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。     

A note on numerical simulation of vortical structures in shock diffraction

In numerical simulation of the Euler equations, the slipstream or shear layer that appears behind a diffracted shock wave may develop small discrete vortices using fine computational meshes. Similar phenomena were also observed in the simulation of a Mach reflection that is accompanied by a shear layer. However, these small vortices have never been observed in any shock-tube experiment, although the wave pattern and the shape of the main vortex agree very well with visualization results. Numerical solutions obtained with coarse grids may agree better with experimental photos than those with very fine grids because of the pollution of the small vortices. This note tries to investigate the effect of viscosity on the small vortices by comparing the solutions of the laminar Navier-Stokes equations and the turbulence model. It is found that the small vortices are still observed in the solution of the laminar Navier-Stokes equations, although they can be suppressed by using the turbulence model. Numerical and experimental factors that are responsible for the deviation of the laminar solutions from experimental results are discussed. The secondary vortex in shock diffraction is successfully simulated by solving the Navier-Stokes equations.Received: 28 March 2003, Accepted: 6 May 2003, Published online: 11 June 2003     

Assessment of WENO-extended two-fluid modelling in compressible multiphase flows

The two-fluid modelling based on an advection-upwind-splitting-method (AUSM)-family numerical flux function, AUSM⁺-up, following the work by Chang and Liou [Journal of Computational Physics 2007;225: 840–873], has been successfully extended to the fifth order by weighted-essentially-non-oscillatory (WENO) schemes. Then its performance is surveyed in several numerical tests. The results showed a desired performance in one-dimensional benchmark test problems: Without relying upon an anti-diffusion device, the higher-order two-fluid method captures the phase interface within a fewer grid points than the conventional second-order method, as well as a rarefaction wave and a very weak shock. At a high pressure ratio (e.g. 1,000), the interpolated variables appeared to affect the performance: the conservative-variable-based characteristic-wise WENO interpolation showed less sharper but more robust representations of the shocks and expansions than the primitive-variable-based counterpart did. In two-dimensional shock/droplet test case, however, only the primitive-variable-based WENO with a huge void fraction realised a stable computation.     

High-order resolution Eulerian–Lagrangian simulations of particle dispersion in the accelerated flow behind a moving shock

This paper presents a computational study of the two-dimensional particle-laden flow developments of bronze particle clouds in the accelerated flow behind a moving normal shock. Particle clouds with a particle volume concentration of 4% are arranged initially in a rectangular, triangular and circular shape. Simulations are performed with a recently developed high-order resolution Eulerian–Lagrangian method that approximates the Euler equations governing the gas dynamics with the improved high order weighted essentially non-oscillatory (WENO-Z) scheme, while individual particles are traced in the Lagrangian frame using high-order time integration schemes. Reflected shocks form ahead of all the cloud shapes. The detached shock in front of the triangular cloud is weakest. At later times, the wake behind the cloud becomes unstable, and a two-dimensional vortex-dominated wake forms. Separated shear layers at the edges of the clouds pull particles initially out of the clouds that are consequently transported along the shear layers. Since flows separated trivially at sharp corners, particles are mostly transported out of the cloud into the flow at the sharp front corner of the rectangular cloud and at the trailing corner of the triangular cloud. Particles are transported smoothly out of the circular cloud, since it lacks sharp corners. At late times, the accelerated flow behind the running shock disperses the particles in cross-stream direction the most for the circular cloud, followed by the rectangular cloud and the triangular cloud.     

Characteristics of unsteady type IV shock/shock interaction

Characteristics of the unsteady type IV shock/shock interaction of hypersonic blunt body flows are investigated by solving the Navier–Stokes equations with high-order numerical methods. The intrinsic relations of flow structures to shear, compression, and heating processes are studied and the physical mechanisms of the unsteady flow evolution are revealed. It is found that the instantaneous surface-heating peak is caused by the fluid in the “hot spot” generated by an oscillating and deforming jet bow shock (JBS) just ahead of the body surface. The features of local shock/boundary layer interaction and vortex/boundary layer interaction are clarified. Based on the analysis of flow evolution, it is identified that the upstream-propagating compression waves are associated with the interaction of the JBS and the shear layers formed by a supersonic impinging jet, and then the interaction of the freestream bow shocks and the compression waves results in entropy and vortical waves propagating to the body surface. Further, the feedback mechanism of the inherent unsteadiness of the flow field is revealed to be related to the impinging jet. A feedback model is proposed to reliably predict the dominant frequency of flow evolution. The results obtained in this study provide physical insight into the understanding of the mechanisms relevant to this complex flow.     

A carbuncle cure for the Harten-Lax-van Leer contact(HLLC)scheme using a novel velocity-based sensor

A hybrid numerical flux scheme is proposed by adapting the carbunclefree modified Harten-Lax-van Leer contact(HLLCM) scheme to smoothly revert to the Harten-Lax-van Leer contact(HLLC) scheme in regions of shear. This hybrid scheme, referred to as the HLLCT scheme, employs a novel, velocity-based shear sensor. In contrast to the non-local pressure-based shock sensors often used in carbuncle cures, the proposed shear sensor can be computed in a localized manner meaning that the HLLCT scheme can be easily introduced into existing codes without having to implement additional data structures. Through numerical experiments, it is shown that the HLLCT scheme is able to resolve shear layers accurately without succumbing to the shock instability.     

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.     

Numerical simulation of shock–vortex interaction in Schardin’s problem

Shock diffraction over a two-dimensional wedge and subsequent shock–vortex interaction have been numerically simulated using the AUSM $+$ + scheme. After the passage of the incident shock over the wedge, the generated tip vortex interacts with a reflected shock. The resulting shock pattern has been captured well. It matches the existing experimental and numerical results reported in the literature. We solve the Navier–Stokes equations using high accuracy schemes and extend the existing results by focussing on the Kelvin–Helmholtz instability generated vortices which follow a spiral path to the vortex core and on their way interact with shock waves embedded within the vortex. Vortex detection algorithms have been used to visualize the spiral structure of the initial vortex and its final breakdown into a turbulent state. Plotting the dilatation field we notice a new source of diverging acoustic waves and a lambda shock at the wedge tip.     

Orbital flow past a cylinder: A numerical approach

Orbital flow past a cylinder is relevant to offshore structures. The numerical scheme presented here is based on a finite-difference solution of the Navier–Stokes equations. Alternating-directional-implicit (ADI) and successive-over-relaxation (SOR) techniques are used to solve the vorticity-transport and stream-function equations. Theoretical simulations to low Reynolds number flows (up to 1000) are discussed for cases involving uniform flow past stationary and rotating cylinders and orbital flow past a cylinder. The separation points for cylinders that are rotating or immersed in an orbital flow are deduced from velocity profiles through the boundary layer using a hybrid mesh scheme. During the initial development of orbital flow surface vorticity on the impulsively started cylinder dominates the flow. A vortex then detaches from behind the cylinder and establishes the flow pattern of the orbit. After some time a collection of vortices circles the orbit and distorts its shape a great deal. These vortices gradually spiral outward as others detach from the cylinder and join the orbital path.     

激波诱导环形SF6气柱演化的机理

基于可压缩多组分Navier-Stokes控制方程,结合5阶加权本质无振荡格式以及网格自适应加密技术和level-set方法,数值模拟了平面激波(Ma=1.23)与环形SF₆气柱(内外半径分别为8和17.5 mm)界面的相互作用过程。相比于之前的实验结果,数值模拟结果揭示了入射激波在界面内4次透射过程中的复杂波系结构,观察到透射激波在内部界面传播时形成自由前导折射结构并向自由前导冯诺依曼折射结构转换的波系演变过程;另外,界面内的复杂激波结构诱导内部下游界面上的涡量发生了3次反向;在界面演化后期,内部界面形成的“射流”结构与下游界面相互作用,诱导界面形成一对主涡、一对次级涡以及一个反向“射流”结构。定量分析了环形界面长度、宽度、位移、环量以及混合率的变化情况,结果表明,内部气柱的存在减弱了前期小涡结构合并形成大涡结构过程中对界面高度与长度的影响,同时提高了重质气体与环境气体的混合率。     

Shock calculation based on second viscosity using local differential quadrature method

Based on the second viscosity, the local differential quadrature (LDQ) method is applied to solve shock tube problems. It is shown that it is necessary to consider the second viscosity to calculate shocks and to simulate shock tubes based on the viscosity model. The roles of the shear viscous stress and the second viscous stress are checked. The results show that the viscosity model combined with the LDQ method can capture the main characteristics of shocks, and this technique is objective and simple.     

Numerical assessment of a shock‐detecting sensor for low dissipative high‐order simulation of shock–vortex interactions

Considering the importance of high‐order schemes implementation for the simulation of shock‐containing turbulent flows, the present work involves the assessment of a shock‐detecting sensor for filtering of high‐order compact finite‐difference schemes for simulation of this type of flows. To accomplish this, a sensor that controls the amount of numerical dissipation is applied to a sixth‐order compact scheme as well as a fourth‐order two‐register Runge–Kutta method for numerical simulation of various cases including inviscid and viscous shock–vortex and shock–mixing‐layer interactions. Detailed study is performed to investigate the performance of the sensor, that is, the effect of control parameters employed in the sensor are investigated in the long‐time integration. In addition, the effects of nonlinear weighting factors controlling the value of the second‐order and high‐order filters in fine and coarse non‐uniform grids are investigated. The results indicate the accuracy of the nonlinear filter along with the promising performance of the shock‐detecting sensor, which would pave the way for future simulations of turbulent flows containing shocks. Copyright © 2014 John Wiley & Sons, Ltd.     

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．     

二维激波与剪切层相互作用的直接数值模拟研究

采用五阶weighed esseritially non-oscillatory (WENO) 格式和三阶total variation diminishing (TVD) Runge-Kutta 格式, 通过求解二维非定常Navier-Stokes 方程, 直接数值模拟了激波与剪切层相互作用, 目的在于揭示激波与剪切层相互作用过程中噪声产生的机理. 研究发现:(1) 当入射激波穿过剪切层时, 剪切层中心位置向下层区域偏移;(2) 入射激波穿过剪切层产生小激波, 在小激波与剪切层接触点处产生声波并向外辐射;(3) 反射激波穿过剪切层后形成了分段弧状激波;(4) 当反射激波穿过剪切层时, 激波在鞍点处泄漏并向外辐射声波, 这是一种激波泄漏机制.     

Implicit high-order discontinuous Galerkin method with HWENO type limiters for steady viscous flow simulations

Two types of implicit algorithms have been improved for high order discontinuous Galerkin (DG) method to solve compressible Navier-Stokes (NS) equations on triangular grids. A block lower-upper symmetric Gauss-Seidel (BLU-SGS) approach is implemented as a nonlinear iterative scheme. And a modified LU-SGS (LLU-SGS) approach is suggested to reduce the memory requirements while retain the good convergence performance of the original LU-SGS approach. Both implicit schemes have the significant advantage that only the diagonal block matrix is stored. The resulting implicit high-order DG methods are applied, in combination with Hermite weighted essentially non-oscillatory (HWENO) limiters, to solve viscous flow problems. Numerical results demonstrate that the present implicit methods are able to achieve significant efficiency improvements over explicit counterparts and for viscous flows with shocks, and the HWENO limiters can be used to achieve the desired essentially non-oscillatory shock transition and the designed high-order accuracy simultaneously.     

一种弹体入水冲击波阴影成像可视化系统及其试验研究

扩大成像视野对于开展充水容器中弹体入水冲击波传播及弥散方面的可视化研究具有重要的实际意义。阴影成像技术适用于大视野实验,且对流场冲击波和扰动的可视化研究具有简单性和通用性,其中直接阴影成像最为简单,但可靠点光源的缺乏是阻碍其发展应用的瓶颈。因此基于国产短弧氙灯管,自制了短弧氙灯点光源,根据阴影成像原理,设计出一种弹体入水冲击波阴影成像可视化系统,详细介绍了其组成和运行原理。利用该系统对高速弹体入水进行了试验研究,获得了弹体入水冲击波的阴影成像和冲击波信号的压力时程曲线,通过阴影成像和冲击波信号相结合分析了弹体入水冲击波的传播特性,并进行了理论验证。结果表明:该弹体入水冲击波阴影成像可视化系统具有可靠性和设计的合理性。弹体高速入水后,初始冲击波的强度最大,随着冲击波的传播,冲击波强度逐渐降低,水中冲击波的传播速度不断降低,球形冲击波的半径逐渐增大。     

Shock wave trapping

This paper presents a two-dimensional investigation into the effectiveness of trapping shock and blast waves in a duct in order to enhance attenuation, by placing an array of opposing wedges in the channel. The concept of the wedge arrangement in the trap is to allow easy shock wave entry, with weak reflected shocks, into the trap, but stronger internal reflected shocks if a wave is re-emering. The internal reflections, including those of vortices shed from earlier shock passage, result in strong shock attenuation. Different wedge placements, wedge angles, and area blockages are investigated numerically, as well as experimentally for a particular case, using pressure measurement and schlieren photography. Received 4 August 1995 / Accepted 12 December 1995     

A third-order compact nonlinear scheme for compressible flow simulations

Reynolds-averaged Navier-Stokes simulations based on second-order numerical methods are widely used by commercial codes and work as dominating tools for most industrial applications. They, however, suffer from limitations in accurate and reliable predictions of skin-friction drag and aerodynamic heating, as well as in simulations of complex flows such as large-scale separation and transition. A remedy for this is the development of high-order schemes, by which numerically induced dissipation and dispersion errors of low-order schemes can be effectively reduced. Weighted compact nonlinear schemes (WCNSs) are a family of high-resolution nonlinear shock-capturing methods. A stencil-selection procedure is introduced in the proposed work with an aim to improve the nonlinear weight of the third-order WCNS. By using the approximate dispersion relation analysis, it is demonstrated that the new scheme has reduced dissipation and dispersion errors, compared with WCNSs using two typical nonlinear weights. Improvements are also achieved by the new scheme in numerical tests such as the double Mach reflection problem and the Rayleigh-Taylor instability simulation, which are characterized by strong shock discontinuities and rich small scales, respectively. The new scheme is therefore highly favored in the simulation of flow problems involving strong discontinuities and multiscales phenomena.     

Demonstration of ultra hi-fi (UHF) methods

Computational aero-acoustics (CAA) requires efficient, high-resolution simulation tools. Most current techniques utilize finite-difference approaches because high order accuracy is considered too difficult or expensive to achieve with finite volume or finite element methods. However, a novel finite volume approach i.e. ultra hi-fi (UHF) which utilizes Hermite fluxes is presented which can achieve both arbitrary accuracy and fidelity in space and time. The technique can be applied to unstructured grids with some loss of fidelity or with multi-block structured grids for maximum efficiency and resolution. In either paradigm, it is possible to resolve ultra-short waves (defined as waves having wavelengths that are shorter than a grid cell). This is demonstrated here by solving the 4th CAA workshop Category 1 Problem 1.     

数值研究平板方舵激波-湍流边界层干扰

数值研究了平板方舵激波-湍流边界层干扰流场。模拟出了分离激波与弓型激波砬撞后形成的“λ”激波结构;消晰地显示了分离区中的旋涡结构,发现流场中会出现二次分离涡,并从理论上分析了流场对称面涡心形态与非定常的关系,得到了涡心为不稳定螺旋点或出现极限环是非定常流动特征的新结论。