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Connecting to this question about various effects of supersonic flow, the question did not mention pressure-velocity dependance for supersonic flow, so I am asking it here now.

For incompressible flow we have the Bernoulli equation:

$$\frac{ v^2}{2}+\frac{p}{\rho}=C\tag1$$

as velocity increases, the pressure decreases and vice versa.

For compressible flow the Bernoulli equation takes the form:

$$\frac{ v^2}{v}+\frac{p}{\rho}+u=C\tag2$$

where $u$ is specific internal energy.

this also seems to have the same inverse relationship as the regular Bernoulli equation, at least regarding $p$ and $v$.

Supersonic flow is compressible, but is equation $(2)$ enough to model the $p-v$ relationship or is something else needed?

I found the following equation:

$$ \frac{dA}{A} = \frac{dp \, (1 - M^2)}{\rho v^2} = \frac{(M^2 - 1) \, dv}{v}\tag3 $$

where $M$ is Mach number.

Question(s):

1.) Is equation $(3)$ correct (I couldn't find a derivation or reference)?

2.) Upon inputting $M>1$ in $(3)$ I obtain that

$$-\frac{dp}{\rho} \propto v\space dv$$

or that pressure is inversely proportional to velocity (same as in incompressible flow)?

Is this conclusion correct?

Qmechanic
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User198
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2 Answers2

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Bernoulli's equation for a compressible flow is written as: $$ \frac{v^2}{2} + \frac{p}{\rho} + u = C. $$ This equation is valid for a steady isentropic flow. I think you can use it together with the relationships of an isentropic process to obtain a $p$-$v$ relationship.

Your equation (3) is correct and it is relevant for so-called quasi-one-dimensional flows. You can find the derivation in any good textbook on compressible flows (for instance "Fundamentals of aerodynamics" for John Anderson).

The relationship $$ dp = -\rho v dv $$ is called Euler's equation. It is valid along a streamline of a steady inviscid flow. It is also valid for a quasi-one-dimensional flow.

Tofi
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Addressing your questions:

  1. equation(s) (3) is "correct" if you need to represent the relation between changes in pressure, velocity and section of a stream tube in a compressible, steady, isentropic (no shock so that differential equations hold, negligible viscosity and heat conduction) quasi-1-dimensional flow, i.e flows whose streamlines have gentle curvature in the streamwise direction. If the flow is isentropic, it's possible to use $dP = c^2 d \rho$, from the definition of the speed of sound $c^2(\rho,s) = \left(\frac{\partial P}{\partial \rho}\right)_s$. You can find a derivation here.

  2. The relation $dP = - \rho v d v$ holds for both subsonic and supersonic flows, as you can easily realize since this simplification doesn't depend on the sign of the term $1-M^2$ in (3). The right conclusion links change in pressure with change in velocity. As you can find in the link I shared above, relation (3) links changes in pressure and velocity with change of the section of a stream tube, and it gives a (maybe) unexpected insight in the difference between compressible subsonic flows (that accelerate while pressure decreases if converging) and supersonic flows (that accelerate while pressure decreases if diverging).

basics
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