Linear non-normal energy amplification of harmonic and stochastic forcing in turbulent channel flow

The linear response to stochastic and optimal harmonic forcing of small coherent perturbations to the turbulent channel mean flow is computed for Reynolds numbers ranging from Re_tau=500 to Re_tau=20000. Even though the turbulent mean flow is linearly stable, it is nevertheless able to sustain large amplifications by the forcing. The most amplified structures consist of streamwise elongated streaks that are optimally forced by streamwise elongated vortices. For streamwise elongated structures, the mean energy amplification of the stochastic forcing is found to be, to a first approximation, inversely proportional to the forced spanwise wavenumber while it is inversely proportional to its square for optimal harmonic forcing in an intermediate spanwise wavenumber range. This scaling can be explicitly derived from the linearised equations under the assumptions of geometric similarity of the coherent perturbations and of logarithmic base flow. Deviations from this approximate power-law regime are apparent in the premultiplied energy amplification curves that reveal a strong influence of two different peaks. The dominant peak scales in outer units with the most amplified spanwise wavelength of $\lambda_z \approx 3.5 h$ while the secondary peak scales in wall units with the most amplified $\lambda_z^+\approx 80$. The associated optimal perturbations are almost independent of the Reynolds number when respectively scaled in outer and inner units. In the intermediate wavenumber range the optimal perturbations are approximatively geometrically similar. Furthermore, the shape of the optimal perturbations issued from the initial value, the harmonic forcing and the stochastic forcing analyses are almost indistinguishable. The optimal streaks corresponding to the large-scale peak strongly penetrate into the inner layer, where their amplitude is proportional to the mean-flow profile. At the wavenumbers corresponding to the large-scale peak, the optimal amplifications of harmonic forcing are at least two orders of magnitude larger than the amplifications of the variance of stochastic forcing and both increase with the Reynolds number. This confirms the potential of the artificial forcing of optimal large-scale streaks for the flow control of wall-bounded turbulent flows

Self-sustained processes in the logarithmic layer of turbulent channel flows

It has recently been shown that large-scale and very-large-scale motions can self-sustain in turbulent channel flows even in the absence of input from motions at smaller scales. Here we show that also motions at intermediate scales, mainly located in the logarithmic layer, survive when motions at smaller scales are artificially quenched. These elementary self-sustained motions involve the bursting and regeneration of sinuous streaks. This is a further indication that a full range of autonomous self-sustained processes exists in turbulent channel flows with scales ranging from those of the buffer layer streaks to those of the large scale motions in the outer layer

A sequence of transcritical bifurcations in a suspension of gyrotactic microswimmers in vertical pipe

Kessler (Nature, vol. 313, 1985, pp. 218-220) first showed that plume-like structures spontaneously appear from both stationary and flowing suspensions of gyrotactic microswimmers in a vertical pipe. Recently, it has been shown that there exist multiple numbers of steady axisymmetric axially uniform solutions to such a system (Bees, M. A. & Croze, O. A., Proc. R. Soc. A., vol. 466, 2010, pp. 2057-2077). In the present study, we generalise this finding by reporting that a countably infinite number of such solutions emerge as Richardson number increases. Linear stability, weakly nonlinear and fully nonlinear analyses are performed, revealing that each of the solutions arises from the destabilisation of uniform suspension. The discrete number of the solutions is due to the finite flow domain, while the transcritical nature of the bifurcation is because of the cylindrical geometry which breaks the horizontal symmetry of the system. It is further shown that there exists a maximum threshold of achievable downward flow rate for each solution if the flow is to remain steady, as varying the pressure gradient can no longer increase the flow rate from the solution. Except the one arising at the lowest Richardson number, all of the solutions found are unstable, implying that they would play a role in the transient dynamics in the route from a uniform suspension to the fully-developed gyrotactic pattern

Near-wall turbulent fluctuations in the absence of wide outer motions

International audienceNumerical experiments that remove turbulent motions wider than lambda(+)(z) similar or equal to 100 are carried out up to Re-tau = 660 in a turbulent channel. The artificial removal of the wide outer turbulence is conducted with spanwise minimal computational domains and an explicit filter that effectively removes spanwise uniform eddies. The mean velocity profile of the remaining motions shows very good agreement with that of the full simulation below y(+) similar or equal to 40, and the near-wall peaks of the streamwise velocity fluctuation scale very well in the inner units and remain almost constant at all the Reynolds numbers considered. The self-sustaining motions narrower than lambda(+)(z) similar or equal to 100 generate smaller turbulent skin friction than full turbulent motions, and their contribution to turbulent skin friction gradually decays with the Reynolds number. This finding suggests that the role of the removed outer structures becomes increasingly important with the Reynolds number; thus one should aim to control the large scales for turbulent drag reduction at high Reynolds numbers. In the near-wall region, the streamwise and spanwise velocity fluctuations of the motions of lambda(+)(z) <= 100 reveal significant lack of energy at long streamwise lengths compared to those of the full simulation. In contrast, the losses of the wall-normal velocity and the Reynolds stress are not as large as those of these two variables. This implies that the streamwise and spanwise velocities of the removed motions penetrate deep into the near-wall region, while the wall-normal velocity and the Reynolds stress do not

On the stability of large-scale streaks in turbulent Couette and Poiseulle flows

The linear secondary stability of large-scale optimal streaks in turbulent Couette flow at Re_τ = 52 and Poiseulle flow at Reτ = 300 is investigated. The streaks are computed by solving the nonlinear two-dimensional Reynolds-averaged Navier-Stokes equations using an eddy-viscosity model. Optimal initial conditions leading the largest linear transient growth are used, and as the amplitude of the initial vortices increases, the amplitude of streaks gradually increases. Instabilities of the streaks appear when their amplitude exceeds approximately 18% of the velocity difference between walls in turbulent Couette flow and 21% of the centerline velocity in turbulent Poiseuille flow. When the amplitude of the streaks is sufficiently large, the instabilities attain significant growth rates in a finite range of streamwise wavenumbers that shows good agreement with the typical streamwise wavenumbers of the large-scale motions in the outer region

Optimal amplification of large-scale structures in plane turbulent couette flow

International audienceIn the present study, we investigate the optimal perturbations in plane turbulent Couette flow with turbulent mean flow and the associated eddy viscosity at Reh = 750. The three canonical types of optimal perturbations are computed: the initial perturbations for transient energy growth, the response to harmonic forcing and the variance to stochastic excitation. In all the cases, the maximum responses are obtained for streamwise uniform perturbations (λx = ∞). The optimal spanwise spacings of the transient growth and the stochastic forcing are λz = 4.2h and λz = 5.2h, respectively. These values are in very good agreement with the spanwise spacing of the large-scale streaks reported in previous studies. Moreover, the velocity field of the responses to the optimal perturbations are strikingly similar to that of large-scale structure obtained with direct numerical simulation. Finally, the optimal response to the harmonic forcing, more related to flow controls, reveals the maximum by steady forcing with larger spanwise wavelength (7.4h)

On the self-sustained nature of large-scale motions in turbulent Couette flow

Large-scale motions in wall-bounded turbulent flows are frequently interpreted as resulting from an aggregation process of smaller-scale structures. Here, we explore the alternative possibility that such large-scale motions are themselves self-sustained and do not draw their energy from smaller-scale turbulent motions activated in buffer layers. To this end, it is first shown that large-scale motions in turbulent Couette flow at Re=2150 self-sustain even when active processes at smaller scales are artificially quenched by increasing the Smagorinsky constant Cs in large eddy simulations. These results are in agreement with earlier results on pressure driven turbulent channels. We further investigate the nature of the large-scale coherent motions by computing upper and lower-branch nonlinear steady solutions of the filtered (LES) equations with a Newton-Krylov solver,and find that they are connected by a saddle-node bifurcation at large values of Cs. Upper branch solutions for the filtered large scale motions are computed for Reynolds numbers up to Re=2187 using specific paths in the Re-Cs parameter plane and compared to large-scale coherent motions. Continuation to Cs = 0 reveals that these large-scale steady solutions of the filtered equations are connected to the Nagata-Clever-Busse-Waleffe branch of steady solutions of the Navier-Stokes equations. In contrast, we find it impossible to connect the latter to buffer layer motions through a continuation to higher Reynolds numbers in minimal flow units

Amplification of coherent streaks in the turbulent Couette flow: an input-output analysis at low Reynolds number

International audienceWe compute the optimal response of the turbulent Couette mean flow to initial conditions, harmonic and stochastic forcing at Re=750. The equations for the coherent perturbations are linearized near the turbulent mean flow and include the associated eddy viscosity. The mean flow is found to be linearly stable but it has the potential to amplify steamwise streaks from streamwise vortices. The most amplified structures are streamwise uniform and the largest amplifications of the energy of initial conditions and of the variance of stochastic forcing are realized by large-scale streaks having spanwise wavelengths of 4.4h and 5.2h respectively. These spanwise scales compare well with the ones of the coherent large-scale streaks observed in experimental realizations and direct numerical simulations of the turbulent Couette flow. The optimal response to the harmonic forcing, related to the sensitivity to boundary conditions and artificial forcing, can be very large and is obtained with steady forcing of structures with larger spanwise wavelength (7.7h). The optimal large-scale streaks are furthermore found proportional to the mean turbulent profile in the viscous sublayer and up to the buffer layer