BASIC CONCEPT OF THE WING IN COMPRESSIBLE FLOW

4- 2-1 Temperature Effects in Compressible Flow

It is a peculiarity of all compressible flows that their aerodynamic processes are always coupled with thermodynamic processes. The pressure changes in the flow, in general, are connected to temperature changes that may be determined from the equation of state [Eq. (1-la)].

Stagnation flow An outstanding station in the flow about a body is, according to Fig. 4-1, the front (upstream) stagnation point, at which the velocity is zero. The flow quantities at the stagnation point will be designated by the index 0. The pressure in the undisturbed flow of velocity is p„, the density £«,, and the temperature Too. At the stagnation point, the velocity is vv0 = 0, and the pressure, density, and temperature are p0, Q0, and T0, respectively. A pressure increase Ap = Pq — p„ takes place on the streamline incident on the stagnation point, which causes a temperature increase AT=T0—T00. The pressure coefficient at the stagnation point is obtained with steady, that is, isentropic compression as

The dependence of the pressure coefficient at the stagnation point cp0 on Маж is shown in Fig. 4-2a. For moderately high Mach numbers of the incident flow,

Figure 4-1 Temperature rise through com­pression.

particularly in the subsonic range, Eq. (4-1) is reduced by binomial expansion to cpо ~ 1 + Маїо, as also shown in Fig. 4-2. Agreement of the approximation with the exact formula, Eq. (4-1), is quite good up to Max = 1. For Ma„ 0, Eq.(4-1) becomes the well-known formula for the stagnation pressure of incompressible flow, Po – P oo = (@=o/2)w«, which is the basis for velocity measurements with the Prandtl impact-pressure tube (pitot tube). Such a tube measures the pressure difference (pо —poo) in compressible flow as well.

At an incident flow of supersonic velocity, the pressure changes from p„ to pQ discontinuously through a shock wave located somewhat upstream of the stagnation point (Fig. 4-2д). The pressure change can be determined in this case by first computing the pressure jump across the shock wave from the equations of the normal shock. In the subsonic flow behind the shock wave, the pressure change is isentropic. The result of this computation is

At very high Mach numbers, Ma„ -» °°, the pressure coefficient approaches a finite value, which for air of 7= 1.405 is cp0 max = 1-84. It can be seen from Fig. 4-2я that for supersonic incident flow, the pressure increase at the stagnation point with unsteady compression (which describes the physical reality) is considerably smaller than that obtained from the computation of steady compression.

Temperature The pressure increase of the stagnation point is always tied to a temperature increase. It is obtained from the energy equation as

This temperature increase at the stagnation point is shown in Fig. 4-26. It is equally valid for steady and for unsteady compression. It increases with the square of the velocity, and therefore, reaches appreciable values in the supersonic incident flow range. Note that a temperature increase according to Eq. (4-3) occurs not only at the stagnation point and its vicinity but, approximately, everywhere along a solid wall. In a thin layer (friction or boundary layer) close to the wall, the kinetic energy of the moving gas is transformed into heat through viscous effects (see Fig.

4- 3). This results in heating of the wall by an amount AT= T — Тж, which can be represented approximately by a relationship similar to Eq. (4-3), it can be realized, therefore, that a “heat cushion” is found over the entire surface of a body immersed in a flow of high velocity. In the immediate vicinity of the stagnation point this heating is produced by compression, and on the remaining portion of the surface by friction.