So I am trying to separate the adjoint advection operator taken from a linearised Navier-Stokes equation. It has been a while since I have delved into these equations, so need some assistance. Would someone be so kind as to check my working? The equation I am trying to solve:
For the current system we also have the conditions,
and , where is a streamwise wavenumber (i.e. -direction, -velocity direction). represents a complex number.
Applying this to the first equation we have (I think),
I get a bit confused with the as I don't really remember covariant derivatives that well. In fact, I didn't even do the tensor calculus, I just remembered some assumptions based on the conditions I stated. Not that sure it's even close to correct.
Thanks for your help in advance!!!!!