Potential flows

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If the Reynolds number is large, and there is no separation, the boundary layers are thin and are close to the solid walls. Then one can neglect the boundary layers entirely and consider the potential flow occupying the entire flow domain.

Typically, one would know the shape of the solid wall, for example, the shape of an aerofoil in an otherwise unbounded domain. Usually, in such a case one would know also the flow velocity far away from the aerofoil. It is convenient to consider such a flow in the frame of reference fixed of the aerofoil itself. This is like looking at the wing from the aircraft window: the wing does not move, but the air moves past the wing. The typical task is then to determine the flow velocity everywhere in the flow domain.

First of all, one needs to understand how the velocity of a potential flow behaves at the surface of the aerofoil, or, more generally, at any solid surface. In a real viscous flow the velocity of the fluid at the wall equals the velocity of the solid wall itself. This boundary condition is usually considered as a sum of two constraints: the no-slip condition and the impermeability condition. No-slip means that the velocity component parallel to the wall is zero. Impermeability means that the velocity component perpendicular to the wall is zero. The no-slip condition is due to friction, while the impermeability condition is due to mass conservation: the fluid cannot flow into or out of a solid wall. It turns out that in general a potential flow cannot satisfy both. In this situation mass conservation takes precedence. This is natural since viscous flows are typically not potential, and the wall-parallel velocity can adjust to become zero at the wall inside the boundary layer. In the potential flow the fluid slides (or slips) along the wall. The case study of the potential flow past a circular cylinder illustrates this feature.

In simple cases the knowledge that the fluid particles do not rotate and do not penetrate the walls combined with knowledge of how they move across the open boundaries of the domain can be sufficient for imagining intuitively the entire flow.

In more complicated cases the superposition principle is helpful. Potential flows have an important property: if we take the some of the velocities of two potential flows and add them together, the result will also be a potential flow. The superposition principle is valid for potential flows because in this case the angular velocities, and hence vorticities, also add, and a sum of zeros is zero, so that a sum of potential flows is a potential flow.

Hence, once can imagine two simple potential flows and then superimpose them. One can also superimpose potential flows that are not so simple. It is worth, therefore, to memorise a number of potential flows, to be used as elements for constructing other potential flows.  Case Studies contain a few examples, and there are numerous examples of potential flows on the Web.

Depending on the circumstances, there might be only a unique potential flow satisfying the boundary conditions, or there might be families of such flows. This important question is considered in the next page.

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