Balasubramanya T. Nadiga

Modeling Mesoscale Turbulence in the Barotropic Double-Gyre Circulation

Abstract This paper presents analytical and numerical results for a class of turbulence closure models called “alpha models,” in which Lagrangian averaging and turbulence closure assumptions modify the Eulerian nonlinearity. The alpha models are investigated in the setting of the barotropic, double-gyre circulation in an ocean basin. Two variants of the alpha models for the barotropic vorticity (BV) equation are found to produce the correct four-gyre configuration for the mean barotropic circulation in numerical simulations performed at a resolution 4 times as coarse as that required in a resolved BV model. These are the BV-α model and the BV-Leray-α model. However, at a resolution 8 times as coarse, only the BV-α model produces the proper four-gyre configuration. Thus, the combination of modified nonlinearity and viscous dissipation (the viscosity is the same in all of the runs) in the BV-α model is found to provide a promising approach to modeling the mean effects of unresolved mesoscale (subgrid scale)...

Global Bifurcation of Shilnikov Type in a Double-Gyre Ocean Model

Abstract The dynamics of an idealized wind-driven double-gyre circulation in an ocean basin are studied from a dynamical systems point of view in an effort to better understand its variability. While previous analyses of this circulation have mostly dealt with local bifurcations of steady states and limit cycles, this study demonstrates the importance of considering global bifurcations as well. In one case, a coherent picture of the global dynamics spanning a range of parameters from where there are only stable steady-state solutions to where there is chaotic eddy shedding is presented. A simple but novel use of power spectra along with dynamical projections of the dynamics suggests that just beyond the regime in which there are only stable steady states, the system exhibits a complicated global bifurcation known as the “Shilnikov phenomenon.”

MOMENT REALIZABILITY AND THE VALIDITY OF THE NAVIER-STOKES EQUATIONS FOR RAREFIED GAS DYNAMICS

We present criteria for monitoring the validity of the Navier–Stokes approximation during the simulation of a rarefied gas. Our approach is based on an underlying kinetic formulation through which one can construct nondimensional non-negative definite matrices from moments of the molecular distribution. We then identify one such 3×3 matrix that can be evaluated intrinsically in the Navier–Stokes approximation. Our criteria are based on deviations of the eigenvalues of this matrix from their equilibrium value of unity. Not being tied to a particular benchmark problem, the resulting criteria are portable and may be applied to any Navier–Stokes simulation. We study its utility here by comparing stationary planar shock profiles computed using the Navier–Stokes equations with those computed using Monte Carlo simulations.

Different approximations of shallow fluid flow over an obstacle

Three different sets of shallow water equations, representing different levels of approximation are considered. The numerical solutions of these different equations for flow past bottom topography in several different flow regimes are compared. For several cases the full Euler solutions are computed as a reference, allowing the assessment of the relative accuracies of the different approximations. Further, the differences between the dispersive shallow water (DSW) solutions and those of the highly simplified, hyperbolic shallow water (SW) equations is studied as a guide to determining what level of approximation is required for a particular flow. First, the Green‐Naghdi (GN) equations are derived as a vertically‐integrated rational approximation of the Euler equations, and then the generalized Boussinesq (gB) equations are obtained under the further assumption of weak nonlinearity. A series of calculations, each assuming different values of a set of parameters—undisturbed upstream Froude number, and the h...

Enhancement of the inverse-cascade of energy in the two-dimensional Lagrangian-averaged Navier-Stokes equations

The recently derived Lagrangian-averaged Navier–Stokes equations model the large-scale flow of the Navier–Stokes fluid at spatial scales larger than some a priori fixed α>0, while coarse-graining the behavior of the small scales. In this communication, we numerically study the behavior of the two-dimensional (2D) isotropic version of this model, also known as the α model. The inviscid dynamics of this model exactly coincide with the vortex blob algorithm for a certain choice of smoothing kernel, as well as the equations of an inviscid second-grade non-Newtonian fluid. While previous studies of this system in 3D have noted the suppression of nonlinear interaction between modes smaller than α, we show that the modification of the nonlinear advection term also acts to enhance the inverse-cascade of energy in 2D turbulence and thereby affects scales of motion larger than α as well. This, we note, (a) may preclude a straightforward use of the model as a subgrid model in coarsely resolved 2D computations, (b) i...

Dispersive–Dissipative Eddy Parameterization in a Barotropic Model

Abstract Recently a new class of coarse-grained equations, known as α models, have been proposed for the mean motion of an ideal incompressible fluid. The use of one such model to represent the time-mean component of a turbulent β-plane circulation characterized by potential vorticity mixing is considered. In particular, the focus is on the wind-driven circulation in a shallow ocean basin, a problem well studied as a prototype of more realistic ocean dynamics. The authors demonstrate the ability of an α model to reproduce qualitatively the structure of a four-gyre circulation that forms (in the time mean) when the barotropic vorticity equation is driven by a symmetric, double-gyre wind forcing, and when the dissipation is weak. This is offered as a first step in assessing the utility of the α-model approach to simulating more complex geophysical flows.

Ocean modelling for climate studies: Eliminating short time scales in long-term, high-resolution studies of ocean circulation

On the decadal to centennial time scale, changes in climate are controlled strongly by changes in ocean circulation. However, because of limitations inherent to the time integration schemes used in present-day ocean models, state-of-the-art climate change simulations resolve the oceans only very coarsely. With an aim to enable long-term simulations of ocean circulation at the high resolutions required for a better representation of global ocean dynamics, we have implemented fully-implicit time integration schemes in a version of the popular ocean general circulation model POP (Parallel Ocean Program), employing Jacobian-free Newton-Krylov techniques. Here, we describe the numerical principles underlying iPOP in some detail and present a few computational results. While there are many advantages to this approach, including a consistent and uniform treatment of the terms in the governing equations, the primary advantage lies in the ability to take time steps that are of relevance to the physical phenomenon that is being studied. The time step is not limited (for stability reasons) by the fastest modes of the system.

Large-eddy simulation, fuel rod vibration and grid-to-rod fretting in pressurized water reactors

Grid-to-rod fretting (GTRF) in pressurized water reactors is a flow-induced vibration phenomenon that results in wear and fretting of the cladding material on fuel rods. GTRF is responsible for over 70% of the fuel failures in pressurized water reactors in the United States. Predicting the GTRF wear and concomitant interval between failures is important because of the large costs associated with reactor shutdown and replacement of fuel rod assemblies. The GTRF-induced wear process involves turbulent flow, mechanical vibration, tribology, and time-varying irradiated material properties in complex fuel assembly geometries. This paper presents a new approach for predicting GTRF induced fuel rod wear that uses high-resolution implicit large-eddy simulation to drive nonlinear transient dynamics computations. The GTRF fluid-structure problem is separated into the simulation of the turbulent flow field in the complex-geometry fuel-rod bundles using implicit large-eddy simulation, the calculation of statistics of the resulting fluctuating structural forces, and the nonlinear transient dynamics analysis of the fuel rod. Ultimately, the methods developed here, can be used, in conjunction with operational management, to improve reactor core designs in which fuel rod failures are minimized or potentially eliminated. Robustness of the behavior of both the structural forces computed from the turbulent flow simulations and the results from the transient dynamics analyses highlight the progress made towards achieving a predictive simulation capability for the GTRF problem. A new approach for predicting grid-to-rod fretting wear in reactor fuel is presented.ILES validation is performed using data for a 5 × 5 fuel rod bundle.Richardson extrapolation of statistical fuel rod forces is used to bound wear work-rates.

An adaptive discrete-velocity model for the shallow water equations

A new approach to solving the shallow water equations is presented. This involves using discrete velocities of an adaptive nature in a finite volume context. The origin of the discrete-velocity space and the magnitudes of the discrete-velocities are both spatially and temporally variable. The near-equilibrium flow method of Nadiga and Pullin is used to arrive at a robust second-order (in both space and time) scheme-the adaptive discrete velocity (ADV) scheme-which captures hydraulic jumps with no oscillations. The flow over a two-dimensional ridge, over a wide range of undisturbed upstream Froude numbers prove the robustness and accuracy of the scheme. A comparison of the interaction of two circular vortex patches in the presence of bottom topography as obtained by the ADV scheme and a semi-Lagrangian scheme more than validates the new scheme in two dimensions.

The equivalence of the Lagrangian-averaged Navier-Stokes-α model and the rational large eddy simulation model in two dimensions

In the Large Eddy Simulation (LES) framework for modeling a turbulent flow, when the large scale velocity field is defined by low-pass filtering the full velocity field, a Taylor series expansion of the full velocity field in terms of the large scale velocity field leads (at the leading order) to the nonlinear gradient model for the subfilter stresses. Motivated by the fact that while the nonlinear gradient model shows excellent a priori agreement in resolved simulations, the use of this model by itself is problematic, we consider two models that are related, but better behaved: The Rational LES model that uses a sub-diagonal Pade approximation instead of a Taylor series expansion and the Lagrangian Averaged Navier-Stokes-� model that uses a regularization approach to modeling turbulence. In this article, we show that these two latter models are identical in two dimensions.