(a) Contour plot of the first y harmonic (q = 1) for the stream function model. This model represents the streamwise constant streaks and vortices commonly observed in DNS and experiments. (b) The stream function computed based on the x-averaged spanwise DNS velocity field, which was integrated to obtain the stream function, i.e., . These plots are reproduced from Ref. 22.
Contour plots of (a) , from and (b) the streamwise velocity component of the x-averaged DNS data. All plots correspond to R = 3000 and have the same contour levels. These plots are reproduced from Ref. 22.
Variation of the 2D/3C (streamwise constant) deviation from laminar, with perturbation amplitude (ɛ); based on input . A version of this figure is further discussed in Ref. 22.
(a) The streamwise energy scales as . (b) The amplification factor Γ ss scales as . The optimal spanwise wavenumber occurs at the maximum Γ ss for each R. The change in peak response with increasing Reynolds number is interesting; however, the exact nature of the optimal input-output response is a function of both k z and ɛ. Therefore, one needs to find the corresponding optimal ɛ in order to determine the true optimal spanwise large-scale feature spacing. (c) and (d) Γ ss for different values of ɛ all at R = 3000. Both Γ ss and the optimal spanwise wavenumber monotonically decrease with increasing ɛ. As ɛ gets small, we approach the linearized equations because . Thus, linear mechanisms dominate for very small ɛ and increasingly linearized equations have a larger input-output response. However, when ɛ = 0, the forced solution (deviation from laminar) is zero and we recover the laminar solution.
(a) The mean velocity profile of the DNS data along with the one computed from the steady-state [Eq. (1a)] for a ψ ss model (Eq. (2)) with q = 1, over a range of ɛ with k z corresponding to the peak Γ ss for each ɛ considered. (b) The mean velocity profile for ɛ = 0.00675 at a number of different k z values compared with DNS data. The data in both (a) and (b) correspond to R = 3000.
The velocity gradient at the wall continues to increase while both Γ ss and the energy peak and then drop off. The solid black line represents the peak Γ ss for each ɛ.
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