(Lectures 16, 17, 18 and 19)

Transcription

1 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. Equations for Reynolds stresses ρui 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 Averaging procedures 3.7. Reynolds form of continuity equation for compressible flow Reynolds form of momentum equations Reynolds form of energy equation References Exercises Dept. of Aerospace Engg., Indian Institute of Technology, Madras 1

2 Appendix - C Basic concepts and equations of fluid dynamics C.1 Introduction ( Material for self study ) 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 + uetc. 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

3 C. 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..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.7x10 19 molecules per cubic centimeter; a cube of side 1/1000 th mm, would contain.7x10 7 molecules. Further, mean free path is of the order of 8x10-8 m and the number of molecules in a cube, the side length of which is one mean free path, is 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 (K n ) defined as: K n = /L where, is the mean free path and L is characteristic length of the flow. For fluid to be treated as continuum, K n <<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

4 assumption. However, the exchange of mass, momentum and heat appear as coefficients of diffusivity, viscosity and thermal conductivity in continuum treatment. C.. 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..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 : ζ xx xy xz ζ yx yy yz ζ zx zy zz (C.) 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

5 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 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

6 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. Hence, in wind tunnel testing or theoretical study in fluid mechanics, the body is kept/assumed at rest and the fluid moves with a uniform velocity past the body. Whereas, in actual practice an airplane moves with a uniform velocity in air at rest. Thus, the forces acting on the airplane moving in air, can be obtained by studying the airplane kept at rest in a wind tunnel. Similarly, the force acting on a submarine moving in water can be obtained by keeping the submarine at rest in a water tunnel. C.3. Description of fluid motion There are two ways of mathematically describing fluid motion viz. Lagrangian and Eulerian methods. C.3..1 Lagrangian method In this method the trajectories of various fluid particles are described. For this purpose a fluid particle is identified by its position r o (or x o, y o, z o,) at time t o. At subsequent instants of time the positions of the same particle are given by: r = r r 0,t Or x = x(x, y, z, t), y = y(x, y, z, t), z = z(x, y, z, t) o o o o o o o o o (C.5) The line along which a particle moves is called a path line. The instantaneous Dept. of Aerospace Engg., Indian Institute of Technology, Madras 6

7 velocity (V) and the acceleration (a) of the particle can be expressed as : V = r / t = V r,t (C.6) r =const. o / t o a = V = r / t = a r,t (C.7) r =const. o r =const. o o The Lagrangian description is found to be less convenient, for mathematical manipulation and the Eulerian method, which is described next, is commonly used. C.3.. Eulerian method In this method the flow properties are described as functions of space and time coordinates. Thus, if b is a flow property, then b (x, y, z, t) or b (r,t) is the value of b when a particle occupies the position r at time t. At a later time, the fluid particle occupying the position (x, y, z) will be different. If the flow is steady then b is a function of x, y and z only. This method of describing fluid motion may be called as cinematographic method, i.e. the complete state of motion is described by a succession of instantaneous states of flow. Considering velocity as a fluid property, it can be written as : V = V r,t or V = V x,y,z,t and U = U x,y,z,t, V = V x,y,z,t,w = W x,y,z,t where U, V and W are the components of velocity V along x, y and z directions. (C.8) Streamline is an important curve in this method. It is an imaginary line drawn in the flow field such that the tangent to it (streamline) at any point P is along the velocity vector of the fluid at that instant of time. Since, an element of length dr along a streamline is tangent to the local velocity, V the equation of a streamline is : V d r = 0 or dx:dy :dz = U: V : W (C. 9) Dept. of Aerospace Engg., Indian Institute of Technology, Madras 7

8 Fig.C. Streamline Remark: Since, the fluid cannot cross a streamline, the mass of fluid passing between a reference point and any point on the streamline is same. This quantity iis denoted by and called stream function. is constant along a stream line. It can be shown that U = ψ y and ψ V = - x C.3..3 Streamtube Consider a simple closed curve lying in the fluid. The streamlines passing through the points on this curve generate a tubular surface called streamtube. Since, there is no component of velocity normal to the surface of the streamtube, the tube is impervious to the fluid. A streamtube of infinitesimal cross section is called a stream filament. Fig.C.3 Streamtube Dept. of Aerospace Engg., Indian Institute of Technology, Madras 8

9 Remarks : (i) A streak line is the locus, at a given instant of time, of all fluid particles that have passed through a fixed point in the fluid. When the motion is steady the streamline, path line and streak line coincide with each other. (ii) In the subsequent discussion the Eulerian method to describe fluid motion is used. C.3..4 Substantial derivative It must be emphasized that, in Eulerian method of describing the fluid motion, the value of the fluid property (e.g. velocity, temperature etc.) at a point is the value of that property of a fluid particle occupying the chosen location. As the time changes, the particles which occupy the chosen point are different. Hence, the time rate of change of the property at a point is not the rate of change of the property of a particle of fixed identity. However, many situation require the rate of change of property of a particle of fixed identity. This rate of change is obtained in the following manner. Consider that at time t a fluid particle occupies a position P. The location of point P is given by r = x i +y j +zk. The property b of the fluid particle has the value b (x, y,z, t). In a small interval of time, t, the same fluid particle moves to a position Q with coordinates r+δ r = x+δx i + y+δy j+ z+δz k. At this point, (i.e. at Q) the fluid property has the value b (x+x, y+y, z+z). Expanding in Taylor series yields: b b b b b r +Δ r,t+δt = b r,t + Δx+ Δy+ Δz+ Δt x y z t Since, a particle of fixed identity is being considered, the displacements x, y and z are not arbitrary. They are actually the distances traveled by the particle in time t along x, y and z directions i.e. x = U t, y = V t, and z = W t. b b b b +Δ,t+Δt -b,t = U +V + W + Δt x y z t Hence, b = br r r Hence, the rate of change of b, for a particle of fixed identity, which is generally denoted by Db/Dt, is : Db = Lt Dt Δt 0 r r r b +Δ,t+Δt -b,t Δt Dept. of Aerospace Engg., Indian Institute of Technology, Madras 9

10 Or Db b = +U b +V b +W b Dt t x y z D = +U +V + W = + Dt t x y z t Or.V (C.10) (C.11) Remark: The derivative D/Dt is called substantial derivative, material derivative or particle derivative. The first term on r.h.s. of Eq.(C.11), i.e. / t is called the local derivative which indicates the unsteady time variation of fluid property at a point. The sum of the last three terms i.e. U + V W x y z is called convective derivative. Since, it indicates the change of property as the particle is convected by the flow. C.3..5 Acceleration of a fluid particle Choosing the velocity (V) as one of the properties, the expression for the rate of change of velocity of a fixed particle i.e. its acceleration (a) can be expressed as: D V V = a = + Dt t.v V (C.1) Or D V V V V V = +U +V +W Dt t x y z (C.13) Noting that a = a x i + a y j + a z k and V = U i + V j + W k the components of the acceleration of a particle occupying a position x, y, z at time t are : DU U U U U a x = = +U + V +W dt t x y z DV V V V V a y = = +U + V +W dt t x y z Dept. of Aerospace Engg., Indian Institute of Technology, Madras 10

11 DW W W W W a z = = +U + V +W dt t x y z Remark: The term.v V can be expressed as - VV V V t Hence, a = + - VV V ; V = U +V +W (C.14) (C.14a) Example C.1 Consider the flow represented by a two-dimensional source wherein the fluid is coming out from points along a line at the mass flow rate of m per unit length. In this case: m=ρπr q per unit length, ρ = fluid density, r = r = x + y ; q = radial velocity = U +V = m/ρπr or q = k / r ; k = m/ πρ. Hence, U = kx / (x + y ); V = ky / (x + y ). Substituting expression for U and V in the set of equations (C.14), gives : -k x x a = Hence, x +y ; -k y y a = x +y - k x i+ y j -k r a = a xi+a y j = = r 4 x + y It is noted that the acceleration is in the radial direction. The negative sign indicates that the flow is decelerating as r increases. C.3..6 Transport and deformation of a fluid particle The normal and shear stresses produced due to the motion of the fluid depend on the rate at which a fluid particle is strained. The rates of strain, when the velocity field is given in Eulerian manner, can be obtained as follows. Dept. of Aerospace Engg., Indian Institute of Technology, Madras 11

12 Consider a particle with the shape of a parallelepiped of sides x, y, and z. Let a corner A have the coordinates x, y, z and the opposite corner B have the coordinates (x + x), (y + y) and (z + z). Further, at time t the velocity components at A be U, V, W and at B be (U + U), (V + V) and (W + W). Then, the motion of B relative to A can be given by Taylor series as: U U U U+ΔU = U+ Δx+ Δy+ Δz x y z V V V V+ΔV = V+ Δx+ Δy+ Δz x y z W W W W+ΔW = W + Δx+ Δy+ Δz x y z Or U U U ΔU = Δx+ Δy+ Δz x y z V V V ΔV = Δx+ Δy+ Δz x y z W W W ΔW = Δx+ Δy+ Δz x y z (C.15) The expressions for U, V and W can be rewritten in the following form : 1 1 ΔU = εxx Δx+εxy Δy+εxz Δz + ηδz- ζ Δy 1 1 ΔV = εxy Δx+εyy Δy+εyz Δz + ζ Δx - ξ Δz C ΔW = εxz Δx+ε yz Δy+εzz Δz + ξ Δy - ηδx where, U V W x y z ε xx = ; ε yy = ; ε zz = (C.17) Dept. of Aerospace Engg., Indian Institute of Technology, Madras 1

13 1 V U 1 W V 1 U W ε xy = + ; ε yz = + ; ε xz = + x y y z z x (C.18) W V U W V U ξ = - ; η = - ; ζ = - y z z x x y (C.19) Thus, the overall motion of a particle occupying a position x, y, z at time t is composed of: (i) pure translation with velocity components U, V and W (ii) rigid body rotation with components of angular velocity ξ, η and ς (iii) dilatation given by the linear strain rates ε xx,ε yy, εzz and (iv) distortion of shape given by the rates of shear strain ε x y,ε yz,εxz. Remarks: i) ε x x,ε y y,ε zz represent extensional rates of strain as can be seen from the following. Let, the flow be such that U xis positive and other derivatives are zero. Then, in a time interval t the point A moves to A and point B moves to B. (Fig.C.4) Fig.C.4 Extensional strain Now, from Fig.C.4, AC = x, DB = x, AD = CB = y A C = D B = x + U+ U/ x Δx Δt -UΔt = Δx+ U/ x Δx Δt But, A D = C B = y. Thus, there is a dilatation of the element in the x-direction. The strain is: Dept. of Aerospace Engg., Indian Institute of Technology, Madras 13

14 Δx+ U/ x Δx Δt-Δx / Δx = U/ x Δt Hence, the rate of strain in the x-direction is U Δt U Lt Δt 0 = = ε x Δt x xx V W Similarly, it can be shown that ε y y = and ε zz = y z The rate of volumetric strain is e= Lt Δt 0 U V W Δx+ Δx ΔtΔy+ Δy ΔtΔz+ Δz Δt- Δx Δy Δz x y z Δx Δy Δz Δt Or U V W e = + + =.V. (C.0) x y z (i) As an example of shear strain, consider that U/ y is positive and other derivatives are zero. Then, in an interval of time t, points A and B of the fluid particle would move to A and B as shown in Fig.C.5. Fig.C.5 Shear strain AC = DB = x, AD = CB = y, A C = x, D B = x+ U/ y Δy Δt. Dept. of Aerospace Engg., Indian Institute of Technology, Madras 14

15 Thus, rotation of the vertical line which is also the shear strain is U/ y Δy Δt /Δy = U/ y Δt. U Δt U Hence, rate of shear strain = Lt Δt0 = y Δt y It may be noted that when the orientation of the diagonal AB of the element is considered it is notice that the element has also undergone a rigid body rotation 1 U / y Δt in the clockwise sense. Similarly, when V / x is alone non-zero, then the shear strain would be V / x. When both U / y and V / x are non-zero the rate of shear strain ε xy would be U V + y x. (iii) When U =- V, the rate of shear strain is zero but the particle has a rate of rotation y x U V equal to ½ -. It may be added that the rate of rotation is taken positive in y x counter clock-wise direction. When U/ y -V / x the particle has a rate of shear strain of U/ y+ V / x and rate of rotation of ½V / y- U/ x. (iv) It may be noted, that the translation and rotation occur without change of shape of the fluid particle. The deformations are given by ε 's. The rate of strain tensor is : ε ε ε ε ε ε ε ε ε xx xy xz xy yy yz xz yz zz (C.1) It may be noted that the rate of strain tensor is symmetric i.e. ε ij = ε ji. The nine quantities involved in the deformations depend on the orientation of the reference axes x,y,z but in a change of axes they follow the transformation laws of tensors. Dept. of Aerospace Engg., Indian Institute of Technology, Madras 15

16 C.3..7 Vorticity Vorticity is a vector denoted by ω and defined as ω = V (C.) If ω = ξ i+η j+ς k, then, (C.3) W V U W V U ξ = - ; η = - ; and ζ = - y z z x x y (C.4) Remarks: (i) Vorticity is twice the angular velocity of the fluid particle. (ii) A flow field in which voriticity is zero is called irrotational flow, i.e. = = =0. (C.5) Further, In such a flow a velocity potential () exists and the flow is called potential flow. The velocity components in this case, can be expressed as: U = - / x ; V = - / y ; W = - / z (C.6) (iii) Vortex line: It is a curve lying in the fluid such that its tangent at any point P gives the direction of vorticity at P at the instant considered. The equation of a vortex line is dx : dy : dz = : :. (C.7) (iv) Vortex tube: Consider a simple closed curve lying in the fluid. Vortex lines through the points on this curve, all drawn at the same instant of time, generate a tubular surface called vortex tube. A vortex tube is also referred to as a vortex. A vortex tube whose cross-section is every where small is called a vortex filament. (v) Vortex sheet is a surface composed of vortex lines. Example C. In a plane couette flow, the flow takes place between two plates separated by a distance h. The bottom plate is stationary and the top plate moves with velocity U e. Then, U = Ue y/h ; V = 0. In this case, shear strain is U e /h and vorticity is U e /h. Dept. of Aerospace Engg., Indian Institute of Technology, Madras 16

17 C.3..8 Circulation The instantaneous line integral of the tangential velocity around any closed curve C is called circulation. It is denoted by and is taken positive when, C is traversed such that the area enclosed by C lies to the left (Fig.C.6). Γ= V.ds C = Udx+V dy+w dz (C.8) C Remarks: Fig.C.6 Circulation (i) It can be shown that the circulation around any curve C, bounding an area A (singly or multiply connected), is the sum of the circulations around all the lesser areas into which the area A, might arbitrarily be divided. (ii) Stokes theorem: Considering an elemental area dxdy in X-Y plane, it can be shown that the circulation around this circuit, Γ is V / x- U/ y dxdy. Generalising this, the relation between vorticity and circulation, known as Stokes theorem, is given by: Γ = V.ds = V.n da. (C.9) C A Where, (a) C is a space curve and A is the area of a surface which has no edge other than C and (b) n is the normal to the elemental area da, positive when pointing outwards from the enclosed volume. Further, V should be continuously differentiable in the area A, and A should be a simply connected region. Thus, circulation around C is equal to the flux of vorticity through the bounded area A. Dept. of Aerospace Engg., Indian Institute of Technology, Madras 17

18 (iii) Strength of a vortex tube is the circulation along a circuit lying on the surface of the vortex tube and passing round it just once. (iv) Helmhotz s vortex theorems: These theorems can be proved using Stokes theorem. Reference C., chapter 3 can be referred to for proofs (a) Strength of a vortex is same throughout the length of the vortex (b) A vortex cannot have an end within the fluid i.e. vortex filament either forms a closed curve or extends to the fluid boundaries. C.3..9 Accelerating reference frame Consider a moving reference frame whose origin accelerates at the rate a o with reference to a fixed frame and which also rotates with an angular velocity Ω relative to the fixed frame. Let V, as before, represent the velocity in the fixed frame, while W refers to velocity in the moving frame. Then, the acceleration relative to the fixed frame is: where, DW dω a = + ao + Ω W + Ω Ω r+ r Dt dt (C.30) D W /Dt is acceleration perceived by an observer in the accelerating frame. The term Ω Wis called Coriolis acceleration and the term Ω Ω r is called centrifugal acceleration. C.4 Forces acting on a fluid particle The forces acting on a fluid particle are body forces, line forces and surface forces. The body forces are proportional to the mass or the volume of the fluid particle. Body force will be denoted by F with components X, Y and Z. i.e. F = X i+ Y j+z k. (C.31) Commonly encountered body force is the gravitational force. The line force is the surface tension force. It does not appear in the equations of motion but in the boundary conditions. Reference C.3, chapter be referred to for details. The surface forces arise due to differences in tractions exerted by the surrounding fluid on the faces of the fluid particle. Since, these forces depend on the surface area it is convenient to work in terms of stresses. The complete state of stress at a point is shown in Fig.C.1. Dept. of Aerospace Engg., Indian Institute of Technology, Madras 18

19 C.5 Laws governing fluid motion When electromagnetic effects are not considered, the laws governing the fluid motion are (i) conservation of mass, (ii) Newton s second law of motion, (iii) the first law of thermodynamics or conservation of energy, and (iv) the second law of thermodynamics. All these laws refer to a system i.e. a collection of material of fixed identity. These laws can be stated as follows: Conservation of mass: It is stated in various ways like the mass of a fluid is conserved or fluid can neither be created nor destroyed in the field of flow or the rate of change of mass of a given system of particles is zero. Mathematically, this can be expressed as : D/Dt dm = 0 (C.3) Newtons second law of motion : The resultant force acting on a fluid particle is equal to the product of the mass of the particle and its acceleration. From this law, the laws of conservation of linear momentum and conservation of angular momentum for a system of particles can be derived. The conservation of linear momentum means that the rate of change of the linear momentum of the system is equal to the sum of the forces acting on the system. i.e. F= D / Dt Vdm (C.33) where, F includes all forces, (body and surface forces), exerted by the outside world on the system. The conservation of angular momentum states that the rate of change of the angular momentum of the system of particles about a fixed axis is equal to the sum of the moments of the external forces about the axis, i.e. F r V r Σ = D / Dt dm (C.34) Conservation of energy : The change of the total energy (i.e. sum of the internal energy, e, kinetic energy and the potential energy, E p ) of a system of fluid particles, in a given interval of time is equal to the work done interval of time i.e. W d on the system plus the heat supplied (Q) to the system during that Dept. of Aerospace Engg., Indian Institute of Technology, Madras 19

20 V D / Dt e+ +E dm = dq / dt-dw / dt p (C.35) d where, V = U + V + W. Second law of thermodynamics: During a thermodynamic process the change of entropy(s) of the system plus that of the surrounding is zero or positive i.e. D /Dt sdm 1 dq T dt (C.36) The equal to sign applies in a reversible process and unequal sign applies in all irreversible processes. Remarks: (i) In Eqs. (C.3) to (C.36) the integrals on which the substantial derivative, (D/Dt), operates are summed up over all the elements of mass of the system. (ii) In addition to the above laws there are subsidiary laws or constitutive relations which apply to specific types of fluid e.g. equation of state for a perfect gas and relationship between stress and strain rate. C.6 Derivation of governing equations The Eqs. (C.3) to (C.36) refer to a system of particles but for many problems it is convenient to think in terms of a control volume fixed in space, through which the fluid flows. Equations (C.3) to (C.36) represent a Lagrangian point of view, while the control volume implies the Eulerian view point. The configuration of the control volume may be chosen to make the analysis most convenient, it may be either infinitesimal or finite in size. Both mass and energy may cross the control surface circumscribing the control volume. Dept. of Aerospace Engg., Indian Institute of Technology, Madras 0

21 C.6.1 Continuity equation The law of conservation of mass expressed in the form of a differential equation is called continuity equation. To derive this, consider a control volume consisting of a small parallelopiped of sides dx, dy, dz. Then, the conservation of mass can be interpreted as follows. The rate of mass leaving an elemental volume minus the mass entering the same elemental volume is equal to the rate of change of mass in the control volume which is equal to the rate of change of density multiplied by the volume of the element. Fig.C.7 Continuity equation With reference to Fig.C.7 one can write: Rate of mass entering the control volume in X-direction : ρudydz Rate of mass leaving in X-direction: ρu+ x ρudxdy dz ρu+ x ρu dx dy dz - ρudy dz = x ρu dx dy dz Rate of net out flow in X-direction = Similarly, the rate of net out flow in Y and Z-directions are : y ρv dx dy dz z and ρw dx dy dz Hence, the rate of outflow from the elemental volume is : ρu + ρv + ρw dxdydz x y z Dept. of Aerospace Engg., Indian Institute of Technology, Madras 1

22 For conservation of mass this must be equal to the rate of decrease of the mass of the control volume i.e. the rate of decrease of density multiplied by the volume of the element or ρ ρu + ρv + ρw dx dy dz = - dx dy dz x y z t (C.37) ρu ρv ρw ρ ρ Or = + ρ V = 0 (C.38) t x y z t Noting the definition of substantial derivative (Eq. C.11), the continuity equation can also be written as : Dρ +ρ V = 0 Dt (C.39) In a steady flow, does not change with time and continuity equation reduces to: ρu + ρv + ρw = 0 (C.40) x y z For an incompressible flow is constant and continuity equation simplifies to: U V W + + = 0 or V = 0 x y z In a two-dimensional incompressible flow : U V + = 0 x y (C.41) (C.4) Remarks: (i) While deriving continuity, it has been tacitly assumed that these are no empty spaces in the flow domain i.e. there is no cavitation. (ii) Since, no assumption has been made about viscosity, the above forms of continuity equation are valid for viscous flow also. (iii) The integral form of continuity equation can be obtained by integrating Eq. (C.38) over a finite volume Q i.e. Q ρ dq + V ρ dq = 0 t Q (C.43) Using Gauss theorem, the second term on the l.h.s. of Eq.(C.43) can be written as : Dept. of Aerospace Engg., Indian Institute of Technology, Madras

23 ρv dq = n.ρv da Q (C.43a) A where n is the unit vector normal to the surface da, positive when pointing outwards from the enclosed volume. Then, Eq. (C.43) becomes: t Q ρdq+ ρ Vn. da = 0 A (C.44) Following Ref.C.4, chapter 1, the conservation law given by Eq.(C.44) can be generalized by stating that the variation per unit time of a scalar quantity S within a volume Q is equal to the net contribution from the incoming fluxes F t through the surface A, plus contributions from sources of quantity S. These sources can be divided into volume source S v and surface source S s. The general form of conservation equation is : ρ F n S n SdQ + t. da = S v dq + s. da t t t Q A Q A (C.45) C.6. Navier-Stokes Equations The set of differential equations obtained by applying Newton s second law to a fluid particle is called Navier-Stokes equations. The forces acting on a fluid particle have already been discussed in section C. 4. The relationship between stresses and rates of strain is discussed below. Dept. of Aerospace Engg., Indian Institute of Technology, Madras 3

24 C.6..1 Relation between stresses and rates of strains In an elastic solid body the stresses are related to strains, but in a fluid the stresses are related to strain rates ε x x,ε y y,ε zz,ε x y,εyz and ε zx. In section 3..6 it is shown that the strain rates are related to velocity field by : U 1 V U 1 W U + + x x y x z ε x x εx y εzx 1 U V V 1 W V ε = εx y εy y ε yz = + + y x y y z εzx ε yz εzz 1 U W 1 V W W + + z x z y z (C.46) For the fluids like air and water the following assumptions can be made to obtain a general relationship between stresses and rates of strain. Such a fluid is called Newtonian fluid. (i)the fluid is isotropic. (ii) Translations and rigid body rotations do not cause stresses. But, they (stresses) are caused by deformations resulting from strain rates. The stresses are linear functions of rates of strains (Newtonian Fluid). (iii) As physical laws do not depend on the choice of coordinate system, the stressstrain relationships are invariant to coordinate transformations and mirror reflections of axes. (iv) When all the velocity gradients are zero the stress components reduce to hydrostatic pressure (-p) which is identical with thermodynamic pressure and decided by equation of state. Under these conditions the behaviour of the fluid can be described in terms of two constants and (Ref.C.1, chapter 3). Dept. of Aerospace Engg., Indian Institute of Technology, Madras 4

25 U V W U ζ xx = -p+λ + + +μ x y z x U V W V ζ yy = -p+λ + + +μ x y z y U V W W ζ zz = -p+λ + + +μ x y z z (C.47) V U W V U W x y = μ + ; yz = μ + ; xz = μ + x y y z z x Here, and are the two constants which depend on the fluid and should be determined experimentally. However, Stokes proposed the hypothesis that = - (/3) where is the familiar coefficient of viscosity. With this hypothesis Eqs. (C.47) reduce to : U ζ xx = -p- μ.v+μ 3 x V ζ yy = -p- μ.v+μ 3 y W ζ zz = -p- μ.v+μ 3 z (C.48) V U W V U W x y = μ + ; yz = μ + ; zx = μ + x y y z z x C.6.. Derivation of Navier-Stokes equations Consider a fluid particle having the shape of a parallelepiped with sides dx, dy, dz. The mass of the particle is dx dy dz; the acceleration is DV/Dt (see section. 3..5); the body force is dx dy dz (Xi + Yj + Zk). Let, the surface force be P. Applying Newton s second law of motion gives : dx dy dz D V /Dt = dx dy dz (Xi + Yj + Zk)+ P. (C.49) Dept. of Aerospace Engg., Indian Institute of Technology, Madras 5

26 The surface force can be obtained as follows: Fig. C.8 Viscous force on a fluid particle In general the stresses vary from point to point. This change produces surface forces on the fluid particle. Figure C.8 shows the change in the stress in the X-direction. The force in the X-direction due to this change is: xx xx xx xx ζ + ζ / x dx dy dz - ζ dy dz = ζ / x dx dy dz Considering also the changes in other stresses, it can be shown that the surface force acting in X-direction is : xy xy zx + + dx dy dz x y z The surface forces acting in Y and Z-directions are : xy ζ yy yz xz yz ζzz + + dx dy dz and + + dx dy dz respectively. x y z x y z Substitute (a) the above expressions for surface forces in Eq.(C.49), and (b) expressions for xx, yy, etc. in terms of the velocity components. On simplification the three scalar equations corresponding to Eq.(C.49) are obtained as : DU p U ρ = ρ X - + μ - V + μ U + V + μ W + U Dt x x x 3 y y x z x z D V p V V W U V ρ = ρy - + μ - V + μ + + μ + Dt y y y 3 z z y x y x (C.50) Dept. of Aerospace Engg., Indian Institute of Technology, Madras 6

27 DW p W V W U V W ρ = ρz - + μ - + μ + + μ + Dt z z z 3 x x z y z y Remarks: (i)these equations are the Navier-StokesN - S equations for an unsteady, compressible, viscous flow. In a compressible flow the changes in temperature and pressure are not small. Hence, and are functions of space coordinates. (ii) N-S Equations for incompressible flow : In this case and can be taken as constants. Further, in this case V 0. As a consequence, the set of Eqs. (C.50) reduces to : U U U U 1 p + U + V + W = X - + v U t x y z ρ x V V V V 1 p t x y z ρ y v + U + V + W = Y - + V C.51 W W W W 1 p +U +V +W = Z - + v W t x y z ρ z where, = + + x y z ; v = μ / ρ; When body force is ignored (e.g. flow of air) the set of Eqs. (C.51) can be written in vector and tensor forms as follows: V 1 V. V + = - p+ v V (C.5) t ρ Ui Ui 1 p Ui +U j = - + t x ρ x x x j i j j (C.53) (iii) For steady two-dimensional incompressible flow the derivatives with respect to t and z are zero. Further, if the body forces are small, then Eqs. (C.51) reduce to : v U U 1 p U U U + V = x y ρ x x y (C.54) Dept. of Aerospace Engg., Indian Institute of Technology, Madras 7

28 V V 1 p V V U + V = x y ρ y x y (iv) Reference C.4, chapter 1 be consulted for the integral form of the momentum equations. C.6.3 Energy equation For an incompressible flow the unknowns are (i) the velocity components U, V and W and (ii) the pressure p. To obtain these four quantities the four equations in the form of continuity and the three Navier-Stokes equations are sufficient. In a compressible flow temperature (T) is an additional unknown; the density () can be calculated using the equation of state when p and T are known. The additional equation for T is obtained by applying the first law of thermodynamics to a fluid particle. This equation is called energy equation. The energy balance is determined by (i) the internal energy, (ii) the conduction of heat, (iii) the convection of heat by the stream, (iv) generation of heat through friction and (v) the work of expansion (or compression) when the volume is changed. At moderate temperature, the contribution of radiation is small and is neglected in the present analysis. Consider a fluid element having the shape of a parallelopipied with sides dx, dy and dz (Fig.C.8). The volume of this element (V*) is dx dy dz. Its mass is m = V*. Let, (a) amount of heat dq be added to the fluid element in time dt (b) increase in the 1 internal energy be V*de, (c) increase in the kinetic energy be ρ ΔV *d U + V + W and (d) dw* be the work performed by the particle. Applying the first law of thermodynamics gives : ' DQ DE t DW = - Dt Dt Dt (C.55) where, D/Dt is the substantial derivative and ' DEt De 1 D = ρ ΔV * + U + V + W Dt Dt Dt (C.56) Dept. of Aerospace Engg., Indian Institute of Technology, Madras 8

29 Note that the addition of heat is assumed to be only due to conduction. According to Fourier s law, the heat flux, q (J / m s), per unit area A and time is proportional to the temperature gradient normal to the surface, i.e. 1 DQ = q = - A Dt T n (C.57) where, ( J/m s K) is the thermal conductivity of the fluid. Hence, the heat transferred into the volume V* through the surface normal to the X-direction is equal to - T/ x dydz. The amount leaving in the X-direction is T T + dx dy dz x x x Thus, the total amount of heat added by conduction during the time dt to the volume V* can be written as: * T T T dq = dt ΔV + + x x y y z z (C.58) To obtain the work done, consider the contribution from the component xx of the stress. * From Fig.C.8 the work done per unit time ( dw ) is : U ζ dw = dy dz -Uζ + U+ dx ζ + dx = ΔV Uζ x x x * xx * ζxx xx xx xx To total work performed by the normal and shear stresses can now be written as: x * dw = ΔV Uζ xx +V xy +Wxz ζxx (C.59) + U xy +V ζ yy +W yz + U zx +V yz +W ζzz y z (C.60) From Eq. (C.48) the stresses ζ xx,ζ yy, ζ zz, xy, yz, zx, can be written in terms of and the velocity components. Substituting Eqs. (C.57), (C.58) and (C.60) in Eq. (C.55) gives the following equation after simplification. De T T T +p. V = + + +μ (C.61) Dt x x y y z z where, called dissipation function, is : Dept. of Aerospace Engg., Indian Institute of Technology, Madras 9

30 U V W V U W V U W U V W = x y z x y y z z x 3 x y z Further, from Eq. (C.39) : U V W 1 Dρ. V = + + = - x y z ρ Dt And the quantity p CpdT = CvdT+d, ρ Cp can also be expressed as : Using Eqs. (C.6) to (C.64), the Eq.(C.61) can be simplified as : DT Dp T T T ρc p = μ Dt Dt x x y y z z C.6.4 The complete set of equations (C.6) (C.63) (C.64) (C.65) Thus, the equations of motion for an unsteady, compressible fluid flow are as follows. Continuity equation: ρ Dρ +. ρ V = +ρ V = 0. (C.66) t Dt Navier-Stokes equations: DU p U U V W U ρ = ρx- + μ -. V + μ + + μ + Dt x x x 3 y y x z x z DV p V V W U V ρ = ρy - + μ -. V + μ + + μ + Dt y y y 3 z z y x y x (C.67) DW p W W U V W ρ = ρz- + μ -. V + μ + + μ + Dt z z z 3 x x z y z y Energy equation: DT Dp T T T ρc p = μ Dt Dt x x (C.68) y y z z Dept. of Aerospace Engg., Indian Institute of Technology, Madras 30

31 where, U V W V U W V U W = x y z x y y z z x 3 Equation of state: p = RT These are the six equations for the six unknowns viz. U, V, W, p, and T. Remarks: V (C.69) (i) For a perfect gas, the following additional relationships exist. e = C T ; h = C T = C /C C = R -1 ; C = R/ -1. (C.70) ; ; / v p p v v p where Cv is the specific heat at constant volume, C p is specific heat at constant pressure, is ratio of specific heats and R is gas constant. For air under standard conditions R = m /s K, C v = J/Kg K; C p = J/kg K and = 1.4. However, at high temperature (above 800 K), C p, C v and become functions of temperature. At moderate temperatures is given by the following Sutherland s formula. = x 10-6 [T 3/ /(T+110.4)] where, T is in K. The thermal conductivity is evaluated from = C p / P r where, P r is Prandtl number, which has a value of 0.7 for air. (ii)the integral form of energy equation is V.n q- W = ρet d vol. + Etρ da t (C.71) where, q is the heat added to the system and W is the work done by the system and 1 E t = e + U + V + W. (iii)though only the momentum equations (Eqs. C.67) are the Navier-Stokes equations, the entire set of equations comprising of continuity, momentum and energy equation is sometimes referred to as Navier-Stokes equations. 7 Equations of motion in conservation and vector forms Dept. of Aerospace Engg., Indian Institute of Technology, Madras 31

32 The integral forms of the equations (e.g. Eq.C.45 for continuity) represent the conservation of scalar and vector quantities. In computational fluid dynamics it is desirable that the differential forms of these equations, when discretised, also preserve the conservation property. Muralidharan and Sundararajan (Ref.C.5, chapter 4) show that the differential forms of equations when expressed in the following form retain this property. U E F G H = 0 (C.7) t x y z where, U, E, F,G and H represent quantities in continuity, momentum and energy equations which are defined later. It may be pointed out that in Eq.(C.7) the various terms in continuity, momentum and energy equation have been recast so that they appear as first order derivatives of t, x, y and z. Following Bertin and Smith (Ref.C.7, Appendix A) Eq. (C.7) can be modified as follows. E -E F -F G -G t x y z U + i v + i v + i v +H i -H v = 0. (C.73) where, the subscript i denotes the terms that are present in the equations of motion for an inviscid flow and subscript v denotes the terms that are unique to the equations of motion when viscous and heat-transfer effects are included. 7.1 Conservation form of continuity equation The continuity equation as given by equation (C.38), is already in the conservation form i.e. ρ + ρu + ρv + ρw = 0. (C.74) t x y z Comparing Eqs. (C.73) and (C.74) yields: U =, E i = U, F i = V, G i = W, H i = 0. (C.75) As the equation is valid for both viscous and inviscid flow. E v = F v = G v = H v = Conservation form of momentum equation Consider next the X-component of momentum equation (Eq.C.50). Neglecting the body force, it can be written as : Dept. of Aerospace Engg., Indian Institute of Technology, Madras 3

33 U U U U p, ρ +ρu +ρv +ρw = - + ζ + + t x y z x x y z xx xy zx (C.76) Note : To obtain Eq (C.76) the normal stress ( xx ) in Equation (C.48) has been split into a sum of (a) pressure p, which is present even when flow is inviscid, and (b) a term ζ xx pertaining to the viscous flow. i.e. U ζ xx = - μ. V +μ 3 x Similarly, yy V V ζ = - μ. +μ and ζ zz = - μ. V+μ W 3 y 3 z The definitions of xy, yz and zx remain the same as in Equation (C.48). Multiplying Eq. (C.74) by U gives: ρ U +U ρu +U ρv +U ρw = 0 t x y z Adding Eqs.(C.79) and (C.76) and rearranging, yields the conservation form of X-momentum equation i.e. ρu p+ρu -ζ xx + ρuv - xy + ρuw - zx = 0 t x y z (C.77) (C.78) (C.79) (C.80) Comparing Equations (C.80) and (C.73) it is noted that for X-component of the momentum equation : U = U, E i = p + U, F i = U V, G i = U W, H i = 0 E v = xx, F v = xy, G v = zx and H v = 0 (C.81) Similar manipulations of Y-momentum equation yield : U = V, E i = U V, F i = p + V, G i = VW, H i = 0 E v = xy, F v = ζ yy, G v = yz and H v = 0 (C.8) The Z-momentum equation is in the conservation form is : U = W, E i = UW, F i = VW, G i = p+w, H i = 0 E v = zx, F v = yz, G v = ζ zz and H v = 0 (C.83) Dept. of Aerospace Engg., Indian Institute of Technology, Madras 33

34 7.3 Conservation form of energy equation To obtain the conservation form of the energy equation, Eq (C.61), it is rewritten as : T T T Up + Uζ + V x x y y z z x x x xx xy + W + U + Vp + Vζ x y z y zx xy yy + W + W + V - Wp + Wζ y z z z z yz zx yz zz Et Et Et Et = ρ +ρu +ρv +ρw t x y z (C.84) 1 Note, that E t = e+ U +V +W (C.85) Multiplying equation (C.74) by E t, adding to Equation (C.84) and rearranging renders the energy equation in conservation form : ρe + t ρe +p U- Uζ +V +W +q t x t xx xy zx x y ρe t +p V - U xy +Vζ yy +W yz +qy ρe t +p W - U zx + V yz + Wζ' zz +q z = 0 z where, q x = T, q y = T, q z = T x y z Comparing Equations (C.86) and (C.73) it is noted that for the energy equation : U=ρE t; E i = ρet -pu; F i = ρe t +pv;g i = ρe t +pw; H = 0; E = Uζ +V +W +q ; i v xx xy zx x F = U +Vζ +W +q ;G = U +V +Wζ +q ; v xy yy yz y v zx yz zz z (C.86) (C.87) H v = 0 (C.88) Dept. of Aerospace Engg., Indian Institute of Technology, Madras 34

35 C.7.4 Strong and weak conservation forms of equations Equations (C.74), (C.80), (C.8), (C.83) and (C.86) are the conservation forms of the continuity, momentum and energy equations. As noted earlier, the various terms in the continuity, momentum and energy equation have been recast so that they appear as first order derivatives of t, x, y and z. This form of equations is also called strong conservation form. However, if only the advective part of the equations is expressed as the first order derivatives of t, x, y and z, then, the form of the equation is called weak conservation form. As an example, the two forms of the x momentum equation for incompressible flow (Eq.C.51) are as follows; body force has been ignored. Weak conservation form : UV UW U U 1 p = - + v U t x y z ρ x Strong conservation form : U p U U U + U + + UV + UW = 0 t x ρ x y y z z C.7.5 Vector form of the equations of motion Equations (C.74), (C.80), (C.8), (C.83) and (C.86) are the conservation forms of the continuity, momentum and energy equations. These, equations can be written in the vector form as follows. Note that E i, E v, F i, F v, G i, G v have the meaning as in Eq. (C.73). ρ ρu ρu p+ρu U = ρv ; E i = ρuv ; E v = ρw ρuw, ζ xx ρe, t +p U xx xy zx x ρe t Uζ + V + W +q 0 xy zx ρv 0 ρ VU xy, F i = p+ρv ; F v = ζ yy ρ V W yz ρe, t +p V Dept. of Aerospace Engg., Indian Institute U xy + V ζ yy + W yz +qy of Technology, Madras 35

36 ρw 0 ρuw zx G i = ρ V W ; G v = yz, p+ρw ζ zz ρe, t +p W U zx + V yz + W ζ zz + qz (C.89) H i and H v are zero in cartesian coordinate system. The vector form as given by Eq.(6.89) is used in computation of the compressible flow. C.8.1 Non-dimensional form of equations It is advantageous to put the equations of motion, Eq.(C.89), in the non-dimensional form. This form brings out the characteristic non-dimensional numbers which ensure dynamic similarity of two flows. In this form, the flow variables are also normalized and their values lie between certain prescribed limits e.g. 0 to 1. A non-dimensional variable is denoted by an asterisk. L is the reference length and free stream conditions are denoted by suffix. The non-dimensional variables are : x* = x/l, y* =y/l, z* = z/l, t* = V t/l, U* = U/ V, V* = V/ V, W* = W/ V, μ * = /, * = /, p* = p/( V ), T* = T/T, e* = e/ V. (C.90) Then, Equation (C.73) in non-dimensional form becomes * * * * U E F G = 0 * * * * t x y z (C.91) The vectors U*, E*, F* and G* are as follows. ρ* ρ *U*,* ρ *U* ρ *U* +p * -ζ xx U* = ρ * V * ; E * = ρ *U* V * - * xy ρ * W * ρ *U* W * - * zx * E *,* * * t ρ *E t +p * U* -U* ζ xx - V * xz - W * xz -q x Dept. of Aerospace Engg., Indian Institute of Technology, Madras 36

37 ρ * V * * ρ *U* V *- xy,* F * = ρ * V * +p *-ζ yy * ρ * V * W *- yz, ρ *E +p * V *-U* - V * ζ - W * -q * *,* * * t xy yy yz y ρ * W * * ρ *U* W *- zx * G* = ρ * V * W *- yz,* ρ * W * +p *-ζ zz ρ *E +p * W *-U* - V * - W * ζ -q where, *,* *,* * t zx yz zz z (C.9),* μ* U* V * W * * μ* U* V * ζ xx = - - ; xy = + 3ReL x * y * z * ReL y * x *,* μ* V * U* W * * μ* V * W * ζ yy = - - ; yz = + 3ReL y * x * z * ReL z * y *,* μ* W * U* V * * μ* U* W * ζ zz = - - ; zx = + 3ReL z * x * y * ReL z * x * μ T V q = ; M = -1 M R Pr x γrt * * * x el * (C.93) * * * μ T ρ VL q y = ; R * el = -1 M R Pr y μ el * * * μ T q Z = ; Pr = μc * p/k -1 M R Pr z el M p * U* + V * + W * p * = -1ρ* e* ; T * = ; E = e *+ ρ * * t Dept. of Aerospace Engg., Indian Institute of Technology, Madras 37

38 Remarks: (i) The steps to obtain the non-dimensional form of equations is illustrated below for the continuity and the X-momentum equations for two-dimensional incompressible flow. The two equations are : U V + = 0 x y U U U 1 p U U +U V = t x y ρ x x y (C.93a) (C.93b) Expressing (a) U as (U / V ) V ; (b) V as (V/V ) V ; (c) x as (x/l)l and (d) y as (y/l)l, Eq.(C.93a) can be rewritten as : U/V V V/V V + = 0 x/l L y/l L Or V U/V V V/V + = 0 L x/l L y/l Dividing throughout by (V /L) gives the non-dimensional form of Eq.(C.93a) as : U* V * + = 0 (C.93c) x * y * t p U Similarly, expressing (a) t as V / L ;(b) p as ρ V ; (c) U as V V /L ρ V V and so on, Eq.(C.93b) can be rewritten as : U/V V / / U U V V U V V V / / / V V V L t V V x/l L y/l L L V p ρv ρv U/V V U/V V 1 =- ρ x L x/l L y/l L L Or V V V + U/ V + V / V U/ V U/ V U/ V L tl L x/l L y/l V Dept. of Aerospace Engg., Indian Institute of Technology, Madras 38