<p>So I spent entirely too much time last night trying to figure out why the momentum equation for compressible flow looks so much like the one for incompressible flow, and further, why you can’t use compressible momentum equation for incompressible flows.</p>
<p>Assumptions:
1-D, inviscid, steady state</p>
<p>Using a control volume (with equal in/out areas), using the surface integral form (negating terms as appropriate from above assumptions), SS[(ρ<em>v dot dS)</em>v] = -SS[P*dS], the derived momentum equation for 1-D compressible flow is:</p>
<p>ρ1<em>(−u1</em>A)<em>u1 + ρ2</em>(u2<em>A)</em>u2 = −(−P1<em>A + P2</em>A)</p>
<p>P1 + ρ1<em>u1^2 = P2 + ρ2</em>u2^2</p>
<p>From Euler’s form, dP = -ρ<em>v</em>dv, for incompressible flow ρ1 = ρ2 and you end up with a 1/2 (from integrating vdv).
P1 + (1/2)ρ<em>v1^2 = P2 + (1/2)ρ</em>v2^2</p>
<p>Now using the control volume derivation I can’t figure out how to get the 1/2 in there with ρ constant to get the incompressible form, and using the Euler form, I can’t figure out how to integrate with ρ as a variable to get the compressible form.</p>
<p>Any aerodynamics gurus care to chime in?</p>
<p>You can use the compressible equations for an incompressible flow, as the incompressible equations are simply the same but with certain terms (e.g. bulk viscosity) zeroed out. Ultimately, the momentum equations for all continuum flows generalize the the Navier-Stokes equations, continuity equation and energy equation. How many of those you use and how complicated the simplified equations become just depends on the assumptions you can justify.</p>
<p>At any rate, the Euler equation can be derived directly from the control volume but it requires both momentum and mass to do it. Basically, before you introduce any specific values like p1 and p2, the control volume approach leaves you with</p>
<pre><code> d(ρuA) = 0
</code></pre>
<p>for conservation of mass and</p>
<pre><code> d(ρu^2A + p) = p*dA
</code></pre>
<p>for momentum. The momentum equation there can be rewritten</p>
<pre><code> ud(ρuA) + ρuAdu + A*dp = 0
</code></pre>
<p>and if you combine the two, you end up with the Euler equation,</p>
<pre><code> ρudu + dp = 0
</code></pre>
<p>from which you already just derived the Bernoulli equation.</p>
<p>For future reference, a question like this is probably more suitable for Physics Forums, as this one isn’t really designed to ask technical questions. They have LaTeX built into their forum software so you can write equations nicely, to boot, and the point of the forum is these types of questions. Hopefully suggesting that isn’t against the terms of service here, but the sites don’t compete so I think I am safe.</p>