Shock-capturing method

In computational fluid dynamics, shock-capturing methods are a class of techniques for computing inviscid flows with shock waves. Computation of flow through shock waves is an extremely difficult task because such flows results in sharp, discontinuous changes in flow variables pressure, temperature, density, and velocity across the shock.

Explanation

In shock-capturing approach the governing equations of inviscid flows (Euler equations) are cast in conservation form and any shock waves or discontinuities are computed as part of the solution. Here, no special treatment is employed to take care of the shocks themselves. This is in contrast to the shock-fitting method, where shock waves are explicitly introduced in the solution using appropriate shock relations (Rankine-Hugoniot relations).

The shock capturing methods are relatively simple compared to the more elaborate shock fitting methods. However, the shock waves predicted by shock-capturing methods are generally not sharp and smear over several grid points. Also, classical shock-capturing methods have the disadvantages that unphysical oscillations (Gibbs phenomenon) may develop in the vicinity of strong shocks.

Euler equation

The Euler equations are the governing equations for inviscid flows. To implement shock-capturing methods, the conservation form of the Euler equations are used. For a flow without external heat transfer and work transfer (isoenergetic flow), the conservation form of the Euler equation in Cartesian coordinate system can be written as

${\frac {\partial {\mathbf {U} }}{\partial t}}+{\frac {\partial {\mathbf {F} }}{\partial x}}+{\frac {\partial {\mathbf {G} }}{\partial y}}+{\frac {\partial {\mathbf {H} }}{\partial z}}=0$

where the vectors U, F, G, and H are given by

${\mathbf {U}}=\left[{\begin{array}{c}\rho \\\rho u\\\rho v\\\rho w\\\rho e_{t}\\\end{array}}\right]\qquad \quad {\mathbf {F}}=\left[{\begin{array}{c}\rho u\\\rho u^{2}+p\\\rho uv\\\rho uw\\(\rho e_{t}+p)u\\\end{array}}\right]\qquad \quad {\mathbf {G}}=\left[{\begin{array}{c}\rho v\\\rho vu\\\rho v^{2}+p\\\rho vw\\(\rho e_{t}+p)v\\\end{array}}\right]\qquad \quad {\mathbf {H}}=\left[{\begin{array}{c}\rho w\\\rho wu\\\rho wv\\\rho w^{2}+p\\(\rho e_{t}+p)w\\\end{array}}\right]\qquad \qquad$

where $e_{t}$ is the total energy (internal energy + kinetic energy + potential energy) per unit mass. That is

$e_{t}=e+{\frac {u^{2}+v^{2}+w^{2}}{2}}+gz$

The Euler equation may be integrated with any of the standard shock-capturing methods available to obtain the solution.

References

• Tannehill, J. C., Anderson, D. A., and Pletcher, R. H., "Computational Fluid Dynamics and Heat Transfer", 2^nd ed., Taylor & Francis (1997).
• Hirsch, C., "Numerical Computation of Internal and External Flows", Vol. I, 2^nd ed., Butterworth-Heinemann (2007).
• Anderson, J. D., "Modern Compressible Flow with Historical Perspective", McGraw-Hill (2004)