Modeling Turbulent Flows

Modeling Turbulent Flows

What is Turbulence? • Unsteady, irregular (aperiodic) motion in which transported quantities (mass, momentum, scalar species) fluctuate in time and space • Identifiable swirling patterns characterizes turbulent eddies. • Enhanced mixing (matter, momentum, energy, etc.) results • Fluid properties exhibit random variations • Statistical averaging results in accountable, turbulence related transport mechanisms. • This characteristic allows for Turbulence Modeling. • Wide range in size of turbulent eddies (scales spectrum). • Size/velocity of large eddies on order of mean flow. • derive energy from mean flow

Is the Flow Turbulent? External Flows where along a surface L = x, D, Dh, etc. around an obstacle Other factors such as free-stream turbulence, surface conditions, and disturbances may cause earlier transition to turbulent flow. Internal Flows Natural Convection where

Choices to be Made Flow Physics Computational Resources Turbulence Model & Near-Wall Treatment Computational Grid Turnaround Time Constraints Accuracy Required

Modeling Turbulence • Direct numerical simulation (DNS) is the solution of the time-dependent Navier-Stokes equations without recourse to modeling. • Mesh must be fine enough to resolve smallest eddies, yet sufficiently large to encompass complete model. • Solution is inherently unsteady to capture convecting eddies. • DNS is only practical for simple low-Re flows. • The need to resolve the full spectrum of scales is not necessary for most engineering applications. • Mean flow properties are generally sufficient. • Most turbulence models resolve the mean flow. • Many different turbulence models are available and used. • There is no single, universally reliable engineering turbulence model for wide class of flows. • Certain models contain more physics that may be better capable of predicting more complex flows including separation, swirl, etc.

Modeling Approaches • ‘Mean’ flow can be determined by solving a set of modified equations. • Two modeling approaches: • (1) Governing equations are ensemble or time averaged (RANS-based models). • Transport equations for mean flow quantities are solved. • All scales of turbulence are modeled. • If mean flow is unsteady, Dt is set by global unsteadiness. • (2) Governing equations are spatially averaged (LES). • Transport equations for ‘resolvable scales.’ • Resolves larger eddies; models smaller ones. • Inherently unsteady, Dt set by small eddies. • Resulting models requires more CPU time/memory and is not practical for the majority of engineering applications. • Both approaches requires modeling of the scales that are averaged out.

u u'i U Ui ui t RANS Modeling - Ensemble Averaging • Imagine how velocity, temperature, pressure, etc. might vary in a turbulent flow field downstreamof a valve that has beenslightly perturbed: • Ensemble averaging may be used to extract the mean flow properties from the instantaneous properties. n identifies the ‘sample’ ID

Deriving RANS Equations • Substitute mean and fluctuating velocities in instantaneous Navier-Stokes equations and average: • Reynolds Averaged Navier-Stokes equations: whereare the Reynolds Stresses. • The transported variables, U,r, p, etc., now represent the mean flow quantities • The Reynolds Stress terms are modeled using functions containing empirical constants and information about the mean flow.

Modeling the Reynolds Stresses • The RANS based turbulence models calculate the Reynolds Stresses by one of two methods: • (1) Using the Boussinesq assumption, the Reynolds stresses are related to the mean flow by a turbulent viscosity, mt: • Strain rate tensor, Sij, described in terms of mean flow. • Isotropic viscosity assumed • (2) Solving individual transport equations for the Reynolds stresses. • Turbulent viscosity is not employed, no assumption of isotropy • Contains more “physics” • More complex and computationally expensive than (1) 2Sij

Calculating mt for Boussinesq Formula • Based on dimensional arguments, mt can be determined from the conservative variables k, e or w. • is the turbulent kinetic energy • is the dissipation rate of k (to thermal energy) • is the specific dissipation rate • mt is calculated differently depending upon the turbulence model. • Spalart-Allmaras • This ‘single equation’ model solves one additional transport equation for a modified viscosity. • Standard k-e, RNG k-e, Realizable k-e • These ‘two equation’ models solve transport equations for k and e. • Standard k-w, SST k-w • These ‘two equation’ models solve transportequations for k and w.

Example: Fluent’s Standard k- Model • Transport equation for k: • Transport equation for e: • Turbulent viscosity: • are empirically defined constants. • Turbulence modeling options account for: • viscous heating (in energy equation) • compressibility effects, YM (activated when ideal gas is used) • buoyancy, Gb • user defined sources, Sk and Se

Pressure/velocity fluctuations Molecular transport Turbulent transport Example: Fluent’s Reynolds Stress Model • Transport equation for Rij: • Generation: • Pressure-StrainRedistribution: • Dissipation: • Diffusion: (computed) (modeled) (related to e) (modeled)

Turbulence Models in Fluent RANS-based models Zero-Equation Models One-Equation Models Spalart-Allmaras Two-Equation Models Standard k-e RNG k-e Realizable k-e Standard k-w SST k-w Reynolds-Stress Model Large-Eddy Simulation Direct Numerical Simulation Increase Computational Cost Per Iteration Available in FLUENT 6

RANS Turbulence Model Descriptions

RANS Turbulence Model Behavior and Usage

Large Eddy Simulation (LES) • Motivation: • Large eddies: • Mainly responsible for transport of momentum, energy, and other scalars, directly affecting the mean fields. • Anisotropic, subjected to history effects, and flow-dependent, i.e., strongly dependent on flow configuration, boundary conditions, and flow parameters. • Small eddies tend to be more isotropic, less flow-dependent, and hence more amenable to modeling. • Approach: • LES resolves large eddies and models only small eddies. • Equations are similar in form to RANS equations • Dependent variables are now spatially averaged instead of time averaged. • Large computational effort • Number of grid points, NLES • Unsteady calculation

Problem Setup for LES • Small eddies defined by grid cell size. • Good starting point for cell size is Taylor length scale, l  (10nk/e)1/2. • You can use a two-equation model on coarse mesh to determine range of k and e. • Time step size can be defined by: tl/U. • Effective stress requires definition of Subgrid Scaleviscosity. • Smagorinsky-Lilly model • constant Cs must be tuned to flow • RNG-based model • Useful for low Re flows where mt m • Post-processing • Statistically time averaged results are available.

y+ u+ Modeling the Near-Wall Region • Accurate near-wall modeling is important for most engineering applications. • Successful prediction of frictional drag, pressure drop, separation, etc., depends on fidelity of local wall shear predictions. • Most k-e and RSM turbulence models will not predict correct near-wall behavior if integrated down to the wall. • Problem is the inability to resolve e. • Special near-wall treatment is required. • Standard Wall Functions • Non-Equilibrium Wall Functions • Enhanced wall treatment • S-A and k-w models are capable ofresolving the near-wall flow provided near-wall mesh is sufficient.

outer layer inner layer Near-Wall Modeling Options • In general, ‘wall functions’ are a collection or set of laws that serve as boundary conditions for momentum, energy, and species as well as for turbulence quantities. • Wall Function Options • The Standard and Non-equilibrium Wall Function options refer to specific ‘sets’ designed for high Re flows. • The viscosity affected, near-wall region is not resolved. • Near-wall mesh is relatively coarse. • Cell center information bridged by empirically-based wall functions. • Enhanced Wall Treatment Option • This near-wall model combines the use of enhanced wall functions and a two-layer model. • Used for low-Re flows or flows with complex near-wall phenomena. • Generally requires a very fine near-wall mesh capable of resolving the near-wall region. • Turbulence models are modified for ‘inner’ layer.

for where Standard and Non-Equilibrium Wall Functions • Standard Wall Function • Momentum boundary condition based on Launder-Spaulding law-of-the-wall: • Similar ‘wall laws’ apply for energy and species. • Additional formulas account for k, e, and ruiuj. • Less reliable when flow departs from conditions assumed in their derivation. • Severe p or highly non-equilibrium near-wall flows, high transpiration or body forces, low Re or highly 3D flows • Non-Equilibrium Wall Function • SWF is modified to account for stronger p and non-equilibrium flows. • Useful for mildly separating, reattaching, or impinging flows. • Less reliable for high transpiration or body forces, low Re or highly 3D flows. • The Standard and Non-Equilibrium Wall functions are options for the k-e and RSM turbulence models.

Enhanced Wall Treatment • Enhanced Wall Treatment • Enhanced wall functions • Momentum boundary condition based on blendedlaw-of-the-wall (Kader). • Similar blended ‘wall laws’ apply for energy, species, and w. • Kader’s form for blending allows for incorporation of additional physics. • Pressure gradient effects • Thermal (including compressibility) effects • Two-layer model • A blended two-layer model is used to determine near-wall e field. • Domain is divided into viscosity-affected (near-wall) region and turbulent core region. • Based on ‘wall-distance’ turbulent Reynolds number: • Zoning is dynamic and solution adaptive. • High Re turbulence model used in outer layer. • ‘Simple’ turbulence model used in inner layer. • Solutions for e and mt in each region are blended, e.g., • The Enhanced Wall Treatment near-wall model are options for the k-e and RSM turbulence models.

Estimating Placement of First Grid Point • Ability for near-wall treatments to accurately predict near-wall flows depends on placement of wall adjacent cell centroids (cell size). • For SWF and NWF, centroid should be located in log-layer: • For best results using EWT, centroid should be located in laminar sublayer: • This near-wall treatment can accommodate cells placed in the log-layer. • To determine actual size of wall adjacent cells, recall that: • The skin friction coefficient can be estimated from empirical correlations: • Flat Plate- • Pipe Flow- • Use post-processing to confirm near-wall mesh resolution.

Near-Wall Modeling RecommendedStrategy • Use SWF or NWF for most high Re applications (Re > 106) for which you cannot afford to resolve the viscous sublayer. • There is little gain from resolving viscous sublayer (choice of core turbulence model is more important). • Use NWF for mildly separating, reattaching, or impinging flows. • You may consider using EWT if: • The characteristic Re is low or if near wall characteristics need to be resolved. • The same or similar cases ran successfully previously with the two-layer zonal model (in Fluent v5). • The physics and near-wall mesh of the case is such that y+ is likely to vary significantly over a wide portion of the wall region. • Try to make the mesh either coarse or fine enough, and avoid putting the wall-adjacent cells in the buffer layer (y+ = 5 ~ 30).

Setting Boundary Conditions • When turbulent flow enters a domain at inlets or outlets (backflow), boundary values for: • k, ,w and/or must be specified • Four methods for directly or indirectly specifying turbulence parameters: • Explicitly input k, ,w, or • This is the only method that allows for profile definition. • Turbulence intensity and length scale • Length scale is related to size of large eddies that contain most of energy. • For boundary layer flows: l  0.4d99 • For flows downstream of grid: l opening size • Turbulence intensity and hydraulic diameter • Ideally suited for duct and pipe flows • Turbulence intensity and turbulent viscosity ratio • For external flows: 1 <mt/m< 10 • Turbulence intensity depends on upstream conditions:

GUI for Turbulence Models DefineModelsViscous... Inviscid, Laminar, or Turbulent Turbulence Model options Near Wall Treatments Additional Turbulence options

Example: Ship Hull Flow • Experiments: KRISO’s 300K VLCC (1998) • Complex, high ReL (4.6  106) 3D Flow • Thick 3D boundary layer in moderate pressure gradient • Streamline curvature • Crossflow • Free vortex-sheet formation (“open separation”) • Streamwise vortices embedded in TBL and wake • Simulation • Wall Functions used to manage mesh size. • y+ 30 - 80 • Hex mesh ~200,000 cells • Experimentally derived contours of axial velocity

Comparing Contour Plots of Axial Velocity • SKO and RSM models capture characteristic shape at propeller plane. RNG RKE SA RSM SKE SKO

Comparing Wake Fraction and Drag • Though SKO (and SST) were able to resolve salient features in propeller plane, not all aspects of flow could be accurately captured. • Eddy viscosity model • RSM models accurately capture all aspects of the flow. • Complex industrial flows provide new challenges to turbulence models.

Summary: Turbulence Modeling Guidelines • Successful turbulence modeling requires engineering judgement of: • Flow physics • Computer resources available • Project requirements • Accuracy • Turnaround time • Turbulence models & near-wall treatments that are available • Modeling Procedure • Calculate characteristic Re and determine if Turbulence needs modeling. • Estimate wall-adjacent cell centroid y+ first before generating mesh. • Begin with SKE (standard k-) and change to RNG, RKE, SKO, or SST if needed. • Use RSM for highly swirling flows. • Use wall functions unless low-Re flow and/or complex near-wall physics are present.

Modeling Turbulent Flows