Flux Vector Splitting Research Articles

Genetic algorithm-based design of transonic aerofoils using Euler equations

The paper describes the adaptation of genetic algorithms (GA) to the design of inviscid transonic aerofoils. GA is effective in optimization strategies for designing transonic aerofoils where the non-linear aerodynamic phenomenon of shock wave results in flow discontinuity. The Euler equations are discretized through the flux-vector splitting method with Poisson's equation-based grid generation. Advanced strategies in the GA such as directed crossover and multi-stage searches are employed in the context of function-based global optimization. The objective is to determine the best combination of weighted parameters of a analytic function for the aerofoil surface, by maximizing the L/ D ratio. The paper first covers the optimization of transonic aerofoils to enhance the baseline design of NACA0012. A two-point design problem is also conducted to accommodate both transonic and subsonic regimes represented by the Mach number and angle of attack. The optimized aerofoils have shown improved aerodynamic performance for both flight regimes.

Computation of non-equilibrium hypersonic flow with artificially upstream flux vector splitting (AUFS) schemes

The AUFS scheme has been presented for solving the Euler equations [Sun, M., Takayama, K., 2003. An artificially upstream flux vector splitting scheme for the Euler equations. Journal of Computational Physics, 189, 305–329]. An extension of this high resolution scheme-based on upwind numerical methods has been developed to calculate a two-dimensional hypersonic viscous flowfield in thermochemical non-equilibrium. The time-dependent Navier–Stokes governing equations are computed by using a multi-block finite volume technique on a structured mesh. The convective fluxes at the interfaces are evaluated using a flux vector splitting (FVS) method with a second-order reconstruction of the interface values and the viscous terms are discretised by second-order central differences. A better evaluation of aerodynamic parameters are obtained with this AUFS scheme and they are also compared to those obtained by previous works. The freestream flow conditions of these computations correspond to high-enthalpy flows with a Mach number range between 6.4 and 25.9. The obtained numerical results indicate that the AUFS scheme is accurate, robust, and efficient for the calculation of hypersonic flow.

Flow Control over a Backward-Facing Step with Application of a Magnetic Field

High-speed flows over a backward-facing step that is subject to an applied magnetic field are numerically simulated. The global domain of computation has been decomposed into upstream and downstream domains from the step location. The low magnetic Reynolds number approximation under a multiblock grid approach is used for modeling the backstep flow. Flux-vector splitting for the convective terms and central differencing for the diffusion terms are used. A time-explicit multistage Runge–Kutta scheme for time integration is implemented. Pressure distribution for the Navier–Stokes analysis is found to be in good agreement with the experimental data. Different typesofmagnetic fielddistributionsareinvestigated.Bothuniformandvariableelectricalconductivitydistributions havebeenconsidered.Ithasbeenobservedthatanincreaseintheseparationzoneanddisplacementofobliqueshock wave toward the exit section occurs subsequent to application of the magnetic field. Comparison of results obtained with uniform and variable electrical conductivities has shown a reduction in magnetic interaction for variable electrical conductivity.

An efficient numerical method for highly loaded transonic cascade flow

In this paper, an efficient numerical method for transonic viscous flow in a highly loaded turbine vane cascade, where the interaction of a shock wave and boundary layer often leads to very complicated flow phenomena, is developed. The numerical code, a modified implicit flux-vector-splitting solver of the Navier–Stokes equations (MIFVS), is extended to simulate such transonic cascade flow. A compressible low-Reynolds-number k–ɛ model, together with a transition-modified damping function, has been implemented into the MIFVS code. With this extended MIFVS solver, the main feature of transonic flow and shock and boundary-layer interactions in the highly loaded transonic turbine vane are efficiently predicted with satisfactory accuracy. The convergence rate is found to be three times faster than that of flux-vector-splitting (FVS) methods.

ON THE NUMERICAL SOLUTION OF THE INVISCID THREE DIMENSIONAL NON CIRCULAR NOZZLES WITH STRUCTURED NONORTHOGONAL GRIDS

A method is presented for solving three dimensional inviscid internal flows, with application to three dimensional noncircular nozzles. Spatial discretization is made on the finite difference formulation of Euler’s equations. The convective terms is differenced using the Van Leer flux vector splitting technique. Time is advanced using the implicit Euler backward time stepping. The resulting system of algebraic equations is solved using ADI method. New numerical treatment of the flow boundary conditions is presented and applied to handle the complicated mixed boundary conditions. Application to solve flow through nozzles shows good agreement with analytical results.

ON THE NUMERICAL SOLUTION OF THE INVISCID TWO DIMENSIONAL INTERNAL FLOWS WITH NON-ORTHOGONAL GRIDS

A new boundary condition treatment strategy is presented in this research for solving the twodimensional inviscid internal flow problems, spatial discretization has made by finite difference formulation using flux vector splitting of Van Leer. The time advancement is made using Euler backward time stepping technique. The aim of this research is to present new non-reflecting boundary condition treatment in the cases of metaphysical flow conditions occurred at the same space point. Such situation occurred in the internal flow problems as the corner points in nozzle., numerical flow simulations is carried out and shows good agreement with the analytical values and with published data.

Developing shock-capturing difference methods

A new shock-capturing method is proposed which is based on upwind schemes and flux-vector splittings. Firstly, original upwind schemes are projected along characteristic directions. Secondly, the amplitudes of the characteristic decompositions are carefully controlled by limiters to prevent non-physical oscillations. Lastly, the schemes are converted into conservative forms, and the oscillation-free shock-capturing schemes are acquired. Two explicit upwind schemes (2nd-order and 3rd-order) and three compact upwind schemes (3rd-order, 5th-order and 7th-order) are modified by the method for hyperbolic systems and the modified schemes are checked on several one-dimensional and two-dimensional test cases. Some numerical solutions of the schemes are compared with those of a WENO scheme and a MP scheme as well as a compact-WENO scheme. The results show that the method with high order accuracy and high resolutions can capture shock waves smoothly.

A characteristic-based shock-capturing scheme for hyperbolic problems

In order to suppress numerical oscillations of linear compact schemes around discontinuities, a characteristic-based flux splitting limited method is introduced instead of ENO/WENO or other shock-capturing algorithms. This method begins with upwind schemes and flux vector splittings. The upwind schemes are projected along characteristic directions in a different way, and their amplitudes are carefully controlled by a special limiter in order to meet entropy condition and to prevent non-physical oscillations. A fifth-order linear compact upwind scheme is modified by this method for solving problems involving discontinuities. The properties of the numerical algorithm are checked on some benchmark problems in one, two and three space dimensions. Numerical results show that it is high-order accurate with high resolution and oscillation-free.

A shock-capturing model based on flux-vector splitting method in boundary-fitted curvilinear coordinates

In this paper, a robust and accurate high-resolution finite-volume scheme is presented which employs flux-vector splitting (FVS) as the building block for solution of shallow water equations in boundary-fitted curvilinear coordinates. Eddy viscosity approach is used to accommodate shear stresses due to turbulence. Splitting of the convective terms is achieved via flux Jacobians whereas Liou–Steffen Splitting (LSS) technique, but in transformed coordinates, is used to split pressure terms. Limited flux gradients are also used to increase the computational accuracy of evaluation of interface fluxes and decrease the excessive numerical dissipation associated with FVS. This will completely remove spurious oscillations in high-gradient regions without introducing too much numerical dissipations. The method is tested for some classic simulations including hydraulic jump, 1D dam break and 2D dam break problems. The results show very satisfactory agreement with experimental data, analytical solutions and other numerical results.

Kinetic Flux Vector Splitting Schemes for Ideal Quantum Gas Dynamics

Novel quantum kinetic flux vector splitting schemes are presented for the computation of ideal quantum gas dynamical flows. The quantum Boltzmann equation approach is adopted and both Bose-Einstein and Fermi-Dirac gases are considered. Formulas for one spatial dimension are first derived, and the resulting kinetic flux vector splitting scheme is tested for shock tube flows. A modified flux vector splitting based on the BGK model is also presented to reduce the numerical diffusion. Implementation of the efficient weighted essentially nonoscillatory method to yield a class of high resolution methods is also devised. Both the classical limit and the nearly degenerate limit are computed. The flow structures can all be accurately captured by the present kinetic flux vector splitting schemes. Results for one-dimensional boson gas under external potential are also included.

An implicit finite volume method for the solution of 3D low Mach number viscous flows using a local preconditioning technique

This paper presents a cell-centered high order finite volume scheme for the solution of the three-dimensional (3D) Navier–Stokes equations with low Mach number. The system of non-linear equations is solved by means of a fully implicit pseudo-transient scheme. Each pseudo-time step is solved by a Newton-GMRes procedure. A local preconditioning technique is used to scale the speed of sound and to improve the system condition number for low Mach number and low cell Reynolds number. This preconditioning is applied to the AUSM+up flux vector splitting function. The method is tested on 2D and 3D low Mach number laminar flows.

High-Order Unstructured Essentially Nonoscillatory and Weighted Essentially Nonoscillatory Schemes for Aerodynamic Flows

In the present work, essentially nonoscillatory schemes and weighted essentially nonoscillatory schemes are implemented in a cell centered finite volume context on unstructured meshes. The two-dimensional Euler equations are considered to represent the flows of interest. The essentially nonoscillatory and weighted essentially nonoscillatory schemes have been developed with the purpose of accurately capturing discontinuities appearing in problems governed by hyperbolic conservation laws. In the aerodynamic studies of interest in the paper, these discontinuities are mainly represented by shock waves. The entire reconstruction process of essentially nonoscillatory and weighted essentially nonoscillatory schemes is described in detail for any order of accuracy with an emphasis on the implementation of second-order and third-order schemes. A flux-difference splitting method and a flux-vector splitting method are tested and compared. The code developed also features an agglomeration multigrid procedure for convergence acceleration, and an adaptive mesh refinement tool. Applications for aerodynamic flows are performed to assess the capability implemented against data available in the literature.

Application of the TVD scheme to the nonlinear instability analysis of a capillary jet

In the past, when either the perturbation-type method or direct-simulation approach was used to analyse capillary jets, the governing equations, which are parabolic in time and elliptic in space, were simplified or linearized. In the present study, the convective derivative term and a full, nonlinear form of the capillary pressure term are retained in the governing equations to investigate nonlinear effects on the break-up of capillary jets. In this work, the TVD (i.e. total variation diminishing) scheme with flux-vector splitting is applied to obtain the solutions of the system of nonlinear equations in a matrix form. Numerical results show that the present nonlinear model predicts longer jet break-up lengths and slower growth rates for capillary jets than the previous linear model does. Comparing with other measurements from past literatures, the nonlinear results are consistent with the experimental data and appear more accurate than the linear analysis. In the past, the classic perturbation-type analyses assumed constant growth rates for the fundamental and all harmonic components. By contrast, the present model is able to capture the local features of growth rates, which are not spatially and temporally constant. Copyright © 2006 John Wiley & Sons, Ltd.

The Runge-Kutta DG finite element method and the KFVS scheme for compressible flow simulations

This article presents a new type of second-order scheme for solving the system of Euler equations, which combines the Runge-Kutta discontinuous Galerkin (DG) finite element method and the kinetic flux vector splitting (KFVS) scheme. We first discretize the Euler equations in space with the DG method and then the resulting system from the method-of-lines approach will be discretized using a Runge-Kutta method. Finally, a second-order KFVS method is used to construct the numerical flux. The proposed scheme preserves the main advantages of the DG finite element method including its flexibility in handling irregular solution domains and in parallelization. The efficiency and effectiveness of the proposed method are illustrated by several numerical examples in one- and two-dimensional spaces. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

Numerical simulation of shock diffraction on unstructured meshes

Shock diffraction over geometric obstacles is performed on two-dimensional unstructured triangular meshes using the AUSM+ flux-vector splitting scheme. Numerical simulations of shock diffraction using structured grids are reviewed in the literature, as are experimental results corresponding to the flow conditions studied. Present unstructured grid results for popular and challenging two-dimensional shock diffraction problems are presented and compared to experimental data and photographs. Benchmark and example test cases were chosen to cover a wide variety of Mach numbers for weak and strong shock waves, and for square and circular geometries. Both single and multiple obstacles are considered, as are obstacles located in the free field and confined in a channel.

Performance of high‐resolution TVD schemes for 1D dam‐break simulations

The performance of high‐resolution total variation diminishing (TVD) schemes for simulating dam‐break problems are presented and evaluated. Three robust and reliable first‐order upwind schemes, namely FVS, Roe and HLLE schemes, are extended to six second‐order TVD schemes using two different approaches, the Sweby flux limiter approach and the direct MUSCL‐Hancock slope limiter. For idealized dam‐break flows, comparisons of the simulated results with the exact solutions show that the flux vector splitting (FVS) scheme coupled with the direct MUSCL‐Hancock (DMH) slope limiter approach has the best numerical performance among the presented schemes. Application of the FVS‐DMH scheme to a dam‐break experiment with sloping dry bed shows that the simulated water depths agree well with the measured.

Simulation of Hydraulic Shock Waves by Hybrid Flux-Splitting Schemes in Finite Volume Method

AbstractIn the framework of the finite volume method, a robust and easily implemented hybrid flux-splitting finite-volume (HFF) scheme is proposed for simulating hydraulic shock waves in shallow water flows. The hybrid flux-splitting algorithm without Jacobian matrix operation is established by applying the advection upstream splitting method to estimate the cell-interface fluxes. The scheme is extended to be second-order accurate in space and time using the predictor-corrector approach with monotonic upstream scheme for conservation laws. The proposed HFF scheme and its second-order extension are verified through simulations of the 1D idealized dam-break problem, the 2D oblique hydraulic shock-wave problem, and the 2D dam-break experiments with channel contraction as well as wet/dry beds. Comparisons of the HFF and several well-known first-order upwind schemes are made to evaluate numerical performances. It is demonstrated that the HFF scheme captures the discontinuities accurately and produces no entropy-violating solutions. The HFF scheme and its second-order extension are proven to achieve the numerical benefits combining the efficiency of flux-vector splitting scheme and the accuracy of flux-difference splitting scheme for the simulation of hydraulic shock waves.

A new flux–vector splitting compact finite volume scheme

A new high resolution finite volume method is proposed here that uses flux–vector splitting as the building block to represent the physical processes. The high accuracy projection in space is obtained here, inter alia, by using some compact schemes of Sengupta et al. [1] and Lele [2] – that have extremely high accuracy in spectral plane. This produces a new class of finite volume (FV) scheme for spatial discretization, that has been analyzed for spectral accuracy, numerical stability and dispersion relation preservation (DRP) property following the full-domain analysis method of [1] using two different time integration strategies. These combination of spatial and temporal methods have been tested for two linear wave problems, reported in [1,3] and the solution of Burgers’ equation is compared with the analytical results presented in Adams and Shariff [4]. The shock capturing method is based on the proposed technique in [5], that does not require the usage of any nonlinear differencing techniques or limiters. To show that the proposed method can also handle irregular grids, the Burgers’ equation is solved using non-uniform grids. Also to demonstrate the utility of the proposed scheme to solve practical problems, the benchmark problem of nonlinear wave propagation in a one-dimensional shock-tube has been solved with the initial data as given in [23] by solving the Euler equation. The analysis and comparison of this flux vector splitting scheme with other well known methods clearly demonstrate the superior scale resolution of the proposed method, at the same time providing a neutrally stable scheme when used with four stage Runge–Kutta scheme. Also the same scheme displays extended ranges of wave numbers and circular frequencies over which DRP property is valid. The computed results of three test cases and their comparison with analytical results clearly reveal that the presented method can be used for practical problems.

Preconditioning method for condensate fluid and solid coupling problems in general curvilinear coordinates

A preconditioned flux-vector splitting (PFVS) scheme in general curvilinear coordinates which can be applied to condensate fluid and solid coupling problems is presented and some typical calculated results are shown to prove the availability of the present method. This method is based on the preconditioning method applied to compressible Navier–Stokes (NS) equations with additional equations and source terms for condensate flows. Since the present PFVS terms composed of the convective and pressure terms of the NS equations are completely reduced to only the pressure terms when the velocities are set to zero, the present scheme can further applied to the calculation not only for a dynamic field but also for a static field including a transitional field from the dynamic region to the static region. In this paper, as a first stage of the present study, coupling problems between a condensate flow in a flow field and heat conduction in a solid structure are simultaneously calculated by using the present method. As numerical examples, transonic and low speed flows around the NACA0012 airfoil, nonequilibrium condensate flows in a nozzle, and natural convection with condensation around a pipe at 1g and zero gravity are simulated with heat conduction in the solid structure.

Prediction of jet flows in the supersonic nozzle and diffuser

AbstractThe authors' recently‐developed code for a needle‐free powdered vaccine delivery device, the epidermal powdered inject system (EPI), is summarized in this paper. The behaviour of supersonic jet flows, which accelerate micron sized powdered vaccines to penetrate human skin or mucosal tissue, is therefore of great importance. A well‐established modified implicit flux vector splitting (MIFVS) solver for the Navier–Stokes equations is extended to study numerically the transient supersonic jet flows of interest. A low Reynolds number k–ε turbulence model, with the compressibility effect considered, is integrated into MIFVS solver to predict the turbulent structures and interactions with inherent shock systems. The results for the NASA validation case NPARC, Venturi and contoured shock tube (CST) of the EPI system are discussed. Copyright © 2005 John Wiley & Sons, Ltd.