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I am running a simulation of fluid in a pipe.

The fluid in this pipe is "swirling" (instead of just moving in one direction, it's rotating as well. Like the Helix below.enter image description here

The fluid is faster on the edges of the pipe. The z direction is the flow of the fluid. X and Y are the fluid flowing.

The Fluid on the edge is moving faster because of an external motor. This adds an extra velocity component to the fluid, particularly the portion of the fluid away from the center. In the plot below the tangential velocity vs. the radius is shown. In the plot you can see the velocity rising as it moves away from the center.

enter image description here

With this plot, I am trying to calculate the Kinetic Energy of the fluid, excluding pressure. Originally, I planned on putting these values into a matrix, summing them, then diving by the number of elements in the matrix. I am not sure if this the appropriate way to get the "overall velocity" to use to find kinetic energy.

What is the appropriate way to calculate the kinetic energy of this fluid?

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  • $\begingroup$ Is the rising velocity away from the center an experimental finding? I ask this because the external motor appears to be imparting vorticity to the fluid flow, and a vortex creates low pressure and rising velocity toward the center. The gradient of vortex pressure forces the fluid to flow more rapidly toward the center. Also, I would expect shear stress where the fluid contacts the walls of the pipe to lower the velocity at the edges of the pipe. $\endgroup$ – Ernie Jul 17 '15 at 15:58
  • $\begingroup$ This is an ideal case, as of now, what you stated above is not being taken into account. I guess all I am trying to ask in this question, is it safe to just sum the velocity points then divide by the number of points? Then use that value to calculate KE? $\endgroup$ – Adam Jul 17 '15 at 16:02
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You can integrate the velocity values over $r$, the radius, using whatever integration technique you would like (trapezoidal rule, Simpson's rule, etc) and use the mean-value theorem to get the average velocity for each radial line. You then have to account for all $2\pi$ radial lines. Assuming the flow is incompressible, you can then find kinetic energy by $K.E. = 1/2 \rho \langle v \rangle^2$.

Alternatively, if it is not incompressible, you would need to find the point-wise kinetic energy and integrate that over the radius and so on.

In other words, you have discrete data points that you measure and you just need to use an integration rule to find the mean value of the function across the surface, whether the function you are integrating is the pointwise velocity or the pointwise kinetic energy is up to you. Although, if the flow is compressible, the choice is made for you.

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