The aim of this test case is to validate the following parameters of compressible steady-state turbulent flow through a de Laval Nozzle subsonic and supersonic flow regimes:
The rhoSimpleFoam solver is used for the subsonic case, while for the supersonic case the sonicFoam solver is employed. The \(k-\omega\ SST\) model was used to model turbulence. Simulaton results of SimScale were compared to analytical results obtained from methods elucidated in 1. The mesh was created locally using the blockMesh tool and then imported on to the SimScale platform.
Import validation project into workspace
A typical configuration of the de Laval nozzle with a non-smooth throat was chosen as the geometry (see Table 1 for coordinates). Since the nozzle is axisymmetric, it was modeled as an angular slice of the complete geometry, with a wedge angle of 18 degrees (see Fig.1.).
| Point | \(x\) | \(y\) | \(z\) |
| A | 0 | 0 | 0 |
| B | 0 | 5.5434 | 35 |
| C | 0 | -5.5434 | 35 |
| D | 68.68 | 0 | 0 |
| E | 68.68 | 3.1677 | 20 |
| F | 68.68 | -3.1677 | 20 |
| G | 238.8 | 6.3354 | 40 |
| H | 238.8 | -6.3354 | 40 |
A non-uniformly-spaced hexahedral mesh was generated using the blockMesh tool (see Fig.2.). Flow near the nozzle wall was resolved using inflation of \(y^+=30\) for the subsonic and \(300\) for the supersonic case. In order to keep the flow two-dimensional, the mesh was designed to have only one layer in the \(y\) direction.
Tool Type : OPENFOAM®
Analysis Type : Compressible Steady-state (Turbulent)
Mesh and Element types :
| Mesh type | Cells in x | Cells in y | Cells in z | Number of nodes | Type |
| blockMesh | 150 | 1 | 100 | 15000 | 2D Hex |
Fluid:
Table 3 encapsulates the properties of fluids used in the subsonic and supersonic case simulations. The need for using a different fluid for supersonic case arises from the courant number restriction.
| Case | \(m\) [\(g/mol\)] | \(c_p\) [\(J/kgK\)] | \(\mu\) [\(N/ms\)] | \(Pr\) |
| Subsonic | 28.9 | 1005 | \(1.79\times10^{-5}\) | 1 |
| Supersonic | 11640.3 | 2.5 | \(1.8\times10^{-5}\) | 1 |
Boundary Conditions:
| Parameter | Inlet (ABC) | Outlet (GHI) | Wall (AEHIFC) | Wedges (AGHEBA + AGIFCA) |
| Velocity | \(7.58\) \(m/s\) | Zero Gradient | \(0.0\) \(m/s\) | Wedge |
| Pressure | Zero Gradient | \(1.3\times10^{5}\) \(N/m^2\) | Zero Gradient | Wedge |
| Temperature | \(300\) \(K\) | Zero Gradient | Zero Gradient | Wedge |
| \(k\) | \(0.862\) \(m^2/s^2\) | Zero Gradient | Wall Function | Wedge |
| \(\omega\) | \(484.269\) \(s^{-1}\) | Zero Gradient | Wall Function | Wedge |
| \(\alpha_t\) | \(0\) Calculated | \(0\) Calculated | Wall Function | Wedge |
| \(\mu_t\) | \(0\) Calculated | \(0\) Calculated | Wall Function | Wedge |
| Parameter | Inlet (ABC) | Outlet (GHI) | Wall (AEHIFC) | Wedges (AGHEBA + AGIFCA) |
| Velocity | Zero Gradient | Zero Gradient | \(0.0\) \(m/s\) | Wedge |
| Pressure | \(1.2999\) \N/m^2\) | Wave Transmissive \(0.0296\) \(N/m^2\) | Zero Gradient | Wedge |
| Temperature | \(1.0388\) \(K\) | Zero Gradient | Zero Gradient | Wedge |
| \(k\) | \3.75\times10^{-5}\) \(m^2/s^2\) | Zero Gradient | Wall Function | Wedge |
| \(\omega\) | \(0.144\) \(s^{-1}\) | Zero Gradient | Wall Function | Wedge |
| \(\alpha_t\) | \(0\) Calculated | \(0\) Calculated | Wall Function | Wedge |
| \(\mu_t\) | \(0\) Calculated | \(0\) Calculated | Wall Function | Wedge |
Results for the subsonic cases are calculated from Bernoulli’s equation and the ideal gas equation as follows:
Subsonic flow does not see a significant temperature rise. So the speed of sound remains almost constant and can be calculated as:
$$c = \sqrt{\frac{\gamma RT}{m}}$$
Here, \(\gamma\), \(T\), \(m\) and \(R\) represent the specific heat ratio, temperature, and molecular weight of the fluid, and the universal gas constant respectively. The Mach number can then be calculated as:
$$M = \frac{A_{in}U_{in}}{Ac}$$
Assuming a stagnation pressure of \(P_{stag} = 0.1301\ MPa\), the static pressure can be computed as:
$$P_{stat} = P_{stag} – \frac{1}{2}\rho u^2$$
The relation between nozzle cross-section area \(A\) and Mach number \(M\) is what governs flow characteristics in supersonic flow through a de Laval nozzle 2:
$$\frac{A}{A_t} = \frac{1}{M}\left[\frac{2}{\gamma + 1}\left(1+\frac{\gamma – 1}{2}M^2\right)\right]^{\frac{\gamma + 1}{2(\gamma – 1)}}$$
Using the known area ratio, the mach number variation is calculated by solving the above equation. The pressure can be calculated using 3:
$$p = p_0\left(1+\frac{\gamma -1}{2}M^2\right)^{\frac{\gamma}{1-\gamma}}$$
A comparison of the Mach number and pressure variation in the nozzle obtained with SimScale with analytical results is given in Fig.3A and 3B for subsonic and Fig.4A and 4B for supersonic flow.
The deviation from analytical results exists because the latter is calculated with a one-dimensional hypothesis. Thus, all parameters are assumed to be uniform in the radial direction. Fig.5. shows that this is in fact not the case – there exists radial variation in flow variables. This is one reason why some deviation is seen between the two.
