Cds 13 Instruct 2

8/13/2019 Cds 13 Instruct 2

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Outline: Part-IIIntroduction: 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


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Introduction: Flow Regimes

Steady and Unsteady Laminar and Turbulent Flow


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Introduction: What is Turbulence?

What is turbulence?

Fluid flow occurs primarily in two regimes: laminar andturbulentflow regimes.

Laminar flow:

smooth, orderly flow restricted (usually) to low values of

key parameters- Reynolds number, Grashof number,

Taylor number, Richardson number.

Turbulent flow:

fluctuating, disorderly (random) motion of fluids


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Turbulent fluid motion is an irregular condition of flow in

which various quantities show a random variation with time

and space coordinates, so that statistically distinct averagevalues can be discerned. (Hinze, 1975)

Beyond the critical values of some dimensionless

parameters (e.g. Reynolds number) the laminar flow

becomes unstable and transitions itself into a more stable

but chaotic mode called turbulencecharacterized by

unsteady, and spatially varying (three-dimensional) randomfluctuations which enhance mixing, diffusion, entrainment,

and dissipation.

Introduction: What is Turbulence?


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Laminar Flow Examples

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


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Turbulence Scales

Velocity (fluctuations): u Length (eddy size):

Time, = /u

Turbulence Reynolds

number Ret= u /

Turbulent kinetic energy:

k~ 3u2/2

Dissipation rate: ~ u

3

/ Kolmogorov scales:

K= (/)1/2

K= (3/)1/4

uK

= ()1/4

Large eddies in a turbulent

boundary layer (Tennekes

and Lumley, 1992):

~ Lt= boundary layer thickness


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Governing Equations

Conservation of Mass:

Conservation of Momentum (Navier-Stokes Equations):

Conservation of Energy:

0)U(xt

i

i

=

+

j

ij

i

ij

i

i

xx

P)UU(

x

)U(

t

+

=

+

direction)-jinflux(heatx

tkq

Dt

DP

x

q

x

)hU()h(

t

j

j

j

j

j

j

=

++

=

+

e,temperaturTenthalpy,h ==


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Governing Equations (Continued)

Stress-strain relation

Viscous dissipation

Equation of state

; Kroneckers delta

j

iij

x

U

=

ij

~~j

i

i

j

ij U3

2

x

U

x

U

+

=

)T,(funcP =

{ jiif1 jiif0ij ==


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U = + u; Notation u = u = fluctuating component of U(x,t)

Time average:

Ensemble average:

Phase Averaging:

t = window width

>====, { } = Favre average

note: = -/< > = - /< > 0,

but = 0.

{uv} = + < uv>/ - /< >2

Averaging Techniques:Favre Averaging

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 = + ; < > = ; > =

= d()/dt; = d ()/dx average of a derivative = derivative of the average

= 0; average of the fluctuations is zero , (not for Favre

averaging)

= + ; 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 extra

terms that need to be calculated separately.

Averaging Rules


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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

;PDiffDt

Dkkk +=

;diffusiondiffusion

TurbulentLaminar

;u'puuu2

1

x

kq

jjiij

kj

=

j

i

j

i

j

i

ji

j

it

ijk

x

u

x

u

x

U

uux

U

P

=

=

=

j

kk

x

qDiff

+=

Production:

Dissipation:


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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 Sijare

aligned (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 scale

uchk1/2, and one length scale, lch .

t

ij

2

3

k


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Turbulent eddy viscosity; (Dimensional analysis)

Diffusion fluxes

(See e.g. Speziale 1995, Hanjalic &Launder, 1972; p. 168 Shyy et al., 1997) + only in the body force term;

g= gT

Additional source terms in the k-equations

Similarly

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

C3= 1.0 (horizontal layers); C3= 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); C3= 0.8

May also use G= C1C3max(Gk, 0)

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

;y

g''vgG tk

==

( )( ) )

kGPRC1CG kki31 ++=

yg''vgGt

k

==


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Two- Equation Models:Variants-4

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

The Curvature Richardson number can be defined (for various definitions

see 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

n

U

n

UR

R

U2Ri

( )[ ] ( )2

2g rU

rW2

rU

rW2Ri

+

=

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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;yk

Re;cc

AReexp1yc;k

ARe

exp1yc;kc

21

t4

3

1

t1

23

t1

21

t

====

==

==

ll

ll

Comments: - Saves computer storage and time, increases robustness

- Avoids solving the troublesome and weakest modeled -equation in the critical near wall regions.


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Two- Equation Models:Examples

0.086 0.0950.2200.1750.122Round Jet

0.100 0.1100.1430.1420.102Plane Jet

0.1150.1090.1000.100Mix. Layer0.3650.3390.3080.256Far Wake

ExperimentalSARNGK-(SST)Flow

(After Menter and Scheurer, 1998)

Spreading rates for free shear flows: )0.5U(Uwidthhalf;dx

d

RateSpread 212

1

===

(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


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

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


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(After Bardina et al., 1997)




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Effects of Compressibility

In compressible flows significant density changes occur even if the

pressure changes are small; D/dt 0; .u0

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 of

dilation, .u, and the second coefficient of viscosity (see e.g.Vandromme, 1995)

The k -equations should be modified to account for the dilatation

dissipation as a function of the Mach number (see e.g. Zeman, 1990)

Wall functions should be modified (see e.g. Wilcox, 1993) to include

density changes near the wall, and the Mach number effects on the log-

law coefficients.

1


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Compressibility Effects:Examples


Fig 5.9: Comparison of computed and measured surface

pressure and heat transfer for Mach 9.2 flow past a 40

cylinder flare. (After Wilcox, 1993)


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Compressibility Effects:Examples

(After Menter and Scheuerer, 1998)


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Two- Equation Models: Assessment-1

Simple, robust, and easy to apply to complex industrial flows. No

restriction other than performance and accuracy concerns.

Eddy viscosity usually improves stability and convergence , but-

equation, especially when used with low Re-corrections can cause

convergence problems.

In general, the results can be rated as good to fair except for some

certain 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.


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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 assumption

which 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 stressesexplicitly using appropriate transport equations

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

0uvxy ==


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Turbulent scalar fluxes and variance:

For problems involving buoyancy effects or density fluctuations (e.g.

combustion, mixture fraction) turbulent fluxes , and variance (or

rms fluctuations) appear in the equations. (Rodi, 1980)

( )

isotropy)localin0(destructionviscous

ll

j

correlationgradientscalar-pressure

i

productionbouyancy

2

i

productionfield-mean

j

ij

j

ji

transportdiffusive

illi

l

transportconvective

l

il

of changetime rate

i

xx

u-

xp

1g-

x

Uu

xuu

1iu

xx

uU

t

u

i=

=

+

+

+

=

+

( )

jj

fieldmeanby the

productionP

j

j

transportdiffusive

2j

j

convective

transport

j

2

j

changeofrate

2

xx2-

xu2u

xxU

t

==

=

+

Destruction of 2

ju

2


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Reynolds Stress Transport Models:Exact Equations

Diff. ij=Diffusion (molecular + turbulent transport)

Pij= Production

ij= redistribution or pressure-strain term

ij = Dissipation (relation to dissipation rate of k , (3/2) ij= ij)

Pij= ; Production rate by the mean flow

k

j

k

iij

x

u

x

u2=

; Dissipation Rate

t

)(t

ij

=+k

t

kx

)(U ij

Diffij+ Pij+ij-ij

+ ijk

k

t

k

Cx

)(

x

ij

jkiikjjikijk pupuuuuC ++=

k

jt

k

it

x

U

x

Uikjk

+

ij

i

j

j

iij ps2

x

u

x

up =

+

=

Diff = ;


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Reynolds Stress Transport Models:Examples.

(Hogg, et. al., 1989)


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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, systemrotation, stratification, sudden changes in mean strain rate, secondary

motion 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)


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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 an

example 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~k

CSSS~k

CS3

2SS~

kCSSS~

kC

3

1~k

CSS~k

CSS3

1SS~

kCS~

kC2a

++

++++

+++

+=

ij

ji

ij k3

2

k

uua

9.0S25.1

32C

++= ( ) ( )

2

21

~

3321y

k2;

S1000

119,15,3C,C,C

=

+=

ijijSS2k

S

=ijij

2k

= ( ) ( )3

7654 C16,16,0,80C,C,C,C =


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Algebraic Stress Models - ASM

The traditional ASMs can be viewed as implicit NLEVM. The most

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

Assumption: Transport of tijis proportional to transport of k

D(tij)/Dt - Diff( tij) = tij/k [ Dk/Dt - Diff ( k ) ] = tij/k [Pk+ G - ]

Result: aij= Fa[ Pij/ - (2/3)P/ij ] + (1-c3)[ Gij/ - (2/3) G/]

Fa= (1- )/[c1- 1 + (P+G)/]

Pij= (tilUj/ xl +

tjlUi/ xl )/; Gij= - [gi +gj]

Pk

= Pii

/2 ; Gk

= Gii

/2

Typical values for the constants are(Rodi, 1980):

= 0.6, c1= 1.8, c3= 0.5

Since tijappears in Pijand aij this equation needs to be solved iteratively.


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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.


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(After Sloan et al., 1986)

Fig. 14. Comparison of predicted and measured velocity

profiles for Case 4 (data from Yoon71; legend supplied

by table 18)

Fig. 15. Case 4 Fig. 16. Comparison of predicted and measured tangentialvelocity profiles for case 4 (data from Yoon71; legend

supplied by table 18).

Fig. 3: Cases 3-5

Fig. 3. Case 6


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(After Wilcox, 1993)

(After Chen et al., 1998)


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NLEUM & k-model:Examples

(After Apsley et al., 1998)


Fig. 1. Plane Channel flow: comparison of solutions with

different 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 profiles

at 82.5% chord; (a) U; (b) -uv; (c) u2; (d) v2.


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NLEUM & RSTM:Examples


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


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NLEUM & RSTM:Examples

(Ref., Apsley et al., 1998)

Figure 6: Plane asymmetric diffuser: development of the mean velocity

profile along the diffuser


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Algebraic Stress Models:Assessment

Have the potential of including the extra strain effects, as well as

anisotropy at some cost less than that of RSTMs 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


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Influence of Inlet Conditions

(After Sloan, et al., 1986)

(After Hogg, et al., 1989)

Fig. 31. Comparison of predicted and measured centerline

axial velocity profiles for Case 7 based on various inlet

conditions (data from Vu and Gouldin30; legend supplied

by Table 19).


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Initial and Boundary ConditionsInlet: Prescribe all unknowns from experiments

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

If k,are not available from experiments:

( ) assumed)orgivenintensitye(turbulencU

uuT;TuU

2

3k

inlet

rms2

inlet ==

1.0Cdiameter;HydraulicD;DC;U

hh

3

rms == ll

or Let ( ) model)-(kkC10-10

t

232t

=

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

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

Symmetry Axis: Zero derivatives normal to the axis.


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Numerical Issues:Iteration Convergence

The CFD solution methodology is usually iterative;

n+1= [A]n+S ; n= number of iterations

Erorr = abs(exact-n) abs[(n+1- n)/(1 -max)]


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Numerical Issues:Grid Convergence

Numerical solutions use finite elements or volumes (cells), called gird

or mesh to discretize the continuum equations (PDEs), to obtain

difference equations (FDEs).

Discretization error = (exact sol. to PDE) - (sol.to FDE)

= exact- num;

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

As h ==> 0, num ==> exact 1st order method: Eh(h- 2h)

2nd order method: Eh(h- 2h)/3

Ehmust 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 through

the sub-layer)

Make sure that grid convergence and iterative convergence are

achieved or characterized. Note thatconvergence of turbulencequantities 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


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Large Eddy Simulation: Introduction


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Large Eddy Simulation


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Large Eddy Simulation: Filtered Equations

LES Examples (channel Flow)


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LES Examples


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

Absolute value of the velocity vectors at 1050crank 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 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 solutionsexist for low Re, simple flows which can be used to bench mark

turbulence models and even experiments!


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LES and DNS Examples


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LES and DNS Examples

Fig. 7: Plane-averaged shear Reynolds stressFig. 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-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. Anassessment 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.


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Concluding Remarks

CFD is still not a mature area which can be used an ordinary software

such asword 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, andgrid convergence errorsmust 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.

Best use for CFD is trend analysis and hence reduction in prototpe

laboratory testing in design improvements and in new design concepts. Some minimal background in the area of fluid mechanics, numerical

methods for partial differential equations, and turbulence, is essential!

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