Isentropic expansion waves

In fluid dynamics, isentropic expansion waves are created when a supersonic flow is redirected along a curved surface.

These waves are studied to obtain a relation between deflection angle and Mach number.

where M is the Mach number immediately before the wave.

Expansion waves are divergent because as the flow expands the value of Mach number increases, thereby decreasing the Mach angle.

In an isentropic wave, the speed changes from v to v + dv, with deflection dθ.

We have oriented the coordinate system orthogonal to the wave.

We write the basic equations (continuity, momentum and the first and second laws of thermodynamics) for this infinitesimal control volume.

Assumptions: The continuity equation is

− ρ v sin ⁡ α

ρ v sin ⁡ α = ( ρ + d ρ ) ( v + d v ) sin ⁡ ( α − d θ )

Now we consider the momentum equation for normal and tangential to shock.

0 = v cos ⁡ α ( − ρ v sin ⁡ α

) + ( v + d v ) cos ⁡ ( α − d θ )

( ρ + d ρ ) ( v + d v ) sin ⁡ ( α − d θ )

v cos ⁡ α = ( v + d v ) cos ⁡ ( α − d θ )

Expanding and simplifying [Using the facts that, to the first order, in the limit as

We skip the analysis of the x-component of the momentum and move on to the first law of thermodynamics, which is

{\displaystyle {\dot {Q}}-{\dot {W}}_{s}-{\dot {W}}_{shear}-{\dot {W}}_{other}={\frac {\partial }{\partial t}}\int \limits _{CV}e\rho dV+\int \limits _{CS}h\rho {\bar {v}}.d{\bar {A}}\qquad \qquad (1.5)}

For our control volume we obtain

( − ρ v sin ⁡ α

( ρ + d ρ ) ( v + d v ) sin ⁡ ( α − d θ )

Expanding and simplifying in the limit to first order, we get

If we confine to ideal gases,

Above equation relates the differential changes in velocity and temperature.

Differentiating (and dividing the left hand side by

{\displaystyle {\frac {dv}{v}}={\frac {dM}{M}}+{\frac {dT}{2T}}}

{\displaystyle {\begin{aligned}{\frac {dv}{v}}&={\frac {dM}{M}}-{\frac {vdv}{2c_{p}T}}\\[2pt]&={\frac {dM}{M}}-{\frac {dv{\frac {v^{2}}{c_{p}T}}}{2v}}\\[2pt]&={\frac {dM}{M}}-{\frac {dv{\frac {M^{2}c^{2}}{c_{p}T}}}{2v}}\\[2pt]&={\frac {dM}{M}}-{\frac {dv{\frac {M^{2}kRT}{c_{p}T}}}{2v}}\\[2pt]&={\frac {dM}{M}}-{\frac {dv[M^{2}(k-1)]}{2v}}\end{aligned}}}

We generally apply the above equation to negative

We can integrate this between the initial and final Mach numbers of given flow, but it will be more convenient to integrate from a reference state, the critical speed (

Leading to Prandtl-Meyer supersonic expansion function,