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

Relative periodic edge orbits in plane channel flow

A branch of genuine relative periodic orbits is found to be an edge state in plane Poiseuille flow in a periodic domain. These periodic solutions correspond to sinuous quasi-streamwise streaks periodically forced by sinuous quasi streamwise vortices in a self-sustained process. The rms-amplitude of the streaks is found to scale as ≈ Re-0.8, while that of the quasi-streamwise vortices scales like ≈ Re-1.6

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

Replacing wakes with streaks in wind turbine arrays

Wind turbine wakes negatively impact downwind turbines in wind farms reducing their global efficiency. The reduction of wake-turbine interactions by actuating control on yaw angles and induction factors is an active area of research. In this study, the capability of spanwise-periodic wind turbine arrays with tilted rotors to reduce negative turbine-wakes interaction is investigated by means of large-eddy simulations. It is shown that by means of rotor tilt it is possible to replace turbine far wakes with high-speed streaks where the streamwise velocity exceeds the freestream velocity at hub height. Considering three aligned arrays of wind turbines, it is found that the global power extracted from the wind can be increased by tilting rotors of upwind turbine arrays similarly to what already known for the case of a single row of aligned turbines. It is further shown that global tilt-induced power gains can be significantly increased by operating the tilted turbines at higher induction rates. Power gains can be further increased by increasing the ratio of the rotor diameters and turbine spacing to the boundary layer thickness. All these findings are consistent with those of previous studies where streamwise streaks were artificially forced by means of arrays of wall-mounted roughness elements in order to control canonical boundary layers for drag-reduction purposes.Comment: revised versio

Optimal amplification of streamwise streaks in plane jets and their stabilizing effect on the inflectional instability

Optimal transient energy growths supported by the plane Bickley jet are computed for a set of spanwise wavenumbers and Reynolds numbers. It is shown that the maximum energy amplification is proportional to the square of the Reynolds number. The computed optimal streamwise vortices are then used to efficiently force finite amplitude streaks that are shown to stabilize the jet's powerful inflectional instability, which is clearly relevant for a number of applications in the control of free shear flows

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