Three-dimensional instability of a flow past a sphere: Mach evolution of the regular and Hopf bifurcations

Abstract

A fully three-dimensional linear stability analysis is carried out to investigate the unstable bifurcations of a compressible viscous fluid past a sphere. A time-stepper technique is used to compute both equilibrium states and leading eigenmodes. In agreement with previous studies, the numerical results reveal a regular bifurcation under the action of a steady mode and a supercritical Hopf bifurcation that causes the onset of unsteadiness but also illustrate the limitations of previous linear approaches, based on parallel and axisymmetric base flow assumptions, or weakly nonlinear theories. The evolution of the unstable bifurcations is investigated up to low-supersonic speeds. For increasing Mach numbers, the thresholds move towards higher Reynolds numbers. The unsteady fluctuations are weakened and an axisymmetrization of the base flow occurs. For a sufficiently high Reynolds number, the regular bifurcation disappears and the flow directly passes from an unsteady planar-symmetric solution to a stationary axisymmetric stable one when the Mach number is increased. A stability map is drawn by tracking the bifurcation boundaries for different Reynolds and Mach numbers. When supersonic conditions are reached, the flow becomes globally stable and switches to a noise-amplifier system. A continuous Gaussian white noise forcing is applied in front of the shock to examine the convective nature of the flow. A Fourier analysis and a dynamic mode decomposition show a modal response that recalls that of the incompressible unsteady cases. Although transition in the wake does not occur for the chosen Reynolds number and forcing amplitude, this suggests a link between subsonic and supersonic dynamics.

Direct numerical simulation of subsonic, transonic and supersonic flow over an isolated sphere up to a Reynolds number of 1000

“…Figure 3 shows the distribution of the flow regime in the Re-M plane under compressible conditions. The results of the previous studies by Nagata et al (2016), Riahi et al (2018) and Sansica et al (2018) are shown for comparison with those of the present study. The result of Nagata et al (2016) is obtained by the three-dimensional DNS with a boundary-fitted coordinate (BFC) grid, the result of Riahi et al (2018) is obtained the three-dimensional DNS with IBM, and the result of Sansica et al (2018) is obtained the three-dimensional GSA, respectively.…”

“…The flow past a sphere under the compressible low-Re flow has numerically been studied by Nagata et al (2016), Riahi et al (2018) and Sansica et al (2018). Nagata et al (2016Nagata et al ( , 2018a used DNS with a body-fitted grid to investigate fundamental characteristics such as aerodynamic force coefficients, flow structures and flow regime, with a stationary adiabatic sphere at 0.3 ≤ M ≤ 2.0 and 50 ≤ Re ≤ 300.…”

“…They found that the wake structure for M = 0.95 is unsteady (alternating hairpin wake) at Re = 600 and the wake for M = 2.0 is steady at Re ≤ 600. Sansica et al (2018) carried out a global stability analysis (GSA) at 0.1 ≤ M ≤ 1.2 and 200 ≤ Re ≤ 370. They examined the effects of Re and M on unsteadiness of the flow field and drew a stability map by tracking the bifurcation boundaries for different Re and M. These studies showed that the flow behind the sphere is stabilized when M increases, and unsteady flow patterns have not been observed at supersonic flows in the numerically investigated Re ranges.…”

Instability of shock train behaviour with incident shocks

“…Meanwhile the real parts of the conjugate eigenvalues increase above zero and STLE instability occurs. The evolution of the conjugate eigenvalues also indicates the occurrence of a Hopf bifurcation ( Sipp et al 2010;Sansica et al 2018). When C f decreases further, the real parts of the conjugate eigenvalues separate at some positions and the imaginary parts disappear.…”

“…2010; Sansica et al. 2018). When decreases further, the real parts of the conjugate eigenvalues separate at some positions and the imaginary parts disappear.…”