Supersonic Flow Through a Converging-Diverging Nozzle

In this application, AcuSolve is used to simulate the high-speed turbulent flow in a converging and then diverging nozzle. The flow within the nozzle enters as subsonic, reaches sonic at the throat and shortly after develops a normal shock. AcuSolve results are compared with experimental results adapted from Bogar and Sajben (1983). The close agreement of AcuSolve results to experimental measurements validates the ability of AcuSolve to simulate internal supersonic flows where normal shocks are present.

Problem Description

The problem consists of a converging and then diverging nozzle represented as an extruded planar surface, as shown in the image below, which is not drawn to scale. The length of the domain is 0.213 m with a throat height of 0.044 m. The stagnation pressure and static temperature are constrained at the inlet to 135,000 Pa and 265.98 K, respectively. The static pressure is defined using the outflow boundary condition, where Pstatic=97216.08 Pa. The flow in the diffuser has an inlet Reynolds number of 6.5x105. The upper and lower walls are set to no-slip and the side walls are set to slip, while the exterior walls are specified to be adiabatic. Images of the numerical mesh are shown below.

The AcuSolve simulation is prepared to model the pseudo-transient state to determine the shock location and corresponding scalar quantities within the diffuser. AcuSolve computes the flow and thermal solution by simulating a planar, two-dimensional flow problem with physical dimensions and input quantities as described in the Sajben Diffuser case presented by Bogar and Sajben (1983). The gas in the simulation domain enters at subsonic conditions and is forced to accelerate beyond Mach=1.0 as it passes through the throat. After reaching supersonic speed a normal shock wave develops causing the flow speed to reduce back below sonic. The pressure along the bottom wall of the diffuser is compared against experimental measurements to demonstrate the solver’s capability to solve supersonic cases.
Figure 1. Parameters for Simulating Compressible Flow within a Converging-Diverging Nozzle

Figure 2. Mesh used for Simulating Compressible Flow in a Converging-Diverging Nozzle

AcuSolve Results

The AcuSolve solution is run for 0.025 seconds (6,250 timesteps with dt=4.0x10-6sec) and converges to a pseudo-steady solution. The change in pressure from the inlet to outlet causes the flow to accelerate within the throat of the diffuser until it reaches approximately Mach=1.35. A normal shock is present just downstream of where the cross-sectional area of the nozzle increases, as shown in the image below. The numerical solution can be compared with the reference after normalizing the pressure along the wall by the inlet stagnation pressure (135 kPa) and the distance along the bottom wall by the nozzle throat height (0.044 m). The image below shows the normalized pressure versus normalized distance in comparison to NASA reference and experimental results. The image shows black circles representing the experimental measurements (Bogar and Sajben 1983), a dashed blue line representing the SA solution as obtained through NASA publication (Mohler 2005) and a solid red line representing the AcuSolve results with the Spalart-Allmaras turbulence model. The non-dimensional pressure within the diffuser agrees well with the reference data, demonstrating the correct physical location of the normal shock and expected moderate flow speed reduction within the constant cross-section diffuser portion of the nozzle.
Figure 3. Contours of Mach Number within a Converging-Diverging Nozzle

Figure 4. Normalized Pressure Versus Normalized Distance in Comparison to NASA Reference and Experimental Results


The AcuSolve solution compares well with experimental and numerical results for the described simulation. In this application, compressibility effects are accounted for using the coupled compressible flow solver with the ideal gas material definition. The thermal and momentum exchange within the nozzle is computed with AcuSolve using the compressible Navier-Stokes solver with the Spalart-Allmaras turbulence model. The experimental values of pressure are presented with the corresponding AcuSolve results and show good agreement, validating the ability to solve compressible cases with the presence of normal shocks. The results indicate that AcuSolve is able to properly predict the peak pressure magnitude and location within the diffuser while maintaining the normal shock when the time-step is selected to result in a maximum CFL of approximately 30.

Simulation Settings for Supersonic Flow Through a Converging-Diverging Nozzle

HyperMesh CFD database file: <your working directory>\diffuser_compressible\


  • Problem Description
    • Flow – Compressible Navier Stokes
    • Analysis type – Transient
    • Turbulence equation – SA
    • Temperature equation – Advective Diffusive
  • Auto Solution Strategy
    • Initial time increment – 4.0E-6
    • Final time – 0.025 sec
    • Convergence Tolerance – 0.001 sec
    • Relaxation factor – 0.5
  • Material Model
    • Air-Ideal Gas
      • Type – Ideal_Gas
      • Viscosity – 1.781e-05 kg/m-sec
      • Specific Heat – 1005 J/kg-K
      • Conductivity – 0.02521 W/m-K
      • Gas Constant – 287.058 J/kg-K


  • Volumes
    • Fluid
      • Material – Air-Ideal Gas
  • Surfaces
    • Inflow
      • Simple Boundary Condition – Inflow
      • Inflow type – Stagnation Pressure
      • Stagnation Pressure – 135000.0 N/m2
      • Turbulence_input_type – Direct
      • Eddy Viscosity – 1e-6 m2/s
      • Temperature – 265.98 K
    • Outflow
      • Simple Boundary Condition – Outflow
      • Pressure – 97216.08 N/m2
    • Wall_1
      • Simple Boundary Condition – Wall
      • Type – auto_wall
    • Wall_2
      • Simple Boundary Condition – Wall
      • Type – auto_wall
    • Slip_1
      • Simple Boundary Condition – Slip
    • Slip_2
      • Simple Boundary Condition – Slip


Bogar, T. J., Sajben, M., and Kroutil, J. C. (1983) "Characteristic Frequencies of Transonic Diffuser Flow Oscillations," AIAA Journal, Vol. 21, No. 9, pp. 1232-1240.

Mohler, S.R., "Wind-US Unstructured Flow Solutions for a Transonic Diffuser," NASA CR-2005-213417, January 2005.