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.

https://hal-polytechnique.archives-ouvertes.fr/hal-01020668/document

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

We 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.

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

A series of direct numerical simulation (DNSs) of the turbulent plane Couette flow were performed with various box lengths, L x = 24h, 32h, 45h and 64h, where h is the channel height. The Reynolds numbers Re w (= U w h/ν) of 3000 and 8600 were chosen, where U w is the relative wall speed between top and bottom walls and ν is the kinematic viscosity. In the core region, the meandering of the large-scale structure (LSS) has been captured with a long box size (L x ≥ 32h). Correspondingly, the streamwise two-point correlation decreases and becomes negative at half the box length. The effect of the computational box size on the statistics, e.g. turbulence intensities, is also examined. Significant Reynolds-number dependence is observed in the streamwise LSS. For the higher Reynolds number, both the visualized instantaneous flow field and the pre-multiplied energy spectra show that the LSS can be present, with a finite wavelength of 21–32h in the streamwise direction and a spanwise spacing of 2.1–2.5h. The stru...

DNS of turbulent Couette flow with emphasis on the large-scale structure in the core region

A new set of numerical simulations of turbulent plane Couette flow in a large box of dimension ( ) at Reynolds number , 180, 250 and 550 is described and compared with simulations at lower Reynolds numbers, Poiseuille flows and experiments. The simulations present a logarithmic near-wall layer and are used to verify and revise previously known results. It is confirmed that the fluctuation intensities in the streamwise and spanwise directions do not scale well in wall units. The scaling failure occurs both near to and away from the wall. On the contrary, the wall-normal intensity scales in inner units in the near-wall region and in outer units in the core region. The spectral ridge found by Hoyas & Jimenez (Phys. Fluids, vol. 18, 2003, 011702) for the turbulent Poiseuille flow can also be seen in the present flow. Away from the wall, very large-scale motions are found spanning through all the length of the channel. The statistics of these simulations can be downloaded from the webpage of the Chair of Fluid Dynamics.

/pdf/turbulent-plane-couette-flow-at-moderately-high-reynolds-2jxudryeve.pdf

Turbulent plane Couette flow at moderately high Reynolds number

We investigate the behaviour of the canonical turbulent Couette flow at computationally high Reynolds number through a series of large-scale direct numerical simulations We achieve a Reynolds number , where is the channel half-height and is the viscous length scale at which some phenomena representative of the asymptotic Reynolds-number regime manifest themselves While a logarithmic mean velocity profile is found to provide a reasonable fit of the data, including the skin friction, closer scrutiny shows that deviations from the log law are systematic, and probably increasing at higher Reynolds numbers The Reynolds stress distribution shows the formation of a secondary outer peak in the streamwise velocity variance, which is associated with significant excess of turbulent production as compared to the local dissipation This excess is related to the formation of large-scale streaks and rollers, which are responsible for a substantial fraction of the turbulent shear stress in the channel core, and for significant increase of the turbulence intermittency in the near-wall region

Turbulence statistics in Couette flow at high Reynolds number

Experimental work to investigate plane Couette flow has been performed in the Reynolds number range of , which is 3 times as long as that in Poiseuille flow.

Experimental study on mean velocity and turbulence characteristics of plane Couette flow : low-Reynolds-number effects and large longitudinal vortical structure

Measurements of time-mean velocity, turbulence intensities (u', v') and Reynolds shear stress in plane turbulent Couette flow have been made in the range of 3000l2hUb/νl20000 (Ub, moving belt speed ; 2h, channel height). The main objective of this work is to study the effects of low-Reynolds number on the turbulent Couette flow. Particular importance is that its effect on the law of the wall can be estimated only from the Couette flow experiment. Low-Reynolds number effects are shown as follows ; additive constant of the law of the wall B, a coefficient of defect law and the maximum turbulence intensity u'/u* (u*, friction velocity) decrease as the Reynolds number decreases. From the skewness-and flatness-factors and four quadrant analysis of the fluctuating velocity, the turbulence structure of the Couette flow can be deduced. Except for close to the wall (y+l12), where the structure is the same as that of the Poiseuille flow, the sweep event becomes more important than in the Poiseuille flow. The central region has a similar structure as a uniformly sheared flow

Low-Reynolds Number Effects on Plane Turbulent Couette Flow.

It has been shown that there exists large longitudinal vortical structure in the core region of turbulent plane Couette flow and they are meandering quasi-periodically. In this paper, we study the fluctuating velocity properties in the core region using power-spectrum and wavelet analysis. Power spectrum in the core region of Couette flow demonstrates the existence of a periodic component in the velocity fluctuation. This periodic component has low-frequency amplitude modulation over the turbulent motion. It is shown that the modulation is caused by the meandering motion of the large longitudinal vortical structure. This meandering motion raises another fluctuating velocity in the core region. The velocity fluctuation component due to the meandering motion does not contribute to Reynolds shear stress. In the Couette flow, sweap motion is more dominant than in Poiseuille flow even near the wall. For the Couette flow, the power spectrum in the core region has ω-1 distribution.

Futoshi Nishimura

Papers

Experimental study on mean velocity and turbulence characteristics of plane Couette flow : low-Reynolds-number effects and large longitudinal vortical structure

Low-Reynolds Number Effects on Plane Turbulent Couette Flow.

Drag Reduction and Its Application. Analisys of Velocity Fluctuation in Turbulent Plane Couette Flow.