Spatial simulation of secondary instability in plane channel flow: comparison of K- and H-type disturbances

Abstract

This study involves a numerical simulation of spatially evolving secondary instability in plane channel flow. The computational algorithm integrates the time-dependent, three-dimensional, incompressible Navier–Stokes equations by a mixed finite-difference/spectral technique. In particular, we are interested in the differences between instabilities instigated by Klebanoff (K-) type and Herbert (H-) type inflow conditions, and in comparing the present spatial results with previous temporal models. It is found that for the present inflow conditions, H-type instability is biased towards one of the channel walls, while K-type instability evolves on both walls. For low initial perturbation amplitudes, H-type instability exhibits higher growth rates than K-type instability while higher initial amplitudes lead to comparable growth rates of both H-and K-type instability. In H-type instability, spectral analysis reveals the presence of the subharmonic two-dimensional mode which promotes the growth of the three-dimensional spanwise and fundamental modes through nonlinear interactions. An intermodal energy transfer study demonstrates that there is a net energy transfer from the three-dimensional modes to the two-dimensional mode. This analysis also indicates that the mean mode transfers net energy to the two-dimensional subharmonic mode and to the three-dimensional modes.

Spanwise-localized solutions of planar shear flows

“…The cross-stream quiver plots of these exact wall-bounded traveling waves, figures 8 and 10 bear intriguing resemblances to the cross-sections of near-wall lambda vortices of developing channel flow shown in Saiki et al (1993) figures 10 and 11. For example, the streamwise mean v, w flow of TW2-1 in figure 10 is strikingly like the temporal-mean v, w of the K-type-generated lambda vortex shown in Saiki et al (1993) figure 10(b).…”

“…For example, the streamwise mean v, w flow of TW2-1 in figure 10 is strikingly like the temporal-mean v, w of the K-type-generated lambda vortex shown in Saiki et al (1993) figure 10(b). The Λ-structure shown in Saiki et al (1993) figure 2(b) is evident in TW2-2 in figure 8(d). Of symmetry α, γ Re c Lx, Lz grid accur.…”

“…Similarly TW1-2 figure 8(b) was found by refining an initial guess formed from windowing TW1 in in its z-phase with σ y , σ z , τ xz symmetry and with windowing parameters a, b = 1.2, 0.4, resulting in a traveling wave solution with σ y , σ z symmetry. TW2-1 and TW2-2, shown in figure 8(c,d), are likely the most interesting solutions presented in this paper, as they represent traveling wave solutions that are spanwise localized and strongly concentrated near a single wall, and as such are the solutions most likely to be relevant to the lambda-vortex coherent structures that occur near the walls of shear flows (Saiki et al (1993)). TW2-1 was formed by windowing TW2 in its z-phase with symmetry τ x σ z , τ xz and windowing parameters a, b = 0.6, 0.4 to get an initial guess with τ x σ z symmetry, and refining that with Newton-Krylov-hookstep.…”

“…course the traveling waves in this paper are exact solutions of different flow conditions than the spatially-developing channel flow of Saiki et al (1993), so some translation of the flow conditions is required to make more precise comparisons. We intend to pursue this in future research.…”

Numerical Simulation of a Cylinder in Uniform Flow: Application of a Virtual Boundary Method

“…The negative constants Ͱ and ͱ are determined tude wave propagation problems (Streett and Macaraeg by observing the response of U once F is applied; Ͱ pro- [31]; Danabasoglu [9]; Danabasoglu et al [10]; Saiki et duces the natural oscillation frequency of the response, al. [26]). …”