Simulation of sonic boom interaction with shear turbulence

We investigate the interaction of pressure waves with the shear turbulence in a supersonic jet flow obtained from the direct numerical integration of the time-dependent, three-dimensional Euler equations. The resulting self-sustaining turbulent jet flow matches closely the relevant time and length scales of atmospheric turbulence. N-waves, characteristic of sonic booms, are simulated by perturbing the pressure profile and following these perturbations in space and time. The results reproduce most of the wave forms reported in laboratory experiments and in flight test data.     

Compressibility effects due to turbulent fluctuations

This paper deals with intrinsic effects of compressibility, i.e. with dilatation fluctuations in response to pressure fluctuations. Three different types of turbulent flows are considered in more detail: homogeneous turbulent shear flow, wall-bounded turbulent shear flow and shock/turbulence interaction. A survey of the present knowledge in this field, mainly based on DNS data, is given. Using the linear inviscid perturbation equations a direct link between fluctuations of dilatation and of velocity in the direction of mean shear is presented for homogeneous shear flow. This relation might form the basis for a more universal pressure-dilatation model. It is conjectured that the insignificance of intrinsic compressibility effects in wall-bounded supersonic shear flow is mainly due to the impermeability constraint of the wall. To this end, a linear stability analysis of supersonic channel flow along cooled, but permeable walls has been performed based on Coleman et al.'s [5] mean flow data. It shows an increase in the moduli of eigenfunctions related to compressibility, like pressure, and in moduli of quantities derived from eigenfunctions such as ‘pressure dilatation’ and squared dilatation. Although these results do not prove our hypothesis they provide hints in this direction. Shock/turbulence interaction is viewed as a source of compressibility. Former DNS data of Hannappel and Friedrich [10] for shock/isotropic turbulence interaction showing the effect of compressibility on the amplification of fluctuations are interpreted based on linear perturbation equations.     

Simulation and Modelling of a Skewed Turbulent Channel Flow

A time-dependent three-dimensionally skewed flow is investigated using direct numerical simulations of the incompressible Navier-Stokes equations. The effect on the instantaneous and mean turbulent field is investigated. Instantaneous flowfields reveal that the skewing has the effect of initially reducing the strength and height of quasi-streamwise vortices of both signs of rotation with respect to the skewing. A mechanism for this process is put forward. The mean flowfields show drops in turbulence quantities such as turbulence kinetic energy. In addition to this, two-equation turbulence modelling of the flow is carried out. This highlights a deficiency, in that the standard turbulence models are unable to capture the drop in turbulence intensity due to the skewing. A modification based on the exact dissipation equation is found to significantly improve the model behaviour for this flow. This revised version was published online in July 2006 with corrections to the Cover Date.     

Large-scale computer simulation of fully developed turbulent channel flow with heat transfer

Recently, with the advent of supercomputers, there has been considerable interest in the use of direct numerical simulation to obtain information about turbulent shear flow at low Reynolds number. This paper presents a pseudospectral technique to solve the full three-dimensional time-dependent Navier-Stokes and advection-diffusion equations without the use of subgrid-scale modelling. The technique has not been previously used for fully developed turbulent channel flow simulation and is based on methods applied in other contexts. The emphasis of this paper is to provide a reasonably detailed account of how the simulation is done rather than to present new calculations of turbulence. The details of an algorithm for turbulent channel flow simulation and the grid and time step sizes needed to integrate through transient behaviour to steady state turbulence have not been published before and are presented here. Results from a Cray-2 simulation of fully developed turbulent flow in a channel with heat transfer are presented along with a critical comparison between experiment and computation. The first- and second-order moments agree well with experimental measurements; the agreement is poor for higher-order moments such as the skewness and flatness near the walls of the channel. Detailed information given about the effects of spatial grid resolution on a computed results is important for estimating the size of the computation required to study various aspects of a turbulent flow.     

Two-equation (k, ϵ) turbulence computations by the use of a finite element model

A finite element technique is presented and applied to some one- and two-dimensional turbulent flow problems. The basic equations are the Reynolds averaged momentum equations in conjunction with a two-equation (k, ?) turbulence model. The equations are written in time-dependent form and stationary problems are solved by a time iteration procedure. The advection parts of the equations are treated by the use of a method of characteristics, while the continuity requirement is satisfied by a penalty function approach. The general numerical formulation is based on Galerkin's method. Computational results are presented for one-dimensional steady-state and oscillatory channel flow problems and for steady-state flow over a two-dimensional backward-facing step.     

A k–ℓ turbulence closure sensitized to non-simple shear flows

This paper introduces a two-equation turbulence model sensitized to deviations from simple shear flows. The closure is topography-parameter-free and is based on solving transport equations for the turbulence kinetic energy (k) and the turbulence length-scale (?). Brief model derivation details are given and test cases are presented to compare the model's performance to other closures and to experimental data. The flow examples demonstrate the advantage of the k–? model in non-simple shear flows.     

Vortex dynamics of a cylinder wake in proximity to a wall

Large-eddy simulations (LES) are used to investigate the modifications of wake dynamics and turbulence characteristics behind a circular cylinder placed near a wall for varying gap-to-diameter (G/D) ratios (where G signifies the gap between the wall and the cylinder, and D the cylinder diameter). The three-dimensional (3-D), time-dependent, incompressible Navier–Stokes equations with a dynamic subgrid-scale model are solved using a symmetry-preserving finite-difference scheme of second-order spatial and temporal accuracy. The immersed boundary (IB) method is employed to impose the no-slip boundary condition on the cylinder surface. Flow visualizations along with turbulence statistics are presented to gain insight into the flow structures that are due to interaction between the shear layers and the approaching boundary layer. Apart from the vortex shedding mechanism, the paper illustrates the physics involving the shear layer transition, stretching, breakdown and turbulence generation, either qualitatively or quantitatively, in the presence of a wall for a Reynolds number of Re=1440 (based on D and the inlet free-stream velocity U_∞).     

Prediction of turbulent flow in annular ducts with differential transport model of turbulence

In general, flow in an annulus will be strongly asymmetric because of unequal curvatures and, possibly, dissimilar texture of the two boundaries. As a result, the turbulent flow structure in such geometries possesses a number of features which can not satisfactorily be accounted for with a simple model of turbulence. The present paper provides predictions based on a more elaborate model, developed by the author, in which differential transport equations are solved for the turbulent shear stress, the turbulent kinetic energy and the rate of the turbulence energy dissipation, simultaneously with the mean momentum equation. The complete model thus consists of a closed set of four coupled non-linear partial differential equations. Predictions of flow in annuli with both smooth and rough cores with various radius ratios display in all cases very good agreement with experimental results of various authors.     

Modeling the Dynamics of Ensemble-Averaged Linear Disturbances in Homogeneous Shear Flow

In order to expand the predictive capability of single-point turbulence closure models to account for the early-stage transition regime, a methodology for the formulation and calibration of model equations for the ensemble-averaged disturbance kinetic energy and energy dissipation rate is presented. The calibration is based on homogeneous shear flow where disturbances can be described by rapid distortion theory (RDT). The relationship between RDT and linear stability theory is exploited in order to obtain a closed set of modeled equations. The linear disturbance equations are solved directly so that the numerical simulation yields a database from which the closure coefficients in the ensemble-averaged disturbance equations can be determined. This revised version was published online in July 2006 with corrections to the Cover Date.     

Transient analysis of diffusive chemical reactive species for couple stress fluid flow over vertical cylinder

The unsteady natural convective couple stress fluid flow over a semi-infinite vertical cylinder is analyzed for the homogeneous first-order chemical reaction effect. The couple stress fluid flow model introduces the length dependent effect based on the material constant and dynamic viscosity. Also, it introduces the biharmonic operator in the Navier-Stokes equations, which is absent in the case of Newtonian fluids. The solution to the time-dependent non-linear and coupled governing equations is carried out with an unconditionally stable Crank-Nicolson type of numerical schemes. Numerical results for the transient flow variables, the average wall shear stress, the Nusselt number, and the Sherwood number are shown graphically for both generative and destructive reactions. The time to reach the temporal maximum increases as the reaction constant K increases. The average values of the wall shear stress and the heat transfer rate decrease as K increases, while increase with the increase in the Sherwood number.     

On stagnation pressure increases in calorically perfect,ideal gases

When stagnation pressure rises in a natural or numerically simulated flow it is frequently a cause for concern, as one usually expects viscosity and turbulence to cause stagnation pressure to decrease. In fact, if stagnation pressure increases, one may suspect measurement or numerical errors. However, this need not be the case, as the laws of nature do not require that stagnation pressure continually decreases. In order to help clarify matters, the objective of this work is to understand the conditions under which stagnation pressure will rise in the unsteady/steady flows of compressible, viscous, calorically perfect, ideal gases. Furthermore, at a more practical level, the goal is to understand the conditions under which stagnation pressure will increase in flows simulated with the Reynolds averaged Navier–Stokes equations and eddy-viscosity turbulence models. In order to provide an improved understanding of increases in stagnation pressure for both these scenarios, transport equations are derived that govern its behavior in the unaveraged and Reynolds averaged settings. These equations are utilized to precisely determine the relationship between changes in stagnation pressure and zeroth, first, and second derivatives of fundamental flow quantities. Furthermore, these equations are utilized to demonstrate the relationship between changes in stagnation pressure and fundamental non-dimensional quantities that govern the conductivity, viscosity, and compressibility of the flow. In addition, based on an analysis of the Reynolds averaged equation (for eddy-viscosity turbulence models), it is shown that stagnation pressure is particularly likely to experience a spurious rise at the outer edges of shear layers that are undergoing convex curvature. Thereafter, numerical experiments are performed which confirm the primary aspects of the theoretical analysis.     

Low-rank solution of an optimal control problem constrained by random Navier-Stokes equations

We develop a low-rank tensor decomposition algorithm for the numerical solution of a distributed optimal control problem constrained by two-dimensional time-dependent Navier-Stokes equations with a stochastic inflow. The goal of optimization is to minimize the flow vorticity. The inflow boundary condition is assumed to be an infinite-dimensional random field, which is parametrized using a finite- (but high-) dimensional Fourier expansion and discretized using the stochastic Galerkin finite element method. This leads to a prohibitively large number of degrees of freedom in the discrete solution. Moreover, the optimality conditions in a time-dependent problem require solving a coupled saddle-point system of nonlinear equations on all time steps at once. For the resulting discrete problem, we approximate the solution by the tensor-train (TT) decomposition and propose a numerically efficient algorithm to solve the optimality equations directly in the TT representation. This algorithm is based on the alternating linear scheme (ALS), but in contrast to the basic ALS method, the new algorithm exploits and preserves the block structure of the optimality equations. We prove that this structure preservation renders the proposed block ALS method well posed, in the sense that each step requires the solution of a nonsingular reduced linear system, which might not be the case for the basic ALS. Finally, we present numerical experiments based on two benchmark problems of simulation of a flow around a von Kármán vortex and a backward step, each of which has uncertain inflow. The experiments demonstrate a significant complexity reduction achieved using the TT representation and the block ALS algorithm. Specifically, we observe that the high-dimensional stochastic time-dependent problem can be solved with the asymptotic complexity of the corresponding deterministic problem.     

Computation of Compressible Flows using a Two-equation Turbulence Model on Unstructured Grids

This paper presents a numerical method for solving compressible turbulent flows using a k - l turbulence model on unstructured meshes. The flow equations and turbulence equations are solved in a loosely coupled manner. The flow equations are advanced in time using a multi-stage Runge-Kutta time stepping scheme, while the turbulence equations are advanced using a multi-stage point-implicit scheme. The positivity of turbulence variables is achieved using a simple change of dependent variables. The developed method is used to compute a variety of turbulent flow problems. The results obtained are in good agreement with theoretical and experimental data, indicating that the present method provides a viable and robust algorithm for computing turbulent flows on unstructured meshes.     

Experimental and numerical investigation of separated flows in subsonic diffusers

The results of a special investigation of the diffuser flowfield are presented for two models of curvilinear diffuser channels with annular and rectangular cross-sections. The flow is visualized and the total pressure fields are measured by means of low-inertia transducers. At the same time, the flows are numerically calculated using commercial programs, together with codes developed by the authors. In these calculations the stationary and time-dependent Reynolds equations closed by different turbulence models, as well as the time-dependent Navier-Stokes equations, were integrated. A considerable difference between the measured data and the results of the numerical calculations in the stationary formulation is found to exist. At the same time, it has been possible to describe the occurrence of spatial inhomogeneities, the flow pattern, and the level of the experimentally observed aerodynamic losses on the basis of the solution of time-dependent problems.     

NUMERICAL SIMULATION OF UNSTEADY VISCOUS FLOWS USING AN IMPLICIT PROJECTION METHOD

In this paper an implicit projection method for the solution of the two-dimensional, time-dependent, incompressible Navier– Stokes equations is presented. The basic principle of this method is that the evaluation of the time evolution is split into intermediate steps. The computational method is based on the approximate factorization technique. The coupled approach is used to link the equations of motion and the turbulence model equations. The standard k-ϵ turbulence model is used. The current methodology, which has been tested extensively for steady problems, is now applied for the numerical simulation of unsteady flows. Several cases were tested, such as plane or axisymmetric channels, a backward-facing step, a square cavity and an axisymmetric stenosis.     

方柱尾流流态对床面冲淤形态的影响

应用平面二维悬沙数学模型对方柱尾流区的泥沙输运及床面调整进行了数值模拟，并对重要物理参数做了分析说明。数值计算采用时间分裂一全隐式有限差分格式，流场的计算基于沿水深平均的RANS方程。通过系统的数值模拟，揭示了不同来流情况下，方柱尾流区的流态变化，及与之相应的床面变化规律，并对不同流态下的泥沙运动作了分析。计算结果表明在方柱尾流区，不同的流场流态对尾流区的床面调整有明显影响。流场较弱的情况下，尾流区中湍动强度相应较弱，此时悬浮泥沙由于流速减小而普遍落淤，床面应力的减小也致使床面冲刷量减小。随着流场强度的逐渐增大，尾流区中湍动强度相应增强，床面应力增大，同时湍流的强烈交换作用增强了对泥沙的输运作用，床面变化不再是简单的淤积状态，部分区域出现了冲刷。     

Investigation of Advanced Turbulence Models for the Flow in a Generic Wing-Body Junction

The predictive performance of several turbulence models, among them formulations based on non-linear stress-strain relationships and on stress-transport equations, is examined in a collaborative university-industry study directed towards a generic wing-body junction. The geometry consists of a variation of the symmetric NACA 0020 aerofoil mounted on a flat plate, with the oncoming stream aligned with the aerofoil's symmetry plane. The dominant feature of this flow is a pronounced horseshoe vortex evolving in the junction region following separation ahead of the aerofoil's leading edge. This case is one of 6 forming a broad programme of turbulence-model validation by UMIST, Loughborough University, BAE Systems, Aircraft Research Association, Rolls-Royce plc and DERA. Key aspects of this collaboration were a high level of interaction between the partners, the use of common grids and boundary conditions, and numerical verifications aimed at maximizing confidence in the validity of the computational solutions. In total, 12 turbulence models were studied by four partners. Model performance is judged by comparing solutions with experimental data for pressure fields on the plane wall and around the aerofoil; for velocity, turbulence energy, shear stress and streamwise normal stress in the upstream symmetry plane; and for velocity, turbulence energy and shear stress in cross-flow planes downstream of the aerofoil leading edge. The emphasis of the study is on the structure of the horseshoe vortex and its effects on the forward flow. The main finding of the study is that, for this particular 3D flow, second-moment closure offers predictive advantages over the other models examined, especially in terms of the far-field structure of the horse-shoe vortex, although no model achieves close agreement with the experimental data in respect of both mean flow and turbulence quantities. This revised version was published online in July 2006 with corrections to the Cover Date.     

高雷诺数流动的基本控制方程体系和扩散跑物化Navier-Stokes方程组的意义和用途

在计算机发达的时代, 高雷诺($Re$)数绕流计算中有无必要使用简化NS方程组, 本文讨论这个问题. 主要内容如下: (1)高$Re$数绕流包含3种基本流动: 所有方向对流占优流动、所有方向对流扩散竞争流动和部分方向对流占优部分方向对流扩散竞争流动(简称干扰剪切流动), 3个基本流动的特征彼此不同且在流场中所占领域大小彼此相差悬殊, NS方程区域很小,它们的最简单控制方程组Euler、Navier-Stokes (NS)和扩散抛物化(DP) NS方程组的数学性质彼此不同, 因此利用Euler-DPNS-NS方程组体系分析计算高$Re$数绕流流动就是一个合乎逻辑的选择, 该法与利用单一NS方程组的常用方法可以彼此检验和补充. (2)流体之间以及流体与外界的动量、能量和质量交换, 流态从层流到湍流的演化主要发生在干扰剪切流动中, 干扰剪切流及其最简单控制方程------DPNS方程组具有基础意义; DPNS方程组笔者在1967年已提出. (3)诸简化NS方程组: DPNS、抛物化(P)NS、薄层(TL)NS、黏性层(VL)NS方程组的发展、相互关系, 它们的历史贡献和今后的用途; 它们的数学性质均为扩散抛物型, 但它们包含的黏性项彼此有所不同; 从流体力学角度来看, 它们中只有DPNS方程组能够准确描述干扰剪切流动. 提出把诸简化NS方程组统一为DPNS方程组的建议. (4)干扰剪切流------DPNS方程组与无干扰剪切流------边界层方程组之间的关系以及进一步研究干扰剪切流的意义.      

Multigrid solution of compressible turbulent flow on unstructured meshes using a two-equation model

The steady state solution of the system of equations consisting of the full Navier-Stokes equations and two turbulence equations has been obtained using a multigrid strategy on unstructured meshes. The flow equations and turbulence equations are solved in a loosely coupled manner. The flow equations are advanced in time using a multistage Runge-Kutta time-stepping scheme with a stability-bound local time step, while the turbulence equations are advanced in a point-implicit scheme with a time step which guarantees stability and positivity. Low-Reynolds-number modifications to the original two-equation model are incorporated in a manner which results in well-behaved equations for arbitrarily small wall distances. A variety of aerodynamic flows are solved, initializing all quantities with uniform freestream values. Rapid and uniform convergence rates for the flow and turbulence equations are observed.     

基于紊流状态的阻尼阀缝隙节流解析计算

提出了一种基于紊流状态的缝隙节流解析计算方法. 运用边界层理论给出紊流状态下流体的三层速度分布表达式,通过积分定理推导出平行平板的缝隙流量解析式,与CFD 软件数值解对比偏差很小,验证了方程的正确性. 以此为基础根据阻尼阀的结构特点又推导出同心环形缝隙的紊流流量公式,并分析不同物性参数对缝隙压差和流量的影响规律,填补了工程设计中没有紊流状态下缝隙节流解析式的空白,为提高阻尼阀的设计精度奠定了基础.