Some Geometric or Kinematic Properties of the Velocity Field

Next we specialize for the velocity vector Q two integral theorems which are valid for any suitably continuous and differentiable vector field.

1. Gauss’ Divergence Theorem. Consider any volume V entirely within the field enclosed by a single closed bounding surface S as in Fig. 1-1.

j^Q-ndS = yyy (V ■ Q)dV. (1-14)

S ■ V

Here n is the outward-directed unit normal from any differential element of area dS. This result is derived and discussed, for instance, in Sections 2.60 and 2.61 of Milne-Thompson (1960). Several alternative forms and some interesting deductions from Gauss’ theorem are listed in Section 2-61. The theorem relates the tendency of the field lines to diverge, or spread out within the volume V, to the net efflux of these lines from the boundary of V. It might therefore be described as an equation of continuity of field lines.

2.

Stokes’ Theorem on Rotation. Now we consider a closed curve C of the sort employed in (1-12) and (1-13), except that the present result is instantaneous so that there is no question of a moving path composed of the same particles. Let S be any open surface which has the curve C as its boundary, as illustrated in Fig. 1-2. The theorem refers to the circulation around the curve C and reads

(1-16)

is called the vorticity and can be shown to be equal to twice the angular velocity of a fluid particle about an axis through its own centroid. The theorem connects the spinning tendency of the particles lying in surface S with the associated inclination of the fluid at the boundary of S to circulate in one direction or the other. Sections 2.50 and 2.51 of Milne-Thompson (1960) provide a derivation and a number of alternative forms.

1-4 The Independence of Scale in Inviscid Flows

Consider two bodies of identical shape, but different scales, characterized by the representative lengths h and /2, moving with the same velocity Qb through two unbounded masses of the same inviscid fluid. See Fig. 1-3. This motion is governed by the differential equations developed in Section 1-1 plus the following boundary conditions: (1) disturbances die out at infinity, (2) Qn = Qb • n at corresponding points on the two surfaces, n being the normal directed into the fluid.

We first treat the case of a steady flow which has gone on for a long period of time so that all derivatives with respect to t vanish. We can then apply the Newtonian transformation, giving the fluid at infinity a uniform motion minus Qb and simultaneously bringing the body to rest. Thus the boundary condition No. 2 becomes Qn = 0 all over the surfaces. We take the differential equations governing these two similar problems and make the following changes of variable: in the first case.

xx	= x/h,	<ci II	Zi = z/h;	(1-17)
in the second case, *2	= x/h,	у	z2 = z/l2-	(1-18)

In the dimensionless coordinate systems the two bodies are congruent with each other. Moreover, the differential equations become identical. That is, for instance, continuity reads, in the first case,

BjpU) , d{pVl, аШ1 = о

dxi dyi dz і

and in the second case,

d(pU) , dJpV) d(pW) =

dx2 dy2 dz2 ‘

We are led to conclude, assuming only uniqueness, that the two flows are identical except for scale. Velocities, gas properties, and all other dependent variables are equal at corresponding points in the two physical fields. This is a result which certainly seems reasonable on grounds of experience.

The same sort of reasoning can be extended to unsteady flows by also scaling time in proportion to length:

Xt = x/h, Ух = у/lx, zx = z/l 1, tx = t/lx, (1-21)

x2 = x/l2, 2/2 = y/h, z2 = z/l2, t2 = t/l2, (1-22)

in the two cases. With regard to boundary conditions, it must also be specified that over all time Qb(h) in the first case and Qbih) are identical functions.

Evidently, it makes no difference what the linear scale of an ideal fluid flow is. In what follows, therefore, we shall move back and forth occasion­ally from dimensional to dimensionless space and time coordinates, even using the same symbols for both. A related simplification, which is often encountered in the literature, consists of stating that the wing chord, body length, etc., will be taken as unity throughout a theoretical develop­ment; the time scale is then established by equating the free-stream to unity as well.

One important warning: the introduction of viscosity and heat conduc­tion causes terms containing second derivatives, the viscosity coefficient, and coefficient of heat conductivity to appear in some of the governing differential equations. When dimensionless variables are introduced, the Reynolds number and possibly the Prandtl number[2] appear as additional
parameters in these equations, spoiling the above-described similarity. The two flows will no longer scale because the Reynolds numbers are differ­ent in the two cases. Even in inviscid flow, the appearance of relaxation or of finite reaction rates introduces an additional length scale that destroys the similarity. It is satisfactory, however, for real gas which remains either in chemical equilibrium or in the “frozen” condition thermody­namically.

If shock waves and/or vortex wakes are present in – the field, it can be reasoned without difficulty that they also scale in the same manner as a flow without discontinuities. An excellent discussion of this whole subject of invariance to scale changes will be found in Section 2.2 of Hayes and Probstein (1959). Finally, it might be observed that differences between the two flows in the ambient state of the fluid at infinity, such as the pres­sure, density, and temperature there, may also be included in the scaling by referring the appropriate state variables to these reference values. The Mach numbers in the two flows must then be the same.