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Modeling Turbulent Flows

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Modeling Turbulent Flows

- 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

- Size/velocity of large eddies on order of mean flow.

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

Flow

Physics

Computational

Resources

Turbulence Model

&

Near-Wall Treatment

Computational

Grid

Turnaround

Time

Constraints

Accuracy

Required

- 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.

- ‘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.

- (1) Governing equations are ensemble or time averaged (RANS-based models).
- Both approaches requires modeling of the scales that are averaged out.

u

u'i

U

Ui

ui

t

- 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

- 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.

- 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)

- (1) Using the Boussinesq assumption, the Reynolds stresses are related to the mean flow by a turbulent viscosity, mt:

2Sij

- 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.

- Spalart-Allmaras

- 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

- Transport equation for Rij:
- Generation:
- Pressure-StrainRedistribution:
- Dissipation:
- Diffusion:

(computed)

(modeled)

(related to e)

(modeled)

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

- 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.

- Large eddies:
- 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

- 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: tl/U.

- Good starting point for cell size is Taylor length scale, l (10nk/e)1/2.
- 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

- Smagorinsky-Lilly model
- Post-processing
- Statistically time averaged results are available.

y+

u+

- 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

- 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.

- The Standard and Non-equilibrium Wall Function options refer to specific ‘sets’ designed for high Re flows.
- 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.

- This near-wall model combines the use of enhanced wall functions and a two-layer model.

for

where

- 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.

- SWF is modified to account for stronger p and non-equilibrium flows.
- The Standard and Non-Equilibrium Wall functions are options for the k-e and RSM turbulence models.

- 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.

- Domain is divided into viscosity-affected (near-wall) region and turbulent core region.
- Solutions for e and mt in each region are blended, e.g.,

- A blended two-layer model is used to determine near-wall e field.

- Enhanced wall functions
- The Enhanced Wall Treatment near-wall model are options for the k-e and RSM turbulence models.

- 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-

- The skin friction coefficient can be estimated from empirical correlations:
- Use post-processing to confirm near-wall mesh resolution.

- 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).

- The characteristic Re is low or if near wall characteristics need to be resolved.

- 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

- Length scale is related to size of large eddies that contain most of energy.
- 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

- Explicitly input k, ,w, or
- Turbulence intensity depends on upstream conditions:

DefineModelsViscous...

Inviscid, Laminar, or Turbulent

Turbulence Model options

Near Wall Treatments

Additional Turbulence options

- 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

- Complex, high ReL (4.6 106) 3D Flow
- Simulation
- Wall Functions used to manage mesh size.
- y+ 30 - 80
- Hex mesh ~200,000 cells

- Wall Functions used to manage mesh size.
- Experimentally derived contours of axial velocity

- SKO and RSM models capture characteristic shape at propeller plane.

RNG

RKE

SA

RSM

SKE

SKO

- 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.

- 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.