The impact of the form of the Euler equations for radial flow in cylindrical and spherical coordinates on numerical conservation and accuracy

Abstract

The radial one-dimensional Euler equations are often rewritten in what is known as the geometric source form. The differential operator is identical to the Cartesian case, but source terms result. Since the theory and numerical methods for the Cartesian case are well-developed, they are often applied without modification to cylindrical and spherical geometries. However, numerical conservation is lost. In this article, AUSM$^+$-up is applied to a numerically conservative (discrete) form of the Euler equations labeled the geometric form, a nearly conservative variation termed the geometric flux form, and the geometric source form. The resulting numerical methods are compared analytically and numerically through three types of test problems: subsonic, smooth, steady-state solutions, Sedov’s similarity solution for point or line-source explosions, and shock tube problems. Numerical conservation is analyzed for all three forms in both spherical and cylindrical coordinates. All three forms result in constant enthalpy for steady flows. The spatial truncation errors have essentially the same order of convergence, but the rate constants are superior for the geometric and geometric flux forms for the steady-state solutions. Only the geometric form produces the correct shock location for Sedov’s solution, and a direct connection between the errors in the shock locations and energy conservation is found. The shock tube problems are evaluated with respect to feature location using an approximation with a very fine discretization as the benchmark. Extensions to second order appropriate for cylindrical and spherical coordinates are also presented and analyzed numerically. Conclusions are drawn, and recommendations are made. A derivation of the steady-state solution is given in the Appendix.

Appendix: Steady-state solutions

In this Appendix, the steady-state solution for radial flows is derived based only on the assumption that the density, pressure, and velocity exist at some point. Therefore, it is representative of all smooth steady-state solutions in radial coordinates. Clain et al. [13] derived a differential equation for pressure under essentially the same assumptions which they then solved numerically. Here, the full solution is obtained analytically. Plots of their solution are qualitatively similar to the results given here.

The steady Euler equations with an axis of symmetry may be written as

$$\begin{aligned}&\frac{\partial }{\partial r} \left( r^\alpha \rho u \right) = 0, \end{aligned}$$

(80)

$$\begin{aligned}&\frac{\partial }{\partial r} \left( r^\alpha \rho u^2 \right) = -r^\alpha \frac{\partial P}{\partial r}, \end{aligned}$$

(81)

$$\begin{aligned}&\frac{\partial }{\partial r} \left( r^\alpha \rho H u \right) = 0, \end{aligned}$$

(82)

where $\alpha =0,\;1,\;2$ for Cartesian, cylindrical, or spherical coordinates, respectively. The variables are scaled by the values at a reference radius $r=r_\mathrm {R}$:

$$\begin{aligned} \begin{array}{ll} {\tilde{r}}=r/r_\mathrm {R},&{}\quad \tilde{u}=u(r)/u_\mathrm {R},\\ \tilde{\rho }=\rho (r)/\rho _\mathrm {R},&{}\quad \tilde{p}=p(r)/p_\mathrm {R}, \end{array} \end{aligned}$$

(83)

where a subscript of $\mathrm {R}$ indicates evaluation at $r=r_\mathrm {R}$. Assuming an ideal gas, the equations become (dropping the tildes immediately):

$$\begin{aligned}&\frac{\partial }{\partial r} \left( r^\alpha \rho u \right) = 0, \end{aligned}$$

(84)

$$\begin{aligned}&\frac{\partial }{\partial r} \left( r^\alpha \rho u^2 \right) + r^\alpha \text {Eu} \frac{\partial P}{\partial r} =0, \end{aligned}$$

(85)

$$\begin{aligned}&\frac{\partial }{\partial r} \left[ r^\alpha \rho u \left( \frac{\gamma }{\gamma -1} \frac{p}{\rho } \text {Eu} + \frac{u^2}{2} \right) \right] = 0, \end{aligned}$$

(86)

subject to $u(1)=1$, $p(1)=1$, $\rho (1)=1$, and where $\text {Eu}=P_\mathrm {R}/(\rho _r u_\mathrm {R}^2)$. Integrating the first and third equation give:

$$\begin{aligned}&r^\alpha \rho u = c_1, \end{aligned}$$

(87)

$$\begin{aligned}&\left( \frac{\gamma }{\gamma -1} \frac{P}{\rho } \text {Eu} + \frac{u^2}{2} \right) = c_2. \end{aligned}$$

(88)

Evaluating at $r=1$:

$$\begin{aligned} c_1 =1, \qquad c_2 = \left( \frac{\gamma }{\gamma -1} \text {Eu} + \frac{1}{2} \right) . \end{aligned}$$

(89)

Solving for P in terms of $c_2$ and using $\rho =1/(r^\alpha u)$:

$$\begin{aligned} P= \frac{\gamma -1}{\gamma \text {Eu}} \frac{1}{r^\alpha u}\left( c_2 -\frac{u^2}{2}\right) . \end{aligned}$$

(90)

Substituting into the momentum equation (85), and using $r^\alpha \rho u=1$ yields

$$\begin{aligned} \frac{\hbox {d}u}{\hbox {d}r} = -r^\alpha \text {Eu} \frac{\hbox {d}}{\hbox {d}r} \left[ \frac{\gamma -1}{\gamma \text {Eu}} \frac{1}{r^\alpha u}\left( c_2 -\frac{u^2}{2}\right) \right] . \end{aligned}$$

(91)

Separating variables, integrating and applying the boundary conditions

$$\begin{aligned} \frac{1}{u} \left( \frac{2c_2-1}{2c_2-u^2}\right) ^{\frac{1}{\gamma -1}} = r^\alpha . \end{aligned}$$

(92)

This gives u implicitly as a function of r. However, r can be found explicitly as a function of u. Hence, for a range of values of u, the corresponding values of r may be found and then the density and pressure are given by

$$\begin{aligned} \rho&= \left( \frac{2c_2-u^2}{2c_2-1}\right) ^{\frac{1}{\gamma -1}}, \end{aligned}$$

(93)

$$\begin{aligned} P&= \frac{\gamma -1}{2\gamma \text {Eu}} \frac{\left( 2c_2-u^2\right) ^{\frac{\gamma +1}{\gamma -1}}}{\left( 2c_2-1\right) ^{\frac{1}{\gamma -1}}}. \end{aligned}$$

(94)

In order to find the interval of existence of the solution, writing (92) as

$$\begin{aligned} u \left( 2c_2-u^2\right) ^{\frac{1}{\gamma -1}} = r^{-\alpha } \left( 2c_2-1\right) ^{\frac{1}{\gamma -1}}, \end{aligned}$$

(95)

and differentiating implicitly led to

$$\begin{aligned}&\frac{\hbox {d}u}{\hbox {d}r}\left( 2c_2-u^2\right) ^{\frac{2-\gamma }{\gamma -1}}\left[ 2c_2-u^2 \left( \frac{\gamma +1}{\gamma -1}\right) \right] \nonumber \\&\quad = \alpha r^{-(\alpha +1)}(2c_2-1)^{\frac{1}{\gamma -1}}. \end{aligned}$$

(96)

The quantity in square brackets is more restrictive on the values of u than the factor before it. Using the definition of $c_2$, we find that (in the unscaled variables)

$$\begin{aligned} u^2 < 2\frac{\gamma -1}{\gamma +1} H_\mathrm {R}=a^*, \end{aligned}$$

(97)

where $H_\mathrm {R}$ is the enthalpy evaluated at $r=R$ (or any other point, since H is constant) and $a^*$ is the critical speed of sound. In particular, $2c_2-u^2 \ne 0$ is equivalent to requiring the internal energy be positive, while

$$\begin{aligned} 2c_2-u^2 \frac{\gamma +1}{\gamma -1} \ne 0, \end{aligned}$$

(98)

is equivalent to requiring u to be subsonic.