A graphical version of this slide is available. In the text only version presented here, * denotes multiplication, / denotes division, ^ denotes exponentiation, ^2 means quantity squared. r is the density, p is the static pressure, T is the temperature, pt is the total pressure. M is the Mach number, a is the wedge angle and s is the shock angle. The equations are specialized for air; the ratio of specific heats is 1.4. Supersonic flow encounters a sharp wedge and a shock is generated. Flow upstream of the shock is denoted by a 0 and flow downstream of the shock is denoted by a 1. The trigonometric functions sine, tangent and cotangent are denoted by sin, tan and cot.

Equation 1. cot(a) = tan(s) * (((6 * M ^2)/(5 * (M ^2 * sin ^2(s) - 1)) - 1)

Equation 2. T1 / T0 = ((7 * M ^2 * sin ^2(s) - 1) * (M ^2 * sin ^2(s) + 5)) / (36 * M ^2 * sin ^2 (s))

Equation 3. p1 / p0 = (7 * M ^2 * sin ^2(s) - 1) / 6

Equation 4. r1 / r0 = (6 * m ^2 * sin ^2(s)) / (M ^2 * sin ^2(s) + 5)

Equation 5. M1 ^2 * sin ^2(s - a) = (M ^2 * sin ^2(s) + 5) / (7 * M ^2 * sin ^2(s) - 1)

Equation 6. pt1 / pt0 = (((6 * m ^2 * sin ^2(s)) / (M ^2 * sin ^2(s) + 5)) ^ (7 / 2)) * ((6 / (7 * M ^2 * sin ^2(s) - 1)) ^ (5 / 2))

As an object moves through a gas, the gas molecules are deflected around the object. If the speed of the object is much less than the speed of sound of the gas, the density of the gas remains constant and the flow of gas can be described by conserving momentum and energy. As the speed of the object increases towards the speed of sound, we must consider compressibility effects on the gas. The density of the gas will vary locally as the gas is compressed by the object.

For compressible flows with little or small flow turning, the flow process is reversible and the entropy is constant. The change in flow properties are then given by the isentropic relations (isentropic means "constant entropy"). But when an object moves faster than the speed of sound, and there is an abrupt decrease in the flow area, the flow process is irreversible and the entropy increases. Shock waves are generated which are very small regions in the gas where the gas properties change by a large amount. Across a shock wave, the static pressure, temperature, and gas density increases almost instantaneously. Because a shock wave does no work, and there is no heat addition, the total enthalpy and the total temperature are constant (the ratio of Tt1 to Tt0 is equal to one). But because the flow is non-isentropic, the total pressure downstream of the shock is always less than the total pressure upstream of the shock; there is a loss of total pressure associated with a shock wave. The ratio of the total pressure is given above. Because total pressure changes across the shock, we can not use the usual (incompressible) form of Bernoulli's equation across the shock. The Mach number and speed of the flow also decrease across a shock wave.

If the shock wave is inclined to the flow direction it is called an oblique shock. On this slide we have listed the equations which describe the change in flow variables for flow past a two dimensional wedge. The equations presented here were derived by considering the conservation of mass, momentum, and energy for a compressible gas while ignoring viscous effects. The equations have been further specialized for a two-dimensional flow (not three dimensional axisymmetric) without heat addition and for a gas whose ratio of specific heats is 1.4 (air). The equations only apply for those combinations of free stream Mach number and wedge angle for which an attached oblique shock occurs. If the Mach number is too low, or the wedge angle too high, the normal shock equations should be used. For the Mach number change across an oblique shock there are two possible solutions; one supersonic and one subsonic. In nature, the supersonic ("weak shock") solution occurs most often. However, under some conditions the "strong shock", subsonic solution is possible.

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byTom Benson
Please send suggestions/corrections to: benson@grc.nasa.gov