Two ideal vortices with circulation Γ are located at (−b, b) and (b, b) at the time of interest in a semi infinite inviscid fluid (see Figure 5.18). A flat impenetrable surface lies at y = 0. Determine the velocities of the point vortices presuming they are free to move

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Each vortex will feel induced velocities from its own image vortex, the other vortex, and the other vortex’s image. For the vortex on the left (no. 1), these contributions lead to:

[latex]\mathbf{u}(-b,b)=\frac{\Gamma}{2\pi\cdot2b}\mathbf{e}_{x}-\frac{\Gamma}{2\pi\cdot2b}\mathbf{e}_{y}+\frac{\Gamma}{2\pi\cdot2\sqrt{2}b}\left(\frac{\mathbf{e}_{x}+\mathbf{e}_{y}}{\sqrt{2}}\right)=\frac{\Gamma}{4\pi b}\left(\frac{3}{2}\mathbf{e}_{x}-\frac{1}{2}\mathbf{e}_{y}\right) [/latex].

For the vortex on the right (no. 2), these contributions are:

[latex]\mathbf{u}(b,b)=\frac{\Gamma}{2\pi\cdot2b}\mathbf{e}_{x}+\frac{\Gamma}{2\pi\cdot2b}\mathbf{e}_{y}+\frac{\Gamma}{2\pi\cdot2\sqrt{2}b}\left(\frac{\mathbf{e}_{x}-\mathbf{e}_{y}}{\sqrt{2}}\right)=\frac{\Gamma}{4\pi b}\left(\frac{3}{2}\mathbf{e}_{x}+\frac{1}{2}\mathbf{e}_{y}\right) [/latex].

These results show that the vortex pair near a flat wall has a higher horizontal velocity than a single vortex at the same distance from the wall. Plus, vortex no. 1 is moving toward the wall while vortex no. 2 is moving away from the wall at the instant shown.

Question 5.6: Two ideal vortices with circulation Γ are located at (−b, b)...

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