On a self-sustaining process in shear flows

Waleffe, Fabian

doi:doi:10.1063/1.869185

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Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow

The growth of the gradient of a scalar temperature in a quasigeostrophic flow is studied numerically in detail. We use a flow evolving from a simple initial condition which was regarded by Constantin et al. as a candidate for a singularity formation in a finite time. For the inviscid problem, we propose a completely different interpretation of the growth, that is, the temperature gradient can be fitted equally well by a double-exponential function of time rather than an algebraic blowup. It seem...

Angular dependence and growth of vorticity in the three-dimensional Euler equations

An investigation of lower bounds on the quantities Wn(t) = ∫ Ω∣ω∣2n dV for n ≥ 1 for the incompressible three-dimensional (3D) Euler equations has led us to consider a set of spatially averaged weighted “eigenvalues,” λS(n)(t) and λP(n)(t), of the strain matrix S and the Hessian matrix of the pressure P = {p,ij}, respectively. It is shown that these obey the simple inequality, S(n)+f(θn)(λS(n))2+λP(n) ≥ 0, where f(θn) = 1−tan2 θn. The θn are spatially averaged weighted angles between the vortic...

A self-sustaining process conjectured to be generic for wall-bounded shear flows is investigated. The self-sustaining process consists of streamwise rolls that redistribute the mean shear to create streaks that wiggle to maintain the rolls. The process is analyzed and shown to be remarkably insensitive to whether there is no-slip or free-slip at the walls. A low-order model of the process is derived from the Navier–Stokes equations for a sinusoidal shear flow. The model has two unstable steady solutions above a critical Reynolds number, in addition to the stable laminar flow. For some parameter values, there is a second critical Reynolds number at which a homoclinic bifurcation gives rise to a stable periodic solution. This suggests a direct link between unstable steady solutions and almost periodic solutions that have been computed in plane Couette flow. It is argued that this self-sustaining process is responsible for the bifurcation of shear flows at low Reynolds numbers and perhaps also for controlling the near-wall region of turbulent shear flows at higher Reynolds numbers. © 1997 American Institute of Physics.