What is Turbulence Modelling?

The flow Reynolds number above a critical value (2300 for internal flow and 5 x 105 for external flow) starts a complicated series of events which eventually leads to a radical change of the flow character. The velocity and all other flow properties vary in a random and chaotic way. The motion of such flow becomes intrinsically unsteady even with constantly imposed boundary conditions. This regime is called turbulent flow.

The random nature of the flow makes it difficult to study the motion of all the fluid particles economically. So, the time-mean behavior of the flow becomes a practical interest. Therefore, the equations for unsteady laminar flow are converted into the time-averaged equations for turbulent flow by an averaging operation in which it is assumed that there are rapid and random fluctuations about the mean value.

In the above figure, the velocity is decomposed into a steady mean value U with a fluctuating component u'(t) superimposed on it: u(t) = U+u'(t). This is called Reynolds decomposition. The velocity u(t) is called an instantaneous velocity. 

The turbulent flow can now be characterized in terms of the mean value of the flow properties(U, V, W, P, etc.) and some statistical properties of their fluctuations(u',v',w',p' etc.). 

 

Description of Turbulent Flow:

The time mean of turbulent velocity U is defined by,

Where T is the averaging period taken as sufficiently longer than the period of fluctuations. 

In time-dependent flows, the mean of a property(velocity, pressure, etc.) at time t is taken to be the average of the instantaneous values of the property over a large number of repeated identical experiments then it is called ensemble average. 

The time average of the fluctuation u' is, by definition, zero:

As the fluctuation is about the mean value( positive and negative side), the average of it will be zero. However, the mean squared of fluctuation is not zero (the square of the negative value is positive) and thus is the measure of turbulent intensity.

In order to calculate the shear stresses in turbulent flow, it is necessary to know the fluctuating components of velocity. So, the Reynolds time-averaging concept is introduced, where the velocity components and pressure are split into mean and fluctuating components.

From now on, we shall not write the instantaneous component as u(t) but, simply u, as written above. 

Averaging of the Navier-Stokes equations:

For the sake of simplicity, lets take the steady NS equations.

Mass Conservation:

Here, the density is constant and u, v & w is taken from the turbulence definition of mean and fluctuating components. After keeping all the values, the equation becomes;

Time averaging the equation and the u term in the x direction takes the form,

after solving in y and z directions, we get

This yields the continuity equation for the mean flow. It is very much similar to the continuity equation for laminar flow except for the fact that the velocity components(instantaneous velocities) are replaced with the mean(time-averaged) values of velocity components of turbulent flow.

Momentum Conservation:

The steady momentum equation is x direction,

Let's put the values of instantaneous values of velocities u,v,w into the above equation,

Again, recall that `bar(u') = 0` and `bar(bar(u))=bar(u)`, etc. After some straightforward simplifications, the above equation becomes,

The averaged velocity in the second term on the left-hand side (LHS) can be included in the partial differential terms by noting that the system is incompressible. In such case, then

Therefore, the turbulent x-momentum becomes

Applying the same operations to the y- and z-momentum equations,

and

If we write the equations in the compact notation, stress is given by,

`"Stress " = (ubrace(μ∇^2u_i)_("shear stress gradient"))-(ubrace(ρ(∂bar(u_i^')bar(u_j^'))/(∂x_j))_("Reynolds Stress gradient"))`

Reynold's stress tensor, which is a component of stress tensor obtained by the averaging operation on momentum equation and represents the turbulent fluctuations, is given by,

 

Reynold's stress tensor for the above equations is expressed in matrix form as:

Prior to the solving momentum equation, we need to model the stress tensor. 

The tensor is symmetric, it represents correlation between fluctuating velocities. It is an additional stress term due to turbulence (fluctuating velocities) and it is unknown. We need to model for stress tensor to close the equation system(x,y,z direction). This is called the closure problem: the number of unknowns (ten: three velocities, pressure, six stresses) is larger than the number of equations (four: the continuity equation and three NS equations). 

There are different levels of approximations involved when closing the equation system. The one-point closure leads to a set of partial differential equations called the Reynolds averaged Navier-Stokes ( RANS) equations. These equations does not lead to a closed set of equations by itself, it requires introduction of approximations referred to as turbulence models. The widely used general type models are explained below in short. 

1) Algebraic Models:

An algebraic equation is used to compute a turbulent viscosity, often called eddy viscosity. The Reynolds stress tensor is then computed using an assumption which relates the Reynolds stress tensor to the velocity gradients and the turbulent viscosity. This assumption is called the Boussinesq assumption. Models which are based on a turbulent (eddy) viscosity are called eddy viscosity models.

2) One-equation models: 

In these models a transport equation is solved for a turbulent quantity (usually the turbulent kinetic energy), and a second turbulent quantity (usually a turbulent length scale) is obtained from an algebraic expression. The turbulent viscosity is calculated from the Boussinesq assumption.

3) Two-equation models:

These models fall into the class of eddy viscosity models. Two transport equations are derived which describe the transport of two scalars, for example, the turbulent kinetic energy k and its dissipation ε. The Reynolds stress tensor is then computed using an assumption that relates the Reynolds stress tensor to the velocity gradients and an eddy viscosity. The latter is obtained from the two transported scalars.

4) Reynolds stress models:

Reynolds stress equation models (RSM), also referred to as second-moment closures, are the most complete classical turbulence model. In these models, the eddy-viscosity hypothesis is avoided and the individual components of the Reynolds stress tensor are directly computed. 

Above, the different types of turbulence models have been listed in increasing order of complexity, ability to model the turbulence, and cost in terms of computational work (CPU time).

Large Eddy Simulation Modelling (and Comparison with RANS and DNS):

The larger the eddy, the higher its non-isotropic nature and the more complex its behavior. The larger eddies obtain their kinetic energy from the bulk fluid energy, contain most of the turbulent kinetic energy (~80%), transfer kinetic energy to the smaller eddies by stretching and breaking them up (“cascading”), and are responsible for the majority of the diffusive processes involving mass, momentum, and energy. For these reasons, the simulation of large eddies is highly desirable. On the other hand, the smaller eddies take the kinetic energy from the larger eddies and transfer their energy back to the fluid through viscous shear. For high Re, the small-scale turbulent eddies are statistically isotropic. Therefore, they are “more universal” and more independent of the boundary conditions and the mean flow velocity than the larger eddies. Thus, simulation of the smaller eddies is also desirable.

So, why not simulate (resolve) the larger eddies and approximate (model) the behavior of the smaller eddies? Based on this premise, large eddy simulation (LES) models have been developed for several decades to capture these important eddy features. 

Above Figures shows the instantaneous velocity based on resolved LES and DNS calculations. Note that LES will capture a significant number of velocity fluctuations associated with the larger eddies, while DNS will capture all the LES fluctuations, as well as the Taylor eddies that were cut off from LES (in the interest of a faster calculation), and all the Kolmogorov eddy fluctuations. In other words, DNS resolves the entire eddy spectrum: integral, Taylor, and Kolmogorov eddies. Hence, the DNS instantaneous velocity is more “jagged” and, of course, synonymous with the fluctuations seen in experimental data. Note that Reynolds-averaged Navier-Stokes (RANS) will not calculate the dynamic eddy behaviour but is instead a representative behaviour of non-dynamic, time-averaged eddy behaviour. As such, the RANS instantaneous velocity as a function of time and space can never have the chaotic velocity wiggles that are resolved significantly by LES and more so by DNS, as shown in the figure below.

A close inspection of the above figure shows significant differences and similarities in velocity distribution. The main idea is that although all three provide useful details of the velocity field, RANS is faster than LES, which is faster than DNS; conversely, DNS is more detailed than LES, which is more detailed than RANS. It is also worthy of mention that theoretically speaking, DNS will calculate all turbulent systems and cases, while LES can do so for most situations, whereas RANS is much more limited in applicability. The choice of turbulence model is ultimately based on financial resources, time constraints, computational resources, and the necessary level of output required of the simulation. In conclusion, we should use the fastest model that gets reasonable accuracy and details for the system of interest.

For the reasons discussed above, LES can be considered as an intermediate methodology between RANS and DNS, as a balance between output and computational effort. In general, LES is about an order or two of magnitude more time intensive

than RANS but is two to three (or more) orders cheaper than DNS. The reason for LES’s computational cost is primarily due to the number of elements used (i.e., a simulation may require refinement up to the smaller eddies within the Taylor scale). On the other hand, RANS elements are much larger because they do not resolve eddy scales. 

So, in conclusion, we can say that the current working horse of the industry is RANS and can not be replaced in coming years.