Conservation Form of Equations

In this section we will formulate the initial-boundary value problem for considered inviscid-viscous flows. It means that the system of equations (14) will be completed by some boundary and initial conditions. Firstly, we rewrite (14) in the so-called conservative form. In problems that will be numerically solved we will neglect the heat sources (i.e. q=0) and the volume forces ( i.e. ${\bf f} =0$). We define the state vector ${\bf w} = (\rho,\rho u_1,\dots,\rho u_d,e)$ and rewrite (14) in the following way:  

$$\frac{\partial {\bf w}}{\partial t} + \sum_{i=1}^{d}\frac... ...i} {\bf R}_i\left(\bf{w,grad\ w}\right) \quad \mbox{in}\ Q_T.$$ (14)

Here

$${\bf f}_i\left({\bf w}\right) := \left(\begin{array} {c} \rh... ...}{\partial x_i} \end{array}\right)\quad i=1,\dots,d. \nonumber$$

The functions ${\bf f}_i({\bf w})$ are called the inviscid Euler fluxes and ${\bf R}_i({\bf w,grad\ w})$ are called the viscous fluxes.

The system (15) gives the conservative form of the complete Navier-Stokes equations for viscous fluids. In the case of inviscid fluids $\lambda=\mu=k=0$ and, hence, ${\bf R}_i = 0$ Therefore, the conservative form of the Euler equations can be written in the form  

$$\frac{\partial {\bf w}}{\partial t} + \sum_{i=1}^{d}\frac... ...al x_i}{\bf f}_i\left({\bf w}\right) = 0 \quad \mbox{in}\ Q_T.$$ (15)

Moreover, the state equations should be added to close the system (15) or (17). Using (9) we get

$$p=(\kappa -1)\left(e-\frac12\rho\left\vert{\bf u}\right\vert^2\right).$$ (16)