A point implicit unstructured grid solver for the euler and Navier–Stokes equations

An upwind finite element technique that uses cell-centred quantities and implicit and/or explicit time marching has been developed for computing hypersonic laminar viscous flows using adaptive triangular grids. The approach is an extension to unstructured grids of the LAURA algorithm due to Gnoffo. A structured grid of quadrilaterals is laid out near a solid surface. For inviscid flows the method is stable at Courant numbers of over 100000. A first-order basic scheme and a higher-order flux-corrected transport (FCT) scheme have been implemented. This technique has been applied to the problem of predicting type III and IV shock wave interactions on a cylinder, with a view to simulating the pressure and heating rate augmentation caused by an impinging shock on the leading edge of a cowl lip of an engine inlet. The predictions of wall pressure and heating rates compare very well with experimental data. The flow features are distinctly captured with a sequence of adaptively generated grids.     

An unstructured/structured multi‐layer hybrid grid method and its application

A multi‐layer hybrid grid method is constructed to simulate complex flow field around 2‐D and 3‐D configuration. The method combines Cartesian grids with structured grids and triangular meshes to provide great flexibility in discretizing a domain. We generate the body‐fitted structured grids near the wall surface and the Cartesian grids for the far field. In addition, we regard the triangular meshes as an adhesive to link each grid part. Coupled with a tree data structure, the Cartesian grid is generated automatically through a cell‐cutting algorithm. The grid merging methodology is discussed, which can smooth hybrid grids and improve the quality of the grids. A cell‐centred finite volume flow solver has been developed in combination with a dual‐time stepping scheme. The flow solver supports arbitrary control volume cells. Both inviscid and viscous flows are computed by solving the Euler and Navier–Stokes equations. The above methods and algorithms have been validated on some test cases. Computed results are presented and compared with experimental data. Copyright © 2006 John Wiley & Sons, Ltd.     

A parallel adaptive numerical method with generalized curvilinear coordinate transformation for compressible Navier–Stokes equations

A fourth‐order finite‐volume method for solving the Navier–Stokes equations on a mapped grid with adaptive mesh refinement is proposed, implemented, and demonstrated for the prediction of unsteady compressible viscous flows. The method employs fourth‐order quadrature rules for evaluating face‐averaged fluxes. Our approach is freestream preserving, guaranteed by the way of computing the averages of the metric terms on the faces of cells. The standard Runge–Kutta marching method is used for time discretization. Solutions of a smooth flow are obtained in order to verify that the method is formally fourth‐order accurate when applying the nonlinear viscous operators on mapped grids. Solutions of a shock tube problem are obtained to demonstrate the effectiveness of adaptive mesh refinement in resolving discontinuities. A Mach reflection problem is solved to demonstrate the mapped algorithm on a non‐rectangular physical domain. The simulation is compared against experimental results. Future work will consider mapped multiblock grids for practical engineering geometries. Copyright © 2016 John Wiley & Sons, Ltd.     

Higher-order Upwind Distribution-formula Scheme for Structured and Unstructured Adaptive Flow Solvers

The development of inviscid and viscous flow solvers for both structured and unstructured meshes is presented in this paper. The solution method is the distribution-formula scheme. This is an explicit, cell-vertex, finite volume method which is essentially second-order accurate in both space and time. The Euler and Navier-Stokes equations are integrated over each finite volume cell to determine the change in flow properties (e.g. density) for the cell. Distribution formulas are then used to distribute such changes to the surrounding vertices. Increments in each vertex (which is a calculation point) thus consist of contributions from the surrounding cells. The original discretization technique involves central differencing and is simple, robust and computationally efficient. In this work, starting with inviscid flow simulations using the original scheme on structured grids, improvements are subsequently made to the scheme by replacing the central differencing portion with MUSCL type higher-order upwind differencing. Numerical investigations with the improved scheme are performed using inviscid flow simulations on structured grids. Upon establishing improved accuracy, stability and excellent shock capturing properties, further extension to viscous flow computations on unstructured adaptive meshes is implemented. Results are presented for laminar, viscous flow over a NACA 0012 airfoil.     

Shock stand-off distance of a solid sphere decelerating in transonic velocity range

The shock stand-off distance of a spherical model flying with transonic speeds is determined through numerical simulations. The model decelerates due to drag forces caused by the pressure and viscous shear stress at the model surface. Two-dimensional axisymmetric numerical codes with numerical grids fixed to the flying spherical model are used in the simulations. Numerically determined shock stand-off distances are compared with experimental data obtained in a previous study as well as with those obtained in our ballistic-range experiments. The numerical results and the experimental data are found to be in good agreement. In addition, the time-dependent shock stand-off distance of a decelerating model is investigated.     

Viscous and inviscid interactions of an oblique shock with a flexible panel

The complex self-sustained oscillations arising from the interaction of an oblique shock with a flexible panel in both the inviscid and viscous regimes have been investigated numerically. The aeroelastic interactions are simulated using either the Euler or the full compressible Navier–Stokes equations coupled to the nonlinear von Karman plate equations. Results demonstrate that for a sufficiently strong shock limit-cycle oscillations emerge from either subcritical or supercritical bifurcations even in the absence of viscous separated flow effects. The critical dynamic pressure diminishes with increasing shock strength and can be much lower than that corresponding to standard panel flutter. Significant changes in panel dynamics were also found as a function of the shock impingement point and cavity pressure. For viscous laminar flow above the panel without a shock, high-frequency periodic oscillations appear due to the coupling of boundary-layer instabilities with high-mode flexural deflections. For a separated shock laminar boundary layer interaction, non-periodic self-excited oscillations arise which can result in a significant reduction in the extent of the time-averaged separation region. This finding suggests the potential use of an aeroelastically tailored flexible panel as a means of passive flow control. Forced panel oscillations, induced by a specified variable cavity pressure underneath the panel, were also found to be effective in reducing separation. For both inviscid and viscous interactions, the significant unsteadiness generated by the fluttering panel propagates along the complex reflected expansion/recompression wave system.     

动态混合网格生成及隐式非定常计算方法

建立了一种基于动态混合网格的非定常数值计算方法. 混合网格由贴体的四边形网格、外场 的多层次矩形网格和中间的三角形网格构成. 当物体运动时，贴体四边形网格随物体运动而 运动，而外场的矩形网格保持静止，中间的三角形网格随之变形；当物体运动位移较大，导 致三角形网格的质量降低，甚至导致网格相交时，在局部重新生成网格. 新网格上的物理量 由旧网格上的物理量插值而得. 为了提高计算效率，采用了双时间步和子迭代相结合的隐式 有限体积格式计算非定常Navier-Stokes方程. 子迭代采用高效的块LU-SGS方法. 利用该 方法数值模拟了NACA0012振荡翼型的无黏和黏性绕流，得到了与实验和他人计算相当一致 的结果.     

ADAPTIVE SOLUTIONS FOR UNSTEADY LAMINAR FLOWS ON UNSTRUCTURED GRIDS

An adaptive finite volume method for the simulation of time-dependent, viscous flow is presented. The Navier–Stokes equations are discretized by central schemes on unstructured grids and solved by an explicit Runge–Kutta method. The essential topics of the present study are a new concept for a local Runge–Kutta time-stepping scheme, called multisequence Runge–Kutta, which reduces the severe stability restriction in unsteady problems, a common grid generation and adaptation procedure and the application of dynamic grids for capturing moving flow structures. Results are presented for laminar, separated flow around an aerofoil with a flap.     

Numerical method for calculation of the incompressible flow in general curvilinear co‐ordinates with double staggered grid

A solution methodology has been developed for incompressible flow in general curvilinear co‐ordinates. Two staggered grids are used to discretize the physical domain. The first grid is a MAC quadrilateral mesh with pressure arranged at the centre and the Cartesian velocity components located at the middle of the sides of the mesh. The second grid is so displaced that its corners correspond to the centre of the first grid. In the second grid the pressure is placed at the corner of the first grid. The discretized mass and momentum conservation equations are derived on a control volume. The two pressure grid functions are coupled explicitly through the boundary conditions and implicitly through the velocity of the field. The introduction of these two grid functions avoids an averaging of pressure and velocity components when calculating terms that are generated in general curvilinear co‐ordinates. The SIMPLE calculation procedure is extended to the present curvilinear co‐ordinates with double grids. Application of the methodology is illustrated by calculation of well‐known external and internal problems: viscous flow over a circular cylinder, with Reynolds numbers ranging from 10 to 40, and lid‐driven flow in a cavity with inclined walls are examined. The numerical results are in close agreement with experimental results and other numerical data. Copyright © 2003 John Wiley & Sons, Ltd.     

脉冲爆震发动机进气道气动性能的数值研究

采用有限体积法计算了脉冲爆震发动机某轴对称超音速进气道在3种 不同出口条件(单个正弦扰动压力、某脉冲爆震发动机爆震室头部表压和进气道出口堵塞) 下的进气道内结尾正激波的运动情况，得出了进气道内结尾正激波运动特性和不同出口条件 的关系. 在计算中，采用了多块结构化网格，控制体积的界面无黏通量采用三阶迎风格 式插值获得，同时采用了minmod通量限制器以确保在激波处的解的物理特性；扩散通量采 用二阶中心差分格式插值获得. 定常计算采用当地时间步法，非定常计算采用双时间步法. 离散的代数方程采用交替方向迭代法求解。     

用鲁棒Riemann求解器和运动重叠网格计算旋翼粘性绕流

发展了一种基于鲁棒Riemann求解器和运动重叠网格技术计算直升机悬停旋翼流场的方法。基于惯性坐标系,悬停旋翼流场是非定常流场,控制方程为可压缩Reynolds平均Navier-Stoke方程,其对流项采用Roe近似Reimann求解器离散,使用改进的五阶加权基本无振荡格式进行高阶重构,非定常时间推进采用含牛顿型LUSGS子迭代的全隐式双时间步方法。为实施旋转运动和便于捕捉尾迹,计算采用运动重叠网格技术。计算得到的桨叶表面压力分布及桨尖涡涡核位置都与实验结果吻合较好。数值结果表明:所发展方法对桨尖涡具有较高的分辨率,对激波具有较好的捕捉能力,该方法可进一步推广到前飞旋翼粘性绕流的计算。     

HLLE++格式在高马赫数空腔流动模拟中的应用

空腔流动存在剪切层运动、涡脱落与破裂,以及激波与激波、激波与剪切层、激波与膨胀波和激波/涡/剪切层相互干扰等现象,流动非常复杂,特别是高马赫数(M>2)时,剪切层和激波更强,激波与激波干扰更严重,对数值格式的要求更高,既需要格式耗散小,对分离涡等有很高的模拟精度,又需要格式在激波附近具有较大的耗散,可以很好地捕捉激波,防止非物理解的出现。Roe和HLLC等近似Riemann解格式在高马赫数强激波处可能会出现红玉现象,而HLLE++格式大大改善了这种缺陷,在捕捉高超声速激波时避免了红玉现象的发生,同时还保持在光滑区域的低数值耗散特性。本文在结构网格下HLLE++格式的基础上,通过改进激波探测的求解,建立了基于非结构混合网格的HLLE++计算方法,通过无粘斜坡算例,验证了HLLE++格式模拟高马赫数流动的能力,并应用于高马赫数空腔流动的数值模拟,开展了网格和湍流模型影响研究,验证了方法模拟高马赫数空腔流动的可靠性和有效性。     

Application of local grid refinement to vortex motion due to a solitary wave passing over a submerged body

A numerical method based on the streamfunction–vorticity formulation is applied to simulate the two‐dimensional, transient, viscous flow with a free surface. This method successfully uses the locally refined grid in an inviscid–viscous model to explore the processes of vortex formation due to a solitary wave passing over a submerged bluff body. The two particular bodies considered here are a blunt rectangular block and a semicircular cylinder. Flow visualization to track dyelines is carried out in the laboratory in order to confirm the validity of the numerical results. Numerical results examined by different grid configurations ensure the locally refined grid to be useful in practical application. Flow phenomena, including the vortex motion and wave patterns during non‐linear wave–structure interaction, are also discussed. Copyright © 2002 John Wiley & Sons, Ltd.     

Simulation of blast wave propagation over a cylinder

A numerical study of the interaction of plane blast waves with a cylinder is presented. Computations are carried out for various blast-wave durations and comparisons are obtained with the corresponding results of planar shock-wave. Both inviscid and viscous results based on the solution of the Euler and Navier-Stokes equations are presented. The equations are solved by an adaptive-grid method and a second-order Godunov scheme. The shock wave diffraction over the cylinder is investigated by means of various contour plots, as well as, pressure and skin-friction histories. The study reveals that the blast-wave duration significantly influences the unsteady flow over the cylinder. The differences between the viscous and inviscid results are also discussed. Received 2 March 1996 / Accepted 28 February 1997     

ON THE FORCE DECOMPOSITIONS OF LIGHTHILL AND MORISON

Lighthill's assertion that the viscous drag force and the inviscid inertia force acting on a bluff body immersed in a time-dependent flow operate independently is not in conformity with the existing exact solutions and experimental facts. The two force components are interdependent as well as dependent on the parameters characterizing the phenomenon: the rate of diffusion of vorticity, relative amplitude of the oscillation, and the surface roughness.     

Three dimensional shock wave/boundary layer interactions

An investigation into a three-dimensional, curved shock wave interacting with a three-dimensional, curved boundary layer on a slender body is presented. Three different nose profiles mounted on a cylindrical body were tested in a supersonic wind tunnel and numerically simulated by solving the Navier–Stokes equations. The conical and hemispherical nose profiles tested were found to generate shock waves of sufficient strength to separate the boundary layer on the cylinder, while the shock wave generated by the ogival profile did not separate the boundary layer. For the separated flow, separation was found to occur predominantly on the windward side of the cylinder with the lee-side remaining shielded from the direct impact of the incident shock wave. A thickening of the boundary layer on the lee-side of all the profiles was observed, and in the conical and hemispherical cases this leads to the re-formation of the incident shock wave some distance away from the surface of the cylinder. A complex reflection pattern off the shock wave/boundary layer interaction (SWBLI) was also identified for the separated flow cases. For comparative purposes, an inviscid simulation was performed using the hemispherical profile. Significant differences between the viscous and inviscid results were noted including the absence of a boundary layer leading to a simplified shock wave reflection pattern forming. The behaviour of the incident shock wave on the lee-side of the cylinder was also affected with the shock wave amalgamating on the surface of the cylinder instead of away from the surface as per the viscous case. Test data from the wind tunnel identified two separation lines present on the cylindrical surface of the hemispherical SWBLI generator. The pair of lines were not explicitly evident in the original CFD simulations run, but were later identified in a high-resolution simulation.     

A domain decomposition approach to compute wave breaking (wave‐breaking flows)

A heterogeneous domain decomposition approach is followed to simulate the unsteady wavy flow generated by a body moving beneath a free surface. Attention being focused on complex free surface configurations, including wave‐breaking phenomena, a two‐fluid viscous flow model is used in the free surface region to capture the air–water interface (via a level‐set technique), while a potential flow approximation is adopted to describe the flow far from the interface. Two coupling strategies are investigated, differing in the transmission conditions. Both the adopted approaches make use of the inviscid velocity field as boundary condition in the Navier–Stokes solution. For validation purposes, two different two‐dimensional non‐breaking flows are simulated. Domain decomposition results are compared with both fully viscous and fully inviscid results, obtained by solving the corresponding equations in the whole fluid domain, and with available experimental data. Finally, the unsteady evolution of a steep breaking wave is followed and some of the physical phenomena, experimentally observed, are reproduced. Copyright © 2003 John Wiley & Sons, Ltd.     

Assessment of gas and liquid velocities induced by an impacting liquid drop

A two-phase flow model using the boundary element method was applied to investigate the physics of a liquid drop impacting onto a solid, dry plate. Xu et al. showed that air pressure plays an important role in splashing: as air pressure was reduced, splashing of an ethanol drop with a Weber number of 838 was suppressed. This remarkable observation provided the motivation for the current modeling effort. We numerically investigate how air pressure affects the behavior of an impacting drop. Surveying both inside and outside the impacting drop, velocities of both the liquid and gas are computed. Simulations show that gas speed, as it is displaced by the falling drop, is more than three times higher than the incoming drop speed. Air entrainment induced by the displaced gas seems to be an important contributor to corona formation, which always precedes any instability, fingering, or splashing of the liquid. To describe drop-impact phenomena, the maximum spreading diameter of the drop and the topology of the impacting fluid are reported as functions of Weber number and gas density.     

网格与高精度差分计算问题

研究NS方程差分求解时来流雷诺数、计算格式精度和计算网格之间的关系．给出了判定空间三个方向上的粘性贡献在给定雷诺数、格式精度和网格下是否能够正确计入的估计方法．指出在NS方程的二阶差分方法的数值模拟中,由于物面法向采用了压缩网格技术,物面附近的网格间距很小,该方向上的粘性贡献可被计入．但是如果流向和周向的网格较粗,相应的差分方程中的粘性贡献可能落入截断误差相同的量级,因此在精度上等于仍是求解略去流向和周向粘性项的薄层近似方程．指出,高阶精度的差分计算格式,可以避免对网格要求苛刻的困难．并进一步讨论了建立高阶精度格式的问题,提出了建立高阶精度格式应该满足的原则：耗散控制原则以及色散控制原则．为了避免激波附近可能出现的微小非物理振荡,建议发展混合高阶精度格式,即在激波区,采用网格自适应的NND格式,在激波以外的区域,采用按上述原则发展的高阶格式．     

Shock capturing viscosities for the general fluid mechanics algorithm

The performance of different shock capturing viscosities has been examined using our general fluid mechanics algorithm. Four different schemes have been tested, both for viscous and inviscid compressible flow problems. Results show that the methods based on the second gradient of pressure give better performance in all situations. For instance, the method constructed from the nodal pressure values and consistent and lumped mass matrices is an excellent choice for inviscid problems. The method based on L₂ projection is better than any other method in viscous flow computations. The residual based anisotropic method gives excellent performance in the supersonic range and gives better results in the hypersonic regime if a small amount of residual smoothing is used. © 1998 John Wiley & Sons, Ltd.