Total Variation Diminishing Research Articles

Development and validation of the hybrid code for numerical simulation of detonations

The present paper describes a numerical code for simulation of detonation flows on hybrid computational clusters (CPU/GPU). The code can solve 1D, 2D and 3D unsteady Euler equations for multispecies chemically reacting flows using high-order shock-capturing TVD schemes and a finite-rate chemistry solver. The implementation is based on the OpenMP, MPI and CUDA technologies allowing for both flexibility and computational efficiency of the code. The program is verified on the analytical Zeldovich-von Neumann-Doering solution for the 1D detonation wave in H2/O2 mixture. Some examples of 2D computations are also given.

The model of transfer of atmosphere pollution in roadside vegetation near the highway

The work is devoted to the development of a model and investigation of the process of pollution transfer in the ground layer of a general-purpose motor road with plantations and terrain. As the basis of the model, a long section of a road with a roadside terrain and plantings is considered. Along the axial line of the road, a linear source is assumed, which releases constant impurity consumption along the road. The impurity is transported in the surface layer of the highway in the conditions of the lateral wind curve, alternating relief of the adjacent terrain and roadside plantations of different densities. The model is based on a grid description of the three-dimensional region under consideration. The motion of air continuous medium is described by the Reynolds averaged Navier-Stokes equations. The air continuous medium is assumed to be incompressible, multicomponent, and chemically nonreactive. To model the turbulent transport effects, two-parameter differential turbulence model with wall functions is used as the basis. Simulation of blocking of leaf space and tree branches is performed on the basis of blocking with a porous medium. The Navier-Stokes equations, as well as transport equations for the parameters of the turbulence model, contain source terms in the right-hand parts in the form of a power-law dependence of the velocity modulus in the porosity regions. This model interprets the influence of vegetation as a homogeneous isotropic resistance of a low-inertial volume; the additional terms in the equations of the turbulence model increase the production of turbulence. The study was carried out using the author's software package MTFS®, in which the basic implicit algorithm is provided by the method of variable directions and TVD scheme of 2/3-th order accuracy. The calculations were performed by the method of establishing the flow from a retarded state to a developed steady-state flow in the middle. The flow outside the calculated region was assumed to be completely turbulent, which was determined by the input boundary conditions. The input wind speed profile was used with allowance for the boundary layer. Simulation of impurity distribution for a long rectilinear section was performed and a satisfactory agreement with experimental data was obtained. A study was carried out in areas with a substantially spatial impurity transport structure. It is shown that the slope of the current lines with respect to the axial line of the road facilitates the demolition of the impurity along the road, while the intensity of the vorticity between the plantations decreases in comparison with the two-dimensional model.Кey words: highway; roadside vegetation; boundary layer; transfer of pollutants.

Modeling and simulation of urban air pollution from the dispersion of vehicle exhaust: A continuum modeling approach

Air pollution has become a serious issue over the past few decades, and the transport sector is an important emission source. In this study, we model and simulate the dispersion of vehicle exhaust in a hypothetical city with a single central business district (CBD) in a complete day, deriving an average daily pollutant concentration, then use the distribution of wind direction over a year, to compute the distribution of average pollutant concentration in the city. All vehicles are assumed to be continuously distributed over the whole city, and the road network is relatively dense and can be approximated as a continuum. The model of Huang et al (2009, Transportation Research Part B, 43(1): 127–141) is used to describe the traffic flow that satisfies the reactive dynamic user equilibrium principle, and the pollution dispersion model is governed by the advection-diffusion equation. The complete model is composed of a coupled system of a conservation law, an eikonal equation and an advection-diffusion equation. The problem is solved by the efficient fifth-order weighted essentially non-oscillatory scheme for the conservation equation and the advection-diffusion equation, and the fast sweeping method for the eikonal equation with third order total variation diminishing (TVD) Runge-Kutta time discretization. The numerical results show a reasonable temporal and spatial distribution of vehicle density and pollution concentration.

Hybrid lattice Boltzmann-TVD simulation of natural convection of nanofluids in a partially heated square cavity using Buongiorno’s model

In this paper, the Buongiorno’s two-phase model is adopted to study natural convection in a partially heated enclosure filled with nanofluids having temperature-dependent properties. In order to solve the governing equations, a hybrid lattice Boltzmann (LB) and total variation diminishing (TVD) scheme is proposed, in which the traditional LB method is employed to handle the flow and temperature fields, while the volume fraction equation is solved by a TVD method due to the fact that the corresponding equation is a type of convection-dominated equation. Additionally, to improve the computational efficiency, the proposed algorithm is executed on the Graphics Processing Unit (GPU) by using “Compute Unified Device Architecture (CUDA)” programming. The effect of several parameters, such as Rayleigh number, nanoparticle diameter, temperature difference between the sidewalls, heating location and heater length on heat transfer rate and nanoparticle distribution are analyzed. It is observed that at low Rayleigh numbers, the heat transfer enhancement increases in nanoparticle volume fraction, while at high Rayleigh numbers, there exists an optimal volume fraction at which the heat transfer performance has a peak. Moreover, the average Nusselt number and the heat transfer enhancement are found to decrease with increasing heater length.

Performance of various shock-capturing-type reconstruction schemes in the Boussinesq wave model, FUNWAVE-TVD

In this study, we assessed the performance of five shock-capturing-type reconstruction schemes in modeling shallow water waves implemented in the public-domain Boussinesq model, FUNWAVE-TVD. The five shock-capturing schemes include two fourth-order Monotonic Upwind Scheme for Conservation Laws - Total Variation Diminishing (MUSCL-TVD) schemes with the Minimod and van-Leer limiters, respectively, a second-order MUSCL-TVD scheme with the van-Leer limiter, a fifth-order Weighted Essentially Non-Oscillatory (WENO) scheme, and a third-order Multi-dimensional Limiting Process (MLP) scheme. Numerical simulations on solitary waves and laboratory measurements showed that for one-dimensional simulations all five shock-capturing schemes maintain non-oscillatory properties, but exhibit different numerical dissipations. Numerical dissipation can be affected by both the order of shock-capturing schemes and choice of limiters. The van-Leer limiter is more dissipative than Minimod when used in fourth-order schemes, which is different from what would be inferred from the linear analysis for the second order. For two-dimensional problems, the shock-capturing schemes can generate either fewer or more high-frequency oscillations, depending on the applied Courant–Friedrichs–Lewy (CFL) number. The most stable results occur in the second-order MUSCL-TVD scheme with the van-Leer limiter, which has large numerical dissipation, and the third-order MLP scheme, which has the ability to control the oscillations in multi-dimensional simulations. The wavelet analysis showed that high-frequency oscillations usually occur at later time and thus affect long-time simulation. Decreasing the CFL number is an effective way of reducing high-frequency oscillations in two-dimensional applications. Models with different shock-capturing schemes can predict similar strength and patterns of wave-induced currents in the breaking wave application. However, the analysis based on the vorticity or enstrophy (integral of the square of the vorticity) calculations show that the schemes with higher numerical dissipation have more significant damping effect on the vorticity field, which may affect the model accuracy in predicting wave-current interaction in a long time simulation.

River Flow Dynamics with Two-dimensional Shallow-water Equations

This study proposes a numerical model to simulate unsteady flows in rivers and associated riverine processes. The mathematical model is based on Saint-Venant equations. The governing equations are descritized by Godunov-type finite volume method. Use of total variation diminishing (TVD) scheme and MUSCL-Hancock technique satisfies second-order temporal and spatial accuracy requirements, respectively. Hydrodynamic flux components associated are computed by Harten-Lax-van Leer (HLL) method. A modified method is introduced for approximating flow features along the shear wave. The proposed scheme is used to model an idealized dam-break case, and a case of dam-break in a channel with a 90° bend which results in bore formation and complex two-dimensional flow; the predictions show excellent agreement with available experimental results and analytical solutions.

Comparison of Numerical Solutions by TVD Schemes in Simulations of Irregular Waves Propagating over a Submerged Shoal Using FUNWAVE-TVD Numerical Model

Comparison of Numerical Solutions by TVD Schemes in Simulations of Irregular Waves Propagating over a Submerged Shoal Using FUNWAVE-TVD Numerical Model

Numerical Modeling and Validation of a Novel 2D Compositional Flooding Simulator Using a Second-Order TVD Scheme

The aim of this paper is to present the latter and develop a numerical simulator aimed at solving a 2D domain porous medium, using the compositional approach to simulate chemical flooding processes. The simulator consists in a two-phase, multicomponent system solved by the IMplicit in Pressure, Explicit in Concentration (IMPEC) approach, which can be operated under an iterative/non-iterative condition on each time-step. The discretization of the differential equations is done using a fully second order of accuracy, along with a Total Variation Diminishing (TVD) scheme with a flux limiter function. This allowed reducing the artificial diffusion and dispersion on the transport equation, improving the chemical species front tracking, decreasing the numerical influence on the recovery results. The new model was validated against both commercial and academic simulators and moreover, the robustness and stability were also tested, showing that the iterative IMPEC is fully stable, behaving as an implicit numerical scheme. The non-iterative IMPEC is conditionally stable, with a critical time-step above which numerical spurious oscillations begin to appear until the system numerically crashes. The results showed a good correspondence in different grid sizes, being largely affected by the time-step, with caused a decrease in the recovery efficiency in the iterative scheme, and the occurrence of numerical oscillations in the non-iterative one. Numerically speaking, the second-order scheme using a flux splitting TVD discretization proved to be a good approach for compositional reservoir simulation, decreasing the influence of numerical truncation errors on the results when compared to traditional, first-order linear schemes. Along with these studies, secondary recoveries in constant and random permeability fields are simulated before employing them in tertiary recovery processes.

High-order upwind and non-oscillatory approach for steady state diffusion, advection–diffusion and application to magnetized electrons

Steady state simulations of magnetized electron fluid equations with strong anisotropic diffusion based on the first-order hyperbolic approach is carried out using cell-centered higher order upwind schemes, linear and weighted essentially non-oscillatory (WENO). Along with the magnetized electrons, the diffusion equation is also simulated to demonstrate the implementation and design order of the accuracy of the approach due to their similar upwind structure. We show the adequacy of linear upwind schemes for diffusion equation and the use of shock-capturing scheme like WENO does not have any adverse effect on the solution, unlike the total-variation diminishing (TVD) methods. We further extended the approach to advection–diffusion equation, and appropriate boundary conditions have obtained a consistent design accuracy of the third and fifth order. We implemented the WENO approach to advection–diffusion equation by using the split hyperbolic method to demonstrate the advantage of non-oscillatory schemes to capture sharp gradients in boundary layer type problems without spurious oscillations. Finally, numerical results for magnetized electrons simulations indicate that with increasing strength of magnetic confinement it is possible to capture sharp gradients without oscillations by WENO scheme.

Modeling wave effects on storm surge and coastal inundation

We present a parametric study of surface wave effects on storm surge and coastal inundation. Hurricane wind forcing terms, atmosphere pressure gradient terms, and radiation open boundary conditions are implemented into an existing quasi-3D nearshore model, NearCoM-TVD, which uses a shock-capturing TVD scheme and can better simulate the wetting-drying process during inundation than a conventional finite difference model. Systematic numerical experiments are carried out in an idealized continental shelf-beach-land system to identify the role of waves in modeling storm surge and inundation under different storm characteristics. Four storm parameters, including storm intensity, storm size, translation speed and incident angle are investigated. Modeling results reveal that the presence of waves can increase the maximum storm surge heights significantly through wave setup, and the contribution of waves varies considerably depending on the storm characteristics. In addition to direct wave forcing, the wave-enhanced bottom stress in the surfzone also promotes higher wave setup and hence higher surge heights. Waves also have a major influence on the maximum inland inundation distance, however, we find that waves do not always favor more inundation as expected. For storms traveling at very slow translation speeds or nearly parallel to the coast, waves surprisingly exert a negative effect on the maximum inundation distance. We argue that wave-enhanced bottom stress combined with longer forcing duration in such storms counteract the positive effect of wave setup, and thus less inundation is predicted in wave-current coupled simulations. Our results highlight the necessity of fully-incorporating wave effects in storm surge and inundation modeling.

Multiscale RBF-based central high resolution schemes for simulation of generalized thermoelasticity problems

In this study, average-interpolating radial basis functions (RBFs) are successfully integrated with central high-resolution schemes to achieve a higher-order central method. This proposed method is used for simulation of generalized coupled thermoelasticity problems including shock (singular) waves in their solutions. The thermoelasticity problems include the LS (systems with one relaxation parameter) and GN (systems without energy dissipation) theories with constant and variable coefficients. In the central high resolution formulation, RBFs lead to a reconstruction with the optimum recovery with minimized roughness on each cell: this is essential for oscillation-free reconstructions. To guarantee monotonic reconstructions at cell-edges, the nonlinear scaling limiters are used. Such reconstructions, finally, lead to the total variation bounded (TVB) feature. As RBFs work satisfactory on non-uniform cells/grids, the proposed central scheme can handle adapted cells/grids. To have cost effective and accurate simulations, the multiresolution–based grid adaptation approach is then integrated with the RBF-based central scheme. Effects of condition numbers of RBFs, computational complexity and cost of the proposed scheme are studied. Finally, different 1-D coupled thermoelasticity benchmarks are presented. There, performance of the adaptive RBF-based formulation is compared with that of the adaptive Kurganov-Tadmor (KT) second-order central high-resolution scheme with the total variation diminishing (TVD) property.

A non‐oscillatory multimoment finite‐volume global transport model on a cubed‐sphere grid using the WENO slope limiter

By using CSL3 multimoment interpolation, a piecewise cubic polynomial for spatial reconstruction can be obtained with four multimoment constraint conditions consisting of two point values at cell boundaries, one volume‐integrated average and one slope parameter at the cell center. The resulting multimoment finite‐volume scheme is of fourth‐order accuracy. A non‐oscillatory scheme can be derived by designing the proper formula to calculate the slope parameter at the cell center. A new strategy was recently proposed, using the Weighted Essentially Non‐Oscillatory (WENO) concept to determine the slope parameter. Using a WENO‐type limiter, the multimoment reconstruction can effectively remove nonphysical oscillations while keeping fourth‐order accuracy in smooth regions. In this study, a WENO‐type slope limiter is proposed and implemented in our multimoment finite‐volume global transport model based on the cubed‐sphere grid. The widely used benchmark tests, including both solid rotation and complicated deformational advection cases, are checked to verify the performance of the proposed global transport model. Numerical results reveal that a WENO‐type slope limiter can greatly improve the accuracy of the multimoment finite‐volume model compared with the former Total Variation Diminishing (TVD)‐type limiter. Furthermore, the proposed limiter is constructed over a compact stencil of only three adjacent cells. Without any user‐defined or problem‐dependent parameters, the present model is very promising for practical applications.

Second-order implicit-explicit total variation diminishing schemes for the Euler system in the low Mach regime

In this work, we consider the development of implicit-explicit total variation diminishing (TVD) methods (also termed SSP: strong stability preserving) for the compressible isentropic Euler system in the low Mach number regime. The proposed scheme is asymptotically stable with a CFL condition independent of the Mach number. In addition, it degenerates, in the low Mach number regime, to a consistent discretization of the incompressible system. Since it has been proved that implicit schemes of order higher than one cannot be TVD (SSP) [30], we construct a new paradigm of implicit time integrators by coupling first-order in time schemes with second-order ones in the same spirit as highly accurate shock-capturing TVD methods in space. For this particular class of schemes, the TVD property is first proved on a linear model advection equation and then extended to the isentropic Euler case. The result is a method which interpolates from the first- to the second-order both in space and time. It preserves the monotonicity of the solution, and is highly accurate for all choices of the Mach number. Moreover, the time step is only restricted by the non-stiff part of the system. We finally show, thanks to one- and two-dimensional test cases, that the method indeed possesses the claimed properties.

Response of a spherical cavity in a fully-coupled thermo-poro-elastodynamic medium by cell-adaptive second-order central high resolution schemes

In this study, fully coupled thermo-poroelastic saturated media are simulated by a grid/cell adaptive central high resolution scheme. The central method corresponds to the second order Kurganov–Tadmor (KT) scheme working on adapted cells with the total variation diminishing (TVD) stability condition. The coupled equations include motion, fluid flow, heat flow, continuity condition, and a constitutive equation. The grid/cell adaptation is performed by the interpolating wavelet transform in the multiresolution framework to capture fine scale responses and to obtain a computationally effective solver. With respect to the use of central schemes, the coupled equations should be re-expressed as a system of coupled first-order hyperbolic-parabolic partial differential equations (PDEs) with possible source (load) terms. The system is initially derived in the Cartesian coordinate system, and it is subsequently modified to consider a spherical cavity in isotropic, symmetric, and saturated media in the spherical coordinate system. It is assumed that the cavity boundary is subjected to sudden time-dependent thermal/mechanical sources. Discontinuous propagating fronts develop in the media due to the aforementioned loading. It is challenging to handle these solutions with numerical methods, and special attention is required to prevent/control numerical dispersion and dissipation. Hence, as previously mentioned, adaptive central high resolution schemes are employed in the present study.

Parallel Numerical Algorithm for Solving Advection Equation for Coagulating Particles

In this work we present a parallel implementation of numerical algorithm solving the Cauchy problem for equation of advection of coagulating particles. This equation describes time-evolution of the concentration f(x, v, t) of particles of size v at the point x at the time-moment t. Our numerical algorithm is based on use of total variation diminishing (TVD) scheme and perfectly matching layers (PML) for approximation of advection operator along spatial coordinate x and utilization of the fast numerical method for evaluation of coagulation integrals exploiting low-rank decomposition of coagulation kernel coefficients and fast FFT-based implementation of convolution operation along particle size coordinate v. In our work we exploit one-dimensional domain decomposition approach along spatial coordinate x because it allows to avoid use of parallel FFT implementations which are very expensive in terms of data exchanges and have poor parallel scalability. Moreover, locality of finite-difference operator from TVD-scheme along x coordinate allows to obtain good scalability even for computing clusters with slow network interconnect due to modest volumes of data necessary for synchronization exchanges between times integration steps.

Computation of Traveling Waves in a Heterogeneous Medium with Two Pressures and a Gas Equation of State Depending on Phase Concentrations

A fine structure theory of shock waves occurring in a gas–particle mixture was developed using an Anderson-type model with allowance for different phase pressures and with an equation of state for the gas component depending on the mean densities of both phases. The conditions for the formation of various types of shock waves based on the different speeds of sound in the phases were indicated. A high-order accurate TVD scheme was developed to prove the stability of some types of shock waves. The scheme was used to implement steadily propagating shock waves found in the stationary approximation, namely, shock waves of dispersive, frozen, and dispersive-frozen structures with one or two fronts.

Suitable diffusion for constructing non-oscillatory entropy stable schemes

In this work, amount of suitable diffusion in entropy stable fluxes is explicitly characterized to construct non-oscillatory schemes in total variation diminishing (TVD) sense. Further, high resolution entropy stable TVD fluxes are constructed and a generic TVD-entropy stable region is given for the flux limiter functions. The non-oscillatory TVD property of proposed fluxes does not depend on the choice of entropy functions and different choices for diffusion matrices are proposed for these fluxes. These fluxes are extendable to the system of higher dimension and resulting entropy stable schemes are used to numerically compute the solution for Burgers and shallow water equations in 1D and 2D case. It is also shown numerically that, the use of proposed diffusion matrices in TECNO schemes can significantly suppress the oscillations exhibited by them while applied with other diffusion matrices. Numerical results show that the resulting schemes capture steady shock exactly and produce non-oscillatory solution profile with high resolution.

A hybrid hydrostatic and non-hydrostatic numerical model for shallow flow simulations

Hydrodynamics of geophysical flows in oceanic shelves, estuaries, and rivers, are often studied by solving shallow water model equations. Although hydrostatic models are accurate and cost efficient for many natural flows, there are situations where the hydrostatic assumption is invalid, whereby a fully hydrodynamic model is necessary to increase simulation accuracy. There is a growing concern about the decrease of the computational cost of non-hydrostatic pressure models to improve the range of their applications in large-scale flows with complex geometries. This study describes a hybrid hydrostatic and non-hydrostatic model to increase the efficiency of simulating shallow water flows. The basic numerical model is a three-dimensional hydrostatic model solved by the finite volume method (FVM) applied to unstructured grids. Herein, a second-order total variation diminishing (TVD) scheme is adopted. Using a predictor–corrector method to calculate the non-hydrostatic pressure, we extended the hydrostatic model to a fully hydrodynamic model. By localising the computational domain in the corrector step for non-hydrostatic pressure calculations, a hybrid model was developed. There was no prior special treatment on mode switching, and the developed numerical codes were highly efficient and robust. The hybrid model is applicable to the simulation of shallow flows when non-hydrostatic pressure is predominant only in the local domain. Beyond the non-hydrostatic domain, the hydrostatic model is still accurate. The applicability of the hybrid method was validated using several study cases.

A coupled level set and volume of fluid method on unstructured grids for the direct numerical simulations of two-phase flows including phase change

In the present study, a coupled level set and volume of fluid (CLSVOF) method is developed for two-dimensional unstructured grids to perform direct numerical simulations of two-phase flows including phase change. The volume fraction is advected using a multi-directional advection algorithm, where the flux polygons are constructed using vertex velocities and a scaling factor based on cell face velocities is used to correct the advected volume fraction. The level set field is advected using a total variational diminishing (TVD) scheme and geometrically reinitialized at the end of each time step. The performance of the proposed CLSVOF method is evaluated in detail on unstructured grids, both qualitatively and quantitatively, prior to simulation of phase change problems. A number of advection test cases and two-phase flow problems are considered for this purpose. Results obtained for film boiling over a horizontal flat plate using an unstructured grid show excellent agreement with results available in the literature. The numerical study of natural convection film boiling over a horizontal cylinder at different wall superheats shows a better agreement with semi-empirical correlations compared to other available numerical results.

Large eddy simulations of the all-speed turbulent jet flow using 3-D CFD code GASFLOW-MPI

Turbulent flow at a wide range of Mach numbers may occur during design based accidents (DBA) or severe accidents in nuclear power plants (NPPs), for instinct, incompressible gas mixing flow (subsonic), gas mixture critical flow (transonic) or detonation (supersonic). All of them need to be simulated in safety analysis of NPP containments, therefore we need the computer code to have the capability to simulate all-speed flow. GASFLOW-MPI is a well-developed parallel all-speed CFD software used to predict gas turbulent mixing, chemical combustion and other related thermal-hydraulic phenomena in containments of NPPs. In order to validate the all-speed turbulent flow capability of GASFLOW-MPI, four turbulent jet cases over a broad range of Mach number have been performed in this paper. The large eddy simulation (LES) turbulent model is employed to capture more details of turbulence and flow features in this study. The standard Smagorinsky sub-grid scale model is utilized to model the unresolved turbulent behaviors at sub-grid scale. Both the instantaneous and time-averaged flow fields are analyzed as well as turbulent statistics to compare with the experimental data sets. The simulation results are in good agreement with the experimental data sets available in the literature. The shear layer instability is observed in the instantaneous flow field which leads to the end of the potential core and then the centerline velocity begins to decay. The size of the potential core and the decay rate of the centerline velocity are consistent well with the experiments. The turbulence intensity in the fully developed region agree well with the experimental data and the velocity power spectrum reveals the -5/3 law. Simulation results show that the compressibility has a significant effect on a free shear layer growing rate which is consistent with the experiments. The complex shock wave structure is also captured exactly by the second-order total variation diminishing (TVD) scheme. GASFLOW-MPI code has been successfully validated using the experimental data of turbulent jet flow at the wide regime of Mach numbers.