation in a turbulent flow field.

( ) ( ) ( ) φφφφρ∂∂

∂ρφ∂φ

∂∂

∂∂φρφρ DissPDiffU

xtxU

tDtD

jjj

j −+=+=

+=

==

jeff

j xxDiffDiffusion

∂∂φ

∂∂

φ ΓΓΓΓ

== prod1

jch1 t/c ;

xuc ;ExactPn Productio; ρφ

∂∂φρφ

diss2 t/cDissnDissipatio ρφφ ==

t

tturblameff σ

ν+=+= lamΓΓΓΓΓΓΓΓΓΓΓΓΓΓΓΓ

144

One Eq. models:

Menter Model

= t ; tprod = [2 Sij(Sij + ij)]-1/2; tdiss = t / l

= 2

2

yU

yU

∂∂

∂∂

l Von Karman length scale

Badwin-Bart Model

= t; model); Menter in as (same Pt=υ

=

j

t

j

t2 xx

cDisst ∂

∂υ∂∂υρυ

Comment: no distance from the surface is required

Prandtl’s Model = k; t = k1/2 l ; Dissk = = cD k3/2 / l ; l has to be supplied

Spalart-Allmaras (1994) Model

t = c k1/2 le.g. chen & Patel (1987):

;ARe

exp1yc yl

−−=

µµl ;

ARe

exp1yc yl

−−=

εεl ll

2/1

y c2A,70A,55.2c,09.0c;ykRe ===== εµµυ(including near-wall correction)

j

iji

j

itk x

UuuxUP

ij ∂∂

∂∂τ −=−=

= t/f ; f = wall function complex two-layer model (see also Bardina et al., 1997)

145

Assessment: One-Equation Models

• Mathematically simple, robust, includes history (or upstream) effects, little numerical overhead.

• Calibrated mostly for aerodynamic boundary-layer tyre flows, or shear dominatedflows.

• The length scale relations are empirical, and not so suitable for more general,complex industrial flows.

• The troublesome -equation is not solved. This is a big advantage if length scale relations are satisfactory.

146

The Standard k-εεεε ModelA phenomenological approach

iji

j

j

it

tij

i

j

j

i

j

itij

j

ijik

jk

t

j)k(

k)k(

k32

xu

xu

xu

,xunDissipatio

xu

xuu,uPoductionPr

xk

xDiffDiffusion

PDiffDt

)k(D

δρ−

∂∂

+∂∂ρν=τ

∂∂

∂∂ν=ε=

∂∂τ=

∂∂ρ−==

∂∂

σµ+µ

∂∂==

ρε−+=ρ

k-equation

3.10.192.1C44.1C09.0Con.optimizati and s,experiment analysis,

asymptotic from determined are constants Empirical

kCViscosityEddy

scale Time k;/k

CD;/kPCP

xxDiff

DPDiffDt

)(D

k21

2

t

2k1

j

t

j)(

)(

=σ=σ===

ε=ν=

=εεε=

ε=

∂ε∂

σµ+µ

∂∂=

ρ−ρ+=ρε

εεεµ

µ

εε

εε

εε

εεε

εεεε-equation

147

Examples: k-εεεε Model

Comparison of the k-ε model with the flat plate boundary layer data of Klebanoff (1955). [After Jones and Launder (1972).]

148

Flow over 35° swept wing; Re = 2.42 × 106. [After Johnston (1987).]

Note: experiments indicate separated flow. All models predict attached flow.

Examples: k-εεεε Model

149

Examples: k-εεεε Model

Flow past a backward-facing step. [After Goldberg (1987).]

150

Examples: k-εεεε Model

Flow in sudden expansion of a pipe. [After Celik, et al (1987)]

151

Assessment of the Standard k-εεεε Model

•Robust, easy to apply, the most commonly used method

•Easily extendible to complex geometries and multi-dimensions

• Fairly good results for many engineering applications,especially good for trend analysis

•Isotropic and inherently assumes “local” equilibrium,hence good for high Reynolds number flows.

•Not good for flows involving significant curvature,rotation, buoyancy, sudden acceleration, separation, andlow Re regions

152

Summary: Part-I•An introduction to fluid flow regimes, governingequations, and turbulence modeling is presented

•The zero- and one-equations models using the mixing-length or eddy-viscosity concept are reviewed.

•These models are good for calculations of boundary orthin shear-layer flows such as wakes, jets, flow overairfoils etc. with simple geometry but without separation.

•The standard k-ε model is introduced using aphenomenological approach. This model is a popularmodel, easy to use and robust, but has manyshortcomings.

•All need further improvements for accurate predictions