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 1
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 2
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. 3
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, 4
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 5
Modeling Turbulent Flows Direct Numerical Simulation All scales of the fluid motion spatial and temporal is resolved by the computation. Largest DNS to date 4096 3 6
Modeling Turbulent Flows Large Eddy Simulation (LES) Only the large scales of the fluid motion is resolved by the computations 7
Modeling Turbulent Flows Reynolds Averaged Navier-Stokes (RANS) The equations are time averaged, and don t resolve the eddies Hybrid LES/ RANS 8
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 9
Reynolds Averaged Navier-Stokes equations (RANS) Reynolds Stress 10
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 11
Reynolds Averaged Momentum Equations The Reynolds Stresses originates from the convective terms They are normally treated together with the diffusive terms 12
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 13
RANS turbulence models Reynolds-stress models Introduces new unknowns (22 new unknowns) 14
RANS turbulence models 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. 15
Algebraic Turbulence Model Prandtls mixing length hypothesis is based on an analogy with momentum transport on a molecular level Molecular transport y U(y) Turbulent transport 16
Prandtl Mixing Length Model The mixing length model closes the equation system The proportionality constant for the mixing velocity c 1 and for the mixing length c 2 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 17
Additions to the basic mixing length model Van Driest (1956) wall damping Clauser (1956) defect layer modification Corrsin and Kistler (1954) intermittency modification 18
Baldwin-Lomax Model Clauser Van Driest Corrsin and Kistler 19
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. 20
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. 21
Second equation 22
The turbulent kinetic energy equation By taking the trace of the Reynolds Stress equation, we get 23
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 24
The k-ε model Eddy viscosity Transport equation for turbulent kinetic energy Transport equation for dissipation of turbulent kinetic energy Constants for the model 25
k-omega SST model 26
k-omega SST model 27
Blending Function F 1 and F 2 28
Constants for k-omega SST model 29
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) 30
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 31
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 32
Boundary Conditions, Log-Law Using the logarithmic profile and the balance between production and 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: 33
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 34
Low Reynolds Number Modification To obtain correct near wall behavior the two equation models are enriched with viscous damping terms 35
Low RE k-omega model 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 36
Inflow conditions 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 37
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 38
Mild adverse pressure gradient 39
Mild adverse pressure gradient 40
Strong adverse pressure gradient 41
Strong adverse pressure gradient 42
Mild adverse pressure gradient 43
Strong adverse pressure gradient 44
Shortcomings of the Boussinesq approximation Flows with sudden changes in mean strain rate 45
Shortcomings of the Boussinesq approximation Flows over curved surfaces So and Mellor, 1972, An Experimental Investigation of Turbuelnt Boundary Layers Along Curved Surfaces 46
Shortcomings of the Boussinesq approximation Flow in ducts with secondary motion Flow in rotating and stratified fluids Three dimensional flows Flows with boundary-layer separation 47
Large Eddy Simulation 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 48
LES, Filtering of the Navier-Stokes equations Again the convective terms generate additional terms Filtering differs from standard averaging in one important respect 49
LES, Filtering of the Navier-Stokes equations The Leonard stresses (L ij ) 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 (C ij ) 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 50
LES, Filtering of the Navier-Stokes equations Smagorinsky model 51
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. 52
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 attached flow regions to LES in separated flow regions. 53
Deep Stall Aerodynamics RANS DES QUICK CDS4 54
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 55