Large-eddy exact coherent solutions (LECS) in the turbulent channel
Résumé
Exact coherent structures (ECS) are invariant solutions of the Navier-Stokes equations and are believed to form the skeleton of turbulence at transitional Reynolds numbers. Large-eddy coherent exact structures (LECS) are invariant solutions of the filtered Navier-Stokes equations used in large-eddy simulations where the effect of residual motions is accounted for by a subgrid model (here, the static Smagorinsky model). LECS solutions typically represent large-scale and intermediate-scale coherent structures in fully developed turbulent flows [3, 4]. Here we show that the recently computed P4 travelling-wave ECS solutions [2] can be continued into large-eddy LECS solutions in the turbulent plane channel by using a 'forward' continuation in the Smagorinsky constant C s of the ECS solution (C s = 0) to increasing values of C s. For Reynolds numbers where both the upper and lower branch ECS solutions exist, these are connected by an associated upper and lower C s-branch separated by a C s-turning point similarly to what found in plane Couette flow [3]. Solutions obtained at finite C s can be then continued in the Reynolds number. The effect of increasing C s is to shift to higher Reynolds numbers the Returning point and to, generally, increase the friction at each selected Reynolds number. At sufficiently high Reynolds numbers this results in the upper branch mean velocity profile further approaching the logarithmic profile of Newtonian turbulence; the frictional Reynolds number Re τ associated with the upper branch also approaches the one of the turbulent flow at the same Reynolds number. These results represent a further indication that LECS can be obtained by continuation from previously known Navier-Stokes invariant solutions. Current work is under way (see [1]) to assess if a relation exits between LECS solutions and Townsend's attached eddies which are believed to form the skeleton of wall-bounded turbulent flows at high Reynolds numbers. 0 0.08 0.16 0.24 1000 1600 2200 2800 3400 4000 2 3 4 5 6 S C s Re S Figure 1. Continuation diagram in the Cs−Re plane in terms of the shear parameter S. The continuation in Reynolds number at Cs = 0 (black line, dashed) shows the P4 Navier-Stokes (ECS) solution branches. The dashed-dotted (red) line represents a continuation in Cs at Re = 1650, while the solid (blue) line represents a continuation at higher Reynolds numbers (see the text for details). The open triangle symbol represents the Navier-Stokes (Cs = 0) Re-saddle point at Re = 1357, the full triangles the Cs-saddle points (found at Cs = 0.13 for Re = 1650 and Cs = 0.29 for Re = 3500). The square and circle symbols are the solutions found at Cs = 0 (Navier-Stokes solutions) used to initiate the Cs continuations (full symbols correspond to upper branch solutions).