Hi Guys,

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:

A = -\left(\mathbf{U} \cdot \nabla \right)\mathbf{u}^* + \left(\nabla \mathbf{U}\right)^{\rm{T}} \cdot \mathbf{u}^*,


\mathbf{U} = U\mathbf{i} + V\mathbf{j} + W\mathbf{k},

\mathbf{u}^*=u^*\mathbf{i} + v^*\mathbf{j} + w^*\mathbf{k}.

For the current system we also have the conditions,

\frac{\partial \mathbf{U}}{\partial z} = U = V = 0 and \frac{\partial \mathbf{u}^*}{\partial z} = ik\mathbf{u}^* , where k is a streamwise wavenumber (i.e. z-direction, W-velocity direction).  i represents a complex number.

Applying this to the first equation we have (I think),

A = -\left[W\frac{\partial\left(u^*\mathbf{i} + v^*\mathbf{j} + w^*\mathbf{k}\right)}{\partial z}\right]\mathbf{k} + \left[\mathbf{k}\frac{\partial W}{\partial x} + \mathbf{k}\frac{\partial W}{\partial y}\right] \cdot \left(u^*\mathbf{i} + v^*\mathbf{j} + w^*\mathbf{k}\right)

A = -\left[i k W \mathbf{u}^*\right]\mathbf{k} + w^*\left(\frac{\partial W}{\partial x}\mathbf{i} + \frac{\partial W}{\partial y}\mathbf{j}\right)

I get a bit confused with the  \nabla \mathbf{U} 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!!!!!