Shock-fitting and shock-capturing simulations of transonic flows in a channel with a half lenticular profile

The present study focuses on particular properties of transonic flows through a planar channel featuring a circular bump on the lower wall. The selected geometry is reminiscent of the region surrounding the trailing edge of an airfoil at zero angle-of-attack and the resulting flow pattern is indeed similar to the fishtail shock-pattern that characterizes airfoils flying at nearly sonic speed. Numerical simulations have been conducted by solving the inviscid Euler equations using both a commercial and an in-house CFD code; discontinuities are modeled using shock- capturing in the former and shock-fitting in the latter. Numerical experiments reveal different shock-patterns obtained by independently varying the pressure ratio defined as the ratio between the static outlet pressure and the inlet total pressure. When shock-interactions occur, shock-polar analysis reveals that the branching point can be modeled using either von Neumann’s three-shock- theory or Guderley’s four-wave-theory, depending on the pressure ratio. Furthermore, for certain values of the pressure ratio, double solutions have been observed.