CFD-FVM-04: Turbulent Flow calculations - RANS

• Reynolds-averaged Navier–Stokes (RANS) equations: attention is focused on the mean flow and the effects of turbulence on mean flow properties.
• Large eddy simulation (LES): tracks the behaviour of the larger eddies. The effects on the resolved flow (mean flow plus large eddies) due to the smallest, unresolved eddies are included by means of a so-called sub-grid scale model
• Direct numerical simulation (DNS): The unsteady Navier–Stokes equations are solved on spatial grids that are sufficiently fine that they can resolve the Kolmogorov length scales

RANS

stands for the density-weighted averaged or Favre-averaged velocity:

Scalar transport equation:

where the overbar indicates a time-averaged variable and the tilde indicates a density-weighted or Favre-averaged variable.

We have already seen in last section that this yields six additional unknowns in the time-averaged momentum equations: the Reynolds stresses , , , , , .

Similarly, time-average scalar transport equations show extra terms containing , , .

Extra transport equations	Name
Zero	Mixing length model
One	Spalart-Allmaras model
Two	model
	model
	Algebraic stress model
Seven	Reynolds stress model

Eddy viscosity and eddy diffusivity

Of the tabulated models the mixing length and models are at present by far the most widely used and validated. They are based on the presumption that there exists an analogy between the action of viscous stresses and Reynolds stresses on the mean flow.

Boussinesq proposed in 1877 that Reynolds stresses might be proportional to mean rates of deformation:

where is the turbulent or eddy viscosity (dimensions ) and is the turbulent kinetic energy per unit mass. The Kronecker delta is 1 if and 0 otherwise.

By analogy turbulent transport of a scalar is taken to be proportional to the gradient of the mean value of the transported quantity:

where is the turbulent or eddy diffusivity.

Since turbulent transport of momentum and heat or mass is due to the same mechanism – eddy mixing – we expect that the value of the turbulent diffusivity is fairly close to that of the turbulent viscosity . This assumption is better known as the Reynolds analogy. We introduce a turbulent Prandtl/Schmidt number defined as follows:

Experiments in many flows have established that this ratio is often nearly constant. Most CFD procedures assume this to be the case and use values of around unity.

Preamble

• Mixing length models attempt to describe the stresses by means of simple algebraic formulae for as a function of position
• models solves two transport equations for the turbulent kinetic energy and the dissipation rate .

They assume an isotropic turbulent viscosity , i.e. the ratio between Reynolds stresses and mean rates of deformation is the same in all directions. This assumption is not valid in many complex flows.

Reynolds stress equation models tries to solve the Reynolds stress equations directly. This is an area of vigorous research.
Algebraic stress models are the most economical form of Reynolds stress model and able to introduce anisotropic turbulence effects.

Mixing length models

On dimensional grounds we assume the kinematic turbulent viscosity can be expressed as a product of a turbulent velocity scale and a turbulent length scale :

where is a dimensionless constant. Then the dynamic turbulent viscosity is given by:

This works well in simple two-dimensional turbulent flows. where the only significant Reynolds stress is and the only significant mean velocity gradient is .

where is a constant. Absorbing the two constants and into a new length scale we have

This is Prandtl’s mixing length model. Noting that is the only significant mean velocity gradient, the turbulent Reynolds stress is described by

Mixing lengths for two-dimensional turbulent flows:

Flow type	Mixing length
Mixing layer		Layer width
Jet		Jet half width
Wake		Wake half width
Axisymmetric jet		Jet half width
Boundary layer
viscous sub-layer and log-law layer		Boundary layer thickness
outer layer		Boundary layer thickness
Pipes and channels (fully developed flow)		Pipe radius or channel half width

The only turbulent transport term:

where and . Rodi (1980) recommended values for of 0.9 in near-wall flows, 0.5 for jets and mixing layers and 0.7 in axisymmetric jets.

The mixing length model is not used on its own in general purpose CFD, but we will find it embedded in many of the more sophisticated turbulence models to describe near-wall flow behaviour as part of the treatment of wall boundary conditions.

Summary

Advantages:

• easy to implement and cheap in terms of computing resources
• good predictions for thin shear layers: jets, mixing layers, wakes and boundary layers
• well established

Disadvantages:

• completely incapable of describing flows with separation and recirculation
• only calculates mean flow properties and turbulent shear stress

The model

Some preliminary definitions are

where is the mean kinetic energy per unit mass and is the turbulent kinetic energy per unit mass.

The rate of deformation and the stresses are defined as

and

We also have , and then

The scalar product of two tensors and is evaluated as

Governing equation for mean flow kinetic energy

Recall that the Reynolds-averaged Navier–Stokes equations are defined as

Multiply x-component by , y-component by and z-component by and sum over all three components to get

Or in words

Rate of change of mean kinetic energy +
Transport of by convection =
Transport of by pressure +
Transport of by viscous stresses +
Transport of by Reynolds stresses -
Rate of viscous dissipation of -
Rate of destruction of due to turbulence production

Governing equation for turbulent kinetic energy

Consider the instantaneous Navier–Stokes equations:




where , , , and . Repeat the same procedure as above and substraction of the two resulting equations gives the equation for turbulent kinetic energy :

Or in words

Rate of change of turbulent kinetic energy + Transport of by convection = Transport of by pressure + Transport of by viscous stresses + Transport of by Reynolds stresses - Rate of dissipation of + Rate of production of

Given the viscous dissipation term , we have the rate of dissipation per unit volume , which is the destruction term in the turbulent kinetic energy equation.

The model equations

We use and to define velocity scale and length scale representative of the large-scale turbulence as follows:

Applying dimensional analysis we can specify the eddy viscosity as follows:

where is a dimensionless constant.

The standard model uses the following transport equations for and :


In words:

Rate of change of or + Transport of or by convection = Transport of or by diffusion + Rate of production of or - Rate of destruction of or

The equations contain five adjustable constants: , , , and . The standard k–ε model employs values for the constants that are arrived at by comprehensive data fitting for a wide range of turbulent flows:





For the interpretation of the constants, please refer to the book.

To compute the Reynolds stresses we use the familiar Boussinesq relationship:

Boundary conditions

The model equations for and are elliptic by virtue of the gradient diffusion term.

For the boundary conditions:

• inlet: distributions of and must be given
• outlet, symmetry axis: and
• free stream: and must be given or and
• solid walls: approach depends on Reynolds number (see below)

Inlet

For inlet, if no information of the and distributions is available, rough approximations can be made from the turbulence intensity and a characteristic length scale :

Free stream

The natural choice of boundary conditions for turbulence-free free stream would seem to be and . Due to the relation of , small, but finite, values are commonly used.

Solid walls

At high Reynolds number the following wall functions relate the local wall shear stress (through ) to the mean velocity:



Von Karman’s constant and wall roughness parameter for smooth walls.

For heat transfer we can use a wall function based on the universal nearwall temperature distribution valid at high Reynolds numbers (Launder and Spalding, 1974)

• = temperature at near-wall point
• = wall temperature
• = fluid specific heat at constant pressure
• = wall heat flux
• = turbulent Prandtl number
• = (laminar or molecular) Prandtl number
• = thermal conductivity

Finally is the pee-function, a correction function dependent on the ratio of laminar to turbulent Prandtl numbers (Launder and Spalding, 1974).

At low Reynolds numbers the log-law is not valid. Wall damping needs to be applied to ensure that viscous stresses take over from turbulent Reynolds stresses at low Reynolds numbers and in the viscous sub-layer adjacent to solid walls.



The constants , and in the standard model are multiplied by wall-damping functions , and , respectively. As an example

Summary

Advantages:

• simplest turbulence model for which only initial and/or boundary conditions need to be supplied
• excellent performance for many industrially relevant flows
• well established, the most widely validated turbulence model

Disadvantages:

• more expensive to implement than mixing length model (two extra PDEs)
• poor performance in a variety of important cases such as:
  1. some unconfined flows
  2. flows with large extra strains (e.g. curved boundary layers, swirling flows)
  3. rotating flows
  4. flows driven by anisotropy of normal Reynolds stresses (e.g. fully developed flows in non-circular ducts)

Reynolds stress equation models (RSM)

Advantages:

• potentially the most general of all classical turbulence models
• only initial and/or boundary conditions need to be supplied
• very accurate calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets, asymmetric channel and non-circular duct flows and curved flows

Disadvantages:

• very large computing costs (seven extra PDEs)
• not as widely validated as the mixing length and models
• performs just as poorly as the model in some flows due to identical problems with the ε-equation modelling (e.g. axisymmetric jets and unconfined recirculating flows)

Advanced turbulence models

Advanced treatment of the near-wall region: two-layer model

• Fully turbulent region, , the standard model is used and the eddy viscosity is computed with the usual relationship .
• Viscous region, , only the -equation is solved and a length scale is specified using for the evaluation of the rate of dissipation with using and the eddy viscosity in this region with and .

In between, to overcome instability issue.

Strain sensitivity: RNG model

Derived based on statistical mechanics. The renormalization group (RNG) represented the effects of the small-scale turbulence by means of a random forcing function in the Navier–Stokes equation

with

Only the constant is adjustable.

Now a number of commercial CFD codes have now incorporated the renormalization group (RNG) version of the k–ε model. The performance of the RNG model is better than the standard model for the expanding duct, but actually worse for a contraction with the same area ratio.

When involving complex geometries, above models may not work. Following models may provide better results

Spalart-Allmaras model

The Reynolds stresses are computed with

The transport equation for kinematic eddy viscosity parameter is

In words

Rate of change of viscosity parameter + Transport of by convection = Transport of by turbulent diffusion + Rate of production of - Rate of dissipation of

Wilcox model

The turbulence frequency . Then, and . The transport equation for and for turbulent flows at high Reynolds is as follows:


In words:

Rate of change of or + Transport of or by convection = Transport of or by turbulent diffusion + Rate of production of or - Rate of dissipation of or

Other models

• Menter SST model
• Algebraic stress equation model
  • Advantages:
    • cheap method to account for Reynolds stress anisotropy
    • potentially combines the generality of approach of the RSM (good modelling of buoyancy and rotation effects possible) with the economy of the model
    • successfully applied to isothermal and buoyant thin shear layers
    • if covection and diffusion terms are negligible the ASM performs as well as the RSM
  • Disadvantages:
    • only slightly more expensive than the model (two PDEs and a system of algebraic equations)
    • not as widely validated as the mixing length and models
    • same disadvantages as RSM apply
    • model is severely restricted in flows where the transport assumptions for convective and diffusive effects do not apply – validation is necessary to define performance limits
• Non-linear models