Transonic Aerodynamics and Shock Waves

Shock Waves

Summary

Classification
Mathematics
Shock Strength

Normal Shock
Oblique Shock

Bomb Blasts
Selected References

The shock waves are strong perturbations in aerodynamics that propagate at supersonic speeds independent of the wave amplitude. Such perturbations occur steady transonic or supersonic flow, lightning strokes, bomb blasts, and contact surfaces in laboratory devices. The following discussion is limited to common physical characteristics.

Classification

Shocks are classified as weak or strong, depending on the value of the pressure jump across the shock; they are also classified as normal or oblique, compression or rarefaction shocks, direct or reflected shocks.

Fig. 1 below shows the progression of the the shock wave and the supersonic pocket on a conventional airfoil in steady transonic flow.

Figure 1: Shock Progression on Airfoil

The shock wave first appears on the suction side and travels slowly toward the trailing edge (Fig 1a), as the supersonic pocket increases in size. The shock wave on the lower side appears later (e.g. at higher Mach numbers), but travels faster and reaches first the trailing edge (Fig 1c). Eventually, also the upper shock wave reaches the trailing edge, and with the lower shock forms a bifurcated trailing edge shock (). The thick boundary layer deflects the external flow and creates compression waves to form the characteristic

Mathematical Aspects

The shock discontinuity corresponds to a change in the type of differential equations that describe the inviscid part of the flow: from elliptic to hyperbolic. The shock wave is the boundary between subsonic and supersonic domains. The importance of this mathematical aspect is this: the domain of influence of a shock waves is limited to downstream points (contrary to acoustic waves).

Shock Strength

A shock is considered weak if the pressure jump is small compared to the pressure ahead of the shock. This is always true at low supersonic speeds. The sign of the pressure jump is of some physical interest, since only positive jumps are possible. Positive jumps mean that only compression shock waves can occur (Thompson, 1972).

Normal Shock

The jumps of the aero-thermodynamic properties of the gas across a normal shock are derived from the conservation equations for mass, momentum and energy (Rankine- Hugoniot).

One approximation that is commonly done is to consider the shock as an insentropic transformation, which is a good approximation for transonic flows. With the further approximation of ideal gas, the jumps can be cast in closed form, whose solution is shown in the graphic below for Mach numbers M < 5.

Figure 2: Normal shock ratios at supersonic speeds

Oblique Shock

Shocks that are incident at an angle (on a wedge or similar situations) are subject to a change of direction of the velocity in addition to the features of the normal shock. The change of direction can be calculated with the full conservation equations, and is found to be dependent on the density ratio shown in Fig. 2. This change occurs only in a direction normal to the shock.

The figure below shows the oblique shock waves past a double wedge airfoil at Mach 1.8. The visualization is a Schlieren photography.

Figure 3: Double wedge airfoil

Shock Reflection

Shock reflection occurs when a shock is intercepted by a wall. This is a peculiar example of aerodynamic interference of special interest in transonic wind tunnels. There is some complication in calculating the reflection, because the flow close to the wall cannot be possibly supersonic. This means that the flow is most likely separated at the wall (shock-induced separation), Shapiro, 1953.

Shock Deflection

Shock deflection occurs when the shock turns through a finite angle. In the weak shock approximation the shock is reduced to a sequence of small shocks. A finite sequence of shocks also occurs in the Prandtl-Meyer Fan, that is when a supersonic flow is forced to change direction at a curved or pointed surface. In the former case shock waves radiate in a form of standing waves. In the latter case the waves are centered at the sharp corner.

Bomb Blast

Blast (or detonation) shock waves are produced through the quick release of a large amount of energy by a chemical reaction or a nuclear explosion.

Related Material

Selected References

Shapiro AH. The Dynamics and Thermodynamics of Compressible Fluid Flow, Pergamon Press, New York, 1953
Thompson PA. Compressible Gas Dynamics, McGraw-Hill, 1972.
Liberman MA, Velikovich AL. Physics of Shock Waves and Plasmas, Springer, 1986.

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