Compressible Flow in a Nozzle using FLUENT

MAE 423 Spring 2007

Problem Specification

Consider air flowing at high-speed through a convergent-divergent nozzle having a circular cross-sectional area, A, that varies with axial distance from the throat, x, according to the formula

A = 0.1 + x2 − 0.5 ≤ x ≤ 0.5

where A is in square meters and x is in meters. The stagnation pressure po at the inlet is 101,325 Pa.The stagnation temperature To at the inlet is 300 K. The static pressure p at the exit is 3,738.9 Pa.

We will calculate the Mach number, pressure and temperature distribution in the nozzle usingFLUENT and compare the solution to quasi-1D nozzle flow results. The Reynolds number for thishigh-speed flow is large. So we expect viscous effects to be confined to a small region close to the wall.So it is reasonable to model the flow as inviscid.

Synopsis of Steps in FLUENT

1. Download the mesh file nozzle.msh fromhttp://www.mae.cornell.edu/swanson/mae423files.html

Launch FLUENT, select 2ddp version and read in the mesh file.

2. Grid → Check

3. Grid → Info → SizeHow many cells and nodes does the grid have?

4. Display → GridHow many nodes are there in the radial direction? Are the nodes clustered towards the wall?Why?

5. Define→ Models → SolverChoose Coupled and Axisymmetric.

6. Define→ Models → ViscousSelect Inviscid.

7. Define→ Models → EnergyThe energy equation needs to be turned on since this is a compressible flow where the energyequation is coupled to the continuity and momentum equations.

8. Define→ MaterialsSelect air under Fluid materials. Select Ideal Gas under Density. This means FLUENT usesthe ideal gas equation of state to relate density to the static pressure and temperature. UseFLUENT’s values for Cp and Molecular Weight for air. Click on Change/Create.

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9. Define → Operating ConditionsWe’ll work in terms of absolute rather than gauge pressures in this example. So set OperatingPressure to zero. It is important that you set the operating pressure correctly in compressibleflow calculations since FLUENT uses it to compute absolute pressure to use in the ideal gas law.

10. Define → Boundary ConditionsSet boundary conditions for the following surfaces: centerline, inlet, outlet, wall. Use thepressure-inlet boundary condition for the inlet surface. Set the total (i.e. stagnation) pressureand temperature at the inlet. For a subsonic inlet, Supersonic/Initial Gauge Pressure is theinitial guess value for the static pressure. Calculate this initial guess value from the 1D solution.Use the pressure-outlet boundary condition for the outlet surface and set the pressure at theoutlet.

11. Solve → Control → SolutionUse defaults.

12. Solve → InitializeSet the initial guess values to the values at the inlet by selecting inlet under Compute From andclicking on Init.

13. Monitors → Residual

14. Solve → IterateWhat does the convergence plot look like? How many iterations does it take to converge? Savecase and data after you have obtained a converged solution.

15. Plot → XY PlotPlot the variation of the Mach number in the axial direction at the axis and wall. The cor-responding variation from 1D theory is provided in the file M 1D.xy. How does the FLUENTsolution compare with the 1D solution?Save the Mach number values at the axis and wall in an “xy” file.

16. Display → ContoursPlot pressure and temperature contours.

Grid Adaption

FLUENT offers the capability of adapting the grid to the solution by enabling the user to increase thegrid resolution in regions of high gradients.To adapt the grid to the pressure gradient, click on Adapt → Gradient. Make sure that Pressure . . .and Static Pressure are selected under Gradients Of. Select Gradient under Method (use Curvatureonly if you want to adapt to the second- rather than first-derivative). The next step is to selectthe Refine Threshold. The grid will be refined in regions where the static pressure gradient is abovethe Refine Threshold. Click on Compute. This will display the minimum and maximum values ofthe adaption variable (pressure gradient). Display the static pressure gradient field by clicking onContour. . . . Select Adaption and Existing Value under Contours Of. Select Filled and deselect NodeValues. Click Display.Deselect Auto Range in the Contours panel. Try various values for Min to determine the RefineThreshold.Close the Contours panel and set the Refine Theshold. Click on Mark. FLUENT will report how manycells have been marked for refinement. Click on Adapt and Yes to Hanging-node mode. FLUENT willadapt the grid and display the old and new grid statistics.Look at the adapted grid using Grid → Display. Iterate until convergence on the adapted grid.

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