Subsonic Flows in a Multi-Dimensional Nozzle

In this paper, we study the global subsonic irrotational flows in a multi-dimensional (n ≥ 2) infinitely long nozzle with variable cross sections. The flow is described by the inviscid potential equation, which is a second order quasilinear elliptic equation when the flow is subsonic. First, we prove the existence of the global uniformly subsonic flow in a general infinitely long nozzle for arbitrary dimension with sufficiently small incoming mass flux and obtain the uniqueness of the global uniformly subsonic flow. Then we show that there exists a critical value of the incoming mass flux such that a global uniformly subsonic flow exists uniquely, provided that the incoming mass flux is less than the critical value. This gives a positive answer to the problem of Bers on global subsonic irrotational flows in infinitely long nozzles for arbitrary dimension (Bers in Surveys in applied mathematics, vol 3, Wiley, New York, 1958). Finally, under suitable asymptotic assumptions of the nozzle, we obtain the asymptotic behavior of the subsonic flow in far fields by means of a blow-up argument. The main ingredients of our analysis are methods of calculus of variations, the Moser iteration techniques for the potential equation and a blow-up argument for infinitely long nozzles.     

Unsteady Turbulent Spiral Flows in a Circular Channel

Turbulent spiral flows of water, developing in a rapidly rotating toroidal channel after abrupt braking, are investigated experimentally. The spiral flow structure is created by fixed-blade diverters located in the channel. The Reynolds number may be as high as 10⁶. A simple model for describing the evolution of the longitudinal and azimuthal velocity components averaged over the channel section is proposed.     

动失速型非定常分离涡结构的控制

本文对动失速型非定常分离涡结构的控制方法，在低速风洞中应用相平均测压技术进行了实验研究。在二元平板模型中部安装一作俯仰振荡的扰流板产生动失速型分离涡，在其上游安装另一用作控制的小扰流板。实验结果表明，应用前置的振荡小扰流板可影响并改变动失速分离涡的强度和对流特性。在最有利的控制相位下，涡吸力峰可降低４８％，涡对流时间可以推迟０．１９周期。对于间歇式振荡扰流板，采用相位提前控制方式比相位滞后控制方式更有效。     

Visualization of Unsteady Creeping Viscous Flows Using Vector Products

A finite element method for analyzing unsteady incompressible creeping flows is presented. Marker particles are introduced to analyze the flow motions. To determine the marker position in the element, vector products are used. By checking the signs of the product, the marker position during the transient analysis can be determined in a simple manner. A benchmark-type problem for which an analytical solution is available and the filling process of a simple axisymmetrical mould shape are solved to illustrate this method.     

Parallel Computation of Unsteady Flows on Network of Workstations

A parallel algorithm is presented for explicit solution of compressible Euler equations. Upwinded forms of the equations are used in conjunction with the finite element formulation. The solution domain is subdivided into overlapped blocks which are distributed to available workstations. Parallelization is achieved by using a machine portable parallel library. This, together with the usage of an explicit time-integration scheme, makes the implementation of the algorithm on distributed systems a rather straight-forward task. Furthermore, it provides a very flexible platform for load balancing of available computers.     

Reduced Basis Method for Optimal Control of Unsteady Viscous Flows

In this article we discuss the reduced basis method (RBM) for optimal control of unsteady viscous flows. RBM is a reduction method in which one can achieve the versatility of the finite element method or another for that matter and gain significant reduction in the number of degrees of freedom. The essential idea in this method is to define a reduced order subspace spanned by few basis elements and then obtain the solution via a Galerkin projection. We present several ways to define this subspace. Feasibility of the approach is demonstrated on two boundary control problems in cavity and wall bounded channel flows. Control action is effected through boundary surface movement on part of the solid wall. Application of RBM to the control problems leads to finite dimensional optimal control problems which are solved using Newton's method. Through computational experiments we demonstrate the feasibility and applicability of the reduced basis method for control of unsteady viscous flows.     

On Plane-Parallel Separated Flows of an Incompressible Fluid near Flat Boundaries

On the basis of Stokes separated flows, examples of separated flows described by the Navier-Stokes equations of a viscous incompressible fluid are constructed. These flows are represented by series convergent in a certain non-zero neighborhood of a flat contour immersed in the flow. In this neighborhood, the series have the same structure as those for the basic Stokes flows. Examples of the regions in which the series segments chosen give only a slight deviation from the numerical solutions of the Navier-Stokes equations are presented. The comparison between inviscid separated flows (without the no-slip condition on the contour) and viscous flows of the same structure (with the no-slip condition) shows that the viscosity does not play a decisive role in the formation of separation or the type of streamline approach to or departure from the contour.     

Unsteady Viscous Flows about Bodies: Vorticity Release and Forces

The force (drag and lift) exerted on a body moving in a viscous fluid is expressed via the free and bound vorticity moments, and the role of vortex shedding is discussed. The formulation encompasses classical, inviscid flows, and leads to efficient computational methods. Numerical results for a few prototype flows are presented.     

非流线体分离流动态载荷的数值模拟方法研究

以圆柱绕流为研究对象,针对圆形边界,采用O型网格对流场进行离散,用二阶精度的中心差分有限体积法作空间离散,用二阶精度的中心差分处理时间问题,用双时间方法求解了二维非定常Navier-Stokes方程,系统研究了计算方法对收敛精度、时间步长和网格数量的依赖性.计算结果表明,对于长时间历程的非定常问题,虽然双时间方法收敛性很好,但对于分离流而言,时间步长的选取并非没有限制;每一步伪时间的推进中,收敛精度也有要求;而要模拟圆柱分离流的非线性气动力现象,计算网格至少要达到260×80的数量.     

A Discrete Adjoint Approach for the Optimization of Unsteady Turbulent Flows

In this paper we present a discrete adjoint approach for the optimization of unsteady, turbulent flows. While discrete adjoint methods usually rely on the use of the reverse mode of Automatic Differentiation (AD), which is difficult to apply to complex unsteady problems, our approach is based on the discrete adjoint equation directly and can be implemented efficiently with the use of a sparse forward mode of AD. We demonstrate the approach on the basis of a parallel, multigrid flow solver that incorporates various turbulence models. Due to grid deformation routines also shape optimization problems can be handled. We consider the relevant aspects, in particular the efficient generation of the discrete adjoint equation and the parallel implementation of a multigrid method for the adjoint, which is derived from the multigrid scheme of the flow solver. Numerical results show the efficiency of the approach for a shape optimization problem involving a three dimensional Large Eddy Simulation (LES).     

A Matched Asymptotic Expansion Analysis of Highly Unsteady Porous Media Flows

The method of matched asymptotic expansions is used to analyse a mixture of wave and diffusive behaviours governing flow in a saturated porous medium inside an elastic pipe that is suddenly subjected to a large hydraulic gradient at its entrance. At early times and near the entrance, the head is a diffusing wave that can be reduced to the linear and non-linear telegrapher equations for the laminar and partially developed turbulent flows, respectively. At later times, laminar flows are diffusive and partially developed turbulent flows follow a ‘fast diffusion’ behaviour. In the case of fully developed turbulence, flows at later times follow a fast diffusion form which is complicated by advection at extremely high gradients. A matched asymptotic expansion approach is used to match flows at early times and near the entrance, with complementary forms that are away from the entrance and which occur at later times.     

A Comparative Study of Turbulence in Ramp-Up and Ramp-Down Unsteady Flows

DNS of a turbulent channel flow subjected to a step change in pressure gradient are performed to facilitate a direct comparison between ramp-up and ramp-down flows. Strong differences are found between behaviours of turbulence in the two flows. The wall shear stress in the ramp-up flow first overshoots, and then strongly undershoots the quasi-steady value in the initial stage of the excursion, before approaching the quasi-steady value. In a strongly decelerating flow, the wall shear stress tends to first undershoot but then overshoot the quasi-steady value. ??Slow?? response of turbulence as well as flow inertia is responsible for these behaviours. In the ramp-up flow, the response of turbulence is similar to that observed in uniformly accelerating flows from previous studies, exhibiting a three-stage development. However, the transition between the various stages is more gradual and the responding stage is much longer and slower in the flows considered here. It has been shown that the delay in the near wall region is longer than that in the buffer layer confirming that turbulence response first occurs at the location of peak turbulence production. In a strongly decelerating flow, the response of turbulence exhibits a two-stage development. In both ramp-up and ramp down flows, the energy distribution in the three components of turbulent kinetic energy deviates from that of the steady flow. In a ramp-up flow, more energy is in $u_1^\prime $ and less in $u_2^\prime $ and $u_3^\prime $ , whereas the trend is reversed in a ramp-down flow. This is a reflection of the redistribution of turbulence from $u_1^\prime $ to $u_2^\prime $ and $u_3^\prime $ .     

非结构动网格在可动边界问题中的应用研究

本文对用于非结构动网格生成的弹簧近似方法进行了研究。通过采用顶点弹簧方法,分析研究了弹簧倔强系数的取值,同时通过引入挤压倔强系数和边界修正,对标准弹簧近似方法进行了改进。改进后的方法可以大大提高网格变形能力和网格质量。应用本文发展的非结构动网格生成方法并通过耦合求解基于(ArbitraryLagrangian-Eulerian ALE)描述的Euler方程,模拟了谐和振动NACA0012翼型及M6机翼的跨音速绕流,计算结果与参考文献提供的结果及实验结果吻合良好。     

Generalized Wall Functions for High-speed Separated Flows Over Adiabatic and Isothermal Walls

We present the extension of our wall-laws developed for low-speed flows to super-and hyper-sonic configurations. In particular, we are interested in flows over isothermal walls and in the modeling of heat transfer. We recall the main steps of the development:

?Obtaining generalized wall functions for low-speed fluids, valid for all y ⁺.

?Taking into account transversal effects in wall-laws.

?Accounting for the compressible feature of the flow on adiabatic walls without using information on the local boundary layer structure, for instance its thickness, but only using information available at the fictitious wall.

?Taking into account thermal effects on isothermal walls. In particular, the heat flux at the real wall is obtained by an a posteriori evaluation using information at the fictitious one.

?Only using information available on unstructured meshes and avoiding the information coming from a Cartesian hypothesis for the mesh in near-wall regions.

These ingredients are validated on hyper-sonic configurations on adiabatic and isothermal walls for expansion and compression ramps.     

跨音速压气机级的三维周期性非定常流动计算

对跨音速压气机级动静叶排相干形成的三维非定常流场进行了数值研究，利用时间推进ＬＵ－ＳＧＳ稳式迭代法求解三维非定常欧拉方程，对流项采用高分辨率ＮＮＤ格式离散。对某压气机第一级动静叶排相干非定常流场的计算结果表明，本文方法不仅在收敛速度上明显地优于一般显式方法，而且保持了流场中激波的高分辨率，适于推广到计算量巨大的的多级轴流压气机三维非定常流场的数值分析问题     

Viscous Standing Asymptotic States of Isentropic Compressible Flows Through a Nozzle

In this work we consider a viscous regularization of a well-known one-dimensional model for isentropic viscous compressible flows through a nozzle. For the existence and multiplicity of standing asymptotic states for a certain type of ducts, a complete analysis in a framework of dynamical systems is provided. As an application of the geometric singular perturbation theory, we show that all standing asymptotic states admit viscous profiles.     

Turbulence Modelling of Unsteady Turbulent Flows Using the Stress Strain Lag Model

This paper reports the application of a recently developed turbulence modelling scheme known as the C _as model. This model was specifically developed to capture the effects of stress-strain misalignment observed in turbulent flows with mean unsteadiness. Earlier work has reported the approach applied within a linear k-ε modelling framework, and some initial testing of it within the k-ω SST model of Menter (AIAA J 32:1598–1605, 1994). The resulting k-ε-C _as and SST-C _as models have been shown to result in some of the advantages of a full Reynolds Stress transport Model (RSM), whilst retaining the computational efficiency and stability benefits of a eddy viscosity model (EVM). Here, the development of the the high-Reynolds-number version of the C _as model is outlined, with some example applications to steady and unsteady homogeneous shear flows. The SST-C _as form of the model is then applied to further, more challenging cases of 2-D flow around a NACA0012 aerofoil beyond stall and the 3-D flow around a circular cylinder in a square duct, both being flows which exhibit large, unsteady, separated flow regions. The predictions returned by a range of other common turbulence modelling schemes are included for comparison and the SST-C _as scheme is shown to return generally good results, comparable in some respects to those obtainable from far more complex schemes, for only moderate computing resource requirements.     

Hybrid Unsteady RANS and PDF Method for Turbulent Non-Reactive and Reactive Flows

A hybrid unsteady Reynolds-averaged numerical simulation (U-RANS) and probability density function (PDF) method is developed for turbulent non-reactive and reactive flows. The resulting modeled equations are solved by a consistent hybrid finite volume and Lagrangian Monte-Carlo particle method. Both turbulent non-reactive and reactive flows in a rectangular channel containing a triangular-shaped bluff-body are simulated. One-step and two-step mechanisms for propane/air combustion are used for the reactive case. The time-averaged results are compared with both experimental data and numerical results from the literature using large eddy simulation (LES) and steady RANS. The results of the present method are in good agreement with the experimental data, and they improve the numerical results available in the literature.     

Subsonic Euler Flows with Large Vorticity Through an Infinitely Long Axisymmetric Nozzle

This paper is a sequel to the earlier work Du and Duan (J Diff Equ 250:813–847, 2011) on well-posedness of steady subsonic Euler flows through infinitely long three-dimensional axisymmetric nozzles. In Du and Duan (J Diff Equ 250:813–847, 2011), the authors showed the existence and uniqueness of the global subsonic Euler flows through an infinitely long axisymmetric nozzle, when the variation of Bernoulli’s function in the upstream is sufficiently small and the mass flux of the incoming flow is less than some critical value. The smallness of the variation of Bernoulli’s function in the upstream prevents the attendance of the possible singularity in the nozzles, however, at the same time it also leads that the vorticity of the ideal flow is sufficiently small in the whole nozzle and the flows are indeed adjacent to axisymmetric potential flows. The purpose of this paper is to investigate the effects of the vorticity for the smooth subsonic ideal flows in infinitely long axisymmetric nozzles. We modify the formulation of the problem in the previous work Du and Duan (J Diff Equ 250:813–847, 2011) and the existence and uniqueness results on the smooth subsonic ideal polytropic flows in infinitely long axisymmetric nozzles without the restriction on the smallness of the vorticity are shown in this paper.     

Existence of Global Solutions for Unsteady Isentropic Gas Flow in a Laval Nozzle

In this paper, we study the motion of isentropic gas in the Laval nozzle. The Laval nozzle is the most important type of nozzle utilized in some turbines. In particular, we consider unsteady flows, including transonic gas flows, and prove the existence of global solutions for the Cauchy problem. In spite of its importance, this problem has received little attention until now. The most difficult point is to obtain bounded estimates for approximate solutions. To overcome this, we introduce a modified Godunov scheme. The corresponding approximate solutions consist of piecewise steady-state solutions of an auxiliary equation and yield a sharper bounded estimate. As a result, we find an invariant region for our solutions. Finally, in order to prove their convergence, we use the compensated compactness framework.