RESUMO
Vortex-induced instability caused by a free-stream vortical excitation is explored here quantitatively with the help of controlled computational results. First, the computed results are compared with experimental results in Lim et al. [Exp. Fluids 37, 47 (2004)10.1007/s00348-004-0783-5] for the purpose of validation of the three-dimensional (3D) computations. Thereafter, the computed results are explained using methods developed to study nonlinear and spatiotemporal aspects of receptivity and instability for incompressible flows. Here a zero-pressure-gradient (ZPG) boundary layer is perturbed by a constant strength vortex traveling at a fixed height, moving with constant speed, as in the cited experiment. The vortex is created by a translating and rotating circular cylinder in the experiment, with absolute control of the physical parameters. The sign of the translating vortex is fixed by the direction of rotation of the translating cylinder. A high accuracy computing method is employed to solve the 3D Navier-Stokes equation (NSE) for different translation speeds and signs of the free-stream vortex. A nonlinear disturbance enstrophy transport equation (DETE) for incompressible flows is used to explain the vortex-induced instability. This equation is exact and explains the instabilities, as governed by the NSE. The DETE approach has been successfully developed to explain two-dimensional (2D) vortex-induced instability in Sengupta et al. [Phys. Fluids 30, 054106 (2018)10.1063/1.5029560], to trace the linear and nonlinear stages of disturbance growth. Apart from quantification of vortex-induced instability, another major goal is to show how the disturbance evolves from an initial 2D to a 3D stage. While the sign of the translating vortex is important in creating the response field, we additionally highlight the distinct differences caused by increased translation speed and strength of the free-stream vortex on the overall instability. These explain creation of small-scale vortices via the instability of an equilibrium flow, even though the excitation is 2D only. For some cases, this causes 3D bypass transition. We also show a case which demonstrates strong unsteady separation with inflectional velocity profiles, yet the disturbance flow remains essentially 2D, which can be termed a bypass transition.
RESUMO
The investigation on grid sensitivity for the bifurcation problem of the canonical lid-driven cavity (LDC) flow results is reported here with very fine grids. This is motivated by different researchers presenting different first bifurcation critical Reynolds number (Re_{cr1}), which appears to depend on the formulation, numerical method, and choice of grid. By using a very-high-accuracy parallel algorithm, and the same method with which sequential results were presented by Lestandi et al. [Comput. Fluids 166, 86 (2018)CPFLBI0045-793010.1016/j.compfluid.2018.01.038] [for (257 × 257) and (513 × 513) uniformly spaced grid], we present results using (1025×1025) and (2049×2049) grid points. Detailed results presented using these grids help us understand the computational physics of the numerical receptivity of the LDC flow, with and without explicit excitation. The mathematical physics of the investigated problem will become apparent when we identify the roles of numerical errors with the ambient omnipresent disturbances in real physical flows as interchangeable. In physical or in numerical setups, presence of disturbances cannot be ignored. In this context, the need for explicit excitation for the used compact scheme arises for a definitive threshold amplitude, below which the flow relaxes back to quiescent state after the excitation is removed in computations. We also implement the present parallel method to show the physical aspects of primary and secondary instabilities to be maintained for other numerical schemes, and we show the results to reflect the complex physics during multiple subcritical Hopf bifurcation. Also, we relate the various sources of errors during computations that is typical of such shear-driven flow. These results, with near spectral accuracy, constitute universal benchmark results for the solution of Navier-Stokes equation for LDC.
