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Potential flow

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A flow in which vorticity is zero is called potential flow, or irrotational flow. Since the vorticity is zero

\omega = \nabla \times u = 0

it implies that the velocity is the gradient of a scalar field called the velocity potential, and usually denoted as \phi

u_i = \frac{\partial \phi}{\partial x_i}

At high Reynolds numbers, flow past slender bodies is attached (no boundary layer separation) and the boundary layers are thin. In such situations vorticity is confined to the thin boundary layers and the rest of the flow is irrotational.

Governing equations

In the case of an incompressible flow, \phi satisfies the Laplace equation.

From the continuity equation, we get:

\frac{\partial^2 \phi}{\partial x_i^2}=0

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