Supersonic Flow

Mach waves

A particle moving in a compressible medium, such as air, emits acoustic disturbances in the form of spherical waves. These waves propagate at the speed of sound (M = 1). If the particle moves at a supersonic speed, the generated waves cannot propagate upstream of the particle. The spherical waves are enveloped in a circular cone called the Mach cone. The generators of the Mach cone are called Mach lines or Mach waves.

Shock waves

When a fluid at a supersonic speed approaches an airfoil (or a high-pressure region), no information is communicated ahead of the airfoil, and the flow adjusts to the downstream conditions through a shock wave. Shock waves propagate faster than Mach waves, and the flow speed changes abruptly from supersonic to less supersonic or subsonic across the wave. Similarly, other properties change discontinuously across the wave. A Mach wave is a shock wave of minimum strength. A normal shock is a plane shock normal to the direction of flow, and an oblique shock is inclined at an angle to the direction of flow. The velocity upstream of a shock wave is always supersonic. Downstream of an oblique shock, the velocity may be subsonic resulting in a strong shock, or supersonic resulting in a weak shock. The downstream velocity component normal to any shock wave is always subsonic. There is no change in the tangential velocity component across the shock.

In a two-dimensional supersonic flow around a blunt body (see illustration), a normal shock is formed directly in front of the body, and extends around the body as a curved oblique shock. At a sufficient distance away, the flow field is unaffected by the presence of the body, and no discontinuity in velocity occurs. The shock then reduces to a Mach wave. See Compressible flow, Fluid flow

Typical normal shock, oblique shock, and Mach wave pattern in supersonic flow past a blunt body

a gas flow such that the speeds v of the gas particles in the region under consideration are higher than the local speed of sound a. A number of important practical problems are associated with the study of supersonic flow. Such problems arise in the designing of aircraft, rockets, and artillery projectiles with supersonic flight speeds. Practical problems of supersonic flow also appear in, for example, the construction of steam turbines, gas turbines, high-pressure turbocompressors, and supersonic wind tunnels.

Figure 1. Mach cones: (COD) downstream Mach cone, (AOB) upstream Mach cone

Specific features. Supersonic gas flows have a number of qualitative differences from subsonic flows. One important difference is a result of the principle that a small disturbance in a gas is propagated at the speed of sound. When a small change in pressure is produced by placing, for example, a body in a uniform supersonic flow, the influence of the disturbance cannot travel upstream. The influence is carried downstream at a speed v > a and stays within what is known as the downstream Mach cone; in Figure 1, the downstream Mach cone is indicated by COD. In addition, a small disturbance can have an influence at the given point O only when the source of the disturbance is located within the cone A OB, which is called the upstream Mach cone and has its vertex at O. The two cones extend in opposite directions and have the same vertex angle. In the case of a nonuniform steady flow, the regions in which a disturbance can have an effect are bounded not by right circular cones but by conoids, or cone-shaped curved surfaces, with vertices at the given point.

In the case of a steady supersonic flow along a wall with a sharp convex corner (Figure 2,a), disturbances occur at the points on the line marking the corner. The influence of the disturbances is limited by the envelope of the Mach cones, which is a plane inclined to the direction of the flow at an angle μ such that sin μ = α/v₁. After passing through this plane, the flow turns and expands in the angular region formed by the set of plane disturbance fronts, or characteristics, which start at the corner. The flow stops turning when it becomes parallel to the direction of the wall after the corner. If, instead of forming a sharp corner, the two straight sections of the wall meet in a smooth curve (Figure 2,b), then the flow turns gradually and passes through a series of straight characteristics that start at each point of the curved section of the wall. In these two types of flows, which are known as Prandtl-Meyer flows, the gas parameters are constant along the straight characteristics.

Waves in a gas that produce an increase in pressure propagate differently from waves that produce a decrease in pressure. A wave producing a pressure increase travels at a speed greater than the speed of sound. The thickness of such a wave can be

Figure 2. Supersonic flow: (a) along a wall with a sharp corner, (b) along a convex curved wall

very small—of the order of the molecular mean free path. In many theoretical studies the wave is replaced by a surface of discontinuity. Such a surface is known as a shock wave. When the gas passes through the shock wave, the velocity, pressure, density, and entropy of the gas abruptly change.

In the case of a supersonic flow around a wedge (Figure 3,a), the translational flow along the side surface of the wedge is separated from the approaching flow by a plane shock wave that starts at the edge of the wedge. For wedge angles greater than a certain limiting angle, the shock wave is curved and is detached from the edge of the wedge. In this case, a zone of subsonic flow is located behind the shock wave. This situation is characteristic of supersonic flow around a blunt-nosed body (Figure 3,b).

When a supersonic stream flows around a plate (see Figure 3 in the article LIFT) at an angle of attack less than that at which the shock wave is detached from the leading edge of the plate, a shock wave extends downward from the leading edge, and a Prandtl-Meyer expansion flow is formed at the leading edge on the upper surface. As a result, the pressure is less on the upper surface of the plate than on the lower surface, and lift and drag appear—that is, the d’Alembert-Euler paradox does not apply here. In contrast to subsonic flow, the supersonic flow of an ideal gas around a body subjects the body to drag. The drag results from the appearance of shock waves and the associated increase in the entropy of the gas when it passes across the shock waves. The greater the disturbances created in the gas by the body, the more intense the shock waves and the larger the resistance to the body’s motion. In order to reduce the drag of wings that is associated with the formation of bow shock waves, swept-back (Figure 4) and delta wings are used at supersonic speeds. The leading edge of such wings forms an acute angle β with the direction of the oncoming flow moving with the speed v. The aerodynamically ideal shape (that is, the shape having a relatively small pressure drag) for supersonic flow is a slender body with pointed ends that moves at small angles of attack. When such bodies move at a moderate supersonic speed (that is, when the flight speed is only a few times greater than the speed of sound), the pressure and density disturbances they produce in a gas and the resulting speeds of the gas particles are small. Linear equations for the motion of a compressible gas can thus be used to determine the aerodynamic characteristics of, for example, airfoil profiles and bodies of revolution.

Figure 3. Supersonic flow: (a) around a wedge, (b) around a blunt body

Numerical methods are used for calculating supersonic flow around bodies of revolution and profiles of appreciable thickness in rocket nozzles, wind-tunnel nozzles, and other cases.

Hypersonic flow. Flows at high supersonic speeds (v ≫ a) are known as hypersonic flows and possess special properties. When a body flies at a hypersonic speed in a gas, the temperature of the gas rises to very high values near the surface of the body. The reason for the temperature increase is that the gas in front of the nose of the moving body is highly compressed and heat is evolved owing to internal friction in the gas carried along by the body. The investigation of hypersonic gas flows consequently requires that the change in the properties of air at high temperatures be taken into account. These changes include the excitation of internal degrees of freedom, the dissociation of the molecules of the gases that make up the air, chemical reactions (such as the formation of nitric oxide), the excitation of electrons, and ionization. In problems where molecular transport phenomena are important—such as in the calculation of surface friction, the heat flow to the surface along which the gas flows, and the temperature of the surface—it is necessary to take into account the change in the air’s viscosity and thermal conductivity. In many cases, the diffusion and the thermal diffusion of the air’s components must also be taken into account.

Figure 4. Diagram of flow around a swept-back wing

Under some conditions of hypersonic flight at high altitudes (seeAERODYNAMICS OF RAREFIED GASES) the processes that occur in a gas cannot be regarded as thermodynamic equilibrium processes. The establishment of thermodynamic equilibrium in a moving “particle” (that is, in a very small volume) of the gas does not occur instantaneously and requires a definite time known as the relaxation time, which is different for different processes. The deviations from thermodynamic equilibrium can have a marked effect on the processes that occur in the boundary layer (in particular, on the magnitude of the heat flow from the gas to the body), on the structure of the shock waves, on the propagation of small disturbances, and on other phenomena. Thus, when the air is compressed in the bow wave, the translational degrees of freedom of the molecules are the most easily excited; these degrees of freedom determine the temperature of the air. The excitation of the vibrational degrees of freedom requires more time. The air’s temperature and radiation in the region behind the shock wave may therefore be much higher than calculated values that do not take into account the relaxation of the vibrational degrees of freedom.

When the temperature is very high (~3000°-4000°K or higher), the air contains a fairly large number of ionized particles and free electrons. The good electrical conductivity of the air close to a body moving at a high supersonic speed makes possible the use of electromagnetic influences on the flow to alter the resistance of the body or to reduce the heat flow from the hot gas to the body. On the other hand, because of the reflection and absorption of radio waves by the ionized gas surrounding the body, the high electrical conductivity complicates radio communication with a vehicle flying through the air at such a speed. Large fluxes of radiant energy can result from the heating of the air compressed in front of the nose of a body moving at a hypersonic velocity. This energy is partially transferred to the body and causes additional difficulties in solving the problem of the cooling of the body.

Figure 5. Values of the drag coefficient for a sphere and for a cylinder with a conical nose; the values do not change appreciably after M = 4

A special situation arises when the speed of the oncoming flow is many times greater than the speed of sound. When small disturbances occur in this case, the rates at which the density and pressure change are no longer small, and nonlinear equations must be used, even when studying the flow around pointed slender bodies. The important role of nonlinear effects is characteristic of hypersonic aerodynamics. Many concepts of the aerodynamics of moderate supersonic speeds are inapplicable at hypersonic speeds. At such speeds, the forces and moments that act on vehicles in flight are different in character from the forces and moments at moderate speeds. In addition, different concepts of vehicle stability and controllability that are applicable at moderate supersonic speeds become inapplicable at hypersonic speeds.

The large values of the Mach number M = v/a for flows at hypersonic speeds make it possible to establish important qualitative characteristics of such flows and to develop nonlinear asymptotic theories for the quantitative analysis of hypersonic flows. Thus, when M is very large, the pressure in the flow approaching a body is negligibly small compared with the pressure in the flow region behind the shock wave that is produced in front of the body. Moreover, the oncoming flow’s heat content, or enthalpy, is unimportant compared with its kinetic energy. Under such conditions, the flow behind the shock wave is independent of the Mach number M of the oncoming flow. This principle underlies the stabilization of the flow around bodies at hypersonic velocities. The stabilization of the flow around blunt bodies occurs at smaller values of M than in the case of pointed slender bodies (Figure 5).

Figure 6. Diagram illustrating the law of plane sections

An important result of the theory of hypersonic flow around pointed slender bodies at a small angle of attack is the law of plane sections. According to this law, when a slender body moves at a hypersonic speed in a gas that is at rest, the gas particles experience almost no longitudinal displacement—that is, the motion of the particles occurs in planes perpendicular to the direction of the body’s movement (Figure 6). From the law of plane sections a similarity law follows that permits, for example, the parameters of motion obtained for one body of revolution at a certain Mach number M to be used to calculate the corresponding parameters for flow around certain other bodies. The other bodies must have the same distribution of relative thickness with respect to length as the first body, and the product Mτ must have the same value, where τ is the largest value of the relative thickness of the body.