What is the exact formulation of compressible Euler equation of gas dynamics in polar coordinates with artificial diffusion in 2D?

Question

The interested equation is advection-diffusion equation. One of the canonical example is Navier-Stokes equations. However, I would like to let the coefficient of diffusion constant goes to zero, $\epsilon \rightarrow 0$, while mesh sizes, $ \Delta x, \Delta dy \rightarrow 0 $, and time steps,$\Delta t \rightarrow 0$, decreasing towards zero. Then, I can obtain inviscid compressible equation which can be an example for the Euler equation of gas dynamics. The following equation is scalar advection-diffusion equation in 2D cartesian coordinates.
$$ \frac{df}{dt} + U\frac{df}{dx} + V\frac{df}{dy} = \epsilon \left( \frac{d^2f}{dx^2} + \frac{d^2f}{dy^2} \right)$$

The main interest is to find exact formulation for the Euler equations of gas dynamics in cylindrical coordinates $(r, \theta)$ with artificial diffusion part.

Edit : The main equation that I am interested in the Euler equations of gas dynamics in 2D. The cartesian vector form can be stated as $$\begin{pmatrix} \rho \\ \rho u \\ \rho v \\ E \end{pmatrix}_t + \begin{pmatrix} \rho u \\ \rho u^2 + p \\ \rho u v \\ u(E + p) \end{pmatrix}_x + \begin{pmatrix} \rho v\\ \rho u v \\ \rho v^2 + p \\ v(E + p) \end{pmatrix}_y = \textbf{0}$$

Here, conservation of mass, momentums in -x and -y directions and energy equations can be seen in a matrix form. However, I would like to add artificial viscosity with small constant factor as in Navier-Stokes equations to smoothen the possible oscillations that will occur during numerical simulation. For this reason, I need to know the vector Laplacian form in cylindrical coordinates for Navier-Stokes equations. Then I can apply the above theory that was stated.

Answer 1

Have a look at Chapter 3 of the bachelor thesis "Modellierung und Simulation von Dispersionen in turbulenten Strömungen" by Manuel Baumann.

It is in German but the equations and pictures can be understood.

In Chapter 3.1.1, the coordinate transformation that in particular affects the convection is explained.

In Chapter 3.2.1, the transformed RANS equations are formulated. Set the right terms to zero and you get the Euler equations (equation (19) on page 29).

For the Euler equation, the main work will be the convective term that in polar coordinates writes as

Note that in polar coordinates, the coordinate unit vectors $e_\theta$, $e_r$ depend on the angular coordinate $\theta$. This means, e.g., $$ \partial _\theta (U_\theta \cdot e_\theta) = \partial _\theta (U_\theta) \cdot e_\theta - U_\theta \cdot e_r $$ what makes the additional terms appear in the last equality of the equation above.

As a reference, I can recommend "Mechanics of Fluid Flow" by Longwell.

Disclaimer: I was supervising the thesis mentioned above.