Reynolds Average Navier-Stokes Equation

Reynolds Average Navier Stokes Equation

Introduction:

Most fluid flows, especially those in pipes and ducts, are turbulent. A characteristic feature of a turbulent flow is a small-scale, high-frequency random fluctuation superimposed on a main flow which has an identifiable direction. Normally, the magnitude of the fluctuation is in the range of 5-10% of the magnitude of the main flow. It is remarkable that such a minor irregularity has such a profound impact on the character of the flow.

Characteristic of Turbulence:

Both diagrams show the instantaneous velocity as a function of time at a fixed position in pipe. In addition, in diagram (b), a horizontal line in brown color is shown to indicate the time independence of a laminar flow. Also, in the same diagram, the fluctuating flow is earmarked with a blue horizontal line which represents the time average of the fluctuating velocities. The averaging process is defined by:

It is standard practice to describe a turbulent velocity field as a superposition of the time averaged velocity U and the fluctuating component u’ as follows:

- Eq 1

Time dependent Navier Stokes Equation:

The unsteady Navier Stokes equation in X direction is given by:

- Eq 2

After substitution of eq (1), the leftmost term in the Navier Stokes equation becomes:

In arriving at the rightmost term of this equation, advantage has been taken of the fact that the time-independent quantity U (u bar) has a zero time-derivative. the time average of is zero since the time average of u’ is zero.

The focus then shifts to the next term in eq (2).

The solution of each right-hand side term can be best understood by viewing the graphical representation of each term displayed below:

The (a) part of the figure represents a quantity that is independent of time, so that its time-average value is equal to the value of the time-independent quantity. The (b) part shows a time-varying quantity which is a multiple of u' . Since the time average of u’ is zero, so is the time average of . The (c) part of figure shows an oscillating function whose value is always positive. Its time average is .

The next term to be considered is

which, upon time averaging, gives rise to:

The same solution results out for the last term in the Navier Stokes equation:

Now, time averaging the remaining terms in the Navier Stokes equation, we get:

Assembly of RANS Equation:

Assembling and rearranging the above terms of the Navier Stokes eqation in X direction, we get:

The terms that appear in the last bracket are called Reynolds Stresses. These “stresses” are the source that creates the turbulent disturbance of the otherwise steady Navier-Stokes equations.

Also, an often-cited form of the RANS equation:

With this rearrangement, the terms involving the fluctuations take on an appearance which supports their designation as stresses. For example, since is the viscous normal stress in a laminar flow, the quantity is termed as the stress due to turbulence.