Extra Source Terms Research Articles

Approximate Riemann solvers for the cosmic ray magnetohydrodynamical equations

We analyze the cosmic-ray magnetohydrodynamic (CR MHD) equations to improve the numerical simulations. We propose to solve them in the fully conservation form, which is equivalent to the conventional CR MHD equations. In the fully conservation form, the CR energy equation is replaced with the CR "number" conservation, where the CR number density is defined as the three fourths power of the CR energy density. The former contains an extra source term, while latter does not. An approximate Riemann solver is derived from the CR MHD equations in the fully conservation form. Based on the analysis, we propose a numerical scheme of which solutions satisfy the Rankine-Hugoniot relation at any shock. We demonstrate that it reproduces the Riemann solution derived by Pfrommer et al. (2006) for a 1D CR hydrodynamic shock tube problem. We compare the solution with those obtained by solving the CR energy equation. The latter solutions deviate from the Riemann solution seriously, when the CR pressure dominates over the gas pressure in the post-shocked gas. The former solutions converge to the Riemann solution and are of the second order accuracy in space and time. Our numerical examples include an expansion of high pressure sphere in an magnetized medium. Fast and slow shocks are sharply resolved in the example. We also discuss possible extension of the CR MHD equations to evaluate the average CR energy.

A conservative semi-Lagrangian HWENO method for the Vlasov equation

In this paper, we propose a high order conservative semi-Lagrangian (SL) finite difference Hermite weighted essentially non-oscillatory (HWENO) method for the Vlasov equation based on dimensional splitting. HWENO was first proposed for solving nonlinear hyperbolic problems by evolving both function values and its first derivative values (Qiu and Shu (2004) [23]). The major advantage of HWENO, compared with the original WENO, lies in its compactness in reconstruction stencils.There are several new ingredients in this paper. Firstly we propose a mass-conservative SL HWENO scheme for a 1-D equation by working with a flux-difference form, following the work of Qiu and Christlieb (2010) [25]. Secondly, we propose a proper splitting for equations of partial derivatives in HWENO framework to ensure local mass conservation. The proposed fifth order SL HWENO scheme with dimensional splitting has been tested to work well in capturing filamentation structures without oscillations when the time step size is within the Eulerian CFL constraint. However, when the time stepping size becomes larger, numerical oscillations are observed for the ‘mass conservative’ dimensional splitting HWENO scheme, as there are extra source terms in equations of partial derivatives. In this case, we introduce WENO limiters to control oscillations. Classical numerical examples on linear passive transport problems, as well as the nonlinear Vlasov–Poisson system, have been tested to demonstrate the performance of the proposed scheme.

Two-dimensional shallow water equations with porosity and their numerical scheme on unstructured grids

In this study, porosity was introduced into two-dimensional shallow water equations to reflect the effects of obstructions, leading to the modification of the expressions for the flux and source terms. An extra porosity source term appears in the momentum equation. The numerical model of the shallow water equations with porosity is presented with the finite volume method on unstructured grids and the modified Roe-type approximate Riemann solver. The source terms of the bed slope and porosity are both decomposed in the characteristic direction so that the numerical scheme can exactly satisfy the conservative property. The present model was tested with a dam break with discontinuous porosity and a flash flood in the Toce River Valley. The results show that the model can simulate the influence of obstructions, and the numerical scheme can maintain the flux balance at the interface with high efficiency and resolution.

TESTING MONDian DARK MATTER WITH GALACTIC ROTATION CURVES

MONDian dark matter (MDM) is a new form of dark matter quantum that naturally accounts for Milgrom's scaling, usually associated with modified Newtonian dynamics (MOND), and theoretically behaves like cold dark matter (CDM) at cluster and cosmic scales. In this paper, we provide the first observational test of MDM by fitting rotation curves to a sample of 30 local spiral galaxies (z approximately 0.003). For comparison, we also fit the galactic rotation curves using MOND, and CDM. We find that all three models fit the data well. The rotation curves predicted by MDM and MOND are virtually indistinguishable over the range of observed radii (~1 to 30 kpc). The best-fit MDM and CDM density profiles are compared. We also compare with MDM the dark matter density profiles arising from MOND if Milgrom's formula is interpreted as Newtonian gravity with an extra source term instead of as a modification of inertia. We find that discrepancies between MDM and MOND will occur near the center of a typical spiral galaxy. In these regions, instead of continuing to rise sharply, the MDM mass density turns over and drops as we approach the center of the galaxy. Our results show that MDM, which restricts the nature of the dark matter quantum by accounting for Milgrom's scaling, accurately reproduces observed rotation curves.

Exponential Runge-Kutta Methods for the Multispecies Boltzmann Equation

AbstractThis paper generalizes the exponential Runge-Kutta asymptotic preserving (AP) method developed in [G. Dimarco and L. Pareschi,SIAM Numer. Anal., 49 (2011), pp. 2057-2077] to compute the multi-species Boltzmann equation. Compared to the single species Boltzmann equation that the method was originally applied on, this set of equation presents a new difficulty that comes from the lack of local conservation laws due to the interaction between different species. Hence extra stiff nonlinear source terms need to be treated properly to maintain the accuracy and the AP property. The method we propose does not contain any nonlinear nonlocal implicit solver, and can capture the hydrodynamic limit with time step and mesh size independent of the Knudsen number. We prove the positivity and strong AP properties of the scheme, which are verified by two numerical examples.

Numerical Simulation of Gas and Liquid Two-Phase Flow in Gas-Stirred Systems Based on Euler–Euler Approach

Based on the Euler–Euler approach, a mathematical model is established to describe gas and liquid two-phase flow in the gas-stirred system for steelmaking, and the influences of the interphase force including turbulent dispersion force, drag force, and lift force are investigated. The modified k–e model with extra source terms to account for the bubble-induced turbulence is adopted to model the turbulence in the system, and the simulation results of gas volume fraction, liquid velocity, and turbulent kinetic energy are compared with the measured data. The results show that the turbulent dispersion force dominates the bubbly plume shape and is responsible for successful prediction of the gas volume fraction. The bubble-induced turbulence has a significant influence on the liquid turbulence, and the conversion coefficient Cb, which denotes the fraction of bubble-induced energy converted into liquid turbulence, should be in the range of 0.8 and 0.9. The drag force also strongly influences the bubbly plume dynamics, and the coefficient model proposed by Kolev performs the best for determining the drag force; however, the lift force and bubble diameter do not have much effect on the current bubbly plume system. For different gas flow rates, the current Euler–Euler approach predictions are more consistent with the measured data than the Euler–Lagrange approach and the early Euler–Euler model.

Numerical Simulations of Inclusion Behavior in Gas-Stirred Ladles

A computation fluid dynamics–population balance model (CFD–PBM) coupled model has been proposed to investigate the bubbly plume flow and inclusion behavior including growth, size distribution, and removal in gas-stirred ladles, and some new and important phenomena and mechanisms were presented. For the bubbly plume flow, a modified k-e model with extra source terms to account for the bubble-induced turbulence was adopted to model the turbulence, and the bubble turbulent dispersion force was taken into account to predict gas volume fraction distribution in the turbulent gas-stirred system. For inclusion behavior, the phenomena of inclusions turbulent random motion, bubbles wake, and slag eye forming on the molten steel surface were considered. In addition, the multiple mechanisms both that promote inclusion growth due to inclusion–inclusion collision caused by turbulent random motion, shear rate in turbulent eddy, and difference inclusion Stokes velocities, and the mechanisms that promote inclusion removal due to bubble-inclusion turbulence random collision, bubble-inclusion turbulent shear collision, bubble-inclusion buoyancy collision, inclusion own floatation near slag–metal interface, bubble wake capture, and wall adhesion were investigated. The importance of different mechanisms and total inclusion removal ratio under different conditions, and the distribution of inclusion number densities in ladle, were discussed and clarified. The results show that at a low gas flow rate, the inclusion growth is mainly attributed to both turbulent shear collision and Stokes collision, which is notably affected by the Stokes collision efficiency, and the inclusion removal is mainly attributed to the bubble-inclusion buoyancy collision and inclusion own floatation near slag–metal interface. At a higher gas flow rate, the inclusions appear as turbulence random motion in bubbly plume zone, and both the inclusion–inclusion and inclusion-bubble turbulent random collisions become important for inclusion growth and removal. With the increase of the gas flow rate, the total removal ratio increases, but when the gas flow rate exceeds 200 NL/min in 150-ton ladle, the total removal ration almost does not change. For the larger size inclusions, the number density in bubbly plume zone is less than that in the sidewall recirculation zones, but for the small size inclusions, the distribution of number density shows the opposite trend.

Galactic rotation curves in brane world models

In the braneworld scenario the four dimensional effective Einstein equation has extra source terms, which arise from the embedding of the 3-brane in the bulk. These non-local effects, generated by the free gravitational field of the bulk, may provide an explanation for the dynamics of the neutral hydrogen clouds at large distances from the galactic center, which is usually explained by postulating the existence of the dark matter. In the present paper we consider the asymptotic behavior of the galactic rotation curves in the brane world models, and we compare the theoretical results with observations of both High Surface Brightness and Low Surface Brightness galaxies. For the chosen sample of galaxies we determine first the baryonic parameters by fitting the photometric data to the adopted galaxy model; then we test the hypothesis of the Weyl fluid acting as dark matter on the chosen sample of spiral galaxies by fitting the tangential velocity equation of the combined baryonic-Weyl model to the rotation curves. We give an analytical expression for the rotational velocity of a test particle on a stable circular orbit in the exterior region to a galaxy, with Weyl fluid contributions included. The model parameter ranges for which the $\chi^2$ test provides agreement (within 1$\sigma$ confidence level) with observations on the velocity fields of the chosen galaxy sample are then determined. There is a good agreement between the theoretical predictions and observations, showing that extra-dimensional models can be effectively used as a viable alternative to the standard dark matter paradigm.

MODELING BIOGEOMORPHOLOGICAL INTERACTIONS IN UNDERWATER NOURISHMENTS

To prevent sandy coastlines from further erosion, nourishments are executed in which sand is usually put underwater at the foreshore. Waves and currents transport the sand on the beach and in this way stabilize the coastal profile. Little is known about the interactions of these so-called shoreface nourishments with the benthic populations inhabiting the coastal strip. Benthos is affected by the nourishments, but benthic populations could in turn affect the morphological evolution of the nourished coast. Monitoring has shown that the benthic community will mainly recovery after ca. 1 year. However, the impact of benthos on the sediment dynamics and hydrodynamics is unknown. In this paper we focus on tube building worms, which have a large abundance in the foreshore, live in patches of several square meters in diameter and protrude several centimeters from the sediment in the water column. Tube building worms are included in a numerical modeling tool (Delft3D), by explicitly accounting for the influence of cylindrical structures on drag and turbulence by an extra source term of friction force in the momentum equation and an extra source term of Total Kinetic Energy (TKE) and turbulent energy dissipation in the k-ε equations respectively. The model is validated against field and flume experiments and it shows a significant influence on flow velocities near the bed, bed shear stress and bed-load transport rates. Moreover, model results reveal that tube building worms are able to stabilize nourishments and slow down the migration of the outer breaker bar. Present model explorations indicate that future research should focus on the measurement of the patchy distribution of bio-engineers in the foreshore and their impact on the sediment dynamics and hydrodynamics. Such knowledge will enable process based modeling of the spatial and temporal variation in biological activity on the morphological development of the coastal profile and also it will lead to validation of the proposed model with field measurements.

Stringy space–time foam, Finsler-like metrics and dark matter relics

We discuss modifications of the thermal dark matter (DM) relic abundances in stringy cosmologies with D-particle space–time foamy backgrounds. As a result of back-reaction of massive DM on the background space–time, owing to its interaction with D-particle defects in the foam, quantum fluctuations are induced in the space–time metric. We demonstrate that these lead to the presence of extra source terms in the Boltzmann equation used to determine the thermal dark matter relic abundances. The source terms are determined by the specific form of the induced metric deformations; the latter depend on the momentum transfer of the DM particle during its interactions with the D-particle defects and so are akin to Finsler metrics. In the case of low string scales, arising from large extra dimensions, our results may have phenomenological implications for the search of viable supersymmetric models.

A Micro-Macro Decomposition-Based Asymptotic-Preserving Scheme for the Multispecies Boltzmann Equation

In this paper we extend the micro-macro decomposition-based asymptotic-preserving scheme developed by Bennoune, Lemou, and Mieussens [J. Comput. Phys., 227 (2008), pp. 3781-3803] for the single species Boltzmann equation to the multispecies problems. An asymptotic-preserving scheme for the kinetic equation is very efficient in the fluid regime where the Knudsen number is small and the collision term becomes stiff. It allows a coarse (independent of the Knudsen number) mesh size and a large time step in the fluid regime. The difficulty associated with multispecies problems is that there are no local conservation laws for each species, resulting in extra stiff nonlinear source terms that need to be discretized properly in order to (1) avoid Newton-type solvers for nonlinear algebraic systems, and (2) to be asymptotic-preserving. We show that these extra nonlinear source terms can be solved using only linear system solvers, and the scheme preserves the correct Euler and Navier-Stokes limits. Numerical examples are used to demonstrate the efficiency and applicability of the schemes for both Euler and Navier-Stokes regimes.

Superposition of local zoom models and simultaneous calibration for 1D–2D shallow water flows

We address the problem of coupling 2D shallow water equations with 1D shallow water equations (St-Venant equations), as applied to river-floodplain flows. Mathematical coupling conditions are derived classicaly from the 3D Navier–Stokes equations by integrating over the vertical wet section, when overflowing occurs. It leads to extra source terms in the 1D equations. Next we assume to be in a variational data assimilation context, then the optimal control process allows to couple both models and assimilate data simultaneously (Joint Assimilation Coupling algorithms). Two different versions of JAC algorithms are presented and compared. In a numerical test case, we superimpose the local 2D model on the 1D global model. The results show the efficiency of the present simultaneous superposition–assimilation approach.

An approximate Riemann solver for sensitivity equations with discontinuous solutions

The sensitivity of a model output (called a variable) to a parameter can be defined as the partial derivative of the variable with respect to the parameter. When the governing equations are not differentiable with respect to this parameter, problems arise in the numerical solution of the sensitivity equations, such as locally infinite values or instability. An approximate Riemann solver is thus proposed for direct sensitivity calculation for hyperbolic systems of conservation laws in the presence of discontinuous solutions. The proposed approach uses an extra source term in the form of a Dirac function to restore sensitivity balance across the shocks. It is valid for systems such as the Euler equations for gas dynamics or the shallow water equations for free surface flow. The method is first detailed and its application to the shallow water equations is proposed, with some test cases such as dike- or dam-break problems with or without source terms. An application to a two-dimensional flow problem illustrates the superiority of direct sensitivity calculation over the classical empirical approach.

One-equation sub-grid scale (SGS) modelling for Euler–Euler large eddy simulation (EELES) of dispersed bubbly flow

In this work, we have presented a one-equation model for sub-grid scale (SGS) kinetic energy and applied it for an Euler–Euler large eddy simulation (EELES) of a bubble column reactor. The one-equation model for SGS kinetic energy shows improved predictions over the state-of-the-art dynamic procedure. With grid refinement, the amount of modelled SGS turbulent kinetic energy diminishes, as one would expect. Bubble induced turbulence (BIT) at the SGS level was modelled with two approaches. In the first approach an algebraic model was used, while in the other approach extra source terms were added in the transport equation for SGS kinetic energy. It was found that the latter approach improved the quantitative prediction of the turbulent kinetic energy. To the best of authors knowledge, this is the first use of a transport equation for SGS kinetic energy in bubbly flows.

CFD simulation of large-scale bubble plumes: Comparisons against experiments

Computational fluid dynamics (CFD) simulations have been carried out to simulate a turbulent, bubble plume in a liquid pool. A two-fluid enhanced k– ɛ model has been used, with the extra source terms introduced to account for the interaction between the bubbles and the liquid and transient calculations have been performed to study the plume growth, the acceleration of the liquid due to viscous drag, and the approach to steady-state conditions. In order to obtain correct spreading of the plume observed experimentally, it was observed that interfacial forces like lift and turbulent dispersion plays important role. The sensitivity analysis for drag coefficient and two different turbulent dispersion forces is presented. The development of the flow variables has been compared with the experimental data. From the CFD predictions obtained in the present work, it can be concluded that a two-dimensional (2D) axisymmetric assumption in this case is justified, with 2D model predictions in good agreement with the experimental data and those of a three-dimensional (3D) model, except for the shear stresses and turbulent kinetic energy. In general, quantitative comparison with the experimental data has revealed that, by applying proper models of inter-phase momentum transfer, and performing simulations based on the two-fluid model, satisfactory predictions of mean flow quantities can be obtained for this application away from the injector.

Evaluation of a Modified Reynolds Stress Model for Turbulent Dispersed Two-Phase Flows Including Two-Way Coupling

A modified Reynolds stress turbulence model for the pressure rate of strain can be derived for dispersed two-phase flows taking into account gas-particle interaction. The transport equations for the Reynolds stresses as well as the equation for the fluctuating pressure can be derived starting from the multiphase Navier–Stokes equations. The unknown pressure rate of strain correlation in the Reynolds stress equations is then modelled by considering the multiphase equation for the fluctuating pressure. This leads to a multiphase pressure rate of strain model. The extra particle interaction source terms occurring in the model for the pressure rate of strain can be constructed easily, with no noticeable extra computational cost. Eulerian–Lagrangian simulation results of a turbulent dispersed two-phase jet are presented to show the differences in results with and without the new two-way coupling terms.

Large eddy simulation of a bubble column using dynamic sub-grid scale model

Euler–Euler simulations of the gas–liquid flow in a square cross-sectioned bubble column with LES (two sub-grid scale models) and the k–e model are presented. The sub-grid scale modeling is based on the Smagorinsky kernel in both its original form and the dynamic procedure of Germano. The attempt has been made to assess the performance of these two sub-grid scale models. The Smagorinsky model with model constant Cs = 0.12 performs quite well, and gives results almost identical to those given by the dynamic procedure of Germano. The SGS models are modified to account for bubble induced turbulence (Sato model) and it is observed that it does not change the results much. Predictions are also compared with the available experimental data. All the non-drag forces (turbulent dispersion force (only for RANS), virtual mass force, lift force) and drag force were incorporated in the model. An extended k–e turbulence model has been used with extra source terms introduced to account for the interaction between the bubbles and the liquid. Though both LES models showed agreement in predictions, the Germano model still can be used to have estimates of Cs value which are not known a priori. Moreover, if objective is to understand the steady and time averaged features, RANS can also perform well. © 2007 Elsevier B.V. All rights reserved.

A Low-Reynolds Number Explicit Algebraic Stress Model

A low-Reynolds number extension of the explicit algebraic stress model, developed by Gatski and Speziale (GS) is proposed. The turbulence anisotropy Πb and production to dissipation ratio P∕ϵ are modeled that recover the established equilibrium values for the homogeneous shear flows. The devised (Πb, P∕ϵ) combined with the model coefficients prevent the occurrence of nonphysical turbulence intensities in the context of a mild departure from equilibrium, and facilitate an avoidance of numerical instabilities, involved in the original GS model. A new near-wall damping function fμ in the eddy viscosity relation is introduced. To enhance dissipation in near-wall regions, the model constants Cϵ(1,2) are modified and an extra positive source term is included in the dissipation equation. A realizable time scale is incorporated to remove the wall singularity. The turbulent Prandtl numbers σ(k,ϵ) are modeled to provide substantial turbulent diffusion in near-wall regions. The model is validated against a few flow cases, yielding predictions in good agreement with the direct numerical simulation and experimental data.

Numerical and Experimental Investigation of Liquid−Liquid Two-Phase Flow in Stirred Tanks

The experimental data on the holdup of the dispersed phase in a Rushton impeller agitated stirred tank are presented. Experimental measurement is performed utilizing the sample withdrawal method to obtain the local dispersed-phase holdup in a laboratory-scale stirred tank under a variety of operating conditions. Three-dimensional turbulent two-phase liquid−liquid flow in the stirred tank is also numerically simulated by solving the Reynolds-averaged Navier−Stokes equations of two phases formulated by the two-fluid model. The turbulence effect is formulated using a simple two-phase extension of the well-known k−ε turbulence model by adding an extra source term generated from the presence of the dispersed phase in the turbulent kinetic energy transport equation of the continuous phase. A modified “inner−outer” iterative procedure is employed to model the interaction of the rotating impeller with the wall baffles. The model-predicted mean velocity, turbulence characteristics of the continuous phase, and holdup profiles of the dispersed phase are compared against the published experimental data and the present measurements to validate the computational procedure, and good agreement is found up to a rather high overall dispersed-phase holdup case (30 vol %).

Modelling the Flow Around Airfoils Equipped with Vortex Generators Using a Modified 2D Navier–Stokes Solver

Vortex generators (VGs) are commonly used for trimming the aerodynamic and aeroelastic performance of wind turbine blades by delaying flow separation. There is therefore a need for the development of reliable, still computationally affordable, models for blade designers to use to predict and enhance the aerodynamic characteristics of airfoils equipped with VGs. Such a model is proposed in the present paper, addressing in particular near-stall and post-stall airfoil performance. Starting from the three-dimensional Navier–Stokes equations that essentially describe the complex flow around a blade/VG configuration, a spanwise averaging procedure is applied, resulting in an equivalent set of two-dimensional equations, enriched with extra source terms. These terms are modelled using elementary vortex flow theory. In turbulent flows, the production term of the turbulent kinetic energy is also augmented by the vorticity induced by the VG. The model is evaluated by studying the flow past a blade section with and without VGs. An analysis of the performance of nine alternative VG configurations is also presented to demonstrate the sensitivity of the airfoil polars to the VG geometric parameters.