Validation Case: Compressible Flow In a de Laval Nozzle
This validation case belongs to fluid dynamics. The aim of this test is to validate the following parameters for a compressible, steady-state turbulent flow through a de Laval Nozzle:
Mach number across the nozzle.
Pressure across the nozzle.
The simulation results from SimScale were compared to analytical results obtained from methods presented in [1].
A typical configuration of the de Laval nozzle with a non-smooth throat that was chosen as the geometry can be seen below:
Figure 1: Geometry of the de Laval Nozzle
The nozzle is axisymmetric and an angular slice of the complete geometry with a wedge angle of 18\(°\) was used. The following table contains the coordinates of the reference points:
The hexahedral mesh was created externally, and then was imported on to the SimScale platform, with an inflation of \(y^+\) = 30 near the walls, as the ‘wall functions’ model was chosen. In order to keep the flow two-dimensional, the mesh was designed to have only one layer in the \(y\) direction.
Mesh type
Cells in the x-direction
Cells inthey-direction
Cells in the z–direction
Number of nodes
Type
blockMesh
150
1
100
15000
2D Hex
Table 2: Mesh Metrics
Figure 2: The mesh used for the SimScale case (zoomed in details at the bottom). It is only one cell thick in the third direction.
\(P_{stag}\) : the stagnation pressure, assumed to be 0.1301 \(MPa\)
\(\rho\) : the density of the fluid
Result Comparison
A comparison of the Mach number and pressure variation in the nozzle obtained with SimScale with analytical results is given below:
Figure 3: Visualization of the Mach number across the nozzle
Figure 4: Visualization of the pressure across the nozzle. Deviation in results is a result of 1D hypothesis assumed in the analytical method.
A source of deviation between the calculated and analytical results exists because the latter is calculated with a one-dimensional hypothesis. This means that all parameters are assumed to be uniform in the radial direction, while in the following figures, it seems that there exists radial variation too in the flow variables.
Figure 5: Velocity contour on the nozzle
One can observe the radial variation causing results to slightly deviate:
Figure 6: Temperature distribution on the de Laval nozzle
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