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 *