20.3 Reduced Models #
Further simplifications to the fluid models provide equations that are easier to solve. But we have to make sure that the reduced models (anything that’s not the full equation system) still capture the relevant physics of the problem.
Let’s assume the (2D) flow is
- irrotational \( (\curl v = 0) \)
- isentropic
- steady-state
This lets us apply potential flow theory. Since the flow is irrotational, the curl of the velocity field is zero, so we can define a potential
\[\vec v = - \grad \phi\]That lets us define the flow direction in each direction in terms of the potential
\[u = - \pdv{\phi}{x} \qquad v = - \pdv{\phi}{y}\]similarly, continuity (conservation of mass) equation:
\[(\rho \phi_x )_x + (\rho \phi_y)_y = 0\]and conservation of momentum
\[\rho \pdv{\vec v}{t} + \rho \vec v \cdot \grad \vec v + \grad p = 0\] \[\rightarrow \rho \left[ \grad \left( \frac{v^2}{2} \right) - \vec v \cross (\curl \vec v) \right] + \grad p = 0\] \[\rightarrow \frac{\rho}{2} \dd (\phi_x ^2 + \phi_y ^2) = - \dd p\]We can define the speed of sound \( a \) for an isentropic flow as
\[a = \sqrt{\dv{p}{\rho}} = \sqrt{\frac{\gamma p}{\rho}}\]so
\[\dd \rho = - \frac{\rho}{2 a^2} \dd ( \phi_x ^2 + \phi _y ^2)\] \[\rightarrow \rho_x = - \frac{\rho}{2 a^2} (\phi_x ^2 + \phi_y ^2) _x = 0 \frac{\rho}{2a^2} ( \phi_x \phi_{xx} + \phi_y \phi_{xy})\]similarly for \( \rho_y \)
\[\rho_y = - \frac{\rho}{2 a^2} (\phi_x ^2 + \phi_y ^2) _y = - \frac{\rho}{2a^2} ( \phi_x \phi_{yx} + \phi_y \phi_{yy})\]Substituting \( \rho_x \) and \( \rho_y \) back into the continuity equation
\[\left( 1 - \frac{\phi_x ^2}{a^2} \right) \phi_{xx} + \left( 1 - \frac{\phi_y ^2}{a^2} \right) \phi_{yy} - \frac{2}{a^2} \phi_x \phi_y \phi_{xy} = 0\]This is the full potential equation that is equivalent to the Euler equations for irrotational, isentropic, steady-state flows. Why would we want to solve this instead of the Euler equations? We’ve taken a 5-vector equation and turned it into a scalar equation! Note, we are no longer in conservation law form, which means we do not get the benefits that come with conservation form, but it is still easier to solve.
Note, for incompressible flow, \( a \rightarrow \infty \) and
\[\phi_{xx} + \phi_{yy} = 0 \qquad (\text{Laplace eq.})\]20.3.2 Transonic Small Disturbance #
If a feature of interest (e.g. wing) does not have a large impact on the bulk flow \( u_{\infty} \) , we can define a perturbed potential to linearize.
\[u = u_{\infty} + u'\] \[= u_{\infty} - \phi_x\] \[v = v' = - \phi_y\]We assume that \( u’, v’ « \infty \), which is the basis of our perturbation theory.
The compressible Bernoulli’s equation tells us
\[\frac{|\vec v|^2}{2} + \frac{\gamma}{\gamma + 1} \frac{p}{\rho} = \text{const.}\] \[\frac{\gamma - 1}{2} |v|^2 + a^2 = \frac{\gamma - 1}{2} u_{\infty}^2 + a_{\infty}^2\]\( a_{\infty} \) is the bulk speed of the flow
\[a_{\infty}^2 = \frac{\gamma p_{\infty}}{\rho}\]Expanding the velocity modulus in our linearization
\[| \vec v | ^2 = u_{\infty}^2 + 2 u' u_{\infty} + (u') ^2 + (v ') ^2\]Substituting into the full potential gives
\[\left[ \frac{1 - M_{\infty}^2}{M_{\infty}^2} - (\gamma + 1) \frac{\phi_x}{u_{\infty}}\right] M_{\infty} ^2 \phi_{xx} + \phi_{yy} = 0\]We call this the transonic small disturbance equation. It is useful when we want to know some property like the lift of a wing, in which we can treat the whole feature as a sort of black box and we want to know what small change in the flow results from its presence.
Now we can compare with the full potential equation, and see that we’ve gotten rid of the \( \phi_y \) and \( \phi_{xy} \) terms. For linear flows (no shocks, completely subsonic or supersonic) we get
\[(1 - M_{\infty}^2 ) \phi_{xx} + \phi_{yy} = 0\]And for sub-sonic flow where \( M_{\infty} \rightarrow 0 \) we’re right back to the equation for incompressible flow.