1. CFD Modeling of Turbulent Flows

2. Overview Properties of turbulence Predicting turbulent flows DNS LES
RANS Models
Summary

3. Properties of Turbulence
Most flows encountered in industrial processes are turbulent.
Turbulent flows exhibit three-dimensional, unsteady, aperiodic motion.
Turbulence increases mixing of momentum, heat and species.
Turbulence mixing acts to dissipate momentum and the kinetic energy in the flow by viscosity acting to reduce velocity gradients.
Turbulent flows contain coherent structures that are deterministic events.
Time u

4. Role of Numerical Turbulence Modeling
An understanding of turbulence and the ability to predict turbulence for any given application is invaluable for the engineer.
Examples:
Increased turbulence is needed in chemical mixing or heat transfer when fluids with dissimilar properties are brought together.
turbulence increases drag due to increased frictional forces.
Historically, experimental measurement of the system was the only option available. This makes design optimization incredibly tedious.

5. Characteristics of the Engineering Turbulence Model
Numerical modeling of turbulence can serve to improve the engineers ability to analyze turbulent flow in design particularly when precise measurements cannot be obtained and when extensive experimentation is costly and time-consuming.
The ideal turbulence model should introduce minimal complexity while capturing the essence of the relevant physics.

6. Modeling Turbulent Flows
Turbulent flows can be modeled in a variety of ways. With increasing levels of complexity they are:
Correlations
Moody chart, Nusselt number correlations
Integral equations
Derive ODE’s from the equations of motion
Reynolds Averaged Navier Stokes or RANS equations
Average the equations of motion over time
Requires closure
Large Eddy Simulation or LES
Solve Navier-Stokes equations for large scale motions of the flow. Model only the small scale motions
DNS
Navier-Stokes equations solved for all motions in the turbulent flow

7. Turbulence Modeling Approaches
Zero-Equation Models
One-Equation Models
Two-Equation Models
Standard k-e
RNG k-e
Second-order closure
Reynolds-Stress Model
Large-Eddy Simulation
Direct Numerical Simulation
Increase
Computational
Cost
Per Iteration
Include
More
Physics
Over the years, many different types of turbulence models have been developed and this slide tries to show you where the turbulence models available in Fluent CFD software fit in the grand scheme of things. The various types of turbulence models are listed in the center in order of increasing sophistication or increasing inclusion of more physics (more physics is supposed to be good !?). Unfortunately, inclusion of more physics usally increases the computational cost. Engineers must find the turbulence modeling approach that satisfies their technical needs at the lowest cost! Fluent Inc. offers several state-of-the-art turbulence modeling options that provide the accuracy that engineers need at reasonable computational expense. These turbulence models include the two-equation and RSM models. We are also working to expand the number of turbulence modeling options available to take advantage of progres in turbulence modeling field that improve the accuracy of one-equation models, develop a k-e model that is fully realizable, and make LES methods more practical.

8. Direct Numerical Simulation (DNS)
Currently DNS is the most exact approach to modeling turbulence since no averaging is done or approximations are made
Since the smallest scales of turbulence are modeled, called the Kolmogoroff scale, the size of the grid must be scaled accordingly.
A DNS simulation scales with ReL3 (u´L/n) where ReL ~ 0.01Re. Turbulent flow past a cylinder would require at least (0.01 x 20,000)3 or 8 million cells
The Kolmogoroff scale is the smallest scale on which viscosity is active It is also the smallest scale on which turbulent kinetic energy is dissipated

9. Direct Numerical Simulation (DNS)
Given the current processing speed and memory of the largest computers, only very modest Reynolds number flows with simple geometries are possible.
Advantages: DNS can be used as numerical flow visualization and can provide more information than experimental measurements; DNS can be used to understand the mechanisms of turbulent production and dissipation.
Disadvantages: Requires supercomputers; limited to simple geometries.

10. Large Eddy Simulation (LES)
DNS u
DNS
LES
LES t
LES is a three dimensional, time dependent and computationally expensive simulation, though less expensive than DNS.
LES solves the large scale motions and models the small scale motions of the turbulent flow.
The premise of LES is that the large scale motions or eddies contain the larger fraction of energy in the flow responsible for the transport of conserved properties while the small.

11. Large Eddy Simulation (LES)
The large scale components of the flow field are filtered from the small scale components using a wavelength criteria related to the size of the eddies
The filter produces the following equation used to model the small scale motions
where
The inequality is then modeled as
tij is called the subgrid scale Reynolds Stress. Different subgrid scale models are available to approximate tij .

12. Large Eddy Simulation (LES)
Mixing plane between two streams with unequal scalar concentrations
Unsteady vortex motions with growing length scale

13. Reynolds Averaged Navier-Stokes (RANS) Models
DNS and LES can produce an overwhelming quantity of detailed information about a flow structure. Generally, in engineering flows, such levels on instantaneous information is not required.
Typical engineering flows are focused on obtaining a few quantitative of the turbulent flow. For example, average wall shear stress, pressure and velocity field distribution, degree of mixedness in a stirred tank, etc.
The approach would be to model turbulence by averaging the unsteadiness of the turbulence.
This averaging process creates terms that cannot be solved analytically but must be modeled.
This modeling approach has been around for 30 years and is the basis of most engineering turbulence calculations.

14. RANS Equations
Velocity or a scalar quantity can be represented as the sum of the mean value and the fluctuation about the mean value as:
Using the above relationship for velocity(let f = u) in the Navier-Stokes equations gives (as momentum equation for incompressible flows with body forces).
The Reynolds Stresses cannot be represented uniquely in terms of mean quantities and the above equation is not closed. Closure involves modeling the Reynolds Stresses.

15. Closure of RANS equations
The RANS equations contain more unknowns than equations.
The unknowns are the Reynolds Stress terms.
Closure Models are:
zero-equation turbulence models
Mixing length model
no transport equation used
one-equation turbulence models
transport equation modeled for turbulent kinetic energy k
two-equation models
more complete by modeling transport equation for turbulent kinetic energy k and eddy dissipation e
second-order closure
Reynolds Stress Model
does not use Bousinesq approximation as first-order closure models

16. Modeling Turbulent Stresses in Two-Equation Models
RANS equations require closure for Reynolds stresses and the effect of turbulence can be represented as an increased viscosity
The turbulent viscosity is correlated with turbulent kinetic energy k and the dissipation rate of turbulent kinetic energy e
Boussinesq Hypothesis:
eddy viscosity model
The RANS equations are closed in two-equation models by assuming that the turbulent stresses are proportional to the mean velocity gradients and that the constant of proportionality is the turbulent viscosity. The constant of proportionality is allowed to vary through out the flow field and has been correlated in terms of the TKE and dissipation rate. Separate transport equations are developed for the TKE and dissipation rate such that the turbulent viscosity and hence the turbulent stresses may be calculated at any point for the RANS equations. The drawbacks to this approach are that the eddy viscosity is the same for each of the Reynolds stresses and that the constant C_mu is found from simple 2DTBL experiments.
Turbulent Viscosity:

17. Turbulent kinetic energy and dissipation
Transport equations for turbulent kinetic energy and dissipation rate are solved so that turbulent viscosity can be computed for RANS equations.
Turbulent
Kinetic Energy:
Dissipation Rate of
Turbulent Kinetic Energy:

18. Standard k- Model Turbulent Kinetic Energy Dissipation Rate
Convection
Generation
Diffusion
Dissipation Rate
Destruction
Convection
Generation
Diffusion
These transport equations represent a basic bookkeepping approach applied to a differential control volume where convection of the parameter is equal to the sum of the generation, diffusion, and dissipation terms. The simplified equations for steady, incompressible flow without body forces are shown here for clarity. The turbulent kinetic energy equation can be derived directly from the transport equations for the velocity fluctuations. The exact equation for the dissipation rate is very complex and virtually all turbulence models that use the dissipation rate use a transport equation that is “modeled” and has the same types of terms as the TKE transport equation.
are empirical constants
(equations written for steady, incompressible flow w/o body forces)

19. Deficiencies of the Boussinesq Approximation
The two-equation models based on the eddy viscosity approximation provide excellent predictions for many flows of engineering interest.
Applications for which the approximation is weak typically are flows with extra rate of strain (due to isotropic turbulent viscosity assumption). Examples of these are:
flows over boundaries with strong curvature
flows in ducts with secondary motions
flows with boundary layer separation
flows in rotating and stratified fluid
strongly three dimensional flows
The RNG k-e model is an improvement over the standard k-e for these classes of flow by incorporating the influence of additional strains rates.
A higher-order closure approximation can be also applied to include wider class of problems including those with extra rates of strain.

20. RNG k- Model Turbulent Kinetic Energy Dissipation Rate where
Convection
Diffusion
Dissipation
Generation
where
Dissipation Rate
Additional term
related to mean strain
& turbulence quantities
Convection
Generation
Diffusion
Destruction
RNG k-e equations have similar form as standard k-e model equations except that the dissipation rate equation has an additonal term related to the mean rate of strain and turbulence quantities that allows it to include more physics. Additionally, the model constants are derived from RNG theory as opposed to being empircally based. Surprisingly, the analytically derived constants are very similar to the empirical constants in the standard k-e model.
are derived using RNG theory
(equations written for steady, incompressible flow w/o body forces)

21. Second-Order Closure models
The Second-Order Closure Models include the effects of streamline curvature, sudden changes in strain rate, secondary motions, etc.
This class of models is more complex and computationally intensive than the RANS models
The Reynolds-Stress Model (RSM) is a second-order closure model and gives rise to 6 Reynolds stress equations and the dissipation equation
Reynolds Stress Transport Equation
Diffusion
Convection
Pressure-Strain
redistribution
Generation
Dissipation

22. Reynolds Stress Model Generation Pressure-Strain Redistribution
(computed)
Pressure-Strain
Redistribution
(modeled)
The RSM attempts to more mechanistically account for the various physical phenomena that contribute to the behavior of the turbulent stresses. The convective and generation/destruction terms are computed directly while the pressure-strain, potentially anisotropic dissipation, and turbulent diffusion terms need to modeled to provide closure. In addtion to the mean momentum equations and pressure equation, the RSM model solves 6 transport equations for the Reynolds stresses and one transport equation for the dissipation rate.
Dissipation
(related to e)
Turbulent
Diffusion
Pressure/velocity
fluctuations
Turbulent
transport
(modeled)
(equations written for steady, incompressible flow w/o body forces)

23. Near Wall Treatment
The RANS turbulence models require a special treatment of the mean and turbulence quantities at wall boundaries and will not predict correct near-wall behavior if integrated down to the wall
Special near-wall treatment is required
Standard wall functions
Nonequilibrium wall functions
Two-layer zonal model

24. Comparison: RNG k-e vs. Standard k-e

25. Summary
Turbulence modeling comes in varying degrees of complexity. Determining the right choice of turbulence model depends on the detail of results expected.
DNS and LES are still far from being engineering tools but in the near future this will be possible.
Two-equation models are widely used for their relatively simple overhead. However, increased complexity of the turbulent flow reduces the adequacy of the models.
Improvements to the two-equation models to incorporate extra strain rates, and the second-order closure RSM model provide the extra terms to model complex engineering turbulent flows.