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。     

Adaptive strategy of transonic flows over vibrating blades with interblade phase angles

An error indicator and a locally implicit scheme with anisotropic dissipation model on dynamic quadri‐ lateral–triangular mesh are developed to study transonic flows over vibrating blades with interblade phase angles. In the Cartesian co‐ordinate system, the unsteady Euler equations with moving domain effects are solved. The error indicator, in which unified magnitudes of dynamic grid speed, substantial derivative of pressure, and substantial derivative of vorticity magnitude are incorporated to capture the unsteady wave behaviours and vortex‐shedding phenomena due to unsteadiness. To assess the accuracy of the locally implicit scheme with anisotropic dissipation model on quadrilateral–triangular mesh, two flow calculations are performed. Based on the comparison with the related numerical and experimental data, the accuracy of the present approach is confirmed. According to the high‐resolutional result on the adaptive mesh, the unsteady pressure wave, shock and vortex‐shedding behaviours are clearly demonstrated. Copyright © 2003 John Wiley & Sons, Ltd.     

Analysis of High-order Explicit LES Dynamic Modeling Applied to Airfoil Flows

A high-order low dissipative numerical framework is discussed to tackle simultaneously the modeling of unresolved sub-grid scale flow turbulence and the capturing of shock waves. The flows around two different airfoil profiles are simulated using a Spectral Difference discretisation scheme. First, a transitional, almost incompressible, low Reynolds number flow over a Selig-Donovan 7003 airfoil. Second, a high Reynolds number flow over a RAE2822 airfoil under transonic conditions. These flows feature both laminar and turbulent flow physics and are thus particularly challenging for turbulence sub-grid scale modeling. The accuracy of the recently developed Spectral Element Dynamic Model, specifically capable of detecting spatial under-resolution in high-order flow simulations, is evaluated. Concerning the test in transonic conditions, the additional complexity due to the presence of shock waves has been handled using an artificial viscosity shock-capturing technique based on bulk viscosity. To mitigate the impact of the shock-capturing on turbulence dissipation, it was necessary to combine the high-order modal-type shock detection with a usual sensor measuring the local flow compressibility.

     

Direct and large eddy simulation of stratified turbulence

Simulations of geophysical turbulent flows require a robust and accurate subgrid-scale turbulence modeling. To evaluate turbulence models for stably stratified flows, we performed direct numerical simulations (DNSs) of the transition of the three-dimensional Taylor–Green vortex and of homogeneous stratified turbulence with large-scale horizontal forcing. In these simulations we found that energy dissipation is concentrated within thin layers of horizontal tagliatelle-like vortex sheets between large pancake-like structures. We propose a new implicit subgrid-scale model for stratified fluids, based on the Adaptive Local Deconvolution Method (ALDM). Our analysis proves that the implicit turbulence model ALDM correctly predicts the turbulence energy budget and the energy spectra of stratified turbulence, even though dissipative structures are not resolved on the computational grid.     

Applications of URANS on predicting unsteady turbulent separated flows

Accurate prediction of unsteady separated turbulent flows remains one of the toughest tasks and a practi cal challenge for turbulence modeling. In this paper, a 2D flow past a circular cylinder at Reynolds number 3,900 is numerically investigated by using the technique of unsteady RANS （URANS）. Some typical linear and nonlinear eddy viscosity turbulence models （LEVM and NLEVM） and a quadratic explicit algebraic stress model （EASM） are evaluated. Numerical results have shown that a high-performance cubic NLEVM, such as CLS, are superior to the others in simulating turbulent separated flows with unsteady vortex shedding.     

Role of the momentum interpolation mechanism of the Roe scheme in shock instability

The shock instability phenomenon is a well‐known problem for hypersonic flow computation by the shock‐capturing Roe scheme. The pressure checkerboard is another well‐known problem for low‐Mach‐number flow computation. The momentum interpolation method (MIM) is necessary for low‐Mach‐number flows to suppress the pressure checkerboard problem, and the pressure‐difference‐driven modification for cell face velocity can be regarded as a version of the MIM by subdividing the numerical dissipation of the Roe scheme. In this paper, MIM has been discovered through analysis and numerical tests to have the most important function in shock instability. MIM should be completely removed for nonlinear flows. However, the unexpected MIM is activated on the cell face nearly parallel to the flow for the high‐Mach‐number flows or low‐Mach‐number cells in numerical shock. Therefore, MIM should be retained for low‐Mach‐number flows and be completely removed for high‐Mach‐number flows and low‐Mach‐number cells in numerical shock. For such conditions, two coefficients are designed on the basis of the local Mach number and a shock detector. Thereafter, the improved Roe scheme is proposed. This scheme considers the requirement of MIM for incompressible and compressible flows, and is validated for good performance of numerical tests. An acceptable result can also be obtained with only the Mach number coefficient for general practical computation. Therefore, the objective of decreasing rather than increasing numerical dissipation to cure shock instability can be achieved with simple modification. Moreover, the mechanism of shock instability has been profoundly understood, in which MIM plays the most important role, although it is not the only factor. Copyright © 2016 John Wiley & Sons, Ltd.     

Prediction of periodic boundary layers

A relatively simple, yet efficient and accurate finite difference method is developed for the solution of the unsteady boundary layer equations for both laminar and turbulent flows. The numerical procedure is subjected to rigorous validation tests in the laminar case, comparing its predictions with exact analytical solutions, asymptotic solutions, and/or experimental results. Calculations of periodic laminar boundary layers are performed from low to very high oscillation frequencies, for small and large amplitudes, for zero as well as adverse time-mean pressure gradients, and even in the presence of significant flow reversal. The numerical method is then applied to predict a relatively simple experimental periodic turbulent boundary layer, using two well-known quasi-steady closure models. The predictions are shown to be in good agreement with the measurements, thereby demonstrating the suitability of the present numerical scheme for handling periodic turbulent boundary layers. The method is thus a useful tool for the further development of turbulence models for more complex unsteady flows.     

A study on the behaviour of high-order flux reconstruction method with different low-dissipation numerical fluxes for large eddy simulation

This work focuses on the numerical dissipation features of high-order flux reconstruction (FR) method combined with different numerical fluxes in turbulence flows. The famous Roe and AUSM+ numerical fluxes together with their corresponding low-dissipation enhanced versions (LMRoe, SLAU2) and higher resolution variants (HR-LMRoe, HR-SLAU2) are incorporated into FR framework, and the dissipation interplay of these combinations is investigated in implicit large eddy simulation. The numerical dissipation stemming from these convective numerical fluxes is quantified by simulating the inviscid Gresho vortex, the transitional Taylor–Green vortex and the homogenous decaying isotropic turbulence. The results suggest that low-dissipation enhanced versions are preferential both in high-order and low-order cases to their original forms, while the use of HR-SLAU2 has marginal improvements and the HR-LMRoe leads to degenerated solution with high-order. In high-order the effects of numerical fluxes are reduced, and their viscosity may not be dissipative enough to provide physically consistent turbulence when under-resolved.     

Gas-kinetic description of shock wave structures by solving Boltzmann model equation

On the basis of the mesoscopic theory of Boltzmann-type velocity distribution function, the modified Boltzmann model equation describing the one-dimensional gas flows from various flow regimes is presented by incorporating the molecular interaction models relating to the viscosity and diffusion cross-sections, density, temperature and the dependent exponent of viscosity into the molecular collision frequency. The gas-kinetic numerical method for directly solving the molecular velocity distribution function is studied by introducing the reduced distribution functions and the discrete velocity ordinate method, in which the unsteady time-splitting method and the NND finite difference scheme are applied. To study the inner flows of non-equilibrium shock wave structures, the one-dimensional unsteady shock-tube problems with various Knudsen numbers and the steady shock wave problems at different Mach numbers are numerically simulated. The computed results are found to give good agreement with the theoretical, DSMC and experimental results. The computing practice has confirmed the good precision and reliability of the gas-kinetic numerical algorithm in solving the highly nonequilibrium shock wave disturbances from various flow regimes.     

On high‐resolution schemes for solving unsteady compressible two‐phase dilute viscous flows

A high‐resolution numerical scheme based on the MUSCL–Hancock approach is developed to solve unsteady compressible two‐phase dilute viscous flow. Numerical considerations for the development of the scheme are provided. Several solvers for the Godunov fluxes are tested and the results lead to the choice of an exact Riemann solver adapted for both gaseous and dispersed phases. The accuracy of the scheme is proven step by step through specific test cases. These simulations are for one‐phase viscous flows over a flat plate in subsonic and supersonic regimes, unsteady flows in a low‐pressure shock tube, two‐phase dilute viscous flows over a flat plate and, finally, two‐phase unsteady viscous flows in a shock tube. The results are compared with well‐established analytical and numerical solutions and very good agreement is achieved. Copyright © 1999 John Wiley & Sons, Ltd.     

Effective eddy viscosities in implicit modeling of decaying high Reynolds number turbulence with and without rotation

We assess the applicability of the numerical dissipation as an implicit turbulence model. The nonoscillatory finite volume numerical scheme MPDATA developed for simulations of geophysical flows is employed as an example of a scheme with an implicit turbulence model. A series of low resolution simulations of decaying homogeneous turbulence with and without Coriolis forces in the limit of zero molecular viscosity are performed. To assess the implicit model the long-time evolution of turbulence in the simulations is investigated and the numerical velocity fields are analyzed to determine the effective spectral eddy viscosity that is attributed to the numerical discretization. The detailed qualitative and quantitative comparisons are made between the numerical eddy viscosity and the theoretical results as well as the intrinsic eddy viscosity computed exactly from the velocity fields by introducing an artificial wave number cutoff. We find that the numerical dissipation depends on the time step and exhibits contradictory dependence on rotation: it is overestimated for rapid rotation cases and is underestimated for nonrotating cases. These results indicate that the numerical dissipation may fail to represent the effects of the physical subgrid scale processes unless the parameters of a numerical scheme are carefully chosen.     

数值耗散对压气机流动分离涡模拟的影响研究

分离涡模拟DES是压气机流动模拟中常用的高保真湍流模式。为了使DES准确解析湍流,数值耗散必须限制在合理范围内。然而,当前的压气机流动DES类研究工作中仍然普遍采用高耗散的迎风格式。本文首先基于DES类方法计算的各向同性衰减湍流结果,定量比较了多种不同数值格式的耗散,证实了高耗散迎风格式严重低估中高波数湍流能量。高阶重构格式可以一定程度上改善该问题,但能量耗散仍然过高。本文在高阶重构的基础上,进一步引入自适应耗散函数修改Riemann求解器,构造了自适应耗散格式。该格式在全波数范围都能准确地预测湍流能谱。将该格式配合DES类方法模拟跨声速离心压气机流动,其预测的压比相比于三阶迎风格式,更加接近实验结果。此外,自适应耗散格式显著提高了中小尺度流动结构的分辨率。分析表明,在使用DES类方法模拟压气机流动时,有必要采用数值耗散较低的离散格式,以准确预测压气机总体性能和流动结构。本文构造的自适应耗散格式是一种良好选择。     

URANS Computations of Shock-Induced Oscillations Over 2D Rigid Airfoils: Influence of Test Section Geometry

The present article deals with recent numerical results from on-going research conducted at ONERA/DMAE regarding the validation of turbulence models for unsteady transonic flows, in which the mechanism of the shock-wave/boundary-layer interaction is important. The main goal is to predict the onset and extent of shock induced oscillations (SIO) that appear over the suction side of two-dimensional rigid airfoils and lead to the formation of unsteady separated areas. Computations are performed with the ONERA object-oriented software "elsA", using the URANS-type approach. In this approach, the unsteady mean turbulent flow is resolved using the standard Reynolds-averaged Navier–Stokes (RANS) equations and closure relationships involving standard transport equation-type models without any explicit modification due to unsteadiness. Applications are provided and discussed for two different test cases, one of which is rather well documented for CFD validation and described by mean-flow, phase-averaged and fluctuating data. Results demonstrate the importance of modelling the upper and lower walls of the test section when trying to capture SIO as precisely as possible with 2D computations, even though the adaptation of wind tunnel walls had been carefully considered. Finally, turbulence validation has been performed using one- and two-transport equation-type models, one of them resulting from in-house investigations for other turbulent flows applications.     

Numerical solution of the Navier-Stokes equations using a finite element method

A finite element algorithm for solving the Navier-Stokes equations is presented for the analysis of high-speed viscous flows. The algorithm uses triangular elements. The unsteady equations are integrated to steady state with a Runge-Kutta time-marching scheme. A postprocessing artificial dissipation term is introduced to stabilize the computations and to dampen dissipation errors. Numerical results are compared with the calculation of uniform flow on a rectangular region which encounters an embedded oblique shock. A shock/turbulent boundary layer problem is also solved and results are compared with experimental data. It is shown that the postprocessing smoothing term and boundary conditions similar to the finite difference method work well in the present numerical studies.     

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.     

Large Eddy simulation for engineering applications

In the field of fluid engineering, controlling turbulent flows remains a crucial problem. This paper presents a basis of numerical methods and turbulence models for the large Eddy simulation. Simulation results include the unsteady analyses of complex flows, such as the vortex dynamics of turbulent jets subject to inlet perturbations and the reacting flow with flame propagation in a gas–turbine combustor flow. Applications employing large Eddy simulation are emerging as one of the most important aspects of the “Frontier Simulation Software for Industrial Science” project for the next generation of fluid dynamic design and development.     

A high‐resolution scheme for compressible multicomponent flows with shock waves

A simple methodology for a high‐resolution scheme to be applied to compressible multicomponent flows with shock waves is investigated. The method is intended for use with direct numerical simulation or large eddy simulation of compressible multicomponent flows. The method dynamically adds non‐linear artificial diffusivity locally in space to capture different types of discontinuities such as a shock wave, contact surface or material interface while a high‐order compact differencing scheme resolves a broad range of scales in flows. The method is successfully applied to several one‐dimensional and two‐dimensional compressible multicomponent flow problems with shock waves. The results are in good agreement with experiments and earlier computations qualitatively and quantitatively. The method captures unsteady shock and material discontinuities without significant spurious oscillations if initial start‐up errors are properly avoided. Comparisons between the present numerical scheme and high‐order weighted essentially non‐oscillatory (WENO) schemes illustrate the advantage of the present method for resolving a broad range of scales of turbulence while capturing shock waves and material interfaces. Also the present method is expected to require less computational cost than popular high‐order upwind‐biased schemes such as WENO schemes. The mass conservation for each species is satisfied due to the strong conservation form of governing equations employed in the method. Copyright © 2010 John Wiley & Sons, Ltd.     

用扰动涡方法模拟叶轮机内动静叶的相互作用

发展了一种研究叶轮机内动、静叶间的相互作用的新方法———扰动涡方法，它利用全三维的定常解为基础解，并由此给出非定常扰动场的初始解．为计算叶片对扰动场的响应过程，采用拉格朗日方法追踪扰动涡团的对流流动过程，用确定性涡方法来描述流体的粘性扩散过程．发展了代数湍流模型（Ｂａｌｄｗｉｎ Ｌｏｍａｘ湍流模型）在尾迹中的应用方法，克服了其它数值方法中无法准确捕捉尾迹中心线的运动轨迹，以及计算出的边界层外的湍流涡粘性系数偏大的缺陷．利用该方法计算轴流叶轮机内由于动、静叶间的相互作用而引起的非定常流动过程，与实验的对照表明，模拟结果与实验数据吻合得相当好，从而说明本文发展的方法是可信的，为更直观地描述尾迹等非定常因素的流动及叶轮机内的掺混问题提供依据．     

Eddy-Viscosity and Stress-Transport Turbulence Models in Application to a Plane Synthetic Jet

A two-dimensional unsteady plane jet is computed using four turbulence models. The strain field generated by the ‘synthetic’ (reciprocating) jet under consideration contains regions of strong normal and shear straining, while vortex pairs formed by the issuing jet are highly rotational. Turbulence closures at the eddy-viscosity level (standard k???ε and two-timescale models) and two stress-transport closures (the ‘basic’ model and a two-component-limit scheme) are examined against experimental data. It is found that the extreme nature of the strain environment presents a severe challenge to the models tested; in particular, fundamental weaknesses in the constitutive equation of linear eddy viscosity models are highlighted. An additional term in the dissipation rate equation of the two-timescale model is found to hinder the formation of clearly defined starting vortex pairs which play a critical role in determining the near-field pattern of synthetic jet development. The stress-transport models are able to reproduce measured turbulence quantities with a greater degree of accuracy, and the two-component-limit formulation shows an improvement over the basic model.     

High‐order shock capturing schemes for turbulence calculations

This work investigates high‐order central compact methods for simulating turbulent supersonic flows that include shock waves. Several different types of previously proposed characteristic filters, including total variation diminishing, monotone upstream‐centered scheme for conservation laws, and weighted essentially non‐oscillatory filters, are investigated in this study. Similar to the traditional shock capturing schemes, these filters can eliminate the numerical instability caused by large gradients in flow fields, but they also improve efficiency compared with classical shock‐capturing schemes. Adding the nonlinear dissipation part of a classical shock‐capturing scheme to a central scheme makes the method suitable for incorporation into any existing central‐based high‐order subsonic code. The amount of numerical dissipation to add is sensed by means of the artificial compression method switch. In order to improve the performance of the characteristic filters, we propose a hybrid approach to minimize the dissipation added by the characteristic filter. Through several numerical experiments (including a shock/density wave interaction, a shock/vortex interaction, and a shock/mixing layer interaction) we show that our hybrid approach works better than the original method, and can be used for future turbulent flow simulations that include shocks. Copyright © 2009 John Wiley & Sons, Ltd.