7.1 Approach for finding potential flow solutions to the Navier-Stokes equations [potential flow top]

For irrotational flows, we can define velocity fields in terms of thegradient of a velocity potential. This leads to a simple approach (Figure 7.1) for satisfying conservation of massand conservation of momentum, but problems when we try to satisfy boundary conditions.

Conservation of mass can be satisfied by solving the Laplace equation given the boundary or freestream conditions, which is numerically and analytically easy as compared to solving the Navier-Stokesequations. The boundary condition we use at walls is the no-penetration condition, i.e., that the velocity normal to a solid boundary is zero. In the freestream or at inlets, the boundary condition is a specified velocity. Solution of the Laplace equation specifies the velocity field everywhere in the flow, including at the boundary. Since the velocity field is then known, the conservation of momentum equations (Navier-Stokes equations) can be satisfied by rearranging the Navier-Stokes equations to solve for the pressure as a function of space, though the pressure is of little use in microscale irrotational flows and is rarely calculated.

Unfortunately, while this solution satisfies the no-penetrationcondition at the walls, it fails in general to satisfy the no-slipcondition at the walls. The Laplace equation solution thus satisfies the Navier-Stokes equations and one of the boundary conditions (the no-penetration condition), but it fails to satisfy the other boundary condition (the no-slip condition). Potential flow solutions are therefore useful for describing flow reasonably far from walls, and these solutions are relevant when the regions ofvorticity are localized to a region we are willing to ignore or treat separately. Chapter 6 illustrates an important example of such flows, since the vorticity was contained inside a thin electrical double layer.

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