Schemes For Conservation Laws Research Articles

On the approximation of dam-break problems using a fuzzified HR-TVD scheme

The construction of the proposed second-order accurate scheme is based on the Monotonic Upstream Schemes for Conservation Laws (MUSCL) strategy. Using some strong notions from fuzzy logic, like fuzzy modifiers and fuzzy inference systems, the work offers a new avenue for optimizing the execution of classical flux-limiting algorithms. A numerical study has also been included to compare the new limiting scheme and the classical Monotonized Central (MC) scheme. The key themes of the research work are:•Development of the background theory and augmented Riemann finite volume (FV) framework required to construct the new fuzzy logic-based limiter for the Shallow Water Equations (SWEs) over flat bottom topography.•The proposed fuzzy flux limiting technique has been employed in a structured one-dimensional FV model to compute the SWEs.•The idealized dam-break flows verify the new fuzzified High-Resolution (HR), Total Variation Diminishing (TVD) technique.

An improved alternative weighted essentially non-oscillatory scheme for conservation laws

An improved alternative weighted essentially non-oscillatory scheme for conservation laws

High resolution compact implicit numerical scheme for conservation laws

We present a novel implicit scheme for the numerical solution of time-dependent conservation laws. The core idea of the presented method is to exploit and approximate the mixed spatial-temporal derivative of the solution that occurs naturally when deriving some second order accurate schemes in time. Such an approach is introduced in the context of the Lax-Wendroff (or Cauchy-Kowalevski) procedure when the second time derivative is not completely replaced by space derivatives using the PDE, but the mixed derivative is kept. If approximated in a suitable way, the resulting compact implicit scheme produces algebraic systems that have a more convenient structure than the systems derived by fully implicit schemes. We derive a high resolution TVD form of the implicit scheme for some representative hyperbolic equations in the one-dimensional case, including illustrative numerical experiments.

Flood Prediction with Two-Dimensional Shallow Water Equations: A Case Study of Tongo-Bassa Watershed in Cameroon

As a result of urbanization, combined with the anthropogenic effects of climate change, natural events such as floods are showing increasingly adverse impacts on human existence. This study proposes a new model, based on shallow water equations, that is able to predict these floods and minimize their impacts. The first-order finite volume method (FVM), the Harten Lax and van Leer (HLL) scheme, and the monotone upwind scheme for conservation laws (MUSCL) are applied in the model. In addition, a virtual boundary cell approach is adopted to achieve a monotonic solution for both interior and boundary cells and flux computations at the boundary cells. The model integrates the infiltration parameters recorded in the area, as well as the Manning coefficient specific to each land-cover type of the catchment region. The results provided were mapped to highlight the potential flood zones and the distribution of water heights throughout the catchment region at any given time, as well as that at the outlet. It has been observed that when standard infiltration and the Manning parameters were selected, the floodable surface increased, as expected, with the increasing rainfall intensity and duration of the simulation. With sufficient infiltration, only a portion of the water tends to stagnate and flow off on the surface toward the outlet. A sensitivity analysis of certain parameters, such as rainfall data and the final infiltration coefficient in the lower watershed of the littoral region, was conducted; the results show that the model simulates well the general character of water flow in the watershed. Finally, the model’s validation using field-collected parameters during the flood of 25 July 2017 and 18 to 22 July 2016 in the Grand Ouaga basin in Burkina reveals Nash–Sutcliffe values of 0.7 and 0.73, respectively.

Numerical Approach of a Coupled Pressure-Saturation Model Describing Oil-Water Flow in Porous Media

Two-phase flow in porous media is a very active field of research, due to its important applications in groundwater pollution, {text {CO}}_2 sequestration, or oil and gas production from petroleum reservoirs, just to name a few of them. Fractional flow equations, which make use of Darcy’s law, for describing the movement of two immiscible fluids in a porous medium, are among the most relevant mathematical models in reservoir simulation. This work aims to solve a fractional flow model formed by an elliptic equation, representing the spatial distribution of the pressure, and a hyperbolic equation describing the space-time evolution of water saturation. The numerical solution of the elliptic part is obtained using a finite-element (FE) scheme, while the hyperbolic equation is solved by means of two different numerical approaches, both in the finite-volume (FV) framework. One is based on a monotonic upstream-centered scheme for conservation laws (MUSCL)-Hancock scheme, whereas the other makes use of a weighted essentially non-oscillatory (ENO) reconstruction. In both cases, a first-order centered (FORCE)-alpha numerical scheme is applied for intercell flux reconstruction, which constitutes a new contribution in the field of fractional flow models describing oil-water movement. A relevant feature of this work is the study of the effect of the parameter alpha on the numerical solution of the models considered. We also show that, in the FORCE-alpha method, when the parameter alpha increases, the errors diminish and the order of accuracy is more properly attained, as verified using a manufactured solution technique.

Lax-Wendroff method for incompressible flow

The Lax-Wendroff scheme is extended to incompressible fluid flow problems within the framework of artificial compressibility method (ACM) utilizing a “cell-centered finite-volume” Δ-approximation on a non-orthogonal non-staggered grid. An ACM induces a transformation between conservative and primitive variables; the physical relevance of ACM is to bring about “matrix preconditionings”, provoking density perturbations. The coupled algorithm is pressure-based; it benefits from enhanced accuracy and greater flexibility using the “monotone upstream-centered schemes for conservation laws (MUSCL)” approach. Numerical experiments in reference to “buoyancy-driven” flows with strong sources illustrate that the entire strategy augments overall damping capability and robustness adhering to the factored pseudo-time integration method. Conclusively, the associated limiter function has little influence on the high resolution for selected test cases.

Numerical study of the thermodynamics and supercavitating flow around an underwater high-speed projectile using a fully compressible multiphase flow model

In this study, the thermodynamic behavior and supercavitating flow around projectiles of high-subsonic to supersonic speeds were numerically analyzed using a fully compressible multiphase flow with phase change. The mathematical model is formulated based on the typical conservation laws of mixtures, including mass, momentum, and energy balance, and by maximizing the thermodynamic properties of water and vapor to enable the analysis of the characteristics of high-speed flows. The model is solved on a body-fitted grid using a finite-volume Riemann solver combined with a monotonic upstream-centered scheme for conservation laws. The numerical method was validated by comparing the supercavitating flow around an underwater projectile with the experimental results available in the literature. The temperature, shock, velocity, and supercavity around the high-speed projectile and the effects of the moving velocities on the physical aspects were investigated. It was observed that the temperature around the stagnation point increased significantly with the increase in the stream velocity, and such high temperatures may soften the steel encasing of the projectile in a manner that the excessive pressure in the area could damage the projectile. The physical context and thermodynamics around the transonic projectile were further observed through numerical investigation.

Improved Wake Velocity Distribution Behind a Rising Bubble for Isothermal and Thermally Stratified Liquid Layers

Abstract This paper aims at (a) improving the vertical velocity distribution in the wake of a rising isolated bubble for isothermal water layers and (b) evaluating the proposed distribution for thermally stratified therminol layers before and after the initiation of vortex shedding. To address these objectives, numerical investigations are performed, for the rise of an isolated bubble in isothermal and thermally stratified liquid layers, with a combination of the monotonic upwind scheme for conservation laws and pressure implicit with splitting of operators numerical scheme. The analysis revealed that the vertical velocity in the wake of a rising isolated bubble, for isothermal and thermally stratified liquid layers, differs remarkably from the Gaussian distribution. Based on the detailed investigations, region-wise wake velocity distribution comprising a linear superposition of Gaussian approximation with Burr distribution is proposed. Furthermore, this distribution is utilized to predict the rise velocity for a chain of bubbles having different frequencies of departure. Thus, the findings will be useful for the design of heat exchangers or cooling devices, which rely on the heat transfer augmentation with rising air bubbles from a heated surface for isothermal (buoyancy suppressed) and thermally stratified (buoyancy assisted) liquid layers.

Application of High-Order WENO Scheme in the CFD/FW–H Method to Predict Helicopter Rotor Blade–Vortex Interaction Tonal Noise

The accurate prediction of helicopter rotor blade–vortex interaction (BVI) noise is challenging. This paper presents an implementation of the seventh-order improved weighted essentially non-oscillatory (WENO-Z) scheme for predicting rotor BVI noise using a high-resolution numerical method based on the Reynolds-averaged Navier–Stokes and the Ffowcs Williams–Hawkings equations. The seventh-order improved WENO-Z scheme is utilized to minimize the inherent numerical dissipation of the reconstruction method in the monotone upstream-centered scheme for conservation laws (MUSCL), thereby improving the rotor wake resolution and the BVI noise-prediction accuracy. The three-layer dummy cell method is used to ensure that the flux at the boundary maintains seventh-order accuracy. The effectiveness of the flow solver and the acoustic solver is validated using the Helishape-7A rotor and the UH-1H rotor, respectively. The flow field and BVI noise characteristics of the OLS rotor obtained from the fifth- and seventh-order WENO-Z schemes are compared with that of the third-order MUSCL for coarse and fine background grids. The wake resolution, noise-prediction accuracy, and computational cost of the three schemes are compared. The results show that the high-order WENO scheme provides higher accuracy for flow field simulation and BVI noise prediction than the MUSCL, but the computational cost of the WENO scheme increases substantially as the grid resolution increases. However, the WENO scheme can predict BVI using a coarser grid than the MUSCL. The computational cost of the WENO scheme is relatively low under the same flow field simulation resolution.

Convergence Rates of Monotone Schemes for Conservation Laws for Data with Unbounded Total Variation

We prove convergence rates of monotone schemes for conservation laws for Hölder continuous initial data with unbounded total variation, provided that the Hölder exponent of the initial data is greater than nicefrac {1}{2}. For strictly {{,mathrm{Lip},}}^+ stable monotone schemes, we prove convergence for any positive Hölder exponent. Numerical experiments are presented which verify the theory.

Numerical investigation of full helicopter with and without the ground effect

In the present work, the aerodynamic performance of the full helicopter PSP in hover flight is investigated using a simplified concept of multiple reference frame (MRF) technique in the context of high-order Monotone Upstream Centred Scheme for Conservation Laws (MUSCL) cell-centred finite volume method. The predictions were obtained for two ground distances and several collective pitch angle at tip Mach number of 0.585. The calculations were made for both out-of-ground-effect (OGE) and in-ground-effect (IGE) cases and compared with experimental data in terms of pressure distribution and integrated thrust and torque and vortex system.

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.

A consistent quasi–second-order staggered scheme for the two-dimensional shallow water equations

Abstract A quasi–second-order scheme is developed to obtain approximate solutions of the two-dimensional shallow water equations (SWEs) with bathymetry. The scheme is based on a staggered finite volume space discretization: the scalar unknowns are located in the discretization cells while the vector unknowns are located on the edges of the mesh. A monotonic upwind-central scheme for conservation laws (MUSCL)-like interpolation for the discrete convection operators in the water height and momentum balance equations is performed in order to improve the accuracy of the scheme. The time discretization is performed either by a first-order segregated forward Euler scheme or by the second-order Heun scheme. Both schemes are shown to preserve the water height positivity under a Courants Friedrichs Lewy (CFL) condition and an important state equilibrium known as the lake at rest. Using some recent Lax–Wendroff type results for staggered grids, these schemes are shown to be LW-consistent with the weak formulation of the continuous equations, in the sense that if a sequence of approximate solutions is bounded and strongly converges to a limit, then this limit is a weak solution of the SWEs; besides, the forward Euler scheme is shown to be LW-consistent with a weak entropy inequality. Numerical results confirm the efficiency and accuracy of the schemes.

High-Order Godunov-Type Scheme for Conservation Laws with Discontinuous Flux Function in Space

The numerical method for the conservation laws with discontinuous flux in space is considered. The difficulty in designing a high-order accurate and maintaining an efficient well-balanced scheme for this system lies in the fact that the flux is discontinuous across the stationary discontinuity, which results in the jump of the unknown function. In order to overcome this difficulty, the Godunov-type numerical flux is written in the positive and negative fluxes for the first time, which is embedded between two discontinuous flux functions. With the Godunov flux-splitting, the high-order accurate scheme is proposed through the WENO reconstruction of flux and the third-order TVD Runge–Kutta time discretization. Some tests are presented to demonstrate that our new high-order scheme is capable of exactly preserving steady-state solutions.

Numerical simulation of compressible multiphase hydrodynamic problems using reduced five-equation model on body-fitted grids

This paper focuses on the numerical simulation of compressible multiphase hydrodynamic problems on body-fitted mapped grids. Assuming the mixture of water and gas with homogeneous mixture properties, the reduced five-equation model (Kapila et al., 2001) is utilized to simulate the flow. Meanwhile, to circumvent the difficulties during computation such as preserving the volume fraction positivity, the pressure relaxation method (Saurel et al., 2009) is adopted to improve the robustness. The method is extended by considering the gravity effect and axisymmetric flows which are absent in the original model. A Godunov method based on the monotone upstream-centered scheme for conservation laws (MUSCL) is utilized to discretize the governing equations and the HLLC approximate Riemann solver is used to compute the numerical flux across a mesh cell face. Source terms representing the effect of gravitational and axisymmetric flow are calculated by the Runge–Kutta method. Several test cases including water–air shock tubes, dam break, underwater explosion, water entry of a hemisphere and an oblique cylinder are presented in the paper. The results obtained agree well with the exact solutions, experimental results, and other numerical computations, demonstrating the capability of the current method to deal with multiphase hydrodynamic problems involving complex geometries.

A discontinuous Galerkin method for shock capturing using a mixed high-order and sub-grid low-order approximation space

This article considers a new discretization scheme for conservation laws. The discretization setting is based on a discontinuous Galerkin scheme in combination with an approximation space that contains high-order polynomial modes as well as piece-wise constant modes on a sub-grid. The high-order modes can continuously be suppressed with a penalty function that is based on a sensor which is intertwined with the approximation space. Numerical tests finally illustrate the performance of this scheme.

Research on Computational Method of Supersonic Inlet/Isolator Internal Flow

In this research, a CFD solver is developed for solving the 2D/3D compressible flow problem: the finite volume method based on multi-block structural grids is used to solve the compressible Reynolds averaged Navier–Stokes equations (RANS). Included in the methodology are multiple high-order reconstruction schemes, such as the 3rd-order MUSCL (Monotone Upstreamcentered Schemes for Conservation Laws), 5th-order WENO (Weight Essentially Non-Oscillatory), and 5th-order MP (Monotonicity-Preserving) schemes. Of the variety of turbulence models that are embedded, this solver is mainly based on the shear stress transport model (SST), which is compatible with OpenMP/MPI parallel algorithms. This research uses the CFD solver to conduct steady-state flow simulation for a two-dimensional supersonic inlet/isolator, incorporating these high-precision reconstruction schemes to accurately capture the shock wave/expansion wave interaction and shock wave/turbulent boundary layer interaction (SWTBLI), among other effects. By comparing the 2D/3D computation results of the same inlet configuration, it is found that the 3D effects of the side wall cannot be ignored due to the existing strong lateral flow near the corner. To obtain a more refined turbulence simulation, the commercial software ANSYS Fluent 18.0 is used to carry out the detached eddy simulation (DES) and the large eddy simulation (LES) of the same supersonic inlet, so as to reveal the flow details near the separation area and boundary layers.

A balanced algorithm of the hybrid large-particle method and its verification on some test problems

In series of numerical experiments involving the Shu-Osher problem of nonlinear acoustics and the Woodward-Colella problem of two interacted blast waves, the author studied the computational properties of a new algorithm for the hybrid large-particle method. The numerical method is a two-step predictor-corrector type in time. Spatial derivatives are split by physical processes. At the first stage of splitting, the gradient and deformation terms of the conservation laws are taken into account, and at the second stage, convective flows are taken into account. The proposed balanced algorithm of the method includes a more dissipative upwind reconstruction of fluxes at the “predictor” step and a centered (non-dissipative on smooth solutions) approximation at the correction step: CDP2-UC (Customizable Dissipative Properties — Upwind-Centered). For a more flexible control of the numerical viscosity, a nonlinear correction of the scheme based on a parametric combination of known limiters is implemented. The numerical scheme has a second-order approximation in space and time on smooth solutions. The balanced algorithm of the hybrid large-particle method demonstrated a monotonic solution with a qualitative resolution of the details of the gas flow in the entire domain of determining the test problems. No spurious oscillations occurred during the process of fining the mesh, and convergence to the reference density profile was observed. The influence of a limiter on the numerical dissipation of the CDP2-UC scheme is analyzed. The results present the comparison with the following variants of the schemes: MUSCL (Monotone Upstream Scheme for Conservation Laws), MUSCL-CABARET with a NOLD limiter (Non-Oscillatory Low-Dissipative), the discontinuous Galerkin method with various forms of nonlinear correction, the hybrid weighted nonlinear scheme of the fourth order of approximation (CCSSR-HW4) and the popular WENO5 scheme (Weighted Essentially Non-Oscillatory Scheme) with fifth order of accuracy. The proposed algorithm successfully competes with modern numerical methods that have a formally higher (fourth and fifth) order of approximation. The hybrid large-particle method has the simplicity, uniformity, and cost-effectiveness of the algorithm, as well as high resolution. The test calculations allowed the author to estimate the range of parametric control of the numerical dissipation of the method for correct numerical modeling of the applied problems with nonlinear wave fields and strong shock waves.

A graphics processing unit-based robust numerical model for solute transport driven by torrential flow condition

Solute transport simulations are important in water pollution events. This paper introduces a finite volume Godunov-type model for solving a 4×4 matrix form of the hyperbolic conservation laws consisting of 2D shallow water equations and transport equations. The model adopts the Harten-Lax-van Leer-contact (HLLC)-approximate Riemann solution to calculate the cell interface fluxes. It can deal well with the changes in the dry and wet interfaces in an actual complex terrain, and it has a strong shock-wave capturing ability. Using monotonic upstream-centred scheme for conservation laws (MUSCL) linear reconstruction with finite slope and the Runge-Kutta time integration method can achieve second-order accuracy. At the same time, the introduction of graphics processing unit (GPU)-accelerated computing technology greatly increases the computing speed. The model is validated against multiple benchmarks, and the results are in good agreement with analytical solutions and other published numerical predictions. The third test case uses the GPU and central processing unit (CPU) calculation models which take 3.865 s and 13.865 s, respectively, indicating that the GPU calculation model can increase the calculation speed by 3.6 times. In the fourth test case, comparing the numerical model calculated by GPU with the traditional numerical model calculated by CPU, the calculation efficiencies of the numerical model calculated by GPU under different resolution grids are 9.8–44.6 times higher than those by CPU. Therefore, it has better potential than previous models for large-scale simulation of solute transport in water pollution incidents. It can provide a reliable theoretical basis and strong data support in the rapid assessment and early warning of water pollution accidents.

Accurate hybrid AUSMD type flux algorithm with generalized discontinuity sharpening reconstruction for two-fluid modeling

This paper presents a single-pressure-field two-fluid model with finite-volume discretization to solve the equations of motion of compressible multiphase flows. To capture the discontinuities caused by shock waves and fluid interfaces, we propose a generalized discontinuity sharpening technique that combines the conventional monotonic upstream scheme for conservation law (MUSCL) and tangent of hyperbola interface capturing (THINC) schemes. In addition, a slope ratio-weighted parameter, ζ, is used to control the proportion of values reconstructed by MUSCL and THINC, and we show that the present method can retain sharp interfaces when the value of the parameter β in the THINC scheme is set ranging from 1.6 to 3.0. Fluxes across various interfaces are evaluated using a hybrid AUSMD-type flux algorithm, where the mass flux and pressure induced on the cell faces are calculated using an approximate Riemann solver. The accuracy and robustness of the proposed method are validated by solving a series of one- and two-dimensional single-phase flows. Furthermore, complex wave patterns arising from two-dimensional shock bubble/water-column interactions are examined, which indicate that compared with the existing schemes applied to two-fluid modeling, the proposed scheme significantly sharpens the interfaces and captures more details of the flow features. Finally, simulations of a three-dimensional example of the liquid jet crossflow are conducted. The proposed scheme shows more details of the fluid interface, including the interfacial instabilities on the windward side of the liquid jet and droplet formation due to the breakup phenomenon in the downstream of the crossflow, than the existing schemes.