A Low Mach Number IMEX Flux Splitting for the Level Set Ghost Fluid Method

Considering droplet phenomena at low Mach numbers,large differences in the magnitude of the occurring characteristic waves are presented.As acoustic phenomena often play a minor role in such applications,classical explicit schemes which resolve these waves suffer from a very restrictive timestep restriction.In this work,a novel scheme based on a specific level set ghost fluid method and an implicit-explicit(IMEX)flux splitting is proposed to overcome this timestep restriction.A fully implicit narrow band around the sharp phase interface is combined with a splitting of the convective and acoustic phenomena away from the interface.In this part of the domain,the IMEX Runge-Kutta time discretization and the high order discontinuous Galerkin spectral element method are applied to achieve high accuracies in the bulk phases.It is shown that for low Mach numbers a significant gain in computational time can be achieved compared to a fully explicit method.Applica-tions to typical droplet dynamic phenomena validate the proposed method and illustrate its capabilities.

TRANSONIC FLOW CALCULATION OF EULER EQUATIONS BY IMPLICIT ITERATING SCHEME WITH FLUX SPLITTING

Three dimensional Euler equations are solved in the finite volume form with van Leer's flux vector splitting technique. Block matrix is inverted by Gauss-Seidel iteration in two dimensional plane while strongly implicit alternating sweeping is implemented in the direction of the third dimension. Very rapid convergence rate is obtained with CFL number reaching the order of 100. The memory resources can be greatly saved too. It is verified that the reflection boundary condition can not be used with flux vector splitting since it will produce too large numerical dissipation. The computed flow fields agree well with experimental results. Only one or two grid points are there within the shock transition zone.

A low-diffusion robust flux splitting scheme towards wide-ranging Mach number flows

This paper develops a low-diffusion robust flux splitting scheme termed TVAP to achieve the simulation of wide-ranging Mach number flows.Based on Toro-Vazquez splitting approach,the new scheme splits inviscid flux into convective system and pressure system.This method introduces Mach number splitting function and numerical sound speed to evaluate advection system.Meanwhile,pressure diffusion term,pressure momentum flux,interface pressure and interface velocity are constructed to measure pressure system.Then,typical test problems are utilized to systematically assess the robustness and accuracy of the resulting scheme.Matrix stability analysis and a series of numerical cases,such as double shear-layer problem and hypersonic viscous flow over blunt cone,demonstrate that TVAP scheme achieves excellent low diffusion,shock stability,contact discontinuity and low-speed resolution,and is potentially a good candidate for wide-ranging Mach number flows.

Further Study on Errors in Metric Evaluation by Linear Upwind Schemes with Flux Splitting in Stationary Grids

The importance of eliminating errors in grid-metric evaluation for highorder difference schemes has been widely recognized in recent years,and it is known from the proof by Vinokur and Yee(NASA TM 209598,2000)that when conservative derivations of grid metric are used by Thomas,Lombard and Neier(AIAA J.,1978,17(10)and J.Spacecraft and rocket,1990,27(2)),errors caused by metric evaluation could be eliminated by linear schemes when flux splitting is not considered.According to the above achievement,central schemes without the use of flux splitting could fulfill the requirement of error elimination.Difficulties will arise for upwind schemes to attain the objective when the splitting is considered.In this study,further investigations are made on three aspects:Firstly,an idea of central scheme decomposition is introduced,and the procedure to derive the central scheme is proposed to evaluate grid metrics only.Secondly,the analysis has been made on the requirement of flux splitting to acquire free-stream preservation,and a Lax-Friedrichs-type splitting scheme is proposed as an example.Discussions about current study with that by Nonomura et al.(Computers and Fluids,2015,107)have been made.Thirdly,for halfnode-or mixed-type schemes,interpolations should be used to derive variables at half nodes.The requirement to achieve metric identity on this situation is analyzed and an idea of directionally consistent interpolation is proposed,which is manifested to be indispensable to avoid violations of metric identity and to eliminate metric-caused errors thereafter.Two numerical problems are tested,i.e.,the free-stream and vortex preservation onwavy,largely randomized and triangular-like grids.Numerical results validate aforementioned theoretical outcomes.

Flux vector splitting solutions for coupling hydraulic transient of gas-liquid-solid three-phase flow in pipelines

The gas-liquid-solid three-phase mixed flow is the most general in multiphase mixed transportation. It is significant to exactly solve the coupling hydraulic transient problems of this type of multiphase mixed flow in pipelines. Presently, the method of characteristics is widely used to solve classical hydraulic transient problems. However, when it is used to solve coupling hydraulic transient problems, excessive interpolation errors may be introduced into the results due to unavoidable multiwave interpolated calculations. To deal with the problem, a finite difference scheme based on the Steger- Warming flux vector splitting is proposed. A flux vector splitting scheme is established for the coupling hydraulic transient model of gas-liquid-solid three-phase mixed flow in the pipelines. The flux subvectors are then discretized by the Lax-Wendroff central difference scheme and the Warming-Beam upwind difference scheme with second-order precision in both time and space. Under the Rankine-Hugoniot conditions and the corresponding boundary conditions, an effective solution to those points located at the boundaries is developed, which can avoid the problem beyond the calculation region directly induced by the second-order discrete technique. Numerical and experimental verifications indicate that the proposed scheme has several desirable advantages including high calculation precision, excellent shock wave capture capability without false numerical oscillation, low sensitivity to the Courant number, and good stability.

Flux Vector Splitting Schemes for Water Hammer Flows in Pumping Supply Systems with Air Vessels

To solve water hammer problems in pipeline systems,many numerical simulation approaches have been developed. This paper improves a flux vector splitting( FVS) scheme whose grid is the same as the fixedgrid MOC scheme. The proposed FVS scheme is used to analyze water hammer problems caused by a pump abrupt shutdown in a pumping system with an air vessel. This paper also proposes a pump-valve-vessel model combining a pump-valve model with an air vessel model. The results show that the data obtained by the FVS scheme are similar to the ones obtained by the fixed-grid method of characteristics( MOC). And the results using the pump-valve-vessel model are almost the same as the ones using both the pump-valve model and the air vessel model. Therefore,it is effective that the proposed FVS scheme is used to solve water hammer problems and the pump-valve-vessel model replaces both the pump-valve model and the air vessel model to simulate water hammer flows in the pumping system with the air vessel.

Kinetic Flux Vector Splitting Method for the Shallow Water Wave Equations

Based on the analogy to gas dynamics, the kinetic flux vector splitting (KFVS) method is used to stimulate the shallow water wave equations. The flux vectors of the equations are split on the basis of the local equilibrium Maxwell-Boltzmann distribution. One dimensional examples including a dam breaking wave and flows over a ridge are calculated. The solutions exhibit second-order accuracy with no spurious oscillation.

Explicit Kinetic Flux Vector Splitting Scheme for the 2-D Shallow Water Wave Equations

Originally, the kinetic flux vector splitting (KFVS) scheme was developed as a numerical method to solve gas dynamic problems. The main idea in the approach is to construct the flux based on the microscopical description of the gas. In this paper, based on the analogy between the shallow water wave equations and the gas dynamic equations, we develop an explicit KFVS method for simulating the shallow water wave equations. A 1D steady flow and a 2D unsteady flow are presented to show the robust and accuracy of the KFVS scheme.

THE SOLUTION FOR TRANSIENT TWO-PHASE FLOW BY SPLIT FLUX VECTOR METHOD

In this paper the transient two-phase flow equations and their eigenvalues are first introduced. The flux vector is then split into subvectors which just contain a specially signed eigenvalue. Using one-sided spatial difference operators finite difference equations and their solutions are obtained. Finally comparison with experiment shows the predicted results produce good agreement with experimental data.

MmB DIFFERENCE SCHEMES FOR TWODIMENSIONAL HYPERBOLIC CONSERVATION LAWS

A new class of second order accuracy semidiscrete difference schemes is presented for the two-dimensional nonlinear scalar hyperbolic conservation laws. It is based on flux splitting, piecewise linear cell-averaged reconstruction and upwind property in the spatial discretization. By using TVD Runge-Kutta time discretization method, the full discrete scheme is obtained and its MmB property is proved. The extension to the two-dimensionalnonlinear hyperbolic conservation law systems is straightforward by using component-wise manner. The main advantage is simple: no Riemann problem is solved, and so field-by-field decomposition is avoided and the complicated computation is reduced. Numerical results of two-dimensional Euler equations of compressible gas dynamics verify the accuracy and robustness of the method.

Characteristic-Based Time Domain Method for Cylindrically Conformal Microstrip Patch Antennas

The characteristics of a cylindrical conformal microstrip patch antenna are analyzed by using the characteristic-based time domain （CBTD） method. A governing equation in the cylindrical coordinate system is formulated directly to facilitate the analysis of cylindrically conformal microstrip patch antennas. The algorithm has second-order accuracy both in time and space domain and has the potential to eliminate the spurious wave reflection from the numerical boundaries of the computational domain, Numerical results demonstrate the important merits and accuracy of the proposed technique in computational electromagnetics,

SIMULATIONS OF FLOWFIELDS AROUND UNDERWATER APPENDED BODIES

The numerical method which is based on flux difference splitting, LUdecomposition, and implicit high-resolution third-order Essentially Non-Oscillatory (ENO) scheme wasconstructed for the efficient computation of steady state solution to three-dimensionalincompressible Navier-Stokes e-quations in general coordinates. The flowfields over underwateraxisymmetric bodies, full-appended axisymmetric bodies and axisymetric bodies with a ring-wing ductwere simulated. The method is proved to be capable of predicting the circumferential-mean velocitydistribution at model scale to the accuracy of around 3% of measured values, and of predicting somedetails of flow features, for example, the wake harmonics.

Numerical simulation of multi-phase combustion flow in solid rocket motors with metalized propellant

Multi-phase flowfield simulation has been performed on solid rocket motor and effect of multi-phases on the performance prediction of the solid rocket motor(SRM) is investigation.During the combustion of aluminized propellant,the aluminum particles in the propellant melt and formliquid aluminum at the burning propellant surface.So the flow within the rocket motor is multi phase or two phase because it contains droplets and smoke particles of Al2O3.Flowsi mulations have been performed on a large scale motor,to observe the effect of the flowfield onthe chamber and nozzle as well.Uniform particles diameters and Rosin-Rammler diameter distribution method that is based on the assumption that an exponential relationship exists betweenthe droplet diameter,dand mass fraction of droplets with diameter greater thandhave been used for the si mulation of different distribution of Al2O3 droplets present in SRM.Particles sizes in the range of 1-100μm are used,as being the most common droplets.In this approachthe complete range of particle sizes is dividedinto a set of discrete size ranges,eachto be defined by single streamthat is part of the group.Roe scheme-flux differencing splitting based on approxi mate Riemann problem has been used to si mulate the effects of the multi-phase flowfeild.This is second order upwind scheme in which flux differencing splitting method is employed.To cater for the turbulence effect,Spalart-All maras model has been used.The results obtained show the great sensitivity of this diameters distribution and particles concentrations to the SRMflowdynamics,primarily at the motor chamber and nozzle exit.The results are shown with various sizes of the particles concentrations and geometrical configurations including models for SRM and nozzle.The analysis also provides effect of multi-phase on performance prediction of solid rocket motor.

FVS SCHEME FOR SEVERE TRANSIENT FLOW IN PIPE NETWORKS

An efficient numerical method with first and second order accuracy is developed by the flux split technology to simulate the water hammer problem in single and multiple pipe networks under severe transient conditions. The finite volume formulation ensures that both schemes conserve mass and momentum and produces physically realizable shock fronts. The conception of the fictitious cell at the junction is developed. The typical water hammer problem and the experi ment with friction and the comprehensive orbicular network with control valve and pressure relief valve and surge tank are implemented to test the numerical method. Strong numerical evidences show that the proposed scheme has several desirable properties, such as, accurate, efficient, robust, high shock resolution, conservative and stable for Courant number.

A Third-Order Upwind Compact Scheme on Curvilinear Meshes for the Incompressible Navier-Stokes Equations

This paper presents a new version of the upwind compact finite difference scheme for solving the incompressible Navier-Stokes equations in generalized curvilinear coordinates.The artificial compressibility approach is used,which transforms the elliptic-parabolic equations into the hyperbolic-parabolic ones so that flux difference splitting can be applied.The convective terms are approximated by a third-order upwind compact scheme implemented with flux difference splitting,and the viscous terms are approximated by a fourth-order central compact scheme.The solution algorithm used is the Beam-Warming approximate factorization scheme.Numerical solutions to benchmark problems of the steady plane Couette-Poiseuille flow,the liddriven cavity flow,and the constricting channel flow with varying geometry are presented.The computed results are found in good agreement with established analytical and numerical results.The third-order accuracy of the scheme is verified on uniform rectangular meshes.