RANS or the Reynolds averaged NS equations govern the mean flow development of a fluid flow.
However, while arriving at these equations we run into what is called the closure problem.
The time averaged momentum equations contain nine additional terms referred
to as the Reynolds stresses which are unknown, and require additional equations to be modeled.
In general, there are two major approaches that have been commonly implemented to provide
closure equations to the RANS system. Reynolds stress transport
A more general method to provide closure to the RANS system is to develop a transport equation
for each of the Reynolds stress terms. This process is referred to as the
Reynolds Stress Modeling approach Though it is a physics based approach
and can offer higher accuracy in modeling the effects of turbulence, it is extremely resource
intensive and hence is not widely used. The Boussinesq Hypothesis
Joseph Boussinesq in early 1900's proposed an ingenious approach of relating the Reynolds
stresses to the mean flow velocity gradients. Like how laminar stresses are proportional to
the mean flow gradients through fluid viscosity, Bousinessq assumed that the Reynolds stresses
are similarly related but instead through a new variable called the eddy viscosity.
The approach has it downsides such as assumption of isotropic turbulence,
which can hurt the models accuracy in complex flows. However, this approach is the most widely
used turbulence modeling technique even to this day because of its robustness.
Let us now get into a bit more detail regarding the Boussinesq approach.
For incompressible flow, the relation Boussinesq formulated is shown here. As previously mentioned,
eddy viscosity is analogous in its effect to molecular viscosity.
However, it is not a property of the fluid. Instead, it is a function of space and time.
Using this approach, the 9 unknown Reynolds stresses can be calculated based on the mean
flow gradients, and by knowing the eddy viscosity and the turbulent kinetic energy. Numerous methods
have been developed in order to model the eddy viscosity and the turbulent kinetic energy.
Quite often, the TKE term is neglected due to the relatively negligible
contribution of the normal stresses compared to the turbulent stresses.
As you already know, there are no analytical solutions to turbulent fluid flows. One has to
depend on numerical studies to solve the equations and obtain solutions to the fluid flow problem.
Turbulence Modeling approaches based on Boussinesq hypothesis
have been the workhorses of the computational codes, and are still widely used even today.
However, a significant drawback is that all the models are limited in their applicability.
There is no single model that can describe a turbulent fluid flow under any given condition.
There is still a significant level of active research going on to either improve
the existing methods or propose more efficient methods. Getting into details of each of these
models is beyond the scope of this course. But instead, we will try to get a birds eye view
of the common philosophy of the eddy viscosity based models.
Let us once again recall the boussinesq hypothesis. In order to calculate the
Reynolds stresses, all we need to know is the eddy viscosity. Proposed models range from
simple algebraic equations to multiple transport equations to predict eddy viscosity. One of
the first successful models was that of Ludwig Prandtl and is known as the Mixing length model.
Prandtl postulated that the eddy viscosity must vary with a parameter referred to as the "mixing
length" using the relation shown here. This mixing length represents the distance over
which fluid parcels retain their original momentum before diffusing in the turbulent boundary layer.
Prandtl calculated this mixing length based on the wall normal distance, as shown here. Based
on multiple experiments, Van Driest later improved the mixing length calculation by adding a damping
function to the original equation. The constant kappa is referred to as the Von Karman constant
and was determined to be 0.4 based on experimental data. Even though the model is fairly simplistic,
it is quite accurate for wall-bounded flows without separation under small pressure gradients.
Algebraic models or the zero equation models, refer to models which calculate eddy viscosity
directly from the flow variables without solving additional equations. Though they are inefficient
for complex flows, they can be useful to a general estimate of the turbulent flow field.
Two of the common algebraic models were proposed by Baldwin-Lomax and Cebeci-Smith.
To improve the efficiency of resolving the turbulent boundary layer, 1-equation turbulence
models were later proposed. Here, a separate transport equation for a turbulence quantity.
One of the most famous and widely used 1 equation turbulence model is the spalart-allmaras model.
This model proposes a transport equation directly for t ushe modeled eddy viscosity as shown here.
This model was originally developed for predicting wall-bounded flows in aerospace applications.
Over the years, this model has shown to yield
accurate results even for boundary layers experiencing adverse pressure gradients.
The workhorses of turbulence modeling methods are the two equation models. IN this approach,
two transport equations are solved for to individual turbulent quantities,
based on which the eddy viscosity of the flow field is computed.
The most popular models are the k-epsilon and k-omega . In the k-epsilon approach,
transport equations for turbulent kinetic energy and the turbulent dissipation rate are solved.
The eddy viscosity is then calculated using the relation shown here. On the other hand,
in the k-omega approach, instead of the turbulent dissipation rate, a transport equation for the
specific rate of dissipation is solved. The eddy viscosity is now calculated using this relation.
The equations shown here are the transport equations for TKE and epsilon
as proposed by Launder and Spalding, which is one of the classical two-equation
model. Here, the sigmas are the effective prandtl numbers, relating the eddy diffusion of k and e
to the momentum equation eddy viscosity. All the constants in these equations are defined
by calibrating the model to match canonical experimental cases involving wall-bounded flows.
Modern CFD makes use of other more robust 2 equation models. However, the original k-e
is still the starting point for those wishing to learn the basics of turbulence modeling.
As mentioned earlier, an alternative to the boussinesq methodology
is the Reynolds Stress modeling technique.
Here, transport equations are derived and solved for each of the individual reynolds stresses.
The advantage of such an approach is that one can preserve the "exactness" of the original RANS
equations and avoid many modeling assumptions. The RSM models are not limited by the assumption of
isotropic turbulence which is the core philosophy of the eddy viscosity based methods. As a result,
these RSM models are capable of predicting complex fluid flows, like swirl-dominated flows, where the
eddy viscosity models may fail. The process of deriving these RSM transport equations is quite
involved and we will not get into it in this lesson. With that, let us wrap it up.