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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Settling of two-way momentum and energy coupled particles subject to Boussinesq and non-Boussinesq heating

This work establishes a procedure to accurately compute heat transfer between an Eulerian fluid and Lagrangian point-particles. Recent work has focused on accurately computing momentum transfer between fluid and particles. The coupling term for momentum involves the undisturbed fluid velocity at the particle location which is not directly accessible in the simulation. Analogously, in the context of thermal coupling, the undisturbed fluid temperature at the particle location is not directly accessible in simulations and must be estimated. In this paper, we develop a scheme to accurately estimate the undisturbed fluid temperature of a point-particle exchanging thermal energy with a surrounding fluid. Furthermore, the temperature disturbance is correlated with the enhanced temperature curvature in the vicinity of the particle and is formally valid in the low heating, low convection limit. We conduct extensive verification of the correction procedure for a settling particle subject to radiation. This setup allows the simultaneous testing of thermal and momentum corrections. By considering equations of drag and Nusselt number extended to finite Péclet and Boussinesq numbers, we establish a large range over which the correction procedure can be applied.

42 ENGINEERING↗
DOE OSTI2023

Statistical Analysis of the Limiting Dynamics of Two dimensional Boussinesq Turbulent Systems

In the current study, we investigate rotational and stratified turbulent systems within the framework of the Boussinesq equations. The effects of rotation and stratification give rise to a variety of dynamical regimes in atmospheric and oceanic turbulence. To develop a better understanding of the fundamental flow properties through a more straightforward mathematical framework, we commence our research with a simplified two-dimensional Boussinesq model that incorporates rotation and stratification effects. We explore two distinct limiting dynamics, considering pure rotation and pure stratification separately. We examine these intriguing dynamics through numerical investigations, applying the exact solution theory of Boussinesq equations and examining small-scale perturbations to the exact solution under rotation and stratification. We conduct an analysis of statistical quantities associated with the turbulent state variables. The results indicate that, under weak rotation at the equilibrium statistical state, the flow field exhibits vortex flow. However, strong rotation transforms the flow into vertical shear flow. We observe both downscale and upscale energy transfers with a decay rate of k −3 in the absence of external forcing and dissipation. Furthermore, we delve into the study of internal gravity wave mode interactions in the simplified two-dimensional Boussinesq system, which is a crucial aspect of geophysical turbulence due to gravity. Our findings reveal that the time series of the mode coefficients exhibit wave-like interactions between wave modes.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Non-Boussinesq subgrid-scale model with dynamic tensorial coefficients

A major drawback of Boussinesq-type subgrid-scale stress models used in large-eddy simulations is the inherent assumption of alignment between large-scale strain rates and filtered subgrid-stresses. A priori analyses using direct numerical simulation (DNS) data have shown that this assumption is invalid locally as subgrid-scale stresses are poorly correlated with the large-scale strain rates [J. Bardina, J. Ferziger, and W. Reynolds, Improved subgrid-scale models for large-eddy simulation, in Proceedings of the 13th Fluid and Plasmadynamics Conference, AIAA (1980); C. Meneveau and K. Katz, Scale-invariance and turbulence models for large-eddy simulation, Ann. Rev. Fluid Mech. 32, 1 (2000)]. In the present work, a new, non-Boussinesq subgrid-scale model is presented where the model coefficients are computed dynamically. Some previous non-Boussinesq models have observed issues in providing adequate dissipation of turbulent kinetic energy [e.g., Bardina et al., Proceedings of the 13th Fluid and Plasmadynamics Conference (1980); R. A. Clark, J. Ferziger, and W.C. Reynolds. Evaluation of subgrid-scale models using an accurately simulated turbulent flow, J. Fluid Mech. 91, 1 (1979); S. Stolz and N. A. Adams, An approximate deconvolution procedure for large-eddy simulation, Phys. Fluids 11, 1699 (1999)]; however, the present model is shown to provide sufficient dissipation using dynamic coefficients. Modeled subgrid-scale Reynolds stresses satisfy the consistency requirements of the governing equations for large-eddy simulation (LES), vanish in laminar flow and at solid boundaries, and have the correct asymptotic behavior in the near-wall region of a turbulent boundary layer. The new model, referred to as the dynamic tensor-coefficient Smagorinsky model (DTCSM), has been tested in simulations of canonical flows: decaying and forced homogeneous isotropic turbulence, and wall-modeled turbulent channel flow at high Reynolds numbers. The results show favorable agreement with DNS data. It has been shown that DTCSM offers similar predictive capabilities as the dynamic Smagorinsky model for canonical flows. In order to assess the performance of DTCSM in more complex flows, wall-modeled simulations of high Reynolds number flow over a Gaussian bump (Boeing speed bump) exhibiting smooth-body flow separation are performed. Predictions of surface pressure and skin friction, compared against DNS and experimental data, show improved accuracy from DTCSM in comparison to existing static coefficient (Vreman) and dynamic Smagorinsky model. The computational cost of performing LES with this model is up to 15% higher than the dynamic Smagorinsky model.

42 ENGINEERING↗
NASA NTRS1990

Recurrence in truncated Boussinesq models for nonlinear waves in shallow water

The rapid spatial recurrence of weakly nonlinear and weakly dispersive progressive shallow-water waves is examined using a numerical integration technique on the discretized and truncated form of the Boussinesq equations. This study primarily examines recurrence in wave fields with Ursell number O(1) and characterizes the sensitivity of recurrence to initial spectral shape and number of allowed frequency modes. It is shown that the rapid spatial recurrence is not an inherent property of the considered Boussinesq systems for evolution distances of 10-50 wavelengths. The main result of the study is that highly truncated Boussinesq models of resonant shallow-water ocean surface gravity waves predict rapid multiple recurrence cycles, but that this is an artifact dependent on the number of allowed modes. For initial conditions consisting of essentially all energy concentrated in a single mode, damping of the recurrence cycles increases as the number of low-power background modes increases. When more than 32 modes are allowed, the recurrence behavior is relatively insensitive to the number of allowed modes.

Elgar, Steve↗
NASA NTRS1996

On the Effect of Feedback Control on Benard Convection in a Boussinesq Fluid

The effect of nonlinear feedback control strategies on the platform of convection in a Boussinesq fluid heated from below is investigated. In the absence of the control, given that non-Boussinesq effects may be neglected, it is well known that convection begins in the form of a supercritical bifurcation to rolls. Non-Boussinesq behaviour destroys the symmetry of the basic state, and through a subcritical bifurcation leads to the formation of hexagonal cells. Here we discuss the influence of regulation of the lower surface temperature by means of a control mechanism, made up of a combination of a proportional linear and nonlinear controller, on the stability of the hexagonal cell pattern.

Shortis, Trudi A.↗
DOE OSTI2025

Assessment of integral models for non-Boussinesq lazy plumes using numerical simulations

Integral modelling of turbulent buoyant plumes is crucial for rapid predictions of plume characteristics. While the governing equations are typically derived using self-similarity and a Boussinesq approximation, these assumptions may not hold for plumes originating from finite-area sources with large density ratios. Here, this work evaluates the accuracy of integral-scale models for non-Boussinesq lazy plumes using high-fidelity numerical simulations of turbulent helium plumes. We analyse the plume kinematics by computing vertical fluxes, plume radius and radial profiles, establishing some disparities between common practice and physical accuracy. We identify how the definition of the plume radius changes the perception of the plume structure when the flow is not self-similar and derive a relationship between the flux-based and threshold-based definitions without requiring self-similarity. We then examine the plume dynamics by evaluating the source terms from the governing plume equations. Our results support neglecting diffusive and viscous effects but emphasise the importance of the mean pressure gradient, even in the self-similar regime. Two coefficients need to be modelled: the well-known entrainment coefficient and the lesser-known momentum correction coefficient, which is a correction required for the momentum equation to account for self-similar and slender approximations. The momentum correction coefficient is found to be approximately constant and slightly greater than the assumed value of 1. The standard entrainment coefficient models perform well up to a local Richardson number three times the asymptotic value but overpredict entrainment for larger Richardson numbers. We propose a correction using the known finite limit of entrainment at infinite Richardson number.

Meehan, Michael Alexander [Sandia National Laborat↗
DOE OSTI2022

Block-structured, equal-workload, multi-grid-nesting interface for the Boussinesq wave model FUNWAVE-TVD (Total Variation Diminishing)

Abstract. We describe the development of a block-structured, equal-CPU-load (central processing unit), multi-grid-nesting interface for the Boussinesq wave model FUNWAVE-TVD (Fully Nonlinear Boussinesq Wave Model with Total Variation Diminishing Solver). The new model framework does not interfere with the core solver, and thus the core program, FUNWAVE-TVD, is still a standalone model used for a single grid. The nesting interface manages the time sequencing and two-way nesting processes between the parent grid and child grid with grid refinement in a hierarchical manner. Workload balance in the MPI-based (message passing interface) parallelization is handled by an equal-load scheme. A strategy of shared array allocation is applied for data management that allows for a large number of nested grids without creating additional memory allocations. Four model tests are conducted to verify the nesting algorithm with assessments of model accuracy and the robustness in the application in modeling transoceanic tsunamis and coastal effects.

Choi, Young-Kwang↗
NASA NTRS1990

Solitons induced by boundary conditions from the Boussinesq equation

The behavior of solitons induced by boundary excitation is investigated at various time-dependent conditions and different unperturbed water depths, using the Korteweg-de Vries (KdV) equation. Then, solitons induced from Boussinesq equations under similar conditions were studied, making it possible to remove the restriction in the KdV equation and to treat soliton head-on collisions (as well as overtaking collisions) and reflections. It is found that the results obtained from the KdV and the Boussinesq equations are in good agreement.

Chou, Ru Ling↗
NASA NTRS1996

Inertial Range Dynamics in Boussinesq Turbulence

L'vov and Falkovich have shown that the dimensionally possible inertial range scaling laws for Boussinesq turbulence, Kolmogorov and Bolgiano scaling, describe steady states with constant flux of kinetic energy and of entropy respectively. These scaling laws are treated as similarity solutions of the direct interaction approximation for Boussinesq turbulence. The Kolmogorov scaling solution corresponds to a weak perturbation by gravity of a state in which the temperature is a passive scalar but in which a source of temperature fluctuations exists. Using standard inertial range balances, the renormalized viscosity and conductivity, turbulent Prandtl number, and spectral scaling law constants are computed for Bolgiano scaling.

Rubinstein, Robert↗
NASA NTRS1997

On the Asymptotic Regimes and the Strongly Stratified Limit of Rotating Boussinesq Equations

Asymptotic regimes of geophysical dynamics are described for different Burger number limits. Rotating Boussinesq equations are analyzed in the asymptotic limit, of strong stratification in the Burger number of order one situation as well as in the asymptotic regime of strong stratification and weak rotation. It is shown that in both regimes horizontally averaged buoyancy variable is an adiabatic invariant for the full Boussinesq system. Spectral phase shift corrections to the buoyancy time scale associated with vertical shearing of this invariant are deduced. Statistical dephasing effects induced by turbulent processes on inertial-gravity waves are evidenced. The 'split' of the energy transfer of the vortical and the wave components is established in the Craya-Herring cyclic basis. As the Burger number increases from zero to infinity, we demonstrate gradual unfreezing of energy cascades for ageostrophic dynamics. The energy spectrum and the anisotropic spectral eddy viscosity are deduced with an explicit dependence on the anisotropic rotation/stratification time scale which depends on the vertical aspect ratio parameter. Intermediate asymptotic regime corresponding to strong stratification and weak rotation is analyzed where the effects of weak rotation are accounted for by an asymptotic expansion with full control (saturation) of vertical shearing. The regularizing effect of weak rotation differs from regularizations based on vertical viscosity. Two scalar prognostic equations for ageostrophic components (divergent velocity potential and geostrophic departure ) are obtained.

Babin, A.↗
DOE OSTI2021

Parallelized POD-based suboptimal economic model predictive control of a state-constrained Boussinesq approximation

Motivated by an energy efficient building application, we want to optimize a quadratic cost functional subject to the Boussinesq approximation of the Navier-Stokes equations and to bilateral state and control constraints. Since the computation of such an optimal solution is numerically costly, we design an efficient strategy to compute a sub-optimal (but applicationally acceptable) solution with significantly reduced computational effort. We employ an economic Model Predictive Control (MPC) strategy to obtain a feedback control. The MPC sub-problems are based on a linear-quadratic optimal control problem subjected to mixed control and state constraints and a convection-diffusion equation, reduced with proper orthogonal decomposition. Finally, to solve each sub-problem, we apply a primal-dual active set strategy. The method can be fully parallelized, which enables the solution of large problems with real-world parameters.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Notes on the non-dimensionalization of the MHD Boussinesq convection equations

The objective of these notes is to demonstrate that the formulation of the MHD Boussinesq convection equations that follows the conventions of the turbulence community used in our previous work is consistent and compatible with the formulation from the convection community used in Busse and Pesch.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Mean Flow from Phase Averages in the 2D Boussinesq Equations

The atmosphere and ocean are described by highly oscillatory PDEs that challenge both our understanding of their dynamics and their numerical approximation. This paper presents a preliminary numerical study of one type of phase averaging applied to mean flows in the 2D Boussinesq equations that also has application to numerical methods. The phase averaging technique, well-known in dynamical systems theory, relies on a mapping using the exponential operator, and then an averaging over the phase. The exponential operator has connections to the Craya–Herring basis pioneered by Jack Herring to study the fluid dynamics of oscillatory, nonlinear fluid dynamics. In this paper, we perform numerical experiments to study the effect of this averaging technique on the time evolution of the solution. We explore its potential as a definition for mean flows. We also show that, as expected from theory, the phase-averaging method can reduce the magnitude of the time rate of change in the PDEs, making them potentially suitable for time stepping methods.

Wingate, Beth A.↗
NASA NTRS1992

Numerical study of the small scale structures in Boussinesq convection

Two-dimensional Boussinesq convection is studied numerically using two different methods: a filtered pseudospectral method and a high order accurate Essentially Nonoscillatory (ENO) scheme. The issue whether finite time singularity occurs for initially smooth flows is investigated. The numerical results suggest that the collapse of the bubble cap is unlikely to occur in resolved calculations. The strain rate corresponding to the intensification of the density gradient across the front saturates at the bubble cap. We also found that the cascade of energy to small scales is dominated by the formulation of thin and sharp fronts across which density jumps.

Weinan, E.↗
NASA NTRS1994

Renormalization Group Theory of Bolgiano Scaling in Boussinesq Turbulence

Bolgiano scaling in Boussinesq turbulence is analyzed using the Yakhot-Orszag renormalization group. For this purpose, an isotropic model is introduced. Scaling exponents are calculated by forcing the temperature equation so that the temperature variance flux is constant in the inertial range. Universal amplitudes associated with the scaling laws are computed by expanding about a logarithmic theory. Connections between this formalism and the direct interaction approximation are discussed. It is suggested that the Yakhot-Orszag theory yields a lowest order approximate solution of a regularized direct interaction approximation which can be corrected by a simple iterative procedure.

Rubinstein, Robert↗
NASA NTRS2011

Deep Ocean Warming Assessed from Altimeters, GRACE, 3 In-situ Measurements, and a Non-Boussinesq OGCM

Observational surveys have shown significant oceanic bottom water warming, but they are too spatially and temporally sporadic to quantify the deep ocean contribution to the present-day sea level rise (SLR). In this study, altimetry sea surface height (SSH), Gravity Recovery and Climate Experiment (GRACE) ocean mass, and in situ upper ocean (0-700 m) steric height have been assessed for their seasonal variability and trend maps. It is shown that neither the global mean nor the regional trends of altimetry SLR can be explained by the upper ocean steric height plus the GRACE ocean mass. A non-Boussinesq ocean general circulation model (OGCM), allowing the sea level to rise as a direct response to the heat added into the ocean, is then used to diagnose the deep ocean steric height. Constrained by sea surface temperature data and the top of atmosphere (TOA) radiation measurements, the model reproduces the observed upper ocean heat content well. Combining the modeled deep ocean steric height with observational upper ocean data gives the full depth steric height. Adding a GRACE-estimated mass trend, the data-model combination explains not only the altimetry global mean SLR but also its regional trends fairly well. The deep ocean warming is mostly prevalent in the Atlantic and Indian oceans, and along the Antarctic Circumpolar Current, suggesting a strong relation to the oceanic circulation and dynamics. Its comparison with available bottom water measurements shows reasonably good agreement, indicating that deep ocean warming below 700 m might have contributed 1.1 mm/yr to the global mean SLR or one-third of the altimeter-observed rate of 3.11 +/- 0.6 mm/yr over 1993-2008.

sea level↗