Len G. Margolin

Direct numerical simulations of the Navier-Stokes alpha model

Abstract We explore the utility of the recently proposed alpha equations in providing a subgrid model for fluid turbulence. Our principal results are comparisons of direct numerical simulations of fluid turbulence using several values of the parameter alpha, including the limiting case where the Navier–Stokes equations are recovered. Our studies show that the large scale features, including statistics and structures, are preserved by the alpha models, even at coarser resolutions where the fine scales are not fully resolved. We also describe the differences that appear in simulations. We provide a summary of the principal features of the alpha equations, and offer some explanation of the effectiveness of these equations used as a subgrid model for three-dimensional fluid turbulence.

On Forward-in-Time Differencing for Fluids: an Eulerian/Semi-Lagrangian Non-Hydrostatic Model for Stratified Flows

ABSTRACT In this paper, we describe a non-hydrostatic anelastic model for simulating stratified flows in terrain-following coordinates. The model is based solely on non-oscillatory forward-in-rime integration schemes, and our primary goal is to demonstrate the utility of such methods for modelling small-scale atmospheric dynamics. We use the formal similarity of the Eulerian and semi-Lagrangian equations of two-time-level approximations to construct a unified model that readily allows selection of either formulation. We apply the model to two test problems of stratified flows past isolated obstacles. We use these tests to validate the forward-in-time approach against a traditional centred-in-time-and-space Eulerian model, and to discuss the relative accuracy and efficiency of the two formulations of our model. One problem illustrates the efficacy of flux-form Eulerian methods, while the other demonstrates strengths of the semi-Lagrangian approach.

Second-order sign-preserving conservative interpolation (remapping) on general grids

An accurate conservative interpolation (remapping) algorithm is an essential component of most arbitrary Lagrangian-Eulerian (ALE) methods. In this paper we describe a local remapping algorithm for a positive scalar function. This algorithm is second-order accurate, conservative, and sign preserving. The algorithm is based on estimating the mass exchanged between cells at their common interface, and so is equally applicable to structured and unstructured grids. We construct the algorithm in a series of steps, clearly delineating the assumptions and errors made at each step. We validate our theory with a suite of numerical examples, analyzing the results from the viewpoint of accuracy and order of convergence.

Large-eddy simulations of convective boundary layers using nonoscillatory differencing

Abstract We explore the ability of nonoscillatory advection schemes to represent the effects of the unresolved scales of motion in numerical simulation of turbulent flows. We demonstrate that a nonoscillatory fluid solver can accurately reproduce the dynamics of an atmospheric convective boundary layer. When an explicit turbulence model is implemented, the solver does not add any significant numerical diffusion. Of greater interest, when no explicit turbulence is employed the solver itself appears to include an effective subgrid scale model. Other researchers have reported similar success, simulating turbulent flows in a variety of regimes while using only nonoscillatory advection schemes to model subgrid effects. At this point there is no theory justifying this success, but we offer some speculations.

A Class of Nonhydrostatic Global Models

Abstract A Cartesian, small- to mesoscale nonhydrostatic model is extended to a rotating mountainous sphere, thereby dispensing with the traditional geophysical simplifications of hydrostaticity, gentle terrain slopes, and weak rotation. The authors discuss the algorithmic design, relative efficiency, and accuracy of several different variants (hydrostatic, nonhydrostatic, implicit, explicit, elastic, anelastic, etc.) of the global model and prepare the ground for a future “global cloud model”—a research tool to study effects of small- and mesoscale phenomena on global flows and vice versa. There are two primary threads to the discussion: (a) presenting a novel semi-implicit anelastic global dynamics model as it naturally emerges from a small-scale dynamics model, and (b) demonstrating that nonhydrostatic anelastic global models derived from small-scale codes adequately capture a broad range of planetary flows while requiring relatively minor overhead due to the nonhydrostatic formulation of the governing...

On balanced approximations for time integration of multiple time scale systems

The effect of various numerical approximations used to solve linear and nonlinear problems with multiple time scales is studied in the framework of modified equation analysis (MEA). First, MEA is used to study the effect of linearization and splitting in a simple nonlinear ordinary differential equation (ODE), and in a linear partial differential equation (PDE). Several time discretizations of the ODE and PDE are considered, and the resulting truncation terms are compared analytically and numerically. It is demonstrated quantitatively that both linearization and splitting can result in accuracy degradation when a computational time step larger than any of the competing (fast) time scales is employed. Many of the issues uncovered on the simple problems are shown to persist in more realistic applications. Specifically, several differencing schemes using linearization and/or time splitting are applied to problems in nonequilibrium radiation-diffusion, magnetohydrodynamics, and shallow water flow, and their solutions are compared to those using balanced time integration methods.

Implicit turbulence modeling for high reynolds number flows.

We continue our investigation of the implicit turbulence modeling property of the nonoscillatory finite volume scheme MPDATA. We start by comparing MPDATA simulations of decaying turbulence in a triply periodic cube with analogous pseudospectral studies. In the regime of direct numerical simulation, MPDATA is shown to agree closely with the pseudospectral model. As viscosity is reduced, the two model results diverge. We study the MPDATA results in the inviscid limit, using a combination of mathematical analysis and computational experiment. We validate the inviscid MPDATA results as representing the turbulent flow in the limit of very high Reynolds number.

A discrete operator calculus for finite difference approximations

In this article we describe two areas of recent progress in the construction of accurate and robust finite difference algorithms for continuum dynamics. The support operators method (SOM) provides a conceptual framework for deriving a discrete operator calculus, based on mimicking selected properties of the differential operators. In this paper, we choose to preserve the fundamental conservation laws of a continuum in the discretization. A strength of SOM is its applicability to irregular unstructured meshes. We describe the construction of an operator calculus suitable for gas dynamics and for solid dynamics, derive general formulae for the operators, and exhibit their realization in 2D cylindrical coordinates. The multidimensional positive definite advection transport algorithm (MPDATA) provides a framework for constructing accurate nonoscillatory advection schemes. In particular, the nonoscillatory property is important in the remapping stage of arbitrary-Lagrangian-Eulerian (ALE) programs. MPDATA is based on the sign-preserving property of upstream differencing, and is fully multidimensional. We describe the basic second-order-accurate method, and review its generalizations. We show examples of the application of MPDATA to an advection problem, and also to a complex fluid flow. We also provide an example to demonstrate the blending of the SOM and MPDATA approaches.

Antidiffusive Velocities for Multipass Donor Cell Advection

Multidimensional positive definite advection transport algorithm (MPDATA) is an iterative process for approximating the advection equation, which uses a donor cell approximation to compensate for the truncation error of the originally specified donor cell scheme. This step may be repeated an arbitrary number of times, leading to successively more accurate solutions to the advection equation. In this paper, we show how to sum the successive approximations analytically to find a single antidiffusive velocity that represents the effects of an arbitrary number of passes. The analysis is first done in one dimension to illustrate the method and then is repeated in two dimensions. The existence of cross terms in the truncation analysis of the two-dimensional equations introduces an extra complication into the calculation. We discuss the implementation of our new antidiffusive velocities and provide some examples of applications, including a third-order accurate scheme.

On Forward-in-Time Differencing for Fluids: Stopping Criteria for Iterative Solutions of Anelastic Pressure Equations

In this note, the authors address the practical issue of selecting appropriate stopping criteria for iterative solutions to the elliptic pressure equation arising in nonoscillatory, forward-in-time Eulerian and semi-Lagrangian anelastic fluid models. Using the simple computational example of 2D thermal convection in a neutrally stratified Boussinesq fluid, it is shown that (a) converging to the machine precision is not necessary for the overall accuracy and stability of the model, and adversely affects the overall model efficiency; and (b) the semi-Lagrangian model algorithm admits fairly liberal stopping criteria compared to the Eulerian flux-form model, unless the latter is formulated in terms of field perturbations.