Global Stability Analysis of JAXA H-II Transfer Vehicle Re-entry Capsule

Abstract

The Japan Aerospace Exploration Agency (JAXA) conducted experimental and numerical investigations on the H-II Transfer Vehicle Re-entry (HTV-R) capsule and showed the existence of strong self-excited oscillations at an angle of attack of $20^\circ$ and Mach 0.4. The present work aims at performing fully three-dimensional global stability analysis on the HTV-R capsule to characterize its wake dynamic instabilities. An implicit pseudo-unsteady approach at high Courant-Friedrichs-Lewy (CFL) number is used to obtain a Reynolds-Averaged-Navier–Stokes (RANS) solution. Separation occurs on the capsule shoulder and a large steady planar-symmetric recirculation bubble appears. The RANS solution is used as baseflow for the global stability and the leading eigen-modes are calculated by coupling a time-stepper technique to an Arnoldi algorithm. A globally unstable mode is found at a non-dimensional frequency of 0.161. Both frequency and spatial structure of the global mode correspond to the dominant hairpin-like vortex shedding mode found by Ohmichi et al. [1] using Dynamic Mode Decomposition (DMD). No evidence of the other two dominant DMD modes, associated with helical vortex shedding and low-frequency recirculation bubble breathing, was found by the global stability analysis.

Appendix

While both nonlinear and linearized solvers have been already successfully verified for 3D compressible laminar flows [13], a validation for a turbulent transonic buffet case on the 2D OAT15A is here given. The flow conditions and geometry correspond to those in [27], for which the Mach number is $M=0.73$ and the Reynolds number based on the chord length (c) is $Re=3.2 \times 10^6$. The same 2D C-type structured grid obtained by normal extrusion around the airfoil has been used for all AoA. The numerical domain extends $30 \times c$ upstream of the profile and $40 \times c$ above, below and downstream. The grid counts about 149,000 cells: 160 in the profile-normal direction (about 60 cells in the boundary-layer and a $y^+_w<5$ for all first points off the profile), 300 along the suction side, 165 along the pressure side and 232 in the horizontal wake direction. The grid has been refined in the shock region between $x/c=0.25$ and $x/c=0.55$ where the grid spacing is about $\Delta _{x}/c=0.002$. The pressure coefficient, $C_p$, distributions of the converged RANS solutions corresponding to the AoA $=3.50, 4.50$ and $5.50^\circ$ are compared against those of [27, 28] in Fig. 5 (left plot) and show good agreement on both airfoil sides and in terms of shock positions. Contours of the dimensional streamwise velocity for the steady solution at AoA $=4.50^\circ$ are plotted in Fig. 5 (right plot) along with the sonic (black) and zero-streamwise velocity (white) iso-lines, showing the supersonic flow region/shock position and separation, respectively. The steady RANS solutions are selected as baseflows for the global stability analysis. For each AoA, an unsteady linearized RANS calculation has been run with a physical time step fixed to $\Delta t = 6.85 \times 10^{-7} \, s$, a Krylov time step of $\Delta T_{Kr} = 2500 \times \Delta t$ and maximum Krylov dimension of 80 for all cases. The evolution of the growth rate of the least stable/most unstable mode as a function of the AoA is reported in Fig. 6 (left plot). The parabolic trend indicates buffet onset and offset at AoA $\approx 3.40^\circ$ and AoA $\approx 5.05^\circ$, respectively. For the AoA $=4.50^\circ$ case, the eigenspectrum is reported in Fig. 6 (middle plot) and shows the existence of an unstable mode at $f \approx 75 \, Hz$. The contours of the real part of the eigen-density corresponding to this unstable mode are plotted in Fig. 6 (right plot). Similarly to [27, 29], this eigenmode corresponds to the buffet instability and is localized on the shock, shock foot and in minorly in the mixing layer.

Fig. 5

Left plot: Pressure coefficient for the AoA $=3.50^\circ$ (blue), AoA $=4.50^\circ$ (red) and AoA $=5.50^\circ$ (black). Present results (solid lines) are compared against those in [28] (empty circle symbols) and [27] (full circle symbols). Right plot: Streamwise velocity contours of the steady solution at AoA $=4.50^\circ$ with sonic (black) and zero-streamwise velocity (white) iso-lines

Fig. 6

Left plot: Growth rate of the least stable/most unstable mode as a function of the AoA. Middle plot: AoA $=4.50^\circ$ case eigenspectrum. Right plot: AoA $=4.50^\circ$ case contours of the real part of the eigen-density corresponding to the unstable (buffet) mode