Present study employed the large eddy simulation and direct numerical simulation techniques to simulate the turbulentCouette-Poiseuille and Couette flows inside a square duct. The numerical framework consists of a finite-volume method with a staggered-grid arrangement of dependent variables. Spatial derivatives are approximated using second-order centered differencing, and a fractional-step method is used for temporal integration. To relieve the computational load, the parallelization of the code together with the direct fast Fourier transform solver is employed for pressure Poisson solver.

Turbulent Couette-Poiseuille and Couette flows at different mean strain rates, (velocity ratio of Couette wall to bulk flow, r= 0.6~3.26), in a square duct at a bulk Reynolds number~10,000 are investigated by large eddy simulation. Simulations are conducted with 128x128x128 grids and validated by computing the Poiseuille flow. Secondary flow near the Couette wall shows a gradual change of vortex size and position as the moving wall velocity increased, where the two clockwise rotating vortices gradually merge in tandem with speed of the moving wall and form a large clockwise vortex. A linear relation is observed to exist between the angle of the two vortices and the parameter r, and a change in slope occurs at r~1.2. Also, together with a small counter-clockwise corner vortex, this vortex pattern is similar to that observed in the corner region of the duct flow with a free surface. The change of the vortex patterns also influences the dominant transport terms in the streamwise vorticty transport equation. Near the moving wall due to the reduction of the streamwise velocity fluctuation at the moving wall, turbulence structure gradually moves towards a rod-like axi-symmetric turbulence, and as r increases beyond 1.2, turbulence reverts to the disk-like structure.

Further study investigates the four different cases (plane Poiseuille (PP), duct Poiseuille (DP), plane Couette (PC), duct Couette (DC)) at the same friction Reynolds number and the turbulent Couette flows at different friction Reynolds numbers (including Retau~257, 360 and 600) inside a square duct using a direct numerical simulation. The influence of side wall and Couette wall on the turbulence structures and energy transport inside a square duct are examined. For the side wall effect along the wall bisector, x/D=0.5, a variation between the duct Couette flow and its plane counterpart is attributed to the emergence of Reynolds
stress <u'w'>. This factor yields an antisymmetric velocity distribution and induces a stronger sweeping process carrying with considerable mean streamwise momentum energy to counterbalance the secondary motion caused by the side wall. With respect to the Couette wall effect, a substantial change is observed from the secondary flow pattern. Two clockwise vortices merge together close
to the moving wall because of a reduction in the corner effect caused by the Couette strain rate. Further, the fast moving wall, together with nearby quiescent fluid, produces the formation of another small clockwise vortex near the corner. As the Couette velocity is gradually increased, an enhanced streamwise momentum is observed, which is associated with a stronger sweeping motion that is transported towards the stationary wall. This makes the turbulence intensity levels approach that of the plane Couette flow. In the corresponding region, the constrained energy results in an increase in the spanwise turbulence intensity.

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