Isentropic expansion waves explained

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. Each wave in this case is a Mach wave, so it is at an angle

\alpha = \sin ^{-1} \tfrac{ 1 }{M}, where 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 to, with deflection . 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.

Relation between θ, M and v

Assumptions:

1. Steady flow.
2. Negligible body forces.
3. Adiabatic flow.
4. No work terms.
5. Negligible gravitational effect.

The continuity equation is$$\backslash frac\backslash int\backslash limits\_\backslash rho\; dV\; +\; \backslash int\backslash limits\_\backslash rho\; \backslash bar\; v\; \backslash ,\; d\backslash bar\; A\; =\; 0\; \backslash qquad\backslash qquad\; (1.1)$$

First term is zero by assumption (1). Now, $$+\; =\; 0$$which can be rewritten as$$\backslash rho\; v\; \backslash sin\backslash alpha\; =\; (\backslash rho\; +\; d\backslash rho)(v\; +\; dv)\; \backslash sin(\backslash alpha\; -\; d\backslash theta)\; \backslash qquad\backslash qquad\; (1.2)$$

Now we consider the momentum equation for normal and tangential to shock. For -component, $$F\_\; +\; F\_\; =\; \backslash frac\backslash int\backslash limits\_v\_y\backslash rho\; dV\; +\; \backslash int\backslash limits\_v\_y\backslash rho\; \backslash bar\; v\; \backslash ,\; d\backslash bar\; A\; \backslash qquad\backslash qquad\; (1.3)$$

Second term of L.H.S and first term of R.H.S are zero by assumption (2) and (1) respectively. Then,

$$0\; =\; v\backslash cos\backslash alpha(-\backslash rho\; v\; \backslash sin\backslash alpha\; A)\; +\; (v+dv)\backslash cos(\backslash alpha\; -\; d\backslash theta)$$

Or using equation 1.1 (continuity),$$v\backslash cos\backslash alpha\; =\; (v+dv)\; \backslash cos(\backslash alpha\; -\; d\backslash theta)$$

Expanding and simplifying [Using the facts that, to the first order, in the limit as <math>d\theta\rightarrow 0</math>, <math>\cos{d\theta}\rightarrow 1</math> and <math>\sin {d\theta}\rightarrow d\theta</math>], we obtain

$$d\backslash theta=\backslash frac$$

But, $$\backslash sin\backslash alpha=\backslash frac$$

so, $$\backslash tan\backslash alpha=\backslash frac$$

and$$d\backslash theta\; =\; -\backslash frac\; \backslash qquad\backslash qquad\; (1.4)$$

Derivation of Prandtl-Meyer supersonic expansion function

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

$$\backslash dot\; -\; \backslash dot\_s\; -\; \backslash dot\_\; -\; \backslash dot\_\; =\; \backslash frac\; \backslash int\backslash limits\_e\backslash rho\; dV\; +\; \backslash int\backslash limits\_\; h\backslash rho\; \backslash bar\; v.d\backslash bar\; A\; \backslash qquad\backslash qquad\; (1.5)$$

First term of L.H.S, next three terms of L.H.S and first term of R.H.S are zero due to assumption (3), (4) and (1) respectively.

where,$$e\; =\; u\; +\; \backslash frac\; +\; gz$$

For our control volume we obtain

$$0\; =\; \backslash left(h\; +\; \backslash frac\backslash right)\; (-\backslash rho\; v\; \backslash sin\; \backslash alpha\; A)\; +\; \backslash left[(h\; +\; dh)\; +\; \backslash frac\{(v\; +\; dv)^2\}\{2\}\backslash right]\backslash bigl((\backslash rho\; +\; d\backslash rho)(v+dv)\backslash sin(\backslash alpha\; -\; d\backslash theta)A\; \backslash bigr)$$

This may be simplified as

$$=\; (h\; +\; dh)\; +\; \backslash frac$$

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

$$dh\; =\; -vdv$$

If we confine to ideal gases,

dh = c _{p} dT , so

$$c\_p\; dT\; =\; -vdv\; \backslash qquad\backslash qquad\; (1.6)$$

Above equation relates the differential changes in velocity and temperature. We can derive a relation between

M and v using v = Mc = M\sqrt{kRT}</Math>. Differentiating (and dividing the left hand side by <math>v and the right by \sqrt{kRT} ),

$$\backslash frac\; =\; \backslash frac\; +\; \backslash frac$$

Using equation (1.6)

Hence,

$$\backslash frac\; =\; \backslash frac\backslash frac\; \backslash qquad\backslash qquad\; (1.7)$$

Combining (1.4) and (1.7)

$$\backslash frac\; =\; -\backslash frac\backslash frac\; \backslash qquad\backslash qquad\; (1.8)$$

We generally apply the above equation to negative

d\theta , let d\omega = d\theta . 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 (M =1 ) to Mach number M , with \omega arbitrarily set to zero at M =1 ,

$$\backslash int\backslash limits\_0^\backslash omega\; d\backslash omega\; =\; \backslash int\backslash limits\_1^M\; \backslash frac\backslash frac$$

Leading to Prandtl-Meyer supersonic expansion function,

$$\backslash omega\; =\; \backslash sqrt\; \backslash tan^\backslash left[\backslash frac\{\backslash sqrt\{k-1\}\}\{\backslash sqrt\{k+1\}\}(M^2-1)\backslash right]\; -\; \backslash tan^(M^2\; -\; 1)$$

References

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Notes and References

1. 'Introduction to Fluid Mechanics' by Robert W. Fox, Philip J. Pritchard and Alan T. McDonald