Vorticity

We start by exploiting the possibility of extracting the vorticity from our 2-D measurement. Vorticity can be estimated by calculating

$$\omega = \frac{\partial V}{\partial X} - \frac{\partial U}{\partial Y}.$$

Several numerical schemes exist for performing this calculation, and three different methods have been implemented in MatPIV. The first is estimation by using Stokes theorem:

$$\omega_{i,j} = \frac{1}{8*\Delta{X} \Delta{Y}} [\Delta{X}(U_{i-1,j-1} +2U_{i,j-1}+U_{i+1,j-1})$$

$$+ \Delta{Y}(V_{i+1,j-1} +2V_{i+1,j}+V_{i+1,j+1})$$

$$- \Delta{X}(U_{i+1,j+1} +2U_{i,j+1}+U_{i-1,j+1})$$

$$- \Delta{Y}(V_{i-1,j+1} +2V_{i-1,j}+V_{i-1,j-1})].$$

This approach integrates to find the circulation around a point. Alternatively we can use a standard differentiation scheme, such as forward or centered differences. MatPIV uses two different and more accurate differential operators, namely least squares and Richardson extrapolation. With the former of these schemes the vorticity can be estimated by

$$\omega_{i,j} = \frac{1}{10\Delta X} (2v_{i+2,j} +v_{i+1,j} -v_{i-1,j} -2v_{i-2,j})$$

$$- \frac{1}{10\Delta Y} (2u_{i,j+2}+u_{i,j+1} -u_{i,j-1} -2u_{i,j-2}),$$

while the latter uses

$$\omega_{i,j} = \frac{1}{12\Delta X} (-v_{i+2,j} +8v_{i+1,j} -8v_{i-1,j} +v_{i-2,j})$$

$$- \frac{1}{12\Delta Y} (-u_{i,j+2}+8u_{i,j+1} -8u_{i,j-1} +u_{i,j-2}).$$

The major difference between the two last operators is that the Richardson extrapolation is designed to produce a smaller truncation error, while the least squares operator reduces the effect of fluctuations. The latter reason is why the least squares operator is often used with PIV measurements and is chosen as the default calculation method in MatPIV.

The file vorticity.m calculates vorticity. Calling should look like:

[vorticity]=vorticity(x,y,u,v,method).

Method should be one of 'centered', 'circulation', 'richardson' or 'leastsq', where the latter is the default. The former of the methods is just an overloaded call to the MATLAB file curl.m (type help curl at the command prompt). This file uses a centered differences approach and will perform well if the velocities are smooth.

If no output argument is included, a figure window will appear showing the result.