Unsteady Simulations of Rotor Stator Interactions Using SBP-SAT Schemes: Status and Challenges

Abstract

Recent developments in the SBP-SAT method have made available high-order interpolation operators (Mattsson and Carpenter, SIAM J Sci Comput 32(4):2298–2320, 2010). Such operators allow the coupling of different SBP methods across nonconforming interfaces of multiblock grids while retaining the three fundamental properties of the SBP-SAT method: strict stability, accuracy, and conservation. As these interpolation operators allow a more flexible computational mesh, they are appealing for complex geometries. Moreover, they are well suited for problems involving sliding meshes, like rotor/stator interactions, wind turbines, helicopters, and turbomachinery simulations in general, since sliding interfaces are (almost) always nonconforming. With such applications in mind, this paper presents an accuracy analysis of these interpolation operators when applied to fluid dynamics problems on moving grids. The classical problem of an inviscid vortex transported by a uniform flow is analyzed: the flow is governed by the unsteady Euler equations and the vortex crosses a sliding interface. Furthermore, preliminary studies on a rotor/stator interaction are also presented.

Appendix

The interpolation weights for the sixth order operator are (see Fig. 1):

$$\displaystyle{ Sol(P) =\sum _{ i=-2}^{3}w_{ i}Sol(N_{i}),\qquad \alpha = (h - h_{0})/h }$$

$$\displaystyle\begin{array}{rcl} w_{-2}& =& -1/120\quad (\alpha -3)(\alpha -2)\quad (\alpha -1)(\alpha )\quad \quad (1+\alpha ) {}\\ w_{-1}& =& +1/24\quad \ \ (\alpha -3)(\alpha -2)\quad (\alpha -1)(\alpha )\quad \quad (2+\alpha ) {}\\ w_{0}& =& -1/12\quad \ \ (\alpha -3)(\alpha -2)\quad (\alpha -1)(1+\alpha )\ (2+\alpha ) {}\\ w_{+1}& =& +1/12\quad \ \ (\alpha -3)(\alpha -2)\quad \quad \ (\alpha )(1+\alpha )\quad (2+\alpha ) {}\\ w_{+2}& =& -1/24\quad \ \ (\alpha -3)(\alpha -1)\quad \quad \ (\alpha )(1+\alpha )\quad (2+\alpha ) {}\\ w_{+3}& =& +1/120\quad \ (\alpha -2)(\alpha -1)\quad \quad \ (\alpha )(1+\alpha )\quad (2+\alpha ) {}\\ \end{array}$$

The interpolation weights for the eighth order operator are (see Fig. 1):

$$\displaystyle{ Sol(P) =\sum _{ i=-3}^{4}w_{ i}Sol(N_{i}),\qquad \alpha = (h - h_{0})/h }$$

$$\displaystyle\begin{array}{rcl} w_{-3}& =& -1/5040\quad (\alpha -4)(\alpha -3)\quad (\alpha -2)(\alpha -1)\quad \quad \ (\alpha )(\alpha +1)\quad \ \ \ (\alpha +2) {}\\ w_{-2}& =& +1/720\quad \ (\alpha -4)(\alpha -3)\quad (\alpha -2)(\alpha -1)\quad \quad \ (\alpha )(\alpha +1)\quad \ \ \ (\alpha +3) {}\\ w_{-1}& =& -1/240\quad \ (\alpha -4)(\alpha -3)\quad (\alpha -2)(\alpha -1)\quad \quad \ (\alpha )(\alpha +2)\quad \ \ \ (\alpha +3) {}\\ w_{0}& =& +1/144\quad \ (\alpha -4)(\alpha -3)\quad (\alpha -2)(\alpha -1)\quad (\alpha +1)\quad (\alpha +2)\quad (\alpha +3) {}\\ w_{+1}& =& -1/144\quad \ (\alpha -4)(\alpha -3)\quad (\alpha -2)(\alpha )\quad \quad \quad (\alpha +1)\quad (\alpha +2)\quad (\alpha +3) {}\\ w_{+2}& =& +1/240\quad \ (\alpha -4)(\alpha -3)\quad (\alpha -1)(\alpha )\quad \quad \quad (\alpha +1)\quad (\alpha +2)\quad (\alpha +3) {}\\ w_{+3}& =& -1/720\quad \ (\alpha -4)(\alpha -2)\quad (\alpha -1)(\alpha )\quad \quad \quad (\alpha +1)\quad (\alpha +2)\quad (\alpha +3) {}\\ w_{+4}& =& +1/5040\quad (\alpha -3)(\alpha -2)\quad (\alpha -1)(\alpha )\quad \quad \quad (\alpha +1)\quad (\alpha +2)\quad (\alpha +3) {}\\ \end{array}$$