Introduction to Turbulence Modeling Techniques

Sourabh Apte

ME569 (Turbulence) ME667 (CFD)

Governing Equations for an Incompressible Flow

Governing Equations for a Compressible Flow

Energydensityperunitwavenumber

Wave number

DNS: simulate all

simulate

Wave number

LES

model

Turbulence Modeling Techniques

• Direct Numerical Simulation (DNS) most expensive (no modeling)

• Large Eddy Simulation (LES) expensive (modeling of unresolved scales)

• Reynolds Averaged Navier Stokes (RANS) cheaper (more equations to solve & code)

- steady

- unsteady (URANS)

Wave number

RANS: model all

Direct Numerical Simulation (DNS)

Measure all flow parameters ateach grid point and then plot theflow parameter using those gridpoints. More details captured bymore number of measurements

• In DNS one solves the complete Navier-Stokes equations on a computational grid without anyapproximations or averaging except the inherent approximations made in numerical discretizationof the governing equations.

• Obviously to “capture” all the details of the flow, your grid resolution should be fine enough tocapture the “smallest” flow structures.

Mixing layer

Problems with DNS

• To ensure that we are capturing all the salient features of the flow, the “computational domain”should be at least as large as the “physical domain” or the “largest turbulent eddy”

• What is a good measure of the largest turbulent eddy?

• What is a good measure of the smallest scale?

• How are they related?L

!~ Re

L

3/4

Integral scale &

Kolmogorov scale

Tennekes and Lumley

Number of grid points !x ~

Lx

", !

y ~

Ly

",!

z ~

Lz

"

Isotropic Turbulence Lx = Ly = Lz; ! x= !

y= !

z= !

Time Step !t ~ !

Total Number of Grid Points !3

Simulation Size Scales as !3 !t ~ ReL

3

To simulate isotropic turbulence of ReL ~ 1000 will require at least 10009/4~5.6 million grid points(based on domain of integral length scale per direction, in practice would require larger domain).

Huge computation for a simple problem.

DNS Simulations

- For Re = 20,000- would require 800 million grid points and 600,000 CPU hours

approximately 7 years on same machine - with new computing power, this can be done in 1 month But still a huge computational effort!

First DNS of turbulent channel flow: Kim, Moin, Moser (1987)

Non-homogeneous flows

- First DNS of Turbulent channel flow (1987)Re = 20002 million grid points and 200 hoursof CPU time on Cray YMP

http://www.stanford.edu/group/ctr/gallery.html

Largest simulations on fast super computers

http://www.psc.edu/science/2005/yeung/taming_the_whirlwind.pdf

Taming the Whirlwind: P.K. Yeung (Pittsburg SC): 2048^3

For Isotropic turbulence this means one can simulate only ReL ~ 26000 i.e.Taylor Microscale Reynolds number of only Reλ ~ 877

DNS is an excellent tool for turbulence research! (Moin and Mahesh 1998)

Formation of planets (Ham & Iaccrino 2006, unpublished)Center for Turbulence Research: 65000 processors 8billion+ CV

RANS

u’

u

Reynolds Averaged Navier-Stokes (RANS)

Time Averaging

Reynolds Decomposition

Some Properties

(Need ensemble averaging for non-stationary flows)

Reynolds StressUnclosed (needs modeling)

Reynolds-averaged NS

Eddy Viscosity Based Models

Simple Zero-Equation Model

Turbulent kinetic energy

Eddy viscosity

Dimensional argumentVelocity scale * Length scale

L = mixing length (function of co-ordinate)

- Specification of L is possible for very simple flows (homogeneous, mixing layers etc)

- not a good model for separated flows

One-Equation Model

Idea is to derive a transport equation for TKE and use TKE for velocity scale

Eddy viscosity

Velocity scale * Length scale

Obtained from solutionof KE equation

and L is modeled as before

Derive TKE equation as follows:

- This gives rise to additional unclosed terms which need to be modeled

One-Equation Model

transport equation for TKE

Eddy viscosity

Velocity scale * Length scale

Requires no model

requires model

requires model

requires model

Models

Scaling argument true for equilibrium flows

Two-Equation & Other Models

Idea is to derive a transport equation for TKE and use TKE for velocity scale

Eddy viscosity

Velocity scale * Length scale

Both Kinetic Energy (or velocity scale) and the Length Scale (L) are obtained bysolving TWO transport equations

- k-ε (Just as the TKE equation derive an equation for ε) L obtained by dimensional analysis

- k-ω (ω is dissipation per unit TKE)

etc…

Several other models v2-f, Reynolds stress etc. have been derived and used

Most of the models need “tuning” of parameters and constants

Large Eddy Simulation (LES)

• Certainly DNS of a complex flow at high Reynolds numbers and in complex configuration is notpossible even today (Is it possible in the next decade? Most likely!)

• Large eddy simulation is a technique where only the large-scale structures of the turbulent flow(the energetic scales) are computed accurately than the small scales.

- thus the grid resolutions are such that only the energetic eddies are resolved- effect of unresolved (or subgrid scales) are modeled through subgrid scale models.- this still is a computationally intensive technique, but much higher Reynolds numbers andmore complex flows (than DNS) can be simulated more accurately.

Note that in LES the filtered velocityfield also depends on time.Only the small scales (or the high frequency)fluctuations are filtered out and should be modeled

Different Filters

In Physical Space

In Fourier Space

Simple top hat typically used inmost LES computations

Effect of Filters on a test function

Filtering the field with a given energyspectrum affects the scalesMagnitude of “large-scale” structures is less affectedbut that of small-scale structures is affected.

LES: A Filtering Operation

Applying the Filter to Governing Equations The Closure Problem

The filtering and differentiation operators commute for most filters

The unclosed term is

Note that the equation can be thought of as written for a new variable

In this case, the correlation that is unknown needs an additional equationor modeling

Models For The Unclosed Term

The basic idea behind modeling is to try to relate the unclosed term in momentumequations to the quantities that are “solved for”

- so relate to

The other approach is to try and derive an additional transport equation for- this, however, leads to additional terms in these equations which are “unclosed”

Eddy Viscosity type models

SGS stress originatingfrom non-linear inertialterms

Viscousstress-liketerm

Remember thattwice the subgrid scale kinetic energy

Energy Transfer Mechanisms in LESResolved scale kinetic energy Sub-grid scale kinetic energy

- Advection and diffusion processes just redistribute KE- Transfer of energy from resolved to subgrid scales give terms with opposite signs in respective eqns- Dissipation terms at resolved scales appear as Production terms in the subgrid scales- Dissipation terms lead to loss of KE (transfer into internal energy)- Addition of the two equations would provide an equation equivalent to “total KE” as obtained in DNS.

dissipation

production

Smagorinsky’s SGS Model

SGS stress originating fromnon-linear inertial terms

Viscous stress-liketerm

Equilibrium Hypothesis: Production = Dissipation

Positive definite

- Cs is obtained by assuming scale-separation Cs~0.1 (Lilley)- Could be too dissipative

Length scale * velocity scale

Need models for

Dynamic Smagorinsky SGS Model

- Still Use Equilibrium Hypothesis: Production = Dissipation

- But obtain Cs Dynamically (depends on space and time)

- Predictive capability, “no tuning parameters” (user-parameter free)

- standard norm in most LES simulations

Energy

Wavenumberk1 2

k

Use thisinformation to model SGS

SGSFirst introduced at Center for Turbulence ResearchStanford University (1990)

Energy

Wall Functions- In wall bounded turbulent flows (boundary layers, channels etc..) the stress at the wall mustbe computed accurately

- With increase in Re, the viscous sublayer of the boundary layer is extremely thin

- Requires very fine grid resolution to resolve the “steep gradients” near the wall

- In addition, at the wall the velocity is zero. However, if the stress is computed directly usingthe finite-differencing and the grid resolution is not fine enough, severe errors result.

- wall models are based on the “existence” of a logarithmic velocity profile in a turbulentboundary layer

Non-dimensional velocity profile

Friction velocity and Constants

von-Karman constant Empirical constantrelated to the thicknessOf the viscous sublayer

Viscous sublayer

Without wall modeling, grid resolution should besuch that first grid point needs to be at n+~1Very fine grids limiting time-steps etc… COSTLY

Wall Functions

required grids “affordable” gridsfor high Re

Zoom in Near the Wall

Non-dimensional velocity profile

Velocity profile in this region is modeledbased on the “law of the wall”

Several versions are possible

(Kalitzin et al. 2004)

ω-5/3

ω/ωst

experiment (Ong & Wallace)

central difference

upwind biased

Need For Discrete Energy Conservation

Flow Around Cylinder @ Re=3,900

- Comparison between two upwind-biased and central differencing

- Central differencing preserves the small scales of turbulence - Upwind schemes are appropriate ONLY for smooth (RANS-type) flow fields

Detrimental Effects of Numerical (Artificial) Dissipation

Some Examples

- Isotropic turbulence, channel

- Some landmark DNS studies in late 80s

- Flow over a backward facing step

- LES in complex geometries

- Future?