Appendix A Dimensional analysis

We want to write down the general equations of fluid dynamics in dimensionless form. Therefore we introduce the following dimensionless quantities

	$r_i^{*}={\displaystyle \frac{r_i}{l_0}}$	$\Rightarrow {\displaystyle \frac{\partial }{\partial r_i}}={\displaystyle \frac{\partial }{\partial r_i^{*}}}{\displaystyle \frac{\partial r_i^{*}}{\partial r_i}}={\displaystyle \frac{1}{l_0}}{\displaystyle \frac{\partial }{\partial r_i^{*}}},$
	$v_i^{*}={\displaystyle \frac{v_i}{v_0}},$
	$t^{*}={\displaystyle \frac{t}{l_0}}$	$\Rightarrow {\displaystyle \frac{\partial }{\partial t}}={\displaystyle \frac{\partial }{\partial t^{*}}}{\displaystyle \frac{\partial t^{*}}{\partial t}}={\displaystyle \frac{1}{t_0}}{\displaystyle \frac{\partial }{\partial t^{*}}},$
	${\rho }^{*}={\displaystyle \frac{\rho }{{\rho }_0}}.$

Inserting these into the continuity equation (2.1) we get

	${\displaystyle \frac{{\rho }_0}{t_0}}{\displaystyle \frac{\partial }{\partial t^{*}}}{\rho }^{*}+{\displaystyle \frac{v_0{\rho }_0}{l_0}}{\displaystyle \frac{\partial }{\partial r_j^{*}}}(v_j^{*}{\rho }^{*})$	$=0|\cdot {\displaystyle \frac{l_0}{{\rho }_0{v}_0}},$
	$\underset{Sr}{\underbrace{{\displaystyle \frac{l_0}{v_0{t}_0}}}}{\displaystyle \frac{\partial }{\partial t^{*}}}{\rho }^{*}+{\displaystyle \frac{\partial }{\partial r_j^{*}}}(v_j^{*}{\rho }^{*})=0.$

This derivation shows that solutions of the continuity equation are similar, if the Strouhal number $Sr=\frac{l_0}{v_0{t}_0}$ is the same. Flows with a Strouhal number $Sr=0$ are so called stationary flows. Nevertheless, the Strouhal number is most often set to one, by assuming $v_o=\frac{l_0}{t_0}$. Using the additional dimensionless quantities $p^{*}=\frac{p}{p_0},{\sigma }_{ij}^{*}=\frac{{\sigma }_{ij}}{{\sigma }_0},g^{*}=\frac{g}{g_0}$ in the momentum equation (2.2) yields

	${\displaystyle \frac{{\rho }_0{v}_0}{t_0}}{\displaystyle \frac{\partial }{\partial t^{*}}}({\rho }^{*}{v}_i^{*})+{\displaystyle \frac{{\rho }_0{v}_0^2}{l_0}}{\displaystyle \frac{\partial }{\partial r_j^{*}}}(v_j^{*}{\rho }^{*}{v}_i^{*})$	$=-{\displaystyle \frac{p_0}{l_0}}{\displaystyle \frac{\partial }{\partial r_i^{*}}}{p}^{*}+{\displaystyle \frac{{\sigma }_0}{l_0}}{\displaystyle \frac{\partial }{\partial r_j^{*}}}{\sigma }_{ij}^{*}+{\rho }_0{g}_0{\rho }^{*}{g}_i^{*}|\cdot {\displaystyle \frac{l_0}{{\rho }_0{v}_0^2}},$
	$\underset{Sr}{\underbrace{{\displaystyle \frac{l_0}{v_0{t}_0}}}}{\displaystyle \frac{\partial }{\partial t^{*}}}({\rho }^{*}{v}_i^{*})+{\displaystyle \frac{\partial }{\partial r_j^{*}}}(v_j^{*}{\rho }^{*}{v}_i^{*})$	$=-\underset{M{a}_{iso}^{-2}}{\underbrace{{\displaystyle \frac{p_0}{{\rho }_0{v}_0^2}}}}{\displaystyle \frac{\partial }{\partial r_i^{*}}}{p}^{*}+\underset{R{e}^{-1}}{\underbrace{{\displaystyle \frac{{\sigma }_0}{{\rho }_0{v}_0^2}}}}{\displaystyle \frac{\partial }{\partial r_j^{*}}}{\sigma }_{ij}^{*}+\underset{F{r}^{-1}}{\underbrace{{\displaystyle \frac{{\rho }_0{g}_0{l}_0}{{\rho }_0{v}_0^2}}}}{\rho }^{*}{g}_i^{*}.$

The occurring dimensionless numbers are the isothermal Mach number $M{a}_{iso}$, which is related to the Euler number $Eu$ or the Ruark number $Ru$ like $M{a}_{iso}^2=Eu=R{u}^{-1}$, the Froude number $Fr$, which is related to the Richardson number $Ri$ like $Fr=R{i}^{-1}$ and the Reynolds number $Re$. All these numbers measure the importance of the term they are related to compared to the nonlinear advection term $\frac{\partial }{\partial r_j^{*}}(v_j^{*}{\rho }^{*}{v}_i^{*})$, e.g. for high Mach numbers, the pressure term $\frac{\partial }{\partial r_i^{*}}{p}^{*}$ becomes less and less important compared to the advection term; for high Reynolds numbers the stress term $\frac{\partial }{\partial r_j^{*}}{\sigma }_{ij}^{*}$ becomes less and less important compared to the advection term, and the equation shows more and more nonlinear behavior. For a newtonian fluid with

${\sigma }_{ij}\simeq \eta {\displaystyle \frac{\partial v_i}{\partial r_j}}={\displaystyle \frac{{\eta }_0{v}_0}{l_0}}{\eta }^{*}{\displaystyle \frac{\partial v_i^{*}}{\partial r_j^{*}}},$

(A.1)

where we introduced the dimensionless quantity ${\eta }^{*}=\frac{\eta }{{\eta }_0}$, we get ${\sigma }_0=\frac{{\eta }_0{v}_0}{l_0}$ and therefore we can express the Reynolds number⁵²⁵²We neglected the second viscosity $\zeta$. In principle there exists a second Reynolds number $R{e}_2=\frac{{\rho }_0{l}_0{v}_0}{{\zeta }_0}$. like

$Re={\displaystyle \frac{l_0{\rho }_0{v}_0^2}{{\eta }_0{v}_0}}={\displaystyle \frac{{\rho }_0{l}_0{v}_0}{{\eta }_0}}.$

(A.2)

Playing the same game with the equation for the internal energy

${\displaystyle \frac{\partial }{\partial t}}\rho e_{int}+{\displaystyle \frac{\partial }{\partial r_j}}{v}_j\rho e_{int}=T\left({\displaystyle \frac{\partial }{\partial t}}\rho s+{\displaystyle \frac{\partial }{\partial r_j}}{v}_j\rho s\right)-p{\displaystyle \frac{\partial }{\partial r_j}}{v}_j$

(A.3)

using $e_{int}^{*}=\frac{e_{int}}{u_0}$ we get

$\underset{Sr}{\underbrace{{\displaystyle \frac{l_0}{v_0{t}_0}}}}{\displaystyle \frac{\partial }{\partial t^{*}}}{\rho }^{*}{e}_{int}^{*}+{\displaystyle \frac{\partial }{\partial r_j^{*}}}{v}_j^{*}{\rho }^{*}{e}_{int}^{*}$

$=-\underset{G{a}_1}{\underbrace{{\displaystyle \frac{p_0}{{\rho }_0{u}_0}}}}{p}^{*}{\displaystyle \frac{\partial }{\partial r_j^{*}}}{v}_j^{*}+\underset{G{a}_2}{\underbrace{{\displaystyle \frac{{\sigma }_0}{{\rho }_0{u}_0}}}}{\sigma }_{ij}^{*}{\displaystyle \frac{\partial }{\partial r_j^{*}}}{v}_i^{*}.$

The new dimensionless quantities that occur in the energy equation seem to have no name in the literature, but we will call them ”Gamma1” (Ga1) and ”Gamma2” (Ga2) for now, since they are related to the adiabatic coefficient. This can be seen by replacing $p_0{p}^{*}$ according to equation

$p_0{p}^{*}=(\gamma -1){\rho }_0{u}_0{\rho }^{*}{e}_{int}^{*},$

(A.4)

which is valid for an an ideal, nonisothermal ($\gamma \ne 1$) gas. Doing this we get⁵³⁵³We cannot get rid of Ga2 in the same way, since therefore we would have to assume an equation relating ${\sigma }_0{\sigma }_{ij}^{*}$ to the internal energy. But this would only be possible, if we would assume that the internal energy is a tensorial quantity, which is not the way how internal energy is defined normally.

$Sr\cdot {\displaystyle \frac{\partial }{\partial t^{*}}}{\rho }^{*}{e}_{int}^{*}+{\displaystyle \frac{\partial }{\partial r_j^{*}}}{v}_j^{*}{\rho }^{*}{e}_{int}^{*}$

$=-(\gamma -1){\rho }^{*}{e}_{int}^{*}{\displaystyle \frac{\partial }{\partial r_j^{*}}}{v}_j^{*}+G{a}_2\cdot {\sigma }_{ij}^{*}{\displaystyle \frac{\partial }{\partial r_j^{*}}}{v}_i^{*}.$

For a selfgravitating fluid we even have on more dimensionless quantity, which appears, when we write down the dimesionless form of the Poisson equation of gravity

$\underset{C_G}{\underbrace{{\displaystyle \frac{g_0}{4\pi G{\rho }_0{l}_0}}}}{\displaystyle \frac{\partial g_i^{*}}{\partial r_i^{*}}}={\rho }^{*}.$

(A.5)

But this quantity $C_G$ also seems to have no name in the literature (e.g. Durst, 2007).