Gradient Reconstruction on Adaptive Meshes

Least squares gradient estimates find application in many fields of computational science, particularly for the purposes of surface fitting and gradient reconstruction in computational fluid dynamics and data visualization. In this paper we derive error bounds for classical and weighted least squares gradient estimates. The bounds reflect how the number of points used in the least squares stencil and the smallest singular value of the least squares matrix impact the accuracy of these estimates. We show how an extrapolation method based on Householder transformations provides substantially tighter bounds. Numerical case studies are presented to elucidate our theory for a data set taken from Franke [Math. Comp., 38 (1982), pp. 181–200].

Past changes in the density and momentum structure of oceanic circulation are an important aspect of changes in the Atlantic Meridional Overturning Circulation and consequently climate. However, very little is known about past changes in the vertical density structure of the ocean, even very extensively studied systems such as the North Atlantic. Here we exploit the physical controls on the settling depth of the dense Mediterranean water plume derived from the Strait of Gibraltar to obtain the first robust, observations-based, probabilistic reconstruction of the vertical density gradient in the eastern North Atlantic during the last 30,000 years. We find that this gradient was weakened by more than 50%, relative to the present, during the last Glacial Maximum, and that changes in general are associated with reductions in AMOC intensity. However, we find only a small change during Heinrich Event 1 relative to the Last Glacial Maximum, despite strong evidence that overturning was substantially altered. This implies that millennial-scale changes may not be reflected in vertical density structure of the ocean, which may be limited to responses on an ocean-overturning timescale or longer. Regardless, our novel reconstruction of Atlantic density structure can be used as the basis for a dynamical measure for validation of model-based AMOC reconstructions. In addition, our general approach is transferrable to other marginal sea outflow plumes, to provide estimates of oceanic vertical density gradients in other locations.

<p indent="0mm">In this paper, using a recently proposed sharp-interface method for compressible multiphase flows with phase change, we numerically study the Richtmyer-Meshkov instability problem of the vapor-liquid interface, both with and without phase change. The wave structures and the interfacial evolution mechanism of a single-mode sinusoidally perturbed interface under the impact of a shock wave are analyzed. First, the classical air-SF6 interface Richtmyer-Meshkov instability and shock water column interaction are simulated in the paper to validate the feasibility of the numerical method. The numerical simulations and experimental results are generally in good agreement, indicating that the present method can accurately capture the interface flow evolution characteristics during the development of Richtmyer-Meshkov instability. Next, we investigate the similarities and differences in Richtmyer-Meshkov instability at the vapor-liquid interface under thirteen conditions: no phase change, evaporation, and condensation with three different Mach numbers. It is shown that, unlike the air-SF6, a double spike structure is formed during the development of interfacial instability due to the large acoustic impedance difference at the vapor-liquid interface. And the phase change process affects the local deformation of the interface, as the evaporation effect inhibits the development of the interfacial instability, while the condensation effect promotes the development of the interfacial instability. Finally, we compare the numerical simulation data with the existing linear model and find that the impulsive model can predict the linear growth of the early stage, while the nonlinear growth of the later stage will be investigated further.

Automatic differentiation approach for solving one-dimensional flow and heat transfer problems

SummaryThe use of equivalent source processing on magnetic datasets is important for the regular gridding and denoising of data before any other processing can occur. The processing technique is setup as an inverse problem and solved for susceptibilities to reproduce the observed data. The drawback to the inverse problem is computation cost and overall speed for large-scale problems. Since aeromagnetics has become common in exploration, it is rare that the datasets acquired are small in data volume or space, and can be handled rapidly on a single workstation. One way to minimize the computational cost is to reduce the number of model parameters. We present an equivalent source processing technique that minimizes the number of cells in the model domain via an adaptive quadtree mesh discretization. The mesh remains coarse where no significant anomalies are present, yet fines on the edges of observed anomalies. The transition from the fine to coarse mesh grid is based on the total-gradient of the dataset, placing smaller cells on the edges of the anomaly where the susceptibilities have the greatest variation spatially. We show that the algorithm can perform over four times as fast as traditional equivalent source processing with a regular cell mesh yet preserves the same accuracy. In this paper, we present a synthetic example for proof of concept.

Extension of an exponential discretization scheme to multidimensional convection–diffusion problems

Strength at extreme pressures (>1 Mbar or 100 GPa) and high strain rates (106-108 s-1) of materials is not well characterized. The goal of the research outlined in this thesis is to study the strength of tantalum (Ta) at these conditions. The Omega Laser in the Laboratory for Laser Energetics in Rochester, New York is used to create such extreme conditions. Targets are designed with ripples or waves on the surface, and these samples are subjected to high pressures using Omega’s high energy laser beams. In these experiments, the observational parameter is the Richtmyer-Meshkov (RM) instability in the form of ripple growth on single-mode ripples. The experimental platform used for these experiments is the “ride-along” laser compression recovery experiments, which provide a way to recover the specimens having been subjected to high pressures. Six different experiments are performed on the Omega laser using single-mode tantalum targets at different laser energies. The energy indicates the amount of laser energy that impinges the target. For each target, values for growth factor are obtained by comparing the profile of ripples before and after the experiment. With increasing energy, the growth factor increased. Engineering simulations are used to interpret and correlate the measurements of growth factor to a measure of strength. In order to validate the engineering constitutive model for tantalum, a series of simulations are performed using the code Eureka, based on the Optimal Transportation Meshfree (OTM) method. Two different configurations are studied in the simulations: RM instabilities in single and multimode ripples. Six different simulations are performed for the single ripple configuration of the RM instability experiment, with drives corresponding to laser energies used in the experiments. Each successive simulation is performed at higher drive energy, and it is observed that with increasing energy, the growth factor increases. Overall, there is favorable agreement between the data from the simulations and the experiments. The peak growth factors from the simulations and the experiments are within 10% agreement. For the multimode simulations, the goal is to assist in the design of the laser driven experiments using the Omega laser. A series of three-mode and four-mode patterns are simulated at various energies and the resulting growth of the RM instability is computed. Based on the results of the simulations, a configuration is selected for the multimode experiments. These simulations also serve as validation for the constitutive model and the material parameters for tantalum that are used in the simulations. By designing samples with initial perturbations in the form of single-mode and multimode ripples and subjecting these samples to high pressures, the Richtmyer-Meshkov instability is investigated in both laser compression experiments and simulations. By correlating the growth of these ripples to measures of strength, a better understanding of the strength of tantalum at high pressures is achieved.

Numerical diffusion can be minimized using fine grid spacing and/or higher-order numerical schemes. In this study, the authors focus on higher-order scalar advection schemes and their effects on simulated cloud fields. A monotonic multidimensional odd-order conservative advection scheme has been implemented, following the approach of Leonard. It has been tested in simulations of idealized scalar fields advected by simple prescribed motion, as well as turbulence fields; large-eddy simulations of turbulent stratocumulus clouds; and simulations of deep convective clouds. New third-, fifth-, and seventh-order schemes are compared with the second-order scheme originally used in the model. For the deep cumulus case, a high-resolution large-eddy simulation with the same domain size is used as a benchmark.The fifth-order scheme shows much less numerical diffusion than the lower-order scheme. The additional improvement with the seventh-order scheme is minor. The higher-order scheme generally produces simulated cloud fields similar to those obtained with a lower-order scheme with a finer grid spacing. This effect is especially noticeable for the updraft-core statistics of the deep cumulus simulation, as compared with the benchmark simulation. The fifth-order scheme with coarse horizontal resolution produces results close to those of the benchmark simulation. Compared to a high-resolution simulation with the low-order scheme, the numerical cost of the fifth-order simulation is smaller than a factor of 10.

A numerical investigation is conducted to characterize the interaction dynamics between planar incident shock waves (Mach numbers ranging from 1.21 to 2.00) and a cylindrical SF6 heavy gas column embedded in ambient air, focusing on Richtmyer–Meshkov instability (RMI) development, interfacial scale evolution, and vorticity-driven material transport mechanisms. From an Eulerian perspective, the motion of several interface characteristic points, the evolution of material line length, and two area parameters associated with SF6 distribution are tracked. Additionally, the SF6 material transport process and trajectories of material points within the gas column are visualized using the Lagrangian particle tracing method. Theoretical models for bubble velocity, vortex core spacing, and material line stretching laws are evaluated. Our results highlight the significant influence of incident shock strength on interface evolution and material transport. The streamwise motion of both the gas column and primary vortex core follows near-linear trajectories, whereas spanwise dynamics exhibit nonlinear behavior driven by hydrodynamic instabilities. The SF6-containing region expands linearly due to the growth of the primary vortex pair. Two constants—vortex core spacing and SF6 volume fraction weighted area—are identified, both dependent on the incident shock strength. Material line length evolution follows a two-stage exponential growth pattern, sequentially driven by RMI and secondary instabilities. Lagrangian analysis reveals a stratified entrainment mechanism, with outer SF6 layers mixing earlier than inner regions. Theoretical models show strong agreement with numerical predictions for bubble velocity and SF6 volume-fraction-weighted area, while discrepancies are observed in vortex core spacing models.

High-order adaptive mesh refinement offers the potential for accurate and efficient resolution of problems in fluid dynamics and other fields where a wide range of length scales is present. A critical requirement for the interface closures used with these methods is stability in the context of hyperbolic systems of partial differential equations. In this study, a class of energy-stable high-order finite-difference interface closures is presented for grids with step resolution changes in one dimension. Asymptotic stability in time for these schemes is achieved by imposing a summation-by-parts condition on the interface closure, which is thus also nondissipative. Interface closures compatible with interior fourth- and sixth-order explicit, and fourth-order implicit centered schemes are presented. Validation tests include linear and nonlinear problems in one and in two dimensions with tensor-product grid refinement. A second class of stable high-order interface closures is presented for two-dimensional cell-centered grids with patch-refinement and step-changes in resolution. For these grids, coarse and fine nodes are not aligned at the mesh interfaces, resulting in hanging nodes. Stability is achieved by again imposing a summation-by-parts condition, resulting in nondissipative closures, at the cost of accuracy at corner interfaces. Interface stencils for an explicit fourth-order finite-difference scheme are presented for each geometry. Validation tests confirm the stability and accuracy of these closures for linear and nonlinear problems. The Richtmyer-Meshkov instability is investigated using a novel first-order perturbation of the two-dimensional Navier-Stokes equations about a shock-resolved base flow. The computational domain is efficiently resolved using the one-dimensional fourth-order interface scheme. Results are compared to analytic models of the instability, showing agreement with predicted asymptotic growth rates in the inviscid range, while significant discrepancies are noted in the transient growth phase. Viscous effects are found to be poorly predicted by existing models.

PurposeThe purpose of the present study is to develop a new numerical framework that can predict the supersonic base flow more accurately, including the development of axisymmetrically separated shear layer and recompression shock. To this end, two aspects are improved and combined, i.e. a newly self-adaptive turbulence eddy simulation (SATES) turbulence modeling method and a high-order discretization numerical scheme. Furthermore, the performance of the new numerical framework within a general-purpose PHengLEI software is assessed in detail.Design/methodology/approachSatisfactory prediction of the supersonic separated shear layer with unsteady wake flow is quite challenging. By using a unified turbulence model called SATES combining high-order accurate discretization numerical schemes, the present study first assesses the performance of newly developed SATES for supersonic axisymmetric separation flows. A high-order finite differencing-based compressible computational fluid dynamics (CFD) code called PHengLEI is developed and several different numerical schemes are used to investigate the effects on shock-turbulence interactions, which include the monotonic upstream-centered scheme for conservation laws (MUSCL), weighted compact nonlinear scheme (WCNS) and hybrid cell-edge and cell-node dissipative compact scheme (HDCS).FindingsCompared with the available experimental data and the numerical predictions, the results of SATES by using high-order accurate WCNS or HDCS schemes agree better with the experiments than the results by using the MUSCL scheme. The WCNS and HDCS can also significantly improve the prediction of flow physics in terms of the instability of the annular shear layer and the evolution of the turbulent wake.Research limitations/implicationsThe small deviations in the recirculation region can be found between the present numerical results and experimental data, which could be caused by the inaccurate incoming boundary layer condition and compressible effects. Therefore, a proper incoming boundary layer condition with turbulent fluctuations and compressibility effects need to be considered to further improve the accuracy of simulations.Practical implicationsThe present study evaluates a high-order discretization-based SATES turbulence model for supersonic separation flows, which is quite valuable for improving the calculation accuracy of aeronautics applications, especially in supersonic conditions.Originality/valueFor the first time, the newly developed SATES turbulence modeling method combining the high-order accurate WCNS or HDCS numerical schemes is implemented on the PHengLEI software and successfully applied for the simulations of supersonic separation flows, and satisfactory results are obtained. The unsteady evolutions of the supersonic annular shear layer are analyzed, and the hairpin vortex structures are found in the simulation.

Numerical study of preburned hydrogen by a high-order numerical scheme for turbulent reactive flows

Simulations of viscous and compressible gas–gas flows using high-order finite difference schemes

Rayleigh-Taylor (RT) instability and Richtmyer-Meshkov (RM) instability are classic problems and important research topics in the field of fluid mechanics, and are widely found in nature and engineering applications. In inertial confinement fusion, these instabilities are crucial for the success of the fusion ignition process. Therefore, gaining a deep understanding of the mechanisms of instability development and establishing corresponding theoretical models not only have profound scientific significance, but also play important theoretical supporting roles in practical engineering applications. This article provides a detailed review of a series of research progress made by the academic community in predicting the evolution of planar single-mode RT instability and RM instability fronts since Layzer proposed the potential flow theoretical model in 1955. The research content covers a variety of key factors affecting the development of such instabilities, including initial disturbance form, fluid density ratio, vorticity, surface tension, viscosity, compressibility, etc. Meanwhile, this article is intended to provide insight into the nonlinear coupling relationships among these effects and the underlying physical mechanisms. Finally, this article also analyzes the additional influence due to geometric effects when the Layzer's potentia flow model is extended to a converging geometry and puts forward prospects for future research directions and priorities.

A simple yet precise relation between the flux gradient and the bed slope source term is presented, which produces a net force within the cell with an inclined water surface, but ensures still water condition when there is no flow across the boundaries. The proposed method consists of calculating the pressure term based on the water depths at the cell vertices, which may be computed by a higher order scheme and the bed slope source term by a centered discretization technique. The methodology is demonstrated with a Godunov-type upwind finite volume formulation. The inviscid fluxes are calculated using Roe’s approximate Riemann solver and a second-order spatial accuracy is obtained by implementing multidimensional gradient reconstruction and slope limiting techniques. The accuracy and applicability of the numerical model is verified with a couple of test problems and a real flow example of tidal water movement in a stretch of River Hooghly in India.