IMPLICIT TVD SCHEMES APPLIED TO GAS-DROPLET DETONATION CALCULATION

In order to investigate parameters of FAE (fuel air explosive) explosion, the two-phase gas-droplet conservation equations with two-dimensional axial symmetry in the Euler coordinate were used. High-resolution implicit TVD ( total variation diminishing) schemes were applied to gas phase equations and MacCormack schemes to liquid equations. The formation and propagation of gas-droplet detonation wave were simulated numerically. The simulation results and the others are compared with a good agreement.

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 Фu in the paper, was validated by the tests to effectively dissipate spurious oscillations.

A CLASS OF COMPACT UPWIND TVD DIFFERENCE SCHEMES

A new method was proposed for constructing total variation diminishing （TVD） upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can ensure the nonlinear compact schemes TVD property. Two compact TVD （CTVD） schemes were tested, one is thirdorder accuracy, and the other is fifth-order. The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems, as well as a twodimensional shock-vortex interaction and a shock-boundary flow interaction. Numerical results show their high-order accuracy and high resolution, and low oscillations across discontinuities.

SOME WEIGHT-TYPE HIGH-RESOLUTION DIFFERENCE SCHEMES AND THEIR APPLICATIONS

By the aid of an idea of the weighted ENO schemes, some weight-type high-resolution difference schemes with different orders of accuracy are presented in this paper by using suitable weights instead of the minmod functions appearing in various TVD schemes. Numerical comparisons between the weighted schemes and the non-weighted schemes have been done for scalar equation, one-dimensional Euler equations, two-dimensional Navier-Stokes equations and parabolized Navier-Stokes equations.

TVD SCHEME WITH CHEMICAL REACTION FLOW AND ITS APPLICATION IN COMBUSTION GAS JETTING FLOW

By conjugating features of combustion gas jetting flows of the solid-rocket and using mathematical methods, a numerical scheme is systematically derived based on Harten′s standard TVD scheme, which fits for the flow with high temperature, pressure and velocity. The rational calculation formula of pressure partial derivation is also given out. By using the chemical kinetics knowledge, problems of multi-component and finite rate chemical reaction contained in combustion gas of the rocket flow field are discussed. The method for solving the mass source term of chemical reaction is clarified. Taking 9 reaction equations with 12 components as an example and utilizing the established calculation program, the free jetting flow field of the rocket is simulated. Numerical results show the correctness of the numerical scheme.

Study of total variation diminishing (TVD) slope limiters in dam-break flow simulation

A two-dimensional （2D） dam-break flow numerical model was developed based on the finite-volume total variation diminishing （TVD） and monotone upstream-centered scheme for conservation laws （MUSCL）-Hancock scheme, which has second-order accuracy in both time and space. A Harten-Lax-van Leer-contact （HLLC） approximate Riemann solver was used to evaluate fluxes. The TVD MUSCL-Hancock numerical scheme utilizes slope limiters, such as the minmod, double minmod, superbee, van Albada, and van Leer limiters, to prevent spurious oscillations and maintain monotonicity near discontinuities. A comparative study of the impact of various slope limiters on the accuracy of the numerical flow model was conducted with several dam-break examples including wet and dry bed cases. The numerical results of the superbee and double minmod limiters agree better with the theoretical solution and have higher accuracy than other limiters in one-dimensional （1D） space. The ratio of the downstream water depth to the upstream water depth was used to select the proper slope limiter. For the 2D numerical model, the superbee limiter should not be used, owing to significant numerical dispersion.

CONSTRUCTION OF HIGH-ORDER ACCURACY IMPLICIT RESIDUAL SMOOTHING SCHEMES

Referring to the construction way of Lax-Wendroff scheme, new IRS (Implicit Residual Smoothing) schemes have been developed for hyperbolic, parabolic and hyperparabolic equations. These IRS schemes have 2nd-order or 3 rd-order time accuracy, and can extend the stability region of basic explicit time-stepping scheme greatly and thus can permit higher CFL number in the calculation of flow field. The, central smoothing and upwind-bias smoothing techniques have been developed tao. Based on one-dimensional linear model equation, it has been found that the scheme is unconditionally stable according to the von-Neumann analysis. The limitation of Dawes' method, which has been applied in turbomachinery widespreadly, has been discussed on solving steady flow and viscous flow. It is shown that stable solution of this method is not completely independent with the value of time step. In the end, numerical results by using IRS schemes and Dawes' method as well as TVD ( total variation diminishing) scheme and four-stage Runge-Kutta technique are presented to verify, the analytical conclusions.

Numerical study on flow separation control over NACA0015 aerofoil using electromagnetic fields

In this study, a flow solver was developed based on the governing RANS equations of compressible flows and was further extended to include the effects of electromagnetic forces namely Lorentz forces. Lorentz forces may be added as a source term in the governing fluid flow equations. Numerical studies were carried out for NACA0015 aerofoil at high angles of incidences from 15° to 30° and compared with some available cases of experimental and incompressible numerical solutions. The hydrodynamics performance was improved using a magnetic momentum coefficient of up to 0.048. The size of flow separation zone was decreased or completely eliminated by increasing this coefficient. The overall drag was not changed considerably, however the overall lift was increased up to 80 percent at stall angles.

Implicit numerical scheme based on SMAC method for unsteady incompressible Navier-Stokes equations

An implicit numerical scheme is developed based on the simplified marker and cell （SMAC） method to solve Reynolds-averaged equations in general curvilinear coordinates for three-dimensional （3-D） unsteady incompressible turbulent flow. The governing equations include the Reynolds-averaged momentum equations, in which contravariant velocities are unknown variables, pressure-correction Poisson equation and k- s turbulent equations. The governing equations are discretized in a 3-D MAC staggered grid system. To improve the numerical stability of the implicit SMAC scheme, the higherorder high-resolution Chakravarthy-Osher total variation diminishing （TVD） scheme is used to discretize the convective terms in momentum equations and k- e equations. The discretized algebraic momentum equations and k- s equations are solved by the time-diversion multiple access （CTDMA） method. The algebraic Poisson equations are solved by the Tschebyscheff SLOR （successive linear over relaxation） method with alternating computational directions. At the end of the paper, the unsteady flow at high Reynolds numbers through a simplified cascade made up of NACA65-410 blade are simulated with the program written according to the implicit numerical scheme. The reliability and accuracy of the implicit numerical scheme are verified through the satisfactory agreement between the numerical results of the surface pressure coefficient and experimental data. The numerical results indicate that Reynolds number and angle of attack are two primary factors affecting the characteristics of unsteady flow.

NUMERICAL MODELING OF THE INITIAL STAGE OF THE GENERATION OF UNSTEADY VORTICES FROM SHARP CORNER IN PLANE COMPRESSIBLE FLOW

The impingement of a plane shock wave in air on a rectangular or triangularobstacle is simulated numerically with high resolution TVD (total variationdiminishing) scheme in finite volume fomulation with Schwarz transformation in meshgeneration. The mesh lines are quite adaptive to the physical features of the unsieadyflow fietd and concentrate locally near the corners. At the initial stage the flow field iscomplex. and the scale of viscous diffusion is very small and the viscosity of fluid incomputation may be neglected. The unsteady generation of concerntrated vorticesdownstream of the sharp corner as the result of the nonnuiformity of both temperatureand entropy fields in plane inviscid compressible fluid, induced by bow shock wave, isshown clearly and in accordance with optical measurements, performed by our request.

Non-Oscillatory Hierarchical Reconstruction for Central and Finite Volume Schemes

This is the continuation of the paper”Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction”by the same authors.The hierarchical reconstruction introduced therein is applied to central schemes on overlapping cells and to finite volume schemes on non-staggered grids.This takes a new finite volume approach for approximating non-smooth solutions.A critical step for high-order finite volume schemes is to reconstruct a non-oscillatory high degree polynomial approximation in each cell out of nearby cell averages.In the paper this procedure is accomplished in two steps:first to reconstruct a high degree polynomial in each cell by using e.g.,a central reconstruction,which is easy to do despite the fact that the reconstructed polynomial could be oscillatory;then to apply the hierarchical reconstruction to remove the spurious oscillations while maintaining the high resolution.All numerical computations for systems of conservation laws are performed without characteristic decomposition.In particular,we demonstrate that this new approach can generate essentially non-oscillatory solutions even for 5th-order schemes without characteristic decomposition.

A TVD SCHEME FOR INCOMPRESSIBLE FLOW COUPLED WITH DIFFERENT TURBULENCE MODLES ON A GROUND-MOUNTED SQUARE-RIB FLOW

A finite-difference Total Variation Diminishing （TVD） numerical simulation model for coupling the Reynolds Averaged Navier-Stokes （RANS） equations, pressure-relative continuity equation and various k-εturbulence models was developed to solve the incompressible flow based on the pseudo-compressibility method. The hyperbolicity of all these equations was studied and the discretization of the fully coupling equations with all the primal variables and source terms were made in this article. Numerical simulation for modeling the flow around a ground-mounted square rib was implemented and validated by comparing with the published wind tunnel experimental data. It is shown that such a numerical simulation method with a proper turbulence model has a very good accuracy to simulate the flow around a surface-mounted rib. It is concluded that the Renormalization Group （RNG） and Chen-Kim k-εturbulence models have much better ability to predict the characteristics of the vortex structure and flow separation than the standard k-εmodel.

NEW APPROACH TO THE LIMITER FUNCTIONS

In this paper we discuss three topics on the designing of the limiter functions. (1) To guarantee the TVD property (2) To maintain enough artificial viscosity. (3) A method to form TVB limiter which can ensure second order accuracy even at the extrema of the solution.

Non-hydrostatic versus hydrostatic modelings of free surface flows

The hydrodynamics of geophysical flows in oceanic shelves, estuaries, and rivers are often studied by solving shallow water equations under either hydrostatic or non-hydrostatic assumptions. Although the hydrostatic models are quite accurate and cost-efficient for many practical applications, there are situations when the fully hydrodynamic models are preferred despite a larger cost for computations. The present numerical model is implemented by the finite volume method （FVM） based on unstructured grids. The model can be efficiently switched between hydrostatic and non-hydrostatic modules. The case study shows that for waves pro- pagating along the bar a criterion with respect to the shallowness alone, the ratio between the depth and the wave length, is insufficient to warrant the performance of shallow flow equations with a hydrostatic approach and the nonlinearity in wave dynamics can be better accounted with a hydrodynamic approach. Besides the prediction of the flows over complex bathymetries, for instance, over asymmetrical dunes, by a hydrodynamic approach is shown to be superior in accuracy to the hydrostatic simulation.

A HIGH RESOLUTION FINITE VOLUME METHOD FOR SOLVING SHALLOW WATER EQUATIONS

A high resolution finite volume numerical method for solving the shallow water equations is developed in this paper. In order to extend finite difference TVD scheme to finite volume method, a new geometry and topology of control bodies is defined by considering the corresponding relationships between nodes and elements. This solver is implemented on arbitrary quadrilateral meshes and their satellite elements, and based on a second order hybrid type of TVD scheme in space discretization and a two step Runge Kutta method in time discretization. Then it is used to deal with two typical dam break problems and very satisfactory results are obtained comparied with other numerical solutions. It can be considered as an efficient implement for the computation of shallow water problems, especially concerning those having discontinuities, subcritical and supercritical flows and complex geometries.

A k-ε Turbulence Model Considering Compressibility in Three-Dimensional Transonic Turbulent Flow Calculation

Based on the standard k-ε turbulence model, a new compressible k-ε model considering the pressure expansion influence due to the compressibility of fluid is developed and applied to the simulation of 3D transonic turbulent flows in a nozzle and a cascade. The Reynolds avenged N-S equations in generalized curvilinear coordinates are solved with implementation of the new model. The high resolution TVD scheme is used to discretize the convective terms. The numerical results show that the compressible k-ε model behaves well in the simulation of transonic internal turbulent flows.

Linear Unsteady Aerodynamic Forces on Vibrating Annular Cascade Blades

The paper presents the formulation to compute numerically the unsteady aerodynamic forces on the vibrating annular cascade blades.The formulation is based on the finite volume method.By applying the TVD scheme to the linear unsteady calculations,the precise calculation of the peak of unsteady aerodynamic forces at the shock wave location like the delta function singularity becomes possible without empirical constants.As a further feature of the present paper,results of the present numerical calculation are compared with those of the double linearization theory(DLT),which assumes small unsteady and steady disturbances but the unsteady disturbances are much smaller than the steady disturbances.Since DLT requires far less computational resources than the present numerical calculation,the validation of DLT is quite important from the engineering point of view.Under the conditions of small steady disturbances,a good agreement between these two results is observed,so that the two codes are cross-validated.The comparison also reveals the limitation on the applicability of DLT.

The Two-Dimensional Supersonic Flow and Mixing with a Perpendicular Injection in a Scramjet Combustor

A numerical investigation has been performed on supersonic mixing of hydrogen with air in a Scramjet (Supersonic Combustion Ramjet) combustor and its flame holding capability by solving Two-Dimensional full Navier-Stokes equations. The main flow is air entering through a finite width of inlet and gaseous hydrogen is injected perpendicularly from the side wall. An explicit Harten-Yee Non-MUSCL Modified-flux-type TVD scheme has been used to solve the system of equations, and a zero-equation algebraic turbulence model to calculate the eddy viscosity coefficient. In this study the enhancement of mixing and good flame holding capability of a supersonic combustor have been investigated by varying the distance of injector position from left boundary keeping constant the backward-facing step height and other calculation parameters. The results show that the configuration for small distance of injector position has high mixing efficiency but the upstream recirculation can not evolved properly which is an important factor for flame holding capability. On the other hand, the configuration for very long distance has lower mixing efficiency due to lower gradient of hydrogen mass concentration on the top of injector caused by the expansion of side jet in both upstream and downstream of injector. For moderate distance of injector position, large and elongated upstream recirculation can evolve which might be activated as a good flame holder.

Numerical Computation of Compressible Viscous Internal Flows

Some implicit time-marching finite-difference solutions of time-averaged Navier-Stokes equations for two- dimensional compressible internal flows are presented.Five numerical examples including subsonic,transonic, supersonic and hypersonic flow fields with steady and unsteady phenomena show validity and flexibility of the present calculation code.The TVD scheme suggested by Harten et al is used to improve the shock resolution, and an algebraic turbulence model suggested by Baldwin and Lomax is introduced to evaluate the viscous effect in the turbulent flows.It is found that the computational results show fairly good agreement with the experi- mental data.