Point-implicit Method Research Articles

A Unified Grid Approach Using Hamiltonian Paths for Computing Aerodynamic Flows

A solution algorithm using Hamiltonian paths is presented as a unified grid approach for rotorcraft applications. Hidden line structures are robustly identified on general two- and three-dimensional unstructured grids with mixed elements, providing a framework for line-based solvers. A pure quadrilateral/hexahedral mesh is a prerequisite for line identification and enables approximate factorisation along the lines. The numerical efficiency obtained using the line-implicit method on various unstructured grids is better than that of the point-implicit method. Both finite-difference and gradient reconstructions are possible regardless of grid type. A combined reconstruction method is applied, which uses different reconstructions simultaneously but for different grid directions. Finally, the solution convergence rate is further improved using a preconditioned generalized minimal residual method (GMRES), where the preconditioning step is performed using the efficient line-implicit method.

A Hydrodynamic-Based Robust Numerical Model for Debris Hazard and Risk Assessment

Debris flow simulations are important in practical engineering. In this study, a graphics processing unit (GPU)-based numerical model that couples hydrodynamic and morphological processes was developed to simulate debris flow, transport, and morphological changes. To accurately predict the debris flow sediment transport and sediment scouring processes, a GPU-based parallel computing technique was used to accelerate the calculation. This model was created in the framework of a Godunov-type finite volume scheme and discretized into algebraic equations by the finite volume method. The mass and momentum fluxes were computed using the Harten, Lax, and van Leer Contact (HLLC) approximate Riemann solver, and the friction source terms were calculated using the proposed splitting point-implicit method. These values were evaluated using a novel 2D edge-based MUSCL scheme. The code was programmed using C++ and CUDA, which can run on GPUs to substantially accelerate the computation. After verification, the model was applied to the simulation of the debris flow process of an idealized example. The results of the new scheme better reflect the characteristics of the discontinuity of its movement and the actual law of the evolution of erosion and deposition over time. The research results provide guidance and a reference for the in-depth study of debris flow processes and disaster prevention and mitigation.

Numerical study of self-sustained oblique detonation in a non-uniform mixture

Self-sustained oblique detonation behind a spherical projectile formed in a non-uniform H2/O2/Ar mixture was numerically investigated. A hypersonic combustible mixture flow around a 4.76 mm diameter body was modeled to be flowing at 2500 m/s and 100 kPa. The concentration gradient was prescribed applying the Gaussian distribution to hydrogen concentration. Axisymmetric Euler equations including a detailed kinetics of 9 species and 27 elementary reactions were solved with an explicit 2nd-order time integration scheme combined with point implicit method for chemical reaction. Oblique detonation was always obtained when the mixture on the centerline was stoichiometric, as it is for a uniform mixture, and a broader range of equivalence ratio could sustain oblique detonation far from the sphere. Local detonation angle was revealed to reasonably match Chapman-Jouguet analytical solutions with a minor difference attributed to curvature, less reactive composition, and the concentration gradient. Also, a strongly fuel-rich region encountered decoupling of the shock-flame, in which an abrupt deflection of the shock front appeared. These decoupling phenomena can be attributed to a slower kinetics of a less reactive mixture. All of interesting findings in this study will also benefit understanding of various form of detonation in non-uniform mixture taking advantage of the analogy between them.

A GPU-based numerical model coupling hydrodynamical and morphological processes

Sediment transport simulations are important in practical engineering. In this study, a graphics processing unit (GPU)-based numerical model coupling hydrodynamical and morphological processes was developed to simulate water flow, sediment transport, and morphological changes. Aiming at accurately predicting the sediment transport and sediment scouring processes, the model resolved the realistic features of sediment transport and used a GPU-based parallel computing technique to the accelerate calculation. This model was created in the framework of a Godunov-type finite volume scheme to solve the shallow water equations (SWEs). The SWEs were discretized into algebraic equations by the finite volume method. The fluxes of mass and momentum were computed by the Harten, Lax, and van Leer Contact (HLLC) approximate Riemann solver, and the friction source terms were calculated by the proposed a splitting point-implicit method. These values were evaluated using a novel 2D edge-based MUSCL scheme. The code was programmed using C++ and CUDA, which could run on GPUs to substantially accelerate the computation. The aim of the work was to develop a GPU-based numerical model to simulate hydrodynamical and morphological processes. The novelty is the application of the GPU techniques in the numerical model, making it possible to simulate the sediment transport and bed evolution in a high-resolution but efficient manner. The model was applied to two cases to evaluate bed evolution and the effects of the morphological changes on the flood patterns with high resolution. This indicated that the GPU-based high-resolution hydro-geomorphological model was capable of reproducing morphological processes. The computational times for this test case on the GPU and CPU were 298.1 and 4531.2 s, respectively, indicating that the GPU could accelerate the computation 15.2 times. Compared with the traditional CPU high-grid resolution, the proposed GPU-based high-resolution numerical model improved the reconstruction speed more than 2.0–12.83 times for different grid resolutions while remaining computationally efficient.

Numerical study of chemically reacting flow in a shock tube using a high-order point-implicit scheme

The shock wave/boundary-layer interaction of chemically reacting flow in a shock tube is studied using a high-order point-implicit solver. The solver employs a high-resolution weighted essentially non-oscillatory (WENO) scheme to capture the complex shock structures, together with a point-implicit method to overcome the stiffness of the chemical production term in the multicomponent Navier–Stokes equations. The numerical code is carefully validated with three benchmark tests, which demonstrates the robustness and good performance of the combined numerical methods. The unsteady interaction process between the shock wave and boundary layer in a two-dimensional shock tube is clearly captured with detailed flow patterns in the simulation. Simulation results show that regular vortex arrangements appear in the flow field for the case of Mach number 2.37, while for the case of Mach number 3.15, the vortex structures break up and chemical nonequilibrium effects become apparent. The influence of real gas effects on shock wave/boundary-layer interaction is further identified on the temperature field and triple point trajectory.

Four Point Implicit Methods for the Second Derivatives of the Solution of First Type Boundary Value Problem for One Dimensional Heat Equation

We construct four-point implicit difference boundary value problem for the first derivative of the solution u(x,t) of the first type boundary value problem for one dimensional heat equation with respect to the time variable t. Also, for the second derivatives of u(x,t) special four-point implicit difference boundary value problems are proposed. It is assumed that the initial function belongs to the Hölder space C8+α,0 < α < 1, the heat source function given in the heat equation is from the Hölder space [see formula in PDF], the boundary functions are from [see formula in PDF], and between the initial and the boundary functions the conjugation conditions of orders q = 0,1,2,3,4 are satisfied. We prove that the solution of the proposed difference schemes converge uniformly on the grids of the order O(h2+τ) (second order accurate in the spatial variable x and first order accurate in time t) where, h is the step size in x and τ is the step size in time. Theoretical results are justified by numerical examples.

Stabilisation of discrete steady adjoint solvers

A new implicit time-stepping scheme which uses Runge–Kutta time-stepping and Krylov methods as a smoother inside FAS-cycle multigrid acceleration is proposed to stabilise the flow solver and its discrete adjoint counterpart. The algorithm can fully converge the discrete adjoint solver in a wide range of cases where conventional point-implicit methods fail due to either physical or numerical instability. This enables the discrete adjoint to be applied to a much wider range of flow regimes. In addition, the new algorithm offers improved efficiency when applied to stable cases for which the conventional Block–Jacobi solver can fully converge. Both stable and unstable cases are presented to demonstrate the improved robustness and performance of the new scheme. Eigen-analysis is presented to outline the mechanism of the adjoint stabilisation effect.

A point implicit time integration technique for slow transient flow problems

We introduce a point implicit time integration technique for slow transient flow problems. The method treats the solution variables of interest (that can be located at cell centers, cell edges, or cell nodes) implicitly and the rest of the information related to same or other variables are handled explicitly. The method does not require implicit iteration; instead it time advances the solutions in a similar spirit to explicit methods, except it involves a few additional function(s) evaluation steps. Moreover, the method is unconditionally stable, as a fully implicit method would be. This new approach exhibits the simplicity of implementation of explicit methods and the stability of implicit methods. It is specifically designed for slow transient flow problems of long duration wherein one would like to perform time integrations with very large time steps. Because the method can be time inaccurate for fast transient problems, particularly with larger time steps, an appropriate solution strategy for a problem that evolves from a fast to a slow transient would be to integrate the fast transient with an explicit or semi-implicit technique and then switch to this point implicit method as soon as the time variation slows sufficiently. We have solved several test problems that result from scalar or systems of flow equations. Our findings indicate the new method can integrate slow transient problems very efficiently; and its implementation is very robust.

Numerical Study of Shock/Boundary Layer Interaction of Chemically Reacting Flow in Shock Tube

In this paper, a high-order Navier-Stokes solver is developed for chemically reacting flow and is applied to investigate the unsteady, viscous interaction between shock wave and boundary layer in an air filled shock tube. The solver employs a high- resolution weighted essentially non-oscillatory (WENO) scheme to capture the complex flow pattern induced by shock/boundary- layer interaction, together with a point-implicit method to overcome the stiffness of chemical production term. The results show the influence of real gas effects and give insights into the mechanism of shock/boundary-layer interaction process.

A stable 2D unstructured shallow flow model for simulations of wetting and drying over rough terrains

This paper proposes a 2D unstructured finite volume Godunov-type shallow water model to simulate complex flows involving wet–dry interfaces over irregular terrains. In this model, the MUSCL linear reconstruction is used to compute the values of variables at cell edges. These values are then modified with a non-negative reconstruction method to preserve non-negative edge water depths and the C-property. The modified values are then employed to evaluate the fluxes and the slope source terms using the Harten, Lax and van Leer approximate Riemann solver with the contact wave restored (HLLC) and the slope flux method, respectively. The friction source terms are solved by a splitting point-implicit method. The novel aspect of the model is an efficient approach to eliminate spurious edge velocities with high local extreme values caused by MUSCL reconstruction of the conservative variables near wet–dry interfaces. This approach is crucial to preserve the numerical stability of the model when simulating wetting and drying. The proposed model is tested against exact solutions, laboratory experiments and a real dam-break case. The results show that the proposed model is accurate and stable for handling complex flows with moving wet–dry interfaces over complex beds on unstructured grids.

Application of a line implicit method to fully coupled system of equations for turbulent flow problems

For unstructured finite volume methods, we present a line implicit Runge–Kutta method applied as smoother in an agglomerated multigrid algorithm to significantly improve the reliability and convergence rate to approximate steady-state solutions of the Reynolds-averaged Navier–Stokes equations. To describe turbulence, we consider a one-equation Spalart–Allmaras turbulence model. The line implicit Runge–Kutta method extends a basic explicit Runge–Kutta method by a preconditioner given by an approximate derivative of the residual function. The approximate derivative is only constructed along predetermined lines which resolve anisotropies in the given grid. Therefore, the method is a canonical generalisation of point implicit methods. Numerical examples demonstrate the improvements of the line implicit Runge–Kutta when compared with explicit Runge–Kutta methods accelerated with local time stepping.

A robust well-balanced model on unstructured grids for shallow water flows with wetting and drying over complex topography

To simulate complex flows involving wet-dry fronts in irregular terrains over arbitrary beds, this paper presents a 2D well-balanced shallow water flow model, based on an unstructured cell-centered finite volume scheme. In this model, hydrostatic reconstruction is applied to reconstruct non-negative water depths at wet-dry interfaces. Harten, Lax and van Leer approximate Riemann solver with the contact wave restored is employed to compute the fluxes of mass and momentum. In addition, the splitting point-implicit method is utilized to solve the friction source terms. The novel aspects of the model include the adaptive method and the slope source term treatment. The former is devised to prevent unphysical high velocities occurring in the part of the domain with varying thin water and topography. The latter converts the slope source terms of a cell into fluxes through its edges and takes account of the influence of wet-dry fronts, so as to satisfy the C-property in any case on unstructured grids. The accuracy and robustness of the proposed model are extensively verified against several benchmark tests as well as a real dam-break event, where its superiority in simulating complex flows with wetting and drying over uneven bed is emphasized.

Application of point implicit Runge-Kutta methods to inviscid and laminar flow problems using AUSM and AUSM+ upwinding

Explicit Runge-Kutta methods preconditioned by a pointwise matrix valued preconditioner can significantly improve the convergence rate to approximate steady state solutions of laminar flows. This h...

Simulation of transient flow caused by pump failure: Point-Implicit Method of Characteristics

A Point-Implicit Method of Characteristics (PIMOC) and an improved pump formulation are presented in this paper for the efficient simulation of transient flows caused by pump failure. The proposed methods can essentially be considered as modifications to the newly introduced Implicit Method of Characteristics (IMOC) by the authors. The IMOC, however, requires more computational effort than the conventional explicit version due to the fact that IMOC requires the solution of a nonlinear system of equations at each time step. To alleviate this problem, a Point-Implicit version of the IMOC is proposed here which requires less computational effort while enjoying all the advantages of the original IMOC. The proposed method is based on the observation that the assembled equations at each interior point of the pipe segments are uncoupled from those of the other points, including nodal points, which are used to define the system layout. These equations can, therefore, be separately solved for the unknown head and flow at these points before the remaining equations are assembled to form the final system of equations. The solution of the resulting system of equations with a size equal to twice as the number of nodes yields the value of head and flow at the remaining nodal points. To further improve the convergence of the method for the simulation of pump failure, an improved formulation of the pump is also proposed. In the proposed formulation, one of the pump characteristic parameters is also used to formulate the behavior of the pump elements. The proposed Point-Implicit Method and improved pump formulation are used to solve a set of numerical examples and the results are compared with those of original IMOC. For this, the element-wise formulations of two new devices namely junction and check-valve are also introduced. The experiments show improved efficiency of the proposed methods compared to the original IMOC.

Unstructured Cartesian Grid Methodology for Inviscid Nonequilibrium Hypersonic Flows

In this work, a solution-adaptive unstructured Cartesian grid solver for thermochemical nonequilibrium hypersonic flows with a two-temperature model is developed. The Cartesian grid methodology offers the convenience of automatic grid generation over three-dimensional complex geometries with minimum user intervention. At the same time, an accurate higher-order flux calculation over unstructured-grid topologies is provided. To avoid convergence problems in the explicit flow solver, a point-implicit method is adopted to discretize the source terms. The AUSMPW+ numerical scheme with MUSCL data reconstruction is used, preventing nonphysical oscillations and carbuncle phenomena. The accuracy of the present methodology was verified against DPLR (NASA Ames Research Center's structured-grid nonequilibrium flow solver), showing a good agreement between the two codes. As expected, the AUSMPW+ scheme effectively resolved the large gradients of the flow properties at the shock waves. In parallel, the point-implicit technique used for the source terms allows the original Courant-Friedrichs- Lewy constraints of the explicit solver to be maintained.

An Efficient Implicit Scheme of MPD Thruster Viscous Flow.

An efficient implicit finite-difference scheme for solving supersonic magnet-plasma dynamic (MPD) viscous flows is proposed. The compressible Navier-Stokes equations and the equation of induction induced from the Maxwell equations and the Ohm's law are simultaneously solved using the LU-SGS approach coupled with the diagonal point implicit method. This approach enables to get the convergence of solution. As numerical examples, two types of the flow through the MPD thruster are calculated. The reliability of the present method is verified to show the convergence history of solutions and the calculated results.

Numerical Study of Scram Accelerator Starting Characteristics

A numerical study is carried out to investigate the ignition and the detonation initiation process in a scram accelerator operating at a superdetonative mode. To simulate the scram accelerator launching process, a conical projectile is considered, injected with an initial velocity of 2500 m/s from the 1 atm air into a 25 atm 2H 2 + O 2 + mN 2 mixture. As a dilution gas, nitrogen is selected and assumed to be inert. The time accurate solutions of Reynolds averaged Navier-Stokes equations for chemically reacting flows are obtained by using a point-implicit method and an upwind-biased third-order scheme with a steady-state solution for airflow as an initial condition. To examine combustion characteristics and ram-accelerator operation limits, mixture compositions are varied from 2H 2 + O 2 + 3.76N 2 to 2H 2 + O 2 + 9N 2 by changing the amount of N 2 . The flowfield results show the detailed ignition mechanism, the initiation process of the oblique detonation, and the staring characteristics of the scram accelerator. The results also identify clearly the combustion characteristics of the operational failures at lower and upper dilution limits that have been observed in experiments

Comparison of Three Navier-Stokes Equation Solvers for Supersonic Open Cavity Simulations

We compare the results obtained with an approximately factored lower-upper triangular solver, an explicit Runge-Kutta scheme, and a point implicit method for a variety of timesteps to assess solver error and timestep issues in computations of unsteady flows

Implicit/Multigrid Algorithms for Incompressible Turbulent Flows on Unstructured Grids

An implicit code for computing inviscid and viscous incompressible flows on unstructured grids is described. The foundation of the code is a backward Euler time discretization for which the linear system is approximately solved at each time step with either a point implicit method or a preconditioned generalized minimal residual (GMRES) technique. For the GMRES calculations, several techniques are investigated for forming the matrix–vector product. Convergence acceleration is achieved through a multigrid scheme that uses nonnested coarse grids that are generated using a technique described in the present paper. Convergence characteristics are investigated and results are compared with an exact solution for the inviscid flow over a four-element airfoil. Viscous results, which are compared with experimental data, include the turbulent flow over a NACA 4412 airfoil, a three-element airfoil for which Mach number effects are investigated, and three-dimensional flow over a wing with a partial-span flap.

Transient simulation of superdetonative mode initiation process in scram-accelerator

A numerical study is carried out to examine the detonation-initiation process of a SCRam accelerator operating in superdetonative mode by considering the injection of a high-speed conical projectile from air into a high-pressure combustible mixture. The time-accurate solutions of Reynolds-averaged Navier-Stokes equations for chemically reacting flows are obtained by using the point-implicit method and the upwind-biased third-order scheme. As a result, two distinctive cases of detonation-initiation processes are observed: a case of unstart by the direct initiation of a detonation and a case of regular ignition to quasi-steady marching. In the former case which is observed in a 2H 2 +air mixture, a detonation initiates on the conical nose of the projectile when a projectile is passing through the diaphragm. The interaction between an oblique shock and a moving normal shock could be one cause of the direct initiation of the detonation. After a sufficient duration. a thernal choking condition, which is the cause of a strong drag, is attained by interactions between the shock train and the boundary layer. In the latter case, which is observed in a 2H 2 +O 2 +5N 2 mixture, an oblique detonation originates in the shock-impinging region where the preburned hot gas boundary layer and oblique shock inteact. After ignition, the oblique detonation propagates forward and settles down on the conical nose. At this stage, the thrust is comparable to the drag, and the projectile moves with nearly constant velocity.