How Is Reynolds Stress Calculated from RANS Equations?

[member 428835]

hi pf!

i had an easy question on how to calculate the reynolds stress from the RANS equations. Reynold's Stress is defined as $\rho \langle v_i v_j \rangle$ where $v_i$ is the residual velocity measurement in the $i$ direction.

my question is, how is the reynolds stress actually calculated? for example, suppose we measure velocity only in the x and y directions. let's say we take 4 measurements and find that $v_1 = 1,2,3,4$ and $v_2=4,5,6,7$.

is the following correct: $$ \langle v_i v_j \rangle = \left| \begin{array}{ccc}
\langle v_1 v_1 \rangle & \langle v_1 v_2 \rangle \\
\langle v_2 v_1 \rangle & \langle v_2 v_2 \rangle \end{array} \right|$$

where $\langle v_1 v_2 \rangle$ would be $1*4+2*5+3*6+4*7$ and hence $\langle v_1 v_2 \rangle = \langle v_2 v_1 \rangle$

i appreciate all help!

 

[member 428835]

i should add, part of my confusion is how to we know that 1 corresponds to 4 and 2 to 5 and so on? i mean, if we are measuring, say with hotwire data, is it possible to measure two velocities at the same time?

 

[boneh3ad]

If you have a suitable hot-wire probe, then yes, you can measure multiple components of the velocity simultaneously provided your data acquisition hardware can sample simultaneously. With enough properly-oriented hot wires, you can even concoct schemes to measure all three velocity and vorticity components simultaneously.

 

[bigfooted]

Your question is a little bit confusing, because you talk about RANS and measurements. The Reynolds Averaging approach gives you an equation for the Reynolds stresses, the Reynolds Stress equations. In the RANS approach, you don't calculate the Reynolds stresses from the instantaneous velocity fluctuations but you calculate them, either using the transport equation for the Reynolds stresses or making more assumptions like assuming isotropic turbulence (and solving an equation for the turbulent kinetic energy).

Also note that when you measure the velocity at distinct instances in time, $v_1$ and $v_2$ are instantaneous velocities and you need instantaneous velocities to construct the mean and variance. The number averaged mean $<v_1>=2.5 m/s$ and the number averaged mean of the squared velocity is $<v^2_1>=7.5 m^2/s^2$
The variance is $<v_1v_1>=<v^2_1> - <v_1>^2=7.5-2.5^2 =1.25 m^2/s^2$

The rules for calculating the covariance can be used to calculate the shear stress <uv>. You can see by interchanging u and v that the covariance <uv> must be equal to <vu>.