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Supersonic Extension

The following methods are described in detail in Ref [1]. The supersonic extension in FlightStream is based on the solution of the Prandtl-Glaurt equation.

(1−Ma2)ϕxx2+ϕyy2+ϕzz2=0(1-Ma^2)\phi_{xx}^2+\phi_{yy}^2+\phi_{zz}^2=0 Eq. (1)

Applying the method of Green’s functions, the perturbation velocity potential at any point in the flow is given by:

κ(r)ϕ(r)=∫∫SGσdS−∫∫Sμ∂G∂ndS\kappa(r)\phi(r)=\int\int_S G \sigma dS - \int\int_S \mu \frac{\partial G}{\partial n}dS Eq. (2)

Where: | σ=∂ϕ∂n−∂ϕi∂n\sigma = \frac{\partial \phi}{\partial n} - \frac{\partial \phi_i}{\partial n} , | μ=ϕ−ϕi\mu = \phi - \phi_i (For other definitions, see Nomenclature).

Equation 2 may be solved approximately by discretizing the surface into, NN surfaces called panels. In order to solve this equation, boundary conditions are required. In FlightStream, modified Dirichlet BCs are used. The first BC is the zero surface mass flux which can be shown to reduce to:

∂ϕ∂n=−c^⋅n\frac{\partial \phi}{\partial n} = -\hat{c} \cdot n Eq. (3)

Boundary Conditions on Superinclined Panels

Superinclined panels are panels inclined to the freestream at an angle greater than the Mach angle. As such, they fall entirely outside of their own domain of influence. Because of this, boundary conditions cannot be imposed on the upstream surfaces of superinclined panels.

Further reading

  1. Implementation of MachLine: A Subsonic/Supersonic, Unstructured Panel Code