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second derivative output

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Old   September 25, 2022, 11:39
Default second derivative output
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David Weyburne
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Is it possible to add an OpenFOAM option that would directly output the second derivative values along with the normal velocities, pressures, temperatures, etc. when using the slice tool, for example? This would have the advantage that the generated second derivatives are of the high order difference scheme used in the actual simulation and present a way to map areas where viscosity is important.

Why would you want the second derivative values? There are many flow simulations where 'viscous' fluid flow along a wall becomes an important factor in the overall flow situation. The momentum conservation equations tell us that viscosity will be important where the second derivative of the velocity is important. I have generated second derivative velocity profiles by taking slices using the slice tool from the wall to a place deep in the bulk flow and then numerically differentiating the values. The thickness and shape of the viscous region can then be determined by using the method of integral mathematical moments of the second derivatives (I wrote several papers on the subject if you want to check it out). This method borrows from the commonly used method to describe probability density functions using integral moments. Thus, you can start to make mean viscous location, viscous width, viscous thickness, and viscous shape plots along the wall using these second derivative-based integral values.

In OpenFOAM, making velocity profile slices is accomplished using the 'slice' tool but that then requires a pass through a numerical derivative module to get the second derivatives. This latter step is eliminated if there was an OpenFOAM option that would directly output the second derivative values along with the normal velocities, pressures, etc. This has the advantage that the generated second derivatives are of the high order difference scheme used in the actual simulation. If second derivative values were available to paraFoam, 2D and 3D 2nd derivative (viscous) contour plots could be generated directly.
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