Supersonic Flow in a 2D Convergent-Divergent Channel

The present example is concerned with the study of compressible flow in a typical design of the nozzle with a fixed convergent section followed by a fixed divergent section. This nozzle configuration is called a convergent-divergent, or CD.

Now let us consider an external supersonic flow of air in a 2D (plane) convergent-divergent channel whose scheme is shown on Figure 1.

Figure 1. Supersonic flow in a 2D convergent-divergent channel.

A uniform supersonic stream of air, having a Mach number M = 3, static temperature of 293.2 K, and static pressure of 1 atm, is specified at the channel inlet between two parallel walls. In the next convergent section (see Figure 2) the stream decelerates through two oblique shocks shown schematically in Figure 1 as lines separating regions 1, 2, and 3. Since the convergent section has a special shape adjusted to the inlet Mach number, so the shock reflected from the upper plane wall and separating regions 2 and 3 comes to the section 3 lower wall edge, a uniform supersonic flow occurs in the next section 3 between two parallel walls. In the following divergent section the supersonic flow accelerates thus forming an expansion waves fan 4. Finally, the stream decelerates in the exit channel section between two parallel walls when passing through another oblique shock.

Figure 2. Dimensions (in m) of the 2D convergent-divergent channel including a reference line for comparing the Mach number.

The geometry model of this 2D channel is shown in Figure 3.

Since the channel was designed for the inviscid flow of an ideal gas, the ideal wall boundary condition has been specified and the laminar only flow has been considered instead of turbulent. The computed Mach number along the reference line and at the reference points (1-5) are compared with the theoretical values in Figure 4.

Figure 3. The model for calculating the 2D supersonic flow in the 2D convergent-divergent channel with Flow Simulation.

To obtain the most accurate results possible with Flow Simulation, the calculations have been performed at result resolution level 6. The predicted Mach number at the selected channel points (1-5) and along the reference line (see Figure 2), are presented in Table 1 and Figure 4 respectively.

Table 1. Mach number values predicted with Flow Simulation with comparison to the theoretical values at the reference points
Point	1	2	3	4	5
X coordinate of point, m	0.0042	0.047	0.1094	0.155	0.1648
Y coordinate of point, m	0.0175	0.0157	0.026	0.026	0.0157
Theoretical M	3.000	2.427	1.957	2.089	2.365
FLOEFD prediction of M	3.000	24.29	19.65	2.106	2.380
Prediction error, %	0.0	0.1	0.4	0.8	0.6

From Table 1 and Figure 4 it can be seen that the Flow Simulation predictions are very close to the theoretical values. In Figure 4 one can see that Flow Simulation properly predicts the abrupt parameter changes when the stream passes through the shock and a fast parameter change in the expansion fan.

Figure 4. Mach number values predicted with Flow Simulation along the reference line (the reference points on it are marked by square boxes with numbers) in comparison with the theoretical values.

To show the full flow pattern, the predicted Mach number contours of the channel flow are shown in Figure 5.

Figure 5. Mach number contours predicted by Flow Simulation.

This example illustrates that Flow Simulation is capable of capturing shock waves with a high degree of accuracy. This high accuracy is possible due to the Flow Simulation solution adaptive meshing capability. Solution adaptive meshing automatically refines the mesh in regions with high flow gradients such as shocks and expansion fans.