In this lesson, we will look at how to obtain governing equations for turbulent flows. You might
be wondering, I already know that Navier-Stokes equations are the one true set of equations to
describe all the flows, why should I worry about another set of governing equations to describe
turbulent flows. As we already know, turbulence is an extremely difficult beast to tame,
which is evident from the fact that there are no exact solutions of the Navier-Stokes equations
for a turbulent flow. You might then ask, I have a super powerful computer, why can't I use
state-of-the-art numerical algorithms to solve the original NS equations.
There is a problem there too. We know that turbulence is a high Reynolds number phenomenon
and that the scales of the turbulent flows increase with increasing Re number.
If you were to resolve every scale of the flow for a practical industrial problem, say flow over
a car, at a Reynolds number of a 1 million, you would require approximately 31.5 trillion cells!
So, it is safe to say that it is practically impossible, even in today's day and age.
It is therefore necessary to identify simplifications that permit a practical study
of turbulent flows. Onee such is Reynolds averaging of the Navier Stokes Equations,
RANS, which we will discuss in this lesson.
Take a general turbulent flow case. If we measure any flow property at any point in the flow,
we should expect to see a graph that looks like below.
The large wavelength oscillations are because of the flow changes,
and the high frequency variation is the impact of the turbulent structures on the mean flow.
For any flow quantity, this separation can be written mathematically as shown here. The mean
quantity is obtained by time averaging the value of the instantaneous property over a large amount
of time. This large amount of time is selected such that time average of all the fluctuations
over this time range is 0. Osborne Reynolds recast the Navier-Stokes equations using this
decomposition, and these equations are called the Reynolds averaged Navier-Stokes equations
or RANS equations in short. These equations describe the mean fluid flow,
and we will soon see why how and why turbulence comes into picture.
Let us quickly redo what Reynolds did to obtain RANS equations. For simplicity, we will consider
the incompressible Navier-Stokes equations with no body forces. Note that, this assumption is
just for our discussion sake. This averaging can be performed for the full NS equations.
Let us decompose just the continuity equation.
If we now time average, because of the fact that time average of fluctuations is zero,
all the fluctuation terms can be dropped and we arrive at the following equation which
is the RANS continuity equation. This form is identical to the original continuity equation,
except that it is now time averaged and valid only for the mean flow.
Let us now apply the same technique to the momentum equation. After removing and
rearranging terms, we arrive at the following equation that is the RANS momentum equation.
Notice the circled term, it is the time average of the product of fluctuating velocities.
It can be thought of as a stress applied
due to the fluctuating velocity field and is generally referred to as the Reynolds stress.
This Reynolds stress is a result of time averaging the momentum equation,
and cannot be neglected in turbulent flows. Let us now rearrange the momentum equation as shown here,
where tau is the total stress tensor which is a sum of the laminar and turbulent stresses.
The Reynolds stress term is a tensor, meaning we have 9 terms, all of which are unknown and were
created as a result of time averaging. Let us take a step back now. Starting from the incompressible
NS equations, where we had 4 equations (1 continuity and 3 momentum) for 4 unknows
(p,u,v,w), we ended up with 4 equations with 13 unknows after time averaging!
Where 9 of them are the Reynolds stresses. In order to make these set of equations well defined,
we either need to have 9 additional governing equations, one each for each of the Reynolds
stress terms or express the Reynolds stress tensor in terms of the mean flow variables,
thereby avoiding adding new equations. This conundrum is referred to as the closure problem
and has been a field of significant research, even to this day. The effect of turbulence and its
impact on the flow-field is a consequence of how well these terms are approximated. As a result,
there is a significant emphasis on modeling these terms, this science or some even refer to it
as art is called Turbulence modeling. Unlike the viscous stresses, which were
related to the strain rate through rigorous physical arguments, no such exacting approach
can be emulated for the Reynolds stress terms. With that, we will wrap up this discussion.