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.      

高浓度固-液两相流紊流的动理学模型

采用分子动理学方法,基于固-液两相流液相分子或颗粒相颗粒的Boltzmann方程,对Boltzmann方程分别取零矩和一次矩,则得到高浓度固-液两相流紊流的连续方程和动量方程,再和较成熟的低浓度两相流连续方程和动量方程比较,取低浓度两相流控制方程中较成熟合理的有关项和高浓度时由动理学方法推导出的颗粒间碰撞项,则得到高浓度固-液两相流紊流的最终控制方程:连续方程和动量方程.     

On the incremental singular value decomposition method to support unsteady adjoint-based optimization

This paper proposes and evaluates an approximation model based on an incremental Singular Value Decomposition (iSVD) algorithm, for unsteady flow field reconstructions, needed for integrating the unsteady adjoint equations backward in time, within a gradient-based optimization loop. Due to the iSVD algorithm, the computational cost of solving the unsteady adjoint equations is reduced considerably, without practically affecting the accuracy of the computed gradient. Approximations to the unsteady flow fields are constructed while solving the time-varying flow equations (moving forward in time) and used to reconstruct these fields during the backward-in-time integration of the continuous adjoint equations. Optimization results obtained using the proposed method are compared to those computed using the binomial checkpointing technique, which acts as the reference method. Test cases for both flow control and shape optimization problems are presented.     

Integrated numerical model for vegetated surface and saturated subsurface flow interaction

The construction of an integrated numerical model is presented in this paper to deal with the interactions between vegetated surface and saturated subsurface flows. A numerical model is built by integrating the previously developed quasi-three-dimensional (Q3D) vegetated surface flow model with a two-dimensional (2D) saturated groundwater flow model. The vegetated surface flow model is constructed by coupling the explicit finite volume solution of 2D shallow water equations (SWEs) with the implicit finite difference solution of Navier-Stokes equations (NSEs) for vertical velocity distribution. The subsurface model is based on the explicit finite volume solution of 2D saturated groundwater flow equations (SGFEs). The ground and vegetated surface water interaction is achieved by introducing source-sink terms into the continuity equations. Two solutions are tightly coupled in a single code. The integrated model is applied to four test cases, and the results are satisfactory.     

Axisymmetric flows of an incompressible fluid between movable rotating disks

Within the Karman family of exact solutions of the Navier-Stokes equations, some non-selfsimilar solutions are considered to the problem of unsteady incompressible flow between two rotating disks one of which moves along a common rotation axis. Three classes of the flow regimes are studied: (i) a flow between the non-rotating disks, (ii) a flow between the disks rotating with identical angular velocities, and (iii) a flow between the disks rotating with opposite velocities. Examples of exact rotationally symmetric solutions for the inviscid-fluid equations, satisfying the no-slip conditions, are given.     

Improved subgrid scale model for dense turbulent solid-liquid two-phase flows

The dense solid-phase governing equations for two-phase flows are obtained by using the kinetic theory of gas molecules. Assuming that the solid-phase velocity distributions obey the Maxwell equations, the collision term for particles under dense two-phase flow conditions is also derived. In comparison with the governing equations of a dilute two-phase flow, the solid-particle‘s governing equations are developed for a dense turbulent solid-liquid flow by adopting some relevant terms from the dilute two-phase governing equations. Based on Cauchy-Helmholtz theorem and Smagorinsky model, a second-order dynamic sub-grid-scale (SGS) model, in which the sub-grid-scale stress is a function of both the strain-rate tensor and the rotation-rate tensor, is proposed to model the two-phase governing equations by applying dimension analyses. Applying the SIMPLEC algorithm and staggering grid system to the two-phase discretized governing equations and employing the slip boundary conditions on the walls, the velocity and pressure fields, and the volumetric concentration are calculated. The simulation results are in a fairly good agreement with experimental data in two operating cases in a conduit with a rectangular cross-section and these comparisons imply that these models are practical.     

Burnett simulations of gas flow in microchannels

The Burnett equations with slip boundary conditions are used to model the gas flow in microchannels in transition flow regime. As the Navier-Stokes equations are not appropriate to model the gas flow in this regime, the higher-order Burnett equations are adopted in the present study. In earlier studies, convergent solutions of the Burnett equations of microPoiseuille flow could only be obtained when Knudsen number is less than 0.2. By using a relaxation method on the boundary values, convergent solutions of the Burnett equations can be obtained even when Knudsen number reaches 0.4. The solutions of Burnett equations agree very well with experimental data and direct simulation Monte Carlo (DSMC) results. The pressure distributions and velocity profiles are then discussed in detail.     

PML absorbing boundary conditions for the linearized and nonlinear Euler equations in the case of oblique mean flow

For the case of uniform mean flow in an arbitrary direction, perfectly matched layer (PML) absorbing boundary conditions are presented for both the linearized and nonlinear Euler equations. Although linear perfectly matched side layers with an oblique mean flow have been studied in previous works, we propose in the present paper a construction of corner layer equations that are dynamically stable. Stability issues are investigated by examining the dispersion relations of linear waves supported by the corner layer equations. For increased efficiency, a pseudo mean flow is included in the derivation of the PML equations for the nonlinear case. Numerical examples are given to support the validity of the proposed equations. Specifically, the linear PML formulation is tested for the case of acoustic, vorticity, and entropy waves traveling with an oblique mean flow. The nonlinear formulation is tested with an isentropic vortex moving diagonally with a constant velocity. Copyright © 2008 John Wiley & Sons, Ltd.     

配置点谱方法-人工压缩法(SCM-ACM)求解同心圆筒内流体流动

发展了配置点谱方法SCM(Spectral collocation method)和人工压缩法ACM(Artificial compressibility method)相结合的SCM-ACM数值方法,计算了柱坐标系下稳态不可压缩流动N-S方程组。选取典型的同心圆筒间旋转流动Taylor-Couette流作为测试对象,首先,采用人工压缩法获得人工压缩格式的非稳态可压缩流动控制方程;再将控制方程中的空间偏微分项用配置点谱方法进行离散,得到矩阵形式的代数方程;编写了SCM-ACM求解不可压缩流动问题的程序;最后,通过与公开发表的Taylor-Couette流的计算结果对比,验证了求解程序的有效性。结果证明,本文发展的SCM-ACM数值方法能够用于求解圆筒内不可压缩流体流动问题,该方法既保留了谱方法指数收敛的特性,也具有ACM形式简单和易于实施的特点。本文发展的SCM-ACM数值方法为求解柱坐标下不可压缩流体流动问题提供了一种新的选择。     

Linear and nonlinear approaches and digital signal processing in aeroacoustics

The linear and nonlinear approaches to the calculation of small acoustic disturbance propagation and evolution in nonuniform flows are compared. In the conventional linear approach it is the linearized equations of time-dependent, ideal (inviscid and non-heat-conducting) or viscous gas flow that are integrated. In the nonlinear approach the original nonlinear equations governing the same time-dependent flow (Euler equations for an ideal gas) are integrated; these are the same equations that, together with time relaxation procedure, are used in the linear approach for calculating the stationary background. It is shown that the application of digital signal processing, widely used in acoustic experiments, makes it possible to isolate the harmonic acoustic waves from the results of integration of the nonlinear equations, though their intensity is smaller than that of the noise due to computational errors, including inadequate attainment of the stationary background.     

Exact solutions of the generalized Navier– Stokes equations for benchmarking

The generalized Navier– Stokes equations for incompressible viscous flows through isotropic granular porous medium are studied. Some analytical classic solutions of the Navier– Stokes equations are generalized to the case of the considered equations. Obtained solutions of generalized equations reduce to classic ones as porosity effect disappears. Average velocity of generalized solutions is calculated and evaluated in two limiting regimes of flow. In the shallow conduit, the generalized flow rate approximates the free (without porous medium) flow rate and in the case of removed boundaries this approaches Darcy's law. The use of the derived exact solutions for benchmarking purposes is described. Copyright © 2002 John Wiley & Sons, Ltd.     

Recursive Conditional Moment Equations for Advective Transport in Randomly Heterogeneous Velocity Fields

Flow and transport parameters such as hydraulic conductivity, seepage velocity, and dispersivity have been traditionally viewed as well-defined local quantities that can be assigned unique values at each point in space-time. Yet in practice these parameters can be deduced from measurements only at selected locations where their values depend on the scale (support volume) and mode (instruments and procedure) of measurement. Quite often, the support of the measurements is uncertain and the data are corrupted by experimental and interpretive errors. Estimating the parameters at points where measurements are not available entails an additional random error. These errors and uncertainties render the parameters random and the corresponding flow and transport equations stochastic. The stochastic flow and transport equations can be solved numerically by conditional Monte Carlo simulation. However, this procedure is computationally demanding and lacks well-established convergence criteria. An alternative to such simulation is provided by conditional moment equations, which yield corresponding predictions of flow and transport deterministically. These equations are typically integro-differential and include nonlocal parameters that depend on more than one point in space-time. The traditional concept of a REV (representative elementary volume) is neither necessary nor relevant for their validity or application. The parameters are nonunique in that they depend not only on local medium properties but also on the information one has about these properties (scale, location, quantity, and quality of data). Darcy's law and Fick's analogy are generally not obeyed by the flow and transport predictors except in special cases or as localized approximations. Such approximations yield familiar-looking differential equations which, however, acquire a non-traditional meaning in that their parameters (hydraulic conductivity, seepage velocity, dispersivity) and state variables (hydraulic head, concentration) are information-dependent and therefore, inherently nonunique. Nonlocal equations contain information about predictive uncertainty, localized equations do not. We have shown previously (Guadagnini and Neuman, 1997, 1998, 1999a, b) how to solve conditional moment equations of steady-state flow numerically on the basis of recursive approximations similar to those developed for transient flow by Tartakovsky and Neuman (1998, 1999). Our solution yields conditional moments of velocity, which are required for the numerical computation of conditional moments associated with transport. In this paper, we lay the theoretical groundwork for such computations by developing exact integro-differential expressions for second conditional moments, and recursive approximations for all conditional moments, of advective transport in a manner that complements earlier work along these lines by Neuman (1993).     

Shallow flow modelling using curvilinear depth-averaged stream function and vorticity transport equations

Most receiving water, such as lakes and open reservoirs, have large plan dimensions with respect to their depth. In such cases, the flow may be nearly two-dimensional and the depth-averaged Reynolds equations are appropriate. This paper presents a new version of the governing equations in curvilinear depth-averaged stream function and vorticity transport (ψ, ω) form appropriate for non-orthogonal computational meshes. The equations are discretized using finite differences and solved using successive over-relaxation for the depth-averaged stream function equation and an alternating direction implicit scheme for the vorticity transport equation. Results from the numerical model are validated against data from flow past a backward facing step and jet-forced flow in a circular reservoir. The results indicate that the (ψ, ω) form of the shallow water equations may be useful for applications where the free surface can either be assumed horizontal, or is know a priori.     

Effect of thermo‐solutal stratification on recirculation flow patterns in a backward‐facing step channel flow

This paper presents results on the combined effect of thermo‐solutal buoyancy forces on the recirculatory flow behavior in a horizontal channel with backward‐facing step and the ensuing impact on heat and mass transfer phenomena. The governing equations for double diffusive mixed convection are represented in velocity–vorticity form of momentum equations, velocity Poisson equations, energy and concentration equations. Galerkin's finite‐element method has been employed to solve the governing equations. Recirculatory flow fields with heat and mass transfer are simulated for opposing and aiding thermo‐solutal buoyancy forces by assuming suitable boundary conditions for energy and concentration equations. The effect of Richardson number (0.1?Ri?10) and buoyancy ratio (?10?N?10) on the recirculation bubble and Nusselt and Sherwood numbers are studied in detail. For Richardson number greater than unity, distinct variations in the gradients of Nusselt number and Sherwood number with buoyancy ratio are observed for flow regimes with opposing and aiding buoyancy forces. Copyright © 2009 John Wiley & Sons, Ltd.     

配置点谱方法-人工压缩法(SCM-ACM)求解不可压缩流体流动

开发了配置点谱方法SCM（spectral collocation method）与人工压缩法ACM（artificial compressibility method）相结合的方法SCM-ACM,用于求解不可压缩粘性流动问题。选取典型的方腔顶盖驱动流为研究测试对象,首先建立人工压缩格式的控制方程,其次采用SCM离散控制方程的空间偏微分项,推导出矩阵形式的代数方程,最后测试了SCM-ACM代码的有效性。结果显示,SCM-ACM能够有效求解不可压缩流动问题,并继承了谱方法的指数收敛特性,且具有ACM求解过程简单及易于实施的特点。     

Accurate 3D viscous incompressible flow calculations with the FEM

A second-order-accurate (in both time and space) formulation is developed and implemented for solution of the three-dimensional incompressible Navier–Stokes equations involving high-Reynolds-number flows past complex configurations. For stabilization, only a fourth-order-accurate artificial dissipation term in the momentum equations is used. The finite element method (FEM) with an explicit time-marching scheme based on two-fractional-step integration is used for solution of the momentum equations. The element-by-element (EBE) technique is employed for solution of the auxiliary potential function equation in order to ease the memory requirements for matrix. The cubic cavity problem, the laminar flow past a sphere at various Reynolds numbers and the flow around the fuselage of a helicopter are successfully solved. Comparison of the results with the low-order solutions indicates that the flow details are depicted clearly even with coarse grids. © 1997 John Wiley & Sons, Ltd.     

Solution of the incompressible Navier–Stokes equations by the method of lines

The numerical method of lines (NUMOL) is a numerical technique used to solve efficiently partial differential equations. In this paper, the NUMOL is applied to the solution of the two‐dimensional unsteady Navier–Stokes equations for incompressible laminar flows in Cartesian coordinates. The Navier–Stokes equations are first discretized (in space) on a staggered grid as in the Marker and Cell scheme. The discretized Navier–Stokes equations form an index 2 system of differential algebraic equations, which are afterwards reduced to a system of ordinary differential equations (ODEs), using the discretized form of the continuity equation. The pressure field is computed solving a discrete pressure Poisson equation. Finally, the resulting ODEs are solved using the backward differentiation formulas. The proposed method is illustrated with Dirichlet boundary conditions through applications to the driven cavity flow and to the backward facing step flow. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical simulation of swirling flow in diffusers

A reduced form of Navier–Stokes equations is developed which does not have the usual minimum axial step size restriction. The equations are able to predict accurately turbulent swirling flow in diffusers. An efficient single sweep implicit scheme is developed in conjunction with a variable grid size domain-conforming co-ordinate system. The present scheme indicates good agreement with experimental results for (1) turbulent pipe flow, (2) turbulent diffuser flow, (3) turbulent swirling diffuser flow. The strong coupling between the swirl and the axial velocity profiles outside of the boundary layer region is demonstrated.     

The application of the Lattice Boltzmann method to the one-dimensional modeling of pulse waves in elastic vessels

The one-dimensional nonlinear equations for the blood flow motion in distensible vessels are considered using the kinetic approach. It is shown that the Lattice Boltzmann (LB) model for non-ideal gas is asymptotically equivalent to the blood flow equations for compliant vessels at the limit of low Knudsen numbers. The equations of state for non-ideal gas are transformed to the pressure-luminal area response. This property allows to model arbitrary pressure-luminal area relations. Several test problems are considered: the propagation of a sole nonlinear wave in an elastic vessel, the propagation of a pulse wave in a vessel with varying mechanical properties (artery stiffening) and in an artery bifurcation, in the last problem Resistor–Capacitor–Resistor (RCR) boundary conditions are considered. The comparison with the previous results shows a good precision.