Continuation of travelling-wave solutions of the Navier–Stokes equations

Abstract

SUMMARYAn e cient way of obtaining travelling waves in a periodic uid system is described and tested. We search for steady states in a reference frame travelling at the wave phase velocity using a ÿrst-order pseudospectral semi-implicit time scheme adapted to carry out the Newton's iterations. The method is compared to a standard Newton-Raphson solver and is shown to be highly e cient in performing this task, even when high-resolution grids are used. This method is well suited to three-dimensional calculations in cylindrical or spherical geometries.

Direct numerical simulation of transitions towards structural vacillation in an air-filled, rotating, baroclinic annulus

“…3, 21 In a companion paper, 14 we have presented and discussed a set of direct numerical simulations ͑DNSs͒ which exhibit in detail a transition to baroclinic chaos via a lowdimensional route involving a succession of modulated traveling waves in an air-filled, rotating annulus. Such studies complement the use of analytical and numerical stability and continuation methods in the context of circularly symmetric systems such as the rotating annulus, [22][23][24][25][26][27] though the latter approaches are not yet able to deal with the full complexity of time-dependent, modulated waves and their nonlinear interactions found in the present work. The use of air as the working fluid is of particular interest since this represents a fluid with Prandtl number Pr= O͑1͒ which has received relatively little attention in previous experimental studies 28 or in numerical simulations of baroclinic flows but, in common with other convective flows, the Prandtl number seems to play an important role in governing the nature of nonlinear interactions in the flow, as well as strongly affecting the form of the advective-diffusive basic state from which baroclinic instabilities grow.…”

Reduced description of exact coherent states in parallel shear flows

“…In the small t limit, P ≈ I and the equation is not preconditioned, while for sufficiently large t, we get the so-called Stokes preconditioner P ≈ t(γ D /γ t )L. This preconditioning method is easy to implement as it only requires a first order implicit Euler time-integration scheme and its use is natural within matrix-free methods, where the Jacobian is not explicitly constructed. The Stokes preconditioner has been widely used in problems that are dominated by diffusion, like coupled convection [47][48][49]. In contrast, most shear flow studies are carried out at large Reynolds numbers and involve weakly diffusive flows.…”

Effect of small inclination on binary convection in elongated rectangular cells

“…Notice that the reflection R can be obtained as R = R 1 •R 2 . The system of equations (2)-(3) and boundary conditions (4)- (5) has been solved numerically using the algorithm IPS described in [24], which can be summarized as follows. To integrate the equations in time, we use a second-order time-splitting method proposed in [25] combined with a pseudospectral method for the spatial discretization, Chebyshev collocation in x and z.…”