Pressure and streamlines

When air is ‘at rest’ — in effect the particles are moving at random — the static pressure on a surface is the ambient pressure at that altitude. The static pressure is the mean force/unit area due to the bombardment of the surface by particles moving at the root mean square of the molecular velocity. That velocity, which corresponds with the velocity of sound at that altitude, may be thought of as the typical limiting velocity at which a pressure pulse may be transmitted through the air. As long as the ambient temperature of the air is unchanged by the passage of a body, the speed of sound remains constant.

The static pressure varies with the head of air above the level at which it is measured. The head of air possesses energy in that it has potential energy due to position, and the static pressure is capable of doing work. If the air is in motion at the pressure level the particles also possess kinetic energy, measured in terms of the dynamic pressure 0.5ρV2. As soon as the air is in motion in a particular direction the motion is no longer random but directed.

Fig. 5.2 (a) and (b) The physical view of the effect upon the static pressure within an elastic tube, caused by particles of air colliding with the walls, when moving about at random with root mean square velocity, a (their local speed of sound, which they are unable to exceed unforced). When set in directed motion at velocity V, their components of motion at right angles to the tube wall are reduced, and also the static pressure caused by their impacts.

Now consider a flexible tube immersed in air with equalized pressures both inside and out as shown in Fig. 5.2(a). If one end of the tube is separated from a low-pressure chamber by a diaphragm which is then punctured, the air within the tube will flow with directed motion towards the low-pressure chamber. If the pressure difference is sufficiently small then the density and temperature of the confined air will be unaffected and the particles will move with a velocity less than the velocity of sound a. One might then draw a vector diagram for a particle in which the vector resultant is equivalent to the velocity of sound (Fig. 5.2(b)). Clearly, the component of velocity, v, normal to the wall of the tube is less than when the motion is random. Therefore the static (normal) pressure is reduced and the walls collapse inwards. The higher the directed velocity V the lower the pressure p₁.

The phenomenon discovered by Leonard Euler (1707—1783) is expressed in a theorem of Daniel Bernoulli (1700—1782) which states that in the flow of an incompressible fluid — if we disregard gravity (i.e. any change in potential energy) and friction — the sum of the pressure head and velocity head is a constant:

2 V 5 . 0 p+ ρ = a constant (5-1)

The argument can be extended by replacing the flexible tube in Fig. 5.2 by a stream tube bounded by streamlines in a directed mass of air — a streamline being an imaginary line along which motion is wholly tangential. When a body is immersed in such a mass of air, particles are displaced at different velocities by different parts of the body surfaces. Where local velocities are increased, the pressure is decreased and the streamlines move together. Where local velocities are decreased, the pressure increases and the streamlines move apart.

In reality air is compressible, but another relationship helps us to visualize the behavior of a stream tube when the density and static pressure of the air are changing. The relationship depends upon the conservative nature of the air, i.e., mass is neither lost nor gained, so that the mass contained within a stream tube and moving with velocity V is

V A

where A is the cross-sectional area of the tube. If the density remains unchanged then a decrease in A is accompanied by an increase in V and vice versa. If the velocity increases greatly and the pressure drop is large enough to decrease the density of the air, then the change will be accompanied by a smaller decrease in the cross-sectional area of the stream tube.

The foregoing explanation presupposes that the mass of air is in directed motion, but, as we said earlier, this has nothing to do with the reality of air being more or less at rest with an aeroplane passing through it. Under these conditions the surfaces of the aeroplane gather up some of the air to a certain extent, and it follows that those particles swept along by the aeroplane impose a reaction against the surfaces of the airframe which is felt as increased pressure.

Eventually the particles free themselves to return to their undisturbed condition and the pressure decreases again. When the particles are swept along they have the lowest velocity relative to the aeroplane and exert the highest pressure. When they are in the process of returning to their undisturbed condition their relative velocity increases, and the pressure begins to drop. The pressure distribution over the surface of the body is therefore a function of the relative velocity, and it is that velocity which is used in aerodynamic calculations.

In this way two quite different situations may be treated in the same way mathematically, but although a fixed body in a wind-tunnel can have the same aerodynamic laws applied to it as an identical body in motion through air, we must not fall into the lazy trap of confusing the one with the reality of the other.

These relationships are the root of a most important design technique: that if we wish to obtain a particular sort of pressure distribution around a part of the airframe at a given design point, then it is possible to calculate the necessary profile to induce it. Two such examples of Computational Fluid Dynamics (CFD) modeling, given by A. Jimenez-Garzon to an IMechE Seminar in 1996, are shown in Fig. 5.2(c) and (d).

Fig. 5.2 (c) and (d) The computer software view, showing computational Fluid Dynamic Methods (CFD) for the EF 2000 Eurofighter. The singularity method (panels) in (c) is used to determine subsonic and supersonic linear flow conditions. The Euler method in (d) is to reproduce non-linear flow conditions in the transonic regime.