Crocco's Theorem

Bernoulli's theorem for the steady, inviscid flow of an ideal gas, in the absence of body forces, implies that, on a given streamline,

$${\cal H} + \frac{1}{2}\,v^{\,2} = {\cal H}_0,$$ (15.85)

where ${\cal H}$ is the specific enthalpy, ${\cal H}_0$ the stagnation enthalpy, and ${\bf v}$ the flow velocity. (See Section 14.5.) The fluid equation of motion, (14.31), reduces to

$$({\bf v}\cdot\nabla)\,{\bf v} = -\frac{\nabla p}{\rho}.$$ (15.86)

However, according to Equation (A.171),

$$({\bf v}\cdot\nabla)\,{\bf v} \equiv\nabla (v^{\,2}/2) -{\bf v}\times \mbox{\boldmath$\omega$} ,$$ (15.87)

where $\omega$ $=\nabla\times {\bf v}$ . Hence, we obtain

$${\bf v}\times \mbox{\boldmath$\omega$} =\nabla(v^{\,2}/2) + \frac{\nabla p}{\rho}.$$ (15.88)

Now, Equation (14.29) implies that

$$\nabla {\cal H} = \frac{\nabla p}{\rho} + T\,\nabla{\cal S},$$ (15.89)

where $T$ is the temperature, and ${\cal S}$ the specific entropy. Moreover, Equation (15.85) yields

$$\nabla {\cal H} +\nabla(v^{\,2}/2) = \nabla {\cal H}_0.$$ (15.90)

Thus, we arrive at Crocco's theorem:

$${\bf v}\times \mbox{\boldmath$\omega$} = \nabla {\cal H}_0 - T\,\nabla{\cal S}.$$ (15.91)

In most aerodynamic flows, the fluid originates from a common reservoir, which implies that the stagnation enthalpy, ${\cal H}_0$ , is the same on all streamlines. Such flow is termed homenergic. It follows from Equation (14.18) that the stagnation temperature, $T_0$ , is the same on all streamlines in homenergic flow. According to Crocco's theorem, an irrotational (i.e., $\omega$ $={\bf0}$ everywhere) homenergic (i.e., $\nabla {\cal H}_0=0$ everywhere) flow pattern is also homentropic (i.e., $\nabla {\cal S}=0$ everywhere). Conversely, a homenergic, homentropic flow pattern is also irrotational (at least, in two dimensions, where ${\bf v}$ and $\omega$ cannot be parallel to one another).