This paper introduces a new algorithm to improve the accuracy of numerical phase-averaging in the oscillatory, multiscale, differential equations that govern geophysical fluid flow. Phase-averaging is a timestepping method that averages a mapped modulation variable in which the equations no longer contain a highly oscillatory linear term. This allows phase-averaging to retain the main contribution of fast waves on the low frequencies without explicitly resolving the rapid oscillations, thus allowing larger timesteps for explicit timesteppers. Whilst the modulation variable mapping is also used in operator splitting methods, and is accurate for asymptotic regimes, the advantage of phase-averaging is that low-frequency oscillations are retained, allowing for accuracy in systems further from the asymptotic limit. However, this comes at the cost of introducing an averaging error when computing phase-averages using a smooth kernel. This paper proposes a method to offset some averaging error through a modified mapping that includes a mean correction term to capture an average measure of nonlinear oscillation. We show that the mean corrected algorithm reduces phase-averaging errors in the swinging spring system and the one-dimensional rotating shallow water equations, which are an important geophysical fluid system for weather and climate applications. We discuss potential new directions for the method based on the outcome of the numerical experiments.
As a starting point to gain an understanding of settling porous objects with complex geometries, such as marine snow and biofouled microplastics, we study the settling of a thin porous mesh. Motivating the theoretical-numerical work presented here are laboratory experiments that show the meshes settle without tumbling and the settling speed is effectively independent of the mesh lateral extent. The numerical simulations model flow through the mesh using a time-relaxation method. This captures the qualitative features of the laboratory experiments and gives insight into the dynamics governing the steady state settling velocity. Specifically it shows that a steep pressure gradient develops across the thickness of the mesh equal in magnitude to the reduced gravity, which itself equals the difference of the fluid and mesh velocity divided by the relaxation time scale. In effect, fluid is sucked through the mesh which provides the balance between buoyancy forces and drag controlled primarily by the mesh thickness, not its lateral extent. This insight justifies a dimensional analysis approach which successfully collapses measurements of the steady state flow across the mesh and mesh settling speeds thus giving semi-empirical predictions for these speeds as they depend upon the reduced gravity, mesh thickness and relaxation time scale.
The efficient and accurate rational approximation of the Schwarz function in the complex plane by means of the recently developed AAA (adaptive Antoulas-Anderson) algorithm is applied here to the dynamics of uniform vorticity patches. The key observation is realising that the AAA algorithm readily computes the poles of the rational approximation, and that this enables a natural partitioning of the Schwarz function in such a way that the velocity field exterior to a patch is given in terms of the poles interior to the patch. In particular, knowing the velocity on the patch boundary and exterior to the patch allows both vortex patch equilibria, including configurations with and without satellite point vortices surrounding the vortex patch, and time-dependent motions to be computed. Examples are presented and the performance of the new numerical approach compared to previously established numerical methods, including the method of contour dynamics and the recently developed AAA-least squares algorithm.
A new vortex-dominated flow problem was defined to reveal some flow physics and to aid the design and assessment of turbulence models. The case is rich in turbulence and vorticity and physically complex although two-dimensional in the mean and free of solid boundaries. The z direction is homogeneous, and used for averages in the DNS so that the turbulence quantities are functions of (x, y, t). The flow is periodic in the x direction. The initial condition is a thin shear layer straddling the (x, z) plane and given a slight wave in the (x, y) plane. It is seeded with random perturbations, and turbulence develops before the Kelvin-Helmholtz instability creates two primary vortices. Each vortex entrains the shear layer into spiral sheets around itself, like vortices do over wings. Most RANS models are inaccurate in these situations, which is confirmed in the present case; they miss how the turbulence rapidly decays in and near the young vortex. The two vortices later collide and merge. The flow is driven by concurrent inviscid and turbulent phenomena, and RANS is more accurate for the former than the latter. A variety of measures of the numerical quality of the DNS are presented. We seek evidence that the vorticity exceeds the bounds of its initial condition. This cannot happen in (2D) laminar flow, but is common in RANS: vorticity of the sign opposite to the initial condition is generated by many models. We see strong evidence that this is not meaningfully present in the DNS, and attribute the small opposite excursions to the finite spatial-averaging interval and residual numerical errors. This matches our intuition, although we know of no theorem stating that opposite vorticity is unphysical. Part II contains comparisons with models, including eddy viscosity with and without rotation corrections and Reynolds-stress transport.
To predict the amplification factor of the Mack second mode instability in a hypersonic boundary layer, an amplification factor transport equation was developed within a RANS framework. Its production term is based on the approximate N-envelope method and modeled using compressible boundary layer profiles and linear stability theory (LST) analysis data. The transport equation is designed to use only local variables and does not require wall temperature as an explicit parameter. Furthermore, it accounts for the influence of wall temperature on the growth of the Mack second mode instability, which is a key contribution of this study. The amplification factor and the modified turbulence intermittency transport equations are coupled with a k- ω SST turbulence model as the transition model. The model was implemented in SU2, and grid experiments were performed. The predicted amplification factor was compared with the LST results with differences typically ranging from 0.4
