In the source tree for Eilmer, there are a number of examples of the simulation code in action. These examples have been collected and maintained over many years. Some are well-tested and exercised, while others are more bleeding edge. The hope is that all of the examples offer launching points for you to start your own compressible flow simulations.

A number of these examples replicate classic test cases in the CFD literature on compressible flows, or classic experiments in high-speed flow. The listing below provides a mapping directories in the source tree to the literature. They are grouped loosely by the type of flow/geometry configuration. This is not a perfect grouping because the world of compressible flow test cases is not always so easy to categorise.

1. External, inviscid flows of perfect gases

1.1. 2D/foil-circular-arc-transonic

foil zoom flow domain mach

This example shows the transonic flow around an airfoil with a circular-arc profile. It appears to be a standard example in the textbooks on Computational Fluid Dynamics, where the solution for the full potential equations are discussed. Note that, effectively, the Euler equations solved here. Also, the image does not show the complete flow domain used for the simulation. The upper boundary is 6 chord lengths above the lower boundary and the inlet and outlet boundaries are each 5 chord lengths from the airfoil.

2. Internal, inviscid flows of perfect gases

2.1. 2D/nozzle-conical-back

nozzle conical back mach field at 4ms

This example shows the acceleration of gas through a converving-diverging nozzle. The left boundary is subsonic and driven by the stagnation condition of a hypothetical reservoir. The case is interesting because there are experimentally-measured wall pressures for the supersonic part of the nozzle.

Reference:

  1. LH Back, PF Massier and HL Gier (1965),
    Comparison of measured and predicted flows through conical supersonic nozzles, with emphasis on the transonic region.
    A.I.A.A. Journal 3(9): pp.1606-1614.

2.2. 2D/forward-facing-step

ffs density 5ms dx 2p5mm

This example shows the flow of gas at Mach 3 over a forward-facing step in a duct. The test case was first proposed by Emery (1968) and is described as a two-dimensional step. However, in the literature, this test case is often cited for its use in the paper by Woodward and Colella (1984) which tested the capabilities of various difference schemes for treating strong shocks. They describe this case as a Mach 3 Wind Tunnel with a Step.

References:

  1. Emery (1968),
    An Evaluation of Several Differencing Methods for Inviscid Fluid Flow Problems,
    Journal of Computational Physics, 2: pp.306-331.

  2. Woodward and Colella (1984),
    The Numerical Simulation of Two-Dimensional Fluid Flow with Strong Shocks,
    Journal of Computational Physics, 54: pp.115-173.

3. Laminar flows of perfect gases

3.1. 2D/flat-plate-hakkinen-swbli

swbli temperature field tindx 4

An oblique shock wave interacts with a laminar boundary layer on a flat flate. The boundary layer can be seen as the warmer temperture gas close to the lower boundary. The oblique shock generated at the leading edge of the upper boundary can be seen to arrive at the plate a little after half-way along the flow domain.

4. Turbulent flows of perfect gases

5. High-temperature gas flows

5.1. 2D/sphere-nonaka

nonaka temperature contours

This example simulates the flow of air over a sphere. It is an example of using the two-temperature air model for high-speed flow. The particular conditions are chosen to replicate the physical experiments performed by Nonaka et al. (2000). Those conditions are Case 18 in Table 1. These conditions were chosen because they align with Figure 10 in their article. Figure 10 is one of the only cases where the full shock shape was reported. Having access to the full shock shape is very useful for validation. The conditions of this case are:

  • sphere radius : 7mm

  • free stream velocity : 3490 m/s

  • free stream pressure : 4850 Pa

  • free stream temperature : 293.0 K (assumed, not stated)

  • free stream composition : f_N2 = 0.767, f_O2 = 0.233

A comparison of the computed shock shape to the experiment is shown here.

nonaka 2T model shock shape

Reference:

  1. Nonaka, S., Mizuno, H., Takayama, K. and Park, C. (2000),
    Measurement of Shock Standoff Distance for Sphere in Ballistic Range,
    Journal of Thermophysics & Heat Transfer, 14(2).