Question 4.7: Vorticity Contours in a Two-Dimensional Flow Consider the CF...
Vorticity Contours in a Two-Dimensional Flow
Consider the CFD calculation of two-dimensional free-stream flow impinging on a block of rectangular cross section, as shown in Figs. 4–34 and 4–35. Plot vorticity contours and discuss.
We are to calculate the vorticity field for a given velocity field produced by CFD and then generate a contour plot of vorticity.
Analysis Since the flow is two-dimensional, the only nonzero component of vorticity is in the z-direction, normal to the page in Figs. 4–34 and 4–35. A contour plot of the z-component of vorticity for this flow field is shown in Fig. 4–48.
The blue region near the upper-left corner of the block indicates large negative values of vorticity, implying clockwise rotation of fluid particles in that region.
This is due to the large velocity gradients encountered in this portion of the flow field; the boundary layer separates off the wall at the corner of the body and forms a thin shear layer across which the velocity changes rapidly. The concentration of vorticity in the shear layer diminishes as vorticity diffuses downstream. The small red region near the top right corner of the block represents a region of positive vorticity (counterclockwise rotation)—a secondary flow pattern caused by the flow separation.
Discussion We expect the magnitude of vorticity to be highest in regions where spatial derivatives of velocity are high (see Eq. 4–30). Close examination reveals that the blue region in Fig. 4–48 does indeed correspond to large velocity gradients in Fig. 4–34. Keep in mind that the vorticity field of Fig. 4–48 is time-averaged.
The instantaneous flow field is in reality turbulent and unsteady, and vortices are shed from the bluff body.
\vec{\zeta } = \left(\frac{∂w}{∂y} – \frac{∂\upsilon }{∂z} \right)\vec{i} + \left(\frac{∂u}{∂z} – \frac{∂w}{∂x} \right)\vec{j} + \left(\frac{∂\upsilon }{∂x} – \frac{∂u}{∂y} \right)\vec{k} (4.30)
