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

$$\frac{\partial}{\partial x}\,(\rho\,u) + \frac{\partial}{\partial y}\,(\rho\,v) = 0,$$ (15.92)

$$u\,\frac{\partial u}{\partial x}+v\,\frac{\partial u}{\partial y} = -\frac{1}{\rho}\,\frac{\partial p}{\partial x},$$ (15.93)

$$u\,\frac{\partial v}{\partial x}+v\,\frac{\partial v}{\partial y} = -\frac{1}{\rho}\,\frac{\partial p}{\partial y},$$ (15.94)

$$\frac{p}{p_0} = \left(\frac{\rho}{\rho_0}\right)^\gamma,$$ (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

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

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

$$\frac{u}{\rho}\,\frac{\partial\rho}{\partial x} + \frac{v}{\rho}\,\frac{\partial \rho}{\partial y} =-\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y},$$ (15.97)

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

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

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

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

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

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

where $c_0$ is the stagnation sound speed.