Upstream-centered Scheme For Conservation Laws Research Articles

Application of Gas-Kinetic BGK Scheme for Solving 2-D Compressible Inviscid Flow around Linear Turbine Cascade

Fluid flows within turbomachinery tend to be extremely complex. Understanding such flows is crucial in the effort to improve current turbomachinery designs. Hence, computational approaches can be used to great advantage in this regard. In this paper, gas-kinetic BGK (Bhatnagar-Gross-Krook) scheme is developed for simulating compressible inviscid flow around a linear turbine cascade. BGK scheme is an approximate Riemann solver that uses the collisional Boltzmann equation as the governing equation for flow evolutions. For efficient computations, particle distribution functions in the general solution of the BGK model are simplified and used for the flow simulations. Second-order accuracy is achieved via the reconstruction of flow variables using the MUSCL (Monotone Upstream-Centered Schemes for Conservation Laws) interpolation technique together with a multistage Runge-Kutta method. A multi-zone H-type mesh for the linear turbine cascades is generated using a structured algebraic grid generation method. Computed results are compared with available experimental data and found to be in agreement with each other. In order to further substantiate the performance of the BGK scheme, another test case, namely a wedge cascade, is used. The numerical solutions obtained via this test are validated against analytical solutions, which showed to be in good agreement.

Investigation of the hydrodynamics of flash floods in ephemeral channels: Scaling analysis and simulation using a shock-capturing flow model incorporating the effects of transmission losses

Flow and infiltration during flash floods in ephemeral channels were investigated through scaling analysis and numerical experiments. Scaling of the equations governing flow has shown that momentum loss due to transmission losses (flow that infiltrates through the channel bed) during flash floods can be of the same order as momentum loss due to channel friction and can significantly affect the flow velocity. Numerical simulations were carried out using a shock-capturing MUSCL (monotonic upstream-centered scheme for conservation laws), which incorporates transmission losses as a sink term in the momentum and continuity equations. The wetted area of the channel bed during floods is the primary control on the volume of water that can infiltrate into the bed. The increased velocity of floods in narrow channels that adds wetted area to the channel bed due to greater flood propagation distance is not sufficient to overcome the reduction in wetted area due to reduced channel width; wider channels transmit a greater percentage of the flood volume that enters the channel reach to the bed sediments. Floods of the same total volume but different hydrograph shapes transmit different proportions of their volume to the bed sediments; the nature and magnitude of the differences will depend on the flood propagation distance. Increasing the total volume of the flood and decreasing the channel width increases the sensitivity of the total infiltration to the hydrograph shape. For reaches of the same bed area but different spatial distributions of channel width, differences in the rate of channel widening affect the spatial distribution but not the total volume of water that infiltrates into the bed sediments.

High-resolution schemes for bubbling flow computations

This paper is concerned with the effects of numerical schemes on the simulation of dense gas-particle two-phase flows. The first-order upwind difference, the central difference, the second-order upwind difference, the central difference plus artificial dissipation, the deferred correction, the quadratic upstream interpolation (for convective kinematics) and the monotone upstream-centered schemes for conservation laws with different flux limiters were all accomplished to simulate the multi-phase flows. It was found that numerical schemes may significantly affect the solution accuracy and numerical convergence. The monotone upstream-centered schemes for conservation laws are the best choice of all, because they can effectively suppress the non-physical oscillations with the introduction of adaptive numerical dissipation into numerical solutions.

THREE-DIMENSIONAL NUMERICAL SIMULATIONS OF SUSPENSION THERMAL ON HORIZONTAL BED

Numerical simulations of 2D suspension thermal as well as axisymmetric suspension thermal on horizontal bed are performed using 3D numerical model. The thermal consists of mixture of nearly uniform sized particles water. The numerical model consists Simplified Marker and Cell (SMAC) method, Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) technique and the treatment of the partilce deposition. The simulated resutls are compared with experimental data of maximum height, propagation speed, buoyancy of the particle thermal as well as deposit profile of the particles. The numerical model shows promissing resutls for simulation of particle thermal on horizontal bed and deposit profile.

EVALUATION OF GAS-KINETIC SCHEMES FOR SOLVING 1D INVISCID COMPRESSIBLE FLOW PROBLEMS

In this paper, two classes of gas-kinetic schemes are investigated for one-dimensional compressible inviscid flow, namely, the Kinetic Flux Vector Splitting (KFVS) scheme and the Bhatnagar-Gross-Krook (BGK) scheme. Second-order high-resolution scheme for both methods are also developed for calculating flows containing discontinuities. This is achieved by means of reconstructing the initial data via Monotone Upstream-Centered Schemes for Conservation Laws (MUSCL) approach. The Total Variation Diminishing (TVD) shock capturing properties of these second-order schemes are achieved through the use of non-linear limiters. In addition, a multistage TVD Runge-Kutta method is employed for the time integration of the finite volume gas-kinetic scheme. Exclusive consideration is focused on the BGK scheme, which yields a better numerical result in comparison with the KFVS scheme. Three typical one-dimensional inviscid flow problems containing shocks, namely, steady flow in a divergent nozzle, the unsteady shock tube problem, and two interacting blast waves problem are analyzed numerically with the gas-kinetic schemes. Numerical results from the first-order and second-order gas-kinetic schemes are presented and compared with the exact solutions. Other computed results such as those from the Steger-Warming's Flux Vector Splitting scheme, Roe's Flux Difference Splitting scheme, MacCormack's scheme, and high-order compact Van Leer's Flux Splitting Scheme are also presented as comparisons to the gas-kinetic schemes. In addition, the effects of grid sizes on the numerical results of the gas-kinetic schemes are also investigated.

NUMERICAL SIMULATIONS OF CONSERVATIVE AND NON-CONSERVATIVE THERMAL ON HORIZONTAL BED

Numerical simulations of 2D conservative and non-conservative thermal on horizontal bed are performed. The thermal consists of either a saline or mixture of nearly uniform sized particles water. The numerical model is constructed with Simplified Marker and Cell (SMAC) method and Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) technique. The simulated resutls are compared with experimental data of maximum height, propagation speed, buoyancy and densimetric Froude number at the front of the thermal. The numerical model shows promissing resutls for simulation of conservative thermal on horizontal bed.

A preliminary investigation on an ELLAM scheme for linear transport equations

AbstractWe present an Eulerian‐Lagrangian localized adjoint method (ELLAM) for linear advection‐reaction partial differential equations in multiple space dimensions. We carry out numerical experiments to investigate the performance of the ELLAM scheme with a range of well‐perceived and widely used methods in fluid dynamics including the monotonic upstream‐centered scheme for conservation laws (MUSCL), the minmod method, the flux‐corrected transport method (FCT), and the essentially non‐oscillatory (ENO) schemes and weighted essentially non‐oscillatory (WENO) schemes. These experiments show that the ELLAM scheme is very competitive with these methods in the context of linear transport PDEs, and suggest/justify the development of ELLAM‐based simulators for subsurface porous medium flows and other applications. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 22–43, 2003

TVD schemes for unstructured grids

A number of approaches have evolved over the last decade for the implementation of total variational diminishing (TVD) schemes within an unstructured grid finite volume method framework. Unfortunately none of these approaches has been comprehensive enough to permit the general implementation of TVD-based schemes in unstructured grids, and/or accurate enough to recover the exact TVD formulation in structured grids. In this paper we propose a simple method that allows the implementation of the full spectrum of TVD schemes in unstructured grids, while recovering their exact formulation on structured grids. Four schemes implemented using this approach, TVD-MINMOD, TVD-MUSCL (monotonic upstream-centered scheme for conservation laws, MUSCL), TVD-SUPERBEE, TVD-OSHER, are tested and compared to Bruner’s TVD formulation [Parallelization of the Euler equations on unstructured grids, AIAA paper 97-1894, 1995], and to the Barth and Jesperson linear reconstruction scheme [The design and application of upwind schemes on unstructured meshes, AIAA paper 89-0366, 1989] by solving four pure advection problems. Results indicate that the Bruner formulation yields, for the same original TVD scheme, overly diffusive results when compared to the current method. The BJ-MUSCL and TVD-MUSCL are shown to be comparable and more accurate than the OSHER scheme. The SUPERBEE performs best though showing tendency for stepping the modeled profile. In all tests the current method is found to retain the behavior of the structured grid TVD formulation.

A Family of Eulerian–Lagrangian Localized Adjoint Methods for Multi-dimensional Advection-Reaction Equations

We develop a family of Eulerian–Lagrangian localized adjoint methods for the solution of the initial-boundary value problems for first-order advection-reaction equations on general multi-dimensional domains. Different tracking algorithms, including the Euler and Runge–Kutta algorithms, are used. The derived schemes, which are fully mass conservative, naturally incorporate inflow boundary conditions into their formulations and do not need any artificial outflow boundary conditions. Moreover, they have regularly structured, well-conditioned, symmetric, and positive-definite coefficient matrices, which can be efficiently solved by the conjugate gradient method in an optimal order number of iterations without any preconditioning needed. Numerical results are presented to compare the performance of the ELLAM schemes with many well studied and widely used methods, including the upwind finite difference method, the Galerkin and the Petrov–Galerkin finite element methods with backward-Euler or Crank–Nicolson temporal discretization, the streamline diffusion finite element methods, the monotonic upstream-centered scheme for conservation laws (MUSCL), and the Minmod scheme.

Convergence of a Finite Volume Extension of the Nessyahu--Tadmor Scheme on Unstructured Grids for a Two-Dimensional Linear Hyperbolic Equation

The nonoscillatory central difference scheme of Nessyahu and Tadmor is a Godunov-type scheme for one-dimensional hyperbolic conservation laws in which the resolution of Riemann problems at the cell interfaces is bypassed thanks to the use of the staggered Lax--Friedrichs scheme. Piecewise linear MUSCL-type (monotonic upstream-centered scheme for conservation laws) cell interpolants and slope limiters lead to an oscillation-free second-order resolution. Convergence to the entropic solution was proved in the scalar case. After extending the scheme to a two-step finite volume method for two-dimensional hyperbolic conservation laws on unstructured grids, we present here a proof of convergence to a weak solution in the case of the linear scalar hyperbolic equation $u_t + \divv(\vec V\,u) = 0$. Since the scheme is Riemann solver--free, it provides a truly multidimensional approach to the numerical approximation of compressible flows, with a firm mathematical basis. Numerical experiments show the feasibility and high accuracy of the method.

Computation of Flowfield About Submarine Rocket Using a High Order Godunov Scheme

In this paper, a Monotone Upstream-Centered Scheme for Conservation Laws(MUSCL) has been employed to solve two dimensional Navier-Stokes equations. Moving boundaries concerned with rocket walls can be considered. A number of simulated results of submarine rocket launch box are obtained and analyzed. Comparing with available test data show excellent agreement.

Riemann Invariant Manifolds for the Multidimensional Euler Equations

A new approach for studying wave propagation phenomena in an inviscid gas is presented. This approach can be viewed as the extension of the method of characteristics to the general case of unsteady multidimensional flow. A family of spacetime manifolds is found on which an equivalent one-dimensional (1-D) problem holds. Their geometry depends on the spatial gradients of the flow, and they provide, locally, a convenient system of coordinate surfaces for spacetime. In the case of zero-entropy gradients, functions analogous to the Riemann invariants of 1-D gas dynamics can be introduced. These generalized Riemann invariants are constant on these manifolds and, thus, the manifolds are dubbed Riemann invariant manifolds (RIM). Explicit expressions for the local differential geometry of these manifolds can be found directly from the equations of motion. They can be space-like or time-like, depending on the flow gradients. This theory is used to develop a second-order unsplit monotonic upstream-centered scheme for conservation laws (MUSCL)-type scheme for the compressible Euler equations. The appropriate RIM are traced back in time, locally, in each cell. This procedure provides the states that are connected with equivalent 1-D problems. Furthermore, by assuming a linear variation of all quantities in each computational cell, it is possible to derive explicit formulas for the states used in the 1-D characteristic problem.

A Fourth-Order MUSCL Finite-Difference Scheme for Solving the Unsteady Compressible Euler Equations.

A fourth-order compact MUSCL (Monotone Upstream-centered Scheme for Conservation Laws) TVD (Total Variation Diminishing) scheme is proposed for solving the unsteady compressible Euler Equations. The fundamental form of the present scheme is based on the second (third) -order accurate MUSCL finite-difference scheme. One of the distinctive features of this scheme is the ability to capture discontinuities such as slip lines or contact surfaces as well as shocks more sharply than the existing TVD schemes in spite of the use of a simpler algorithm than that of the so-called ENO scheme. Therefore, this scheme can be easily applied to any ordinary numerical solvers based on the second (third) -order MUSCL scheme. We also adopt this scheme into the existing Euler solver developed by the authors which can simulate unsteady inviscid flows accurately by means of the Newton iteration and the Crank-Nicholson method. In order to verify the reliability of the present scheme, an unsteady inviscid supersonic flow having oblique shocks and a slip line is computed. The results show that the slip line as well as oblique shocks can be captured completely.

Local solution acceleration method for the Euler and Navier-Stokes equations

The solution of the compressible Euler and Navier-Stokes equations via an upwind finite volume scheme is obtained. For the inviscid fluxes, the monotone upstream-centered scheme for conservation laws (MUSCL) has been incorporated into a Riemann solver. The MUSCL scheme is used for the unfactored implicit equations that are solved by a Newton form, and relaxation is performed via Gauss-Seidel relaxation technique. The solution on the fine grid is obtained by iterating first on a sequence of coarser grids and then interpolating the solution up to the next refined grid. Since the distribution of the numerical error is nonuniform, the local solution of the equations can be obtained in regions where the numerical errors are large. The construction of the partial meshes, in which the iterations will be continued, is determined by an adaptive procedure taking into account some convergence criteria. Reduction of the computational work units for two-dimensional problems is obtained via the local adaptive mesh solution which is expected to be more effective in three-dimensional complex flow computations.