Turbulence Modeling

Numerical Methods in AerodynamicsLecture 5: Turbulence modeling

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Turbulence Modeling

Niels N. SrensenProfessor MSO, Ph.D.

Department of Civil Engineering, Alborg University &Wind Energy Department, Ris National Laboratory

Technical University of Denmark


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Outline of lecture

Characteristics of turbulence What is the problem of modeling turbulence

Reynolds Averaging and Reynolds stresses RANS Turbulence Models

Boussinesq approximation Boundary Conditions

Log-law Low Reynolds Number Modifications

Example of RANS comp. Shortcomings of RANS models Large Eddy Simulation models

Filtering Hybrid models


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The Nature of turbulence (I)

Irregularity Turbulence is irregular or random.

Diffusivity Turbulent flows causes rapid mixing, increases heat transfer and flow

resistance. This is the single most important aspect of turbulence from a engineering point of view.

Three-dimensional vorticity fluctuations (rotational) Turbulence is rotational, and vorticity dynamics plays an important role.

Energy is transferred from large to small scale by the interaction of vortices.


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The Nature of Turbulence (II)

Dissipation Turbulent flows are always dissipative. Viscous shear stresses perform

deformation work which increases the internal energy of the fluid at the expense of kinetic energy of turbulence.

Continuum The smallest scale of turbulence are ordinary far larger than any

molecular length scale Flow feature

Turbulence is a feature of the flow not of the fluid,


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How Does Turbulence Look

The Onset of Two-Dimensional Grid Generated Turbulence in Flowing Soap Films Maarten A. Rutgers, Xiao-lun Wu, and Walter I. Goldberg


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Direct Numerical Simulation All scales of the fluid motion spatial and temporal is resolved by the

computation. Largest DNS to date 40963

Modeling Turbulent Flows


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Large Eddy Simulation (LES) Only the large scales of the fluid motion is resolved by the computations



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Reynolds Averaged Navier-Stokes (RANS) The equations are time averaged, and dont resolve the eddies

Hybrid LES/ RANS


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Derivation of the Reynolds Averaged Navier-Stokes eqns.

1) Introduce the Reynolds Decomposition of the variables

2) Insert the Reynolds Decomposition in the flow equations

3) Perform time averaging


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Reynolds Averaged Navier-Stokes equations (RANS)

Reynolds Stress


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Reynolds stresses

Performing the Reynolds Averaging Process, new terms has arisen,namely the Reynolds-stress tensor:

This brings us at the turbulent closure problem, the fact that we have more unknowns than equations. Three velocities + pressure + six Reynolds-stresses Three momentum equations + the continuity equation

To close the problem, we need additional equations to model the Reynolds-stresses


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Reynolds Averaged Momentum Equations

The Reynolds Stresses originates from the convective terms

They are normally treated together with the diffusive terms


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RANS turbulence models

Algebraic turbulence models Prandtl Mixing Length Model Cebeci-Smith Model Baldwin-Lomax Model

One equation turbulence models Spalart-Allmaras Baldwin-Barth

Two equation turbulence models k-epsilon model k-omega model k-tau model

Reynolds stress models


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Reynolds-stress models

Introduces new unknowns (22 new unknowns)



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Eddy-viscosity models Compute the Reynolds-stresses from explicit expressions of the mean

strain rate and a eddy-viscosity, the Boussinesq eddy-viscosity approximation

The k term is a normal stress and is typically treated together with the pressure term.



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Prandtls mixing length hypothesis is based on an analogy with momentum transport on a molecular level

Algebraic Turbulence Model

yU(y)

Molecular transport

Turbulent transport


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Prandtl Mixing Length Model

The mixing length model closes the equation system

The proportionality constant for the mixing velocity c1 and for the mixing length c2 needs to be specified

The equation for the turbulent eddy viscosity is a part of the flow solutions, as it depends on the mean flow gradient

Turbulence is not a fluid property but a property of the flow


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Additions to the basic mixing length model

Van Driest (1956) wall damping Clauser (1956) defect layer modification Corrsin and Kistler (1954) intermittency modification


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Baldwin-Lomax Model

Clauser Van Driest

Corrsin and Kistler


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Algebraic Models

Gives good results for simple flows, flat plate, jets and simple shear layers

Typically the algebraic models are fast and robust Needs to be calibrated for each flow type, they are not very general They are not well suited for computing flow separation Typically they need information about boundary layer properties,

and are difficult to incorporate in modern flow solvers.


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One and Two Equation Turbulence Models

The derivation is again based on the Boussinesq approximation

The mixing velocity is determined by the turbulent turbulent kinetic energy

The length scale is determined from another transport equation ex.


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Second equation


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The turbulent kinetic energy equation

By taking the trace of the Reynolds Stress equation, we get


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Dissipation of turbulent kinetic energy

The equation is derived by the following operation on the Navier-Stokes equation

The resulting equation have the following form


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The k- model

Eddy viscosity

Transport equation for turbulent kinetic energy

Transport equation for dissipation of turbulent kinetic energy

Constants for the model


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k-omega SST model


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k-omega SST model


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Blending Function F1 and F2


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Constants for k-omega SST model


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Boundary Conditions

Inflow conditions Mean flow velocities, turbulence intensity, length scale

Wall conditions Bridging the near wall region (log-law) (30 < y+ < 100) Resolving the near wall region (y+ < 2)


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Boundary Conditions, Log-Law

The flow is assumed to be a one-dimensional Couette flow, steady and with zero development in the flow direction, and with constant shear stress in the near wall region.

The momentum equations are not abandoned in the wall cell, instead the viscous stresses at the wall is substituted by the following expression derived from the log law


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The Couette flow assumption reduces the turbulent kinetic energy equation to a simple balance between production and dissipation.Zero diffusion to the wall is assumed for the turbulent kinetic energy, and the production and dissipation terms are computed from the mean flow assumption, using


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Using the logarithmic profile and the balance between productionand dissipation the following expression for dissipation can be derived, the dissipation equation is abandoned in the wall cell and the dissipation is fixed to the value given below:


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Low Reynolds Number Modification

The turbulence equations are derived under high Reynolds Number assumptions

We need to assure that the equations has the correct near wall behavior, the so called asymptotically consistent behavior


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Low Reynolds Number Modification

To obtain correct near wall behavior the two equation models areenriched with viscous damping terms


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The k- model do not need any modification to have nearly the correct near wall behavior, and is often used in the default version.

The boundary conditions are relatively simple to apply

The model is robust in the low Re version

Low RE k-omega model


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Typically the inflow turbulence intensity is known:

For aerodynamic applications where the flow is nearly laminar in the farfield we have

For cases with a wall, the eddy viscosity in the inlet region can often be specified by the mixing length hypotesis assuming a velocity profile

Inflow conditions


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Driver, D. M., "Reynolds Shear Stress Measurements in a Separated Boundary Layer," AIAA Paper 91-1787, 1991.

Performance of Popular Turbulence Models for Attached and Separated Advedrse Pressure Gradient Flows. Menter, F.R. AIAA Journal 1992 vol. 30 no. 8


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Mild adverse pressure gradient


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Strong adverse pressure gradient


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Shortcomings of the Boussinesq approximation

Flows with sudden changes in mean strain rate


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Flows over curved surfacesSo and Mellor, 1972, An Experimental Investigation of

Turbuelnt Boundary Layers Along Curved Surfaces


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Flow in ducts with secondary motion Flow in rotating and stratified fluids Three dimensional flows Flows with boundary-layer separation


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Filtering of the Navier-Stokes equations, splitting the velocities in the resolvable-scale filtered velocity and the subgrid scale (SGS) velocity

A typical filter used could be the volume-averaged box filter

Large Eddy Simulation


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LES, Filtering of the Navier-Stokes equations

Again the convective terms generate additional terms

Filtering differs from standard averaging in one important respect


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The Leonard stresses (Lij) are of the same order as the truncation error when a finite-difference scheme of second-order accuracy is used, and are normally not considered

The cross-term stress tensor (Cij) are typically modeled together with the Reynolds stresses

The first model for the subgrid scale stresses (SGS) was the model by Smagorinsky (1963) based again on gradient-diffusion process



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Smagorinsky model


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LES modeling

LES models are by nature unsteady LES models are by nature full three dimensional They resolve the large scales and only model the isotropic small

scales The standard SGS model needs damping of the eddy viscosity near

solid wall similar to the van Driest damping used for mixing length models

Resolving the anisotropic eddies in the near wall region where the cells are small may require a very fine computational mesh

LES models can be combined with approximate wall boundary conditions, or even zero, one or two equation models for the near wall region.


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Hybrid models

Hybrid models are combinations of RANS and LES models One example is zonal models where regions are flagged to use

either RANS or DES models The Detached Eddy Simulation technique of Spalart et al. is another

example, where the model it self switches from RANS for attachedflow regions to LES in separated flow regions.


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Deep Stall Aerodynamics

RANS

DES QUICK CDS4


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What have we learned

The RANS or LES equations are derived by an averaging or filtering process from the Navier-Stokes equations.

The averaging process results in more unknown that equations, the turbulent closure problem

Additional equations are derived by performing operation on the Navier-Stokes equations

Non of the model are complete, all model needs some kind of modeling

Special care may be need when integrating the model all the way to the wall, low-Reynolds number models and wall damping terms

Log-law boundary conditions, can be used to limit the necessary resolution, but are not well suited for separation reattachment

The LES models are one way to circumvent some of the inherent problems of the RANS models

Turbulence Modeling

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