Exact and Asymptotic Conditions on Traveling Wave Solutions of the Navier-Stokes Equations

We derive necessary conditions that traveling wave solutions of the Navier-Stokes equations must satisfy in the pipe, Couette, and channel flow geometries. Some conditions are exact and must hold for any traveling wave solution irrespective of the Reynolds number ($Re$). Other conditions are asymptotic in the limit $Re\to\infty$. The exact conditions are likely to be useful tools in the study of transitional structures. For the pipe flow geometry, we give computations up to $Re=100000$ showing the connection of our asymptotic conditions to critical layers that accompany vortex structures at high $Re$

High-frequency asymptotic compression of dense BEM matrices for general geometries without ray tracing

Wave propagation and scattering problems in acoustics are often solved with boundary element methods. They lead to a discretization matrix that is typically dense and large: its size and condition number grow with increasing frequency. Yet, high frequency scattering problems are intrinsically local in nature, which is well represented by highly localized rays bouncing around. Asymptotic methods can be used to reduce the size of the linear system, even making it frequency independent, by explicitly extracting the oscillatory properties from the solution using ray tracing or analogous techniques. However, ray tracing becomes expensive or even intractable in the presence of (multiple) scattering obstacles with complicated geometries. In this paper, we start from the same discretization that constructs the fully resolved large and dense matrix, and achieve asymptotic compression by explicitly localizing the Green's function instead. This results in a large but sparse matrix, with a faster associated matrix-vector product and, as numerical experiments indicate, a much improved condition number. Though an appropriate localisation of the Green's function also depends on asymptotic information unavailable for general geometries, we can construct it adaptively in a frequency sweep from small to large frequencies in a way which automatically takes into account a general incident wave. We show that the approach is robust with respect to non-convex, multiple and even near-trapping domains, though the compression rate is clearly lower in the latter case. Furthermore, in spite of its asymptotic nature, the method is robust with respect to low-order discretizations such as piecewise constants, linears or cubics, commonly used in applications. On the other hand, we do not decrease the total number of degrees of freedom compared to a conventional classical discretization. The combination of the ...Comment: 24 pages, 13 figure

Conductances between confined rough walls

Two- and three-dimensional creeping flows and diffusion transport through constricted and possibly rough surfaces are studied. Asymptotic expansions of conductances are derived as functions of the constriction local geometry. The validity range of the proposed theoretical approximations is explored through a comparison either with available exact results for specific two-dimensional aperture fields or with direct numerical computations for general three-dimensional geometries. The large validity range of the analytical expressions proposed for the hydraulic conductivity (and to a lesser extent for the electrical conductivity) opens up interesting perspectives for the simulation of flows in highly complicated geometries with a large number of constrictions

A numerical investigation of the solution of a class of fourth-order eigenvalue problems

This paper is concerned with the accurate numerical approximation of the spectral properties of the biharmonic operator on various domains in two dimensions. A number of analytic results concerning the eigenfunctions of this operator are summarized and their implications for numerical approximation are discussed. In particular, the asymptotic behaviour of the first eigenfunction is studied since it is known that this has an unbounded number of oscillations when approaching certain types of corners on domain boundaries. Recent computational results of Bjorstad & Tjostheim, using a highly accurate spectral Legendre-Galerkin method, have demonstrated that a number of these sign changes may be accurately computed on a square domain provided sufficient care is taken with the numerical method. We demonstrate that similar accuracy is also achieved using an unstructured finite-element solver which may be applied to problems on domains with arbitrary geometries. A number of results obtained from this mixed finite-element approach are then presented for a variety of domains. These include a family of circular sector regions, for which the oscillatory behaviour is studied as a function of the internal angle, and another family of (symmetric and non-convex) domains, for which the parity of the least eigenfunction is investigated. The paper not only verifies existing asymptotic theory, but also allows us to make a new conjecture concerning the eigenfunctions of the biharmonic operator

Rossby and Magnetic Prandtl Number Scaling of Stellar Dynamos

Rotational scaling relationships are examined for the degree of equipartition between magnetic and kinetic energies in stellar convection zones. These scaling relationships are approached from two paradigms, with first a glance at scaling relationship built upon an energy-balance argument and second a look at a force-based scaling. The latter implies a transition between a nearly-constant inertial scaling when in the asymptotic limit of minimal diffusion and magnetostrophy, whereas the former implies a weaker scaling with convective Rossby number. Both scaling relationships are then compared to a suite of 3D convective dynamo simulations with a wide variety of domain geometries, stratifications, and range of convective Rossby numbers.Comment: 15 pages, 6 figures, accepted in Ap