An implicit,conservative, zonal-boundary scheme for Euler equation calculations

A “zonal”, or “patched-grid”, approach is one in which the flow region of interest is divided into subregions which are then discretized independently, using existing grid generators. The equations of motion are integrated in each subregion in conjunction with zonal-boundary schemes which allow proper information transfer across interfaces that separate subregions. The zonal approach greatly simplifies the treatment of complex geometries and also the addition of grid points to selected regions of the flow. In this study a conservative, zonal-boundary condition that could be used with explicit schemes has been extended so that it can be used with existing second-order accurate implicit integration schemes such as the Beam-Warming and Osher schemes. In the test case considered, the implicit schemes increased the rate of convergence considerably (by a factor of about 30 over that of the explicit scheme). Results demonstratiting the time-accuracy of the zonal scheme and the feasibility of performing calculations on zones that move relative to each other are also presented.     

A weighted ENO-flux limiter scheme for hyperbolic conservation laws

We present a new scheme that combines essentially non-oscillatory (ENO) reconstructions together with monotone upwind schemes for scalar conservation laws interpolants. We modify a second-order ENO polynomial by choosing an additional point inside the stencil in order to obtain the highest accuracy when combined with the Harten–Osher reconstruction-evolution method limiter. Numerical experiments are done in order to compare a weighted version of the hybrid scheme to weighted essentially non-oscillatory (WENO) schemes with constant Courant–Friedrichs–Lewy number under relaxed step size restrictions. Our results show that the new scheme reduces smearing near shocks and corners, and in some cases it is more accurate near discontinuities compared with higher-order WENO schemes. The hybrid scheme avoids spurious oscillations while using a simple componentwise extension for solving hyperbolic systems. The new scheme is less damped than WENO schemes of comparable accuracy and less oscillatory than higher-order WENO schemes. Further experiments are done on multi-dimensional problems to show that our scheme remains non-oscillatory while giving good resolution of discontinuities.     

高超声速流动CFD并行计算研究

并行计算是CFD技术发展的必然趋势。本文从高超声速流动的特点出发,研究多分区结构网格下CFD并行计算方法,重点解决了区域之间流场信息的数据交换问题和边界处理问题,以保证流场的连续性。本文采用有限体积法求解高超声速流场,空间离散格式为Osher-Chakravarthy TVD格式,利用MPI消息传递模式完成数据交换,在自主搭建的PC集群上进行算例考核,验证了算法的可行性和正确性。     

A Hybrid Method to Solve Shallow Water Flows with Horizontal Density Gradients

In this work, a model for shallow water flows that accounts for the effects of horizontal density fluctuations is presented and derived. While the density is advected by the flow, a two-way feedback between the density gradients and the time evolution of the fluid is ensured through the pressure and source terms in the momentum equations. The model can be derived by vertically averaging the Euler equations while still allowing for density fluctuations in horizontal directions. The approach differs from multi-layer shallow water flows where two or more layers are considered, each of them having their own depth, velocity and constant density. A Roe-type upwind scheme is developed and the Roe matrices are computed systematically by going from the conservative to the quasi-linear form at a discrete level. Properties of the model are analyzed. The system is hyperbolic with two shock-wave families and a contact discontinuity associated to interfaces of regions with density jumps. This new field is degenerate with pressure and velocity as the corresponding Riemann invariants. We show that in some parameter regimes numerically recognizing such invariants across contact discontinuities is important to correctly compute the flow near those interfaces. We present a numerical algorithm that correctly captures all waves with a hybrid strategy. The method integrates the Riemann invariants near contact discontinuities and switches back to the conserved variables away from it to properly resolve shock waves. This strategy can be applied to any numerical scheme. Numerical solutions for a variety of tests in one and two dimensions are shown to illustrate the advantages of the strategy and the merits of the scheme.     

Computation of physiological bifurcation flows using a patched grid

A finite volume method in a boundary-fitted coordinate system together with a zonal grid method is employed to compute the flow field of a real-shape two-dimensional aortic bifurcation. The steady terms in the governing equations are treated by a fully explicit scheme. The zonal gridding procedure is discussed in detail. The numerical method is first tested in a laminar backward facing step flow to demonstrate the features of the method. The effect of the interface treatment on the flow fields can be significant. A 90° T-junction is then computed. The results are in good agreement with the available experimental data. The method is then applied to simulate the flows of an atherosclerotic human aorta. Both the steady and pulsatile flows are considered. It is shown that the mean shear stresses in recirculation regions of a pulsatile flow cannot be adequately described by a corresponding steady flow with a mean Reynolds number. In pulsatile flows, a sinusoidal input pulse and a realistic input pulse are both used in the computations. It is found that the “averaged” flow behavior is similar in both cases. However, the details of the flow field are significantly different. During pulsatile flow, permanent eddies are not present. That is, for a certain period in a cycle, the entire wall is free from eddies. On the other hand, in another period, the overall wall is almost completely in the reversing flow near the walls. These phenomena have been observed by other authors experimentally. Distributions of wall shear stresses and locations of recirculation zones in a realistic flow are shown and discussed briefly.     

Zonal finite-volume computations of incompressible flows

The paper addresses convection modelling and flow field zoning procedures destined for finite-volume methods for incompressible steady-state flows with irregular boundaries. A composite oscillation-damping algorithm which is capable of yielding bounded and low diffusive solutions is used to approximate the convection terms of transport equations. To deal with complex geometries, a zonal procedure is introduced into an advanced finite-volume method that uses general non-orthogonal and non-staggered grids. The solution domain is divided into simple subregions each covered by a separate mesh. The flow is solved for concurrently in each of the different zones, and information exchange among the zones is realized through overlapping the grids in the vicinity of the zonal interfaces. The zonal procedure enables the finite-volume method to handle those domains which are extremely difficult to cover with a single grid. Applications to a range of laminar elliptic problems demonstrate the capability of the method for predicting complex flows.     

Separation-bubble flow solution using Euler/Navier-Stokes zonal approach with downstream compatibility conditions

An Euler/Navier-Stokes zonal scheme is developed to numerically simulate the two-dimensional flow over a blunt leading-edge plate. The computational domain has been divided into inner and outer regions where the Navier-Stokes and Euler equations are used, respectively. On the downstream boundary, compatibility conditions derived from the boundary-layer equations are used. The grid is generated by using conformal mapping and the problem is solved by using a compressible Navier-Stokes code, which has been modified to treat Euler and Navier-Stokes regions. The accuracy of the solution is determined by the reattachment location. Bench-mark solutions have been obtained using the Navier-Stokes equations throughout the optimum computational domain and size. The problem is recalculated with sucessive decrease of the computational domain from the downstream side where the compatibility conditions are used, and with successive decrease of the Navier-Stokes computational region. The results of the zonal scheme are in excellent agreement with those of the benchmark solutions and the experimental data. The CPU time saving is about 15%.     

A zonal grid algorithm for DNS of turbulent boundary layers

A zonal grid algorithm for direct numerical simulation (DNS) of incompressible turbulent flows within a Finite-Volume framework is presented. The algorithm uses fully coupled embedded grids and a conservative treatment of the grid-interface variables. A family of conservative prolongation operators is tested in a 2D vortex dipole and a 3D turbulent boundary layer flow. These tests show that both, first- and second-order interpolation conserves the overall second-order spatial accuracy of the scheme. The first-order conservative interpolation has a smaller damping effect on the solution but the second-order conservative interpolation has better spectral properties. The application of this algorithm in boundary layer flow separating and reattaching due to the presence of a streamwise pressure gradient reveals the power and usefulness of the presented algorithm. This simulation has been made possible by the zonal grid algorithm by reducing the required number of grid points from about 500 × 10⁶ to 130 × 10⁶ grid cells.     

A flux-coordinate-splitting technique for flows with shocks and contact discontinuities

The two-dimensional gasdynamic equations are solved everywhere in the flow field except in regions surrounding the contact discontinuites. A flux-vector-splitting (FVS) technique is applied to the Euler equations so that the directions of propagation of the signals and hence the shocks in the flow can be correctly captured. The split flux equations are solved using conventional second-order-accurate finite difference methods. In the regions surrounding the contact discontinuities, the gasdynamic equations are split into a set of one-dimensional equations. These are transformed in such a way that the density does not appear explicitly in the spatial derivatives of the resultant equations, which are of the Langrangian form. The equations are then solved using second-order-accurate finite difference schemes and numerical smearing of the contact discontinuities is avoided because the dependent variables are continuous across the discontinuities. Consequently, both shocks and contact discontinuities in a two-dimensional gasdynamic flow are accurately resolved. This flux-coordinate-splitting technique is used to calculate the gasdynamic flow in a shock tube, a converging cylindrical shock and the mixing of two supersonic streams. The results are compared with exact solutions and with those deduced from proven numerical techniques. Good correlations are obtained, especially in the sharp definition of contact discontinuities. Therefore, the proposed coordinate-splitting technique improves the resolution of contact discontinuities without affecting the overall calculations of the flow field. In view of this, the coordinate-splitting technique can also be used with other shock capturing techniques besides FVS to achieve the same results.     

A block LU-SGS implicit dual time-stepping algorithm for hybrid dynamic meshes

A block lower-upper symmetric Gauss-Seidel (BLU-SGS) implicit dual time-stepping method is developed for moving body problems with hybrid dynamic grids. To simulate flows over complex configurations, a hybrid grid method is adopted in this paper. Body-fitted quadrilateral (quad) grids are generated first near solid bodies. An adaptive Cartesian mesh is then generated to cover the entire computational domain. Cartesian cells which overlap the quad grids are removed from the computational domain, and a gap is produced between the quad grids and the adaptive Cartesian grid. Finally triangular grids are used to fill this gap. With the motion of moving bodies, the quad grids move with the bodies, while the adaptive Cartesian grid remains stationary. Meanwhile, the triangular grids are deformed according to the motion of solid bodies with a ‘spring’ analogy approach. If the triangular grids become too skewed, or the adaptive Cartesian grid crosses into the quad grids, the triangular grids are regenerated. Then the flow solution is interpolated from the old to the new grid. The fully implicit equation is solved using a dual time-stepping solver. A Godunov-type scheme with Roe’s flux splitting is used to compute the inviscid flux. Several sub-iteration schemes are investigated in this study. Both supersonic and transonic unsteady cases are tested to demonstrate the accuracy and efficiency of the method.     

High-efficiency design of a mixed-flow pump

High-efficiency design of a mixed-flow pump has been carried out based on numerical analysis of a three-dimensional viscous flow.For analysis,the Reynolds-averaged Navier-Stokes equations with a shear stress transport turbulence model were discretized by finite-volume approximations.Structured grid system was constructed in the computational domain,which has O-type grids near the blade surfaces and H/J-type grids in other regions.The numerical results were validated with experimental data for the heads and ...     

An implicit low-diffusive HLL scheme with complete time linearization: Application to cavitating barotropic flows

A numerical method for generic barotropic flows is presented, together with its application to the simulation of cavitating flows. A homogeneous-flow cavitation model is indeed considered, which leads to a barotropic state equation. The continuity and momentum equations for compressible flows are discretized through a mixed finite-element/finite-volume approach, applicable to unstructured grids. P1 finite elements are used for the viscous terms, while finite volumes for the convective ones. The numerical fluxes are computed by shock-capturing schemes and ad-hoc preconditioning is used to avoid accuracy problems in the low-Mach regime. A HLL flux function for barotropic flows is proposed, in which an anti-diffusive term is introduced to counteract accuracy problems for contact discontinuities and viscous flows typical of this class of schemes, while maintaining its simplicity. Second-order accuracy in space is obtained through MUSCL reconstruction. Time advancing is carried out by an implicit linearized scheme. For this HLL-like flux function two different time linearizations are considered; in the first one the upwind part of the flux function is frozen in time, while in the second one its time variation is taken into account. The proposed numerical ingredients are validated through the simulations of different flow configurations, viz. the Blasius boundary layer, a Riemann problem, the quasi-1D cavitating flow in a nozzle and the flow around a hydrofoil mounted in a tunnel, both in cavitating and non-cavitating conditions. The Roe flux function is also considered for comparison. It is shown that the anti-diffusive term introduced in the HLL scheme is actually effective to obtain good accuracy (similar to the one of the Roe scheme) for viscous flows and contact discontinuities. Moreover, the more complete time linearization is a key ingredient to largely improve numerical stability and efficiency in cavitating conditions.     

Implementation of quadratic upstream interpolation schemes for solute transport into HYDRUS-1D

Numerical solution of the advection–dispersion equation, used to evaluate transport of solutes in porous media, requires discretization schemes for space and time stepping. We examine use of quadratic upstream interpolation schemes QUICK, QUICKEST, and the total variation diminution scheme ULTIMATE, and compare these with UPSTREAM and CENTRAL schemes in the HYDRUS-1D model. Results for purely convective transport show that quadratic schemes can reduce the oscillations compared to the CENTRAL scheme and numerical dispersion compared to the UPSTREAM scheme. When dispersion is introduced all schemes give similar results for Peclet number Pe < 2. All schemes show similar behavior for non-uniform grids that become finer in the direction of flow. When grids become coarser in the direction of flow, some schemes produce considerable oscillations, with all schemes showing significant clipping of the peak, but quadratic schemes extending the range of stability tenfold to Pe < 20. Similar results were also obtained for transport of a non-linear retarded solute transport (except the QUICK scheme) and for reactive transport (except the UPSTREAM scheme). Analysis of transient solute transport show that all schemes produce similar results for the position of the infiltration front for Pe = 2. When Pe = 10, the CENTRAL scheme produced significant oscillations near the infiltration front, compared to only minor oscillations for QUICKEST and no oscillations for the ULTIMATE scheme. These comparisons show that quadratic schemes have promise for extending the range of stability in numerical solutions of solute transport in porous media and allowing coarser grids.     

Formulation of Kinetic Energy Preserving Conservative Schemes for Gas Dynamics and Direct Numerical Simulation of One-Dimensional Viscous Compressible Flow in a Shock Tube Using Entropy and Kinetic Energy Preserving Schemes

This paper follows up on the author’s recent paper “The Construction of Discretely Conservative Finite Volume Schemes that also Globally Conserve Energy or Enthalpy”. In the case of the gas dynamics equations the previous formulation leads to an entropy preserving (EP) scheme. It is shown in the present paper that it is also possible to construct the flux of a conservative finite volume scheme to produce a kinetic energy preserving (KEP) scheme which exactly satisfies the global conservation law for kinetic energy. A proof is presented for three dimensional discretization on arbitrary grids. Both the EP and KEP schemes have been applied to the direct numerical simulation of one-dimensional viscous flow in a shock tube. The computations verify that both schemes can be used to simulate flows with shock waves and contact discontinuities without the introduction of any artificial diffusion. The KEP scheme performed better in the tests.     

Simulation of the head-disk interface gap using a hybrid multi-scale method

We present a hybrid multi-scale method that provides a capability to capture the disparate scales associated with modelling flow in micro- and nano-devices. Our model extends the applicability of an internal-flow multi-scale method by providing a framework to couple the internal (small scale) flow regions to the external (large scale) flow regions. We demonstrate the application of both the original methodology and the new hybrid approach to model the flow field in the vicinity of the head-disk interface gap of a hard disk drive enclosure. The internal flow regions within the gap are modelled by an extended internal-flow multi-scale method that utilises a finite-difference scheme for non-uniform grids. Our proposed hybrid multi-scale method is then employed to couple the internal micro-flow region to the flow external to the gap, to capture entrance/exit effects. We also demonstrate the successful application of the method in capturing other localised phenomena (e.g. those due to localised wall heating).     

A staggered scheme for hyperbolic conservation laws applied to unsteady sheet cavitation

We demonstrate the advantages of discretizing on a staggered grid for the computation of solutions to hyperbolic systems of conservation laws arising from instationary flow of an inviscid fluid with an arbitrary equation of state. Results for a highly nonlinear, nonconvex equation of state obtained with the staggered discretisation are compared with those obtained with the Osher scheme for two different Riemann problems. The staggered approach is shown to be superior in simplicity and efficiency, without loss of accuracy. The method has been applied to simulate unsteady sheet cavitation on a NACA0012 hydrofoil. Results show good agreement with those obtained with a cavity interface tracking method. Received: 25 February 1999 / Accepted: 17 June 1999     

前机身/进气道流场的一体化数值模拟

该文针对飞行器设计中的机体和进气道的一体化问题,采用分区结构搭接网格技术,对机体外流场采用有限体积法求解Euler方程,进气道内流场采用有限体积法求解N—S方程,发展了求解三维超音速前机身／进气道内外流的Euler／N—S方程分区处理程序,同时对某型飞机的前机身／进气道流场进行了一体化模拟。通过分析计算出的附加阻力和畸变系数等结果的合理性验证了计算程序的正确性。同时通过对计算域采用不同的密度网格进行计算,分析研究了网格疏密对计算结果的影响。     

A local mesh-refinement technique for incompressible flows

As part of a Multi-Grid scheme for the solution of the Navier-Stokes equations in primitive variables, we introduce a local mesh refinement procedure. New cartesian sub-grids are introduced into regions where the estimated truncation errors are too large. Through the Multi-Grid processing, informations is transferred among the grids in a stable and efficient manner. A simple pointer system allows the storage of the dependent variables, without increasing in the required computer memory. Two computed examples of incompressible flow problems are discussed.     

Third-order accurate finite volume schemes for Euler computations on curvilinear meshes

After looking for a convenient definition of accuracy for finite-volume schemes on structured meshes, a high-order accurate scheme is constructed for the Euler equations. Thanks to suitably weighted discretization operators, the proposed scheme is third-order on mildly deformed grids and second-order on highly deformed grids. The influence of mesh deformations on the scheme accuracy is studied theoretically and numerically. Numerical results are shown for a Lamb vortex, subsonic flow past a cylinder and transonic flow past a NACA0012 airfoil.