23.1 Boundary layer equations
Let’s derive the energy equation under the assumption of very thin boundary layer for very high Reynolds number hypersonic flows. We know that the non-dimensional energy equation is given by Eq. 20.8. This energy equation is for total energy which means the summation of kinetic energy and internal energy. We have neglected the potential energy of the fluid particle. Therefore let us derive the energy equation for kinetic energy alone. Consider the X-momentum and Y-momentum equations given by Eq. (20.2) and (20.3). in the dimensional form. Multiplying the X momentum equation by u velocity, we get;

Similarly multiply Y momentum equation by v velocity, we get;

Adding the above both the equations we have;

But the energy equation given by Eq. (20.4),