Total Variation Diminishing Research Articles

Numerical Simulations Using TVD Schemes of Two-Dimensional Supersonic Flow in Chemical Equilibrium

A numerical scheme for the solution of both unsteady and steady-state, two-dimensional Euler equations considering gas in chemical equilibrium, is presented. Three alternatives of the Total Variation Diminishing (TVD) Harten–Yee scheme are implemented. One of them is a technique based on the adaptive use of different limiter functions in each wave of the inter-cell Riemann problem. With this technique, the undesirable effects of the artificial viscosity on the capture of contact discontinuities are reduced, without loss of robustness in nonlinear waves resolution. In order to verify the accuracy of the proposed scheme, results of the unsteady flow in cylindrical explosions, and of the steady-state solution of hypersonic flow over a blunt body, are presented. Finally, comparisons considering accuracy of results and convergence properties between the three Harten–Yee schemes are carried out.

Transient evolution of suspended and benthic algae in a riverine ecosystem: A numerical study

It is a well-established fact that there is no direct resource competition between pelagic and benthic primary producers when their habitats are located in physically separate areas. However, the benthic habitat located at the bottom of the water column is affected by the light rays that gets attenuated as it passes through the pelagic zone and is obstructed by the algal cells and suspended material present in the water column. Therefore, benthic primary production is very much influenced by the conditions and concentrations in the pelagic zone. Another level of complexity is added to the system while considering the impact of flowing water in a riverine environment. In the research presented here a numerical model is developed to examine the impact of flow and nutrients on pelagic and benthic primary producers. The aforementioned numerical model solves advection diffusion and reaction (ADR) equation through TVD (total variation diminishing) − MacCormack scheme. The diffusion term is solved through central difference scheme. The model results are first validated by comparing the results with analytical solutions for the simplified case. The validated model is then applied to a 30kms stretch of the Bode River and simulations are conducted for multiple flow conditions. Impact of transient flow and nutrient boundary conditions on the evolution of algal trait is examined. Our simulation exercise highlights the importance of residence time in the temporal evolution of algae in pelagic and benthic zone and further identifies the most sensitive parameter influencing the evolution of the algal community modelled. Our research also reveals that largest uncertainty in the modelling stems from the wide range of entrainment rate that could be used for modelling resuspension of benthic algae in pelagic zone. The use of higher entrainment rate increased the algal concentration in the water column by 48% for identical flow conditions.

Explicit and Implicit TVD Schemes for Conservation Laws with Caputo Derivatives

In this paper, we investigate numerical approximations of the scalar conservation law with the Caputo derivative, which introduces the memory effect. We construct the first order and the second order explicit upwind schemes for such equations, which are shown to be conditionally $$\ell ^1$$l1 contracting and TVD. However, the Caputo derivative leads to the modified CFL-type stability condition, $$ (\Delta t)^{\alpha } = O(\Delta x)$$(Δt)ź=O(Δx), where $$\alpha \in (0,1]$$źź(0,1] is the fractional exponent in the derivative. When $$\alpha $$ź is small, such strong constraint makes the numerical implementation extremely impractical. We have then proposed the implicit upwind scheme to overcome this issue, which is proved to be unconditionally $$\ell ^1$$l1 contracting and TVD. Various numerical tests are presented to validate the properties of the methods and provide more numerical evidence in interpreting the memory effect in conservation laws.

Implementation and Comparison of High-Resolution Spatial Discretization Schemes for Solving Two-Fluid Seven-Equation Two-Pressure Model

As compared to the two-fluid single-pressure model, the two-fluid seven-equation two-pressure model has been proved to be unconditionally well-posed in all situations, thus existing with a wide range of industrial applications. The classical 1st-order upwind scheme is widely used in existing nuclear system analysis codes such as RELAP5, CATHARE, and TRACE. However, the 1st-order upwind scheme possesses issues of serious numerical diffusion and high truncation error, thus giving rise to the challenge of accurately modeling many nuclear thermal-hydraulics problems such as long term transients. In this paper, a semi-implicit algorithm based on the finite volume method with staggered grids is developed to solve such advanced well-posed two-pressure model. To overcome the challenge from 1st-order upwind scheme, eight high-resolution total variation diminishing (TVD) schemes are implemented in such algorithm to improve spatial accuracy. Then the semi-implicit algorithm with high-resolution TVD schemes is validated on the water faucet test. The numerical results show that the high-resolution semi-implicit algorithm is robust in solving the two-pressure two-fluid two-phase flow model; Superbee scheme and Koren scheme give two highest levels of accuracy while Minmod scheme is the worst one among the eight TVD schemes.

Numerical study of galloping oscillation of a two-dimensional circular cylinder attached with fixed fairing device

Galloping oscillation has been observed in the practical applications of the fairing devices on offshore oil and gas risers. The ability of the riser movement prediction codes to predict the galloping oscillation is important since major structure failures due to poor design may be avoided. Few reports have been found on the numerical simulation of the galloping oscillation of the faired cylinders. In the present work, a numerical investigation is performed on the oscillation of a two-dimensional circular cylinder attached with a short fairing device. The cylinder is free to move in the cross-flow direction, supported by a spring and a damper and the fairing device is firmly attached to the cylinder. The flow field is simulated with an unsteady Reynolds-averaged Navier–Stokes (URANS) model corrected with the arbitrary Lagrange Euler (ALE) method and the governing equations are solved with a total variation diminishing (TVD) scheme. The SST turbulence model is used. Under the reduced velocities varying from 3 to 24, the simulation results of the fluid–solid interaction process are obtained and severe galloping oscillations are captured. The mechanism behind the galloping oscillation of the faired device is analyzed combining the cylinder movements and the corresponding vorticity fields.

TVD scheme for stiff problems of wave dynamics of heterogeneous media of nonhyperbolic nonconservative type

A finite-difference TVD scheme is presented for problems in nonequilibrium wave dynamics of heterogeneous media with different velocities and temperatures but with identical pressures of the phases. A nonlinear form of artificial viscosity depending on the phase relaxation time is proposed. The computed solutions are compared with exact self-similar ones for an equilibrium heterogeneous medium. The performance of the scheme is demonstrated by numerical simulation with varying particle diameters, grid sizes, and particle concentrations. It is shown that the scheme is efficient in terms of Fletcher’s criterion as applied to stiff problems.

Towards the mechanical characterization of abdominal wall by inverse analysis

The aim of this study is to characterize the passive mechanical behaviour of abdominal wall in vivo in an animal model using only external cameras and numerical analysis. The main objective lies in defining a methodology that provides in vivo information of a specific patient without altering mechanical properties. It is demonstrated in the mechanical study of abdomen for hernia purposes. Mechanical tests consisted on pneumoperitoneum tests performed on New Zealand rabbits, where inner pressure was varied from 0mmHg to 12mmHg. Changes in the external abdominal surface were recorded and several points were tracked. Based on their coordinates we reconstructed a 3D finite element model of the abdominal wall, considering an incompressible hyperelastic material model defined by two parameters. The spatial distributions of these parameters (shear modulus and non linear parameter) were calculated by inverse analysis, using two different types of regularization: Total Variation Diminishing (TVD) and Tikhonov (H1). After solving the inverse problem, the distribution of the material parameters were obtained along the abdominal surface. Accuracy of the results was evaluated for the last level of pressure.Results revealed a higher value of the shear modulus in a wide stripe along the craneo-caudal direction, associated with the presence of linea alba in conjunction with fascias and rectus abdominis. Non linear parameter distribution was smoother and the location of higher values varied with the regularization type. Both regularizations proved to yield in an accurate predicted displacement field, but H1 obtained a smoother material parameter distribution while TVD included some discontinuities. The methodology here presented was able to characterize in vivo the passive non linear mechanical response of the abdominal wall.

Derivative artificial compression method for conservation laws with multi-dimensional extension

ABSTRACTA new resolution-enhancing technique called derivative artificial compression method is developed with multi-dimensional extension. The method is constructed via applying high-resolution difference schemes on derivative equations of conservation laws. In this way, one could overcome the defect of accuracy decay at extreme points that has plagued almost all high-resolution schemes. The new method has high resolution, low dissipation and low diffusion properties, and could enhance the resolution (of numerical solution) both at discontinuities and at extreme points. Numerical experiments are implemented using initial value problems of single conservation law, one-dimensional shock-tube problem, two-dimensional Riemann problems, double Mach reflection problem, and a shock reflection from a wedge. Resolutions of discontinuities, extremes and fine structures are compared between the original TVD scheme, TVD scheme with artificial compression method and TVD scheme with derivative artificial compression method.

Efficiency benchmarking of seventh-order tri-diagonal weighted compact nonlinear scheme on curvilinear mesh

ABSTRACTEfficiency improvements of high-order weighted compact nonlinear scheme (WCNS) are verified using a series of benchmark cases, proposed at the International Workshop on High-Order CFD Methods. A seventh-order tri-diagonal compact one of WCNSs (WCNS-E8T7), constructed in recent years, is investigated as a basic scheme, and compared to a typical fifth-order explicit WCNS (WCNS-E6E5) and a traditional second-order TVD scheme MUSCL. Among these tests, a symmetrical conservative metric method (SCMM) is adopted to ensure the accuracy and robustness of WCNSs when solving cases in the curvilinear coordinates. The computational efficiency of schemes is evaluated based on a non-dimensional cost in achieving the same level of accuracy. Related results show that WCNS-E8T7 has a better performance than WCNS-E6E5 with the same interpolation stencils. Moreover, the opinion that high-order methods can obtain higher computational efficiency than second-order methods is demonstrated on the cases ranging from academic problems to real-life computations.

A Novel Explicit-Implicit Coupled Solution Method of SWE for Long-term River Meandering Process Induced by Dambreak

Large amount of sediment deposits in the reservoir area can cause dam break, which not only leads to an immeasurable loss to the society, but also the sediments from the reservoir can be transported to generate further problems in the downstream catchment. This study aims to investigate the short-to-long term sediment transport and channel meandering process under such a situation. A coupled explicit-implicit technique based on the Euler-Lagrangian method (ELM) is used to solve the hydrodynamic equations, in which both the small and large time steps are used separately for the fluid and sediment marching. The main feature of the model is the use of the Characteristic-Based Split (CBS) method for the local time step iteration to correct the ELM traced lines. Based on the solved flow field, a standard Total Variation Diminishing (TVD) finite volume scheme is applied to solve the sediment transportation equation. The proposed model is first validated by a benchmark dambreak water flow experiment to validate the efficiency and accuracy of ELM modelling capability. Then an idealized engineering dambreak flow is used to investigate the long-term downstream channel meandering process with non-uniform sediment transport. The results showed that both the hydrodynamic and morphologic features have been well predicted by the proposed coupled model.

SIMULATIONS OF VISCOUS ACCRETION FLOW AROUND BLACK HOLES IN A TWO-DIMENSIONAL CYLINDRICAL GEOMETRY

ABSTRACT We simulate shock-free and shocked viscous accretion flows onto a black hole in a two-dimensional cylindrical geometry, where initial conditions were chosen from analytical solutions. The simulation code used the Lagrangian total variation diminishing plus remap routine, which enabled us to attain high accuracy in capturing shocks and to handle the angular momentum distribution correctly. The inviscid shock-free accretion disk solution produced a thick disk structure, while the viscous shock-free solution attained a Bondi-like structure, but in either case, no jet activity nor any quasi-periodic oscillation (QPO)-like activity developed. The steady-state shocked solution in the inviscid as well as in the viscous regime matched theoretical predictions well. However, increasing viscosity renders the accretion shock unstable. Large-amplitude shock oscillation is accompanied by intermittent, transient inner multiple shocks. This oscillation of the inner part of the disk is interpreted as the source of QPO in hard X-rays observed in micro-quasars. Strong shock oscillation induces strong episodic jet emission. The jets also show the existence of shocks, which are produced as one shell hits the preceding one. The periodicities of the jets and shock oscillation are similar; the jets for the higher viscosity parameter appear to be stronger and faster.

Numeric simulation of wet-steam two-phase condensing flow in a steam turbine cascade

Research on the wet-steam condensing flow problem is important for improving steam turbine efficiency. This paper derived a total variation diminishing (TVD) discrete method for a two-dimensional curvilinear coordinate system, which discretizes a control equation via a difference scheme that can accurately describe the transonic flow of compressible gas and capture abrupt parameter changes. This method can be applied to improve the shock wave capture accuracy in a cascade passage. A Cascade in White test is chosen as the study object to obtain test data to verify the accuracy of the proposed model. The calculation results indicate that the condensation process has a significant impact on the local flow parameters and that the distribution gradient of the parameters for the liquid phase is relatively large in the condensation position. The degree of subcooling of wet steam is the primary factor that influences the parameter distribution for the liquid phase in the main flow.

Error analysis of flux limiter schemes at extrema

Total variation diminishing (TVD) schemes have been an invaluable tool for the solution of hyperbolic conservation laws. One of the major shortcomings of commonly used TVD methods is the loss of accuracy near extrema. Although large amounts of anti-diffusion usually benefit the resolution of discontinuities, a balanced limiter such as Van Leer's performs better at extrema. Reliable criteria, however, for the performance of a limiter near extrema are not readily apparent. This work provides theoretical quantitative estimates for the local truncation errors of flux limiter schemes at extrema for a uniform grid. Moreover, the component of the error attributed to the flux limiter was obtained. This component is independent of the problem and grid spacing, and may be considered a property of the limiter that reflects the performance at extrema. Numerical test problems validate the results.

Numerical investigation of hydrogen detonation suppression with inert particle in pipelines

To numerically study the propagation process of detonation of H2O2N2 mixture and explosion suppression process with inert particle explosion, this paper developed theoretical models for detonation of one-dimensional combustible system and explosion suppression with inert particle based on two-fluid model by considering elementary chemical reactions, shock wave as well as momentum and energy transport effects between gas and particle phases. Meanwhile, Riemann invariant was derived on the basis of the one-dimensional problem. Gas and particle phase equations were solved using TVD and MacCormack schemes, respectively, Stiffness of equations was addressed by time-splitting scheme. Runge-Kutta method was employed to solve chemical reaction. Based on these equations, numerical method was constructed to analyze detonation of H2O2N2 mixture and explosion suppression process. The results showed that under certain initial conditions, high-temperature fireball induced shock wave formed and subsequently developed into detonation. After loading suitable concentration of inert particle into a certain explosion suppression area, combustion flame decelerated due to the transfer of momentum and energy between gas and particle phases, and distance between shock wave and flame increased gradually to lose energy support. Meanwhile, inert particles also absorbed shock wave energy. The two effects proceeded simultaneously and mutually fed back to rapidly attenuate shock waves, thereby successfully suppressing explosion. The inert particle density, concentration and diameter were critical important to explosion suppressant. Besides the product, reactant and some mainly radical components affected the detonation velocity. Especially, OH radicals were present almost only in the combustion reaction zone, indicating their significant impact on the chemical reaction rates.

An improved accurate monotonicity-preserving scheme for the Euler equations

The accurate monotonicity-preserving (MP) scheme of Suresh and Huynh (1997) [5] is a high-order and high-resolution method for hyperbolic conservation laws. However, the robustness of the MP scheme is not very high. In this paper, a detailed analysis on this scheme is performed, and two potential causes which may account for the weak robustness are revealed. Furthermore, in order to enhance the robustness of the MP scheme, an improved version of the MP scheme is presented, in which a strict continuous total-variation-diminishing (TVD) numerical flux is used at a disturbed discontinuity so that oscillations cannot grow indefinitely without violating the TVD condition. Without destroying the very high resolution property, numerical tests show that the improved scheme shares a strong robustness in simulating extreme numerical tests.

Random dynamic analysis of a train-bridge coupled system involving random system parameters based on probability density evolution method

The development of high-speed railway has made it important to clarify the influence of random system parameters (i.e. vehicle load, elastic modulus, damping ratio, and mass density of bridge) on train-bridge dynamic interactions. The probability density evolution method (PDEM), a newly developed theory which is applicable to train-bridge systems, can capture instantaneous probability density functions of dynamic responses. In this study, PDEM is employed to implement random dynamic analysis of a 3D train-bridge system subjected to random system parameters. The number theory method (NTM) is employed to choose the representative point sets of random parameters, whose initial probability distribution is divided by Voronoi cells., MATLAB® software is prepared for calculation, the Newmark-β integration method and the bilateral difference method of TVD (total variation diminishing) are adopted for solution. A case study is presented in which the train travels on a three-span simply supported high-speed railway bridge. The calculation accuracy and computational efficiency of the PDEM has been verified and some conclusions are provided. Furthermore, the influence of train speed under various combinations of random parameters is beyond discuss.

High order accurate dual-phase-lag numerical model for microscopic heating in multiple domains

In this article, a characteristic-based dual-phase-lag numerical model based on finite difference method has been developed to predict the microscopic heating response in time as well as consideration of the micro-structured effect. High-order TVD (Total Variation Diminishing) schemes being oscillation-free can yield high-order accurate solutions without introducing wiggles and therefore are utilised in this work. A multi-domain approach integrated within the dual-phase-lag numerical model allows the computation of microscopic conjugate heat transfer problems. Effects of different phase-lag values on the behaviour of heat transfer are investigated. The model is capable of predicting temperature patterns transiting from the wave nature of heat propagation to additional diffusion being experienced within different solid regions via phonon–electron interaction or phonon scattering.

Time-Domain Numerical Modeling of Brass Instruments Including Nonlinear Wave Propagation, Viscothermal Losses, and Lips Vibration

A time-domain numerical modeling of brass instruments is proposed. On one hand, outgoing and incoming waves in the resonator are described by the Menguy-Gilbert model, which incorporates three key issues: nonlinear wave propagation, viscothermal losses, and a variable section. The non-linear propagation is simulated by a TVD scheme well-suited to non-smooth waves. The fractional derivatives induced by the viscothermal losses are replaced by a set of local-in-time memory variables. A splitting strategy is followed to couple optimally these dedicated methods. On the other hand, the exciter is described by a one-mass model for the lips. The Newmark method is used to integrate the nonlinear ordinary differential equation so-obtained. At each time step, a coupling is performed between the pressure in the tube and the displacement of the lips. Finally, an extensive set of validation tests is successfully completed. In particular, self-sustained oscillations of the lips are simulated by taking into account the nonlinear wave propagation in the tube. Simulations clearly indicate that the nonlinear wave propagation has a major influence on the timbre of the sound, as expected. Moreover, simulations also highlight an influence on playing frequencies, time envelopes and on the playability of the low frequencies in the case of a variable lips tension.

Comparison and modification: TVD schemes for scalar transport on an unstructured grid

In most TVD schemes, the r-factors were proposed according to the cell-centered (CC) finite volume method (FVM) framework for the numerical approximation to the convective term. However, it is questionable whether those r-factors would be appropriate and effective for the vertex-centered (VC) FVM. In the paper, we collected five kinds of r-factor formulae and found out that only three of those, respectively by Bruner (1996), Darwish and Moukalled (2003) and Cassuli and Zanolli (2005) can be formally extended to a context of the VC FVM. Numerical tests indicate that the TVD schemes and r-factors, after being extended and introduced to a context of the VC FVM, maintained their similar characteristics as in a context of the CC FVM. However, when the gradient-based r-factors and the SUPERBEE scheme were applied simultaneously, non-physical oscillations near the sharp step would appear. In the transient case, the oscillations were weaker in a context of the VC FVM than those in a context of the CC FVM, while the effect was reversed in the steady case. To eliminate disadvantages in the gradient-based r-factor formula, a new modification method by limiting values on the virtual node, namely ϕU in the paper, was validated by the tests to effectively dissipate spurious oscillations.

NUMERICAL STUDY CONCERNING THERMAL RESPONSES OF NANOFILMS UNDER THE THERMOMASS THEORY

he Thermomass theory is based on the relationship mass-energy of Einstein, i.e., the heat has mass-energy duality, behaving as energy in processes where its conversion occurs in another form of energy, and behaving as mass in heat transfer processes. The mathematical model stablished by the Thermomass model falls within the class of problems called models non-Fourier heat conduction. The present work aims to analyze the thermal responses provided by Thermomass theory of nanofilms submitted to a very fast heating process using two different heat sources (laser pulses). During the process of analysis, the equations are written in conservation law, put into dimensionless form and discretized in the way that a high-order TVD scheme is used on to provide accurate and reliable numerical simulations for obtaining the thermal responses predicted by the Thermomass model. The results show that the Thermomass theory predicted a heterogeneous temperature distribution with elevated temperature peaks. The thermal responses obtained from this model may prevent the thermal damage caused by technologies of the processing and manufacturing of elements based on high-power laser applications.