Cart3D Sharp Wedge Study

This example highlights the use of PySAGAS on a CFD solution. Here, a diamond wedge geometry has been simulated in Cart3D, though the flow solution from any solver could be used instead of Cart3D.

Problem Defintion

The geometry for this example was generated using the parameteric geometry generation tool HyperVehicle. This tool provides the capability of generating cell vertex sensitivities to geometric parameters.

Wedge Geometry

Free Stream Conditions

The wedge is simulated at Mach 6, with a 3-degree angle of attack. Since Cart3D is an inviscid flow solver, this is all that is required to define the non-dimensional flow state.

Parameterisation

To simplify this case study, a single parameter of wedge thickness is used to alter the wedge geometry.

Wedge with thickness variations

Cart3D Solution

Wedge flow visualisation

Parameter Sensitivities via Finite Differencing

Running a series of simulations in Cart3D for geometric perturbations of the wedge thickness about the nominal value, the following sensitiviites can be generated using finite differencing. Note that the values are reported for sensitivities in the non-dimensional coefficients \(C_x, C_y\) and \(C_z\).

Parameter	\(\partial C_x/ \partial \theta\)	\(\partial C_y/ \partial \theta\)	\(\partial C_z/ \partial \theta\)
Thickness	0.14517	0.126153	0.0000

PySAGAS Solution

Using the nominal geometry’s Cart3D solution, the following sensitivities can be generated using PySAGAS. Note, the error of each sensitivitiy, as calculated using the Cart3D solution for reference, is shown in brackets.

Parameter	\(\partial C_x/ \partial \theta\)	\(\partial C_y/ \partial \theta\)	\(\partial C_z/ \partial \theta\)
Thickness	0.15496 (5.0%)	0.11912 (-5.6%)	0.00312 (-)

Special Considerations

Having a coarse geometry mesh will impact the accuracy of the Cart3D solution. In terms of computational expense, there is little reason to use a coarse geometry mesh, since this is loaded by Cart3D just once. A coarse geometry mesh may also not accurately capture the features of the geometry.
The computational expense of PySAGAS scales with the number of cells which must be transcribed from the Cart3D solution, so the resolution of the geometry mesh should not be excessive.