Total Variation Diminishing Research Articles

Investigation of a non-Newtonian MR fluid flow between parallel plates by developed CFD code for different numerical schemes

Magnetorheological (MR) fluids are composed of a solution of micron-sized magnetizable particles dispersed in a carrier fluid, typically oil. Thus, the MR fluids are classified as smart fluids by showing variable apparent viscosity under a precisely controllable external magnetic field. This feature of the MR fluids gave the possibility to use them in the MR dampers to control variational damping forces. In this work, an MR damper on relatively small scales is considered to investigate the MR fluid flow behavior through the piston annuli where the magnetic field exists. This work mainly focuses on the numerical solutions of the flow variables in a magnetically excited non-Newtonian flow medium by comparing analytical results for various Reynolds numbers. The Herschel–Bulkley (HB) viscous model is used for the non-Newtonian characteristic of the MR fluid. For the numerical modeling of the HB viscous model, the regularized approaches are used to avoid numerical errors. However, contrary to the actual HB model, the regularized models can be incapable to give true apparent viscous values at very low shear rates depending on the regularization parameters. Thus, the present study aims to give a better understanding of choosing the optimal regularization parameters for the studied flow conditions. The second part of the study discusses the numerical discretization schemes aiming to present their performances while changing the Reynolds number between 0.002 and 1. In this manner, a Computational Fluid Mechanics (CFD) solver has been developed for two-dimensional geometries which are meshed with structured grids by using six different discretization schemes including two of the most known total variation diminishing (TVD) schemes. For simplicity of the problem, a two-dimensional parallel plates geometry and a constant magnitude of the magnetic field intensity along the piston annuli are assumed. The study also summarizes the CFD technique by evaluating the physical meaning of the flow field variables.

Doubly Monotonic Constraint on Interpolators: Bridging Second-Order to Singularity Preservation to Cancel “Numerical Wetting” in Transport Schemes

Monotonic interpolation and its avatars are major ingredients of many numerical schemes for solving partial differential equations (PDEs) under total variation diminishing (TVD) or similar constraints. However, despite over forty years of extensive study of principles and applications, a key aspect of monotonic interpolant design can still appear somewhat empirical: how does a monotonic interpolator connect the limiting cases of smooth (differentiable) and singular (limited) functions in a consistent and possibly canonical way? The present study aims at providing understanding in the basic but important case of per-cell monotonic one-dimensional scalar reconstruction and at applying it to second-order accurate transport. First, a general mapping of bounded monotonic functions in elliptic coordinates is built. Then, the usual “single-slope” second-order monotonic interpolants are continued into “slope-and-bound” monotonic interpolants. Finally, a critical constraint is introduced, the “double monotonicity,” in order to build various slope-and-bound monotonic interpolators from this set of interpolants. With these slope-and-bound interpolators, standard numerical tests show a complete cancellation of the “numerical wetting” that usual TVD transport schemes produce. When transporting scalar fields of compact support, this effect---not to be confused with usual numerical diffusion---is the low-level contamination that spreads linearly in time over all the regions of the computational domain where nonvanishing transport is present. Removal of numerical wetting is of particular importance in many industrial and academic applications, notably at “phase disappearance” episodes in multiphase flows or “wet-dry” transitions in shallow water flows. Improvement of the “numerical erosion” of extrema is also observed. The general principles exposed here can be extended to multidimensional settings, high-order schemes, and other PDEs.

Development of a new OpenFOAM solver for plasma cutting modeling

A new OpenFOAM solver is presented, for the simulation of plasma cutting torches. The mathematical model that is introduced is based on the compressible Navier–Stokes equations coupled via source terms to the electric current conservation equation. Due to the conservative and hyperbolic nature of the model, a Godunov-type scheme is used for the first time in the context of plasma cutting simulation. The numerical method consists of a second-order Total Variation Diminishing (TVD) integration with flux Harten–Lax–van Leer-Contact (HLLC) Riemann solver for the flow conservation equations, coupled with a Laplace solver for the current conservation equation. An efficient formulation for the equation of state, accurately taking into account the plasma properties, is also presented. The solver is validated through a set of canonical test cases (shock tubes and 2D Riemann problems) and it is used to simulate a three-dimensional plasma cutting torch. Good agreement is found with the literature, with an improvement in the ability to deal with the shocks occurring during plasma cutting.

A heterogeneous non-overlapping domain decomposition explicit finite volume method for a real-time hybrid process-state estimator of 3D unsteady advection-diffusion fields

A heterogeneous, nonoverlapping domain-decomposition, explicit, finite volume method (HT-NODDE-FVM) is developed and applied to a real-time, hybrid, process-state estimator of 3D unsteady, advection-diffusion concentration fields produced by stationary or moving gas sources. The hybrid Luenberger-naive estimator uses a single, pointwise concentration measurement of the field taken onboard a sensing aerial vehicle (SAV) and computes in real-time the concentration field in the entire domain, while guiding the SAV to locations that provide optimal information to the estimator. The computational domain of interest is divided into multiple non-overlapping subdomains using a structured and uniform grid. A Luenberger estimator (or observer) in the form of a 3D advection-diffusion partial differential equation (PDE) is developed for the subdomain where the SAV resides, and a naive estimator of a similar form is developed for the remaining subdomains. The transmission conditions are used on the interfaces between adjacent subdomains for data communication. The hybrid Luenberger-naive estimator is discretized in space by the HT-NODDE-FVM with Total Variation Diminishing (TVD) and the resulting semi-discrete equations are integrated by a fourth order Runge-Kutta scheme. Continuity and flux balance transmission conditions are enforced at the interfaces of adjacent subdomains. The HT-NODDE-FVM hybrid estimator is implemented in parallel using OpenMP. Verification and error analysis of the NODDE-FVM are performed with benchmark tests that include the 3D advection PDE for various initial density distributions and the 3D advection-diffusion PDE for instantaneous gas releases under constant wind speed and eddy diffusivities for a range of Peclet numbers. Additional benchmark tests provide verification and error analysis of the HT-NODDE-FVM hybrid estimator for an instantaneous release by a stationary gas source in a large domain with constant atmospheric properties. The tests examine the impact of grid resolution, sensor model, estimation gain, and numerical data, on the L1, L2, and L∞ norms of the estimation error. Parallelization efficiency analysis of the OpenMP implementation of the hybrid estimator is also presented. The hybrid estimator is applied to the simulation of gaseous plumes from stationary and moving sources in km-scale computational domains under realistic atmospheric conditions and the results show that it achieves real-time estimation of the concentration field.

Adaptive total variation stable local timestepping for conservation laws

This paper proposes a first-order total variation diminishing (TVD) treatment for coarsening and refining of local timestep size in response to dynamic local variations in wave speeds for nonlinear conservation laws. The algorithm is accompanied with a proof of formal correctness showing that given a sufficiently small minimum timestep the algorithm will produce a TVD solution for nonlinear scalar conservation laws. A key feature of the algorithm is its formulation as a discrete event simulation, which allows for easy and efficient parallelization using existing software. Numerical results demonstrate the stability and adaptivity of the method for the shallow water equations. We also introduce a performance model to load balance and explain the observed performance gains. Performance results are presented for a single node on Stampede2's Skylake partition using an optimistic parallel discrete event simulator. Results show the proposed algorithm recovering 59%-77% of the theoretically achievable speed-up with the remainder being attributed to the cost of computing the CFL condition and load imbalance.

A comparative study of high‐resolution methods for nonlinear hyperbolic problems

AbstractThis paper reviews some of the established high‐resolution (HR) schemes for computing solutions to hyperbolic systems by means of a finite volume (FV) framework. The objective of this research is to assess the ability of various HR flux limiting techniques to capture intense shocks and discontinuities. A crucial design factor in developing these methods is selecting a suitable flux‐limiter, so a review of different flux‐limiters for standard explicit FV frameworks has also been included. A brief discussion of the HR total variation diminishing (TVD) numerical schemes, starting from some historical introduction, going on to exploring the design principles behind the HR, second‐order, oscillation free schemes, turning towards high order nonlinear techniques and ending with an overview of some of the most significant developments and applications in the last decades has been covered. Several one‐dimensional and two‐dimensional benchmark examples with severe and challenging wave configurations have been utilized to compare the various numerical approaches. These nonlinear test cases have been selected such that the computational setups are as basic as possible.

On the investigation of oblique shock‐wave/turbulent boundary‐layer interactions with a high‐order discontinuous Galerkin method

AbstractShock‐wave/turbulent boundary‐layer interactions are still a challenge for numerical simulation. The shock capturing needs dissipation to avoid spurious oscillations while turbulence will be falsified by introducing dissipation. Especially, an accurate prediction of quantities such as the skin‐friction coefficient inside the interaction area of shock wave and turbulent flow is a critical point. In this article, we investigate a wall‐resolved large eddy simulation of oblique shock‐wave/turbulent boundary‐layer interactions by a high‐order discontinuous Galerkin scheme. The high‐order scheme handles the turbulent flow very well. The shock‐capturing is confined to the near shock region by switching locally to a finite volume second‐order TVD scheme on subcells. This strategy is completed with the application of a shock indicator to a filtered flow field. A global spanwise filter is applied to avoid switching on the shock‐capturing procedure in regions of under‐resolved turbulent structures. We validate our numerical results first at shock‐wave/laminar boundary‐layer interaction. The main simulation under consideration is a Mach turbulent boundary‐layer with an inlet momentum‐thickness Reynolds number of , interacting with an oblique shock that deflects the incoming flow by . We employ a reformulated synthetic eddy method at the inlet to avoid the influence of recycling‐based turbulence generating schemes on the low‐frequency unsteadiness. The anisotropic linear forcing technique is adopted to further reduce the turbulence recovery length. Through the spectral analysis of wall pressure probes, a typical Strouhal number of around is observed. We attribute the discrepancies between an experimental scaling law and our computation to the three‐dimensional sidewall effects in the experiment. With the assistance of numerical results from this article and other authors, a new scaling law for the spanwise‐periodic computations is suggested to quantify the difference between experimental and computed data.

Effects of Different Slope Limiters on Stratified Shear Flow Simulation in a Non-hydrostatic Model

To simulate the dynamical structures of stratified shear flows, the high-resolution Total Variation Diminishing (TVD) method is necessary and widely-used due to its high-order spatial accuracy, oscillation control, and ability to capture the well-defined structures of vortices. Lack of understanding the TVD slope limiters usually results in inaccurate numerical simulation on stratified shear flows in terms of shear instability and spatiotemporal variations of mixing. In this study, the performances of four typical TVD slope limiters, namely the minmod, van Leer, Monotonized Central (MC), and superbee limiters, were investigated on modelling stratified shear flows based on the open-source non-hydrostatic model, NHWAVE. The four slope limiters are all commonly-used and have the typical numerical characteristics. All the limiters were respectively applied in two classical test cases, namely, shear instability and lock-exchange problem. The simulation results showed that the effects of slope limiters were correlated with their characteristics of numerical dissipation (or anti-dissipation), which can influence notably the model predictions of the generation of shear instability, the development of interfacial structures, and the mixing process. In the test cases, MC limiter’s performance was the best, because it could simulate the well-defined structures of instability while not introducing noticeable error. Minmod has an excessively large dissipation, which introduced noticeable numerical errors that can influence the model accuracy and can even suppress or omit the generation of interfacial vortices. Superbee limiter, the most anti-dissipative one, usually over-predicted the instability and mixing effects in time and space domain, and was likely to cause computational instability in some cases. The performances of van Leer and MC were similar, but their predictions of the evolutions of interfacial structures and mixing could be significantly different. Besides, the co-effects of grid resolution and slope limiters were also investigated; it was found that the refinement of grids may not help to reproduce a higher-quality result with a specific slope limiter.

GPU implementation of Explicit and Implicit Eulerian methods with TVD schemes for solving 2D solute transport in heterogeneous flows

In this work we present an efficient implementation of Eulerian TVD methods. We apply parallelization strategies based entirely on GPU for the solution of the 2D transport equation in heterogeneous porous media. Additionally, a parallel strategy is proposed for the generation of exponentially correlated lognormally distributed permeability fields in GPU. The programs are developed using C++/CUDA. The implemented methods are used to solve advective dominant problems, in a context of Monte Carlo type simulations to numerically determine the longitudinal and transversal macrodispersion coefficients averaging over 100 simulations for permeability fields for a large range of variances. The following types of transport are considered for testing: pure advection, advection-diffusion and advection-dispersion. The performance in terms of the computation time of explicit and implicit methods are compared. We show that the implemented algorithms allow to efficiently solve problems in computational domains of up to 134.5 million cells in a single GPU.

An augmented HLLEM ADER numerical model parallel on GPU for the porous Shallow Water Equations

This paper presents a novel explicit finite volume scheme for the solution of the two-dimensional Shallow Water Equations (SWEs) with porosity based on the use of an augmented Riemann problem. Bottom and porous source terms are treated as non-conservative fluxes by including two additional equations to the porous SWEs. The HLLEM approximate Riemann solver is chosen to compute the fluxes of the non-conservative system of equations in order to ensure robustness, computational efficiency and entropy enforcement also in the presence of significant discontinuities in the porosity field. Moreover, the proposed solver is extended to second order in time and space adopting the ADER time-integration scheme, together with a Total Variation Diminishing (TVD) spatial reconstruction. The model is implemented in CUDA (Compute Unified Device Architecture) language to exploit the computational power of modern Graphic Processing Units. The theoretical and numerical verification of the C-property is achieved also when discontinuities characterize the porosity and bottom fields. The scheme accurately reproduces Riemann problems generated by the presence of porous discontinuities. Numerical tests also confirm that the proposed algorithm retains roughly the same numerical efficiency of an alternative formulation based on a non-augmented Riemann solver, while providing more accurate results.

Comparison of Pressure Distribution of Naca 0012 Between CFD Code and Experimental

Computational Fluid Dynamics (CFD) is a tool to solve engineering problem. Commercial CFD code are being used most engineering problem but rarely for external aerodynamic problem due some limitation of features. Compare to in house CFD code that able to written in certain area of interest and also implement specific discretization scheme to increase accuracy. In this work, the CFD code are developed by using high resolution scheme (ROE scheme and TVD scheme) for flow past through airfoil NACA 0012. Experimental result of wind tunnel is obtained from literature which are from Gregory and O’Reilly at Mach number M = 0.13 for different values of angle of attack at α= 0° and α = 10°. Another wind tunnel result from Haris at Mach number = 0.8 for different angles of attack α= 0° and α= 3.86°. Comparison results from the both codes indicate that developed CFD code by using TVD scheme able to give the closest result for both experimental.

MPS modeling of cross-sectional averaged shallow water flows with open boundaries using TVD-MacCormack predictor-corrector

The present study utilizes a moving particle simulation (MPS) method to solve cross-sectional averaged shallow water equations applicable for non-prismatic open channels. The well-known MacCormack predictor–corrector scheme is extended to the Lagrangian framework, ensuring the second accuracy in time and space. For this purpose, upwind and downwind MPS gradient operators are introduced. The total variation diminishing (TVD) flux-limiter scheme is developed to the MPS formulation to preserve monotonicity, which does not require characteristic speeds of the system. In mesh-based methods, the non-conservative form of the momentum equation is problematic in scenarios with discontinuities. However, the non-conservative convective flux term does not exist under the Lagrangian framework in preserving the momentum conservation. Moreover, the cross-sectional area is computed by the algebraic density-ratio equation in the present work since the mass is automatically conserved. This advantage is beneficial for open boundary treatment resulting in less computational effort than mesh-based methods because the partial differential continuity equation is omitted as a boundary value problem. The present high-order solver has positivity-preserving and well-balancing properties without the need for any threshold value for treating the dry bed and friction term. Additionally, it can model free-surface flows in prismatic and non-prismatic open channels with various open and/or closed boundary conditions.

Fully compressible multiphase model for computation of compressible fluid flows with large density ratio and the presence of shock waves

We propose a fully compressible multiphase model for simulating compressible interfacial flows. The mathematical model is formulated on the basis of the typical conservation laws for mixtures, including mass, momentum, and energy balance. The mass balance equation in each phase is considered to discretely conserve the mass of each phase; therefore, it is always mass conservative for each phase. The numerical solution procedure for this model is developed using a high-resolution shock-capturing finite-volume method based on the Harten–Lax–van Leer-contact (HLLC) Riemann solver and total variation diminishing (TVD) time-integration schemes for simulating unsteady multiphase flows and capturing strong shocks. The finite-volume Riemann solver is combined with a monotonic upstream-centered scheme for conservation laws (MUSCL) and compressive limiters to maintain a sharp interface. The numerical framework yields a conceptually simple but robust and versatile numerical procedure. Numerical results of benchmark Riemann problems demonstrate the ability of the proposed method to obtain sharp interfaces, free of spurious oscillations, high accuracy, conservation properties, and thermodynamic consistency over various Mach numbers.

Investigation on flow field characteristics in an open-pit coal mine.

In this paper, the atmospheric flow field characteristics in a deep open-pit mine are investigated numerically and theoretically. A theoretical model on the recirculation length based on the energy equation is used, and a total variation diminishing (TVD) difference scheme with second-order accuracy is used to solve the NS equations with a standard two-equation k-ε turbulence model. The effects of elevated inner dump on the flow field characteristics of the open-pit mine are analyzed detailedly with the same inflow parameters. The results show that a recirculation zone exists in the open-pit mine due to the reflux from the high-pressure zone to the low-pressure zone. As the height of the inner dump increases, the flow becomes more complicated, and the low-pressure zone and the recirculation zone become bigger. The elevated inner dump makes it difficult for the internal fluid to flow to the outside, which results in the increase of the dust concentration. At last, the influences of key parameters on flow filed are conducted by normalizing the depth of the lowest direct current flow on the windward slope. The sensitivity analysis is done by study each influencing factor. This paper offers an effective way to study the flow field characteristics in an open-pit coal mine, which is essential to the dust pollution control of open-pit mine.

A Realizable Second-Order Advection Method with Variable Flux Limiters for Moment Transport Equations

A second-order total variation diminishing (TVD) method with variable flux limiters is proposed to overcome the non-realizability issue, which has been one of major obstacles in applying the conventional second-order TVD schemes to the moment transport equations. In the present method, a realizable moment set at a cell face is reconstructed by allowing the flexible selection of the flux limiter values within the second-order TVD region. Necessary conditions for the variable flux limiter scheme to simultaneously satisfy the realizability and the second-order TVD property for the third-order moment set are proposed. The strategy for satisfying the second-order TVD property is conditionally extended to the fourth- and fifth-order moments. The proposed method is verified and compared with other high-order realizable schemes in one- and two-dimensional configurations, and is found to preserve the realizability of moments while satisfying the high-order TVD property for the third-order moment set and conditionally for the fourth- and fifth-order moments.

A Framework of Runge–Kutta, Discontinuous Galerkin, Level Set and Direct Ghost Fluid Methods for the Multi-Dimensional Simulation of Underwater Explosions

This work describes the development of a hybrid framework of Runge–Kutta (RK), discontinuous Galerkin (DG), level set (LS) and direct ghost fluid (DGFM) methods for the simulation of near-field and early-time underwater explosions (UNDEX) in early-stage ship design. UNDEX problems provide a series of challenging issues to be solved. The multi-dimensional, multi-phase, compressible and inviscid fluid-governing equations must be solved numerically. The shock front in the solution field must be captured accurately while maintaining the total variation diminishing (TVD) properties. The interface between the explosive gas and water must be tracked without letting the numerical diffusion across the material interface lead to spurious pressure oscillations and thus the failure of the simulation. The non-reflecting boundary condition (NRBC) must effectively absorb the wave and prevent it from reflecting back into the fluid. Furthermore, the CFD solver must have the capability of dealing with fluid–structure interactions (FSI) where both the fluid and structural domains respond with significant deformation. These issues necessitate a hybrid model. In-house CFD solvers (UNDEXVT) are developed to test the applicability of this framework. In this development, code verification and validation are performed. Different methods of implementing non-reflecting boundary conditions (NRBCs) are compared. The simulation results of single and multi-dimensional cases that possess near-field and early-time UNDEX features—such as shock and rarefaction waves in the fluid, the explosion bubble, and the variation of its radius over time—are presented. Continuing research on two-way coupled FSI with large deformation is introduced, and together with a more complete description of the direct ghost fluid method (DGFM) in this framework will be described in subsequent papers.

Comparison of finite-volume method and method of characteristics for simulating transient flow in natural-gas pipeline

In this study, two numerical methods of modeling transient flow in a single natural-gas pipeline are analyzed and compared. Specifically, the pressure-based finite-volume method (FVM)—which uses a collocated mesh with the total variation diminishing (TVD) scheme—and the conventional implicit method of characteristics (iMOC)—which uses an inertial multiplier—are considered. Network analysis is an essential part of the design and operation of natural-gas pipelines. Studies have been conducted on various numerical methods to analyze the transient flow of natural-gas pipelines, but comparative studies on the characteristics and relative advantages and disadvantages of the methods are inadequate; therefore, it is necessary to analyze and compare the underlying numerical methods to ensure their reliability. Accordingly, two test cases are considered: slow transient flow validated with experimental data, and fast transient flow with an artificial problem including valve stroking. The two methods are examined quantitatively and qualitatively. The order of accuracy, determined using the grid-convergence index, and the computational cost are evaluated for quantitative analysis. Overall, the pressure-based FVM with the TVD scheme outperforms iMOC for both slow and fast transient problems.

Some aspects in Kelvin-Helmholtz instability with and without Boussinesq approximation

The topic of this paper is the Kelvin-Helmholtz instability, a phenomenon which occurs on the interface of a stratified fluid, in the presence of a parallel shear flow, when there is a velocity and density difference across the interface of two adjacent layers. This paper focuses on a numerical simulation modelled by the Taylor-Goldstein equation, which represents a more realistic case compared to the basic Kelvin-Helmholtz shear flow. The Euler system is solved with new modelled smooth velocity and density profiles at the interface. The flux at cell boundaries is reconstructed by implementing a third order WENO (Weighted Essentially Non-Oscillatory) method. Next, a Riemann solver builds the fluxes at cell interfaces. The use of both Rusanov and HLLC solvers is investigated. Temporal discretization is done by applying the second order TVD (total variation diminishing) Runge-Kutta method on a uniform grid. Numerical simulations are performed comparatively for both Kelvin-Helmholtz and Taylor-Goldstein instabilities, on the same simulation domains. We find that increasing the number of grid points leads to a better accuracy in shear layer vortices visualization. Thus, we can conclude that applying the Taylor-Goldstein equation improves the realism in the general fluid instability modelling.

Wave propagation in generalized thermo-poro-elastic media via wavelet-based cell-adaptive central high resolution schemes using UNO limiters

The uniformly non-oscillatory (UNO) limiters with second-order accuracy are updated to handle nonuniform cell obtained from wavelet-based cell adaptation. These limiters are then used in the cell-adaptive second-order Kurganov-Tadmor (KT) central high resolution scheme. Over the adapted cells, the performance of the UNO limiters are compared with the generalized MINMOD (GMM) limiter with the total variation diminishing (TVD) feature, by measuring error of reconstruction of some functions and also by numerical solution of some benchmark partial differential equations (PDEs). Then, unified forms of generalized fully-coupled saturated thermo-poro-elastic systems are presented as first-order hyperbolic-parabolic systems in the 2D Cartesian and the asymmetric spherical coordinates. For these systems, corresponding flux, source (load), diffusion and nonlinear terms are derived. It is shown that in the case of the asymmetric spherical coordinate, due to the geometry, considerable complexities are introduced in the load (source) and flux terms. Finally, the coupled 1D and 2D systems are simulated by the adaptive central KT scheme with the GMM and UNO limiters.

Aerodynamic thermal environment around transonic tube train in choked/unchoked flow

The aerodynamic thermal environment in an evacuated tube transport (ETT) system is an important factor in ensuring the operational safety of tube trains, where choking can further worsen air flow and aerodynamic heating. A compressible flow solver based on total variation diminishing (TVD) schemes was used to calculate the transonic aerodynamic behaviour of a capsule train in a confined space under low pressure conditions, and the flow fields and aerodynamic heating effect on the train were obtained. The results showed that the flow state around the train could be classified into choked and unchoked flow according to the blockage ratio (BR) and train speed based on the Kantrowitz limit. The wall viscosity caused a difference in the boundary-layer flow and potential flow in the annular space between the train and the tube with an increase in the BR. The choked flow was driven forward by the train, passing through its throat at the speed of sound. Owing to the complicated compressible flow in the tube, the thermal environment around the train gave rise to extreme temperature changes on its surface. In transonic choked flow, the temperature rise of the train head reached a maximum of 525 K, whereas local cooling could occur in the afterbody, causing the surface temperature to fall below the ambient temperature under certain conditions. The findings can be used to guide the design of ETT systems.