Modeling turbulent flows
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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.

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Modeling Turbulent Flows

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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


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


Choices to be Made





Turbulence Model


Near-Wall Treatment








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.







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)


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







Example: Fluent’s Reynolds Stress Model

  • Transport equation for Rij:

  • Generation:

  • Pressure-StrainRedistribution:

  • Dissipation:

  • Diffusion:



(related to e)


Turbulence Models in Fluent



Zero-Equation Models

One-Equation Models


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




Per Iteration



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.



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.



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


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.







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.

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