Turbulence Prof. E.G. Tulapurkara Chapter-3 Chapter 3 Equations for turbulent flows (Lectures 16, 17, 18 and 19) Keywords: Deviation of Reynolds averaged Navier-Stokes (RANS) equation; equation for Reynolds stress, kinetic energy of mean and turbulent motions; boundary layer equations for turbulent flow; momentum integral equation. Topics Study of Appendix ‘C’ on self study basis 3.1 Reynolds averaged Navier-Stokes (RANS) equations 3.2 Equations for Reynolds stressesρui u j 3.3 Equations for kinetic energy of mean and turbulent motion 3.4 Energy transfer in laminar and turbulent flows 3.5 Boundary layer equations for turbulent flow 3.6. Momentum integral equation for turbulent boundary layer 3.7 Reynolds Average equations for compressible flow 3.7.1 Averaging procedures 3.7.2 Reynolds form of continuity equation for compressible flow 3.7.3 Reynolds form of momentum equations 3.7.4 Reynolds form of energy equation References Exercises Dept. of Aerospace Engg., Indian Institute of Technology, Madras 1 Turbulence Prof. E.G. Tulapurkara Chapter-3 Appendix - C Basic concepts and equations of fluid dynamics ( Material for self study ) C.1 Introduction The turbulent flows are governed by the Navier-Stoke (N-S) equations. In the approach called “Direct Numerical Simulation (DNS)” the three-dimensional time dependent N-S equations are solved using a very fine grid. In another approach, the flow variables are expressed as sum of the time averaged value plus the fluctuating part e.g. U=U+u ,V=V+u etc. This is known as Reynolds decomposition. These are substituted in the N-S equations and time average is taken. The resulting equations involve unknown correlation. Models of turbulence are needed to make the equations a closed set. To appreciate both these approaches the knowledge of the derivation of the N-S equation is required. Further, derivation of N-S; equations presupposes many basic concepts. Hence, the background material and the equations of fluid flow are recapitulated in this Appendix. The next section begins with explanation of the basic concepts. Then, the kinematics and the laws of fluid motion, which lead to the equations of motions, are discussed. After deriving the equations, their representations in different forms and special cases are dealt with. The aim of this Appendix is to clarify the basic concepts and the equations needed for study of chapter 3 of the main text. Students familiar with the material in this appendix, can skip it or revert to it in case of doubts during the study of chapter 3. Dept. of Aerospace Engg., Indian Institute of Technology, Madras 2 Turbulence Prof. E.G. Tulapurkara Chapter-3 C.2 Basic concepts and definitions A fluid is considered as an isotropic substance the individual particles of which continue to deform under the influence of applied surface stresses. The deformations imply changes in shape and size. There is no shear stress in a fluid at rest. Fluids comprise both liquids and gases. C.2.1 Continuum model of a fluid A fluid consists of a large number of molecules, each of which has a certain position, velocity, and energy which vary as a result of collision with other molecules. However, in substantial part of fluid mechanics, one is not interested in the motion of individual molecules but their average behavior i.e. distribution of physical quantities like pressure, density, temperature etc. as functions of position and time. In this context when one speaks of the value of a physical quantity at a point, it implies an average value over a small region of volume v* around the point. A typical length scale of the region would be very small on macroscopic level but is large compared to molecular dimensions and hence contains a great number of molecules. Thus, the number of molecules entering or leaving the region does not significantly change the value of the physical variable. It may be pointed out that air, at normal temperature and pressure, contains 2.7x1019 molecules per cubic centimeter; a cube of side 1/1000thmm, would contain 2.7x107 molecules. Further, mean free path is of the order of 8x10-8m and the number of molecules in a cube, the side length of which is one mean free path, is 15000. Density of this cube fluctuates only by 0.8% on the average. When the fluid (or solid) is treated as a continuous distribution of matter, it is called continuum and the analysis is called continuum mechanics. If fluid is not treated as a continuum, then the terms like temperature, density etc. at a point, would loose their meaning. Remarks: (i) To decide as to when to treat a fluid as continuum, one uses Knudsen number (Kn) defined as: Kn = /L where, is the mean free path and L is characteristic length of the flow. For fluid to be treated as continuum, Kn <<1. (ii) The random motion of molecules causes, over a period of time, exchange of mass, momentum and heat. These phenomenons cannot be treated by continuum Dept. of Aerospace Engg., Indian Institute of Technology, Madras 3 Turbulence Prof. E.G. Tulapurkara Chapter-3 assumption. However, the exchange of mass, momentum and heat appear as coefficients of diffusivity, viscosity and thermal conductivity in continuum treatment. C.2.2 Fluid particle The smallest lump of fluid having sufficient number of molecules to permit continuum interpretation on a statistical basis, is called a fluid particle. The average properties of the fluid particle are, in the limit, assigned to a point, thus making possible a field representation of properties. For example, the field of a property “b” can be described by an equation of the form: b = br,t or b = (x, y, z, t) ; bold letters indicate a vector (C.1) In fluid dynamics one deals with scalar fields (e.g. density), vector fields (e.g. velocity) and tensor fields (e.g. stress tensor). C.2.3 Stress at point Consider an area A* lying in some plane through the point P and including the point P. The dimensions of A* correspond to the dimensions of fluid particle having the volume v*. The fluid on the two sides of the surface A* appear, on the macroscopic scale, to exert equal and opposite force, F*. The ratio F*/A* is called the surface stress at point P. The surface stress at a point may be resolved into a normal component and a tangential (shear) component. Moreover, there will be different surface stresses at P for each different orientation of the plane. Accordingly, the state of stress at a point is characterized by nine Cartesian components. Furthermore, these nine quantities obey the transformation laws of a tensor. The stress tensor is represented by : ζxx xy xz yxζ yy yz (C.2) zx zyζ zz in which xx is the normal stress acting on a face normal to x-axis (Fig. C.1); xy is a shear stress acting in the y-direction on a face normal to x. The various stresses are positive, when they have the directions as shown in Fig.C.1. Dept. of Aerospace Engg., Indian Institute of Technology, Madras 4 Turbulence Prof. E.G. Tulapurkara Chapter-3 Fig.C.1 State of stress It can be shown, from considerations of angular momentum, that the stress tensor is symmetric i.e. xy = yx etc. Reference C.1, chapter 3 be referred to for the proof. Thus six, rather than nine, quantities suffice to determine the state of stress at a point. In a fluid at rest, all shear stresses vanish and then it can be shown, from equilibrium consideration, that the normal stress at a point is the same in all directions. The stress tensor then reduces to : -p 0 0 0 -p 0 (C.3) 0 0 -p where, p is the hydrostatic pressure which is same in all directions. C.3. Kinematics In kinematics the motion of fluid particles is studied without considering the forces that cause the motion. Kinematics of fluids is more complicated than that of rigid bodies because the distance between two fluid particles does not remain the same during the motion of fluid. However, there is the constraint that no two particles can occupy the same position at the same time. C.3.1 Steady and unsteady flows When the fluid properties at a given position in space vary with time, the flow is said to be unsteady. Sometimes, the fluid properties at any fixed position in space do not change with time as successive fluid particles come to occupy the point. The flow is Dept. of Aerospace Engg., Indian Institute of Technology, Madras 5 Turbulence Prof. E.G. Tulapurkara Chapter-3 then described as steady. The description of the flow field, i.e. Eq.(C.1), then takes the simpler form as : b = b(r) or b = b(x, y, z) (C.4) Remark: If a reference frame „A‟ moves with a constant velocity with reference to another frame „B‟, then the acceleration of the fluid particle is the same in both frames. The dynamic laws are identical in the two reference frames. When a body moves with a constant velocity through a stationary infinite fluid, the flow appears as unsteady to an observer in a reference frame which is attached to the fluid. However, with respect to a reference frame attached to the moving body, the flow would appear steady. In other words, the force acting on a body is same whether (a) the body moves with a uniform velocity in a fluid at rest or (b) the fluid moves with a uniform velocity past a body at rest.