Total Variation Diminishing Research Articles

Течія рідини в циліндричному каналі з діафрагмами прямокутного профілю

The flow of a viscous incompressible liquid in a cylindrical duct with two serial diaphragms of a rectangular profile is studied by the numerical solution of the unsteady Navier-Stokes equations. The discretization procedure is based on the finite volume method using the TVD scheme for the discretization of the convective terms and second order accurate in both space and time difference schemes. The resulting system of non-linear algebraic equations is solved by the PISO algorithm. It is shown that the fluid flow in the region between the diaphragms is non-stationary and is characterized by the presence of an unstable shear layer under the certain parameters. A series of ring vortices is formed in the shear layer that causes quasi-periodic self-sustained oscillations of the velocity field in the vicinity of the orifice of the second diaphragm. In comparison with the case of rounded diaphragms, an increase in the maximum jet velocity is observed, which in turn leads to an increase in the frequency of self-sustained oscillations and a decrease in the Reynolds numbers at which quasi-periodic oscillations are excited.

IMPURITY TRANSFER MODEL OF TRANSPORT ORIGIN IN THE STRUCTURE OF THE STREETS-CANYONS OF THE CENTRAL PART OF KHARKOV

The work is devoted to the development of a model and the study of the characteristics of the process of transporting pollutants in the surface layer of the quarters of the central part of the city in the presence of plantings and terrain. The micro-district of the central part of the city of Kharkov is considered as the basis of the model. On this basis, the LAWEP approach was applied [17], which consists in preliminary calculation of advection for the general center model on a relatively coarse grid. A linear source is provided along the centerline of the streets, emitting a constant in time consumption of impurities. Methodology. The motion of the air is described by the Navier-Stokes equations averaged by Reynolds. To model turbulent transport effects, the High-Reynolds two-parameter differential turbulence model with wall functions was used as the base one. Modeling of space canopy with leaves and tree branches is carried out on the basis of a porous medium model. The study was carried out using the authors MTFS® software package, with TVD scheme of 2/3 order of accuracy. The Results. Satisfactory agreement with experimental data was obtained. A thermal island phenomenon has been detected. The influence of the structure of canyon streets on the distribution of impurities from vehicles is shown.

Vorticity Confinement Applied to Accurate Prediction of Convection of Wing Tip Vortices and Induced Drag

The vorticity confinement (VC) method was applied to tip vortices shed by edges of wings in order to predict induced drag using far-field integration. The optimal VC parameter was determined by its application to tip vortices shed by 3-D stationary and rotating wings. The VC was used with a total variation diminishing (TVD) approach to reduce the needed confinement. The TVD minmod and differentiable flux limiters were tested to evaluate their effect on the VC parameter. The 3-D inviscid simulations were post-processed using the wake-integral technique to determine lift-induced drag force. Dependence of the VC parameter on the flight Mach number and the angle of attack was evaluated. Grid convergence studies were conducted for induced drag generated by 3-D wing. The VC approach was combined with the Reynolds stress equation turbulence model, and the results were compared to experimental data of tip vortex evolution.

A conservative flux‐splitting method for steady shock capturing

AbstractA novel flux‐splitting method that can preserve the flux conservation properties for the steady shock waves is proposed in present paper. The proposed method is based on the total variation diminishing (TVD) scheme, and the dissipative flux term is constructed by the flux directly instead of conservative variables. Similar transformations are imposed on the anti‐dissipation term, which is formulated by utilizing the limiter function. Numerical experiments were performed on the one, quasi one‐ and two‐dimensional shock wave problems. The computed results show that, the steady shock waves can be accurately captured within two grid points by the proposed conservative flux‐splitting (CFS) method, and numerical oscillations are successfully eliminated near the flow discontinuities. The conservation and accuracy of present method are demonstrated by comparisons with the TVD, flux vector splitting and flux difference splitting methods.

An improved r‐factor algorithm for total variational diminishing (TVD) schemes on two‐dimension non‐uniform unstructured grids

AbstractFor advection simulation, an improved r‐factor algorithm for total variational diminishing (TVD) schemes is proposed to extend the TVD schemes to non‐uniform unstructured grids. In the new algorithm, the further upwind node (called node U) location is modified based on the size differences of the related grids, that is, the formula of the relationship between two length ratios, LUC/LCD = LCf/LfD, should be maintained to reveal the meaning of r‐factor correctly on non‐uniform unstructured grids. After that, the inverse‐distance weighting average method rather than the Gauss theory method is adopted to estimate the value of node U. Furthermore, the deviations between the value of cell centroids and of their corresponding auxiliary points are exploited to complete the algorithm. The new algorithm is utilized in two pure convection cases, including a double‐step profile and a sinusoidal profile. The algorithm was compared against the Hou's r‐factor algorithm by using Superbee and Van Leer limiters on two‐dimension non‐uniform unstructured grids. The results indicate that a monotonous behavior and a higher accuracy result can be obtained by using the new algorithm.

A low-dissipation shock-capturing framework with flexible nonlinear dissipation control

In this work, a framework to construct arbitrarily high-order low-dissipation shock-capturing schemes with flexible and controllable nonlinear dissipation for convection-dominated problems is proposed. While a set of candidate stencils of incremental width is constructed, each one is indicated as smooth or nonsmooth by the ENO-like stencil selection procedure proposed in the targeted essentially non-oscillatory (TENO) scheme (Fu et al. 2016 [9]). Rather than being discarded directly as with TENO schemes, the nonsmooth candidates are filtered by an extra nonlinear limiter, such as a monotonicity-preserving (MP) limiter or a total variation diminishing (TVD) limiter. Consequently, high-order reconstruction is achieved by assembling candidate fluxes with optimal linear weights since they are either smooth reconstructions or filtered ones which feature good non-oscillation property. A weight renormalization procedure as with the standard TENO paradigm is not necessary. This new framework concatenates the concepts of TENO, WENO and other nonlinear limiters for shock-capturing, and provides a new insight to designing low-dissipation nonlinear schemes. Through the adaptation of nonlinear limiters, nonlinear dissipation in the newly proposed framework can be controlled separately without affecting the performance in smooth regions. Based on the proposed framework, a family of new six- and eight-point nonlinear schemes with controllable dissipation is proposed. A set of critical benchmark cases involving strong discontinuities and broadband fluctuations is simulated. Numerical results reveal that the proposed new schemes capture discontinuities sharply and resolve the high-wavenumber fluctuations with low dissipation, while maintaining the desired accuracy order in smooth regions.

Debris Flow Analyst (DA): A debris flow model considering kinematic uncertainties and using a GIS platform

Although the main driving force of debris flows is simply gravity, the dynamic processes are very complex. Kinematic uncertainties during motion make it difficult to exactly simulate debris flows. In traditional numerical simulation models of debris flows, the kinematic uncertainties are generally not considered in the Lagrangian description. In addition, the challenge of obtaining accurate simulation parameters further suggests the necessity of probabilistic modeling of debris flows. In this study, a novel statistical debris flow simulation model (Debris Flow Analyst model), is proposed in the Lagrangian description, in which the inherent stochasticity in the motion of a debris flow is described by stochastic differential equations (SDEs). In this model, the accelerations of moving debris are calculated by Newton's second law, and their locations are estimated based on Monte Carlo simulation. According to case studies, the proposed model provides results close to the analytical solution in a 1-D simulation of dam breaks in the Eulerian description and reduces some numerical oscillations. Additionally, the Debris Flow Analyst model can produce reasonable results based on a single-phase rheological model in 2-D simulations of a debris flow in Tibet and a flow-like landslide triggered by storms on 4–5 November 1993 in Hong Kong. From the theoretical perspective, our model is equivalent to the total variation diminishing (TVD) scheme in the Eulerian description. In addition, the statistical model provides a new tool to estimate kinematic uncertainties in the hazard analysis of debris flows on GIS platforms.

Parallel discontinuous Galerkin finite element method for computing hyperbolic conservation law on unstructured meshes

PurposeThe discontinuous Galerkin finite element method (DGFEM) is very suited for realizing high order resolution approximations on unstructured grids for calculating the hyperbolic conservation law. However, it requires a significant amount of computing resources. Therefore, this paper aims to investigate how to solve the Euler equations in parallel systems and improve the parallel performance.Design/methodology/approachDiscontinuous Galerkin discretization is used for the compressible inviscid Euler equations. The multi-level domain decomposition strategy was used to deal with the computational grids and ensure the calculation load balancing. The total variation diminishing (TVD) Runge–Kutta (RK) scheme coupled with the multigrid strategy was employed to further improve parallel efficiency. Moreover, the Newton Block Gauss–Seidel (GS) method was adopted to accelerate convergence and improve the iteration efficiency.FindingsNumerical experiments were implemented for the compressible inviscid flow problems around NACA0012 airfoil, over M6 wing and DLR-F6 configuration. The parallel acceleration is near to a linear convergence. The results indicate that the present parallel algorithm can reduce computational time significantly and allocate memory reasonably, which has high parallel efficiency and speedup, and it is well-suited to large-scale scientific computational problems on multiple instruction stream multiple data stream model.Originality/valueThe parallel DGFEM coupled with TVD RK and the Newton Block GS methods was presented for hyperbolic conservation law on unstructured meshes.

Construction and application of several new symmetrical flux limiters for hyperbolic conservation law

The construction of limiter functions is a crucial factor in total-variation-diminishing (TVD) schemes to achieve high resolution and numerical stability. In this paper, three symmetrical limiter functions are constructed by introducing the MAX function into the classical van Albada, van Leer, and PR-κ limiters. The analysis and numerical results demonstrate that the MUSCL scheme equipped with the proposed limiters satisfies the sufficient conditions for second-order convergence in smooth regions and exhibits lower dissipation and better resolution than the MUSCL scheme utilizing classic limiters corresponding to both smooth and discontinuous solutions.

Coastal pollutant transport modeling using smoothed particle hydrodynamics with diffusive flux

The Smoothed Particle Hydrodynamics (SPH) method is used in this paper to simulate the transport process of coastal pollutant by solving the two-dimensional (2D) depth averaged advection-diffusion equation in Lagrangian framework. To avoid directly discretizing the second-order derivatives, the diffusion terms are decomposed into two first-order derivatives by introducing the diffusive flux. To verify the proposed Lagrangian particle model, numerical experiments on four typical tests are performed, including the discontinuity, pure advection, grid anisotropy and large deforming flow. The high-resolution Total Variation Diminishing (TVD) scheme - the modified TVD Lax–Friedrichs method with Superbee limiter (MTVDLF-Superbee) - is also implemented for detailed comparison. The simulation results indicated that the present SPH-based model has better performance than MTVDLF-Superbee. In addition, the stability, convergence and efficiency of the proposed model are numerically analyzed. The Lagrangian particle model with diffusive flux gives satisfactory results with big time step and hence allows large Courant number, guaranteeing the computational efficiency.

Inertia-driven and elastoinertial viscoelastic turbulent channel flow simulated with a hybrid pseudo-spectral/finite-difference numerical scheme

Numerical simulation of viscoelastic flows is challenging because of the hyperbolic nature of viscoelastic constitutive equations. Despite their superior accuracy and efficiency, pseudo-spectral methods require the introduction of artificial diffusion (AD) for numerical stability in hyperbolic problems, which alters the physical nature of the system. This study presents a hybrid numerical procedure that integrates an upwind total variation diminishing (TVD) finite-difference scheme, which is known for its stability in hyperbolic problems, for the polymer stress convection term into an overall pseudo-spectral numerical framework. Numerically stable solutions are obtained for Weissenberg number well beyond O(100) without the need for either global or local AD. Side-by-side comparison with an existing pseudo-spectral code reveals the impact of AD, which is shown to differ drastically between flow regimes. Elastoinertial turbulence (EIT) becomes unphysically suppressed when AD, at any level necessary for stabilizing the pseudo-spectral method, is used. This is attributed to the importance of sharp stress shocks in its self-sustaining cycles. Nevertheless , in regimes dominated by the classical inertial mechanism for turbulence generation, there is still an acceptable range of AD that can be safely used to predict the statistics, dynamics, and structures of drag-reduced turbulence. Detailed numerical resolution analysis of the new hybrid method, especially for capturing the EIT states, is also presented.

Numerical Approach of the Equilibrium Solutions of a Global Climate Model

We consider a coupled model surface-deep ocean effect, where an Energy Balance Model (EBM) is used for modelling the surface temperature and a two-dimensional heat equation represents the evolution of the temperature of the deep ocean. Although the model under study is based on that proposed by Watts & Morantine (1990), here we consider a modified model that incorporates other processes, such as the nonlinear diffusion and the action of coalbedo, depending on the temperature. The stationary states of the model under study, taking the solar constant as the parameter, are numerically attained. The results of the simulation are depicted in a {(Q,u)} plot where u is the temperature in the surface and Q is the solar constant. The numerical solution is achieved by means of a finite volume scheme with Weighted Essentially Non-Oscillatory (WENO) reconstruction in space and third order Runge-Kutta scheme, which verifies the Total Variation Diminishing (TVD) property, for time integration. The equilibrium states are accomplished by evolving in time the numerical solution until the stationary solutions are reached. The main novel results of this work concern the numerical obtention of the stationary solutions of both the EBM and the coupled model EBM-deep ocean and the agreement of these results with the theoretically obtained in previous works, where an interval of values of the solar constant Q was obtained with the existence of at least three stationary solutions. In this work, we have numerically obtained more than three stationary solutions for such interval of Q.

Hybrid scheme for complex flows on staggered grids and application to multiphase flows

A hybrid scheme is developed that enables advanced simulations, such as multiphase flows, in complex flow setups with compressible flow solvers using a staggered arrangement of flow variables. For this purpose, a weighted essentially non-oscillatory (WENO) scheme is adapted for staggered grids and combined with a finite difference central scheme. Hybrid weights that maintain continuity in time are computed based on a limiter constructed from the deviation of the local weights from the optimal weights. A total variation diminishing (TVD) interpolation is introduced to avoid numerical oscillations from the flow field. Additionally, the finite difference operators are reformulated in a flux-based finite volume form for convective terms in order to avoid spurious oscillations for turbulent configurations. For demonstrating the accuracy of the newly developed scheme and its improvements compared to the existing schemes, a modified wavenumber analysis is shown. Furthermore, the new scheme is applied to single and multiphase test cases. Finally, it is used for simulating the needle-opening process of a gasoline direct injector (GDI) and predicting the amount of gas inside the nozzle.

Combined Vorticity Confinement and TVD Approaches for Accurate Vortex Modelling

The vorticity confinement (VC) method was used with total variation diminishing (TVD) schemes to reduce possible over-confinement and applied to convected and stationary vortices. The optimal VC parameter was determined by application to 2-D vortices. The conservativity of the scheme is investigated in terms of kinetic energy and momentum. Grid convergence study was conducted for the range of grid steps. The log–log plot of the error shows that the second-order upwind scheme with the VC nearly reaches the 4th order of accuracy for the wide range of grid cell sizes. VC was used with TVD minmod and differentiable flux limiters to evaluate their effect on the VC method and prevention of over-confinement.

Calculation of Waves in an Elastic-Plastic Body Based on ENO Modifications of the Godunov Method

The efficiency of two ENO modifications of the Godunov method for calculation axisymmetric problems of the dynamics of elastic-plastic bodies, namely a UNO scheme of the second-order of accuracy and a TVD scheme of the second order of accuracy outside the solution extremes where it decreases to the first order, has been numerically studied. The efficiency of the modifications is estimated by comparing the results of calculation of a number of problems on the propagation of spherical and cylindrical waves in a body with the results of reference solutions and the Godunov method. The exact and numerical solutions are used as reference solutions, the latter are obtained on fine grids. It is shown that for all the problems considered, the solutions by the TVD and UNO schemes are much closer to the reference ones than those by the Godunov method. The TVD and UNO schemes are characteristic of significantly less smearing of sharp wave fronts and a more accurate resolution of extrema. At the same time, the wave profiles and the extrema are reproduced by the UNO scheme somewhat better than by the TVD scheme.

Comparative Study on Various Numerical Schemes for Solving Nozzle Flow Problems

The inviscid compressible flows represent the flow model of high-speed flow with the viscous effects can be ignored. Such flow phenomena appear when a high Reynolds number flow past trough a streamlined body at relative a low angle attack. The most commercial passenger airplane is operated at such flow condition, as results the aircraft industries have already put their efforts to develop the capability for solving the governing equation of inviscid compressible flow. The governing equation of inviscid compressible flow which is called as the Euler Equation can solved numerically. There are various numerical schemes had been developed and the most of them are dedicated for capturing the presence of shock wave phenomena which is normally appearing at high speed flow. The present work presents a comparative study over various numerical schemes which they are already been developed for solving the Euler equations. Those numerical schemes are the MacCormack Scheme, the Fourth Order Runge Kutta Scheme, The First and Second Order Flux Splitting Steger Warming Scheme, Flux Splitting Bram Van Leer Scheme, the Harten Yee TVD scheme, Roe Sweby TVD scheme, Davies – Yee TVD scheme and the Modified Four Order Runge Kutta and Harten Yee TVD scheme. These nine numerical schemes are applied to the case of flow past through nozzle. The flow phenomena may appear along the nozzle may be the form completely as an isentropic flow or the flow with shock wave. These two flow phenomena are depended the boundary condition of the flow problem under considered. The implementation to case of isentropic flow along the nozzle, those nine numerical schemes works well, and the number of iterations required nearly the same between one with the other except for the Fourth Order Runge Kutta Scheme. In the case of flow with shock wave, the Fourth Order Runge Kutta Scheme, Second Order Steger Warming and the Bram Van Leer Scheme are failing. The presence of shock wave, each numerical scheme requires more iteration number to achieve a convergent solution. The Modified Runge Kutta - Harten Yee scheme may represent an appropriate a numerical scheme for solving the flow problem with the presence of shock wave.

Effect of hydrophobic patch on the modulation of electroosmotic flow and ion selectivity through nanochannel

In this article we made a systematic study of the electrokinetically driven flow through a long nanochannel patterned with charged hydrophobic patch embedded along the walls. The hydrophobic patch possess either similar or opposite charge to that of the hydrophilic portion of the channel wall. The hydrodynamic slip length is considered to be a function of the surface charge distributed along the patch. We consider the aqueous medium as binary symmetric electrolyte solution with constant property Newtonian fluid. We adopt a nonlinear model based on the Poisson-Nernst-Planck equations coupled with Navier–Stokes equations. The coupled set of governing equations are solved using a finite volume method. We use a higher order upwind based total variation diminishing (TVD) scheme to discretize the convective and electromigration terms (i.e., hyperbolic terms) and the central difference scheme is used to discretize the diffusion term. We have identified several interesting key features of the modulation of electroosmotic flow (EOF) through such a nanochannel. An enhancement or a reduction in average flow rate may be achieved depending on the polarity of the charges distributed along the hydrophobic or hydrophilic portion of the channel wall. We have also shown that the reversal in EOF may occur depending on the critical choice of the pertinent parameters. We have further studied the selectivity of mobile ions by regulating the charge properties of the channel walls.

Numerical 2D MHD simulations of the collapse of magnetic rotating protostellar clouds with the Enlil code

AbstractWe numerically investigate the gravitational collapse of rotating magnetic protostellar clouds. The simulations are performed using 2D MHD code ‘Enlil’. The code is based on TVD scheme of increased order of accuracy. We developed a model of the initially non-uniform cloud, which self-consistently treats gas density and large-scale magnetic field distribution. Simulation results for the typical parameters of a solar mass cloud are presented. In agreement with our previous results for the uniform cloud, the isothermal collapse of the non-uniform cloud results in formation of hierarchical structure of the cloud, consisting of flattened envelope and thin quasi-magnetostatic primary disk near its equatorial plane. The non-uniform cloud collapses longer than the uniform one, since the magnetic field is dynamically stronger at the periphery of the cloud in the former case.

Compressive properties of Min-mod-type limiters in modelling shockwave-containing flows

The long-ignored compressive properties of Min-mod-type limiter is investigated in this manuscript by demonstrating its potential in numerically modelling shockwave-containing flows, especially in shock wave/boundary layer interaction (SWBLI) problems. Theoretical studies were firstly performed based on Sweby’s total variation diminishing (TVD) limiter region and Spekreijse’s monotonicity-preserving limiter region to indicate Min-mod-type limiters’ compressive properties. The influence of limiters on the solution accuracy was evaluated using a hybrid-order analysis method based on the grid-independent study in three typical shockwave-containing flows. The conclusions are that, Min-mod-type limiter can be utilized as a dissipative and/or compressive limiter, but depending on the reasonable value of the compression parameter. The compressive Min-mod limiter tends to be more attractive in modelling shockwave-containing flows as compared to other commonly preferred limiters because of its stable computational process and its high-resolution predictions. However, the compressive Min-mod limiter may suffer from its slightly poor convergence, as that observed in other commonly accepted smooth limiters in modelling SWBLI problems.

Catchment-scale multi-process modeling with local time stepping

Numerical modeling is an important tool for prediction, analysis and understanding of the dynamics of land surface processes. To increase the usage and impact of such tools, it is crucial to decrease runtime by increasing computational efficiency. Dynamic processes such as water flow are typically described by higher-order differential equations. Solving these accurately requires numerical integration over time, where numerical errors depend on the time steps taken. Typically, flow simulation use the smallest required time steps in a model’s domain to simulate flow. In this paper, we analyze the usage of local time stepping, for catchment-scale simulation of land surface processes such as water flow, infiltration, slope stability and landslide runout. In such a scheme, temporal integration is cell specific, allowing for higher numerical efficiency. The implemented scheme works with fully free local time steps that are synchronized only for visualization. We implement this method in a monotonic upwind scheme for conservation laws (MUSCL). We investigate the influence on stability and the resulting changes in computation time and accuracy in a hydrology-coupled, catchment-scale flood simulation. Results show that local time stepping can be implemented in a total variation diminishing (TVD) numerical scheme that is second-order spatially accurate. Simulation results in both 1D dam-break scenarios and catchment-scale flash flood scenarios show insignificant changes in modeling result, while computation time reduces with over 50%. Finally, the method is successfully implemented in a multi-process lands surface model with hydrology, flooding, slope failure, and runout. The implementation of a local time stepping for computation of dynamic land surface processes could be implemented widely for increased computational efficiency without significant loss of accuracy.