Flow Statistics in the Transitional Regime of Plane Channel Flow

The transitional regime of plane channel flow is investigated above the transitional point below which turbulence is not sustained, using direct numerical simulation in large domains. Statistics of laminar-turbulent spatio-temporal intermittency are reported. The geometry of the pattern is first characterized, including statistics for the angles of the laminar-turbulent stripes observed in this regime, with a comparison to experiments. High-order statistics of the local and instantaneous bulk velocity, wall shear stress and turbulent kinetic energy are then provided. The distributions of the two former quantities have non-trivial shapes, characterized by a large kurtosis and/or skewness. Interestingly, we observe a strong linear correlation between their kurtosis and their skewness squared, which is usually reported at much higher Reynolds number in the fully turbulent regime.

Restricted nonlinear model for high- and low-drag events in plane channel flow

A restricted nonlinear (RNL) model, obtained by partitioning the state variables into streamwise-averaged quantities and superimposed perturbations, is used in order to track the exact coherent state in plane channel flow investigated by Toh & Itano (J. Fluid Mech., vol. 481, 2003, pp. 67–76). When restricting nonlinearities to quadratic interaction of the fluctuating part into the streamwise-averaged component, it is shown that the coherent structure and its dynamics closely match results from direct numerical simulation (DNS), even if only a single streamwise Fourier mode is retained. In particular, both solutions exhibit long quiescent phases, spanwise shifts and bursting events. It is also shown that the dynamical trajectory passes close to equilibria that exhibit either low- or high-drag states. When statistics are collected at times where the friction velocity peaks, the mean flow and root-mean-square profiles show the essential features of wall turbulence obtained by DNS for the same friction Reynolds number. For low-drag events, the mean flow profiles are related to a universal asymptotic state called maximum drag reduction (Xi & Graham, Phys. Rev. Lett., vol. 108, 2012, 028301). Hence, the intermittent nature of self-sustaining processes in the buffer layer is contained in the dynamics of the RNL model, organized in two exact coherent states plus an asymptotic turbulent-like attractor. We also address how closely turbulent dynamics approaches these equilibria by exploiting a DNS database associated with a larger domain.

Artificial boundary conditions for the ILES modeling of plane channel flow using the Cabaret scheme

Представлены результаты ILES-моделирования классической задачи течения вязкой несжимаемой жидкости в плоском канале по схеме Кабаре. Рассматривается возможность модификации расчета течения возле стенки для более точного определения средних характеристик. Предложено введение "искусственных" граничных условий путем использования в первом слое ячеек вблизи стенки специальной модели вихревой вязкости для корректного учета сдвиговых эффектов. Приводится сравнение результатов расчета течения в плоском канале по схеме Кабаре с предложенными искусственными граничными условиями и без в широком диапазоне чисел Рейнольдса. Показано, что введенные модификации в пристеночном слое позволяют повысить точность определения средних характеристик течения, в особенности вторых моментов. Полученные данные также сравниваются с результатами LES-моделирования с использованием псевдоспектрального метода и с данными прямого численного моделирования. Some results of ILES modeling of the plane channel flow of a viscous incompressible fluid using the Cabaret scheme are discussed. The possibility of modifying the calculation of flow near the wall is considered to determine the average characteristics more accurately. The "artificial" boundary conditions are introduced by using a special eddy viscosity model in the first layer of cells near the wall to correctly account for shear effects. The results of numerical simulation of plane channel flow obtained using the Cabaret scheme with and without artificial boundary conditions are compared in a wide range of Reynolds numbers. It is shown that the introduced modifications in the near-wall layer improve the accuracy of determining the average flow characteristics, especially the second moments. The obtained data are also compared with the results of LES modeling by the pseudospectral method and with the data from direct numerical simulation.

Dominant modes in hydromagnetic stability of channel flow with porous walls

A linear stability analysis of a plane channel flow with porous walls under a uniform cross-flow and an external transverse magnetic field is explored. The physical problem is governed by a system of combined equations of the hydrodynamic and those of Maxwell. The perturbed problem of base state leads to a modified classical Orr-Sommerfeld equation which is solved numerically using the Chebyshev spectral collocation method. The combined effects of the cross-flow Reynolds number and the Hartmann number on the dangerous mode of hydromagnetic stability are investigated.The study shows that, the magnetic field tends to suppress the instability occurred by cross-flow. This stabilizing effect becomes perceptible when the magnetic field produces a mode transition from walls mode to that of the center.