Reynolds equation and applications applied tribology. The general form of reynolds equation mit opencourseware. Understand the use and limitations of the bernoulli equation, and apply it to solve a variety of fluid flow problems. Cope 2 modified the reynolds equation by including viscosity and density variation along the fluid film. Derivation of twodimensional nonnewtonian reynolds equation. Reynolds transport theorem derivation linear momentum. I have been trying to reconcile two different forms of the reynolds transport theorem rtt that i have seen in textbooks. Now, coming back to the reynolds equation, we want to find out the various variations, we derive this equation with a number of assumptions. A necessary concept for the derivation of the conservation of momentum equations is that of the material derivative. If heat transfer is occuring, the ns equations may be coupled to the first law of thermodynamics conservation of energy. The equation is shown to retain the important structures of the counterparts in the conventional second. Recognize various forms of mechanical energy, and work with energy conversion efficiencies. Mod01 lec08 reynolds transport theorem and the equation of continuity.
The reynolds number is the ratio of inertial forces to viscous forces and is a convenient parameter for predicting if a flow condition will be laminar or turbulent. The following is not a rigorous derivation of reynolds number, for that would defeat the purpose for the paper. It then reflects tremendous insight into fluid behavior in bearing lubricant films, which then gave birth to the science of hydrodynamic lubrication. Moreover, in this topic, we will examine a property of fluid flow known as the reynolds number and the reynolds number formula. Two case studies, related to pinondisk and cmp, are presented and results from the derived form of reynolds equation are compared with results from the traditionally used form.
Reynolds transport theorem applied to classical thermodynamics david calamas1, alan hewitt2, john baker3, beth todd 4 abstract the reynolds transport theorem is often used in undergraduate fluid mechanics courses to transform governing equations from a lagrangian to an eulerian coordinate system. View enhanced pdf access article on wiley online library html view download pdf for offline viewing. Hello, reading a text book about the derivation of the reynold s equation i did not understand the following point. That is when the force is acting on the system through incoming mass the system is acting with force to its surroundings through outcoming mass. This does not reduce the equation to dimensionless form, so you have no basis for comparing the magnitudes of the various terms. Reynolds transport theorem and continuity equation 9. Publishers pdf, also known as version of record link back to dtu orbit citation apa. Journal of the brazilian society of mechanical sciences and engineering, 373. Air and liquids are fluids because they move and their movement is flow.
Contrary to 4 where, starting from the full navierstokes equations, we derived rigorously the classical reynolds equation for lubrication of a rotating shaft. A solution method for the reynoldsaveraged navierstokes equation t. The uid is homogeneous, incompressible and newtonian. Reynolds number is a dimensionless value which is applied in fluid mechanics to represent whether the fluid flow in a duct or pat a body is steady or turbulent.
Pdf derivation of the reynolds equation for lubrication. Contrary to 4 where, starting from the full navierstokes equations, we derived rigorously the classical reynolds equation for lubrication of a rotating shaft, in this paper we consider temporal. This renormalization procedure leads to a model reynolds. The validity and accuracy of the compressible reynolds equation in predicting the local pressure in a gaslubricated, textured parallel slider bearing is studied by comparing the local bearing pressure obtained with the reynolds equation and the navierstokes equations for a single dimple textured bearing model. Equation of motion derivation of the equation of motion of the simple pendulum with a linear drag force is trivial, however, we present it here for completeness of the discussion. Since then many works on the solutions of the equation have been published. Deriving the kuttajoukowsky equation and some of its. Reynolds equation is the basis of lubrication theory. Figure 1b shows a simple pendulum with a bob of mass m and a total length l. Derivation of reynolds equation in cylindrical coordinates. Hello, reading a text book about the derivation of the reynolds equation i did not understand the following point.
Reynolds equation is one of the fundamental equations of lubrication theory and tribology. When an object is placed in an initially uniform flow, a portion of the fluid approximate volume l 3 is forced to redirect around the object, causing the flow to accelerate upstream and. A normal derivative is the rate of change of of an intensive property at a point. In addition to the constraints, the continuity equation conservation of mass is frequently required as well. Derivation of reynolds equation cfd online discussion forums. May 10, 2017 reynolds transport theorem derivation linear momentum. A solution method for the reynoldsaveraged navierstokes. Apr 01, 2014 the validity and accuracy of the compressible reynolds equation in predicting the local pressure in a gaslubricated, textured parallel slider bearing is studied by comparing the local bearing pressure obtained with the reynolds equation and the navierstokes equations for a single dimple textured bearing model. Analytical solutions of reynolds equation exist only for certain assumptions. As stated in the introduction to the rans approach, to lead to the closure of the equation system, an approximation for the reynolds. The brinkmanextended darcy equations and stokes equations are utilized to model the flow in the porous region and fluid film region, respectively. Thus, study of hydrodynamic lubrication is from a mathematical. Reynolds transport theorem derivation question physics.
Reynolds equation article about reynolds equation by the. Consider a liquid flowing through a thin film region separated by two closely spaced moving surfaces. Finite difference method to solve reynolds equation youtube. Fast methods to numerically integrate the reynolds. Jun 05, 2009 a form of reynolds equation is derived in this manuscript from the navierstokes equations without entrainment velocity assumptions.
Integration of this equation over the length of the string, l, gives the total torque on the string, 3 2 0 d 3 l s kl d. The instataneous quantities of u, derivation of reynolds equation cfd online discussion forums. Aug 03, 2015 reynolds transport theorem energy bernoulli equation. Reynolds equation is a partial differential equation that describes the flow of a thin lubricant film between two surfaces. Another one is for reynolds transport theorem and its derivation and how to use this eqaution and where are we use. The fluid pressure does not vary across the film thickness and fluid inertia effects are ignored. Comparison of perturbed reynolds equation and cfd models. The navierstokes equations classical mechanics classical mechanics, the father of physics and perhaps of scienti c thought, was initially developed in the 1600s by the famous natural philosophers the codename for physicists of the 17th century such as isaac newton. Lecture tubular laminar flow and hagen poiseuille equation. In 1886, within the wake of a classical experiment by beauchamp tower 7, he developed an equation celebrated as.
The interpretation of inertial to viscous forces comes from looking at the nondimensional equation. Reynolds equation is the governing equation that describes the pressure distribution in a hydrodynamic film. Derivation of reynolds equation in three dimensions. Finite element analysis of the reynolds lubrication. In the simplest case, it considers incompressible and newtonian lubricant. A form of reynolds equation is derived in this manuscript from the navierstokes equations without entrainment velocity assumptions. It directly relates the rate of change of a property within a material volume mv to some integrals of an associated control volume cv. Fast methods to numerically integrate the reynolds equation. The left hand side of the reynolds stress equation can easily be recognized as the rate of change of the reynolds stress following the mean motion. Since the twodimensional nonnewtonian reynolds equation of sliding bearings is important in investigating the lubrication performances of hydrodynamic bearings, a further derivation is useful for engineering application.
Derivation of modified reynolds equationa porous media model. The reynolds number re is an important dimensionless quantity in fluid mechanics used to help predict flow patterns in different fluid flow situations. Trouble with the derivation of the reynolds transport theorem. Reynolds transport theorem energy bernoulli equation. Dec 28, 2019 hagen poiseuille equation derivation pdf from the velocity gradient equation above, and using the empirical velocity gradient limits, an integration can be made to get an expression for the. At low reynolds numbers, flows tend to be dominated by laminar sheetlike flow, while at high reynolds numbers turbulence results from differences in the fluids speed and direction, which may sometimes intersect or even move counter to the. However, in many cases, the newtonian cannot describe lubricant friction. In deriving this equation the thermal, compressibility, viscosity variation, slip at the surface, inertia and surface roughness effects were ignored. The applied load causes the center line of the journal to be displaced from the center line.
Pdf this paper presents a derivation of the empirical colburn analogy and discusses its implications. The demonstration includes isolated bod ies, infinite cascades with application to rotating fluid machines, and pairs of identical or mirrorimage bodies. Since the term only appears due to the reynolds average, it is called reynolds stress or apparent turbulent shear stress. It can be interpreted that when the viscous forces are dominant slow flow, low re they are sufficient enough to keep all the fluid particles in line, then the flow is laminar. The finished article is a perfectly generalised version of reynolds equation. Harish hirani, department of mechanical engineering, iit delhi. This is the socalled reynolds stress equation which has been the primary vehicle for much of the turbulence modelling efforts of the past few decades. This is accomplished with the reynolds transport theorem rtt. Thus, study of hydrodynamic lubrication is from a mathematical point of view is in fact, the study of a particular form of navierstokes equations compatible with the system. The stress jump boundary condition at the porous mediafluid film interface and effects of viscous shear are included in deriving the modified reynolds equation. Efficient solution of the nonlinear reynolds equation for.
Different phases of reynolds equation international journal of. Solution of reynolds equation for the short journal bearings rotation system 119 in totally hydrodynamic or fullfilm lubrication, the moving surface of the journal is totally separated from the bearing surface by a really thin film of lubricants. The reynolds equation is a partial differential equation governing the pressure distribution of thin viscous fluid films in lubrication theory. Pdf derivation of the reynolds equation for lubrication of. Reynolds 1886 first obtained the differential equation governing pressure distribution in a newtonian lubricant film. Navier stoke equation and reynolds transport theorem. The result is a reynolds equation applicable to mhd. The treatment closely follows that in eckert and drake 1959. Fast methods to numerically integrate the reynolds equation for gas fluid films jt florin dimofte national aeronautics and space administration lewis research center cleveland, ohio 445 summary the alternating direction implicit adi method is adopted, modified, and applied to the reynolds equation for thin, gas fluid films. Pdf derivation of the reynolds equation for lubrication of a. For numerical solutions, the variational problem is transformed into a penalty problem and discretized with the nite element method. Damping of a simple pendulum due to drag on its string. This is important, because increased mixing and shearing occur in. Reynolds equation the uid ow between two solid surfaces is modeled with the reynolds equation for a incompressible uid, which is derived from the navierstokes equations using the orderofmagnitude analysis with the following assumptions.
Very near the wall, the fluid motion is smooth and laminar, and molecular conduction and shear are important. Bernoullis equation derivation from eulers equation of. The first form comes from a finite volume method cfd textbook. Reynolds transport theorem derivation question physics forums. Lecture material environmental hydraulic simulation page 66 2. Another tacit assumption in the original derivation of reynolds equation is that surfaces are smooth. To keep the minimum pressure at the cavitation limit, the problem is represented as a variational inequality. Likewise, consistent with these effects and the requirement of the particular bearing problems, it may become necessary to relax few of the assumptions used for derivation of the reynolds equation. Pressures separate surfaces o normal loads on bodies o convergent profile between surfaces o tangential motion between surfaces o viscous effects generate shear stresses o pressures equilibrate shear stresses o surfaces lift apart. The equation is solved with the assumption that there is no lubricant flow in the axial direction infinitely long bearing assumption. Generally, the conventional reynolds equation cannot be. Aug 09, 2011 hi, im struggling to understand one step in the derivation of the reynold s transport theorem.
It should not be confused with osborne reynolds other namesakes, reynolds number and reynoldsaveraged navierstokes equations. There is a direct analogy between the transformation from lagrangian to eulerian descriptions for differential analysis using infinitesimally small fluid elements and the transformation from systems to control volumes for integral analysis using large, finite flow fields. If you inspect the magnitude of the terms, namely the convective or inertial term and the viscous term, the role of the number should be obvious. Derivation of modified reynolds equationa porous media. On the derivation of reynolds equation emerald insight. Bernoulli equation is also useful in the preliminary design stage. Derivation of reynolds equation cfd online discussion. A solution method for the reynolds averaged navierstokes equation t. Efficient solution of the nonlinear reynolds equation for compressible fluid using the finite element method. Also, its value will determine whether the fluid is turbulent or laminar. Reynolds number formula definition, formula and solved examples. Learn more about reynolds number formula and solved examples. A precise study of asymptotic behavior of the newtonian flow in a thin domain between a rotating shaft and lubricated support was given in 4 and the difference between the solution.
Steadystate, laminar flow through a horizontal circular pipe. It then reflects tremendous insight into fluid behavior in. Apply the conservation of mass equation to balance the incoming and outgoing flow rates in a flow system. It may be derived from the full navierstokes ns equations see cameron, 1967 by making simplifying assumptions at appropriate points in the analysis. Pdf in this paper, using the asymptotic expansion, we prove that the reynolds lubrication equation is an approximation of the full navierstokes. Lee mechanical and aerospace engineering, semte, arizona state university, tempe, az, 85287 abstract using the lagrangian transport of momentum, the reynolds stress can be expressed in terms of basic turbulence parameters.
Close form solution of reynolds equation can not be obtained therefore finite elements method is used to solve it. This derivation, employing the continuity equation as obtained from consideration of an elementary cube of the fluid, is probably the most direct from a. Reynolds equation an overview sciencedirect topics. Hi, im struggling to understand one step in the derivation of the reynolds transport theorem. V d q m v d a m a v v d m d the reynolds re number is a quantity which engineers use to estimate if a fluid flow is laminar or turbulent. The accuracy of the compressible reynolds equation for. Chapter 2 is devoted to the derivation of the homogenized equations associated with the stationary forms of the compressible and incompressible. The subject is mainly considered in the limit of incompressible flows with. The lecture presents the derivation of the reynolds equation of classical lubrication theory. In nonideal fluid dynamics, the hagenpoiseuille equation, also known as the the theoretical derivation of a slightly different form of the law was made. Derivation of twodimensional nonnewtonian reynolds. Apr 25, 2016 this is ppt on navier stoke equation,how to derive the navier stoke equation and how to use,advantage. The second integral is nonzero when the mass incoming or outcoming from the control volume has nonzero accelaration. It is derived from the navierstokes equations and is one of the fundamental equations of the classical lubrication theory.
This model is sufficient to describe lubricant friction in some cases relatively low loads and sliding speeds, no temperature rise. Contrary to 4 where, starting from the full navierstokes equations, we derived rigorously the classical reynolds equation for lubrication of a. Introduction to turbulencereynolds averaged equations cfd. There is one way to do that, which ends up being the reynolds number. Modified reynolds equation and analytical analysis of. Consider a stream tube in an onedimensional flow as sketched. This chapter begins by presenting the derivation of the classical reynolds equation based on the work of osborne reynolds. Solution of reynolds equation for the short journal. The governing equation of hydrodynamic lubrication, the reynolds equation, was established in 1886 1.
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