next up previous
Next: Small-Perturbation Theory Up: Two-Dimensional Compressible Inviscid Flow Previous: Crocco's Theorem

Homenergic Homentropic Flow

Consider a steady, two-dimensional, homenergic, homentropic flow pattern, in the absence of body forces. Suppose that all quantities are independent of the Cartesian coordinate $ z$ , and that the flow velocity, $ {\bf q}$ , is confined to the $ x$ -$ y$ plane. Let $ {\bf q} = u\,{\bf e}_x+v\,{\bf e}_y$ . Equations (14.25), (14.30), and (14.31) reduce to

$\displaystyle \frac{\partial}{\partial x}\,(\rho\,u) + \frac{\partial}{\partial y}\,(\rho\,v)$ $\displaystyle = 0,$ (15.92)
$\displaystyle u\,\frac{\partial u}{\partial x}+v\,\frac{\partial u}{\partial y}$ $\displaystyle = -\frac{1}{\rho}\,\frac{\partial p}{\partial x},$ (15.93)
$\displaystyle u\,\frac{\partial v}{\partial x}+v\,\frac{\partial v}{\partial y}$ $\displaystyle = -\frac{1}{\rho}\,\frac{\partial p}{\partial y},$ (15.94)
$\displaystyle \frac{p}{p_0}$ $\displaystyle = \left(\frac{\rho}{\rho_0}\right)^\gamma,$ (15.95)

where $ p_0$ and $ \rho_0$ are the uniform stagnation pressure and density, respectively. (Note that $ p_0$ and $ \rho_0$ must be uniform because the stagnation specific entropy, $ {\cal S}_0\propto p_0/\rho_0^{\,\gamma}$ , and the stagnation temperature, $ T_0\propto p_0/\rho_0$ , are both uniform.) Equation (15.95) implies that

$\displaystyle \nabla p= c^{\,2}\,\nabla\rho,$ (15.96)

where $ c=\sqrt{\gamma\,p/\rho}$ is the sound speed. Hence, Equations (15.92)-(15.94) yield

$\displaystyle \frac{u}{\rho}\,\frac{\partial\rho}{\partial x} + \frac{v}{\rho}\,\frac{\partial \rho}{\partial y}$ $\displaystyle =-\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y},$ (15.97)
$\displaystyle u^{\,2}\,\frac{\partial u}{\partial x} + u\,v\,\frac{\partial u}{\partial y}$ $\displaystyle = -\frac{c^{\,2}}{\rho}\,u\,\frac{\partial \rho}{\partial x},$ (15.98)
$\displaystyle u\,v\,\frac{\partial v}{\partial x} + v^{\,2}\,\frac{\partial v}{\partial y}$ $\displaystyle = -\frac{c^{\,2}}{\rho}\,v\,\frac{\partial \rho}{\partial y}.$ (15.99)

Summing the previous two equations, and then making use of Equation (15.97), we obtain

$\displaystyle \left(u^{\,2}-c^{\,2}\right)\,\frac{\partial u}{\partial x} + \le...
...\,v\left( \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)=0.$ (15.100)

Finally, given that $ c\propto \sqrt{T}$ , Equation (14.58) implies that

$\displaystyle c^{\,2} = c_0^{\,2} -\frac{1}{2}\,(\gamma-1)\left(u^{\,2}+v^{\,2}\right),$ (15.101)

where $ c_0$ is the stagnation sound speed.


next up previous
Next: Small-Perturbation Theory Up: Two-Dimensional Compressible Inviscid Flow Previous: Crocco's Theorem
Richard Fitzpatrick 2016-03-31