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 pressure, T is the temperature, pt is the total pressure and Tt is the total temperature. M is the Mach number, and a is the wedge angle. The equations are specialized for air; the ratio of specific heats is 1.4. Supersonic flow encounters a 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.

Equation 1. Only use these equations for: a > .3208 * ((M ^2 - 1) ^3/2) / M ^2

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

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

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

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

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

Equation 7. Tt1 / Tt0 = 1

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 perpendicular to the flow direction it is called a **normal**
shock. On this slide we have listed the equations which describe the change
in flow variables for flow across a normal shock.
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 one-dimensional flow
without heat addition and
for a gas whose ratio of
specific heats is 1.4 (air). The equations can be applied to the
two dimensional flow past a wedge for the combination of
free stream Mach number and wedge angle listed in blue.
If the wedge angle is less than the angle shown on the slide, an attached
oblique shock
occurs and the equations are slightly modified.

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