Numerical Methods in Aerodynamics Lecture 5: Turbulence modeling 1 Turbulence Modeling Niels N. Sørensen Professor MSO, Ph.D. Department of Civil Engineering, Alborg University & Wind Energy Department, Risø National Laboratory Technical University of Denmark
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Numerical Methods in AerodynamicsLecture 5: Turbulence modeling
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Turbulence Modeling
Niels N. SørensenProfessor MSO, Ph.D.
Department of Civil Engineering, Alborg University &Wind Energy Department, Risø National Laboratory
Technical University of Denmark
Numerical Methods in AerodynamicsLecture 5: Turbulence modeling
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Outline of lecture
Characteristics of turbulenceWhat is the problem of modeling turbulence
Reynolds Averaging and Reynolds stressesRANS Turbulence Models
Boussinesq approximationBoundary Conditions
Log-lawLow Reynolds Number Modifications
Example of RANS comp.Shortcomings of RANS modelsLarge Eddy Simulation models
FilteringHybrid models
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The Nature of turbulence (I)
IrregularityTurbulence is irregular or random.
DiffusivityTurbulent 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)
DissipationTurbulent 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.
ContinuumThe smallest scale of turbulence are ordinary far larger than any molecular length scale
Flow featureTurbulence 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 SimulationAll 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
Modeling Turbulent Flows
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Modeling Turbulent Flows
Reynolds Averaged Navier-Stokes (RANS)The equations are time averaged, and don’t 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-stressesThree 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 modelsPrandtl Mixing Length ModelCebeci-Smith ModelBaldwin-Lomax Model
One equation turbulence modelsSpalart-AllmarasBaldwin-Barth
Two equation turbulence modelsk-epsilon modelk-omega modelk-tau model
Reynolds stress models
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Reynolds-stress models
Introduces new unknowns (22 new unknowns)
RANS turbulence models
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Eddy-viscosity modelsCompute 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.
RANS turbulence models
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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 specifiedThe equation for the turbulent eddy viscosity is a part of the flow solutions, as it depends on the mean flow gradientTurbulence 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 layersTypically the algebraic models are fast and robustNeeds to be calibrated for each flow type, they are not very generalThey are not well suited for computing flow separationTypically 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-ε 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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Wall conditionsBridging 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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Boundary Conditions, Log-Law
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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Boundary Conditions, Log-Law
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 assumptionsWe 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-ω 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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Mild adverse pressure gradient
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Strong adverse pressure gradient
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Strong adverse pressure gradient
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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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Shortcomings of the Boussinesq approximation
Flows over curved surfacesSo and Mellor, 1972, An Experimental Investigation of
Turbuelnt Boundary Layers Along Curved Surfaces
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Shortcomings of the Boussinesq approximation
Flow in ducts with secondary motionFlow in rotating and stratified fluidsThree dimensional flowsFlows 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 consideredThe 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
LES, Filtering of the Navier-Stokes equations
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LES, Filtering of the Navier-Stokes equations
Smagorinsky model
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LES modeling
LES models are by nature unsteadyLES models are by nature full three dimensionalThey resolve the large scales and only model the isotropic smallscalesThe 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 meshLES 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 modelsOne example is zonal models where regions are flagged to use either RANS or DES modelsThe 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 problemAdditional equations are derived by performing operation on the Navier-Stokes equationsNon of the model are complete, all model needs some kind of modelingSpecial care may be need when integrating the model all the way to the wall, low-Reynolds number models and wall damping termsLog-law boundary conditions, can be used to limit the necessary resolution, but are not well suited for separation reattachmentThe LES models are one way to circumvent some of the inherent problems of the RANS models