Modified Reynolds Equation for Squeeze-Film Air Damping of Slotted Plates in MEMS Devices

A differential equation for calculating squeeze-film air damping in slotted plates is developed by modifying the Reynolds equation. A term is added to account for the effect of airflow through the slots on the air damping of the plate. The end effect of the airflow in the slots is also treated by substituting an effective channel length for the geometric channel length （i. e. the thickness of the plate）..The damping pressure distribution, damping force, and damping force coefficient of the slotted plates can be found by solving the equation under appropriate boundary conditions. With restrictions on the thickness and the lateral dimensions of the slotted plate removed,the equation provides a useful tool for analysing the squeeze-film air damping effect of slotted plates with finite thickness and finite lateral dimensions. For a typical slotted plate structure, the damping force coefficient obtained by this equation agrees well with that generated by ANSYS.

MODELING THE INTERACTION OF SOLITARY WAVES AND SEMI-CIRCULAR BREAKWATERS BY USING UNSTEADY REYNOLDS EQUATIONS

A vertical 2-D numerical wave model was developed based on unsteady Reynolds equations. In this model, the k-epsilon models were used to close the Reynolds equations, and volume of fluid(VOF) method was used to reconstruct the free surface. The model was verified by experimental data. Then the model was used to simulate solitary wave interaction with submerged, alternative submerged and emerged semi-circular breakwaters. The process of velocity field, pressure field and the wave surface near the breakwaters was obtained. It is found that when the semi-circular breakwater is submerged, a large vortex will be generated at the bottom of the lee side wall of the breakwater; when the still water depth is equal to the radius of the semi-circular breakwater, a pair of large vortices will be generated near the shoreward wall of the semi-circular breakwater due to wave impacting, but the velocity near the bottom of the lee side wall of the breakwater is always relatively small. When the semi-circular breakwater is emerged, and solitary wave cannot overtop it, the solitary wave surface will run up and down secondarily during reflecting from the breakwater. It can be further used to estate the diffusing and transportation of the contamination and transportation of suspended sediment.

A GENERALIZED REYNOLDS EQUATION BASED ON NON-NEWTONIAN FLOW IN LUBRICATION MECHANICS

A generalized Reynolds equation based on non-Newtonian flow is derived in this paper.This equation is suitable for a number of non-Newtonian flow models and can be solved numerically to obtain pressure fields in thermalhydrodynamically or elastohydrodynamically lubricated fluid films.A mathematical ap- proach is given for solving simultaneously the shearing stress,shearing rate,velocity and equivalent viscosity.To show the application of this equation,two rheological models which have been widely used in lubrication mechnaics are incorporated into this equation to obtain numerical solutions to the line contact thermal elastohydrodynamic lubrication problem.

Travelling Waves: Interplay of Low to High Reynolds Number and Tan-Cot Function Method to Solve Burger’s Equations

We study the nonlinear parabolic equations for travelling wave solutions of Burger’s equations. The purpose of the present work is to study various types of Burger’s equations describing waves and those are based on nonlinear equations. We focus on to describe the analytic solution in the special pattern of travelling wave solutions using tan-cot function method. We discuss about inviscid and viscous version of Burger’s equation for fluid flow and investigate the effects of internal friction of a fluid via Reynolds number. By changing the velocity amplitude, the nature of flows with shock wave and disturbance are observed. For numerical solutions, the Crank-Nicolson scheme is introduced to establish the wave solutions.

Water Profile over an Ogee-Shaped Crest and the Reynolds Fractional Equations

Weir crest must have the correct shape in the concave side of an ogee-shaped crest to diminish erosion. This shape can be obtained using an approximation of the fractional Reynolds equations when the water interacts with the surface. A model is introduced for the Reynolds stresses complemented with a closure relation of fractional origin. A power type solution is obtained for the main velocity and stress. Velocity profile is found based on the assumption of a steady flow and the energy conservation equation. A Froude number and the cubic equation of the weir are built. The dimensionless upstream velocity head is also determined which allow graphically showing the exponent and coefficient of the water-profile over an ogee-shaped crest. It is possible to estimate the occupied-space index trough an exponents＇ ratio of profile over the velocity head.

Reynolds方程在纹理表面动压润滑计算中的有效性评价

从理论上阐述了纹理表面动压润滑计算中决定Reynolds方程有效性的两个关键因素为油膜厚度与纹理特征长度的比值h/L和缩减的雷诺数re;只有当h/L和re同时趋近于零时Reynolds方程才能够获得准确的结果,并由此在h/L-Re平面上标注了Reynolds方程的适用范围.继而以二维矩形沟槽为实例,采用数值方法计算了h/L和re对Reynolds方程误差的影响规律;分析了Reynolds方程在不同条件下的失效机制;分析了矩形沟槽纹理表面Reynolds方程有效性的评价标准:当缩减的雷诺数re小于0.20,并且h/L小于0.015时能够保证Reynolds方程的误差在10%以下.

Reynolds控制方程下粗糙节理渗流空腔模型研究

在研究节理的渗流时,渗流控制方程对节理渗流分析结果具有显著影响。首先介绍了节理渗流分析中的控制方程:Navier-Stokes方程、Stokes方程、Reynolds方程和立定定理,并分析了各控制方程在节理渗流分析中的适用性。以Reynolds方程作为渗流分析控制方程,建立了粗糙节理渗流空腔模型。然后以节理试件为研究对象,在实测节理三维表面形貌并计算隙宽分布后,分别进行相同渗流边界条件下的室内渗流试验和空腔模型计算,得到节理在不同接触状态下的渗流量实测值和计算值,并分别将计算结果与立方定理下的空腔模型、将整个节理简化为光滑平行板模型的立方定理以及速宝玉经验公式的计算结果进行比较,结果表明,Reynolds方程下的节理渗流空腔模型计算结果与实测值最为吻合,可以较为准确地反映节理的渗流情况。同时,根据Reynolds方程下空腔模型得到的渗流流量分布可以呈现节理渗流的曲折现象,为从本质上研究节理渗流特性奠定了理论基础。

润滑力学中非牛顿流动的普遍Reynolds方程

本文导出了润滑力学中关于非牛顿流动的普遍 Reynolds 方程。这一方程适用于多种非牛顿流动模型,可以用于解算热流体动力润滑或热弹性流体动力润滑膜的压力分布。本文给出了一种同时求出剪应力、剪切率、速度和等效粘度的解法,并以两种润滑力学中常用的流变模型为例,应用这一方程得到了线接触热弹性流体动力润滑问题的数值解。

考虑气体稀薄效应修正Reynolds方程的新模型及其数值分析

硬盘中磁头和磁盘之间的气膜厚度已接近或小于分子平均自由路径,为了更精确地模拟气膜的压力分布,Reynolds方程必须要考虑气体稀薄效应的影响。基于硬盘中常用的修正Reynolds方程——FK模型,提出一种修正Reynolds方程的新模型,并且通过最小二乘有限差分法求解该Reynolds方程。分别利用FK模型和新模型,研究气膜数对承载力和压力中心的影响,并比较2种模型的计算时间。结果表明,2种模型所得的数值模拟结果具有很好的一致性,但新模型的计算效率高于FK模型。

An iterative data-driven turbulence modeling framework based on Reynolds stress representation

Data-driven turbulence modeling studies have reached such a stage that the basic framework is settled,but several essential issues remain that strongly affect the performance.Two problems are studied in the current research:(1)the processing of the Reynolds stress tensor and(2)the coupling method between the machine learning model and flow solver.For the Reynolds stress processing issue,we perform the theoretical derivation to extend the relevant tensor arguments of Reynolds stress.Then,the tensor representation theorem is employed to give the complete irreducible invariants and integrity basis.An adaptive regularization term is employed to enhance the representation performance.For the coupling issue,an iterative coupling framework with consistent convergence is proposed and then applied to a canonical separated flow.The results have high consistency with the direct numerical simulation true values,which proves the validity of the current approach.

Local projection stabilized finite element method for Navier-Stokes equations

This paper extends the results of Matthies, Skrzypacz, and Tubiska for the Oseen problem to the Navier-Stokes problem. For the stationary incompressible Navier- Stokes equations, a local projection stabilized finite element scheme is proposed. The scheme overcomes convection domination and improves the restrictive inf-sup condition. It not only is a two-level approach but also is adaptive for pairs of spaces defined on the same mesh. Using the approximation and projection spaces defined on the same mesh, the scheme leads to much more compact stencils than other two-level approaches. On the same mesh, besides the class of local projection stabilization by enriching the approximation spaces, two new classes of local projection stabilization of the approximation spaces are derived, which do not need to be enriched by bubble functions. Based on a special interpolation, the stability and optimal prior error estimates are shown. Numerical results agree with some benchmark solutions and theoretical analysis very well.

A two-grid algorithm based on Newton iteration for the stream function form of the Navier-Stokes equations

In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinear system on the coarse mesh space and two similar linear systems （with same stiffness matrix but different right-hand side） on the fine mesh space. The convergence analysis and error estimation of the algorithm are given for the case of conforming elements. Furthermore, the Mgorithm produces a numerical solution with the optimal asymptotic H^2-error. Finally, we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the Navier-Stokes equations.

A new full discrete stabilized viscosity method for transient Navier-Stokes equations

A new full discrete stabilized viscosity method for the transient Navier-Stokes equations with the high Reynolds number （small viscosity coefficient） is proposed based on the pressure projection and the extrapolated trapezoidal rule. The transient Navier-Stokes equations are fully discretized by the continuous equal-order finite elements in space and the reduced Crank-Nicolson scheme in time. The new stabilized method is stable and has many attractive properties. First, the system is stable for the equal-order combination of discrete continuous velocity and pressure spaces because of adding a pres- sure projection term. Second, the artifical viscosity parameter is added to the viscosity coefficient as a stability factor, so the system is antidiffusive. Finally, the method requires only the solution to a linear system at every time step. Stability and convergence of the method is proved. The error estimation results show that the method has a second-order accuracy, and the constant in the estimation is independent of the viscosity coefficient. The numerical results are given, which demonstrate the advantages of the method presented.

Finite difference streamline diffusion method using nonconforming space for incompressible time-dependent Navier-Stokes equations

This paper proposes a new nonconforming finite difference streamline diffusion method to solve incompressible time-dependent Navier-Stokes equations with a high Reynolds number. The backwards difference in time and the Crouzeix-Raviart （CR） element combined with the P0 element in space are used. The result shows that this scheme has good stabilities and error estimates independent of the viscosity coefficient.

New Dynamic Equations of Aerosol in Air of Certain Type

A new dynamic equation of aerosol in air is derived, using a model-in-model, by equilibrium of buoyancy, gravity and pressure, together with conservation laws of mass, momentum and energy via Reynolds transport theorem and supplemented by corresponding scientific laws for related properties of air and aerosols. This new dynamic equation of aerosol in air is a set of non-linear partial differential equations involved six unknown functions of mass densities, pressure, air and aerosol speeds and temperature. It has features: 1, it belongs to certain type;2, it emphases the effect of buoyancy in equilibrium and potential energy, and the Archimedes principle of buoyancy is firstly extended to lateral directions based on logical deduction, the phenomenon of stirring a glass of oil-water mixture and the recorded of Hurricane Isabel (2003) from space station. The later shows the evidence of existence of lateral buoyancy;3, the mass densities of air and aerosol of a point in our model are varied in different directions due to traction and are treated as vectors, and they have been used in the calculation of lateral buoyancy.

Derivation of a second-order model for Reynolds stress using renormalization group analysis and the two-scale expansion technique

With the two-scale expansion technique proposed by Yoshizawa,the turbulent fluctuating field is expanded around the isotropic field.At a low-order two-scale expansion,applying the mode coupling approximation in the Yakhot-Orszag renormalization group method to analyze the fluctuating field,the Reynolds-average terms in the Reynolds stress transport equation,such as the convective term,the pressure-gradient-velocity correlation term and the dissipation term,are modeled.Two numerical examples：turbulent flow past a backward-facing step and the fully developed flow in a rotating channel,are presented for testing the efficiency of the proposed second-order model.For these two numerical examples,the proposed model performs as well as the Gibson-Launder （GL） model,giving better prediction than the standard k-ε model,especially in the abilities to calculate the secondary flow in the backward-facing step flow and to capture the asymmetric turbulent structure caused by frame rotation.

An approach for choosing discretization schemes and grid size based on the convection-diffusion equation

A new approach for selecting proper discretization schemes and grid size is presented. This method is based on the convection-diffusion equation and can provide insight for the Navier-Stokes equation. The approach mainly addresses two aspects, i.e., the practical accuracy of diffusion term discretization and the behavior of high wavenum- ber disturbances. Two criteria are included in this approach. First, numerical diffusion should not affect the theoretical diffusion accuracy near the length scales of interest. This is achieved by requiring numerical diffusion to be smaller than the diffusion discretization error. Second, high wavenumber modes that are.much smaller than the length scales of interest should not be amplified. These two criteria provide a range of suitable scheme combinations for convective flux and diffusive flux and an ideal interval for grid spacing. The effects of time discretization on these criteria are briefly discussed.

Developed Numerical Simulation of Falling and Moving Objects in Viscous Fluids under the Action of a Reynolds Lubrication Theory and Low Reynolds Numbers

The development work focuses on the numerical simulations of free body movement in viscous fluid. The aim is to make the simulation of very slow motion of the small body in viscous fluid. We developed bodies’ immersed dynamics simulations in viscous fluid by seeking numerical solutions for appropriate field variables. We developed the methods for vertically and spherically cylindrical objects’ motions, the forces on bodies close to a plane stationary wall are computed from the velocity and pressure fields using the Stokes equation through COMSOL Multiphysics finite element software. The Navier-Stokes equation is reduced to Stokes equation there is independence of time which means object will have an effect only on the motion and the slightly compressible flow assumption is made in order to obtain smooth solution numerically. The forces on an object in slightly compressible Stokes flow have been exerted on the falling objects. The resulting forces have compared with analytical results from the Reynolds Lubrication Theory, and achieved significant results from the development method in Matlab and achieved significant numerical simulations in COMSOL. In addition, an investigation has been made to an object swimming at low Reynolds number. At low Reynolds number moving is possible when object scale is small and flow pattern is slow and sticky. We have developed a system for a thin two-dimensional (2D) worm-like object wiggle that is passing a wave along its centreline and its motion has simulated by the Ordinary Differential Equations (ODE) system and by the Arbitrary Lagrangian-Eulerian (ALE) moving mesh technology. The development method result shows that it is possible for the small object to have a motion from one position to another through small amplitudes and wavelengths in viscous fluid.

Simulation of Non-Isothermal Turbulent Flows Through Circular Rings of Steel

This article is intended to examine the fluid flow patterns and heat transfer in a rectangular channel embedded with three semi-circular cylinders comprised of steel at the boundaries.Such an organization is used to generate the heat exchangers with tube and shell because of the production of more turbulence due to zigzag path which is in favor of rapid heat transformation.Because of little maintenance,the heat exchanger of such type is extensively used.Here,we generate simulation of flow and heat transfer using nonisothermal flow interface in the Comsol multiphysics 5.4 which executes the Reynolds averaged Navier stokes equation(RANS)model of the turbulent flow together with heat equation.Simulation is tested with Prandtl number(Pr=0.7)with inlet velocity magnitude in the range from 1 to 2 m/sec which generates the Reynolds number in the range of 2.2×10^(5) to 4.4×10^(5) with turbulence kinetic energy and the dissipation rate in ranges(3.75×10^(−3) to 1.5×10^(−2))and(3.73×10^(−3)−3×10^(−2))respectively.Two correlations available in the literature are used in order to check validity.The results are displayed through streamlines,surface plots,contour plots,isothermal lines,and graphs.It is concluded that by retaining such an arrangement a quick distribution of the temperature over the domain can be seen and also the velocity magnitude is increasing from 333.15%to a maximum of 514%.The temperature at the middle shows the consistency in value but declines immediately at the end.This process becomes faster with the decrease in inlet velocity magnitude.

Multidimensional Stability of Subsonic Phase Transitions in a Non-Isothermal Van Der Waals Fluid

We show the multidimensional stability of subsonic phase transitions in a non-isothermal van der Waals fluid. Based on the existence result of planar waves in our previous work [1], a jump condition is posed on non-isothermal phase boundaries which makes the argument possible. Stability of planar waves both in one dimensional and multidi-mensional spaces are proved.