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At least 1,099 records · Page 61

Two-dimensional Euler and Navier-Stokes Time accurate simulations of fan rotor flows

Two numerical methods are presented which describe the unsteady flow field in the blade-to-blade plane of an axial fan rotor. These methods solve the compressible, time-dependent, Euler and the compressible, turbulent, time-dependent, Navier-Stokes conservation equations for mass, momentum, and energy. The Navier-Stokes equations are written in Favre-averaged form and are closed with an approximate two-equation turbulence model with low Reynolds number and compressibility effects included. The unsteady aerodynamic component is obtained by superposing inflow or outflow unsteadiness to the steady conditions through time-dependent boundary conditions. The integration in space is performed by using a finite volume scheme, and the integration in time is performed by using k-stage Runge-Kutta schemes, k = 2,5. The numerical integration algorithm allows the reduction of the computational cost of an unsteady simulation involving high frequency disturbances in both CPU time and memory requirements. Less than 200 sec of CPU time are required to advance the Euler equations in a computational grid made up of about 2000 grid during 10,000 time steps on a CRAY Y-MP computer, with a required memory of less than 0.3 megawords.

Boretti, A. A.↗

Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations

Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one-dimensional convection-diffusion-reaction (CDR) equations. First, accuracy, stability, conservation, and dense output are considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, N = 2, additive Runge-Kutta ARK2 methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms are of equal order to those of the elemental methods. Derived ARK2 methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z(exp [I]) goes to infinity, and retain high stability efficiency in the absence of stiffness, z(exp [I]) goes to zero. Extrapolation-type stage-value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK2 error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK2 methods while offering error control and stage-value predictors.

Kennedy, Christopher A.↗

Efficient implementation of essentially non-oscillatory shock capturing schemes, 2

Earlier work on the efficient implementation of ENO (essentially non-oscillatory) shock capturing schemes is continued. A new simplified expression is provided for the ENO construction procedure based again on numerical fluxes rather than cell averages. Also considered are two improvements which are labeled ENO-LLF (local Lax-Friedrichs) and ENO-Roe, which yield sharper shock transitions, improved overall efficiency, and lower computational cost than previous implementation of the ENO schemes. Two methods of sharpening contact discontinuities, i.e., the subcell resolution idea of Harten and the artificial compression idea of Yang, which those authors used originally in the cell-average framework, are supplied to the current ENO schemes using numerical fluxes and TVD Runge-Kutta time discretizations. The implementation for nonlinear systems and multi-dimensions is given. Finally, many numerical examples, including a compressible shock turbulence interaction flow calculation, are given.

Shu, Chi-Wang↗

Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations

Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one- dimensional convection-diffusion-reaction (CDR) equations. Accuracy, stability, conservation, and dense-output are first considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, (N = 2), additive Runge-Kutta (ARK(sub 2)) methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms of the partitioned method are of equal order to those of the elemental methods. Derived ARK(sub 2) methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z['] yields -infinity, and retain high stability efficiency in the absence of stiffness, z['] yield 0. Extrapolation-type stage- value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK(sub 2) error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK(sub 2) methods.

Kennedy, Christopher A.↗

Numerical approximation of boundary conditions with applications to inviscid equations of gas dynamics

A comprehensive overview of the state of the art of well-posedness and stability analysis of difference approximations for initial boundary value problems of the hyperbolic type is presented. The applicability of recent theoretical development to practical calculations for nonlinear gas dynamics is examined. The one dimensional inviscid gas dynamics equations in conservation law form are selected for numerical experiments. The class of implicit schemes developed from linear multistep methods in ordinary differential equations is chosen and the use of linear extrapolation as an explicit or implicit boundary scheme is emphasized. Specification of boundary data in the primitive variables and computation in terms of the conservative variables in the interior is discussed. Some numerical examples for the quasi-one-dimensional nozzle are given.

Yee, H. C.↗

A controlled variation scheme for convection treatment in pressure-based algorithm

Convection effect and source terms are two primary sources of difficulties in computing turbulent reacting flows typically encountered in propulsion devices. The present work intends to elucidate the individual as well as the collective roles of convection and source terms in the fluid flow equations, and to devise appropriate treatments and implementations to improve our current capability of predicting such flows. A controlled variation scheme (CVS) has been under development in the context of a pressure-based algorithm, which has the characteristics of adaptively regulating the amount of numerical diffusivity, relative to central difference scheme, according to the variation in local flow field. Both the basic concepts and a pragmatic assessment will be presented to highlight the status of this work.

Shyy, Wei↗

A New Low Dissipative High Order Schemes for MHD Equations

The goal of this talk is to extend our recently developed highly parallelizable nonlinear stable high order schemes for complex multiscale hydrodynamic applications to the viscous MHD equations. These schemes employed multiresolution wavelets as adaptive numerical dissipation controls to limit the amount and to aid the selection and/or blending of the appropriate types of dissipation to be used. The new scheme is formulated for both the conservative and non-conservative form of the MHD equations in curvilinear grids.

Yee, H. C.↗

Two-dimensional atmospheric transport and chemistry model - Numerical experiments with a new advection algorithm

Extensive testing of the advective scheme proposed by Prather (1986) has been carried out in support of the California Institute of Technology-Jet Propulsion Laboratory two-dimensional model of the middle atmosphere. The original scheme is generalized to include higher-order moments. In addition, it is shown how well the scheme works in the presence of chemistry as well as eddy diffusion. Six types of numerical experiments including simple clock motion and pure advection in two dimensions have been investigated in detail. By comparison with analytic solutions, it is shown that the new algorithm can faithfully preserve concentration profiles, has essentially no numerical diffusion, and is superior to a typical fourth-order finite difference scheme.

Shia, Run-Lie↗

Protection Scheme for Multiterminal HVDC System Based on Wavelet Transform Modulus Maxima

Fault protection is a major issue in the development of multiterminal high‐voltage direct current (HVDC) system based on voltage‐source converter. Owing to the rapid increase in fault current under DC faults, protection requires a fast fault detection method to isolate the fault current. However, the selectivity and reliability of the fast protection scheme should be primarily guaranteed. Therefore, a new protection strategy based on modulus maximum under wavelet transform is proposed in this paper. Fault detection and discrimination are achieved using single‐end signals of fault current. The external and internal faults can be discriminated via local information. The reliability and robustness of the proposed scheme against potential factors are verified. Numerous simulation cases are performed in PSCAD/EMTDC under different fault scenarios, and signal data are analyzed in MATLAB to demonstrate the effectiveness of the proposed fault protection scheme. © 2020 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Huai, Qing↗

Influence of boundary approximations and conditions on finite difference solutions

Numerical representations of boundary approximations and conditions for three problems are investigated to determine the resulting global accuracy of the steady state solution. Numerical accuracy with various boundary approximations is determined for quasi-one-dimensional inviscid flow in a duct with the interior grid points evaluated using the MacCormack scheme. When an extrapolation approximation with first order local truncation error is used, the global second order accuracy of the difference scheme can be destroyed. For one dimensional flow in a porous medium, an implicit midpoint difference scheme which is consistent with the boundary conditions is developed without the need of boundary approximations. A dissipative model problem is solved with the boundary conditions discretized with first and second order accuracy. The overall second order accuracy of the difference scheme is destroyed if first order numerical representation of one of the boundary conditions is used. With a boundary approximation, the second order global accuracy of the model problem is retained if either second order extrapolation or first order representation of the governing equation is used.

Blottner, F. G.↗

Computation of Feedback Aeroacoustic System by the CE/SE Method

It is well known that due to vortex shedding in high speed flow over cutouts, cavities, and gaps, intense noise may be generated. Strong tonal oscillations occur in a feedback cycle in which the vortices shed from the upstream edge of the cavity convect downstream and impinge on the cavity lip, generating acoustic waves that propagate upstream to excite new vortices. Numerical simulation of such a complicated process requires a scheme that can: (1) resolve acoustic waves with low dispersion and numerical dissipation, (2) handle nonlinear and discontinuous waves (e.g. shocks), and (3) have an effective (near field) nonreflecting boundary condition (NRBC). The new space time conservation element and solution element method, or CE/SE for short, is a numerical method that meets the above requirements.

Loh, Ching Y.↗

A multiscale phase field fracture approach based on the non-affine microsphere model for rubber-like materials

Rubber-like materials have a broad scope of applications due to their unique properties like high stretchability and increased toughness. Hence, computational models for simulating their fracture behavior are paramount for designing them against failures. In this study, the phase field fracture approach is integrated with a multiscale polymer model for predicting the fracture behavior in elastomers. At the microscale, damaged polymer chains are modeled to be made up of a number of elastic chain segments pinned together. Using the phase field approach, the damage in the chains is represented using a continuous variable. Both the bond stretch internal energy and the entropic free energy of the chain are assumed to drive the damage, and the advantages of this assumption are expounded. A framework for utilizing the non-affine microsphere model for damaged systems is proposed here by considering the minimization of a hypothetical undamaged free energy, ultimately connecting the chain stretch to the macroscale deformation gradient. At the macroscale, a thermodynamically consistent formulation is derived in which the total dissipation is assumed to be mainly due to the rupture of molecular bonds. Using a monolithic scheme, the proposed model is numerically implemented and the resulting three-dimensional simulation predictions are compared with existing experimental data. The capability of the model to qualitatively predict the propagation of complex crack paths and quantitatively estimate the overall fracture behavior is verified. Additionally, the effect of the length scale parameter on the predicted fracture behavior is studied for an inhomogeneous system.

97 MATHEMATICS AND COMPUTING↗

Data reduction considerations for the burning velocity of spherical constant volume flames of R32 (CH 2 F 2 ) with air

Here, the present work explores data reduction techniques for the measurement of the laminar burning velocities of R32(CH 2 F 2 )-air mixtures using a constant volume combustion device, in which the pressure-time history is the only measured parameter. To allow clear assessment of the accuracy of the data reduction methods, the pressure-time histories used for analysis are synthetically generated via a detailed numerical simulation employing full kinetics and with and without an optically-thin radiation model. Various data reduction models are employed, including a two-zone model and two multi-zone models, and these are compared with the results from the burning velocity obtained from the output of the numerical simulation. The data reduction schemes are shown to be accurate if the same radiation model is employed in the data reduction as was used in the flame simulation to generate the pressure trace used for post-processing. If the incorrect radiation model is employed, however, the errors can be quite large. The effects of stretch, radiation, and different data post-processing methodologies are explored and the errors quantified. Stretch is shown to be important for the early stages and the selected data range that is used for extrapolation has a significant effect on the extrapolated burning velocity. However, with an appropriate choice of data considered for extrapolation, the prediction of the unstretched burning velocity can be quite accurate.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Entropy stable modal discontinuous Galerkin schemes and wall boundary conditions for the compressible Navier-Stokes equations

Entropy stable schemes ensure that physically meaningful numerical solutions also satisfy a semi-discrete entropy inequality under appropriate boundary conditions. Here in this work, we describe a discretization of viscous terms in the compressible Navier-Stokes equations which enables a simple and explicit imposition of entropy stable no-slip and reflective (symmetry) wall boundary conditions for discontinuous Galerkin (DG) discretizations. Specifically, we derive methods for imposing adiabatic no-slip and reflective (symmetry) boundary conditions for modal entropy stable DG formulations which preserve a semi-discrete entropy inequality. Numerical results confirm the robustness and accuracy of the proposed approaches.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Calibration of parallel bond parameters in bonded particle models via physics-informed adaptive moment optimisation

This study proposes an automated calibration procedure for bond parameters in bonded discrete element modelling. By exploring the underlying physical correlations between microscopic parameters of bonds and macroscopic strength parameters of the continuum to be modelled, the microscopic shear strength and tensile strength are identified as independent variables for calibration purpose. Then a physics-informed iterative scheme is proposed to automatically approximate the bond parameters by viewing the micro-macro relation as an implicitly defined mathematical mapping function. As a result of highly non-convex features of this implicit mapping, the adaptive moment estimation (Adam), which is especially suitable for problems with noisy gradients, is adopted as the basic iterative scheme, in conjunction with other numerical techniques to approximately evaluate the partial derivatives involved. The whole procedure offers a simple and effective framework for bond parameter calibration. Finally, a numerical example of SiC ceramic is provided for validation. By compared with some existing calibration methods, the proposed method shows significant advantages in terms of calibration efficiency and accuracy.

42 ENGINEERING↗

A moment-conserving discontinuous Galerkin representation of the relativistic Maxwellian distribution

Kinetic simulations of relativistic gases and plasmas are critical for understanding diverse astrophysical and terrestrial systems, but the accurate construction of the relativistic Maxwellian, the Maxwell–Jüttner distribution, on a discrete simulation grid is challenging. Difficulties arise from the finite velocity bounds of the domain, which may not capture the entire distribution function, as well as errors introduced by projecting the function onto a discrete grid. Here, we present a novel scheme for iteratively correcting the moments of the projected distribution applicable to all grid-based discretizations of the relativistic kinetic equation. In addition, we describe how to compute the needed nonlinear quantities, such as Lorentz boost factors, in a discontinuous Galerkin scheme through a combination of numerical quadrature and weak operations. The resulting method accurately captures the distribution function and ensures that the moments match the desired values to machine precision.

astrophysical plasmas↗

Water Mass Transformation Budgets in Finite‐Volume Generalized Vertical Coordinate Ocean Models

Water Mass Transformation (WMT) theory provides conceptual tools that in principle enable innovative analyses of numerical ocean models; in practice, however, these methods can be challenging to implement and interpret, and therefore remain under-utilized. Our aim is to demonstrate the feasibility of diagnosing all terms in the water mass budget and to exemplify their usefulness for scientific inquiry and model development by quantitatively relating water mass changes, overturning circulations, boundary fluxes, and interior mixing. We begin with a pedagogical derivation of key results of classical WMT theory. We then describe best practices for diagnosing each of the water mass budget terms from the output of Finite-Volume Generalized Vertical Coordinate (FV-GVC) ocean models, including the identification of a non-negligible remainder term as the spurious numerical mixing due to advection scheme discretization errors. We illustrate key aspects of the methodology through the analysis of a polygonal region of the Greater Baltic Sea in a regional demonstration simulation using the Modular Ocean Model v6 (MOM6). We verify the convergence of our WMT diagnostics by brute-force, comparing time-averaged (“offline”) diagnostics on various vertical grids to timestep-averaged (“online”) diagnostics on the native model grid. Finally, we briefly describe a stack of xarray-enabled Python packages for evaluating WMT budgets in FV-GVC models (culminating in the new xwmb package), which is intended to be model-agnostic and available for community use and development.

54 ENVIRONMENTAL SCIENCES↗

Complex Dependence of Calcite Crack Kinetics on Salinity: The Role of DLVO and Hydration Forces

Abstract Subcritical crack growth (SCG) plays an important role in many geological processes such as delayed earth rupture and rock weathering. The complex dependency of SCG on the in‐crack fluid chemistry, however, is still poorly understood. In this study, we utilize the newly developed surface force‐based fracture theory (SFFT) to elucidate the relative contributions of surface forces and solute transport to the crack growth kinetics of calcite in NaCl solutions. Expanding on Barenblatt's cohesive crack model, SFFT introduces an effective stress intensity at the crack tip that encompasses all the relevant intermolecular forces across the crack in addition to the external far‐field stresses. The nonlinear system of equations portraying the crack opening profile, the solute distribution in a propagating crack, and the crack growth velocity are numerically solved via an implicit scheme. After carefully calibrating the model for calcite‐water systems, the SFFT is used to predict the SCG response of calcite at different NaCl concentrations, based on various hypotheses. These predictions are then compared to existing SCG data from the literature. We demonstrate that the experimentally observed variation of SCG rate with NaCl concentration cannot be explained solely by DLVO forces (electrostatic and Van der Waals interactions). This can be remediated by introducing an exponentially decaying hydration force with a nonlinear, nonmonotonic dependence on NaCl concentration. Furthermore, we demonstrate that accounting for both diffusive and advective transport of ions is important in explaining the absence of a stage‐II SCG response for calcite in electrolyte solutions. Plain Language Summary Subcritical crack growth (SCG) refers to the slow propagation of cracks in materials under a stress below the threshold for catastrophic failure. SCG is a key process in many geological events, for example, delayed earth ruptures and rock weathering. New initiatives such as underground CO 2 and H 2 storage in carbonate reservoirs further call for better understanding of SCG in carbonate minerals subjected to varying fluid chemistry. This study examines the SCG of calcite, a key mineral found in carbonate rocks, intergranular cement in sandstones, and filling material in mineral veins and faults, determining their deformation and strength. A mathematical model is developed to describe how the crack opens and propagates, how solutes (like salts) distribute within the crack, and how the crack surfaces interact with each other. We used the model to predict calcite SCG in water at different salt concentrations and compared it with experimental data. Our results revealed that the hydration force is the dominating factor in determining the complex, non‐linear dependency of SCG on salinity. We also found that both the movement of ions by diffusion and by bulk water flow are crucial for explaining the SCG rates, especially when the cracks grow quickly. Key Points Surface Force‐Based Fracture Theory predicts the complex subcritical crack growth patterns of calcite crystals immersed in NaCl solutions Results highlight the dominant role of hydration forces in altering the fracture behavior of calcite compared to VdW and electric double‐layer forces Advective solute transport explains the absence of stages‐II and ‐III subcritical crack growth responses in solid‐liquid systems

DLVO↗