# Defining appropriate test function spaces for the finite element solution of Euler's fluid equations

I have the following coupled equations for the conservation of mass and momentum of a compressible fluid

$$$$\rho_t + (\rho u)_z = 0,$$$$

$$(\rho u)_t + (\rho u^2)_z + \tfrac{1}{k}\rho_z = \left(\frac{dp}{dz}\right)_f + \rho g$$ over the domain $$[0, 2L]\times[0, \infty)$$ for density and velocity $$\rho = \rho(z, t)$$, $$u = u(z, t)$$ respectively. Note that $$g = g(z) = \begin{cases} 9.81 & \text{if } 0 \le z \le L \\ -9.81 & \text{if } L < z \le 2L \end{cases}$$ and that $$\left(\frac{dp}{dz}\right)_f$$ is a frictional pressure term that is non-linear in $$\rho$$ and $$u$$.

The weak variational problem is: $$\textit{find u\in V_1, \rho\in V_2 such that}$$ $$$$\int_0^{2L} v\rho_t - v_z\rho u dz = 0$$$$ $$\int_0^{2L}q(\rho u)_t - q_z\rho u^2 - \tfrac{1}{k}q_z\rho - q\left(\frac{dp}{dz}\right)_f - q\rho gdz = 0$$ $$\textit{for all test functions v, q in spaces V_1, V_2 respectively.}$$

I am then discretising these equations in time using an implicit backward Euler discretisation for implementation in the finite element solver software Firedrake (this is a standard approach when using Firedrake).

The problem I'm having is that the Firedrake solver does not converge with the inclusion of $$\left(\frac{dp}{dz}\right)_f$$, and gives the wrong solution when $$\left(\frac{dp}{dz}\right)_f$$ is excluded.

So in order to narrow down the causes of this divergence, my question is: are there specific spaces for $$V_1$$ and $$V_2$$ that should be used for this kind of variational problem?

• Hi, It is interesting. I want to know, 1. Why you are left with frictional pressure term without viscous term ($\mu \nabla^2 u$)? 2. If it is not frictional pressure term and, it is compressible, will there be issues in using equation of state ($p =\rho RT$ )? Feb 4 '20 at 8:18
• Hi mustang, I'm not entirely sure. What I do know is that for fluid moving at high speeds, the Navier-Stokes equations can be simplified to the Euler equations. To the best of my knowledge, this results in the Laplacian term being dropped. In terms of where the frictional term described above comes in to play, that may just be a constitutive term specific to the fluid being studied in my PhD. Feb 13 '20 at 12:19