IRROTATIONAL FLOW AND THE VELOCITY POTENTIAL

It has been shown that the vorticity in the high Reynolds number flowfields that are being studied is confined to the boundary layer and wake regions where the influence of viscosity is not negligible and so it is appropriate to assume an irrotational as well as inviscid flow outside these confined regions. (The results of Sections 2.2 and 2.3 will be used when it is necessary to model regions of vorticity in the flowfield.)

Consider the following line integral in a simply connected region, along the line C:

/ q • dl = і udx + vdy + wdz

Jc ‘c

If the flow is irrotational in this region then udx + vdy + wdz is an exact differential (see Kreyszig,2 1 p. 741) of a potential Ф that is independent of the integration path C and is a function of the location of the point P(x, y, z):

Ф(x, y, z)=f udx + vdy + wdz (2.18)

JPo

where P0 is an arbitrary reference point. Ф is called the velocity potential and the velocity at each point can be obtained as its gradient

The substitution of Eq. (2.19) into the continuity equation (Eq. (1.23)) leads to the following differential equation for the velocity potential

V • q = V • УФ = У2Ф — 0 (2.21)

which is Laplace’s equation (named after the French mathematician Pierre S. De Laplace (1749-1827)). It is a statement of the incompressible continuity equation for an irrotational fluid. Note that Laplace’s equation is a linear
differential equation. Since the fluid’s viscosity has been neglected, the no-slip boundary condition on a solid-fluid boundary cannot be enforced and only Eq. (1.28a) is required. In a more general form, the boundary condition states that the normal component of the relative velocity between the fluid and the solid surface (which may have a velocity qB) is zero on the boundary:

■•(q-4n) = 0 (2.22)

This boundary condition is physically reasonable and is consistent with the proper mathematical formulation of the problem as will be shown later in the chapter.

For an irrotational inviscid incompressible flow it now appears that the velocity field can be obtained from a solution of Laplace’s equation for the velocity potential. Note that we have not yet used the Euler equation, which connects the velocity to the pressure. Once the velocity field is obtained it is necessary to also obtain the pressure distribution on the body surface to allow for a calculation of the aerodynamic forces and moments.