Rigorous bound on the plane-shear-flow dissipation rate

Linear stability analysis in a liquid layer with a surface velocity gradient

A case of combined planar Couette-Poiseuille flow corresponding to vanishing horizontal flux has been generalized by the introduction of a model for the surface velocity gradient. A relation corresponding to the Orr-Sommerfeld equation has been derived for this model. The critical value of the surface velocity gradient has been obtained. At the critical point, the corresponding critical Reynolds number equals infinity. Using an approximated method we estimated the behavior of the critical Reynolds number for a slightly overcritical surface velocity gradient.

Periodically kicked turbulence

Periodically kicked turbulence is theoretically analyzed within a mean-field theory. For large enough kicking strength A and kicking frequency f the Reynolds number grows exponentially and then runs into some saturation. The saturation level Re(sat) can be calculated analytically; different regimes can be observed. For large enough Re we find Re(sat) approximately Af, but intermittency can modify this scaling law. We suggest an experimental realization of periodically kicked turbulence to study the different regimes we theoretically predict and thus to better understand the effect of forcing on fully developed turbulence.

Variational principle for the Navier-Stokes equations

A variational principle is presented for the Navier-Stokes equations in the case of a contained boundary-driven, homogeneous, incompressible, viscous fluid. Based upon making the fluid's total viscous dissipation over a given time interval stationary subject to the constraint of the Navier-Stokes equations, the variational problem looks overconstrained and intractable. However, introducing a nonunique velocity decomposition, u(x,t)=phi(x,t) + nu(x,t), "opens up" the variational problem so that what is presumed a single allowable point over the velocity domain u corresponding to the unique solution of the Navier-Stokes equations becomes a surface with a saddle point over the extended domain (phi,nu). Complementary or dual variational problems can then be constructed to estimate this saddle point value strictly from above as part of a minimization process or below via a maximization procedure. One of these reduced variational principles is the natural and ultimate generalization of the upper bounding problem developed by Doering and Constantin. The other corresponds to the ultimate Busse problem which now acts to lower bound the true dissipation. Crucially, these reduced variational problems require only the solution of a series of linear problems to produce bounds even though their unique intersection is conjectured to correspond to a solution of the nonlinear Navier-Stokes equations.