Unsteady Aerodynamic Flows

This chapter will examine the general low-speed unsteady potential flow problem. Specifically, we will revisit flow-field modeling in the unsteady case, and also the unsteady Bernoulli equation for the unsteady pressure. The special case of unsteady airfoil flows will be examined in more detail.

3.1 Unsteady Flow-Field Representation

Chapter 2 defined the source density and vorticity as the divergence and curl of the velocity field.

a = V-V ш = VxV

These involve only spatial derivatives and hence apply instantaneously even if V is changing in time. Simi­larly, representation of the velocity field via its source, vorticity, and boundary contributions

V(r, t) = V + V + Vb

involves only spatial integrations, and likewise applies instantaneously. An unsteady flow can therefore be represented in the same manner as a steady flow, but all the relevant quantities will now depend on time as

well as space.

V(r, t)

= 4п111а{Г’Л) r-rfdX’dy’dZ’


Vw(r, t)

– 4vr US dx’dy’dz’



= V» (observer moving steadily in airmass)



= 0 (observer fixed in airmass)


We see that the unsteadiness of the velocity field is captured entirely by the time dependence of the source and vorticity fields a(r, t), ш(r, t). Furthermore, since the lumping process is strictly spatial, as in the steady case the integrals above can be simplified using the lumped unsteady sheet strengths X(s, e,t), 7(s, e,t), line strengths Л(г, і), V(e, t), or point strengths £(t). For free vorticity such as in trailing wakes, an alternative approach is to move the vortex points rather than change the singularity strengths. In that case (7.2) is still valid, but now the integration points Г are functions of time.

In typical steady flow applications so far we have assumed that the body is fixed, so that the last boundary – condition component Vb is the freestream velocity V». In unsteady flow applications it is often more convenient to represent the body motion explicitly via its velocity U(t) and rotation rate O(t). In this case we choose Vb = 0, and V then represents the “perturbation velocity,” which is what’s seen by an observer stationary with respect to the airmass.