Stream function and vorticity The rotation of a threedimensional velocity field V = ⟨u, v, w⟩ is measured by the vorticity ω = ∇ x V. If ω = 0 at all points in the domain, the flow is irrotational.a. Which of the following velocity fields is irrotational:V = ⟨2, -3y, 5z⟩ or V = ⟨y, x - z, -y⟩?b. Recall that for a two-dimensional source-free flow V = ⟨u, v, 0⟩ , a stream function ψ(x, y) may be defined such that u = ψy and v = -ψx. For such a two-dimensional flow, let ζ = k ⋅ ∇ x V be the k-component of the vorticity. Show that ∇2ψ = ∇ ⋅ ∇ψ = -ζ.c. Consider the stream function ψ(x, y) = sin x sin y on the square region R = {(x, y): 0 ≤ x ≤ π, 0 ≤ y ≤ π}. Find the velocity components u and v; then sketch the velocity field.d. For the stream function in part (c), find the vorticity function ζ as defined in part (b). Plot several level curves of the vorticity function. Where on R is it a maximum? A minimum?
Stream function and vorticity The rotation of a threedimensional velocity field V = ⟨u, v, w⟩ is measured by the vorticity ω = ∇ x V. If ω = 0 at all points in the domain, the flow is irrotational.
a. Which of the following velocity fields is irrotational:
V = ⟨2, -3y, 5z⟩ or V = ⟨y, x - z, -y⟩?
b. Recall that for a two-dimensional source-free flow V = ⟨u, v, 0⟩ , a stream function ψ(x, y) may be defined such that u = ψy and v = -ψx. For such a two-dimensional flow, let ζ = k ⋅ ∇ x V be the k-component of the vorticity. Show that ∇2ψ = ∇ ⋅ ∇ψ = -ζ.
c. Consider the stream function ψ(x, y) = sin x sin y on the square region R = {(x, y): 0 ≤ x ≤ π, 0 ≤ y ≤ π}. Find the velocity components u and v; then sketch the velocity field.
d. For the stream function in part (c), find the vorticity function ζ as defined in part (b). Plot several level curves of the vorticity function. Where on R is it a maximum? A minimum?
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