层流射流流动的简化Navier-Stokes方程(SNSE)解

本文利用十一种简化 Navier-Stokes 方程(SNSE) 求解已知Navier-Stokes(NS)方程准确解的层射流流动,表明:多数SNSE~([1-6])的解与NS方程的准确解不一致;少数SNSE~([7,8])的解与NS方程的准确解一致,文中在射流的喉部和拐点位置,给出几种SNSE解与准确解的相对偏差,并把粘性及惯性诸项加以定量比较,强调指出:按照边界层理论量级分析为Re~(1/2)和Re~1量级的惯性项以及Re~(-1/2)量级的粘性项具有重要影响;据此从力学角度论证了简化 NS 方程时,保留全部惯性项和合理取舍粘性项的必要性。     

A new method for obtaining the exact solutions of the Navier-Stokes (NS) equations and its applications

In this paper we propose a new method for obtaining the exact solutions of the Mavier-Stokes (NS) equations for incompressible viscous fluid in the light of the theory of simplified Navier-Stokes (SNS) equations developed by the first author^[1,2], Using the present method we can find some new exact solutions as well as the well-known exact solutions of the NS equations. In illustration of its applications, we give a variety of exact solutions of incompressible viscous fluid flows for which NS equations of fluid motion are written in Cartesian coordinates, or in cylindrical polar coordinates, or in spherical coordinates. The project supported by National Natural Science Foundation of China.     

A uniformly valid asymptotic solution of the Navier-Stokes equations

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Viscous-inviscid interacting flow theory

In this paper a viscous-inviscid interacting flow theory (IFT) is developed for an incompressible, two—dimensional laminar flow. IFT's main points are as follows. (1) By introducing a concept of interaction layer where the normal momentum exchange is dominating, a new three-layer structure is established. (2) Through the conventional manipulations and by introducing an interaction model, both the streamwise and normal length scales are proved to be functions of a single parameterm, which is related to the streamwise pressure gradient and Reynolds number. (3) The approximate equations governing the flow of each layer as well as the whole interaction flow are derived. The present IFT is applicable to both attached and attached-separation bubble—reattached flows. The classical boundary layer theory^[1] and Triple-deck theory^[2] are shown to be two special cases of the present theory underm=0 and 1/4, respectively. Furthermore IFT provides new distinctions of both the normal and streamwise length scales for flow-field numerical computation and also gives a new approach to developing the simplified Navier-Stokes (SNS) equations. The project is supported by the National Natural Science Foundation of China.     

Pulsatile flow in arteries with external restraint of visco-elastic tissue

The presence of a longitudinal constraint must be conceived physiologically as due largely to the connective tissue attachments on the outside of the artery. Living tissues are viscoelastic bodies. In order to analyse the effect of external viscoelastic tissue to pulsatile flow in arteries, in this paper the external connective tissue of artery will be considered as a Voigt visco-elastic body, and the expressions of pulse wave velocity and the velocity of pulsatile flow will be found by the velocity of pulsatile flow will be found by the blood motion equations (Navier-Stokes equation) as well as wall motion equations (Lamb equation). The results of free elastic tube^[4] and those of external elastic restraint^[6] by Womersley can be considered as a particular case and can be covered in this paper.     

The method of matrices conjoint multiplication for the problem of circular arc corrugated diaphragm

The circular are corrugated diaphragins are taken in this paper and structures of several sections of the ring shells and a central cireular plale I matrices and link matrices are derived by using Prof. Chuen Hei-zang’s general the ring shell[1] and perturbation theory of the circular thin plates. Throngh the meined of matrices conjoint multiplication, the linear exact solution and nonlinear soluaen are obtained. The resutts agree with that of the experiments presented by W. A. Wildhack..     

The numerical solution of the unsteady natural convection flow in a square cavity at high Rayleigh number using SADI method

The unsteady natural convection flow in a square cavity at high Rayleigh number Ra=10 ⁷ and 2×10 ⁷ has been computed using cubic spline integration. The required solutions to the two dimensional Navier-Stokes and energy equations have been obtained using two alternate numerical formulations on non-uniform grids. The main features of the transient flow have been briefly discussed. The results obtained by using the present method are in good agreement with the theoretical predictions ^[1,2].The steady state results have been compared with accurate solutions presented recently for Ra=10 ⁷.     

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.     

Pulsatile flow between permeable beds

Pulsatile flow of a viscous fluid between two permeable beds is analyzed. The flow between and through the permeable beds are governed by the Navier-Stokes equations and Darcy's law, respectively. The velocity field and the volume flux are obtained for several cases and discussed. Further, when the permeability parameter k→0, the results agree with those of Wang (J. Appl. Mech. 38 (1971) 553).     

Numerical investigation of mixed convection heat transfer past an in-line bundle of cylinders

Two dimensional unsteady Navier-Stokes and the energy equations are solved using finite element method for the case of flow past five row deep in-line bundle of circular cylinders with pitch to diameter ratios (PDR) of 1.5 and 2.0. Numerical solutions of governing equations have been obtained using Euler's explicit algorithm. Analysis have been made for Reynolds number of 100 and Prandtl number of 0.71. The effect of Richardson number (Ri=Gr/Re ²) on the flow and heat transfer have been investigated forRi=?1.0, ?0.5, 0.0, +0.5 and +1.0. Streamlines, isovorticity lines, pressure and temperature contours, local and average Nusselt numbers, pressure and shear stress distribution around the cylinders are presented. Results obtained for forced convection (Ri=0.0) agree well with the available experimental and numerical results. There is considerable effect of buoyancy over tube bundles both in buoyancy aiding and opposing flows.     

坐标变换法数值求解微通道行波电场电渗流

采用坐标变换法数值求解了耦合的Poisson-Nernst-Planck (PNP)方程和Navier-Stokes(NS)方程, 研究二维狭窄微通道行波电场电渗流数值解. 数值结果表明,坐标变换法能有效降低电渗流解数值解在双电层的高梯度, 有效改善数值解的收敛性和稳定性. 坐标变换的电渗流数值解和原始坐标下的数值解完全一致. 坐标变换后采用简单的网格也能得到和原始坐标下复杂网格相同的解. 给出了滑移边界的近似解与完整的PNP-NS数值解的比较. 在双电层厚度与微通道深度比值(λ/H)很小的情况下(相对深通道), 两者的解基本一致. 但在λ/H较大时(相对浅通道)滑移边界的解高于电渗流速度.      

Low order nonconforming mixed finite element method for nonstationary incompressible Navier-Stokes equations

This paper studies a low order mixed finite element method (FEM) for nonstationary incompressible Navier-Stokes equations. The velocity and pressure are approximated by the nonconforming constrained Q ₁ ^rot element and the piecewise constant, respectively. The superconvergent error estimates of the velocity in the broken H ¹-norm and the pressure in the L ²-norm are obtained respectively when the exact solutions are reasonably smooth. A numerical experiment is carried out to confirm the theoretical results.     

Numerical simulation of a solitary wave interaction with submerged multi-bodies

The problems of a solitary wave passing over rectangular cylinders have been analysed. The numerical simulation is based on the full nonlinear two-dimensional Navier-Stokes equations which are solved by the finite difference method. The free surface is dealt with by the Volume of Fluid method (VOF). Results for a solitary wave passing over a single cylinder are compared with the experimental data of Seabra-Santos, Penouard and Temperville^[2] and better agreement is obtained than those obtained from the long wave equation based on the potential flow theory. Results are also given for two cylinders with different gaps. The project supported by the National Natural Science Foundation of China and the Development Foundation of Science and Technology of Shanghai Education Committee and the Royal Society.     

Numerical study of buoyancy- and thermocapillary-driven flows in a cavity

Thermocapillary- and buoyancy-driven convection in open cavities with differentially heated endwalls is investigated by numerical solutions of the two-dimensional Navier-Stokes equations coupled with the energy equation. We studied the thermocapillary and buoyancy convection in the cavities, filled with low-Prandtl-number fluids, with two aspect-ratiosA=1 and 4, Grashof number up to 10⁵ and Reynolds number ⋎Re⋎≤10⁴. Our results show that thermocapillary can have a quite significant effect on the stability of a primarily buoyancy-driven flow, as well as on the flow structures and dynamic behavior for both additive effect (i.e., positiveRe) and opposing effect (i.e., negativeRe).     

On the unsteady unidirectional flows generated by impulsive motion of a boundary or sudden application of a pressure gradient

Some exact solutions of the time-dependent Navier-Stokes equations are discussed for flows impulsively started from rest by the motion of a boundary or two boundaries or by sudden application of a pressure gradient. It is shown that the expressions of the quantities such as velocity, flux and skin friction obtained in a series form for large times can also be used to find their values for small times or vice versa. Furthermore, conditions for which the values of the quantities at large times are nearly the same with those at small times are derived.     

Calculation of stresses and deformations of bellows by initial parameter method of numerical integration

By reducing the boundary value problem in stress analysis of bellows into initial value problem, this paper presents a numerical solution of stress distribution in semi-circular arc type bellows based upon the toroidal shell equation of V. V. Novozelov^[8]. Throughout the computation, S. Gill’s method^[1O] of extrapolation is used. The stresses and deformations of bellows under axial load and internal pressure are c-alculated, the results of which agree completely with those derived from the general solution of Prof. Chien Wei-zang^[1-4]. The extrapolation formula presented in this paper greatly promotes the accuracy of discrete calculation.The computer program in BASIC language of Wang 2200 VS computer is included in the appendix.     

The experimental results of micro-structure in grid-produced turbulence

The variation of main turbulent quantities in an isotropic turbulent flow, such as the decay of turbulent energy and the variation of Taylor microscale of turbulence with time are obtained, by employing a hot-wire anemometer and a nearly isotropic turbulent flow which is produced by a gridscreen located at the entrance of the test section in a low-level turbulence and low-speed wind tunnel in Peking University. The experimental results of the decay of turbulent energy and the variation of Taylor microscale of turbulence with time at the whole period from initial to final stage, normalized in an non-dimensional form, are consistent quite well with the computational results by the theory of the statistical vorticity structure^[1]. The experimental results presented in this paper also agree with Townsend's results obtained in earlier years^[2] as well as with Bennett's in the seventy's^[3].     

Flow of a viscous fluid in an annular gap with intensive blowing from the walls

Self-similar solutions are obtained in [1, 2] to the Navier-Stokes equations in gaps with completely porous boundaries and with Reynolds number tending to infinity. Approximate asymptotic solutions are also known for the Navier-Stokes equations for plane and annular gaps in the neighborhood of the line of spreading of the flow [3, 4]. A number of authors [5–8] have discovered and studied the effect of increase in the stability of a laminar flow regime in channels of the type considered and a significant increase in the Reynolds number of the transition from the laminar regime to the turbulent in comparison with the flow in a pipe with impermeable walls. In the present study a numerical solution is given to the system of Navier-Stokes equations for plane and annular gaps with a single porous boundary in the neighborhood of the line of spreading of the flow on a section in which the values of the local Reynolds number definitely do not exceed the critical values [5–8]. Generalized dependences are obtained for the coefficients of friction and heat transfer on the impermeable boundary. A comparison is made between the solutions so obtained and the exact solutions to the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 21–24, January–February, 1987.     

A class of exact solutions for the incompressible viscous magnetohydrodynamic flow over a porous rotating disk

The present paper is concerned with a class of exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous electrically conducting fluid flow due to a porous disk rotating with a constant angular speed.The three-dimensional hydromagnetic equations of motion are treated analytically to obtained exact solutions with the inclusion of suction and injection.The well-known thinning/thickening flow field effect of the suction/injection is better understood from the constructed closed form velocity equations.Making use of this solution,analytical formulas for the angular velocity components as well as for the permeable wall shear stresses are derived.Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation.The temperature field is shown to accord with the dissipation and the Joule heating.As a result,exact formulas are obtained for the temperature field which take different forms corresponding to the condition of suction or injection imposed on the wall.     

Analytic solutions of incompressible laminar flow in the inlet region of a pipe

The laminar analytic solutions of velocities and pressure in the central zone of the inlet region of pipe flow are given under the condition of uniform inflow, based on the Navier-Stokes equations of incompressible viscous flow.