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
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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.
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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
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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.
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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
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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. 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.3.2 Description of fluid motion
There are two ways of mathematically describing fluid motion viz. Lagrangian and Eulerian methods.
C.3.2.1 Lagrangian method In this method the trajectories of various fluid particles are described. For this purpose a fluid particle is identified by its position ro (or xo, yo, zo,) at time to. At subsequent instants of time the positions of the same particle are given by: r= r r ,t 0 (C.5) Or x = x(xo , y o , z o , t), y = y(x o , y o , z o , t), z = z(x o , y o , z o , t) The line along which a particle moves is called a path line. The instantaneous
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Turbulence Prof. E.G. Tulapurkara Chapter-3 velocity (V) and the acceleration (a) of the particle can be expressed as :
V= r / t = V ro ,t (C.6) ro =const. a= V =2 r / t = a r ,t / tr =const. o (C.7) o ro =const. The Lagrangian description is found to be less convenient, for mathematical manipulation and the Eulerian method, which is described next, is commonly used.
C.3.2.2 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 (C.8) 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. 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 : Vr d =0or dx:dy:dz=U:V:W (C. 9)
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Turbulence Prof. E.G. Tulapurkara Chapter-3
Fig.C.2 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 ψ ψ U = and V = - y x
C.3.2.3 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
Turbulence Prof. E.G. Tulapurkara Chapter-3 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.3.2.4 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 +z k. 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 rr+Δ = x+Δxi + 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+rΔ,t+Δt r =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 Hence, b = br+Δ,t+Δt-b r r ,t=U +V +W + Δt x y z t Hence, the rate of change of b, for a particle of fixed identity, which is generally denoted by Db/Dt, is : Db b+rΔ r ,t+Δt -b r ,t = Lt Dt Δt 0 Δt
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Turbulence Prof. E.G. Tulapurkara Chapter-3
Db b b b b Or = +U +V +W (C.10) Dt t x y z
D Or = +U +V + W = +.V (C.11) Dt t x y z t
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 is called „convective derivative‟. Since, it x y z indicates the change of property as the particle is convected by the flow.
C.3.2.5 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:
DVV =a = + .V V (C.12) Dt t
Or
DVVVVV = +U +V +W (C.13) Dt t x y z
Noting that a = ax i + ay j + az 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 = = +U + V +W x dt t x y z
DV V V V V a = = +U + V +W y dt t x y z
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Turbulence Prof. E.G. Tulapurkara Chapter-3 DW W W W W a = = +U + V +W (C.14) z dt t x y z
Remark: V 2 The term .V V can be expressed as - VV ; V 2 = U2 +V 2 +W 2 2 2 V V Hence, a = +-VV (C.14a) t2
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=ρ2πr q per unit length, ρ = fluid density , r = r = x22 +y ; q = radial velocity = U22 +V = m/ρ2πr or q = k / r ; k = m/2πρ. Hence, U = kx / (x2 + y2); V = ky / (x2 + y2).
Substituting expression for U and V in the set of equations (C.14), gives :
-k2 x -k2 y a=x ; a=y 222 222 x +y x +y
- k2 xij +y -k2 r Hence, a= axy i +a j =2 = 4 x22 +y r
It is noted that the acceleration is in the radial direction. The negative sign indicates that the flow is decelerating as „r‟ increases.
C.3.2.6 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.
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Turbulence Prof. E.G. Tulapurkara Chapter-3 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: UUU U+ΔU = U+ Δx+ Δy+ Δz x y z VVV V+ΔV = V+ Δx+ Δy+ Δz x y z
WWW W+ΔW = W + Δx+ Δy+ Δz x y z Or UUU ΔU = Δx+ Δy+ Δz x y z VVV ΔV = Δx+ Δy+ Δz x y z
WWW ΔW = Δx+ Δy+ Δz (C.15) x y z
The expressions for U, V and W can be rewritten in the following form :
11 ΔU = εxx Δx+ε xy Δy+ε xz Δz + ηΔz- ζ Δy 22
11 ΔV = εxy Δx+ε yy Δy+ε yz Δz + ζ Δx - ξ Δz C.16 22
11 ΔW= εxz Δx+ε yz Δy+ε zz Δz + ξΔy- ηΔx 22 where,
UVW εxx = ; ε yy = ; ε zz = (C.17) x y z
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Turbulence Prof. E.G. Tulapurkara Chapter-3 1 V U 1 W V 1 U W ε=xy +;ε= yz +;ε= xz + (C.18) 2 x y 2 y z 2 z x
WVUWVU ξ= - ;η= - ;ζ= - (C.19) y z z x x y
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 1 1 1 components of angular velocity ξ, η and ς (iii) dilatation given by the linear strain 2 2 2 rates εxx ,ε yy , ε zz and (iv) distortion of shape given by the rates of shear strain εx y ,ε yz ,ε xz .
Remarks: i) εx x ,ε y y,ε zz represent extensional rates of strain as can be seen from the following. Let, the flow be such that Uxis 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/ xΔx 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:
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Turbulence Prof. E.G. Tulapurkara Chapter-3 Δx+ U/ x ΔxΔt-Δx /Δx= U/ x Δt Hence, the rate of strain in the x-direction is U Δt U Lt = = εxx Δt 0 x Δt x VW Similarly, it can be shown that εyy = and ε zz = yz The rate of volumetric strain is
UVW Δx+ Δx Δt Δy+ Δy Δt Δz+ Δz Δt - Δx Δy Δz x y z e=Lt Δt 0 Δx Δy Δz Δt
UVW Or e = + + = .V. (C.20) 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/ yΔy Δt .
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Turbulence Prof. E.G. Tulapurkara Chapter-3 Thus, rotation of the vertical line which is also the shear strain is U/ yΔy Δt /Δy = U/ y Δt . U Δt U Hence, rate of shear strain = = Lt Δt 0 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. 2 Similarly, when V / x is alone non-zero, then the shear strain would be V / x . When UV both U/ y and V / x are non-zero the rate of shear strain ε xy would be+ . yx
UV (iii) When =- , the rate of shear strain is zero but the particle has a rate of rotation yx UV equal to ½ -.It may be added that the rate of rotation is taken positive in yx 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 :
εεε xx xy xz εεεxy yy yz (C.21) ε ε ε xz yz zz
It may be noted that the rate of strain tensor is symmetric i.e. εij = 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.
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Turbulence Prof. E.G. Tulapurkara Chapter-3 C.3.2.7 Vorticity
Vorticity is a vector denoted by ω and defined as
ωV=× (C.22) If ω = ξ i +η j +ς k , then, (C.23)
WVUWVU ξ = - ; η = - ; and ζ = - (C.24) y z z x x y
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.25) 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.26) (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.27) (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.2 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
Ue . Then,
U = Ue y/h ; V = 0. In this case, shear strain is Ue /h and vorticity is –Ue /h.
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Turbulence Prof. E.G. Tulapurkara Chapter-3 C.3.2.8 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+Vdy+Wdz (C.28) C
Fig.C.6 Circulation
Remarks:
(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.29) CA 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
Turbulence Prof. E.G. Tulapurkara Chapter-3 (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.2, 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.3.2.9 Accelerating reference frame
Consider a moving reference frame whose origin accelerates at the rate ao 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: DdW Ω aa= + +2Ω × W + Ω × Ω × r + × r (C.30) Dto dt where, DW /Dt is acceleration perceived by an observer in the accelerating frame. The term 2×ΩWis 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 2 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
Turbulence Prof. E.G. Tulapurkara Chapter-3 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.32) 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. FV= D / Dt dm (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. ΣF× r = D / Dt V × r 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, Ep) of a system of fluid particles, in a given interval of time is equal to the work done Wd on the system plus the heat supplied (Q) to the system during that interval of time i.e.
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Turbulence Prof. E.G. Tulapurkara Chapter-3 ||V 2 D/Dt e+ +E dm= dQ/dt-dW/dt (C.35) pd 2 where, |V|2 = U2 + V2 + W2. 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. 1 dQ D/Dt sdm (C.36) T dt The equal to sign applies in a reversible process and unequal sign applies in all irreversible processes. Remarks:
(i) In Eqs. (C.32) 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.32) 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.32) 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.
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Turbulence Prof. E.G. Tulapurkara Chapter-3 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+ ρU dx dy dz x
Rate of net out flow in X-direction = ρU+ ρU dx dydz-ρUdydz= ρU dxdydz xx Similarly, the rate of net out flow in Y and Z-directions are : ρV dx dy dz and ρW dx dy dz y z 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 21
Turbulence Prof. E.G. Tulapurkara Chapter-3 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 dxdydz=- dxdydz (C.37) x y z t
ρρρU ρV ρW Or + + + = +ρV = 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 (C.39) Dt
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:
UVW + + = 0 or V = 0 (C.41) x y z
UV In a two-dimensional incompressible flow : + = 0 (C.42) xy
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. ρ dQ + ρV dQ = 0 (C.43) QQt
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 22
Turbulence Prof. E.G. Tulapurkara Chapter-3 ρV dQ = n .ρ V dA Q A (C.43a) where n is the unit vector normal to the surface dA, positive when pointing outwards from the enclosed volume. Then, Eq. (C.43) becomes:
ρdQ+ ρVn . dA = 0 (C.44) t QA 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 Ft through the surface A , plus contributions from sources of quantity S. These sources can be divided into volume source Sv and surface source Ss. The general form of conservation equation is : ρ SdQ+F .dA= n SdQ+ S .dA n (C.45) t v s tQAQA t t
C.6.2 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 23
Turbulence Prof. E.G. Tulapurkara Chapter-3 C.6.2.1 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 εxx ,ε yy ,ε zz ,ε xy ,ε yz andε zx . In section 3.2.6 it is shown that the strain rates are related to velocity field by : U 1 V U 1 W U ++ x 2 x y 2 x z εx x ε x y ε zx 1 U V V 1 W V ε = εx y ε y y ε yz = + + 2 y x y 2 y z (C.46) ε ε ε zx yz zz 1 U W 1 V W W ++ 2 z x 2 z y z
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 stress- strain 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 24
Turbulence Prof. E.G. Tulapurkara Chapter-3 UVWU ζxx =-p+λ + + +2μ x y z x
UVWV ζyy =-p+λ + + +2μ x y z y (C.47) UVWW ζzz =-p+λ + + +2μ x y z z
VUWVUW 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 = - (2/3) where is the familiar coefficient of viscosity. With this hypothesis Eqs. (C.47) reduce to :
2U ζ = -p- μ.V +2μ xx 3x
2V ζ = -p- μ.V +2μ yy 3y (C.48) 2W ζ = -p- μ.V +2μ zz 3z
VUWVUW x y= μ + ; yz = μ + ; zx = μ + x y y z z x
C.6.2.2 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. 3.2.5); 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 DV /Dt = dx dy dz (Xi + Yj + Zk)+ P . (C.49) Dept. of Aerospace Engg., Indian Institute of Technology, Madras 25
Turbulence Prof. E.G. Tulapurkara Chapter-3
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: ζ + ζ / x dx dydz - ζ dydz = ζ / x dxdydz xx xx xx xx 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 yz ζ + + dxdydzand xz + + zz dxdydz 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 2 U V W U ρ=ρX-+μ2-V +μ+ + μ + Dt xx x3 yyx zxz
DV p V 2 V W U V ρ=ρY-+μ2-V +μ + +μ+ Dt yy y3 zzyxyx (C.50)
Dept. of Aerospace Engg., Indian Institute of Technology, Madras 26
Turbulence Prof. E.G. Tulapurkara Chapter-3
DW p W 2 W U V W ρ =ρZ-+μ2 -V +μ + +μ + Dt zz z3 xxzyzy
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- +v2 U t x y z ρx
V V V V 1 p + U + V + W = Y - +v2 V C.51 t x y z ρy
W W W W 1 p +U +V +W =Z- +v2 W t x y z ρz
2 2 2 where, 2 = + + ; v = μ / ρ; xyz2 2 2 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.VV = - p+v2 (C.52) t ρ
2 UUUi i1p i +Uj = - + (C.53) tx jρ x i x j x j
(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 : U U 1 p 22 U U U + V = - +v22 + xy ρ x x y (C.54)
Dept. of Aerospace Engg., Indian Institute of Technology, Madras 27
Turbulence Prof. E.G. Tulapurkara Chapter-3 V V 1 p 22 V V U + V = - + 22 + xy 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 ρΔV *d U2 + V 2 + W 2 2 and (d) dW* be the work performed by the particle. Applying the first law of thermodynamics gives :
DQDE' DW =-t (C.55) Dt Dt Dt where, D/Dt is the substantial derivative and
' DEt De 1 D 2 2 2 = ρ ΔV * + U + V + W (C.56) Dt Dt 2 Dt
Dept. of Aerospace Engg., Indian Institute of Technology, Madras 28
Turbulence Prof. E.G. Tulapurkara Chapter-3 Note that the addition of heat is assumed to be only due to conduction. According to Fourier‟s law, the heat flux, q (J / m2s), per unit area A and time is proportional to the temperature gradient normal to the surface, i.e. 1 DQ T = q = - (C.57) A Dt n 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
TT + dx dydz x x x Thus, the total amount of heat added by conduction during the time dt to the volume V* can be written as:
* TTT dQ = dt ΔV + + (C.58) x x y y z z
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 : ζxx
U ζ dW** = dydz -Uζ+U+ dx ζ+xx dx =ΔV Uζ (C.59) ζxx xx xx xx x x x To total work performed by the normal and shear stresses can now be written as:
* dW = ΔV Uζxx +V xy +W xz x + Uxy +V ζ yy +W yz + U zx +V yz +W ζ zz (C.60) yz
From Eq. (C.48) the stresses ζxx ,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 29
Turbulence Prof. E.G. Tulapurkara Chapter-3 22 2 2 2 2 2 U V W VU WV UW2UVW =2 + + ++++++- ++ x y z xy yz zx3xyz (C.62) Further, from Eq. (C.39) : U V W 1 D ρ .V = + + = - (C.63) x y z ρ Dt
And the quantity Cp can also be expressed as :
p Cpv dT = C dT+d, ρ (C.64) Using Eqs. (C.62) to (C.64), the Eq.(C.61) can be simplified as : DT Dp T T T ρC = + + + +μ (C.65) p Dt Dt x x y y z z C.6.4 The complete set of equations Thus, the equations of motion for an unsteady, compressible fluid flow are as follows. Continuity equation: ρ Dρ +.ρVV = +ρ = 0 . (C.66) t Dt
Navier-Stokes equations:
DU p U 2 U V W U ρ =ρX-+ μ2-. V + μ + +μ + Dt xx x3 yyxzxz
DV p V 2 V W U V ρ =ρY-+ μ2-. V + μ + + μ + (C.67) Dt y y y 3 z z y x y x
DW p W 2 W U V W ρ =ρZ-+μ2 -.+μ V + +μ + Dt z z z 3 x x z y z y Energy equation:
DT Dp T T T ρC = + + + +μ (C.68) p Dt Dt x x y y z z Dept. of Aerospace Engg., Indian Institute of Technology, Madras 30
Turbulence Prof. E.G. Tulapurkara Chapter-3 where, 22 2 2 2 2 U V W V U W V U W 2 2 =2 + + ++++++- V x y z x y y z z x 3 Equation of state: p = RT (C.69) These are the six equations for the six unknowns viz. U, V, W, p, and T. Remarks:
(i) For a perfect gas, the following additional relationships exist. e=CT;h=CTv p;;/ =C/C p v C v =R -1;C p = R/ -1 . (C.70) where Cv is the specific heat at constant volume, Cp is specific heat at constant pressure, is ratio of specific heats and R is gas constant. For air under standard 2 2 conditions R = 287.04 m /s K, Cv = 717.65 J/Kg K; Cp = 1004.7 J/kg K and = 1.4.
However, at high temperature (above 800 K), Cp , Cv and become functions of temperature. At moderate temperatures is given by the following Sutherland‟s formula. = 1.458 x 10-6 [T3/2/(T+110.4)] where, T is in K. The thermal conductivity is evaluated from
= Cp / Pr where, Pr is Prandtl number, which has a value of 0.72 for air. (ii)The integral form of energy equation is q- W = ρE d vol. + E ρV.n dA (C.71) t tt where, q is the heat added to the system and W is the work done by the system and
1 2 2 2 Et = e + U + V + W . 2 (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
Turbulence Prof. E.G. Tulapurkara Chapter-3 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. UEFG + + + +H = 0 (C.72) 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.72) 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.72) can be modified as follows. U E-EF-FG-G +i v + i v + i v +H -H = 0 . (C.73) t x y z iv 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 = , Ei = U, Fi= V, Gi = W, Hi = 0. (C.75)
As the equation is valid for both viscous and inviscid flow. Ev = Fv = Gv = Hv = 0. 7.2 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 32
Turbulence Prof. E.G. Tulapurkara Chapter-3
U U U U p , ρ +ρU +ρV +ρW =- + ζxx + xy + zx (C.76) t x y z x x y z
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 ζxx pertaining to the viscous flow. i.e. 2U ζ = - μ .V +2μ (C.77) xx 3x
2 V 2 W Similarly, ζ =- μ.VV +2μ andζ =- μ. +2μ (C.78) yy3 y zz 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 (C.79) t x y z Adding Eqs.(C.79) and (C.76) and rearranging, yields the conservation form of X-momentum equation i.e.
2 ρU p+ρU-ζ xx + ρUV- xy + ρUW- zx =0 (C.80) t x y z Comparing Equations (C.80) and (C.73) it is noted that for X-component of the momentum equation : 2 U = U, Ei = p + U , Fi = U V, Gi = U W, Hi = 0
Ev = ‟xx, Fv = xy, Gv = zx and Hv = 0 (C.81)
Similar manipulations of Y-momentum equation yield :
2 U = V , Ei = U V, Fi = p + V , Gi = VW, Hi = 0
Ev = xy , Fv = ζyy , Gv = yz and Hv = 0 (C.82)
The Z-momentum equation is in the conservation form is : 2 U = W, Ei = UW, Fi = VW, Gi = p+W , Hi = 0
Ev = zx, Fv = yz, Gv = ζzz and Hv = 0 (C.83)
Dept. of Aerospace Engg., Indian Institute of Technology, Madras 33
Turbulence Prof. E.G. Tulapurkara Chapter-3 7.3 Conservation form of energy equation To obtain the conservation form of the energy equation, Eq (C.61), it is rewritten as :
TTT + + - Up + Uζxx + V xy x x y y z z x x x
+ Wzx + U xy + Vp + Vζ yy x y z y
+W+W+V-Wp+W yz zx yz ζ zz y z z z z
EEEE = ρt +ρU t +ρV t +ρW t (C.84) t x y z
1 2 2 2 Note, that Et = e+ U +V +W (C.85) 2
Multiplying equation (C.74) by Et, adding to Equation (C.84) and rearranging renders the energy equation in conservation form : ρEt + ρE+pU-Uζ t xx +V xy +W zx +q x tx
ρE+pV-Ut xy +Vζ yy +W yz +q y y ρEt +p W - U zx +V yz +Wζ' zz +q z = 0 (C.86) z TTT where, q = , q = , q = (C.87) xx y y z z Comparing Equations (C.86) and (C.73) it is noted that for the energy equation :
U=ρE;Et i = ρE-p t U;F= i ρE+p t V;G i = ρE+p t W;
Hi = 0; E v = Uζ xx +V xy +W zx +q x ;
Fv = U xy +Vζ yy +W yz +q;G=U y v zx +V yz +Wζ zz +q; z
Hv = 0 (C.88)
Dept. of Aerospace Engg., Indian Institute of Technology, Madras 34
Turbulence Prof. E.G. Tulapurkara Chapter-3
C.7.4 Strong and weak conservation forms of equations
Equations (C.74), (C.80), (C.82), (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 : U U2 UV UW 1 p + + + = - +v2 U t x y z ρx
Strong conservation form :
U 2 p U U U +U+ +UV +UW =0 tx ρ x y y z z
C.7.5 Vector form of the equations of motion Equations (C.74), (C.80), (C.82), (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 Ei, Ev, Fi, Fv, Gi, Gv have the meaning as in Eq. (C.73). ρ ρU 0 ρU p+ρU2 , ζ xx U =ρV ; Ei = ρUV ; Ev = xy ρW ρUW zx , ρEt ρEt +p U Uζ +V +W +q xx xy zx x
ρV 0 ρVU xy , F =p+ρV2 ; F = ζ i v yy ρV W yz , ρEt +p V U + V ζ + W +q Dept. of Aerospace Engg., Indian Institute ofxy Technology, yy yzMadras y 35
Turbulence Prof. E.G. Tulapurkara Chapter-3
ρW 0 ρUW zx G =ρ V W ; G = (C.89) i v yz 2 , p+ρW ζ zz , ρEt +p W U + V + W ζ + q zx yz zz z
Hi and Hv 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, 2 W* = W/ V, μ * = / , * = /, p* = p/( V ), T* = T/T,
2 e* = e/ V . (C.90) Then, Equation (C.73) in non-dimensional form becomes UEFG**** + + + = 0 (C.91) t**** x y z The vectors U*, E*, F* and G* are as follows.
ρ* ρ *U* ρ *U* ρ *U*2 +p *-ζ ,* xx U* = ρ*V* ;E*= ρ*U*V*-* xy ρ*W* ρ*U*W*-* zx * *,*** Et ρ*E+p*U*-U*ζ -V* -W* -q t xx xz xz x
Dept. of Aerospace Engg., Indian Institute of Technology, Madras 36
Turbulence Prof. E.G. Tulapurkara Chapter-3
ρ* V * * ρ*U* V *- xy 2 ,* 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 (C.92) ρ* W *2 +p *-ζ ,* zz ρ*E+p*W*-U** ,* -V* * -W*ζ ,* -q * t zx yz zz z where,
,*2μ* U* V*W* * μ* U* V* ζxx = 2 - - ; xy = + 3ReL x* y* z* R eL y* x*
,*2μ* V* U*W* * μ* V* W* ζyy = 2 - - ; yz = + 3ReL y* x*z* R eL z* y*
,*2μ* W* U*V* * μ* U* W* ζzz = 2 - - ; zx = + 3ReL z* x*y* R eL z* x*
μT** V q* = ; M = (C.93) x -1 M2* R Pr x γRT eL
** * μT ρ V L qy =2* ; R eL = -1 M ReL Pr y μ
** * μT qZp =2* ; Pr = μC /k -1 M ReL Pr z
M2 p * U*2 + V * 2 + W * 2 p * = -1ρ*e*;T*= ;E=e*+* ρ *t 2
Dept. of Aerospace Engg., Indian Institute of Technology, Madras 37
Turbulence Prof. E.G. Tulapurkara Chapter-3 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 :
UV + = 0 (C.93a) xy U U U 1 p 22 U U +U V = - + + (C.93b) t x y ρ x x22 y
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
VVU/V V/V Or + = 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 p2 U Similarly,expressing (a) tas V /L ;(b) pas 2 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 VVVV / L V x/L L y/L L t// V L V
p 2 2 ρV 2 2 1 ρV U/V V U/V V =- 2222 ρ x x/L L y/L L L L
Or
22 2 VVVUVUVUV/// + U// V + V V LtL L x/L L y/L V
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Turbulence Prof. E.G. Tulapurkara Chapter-3 2 p / ρV2 22 VV UVUV// = 2 22 L x/L L x/ L y/L
2 Dividing throughout by V /: L gives
* * * * 2 * 2 * U** U U p U U ****+U + V = - + 22 t x y x LV x y
Or
22 U* U* U* p*1 U* U* VL +U* + V * = - +22 +; ReL = C.93d t* x* y* x*ReL x* y* This is the non-dimensional form of the X-momentum equation for 2-D incompressible flow. When this process of non-dimensionalization is carried out for 3-D, compressible flow, two additional parameters viz. M & Pr appear. Equation (C.92) is the set of non- dimensional equation, in vector form.
(ii)The non-dimensional parameters are Reynolds number (ReL), Mach number (M) and Prandtl number (Pr). (iii)Equations (C.89) and (C.92) differ only by an asterisk. Hence, the asterisk can be dropped from the non-dimensional equations. However, the definitions of ‟*, * , q*, p*,
* T* and E t given by Eq. (C.93) be kept in mind. (iv)Reference C.1 chapter 3 gives equations of motion in cylindrical and spherical polar coordinates. Reference C.6, chapter 5 gives these equations in other coordinate systems.
C.9 Simplified forms of equations
The Navier-Stokes equations are non-liner, partial differentional equations. Exact solutions are available at low Reynolds number. In 1904 Prandtl showed that for sufficiently high Reynolds number the important viscous effects are confined to a thin layer, called boundary layer near the surface of a solid boundary. Outside this layer the products of the velocity gradient and the coefficient of viscosity i.e. the viscous terms are small as compared to other terms and hence viscous terms can be neglected from the equations of motion.Such a treatment is called inviscid flow and the simplified equations are called Euler equations in honour of L.Euler who derived them in 1755 considering the fluid as inviscid.
Dept. of Aerospace Engg., Indian Institute of Technology, Madras 39
Turbulence Prof. E.G. Tulapurkara Chapter-3 The thinness of the boundary layer allows certain simplifications and the Navier-Stokes equations can be simplified in some other manner. These equations are called boundary layer equations. The Euler equations are described first.
C.9.1 Euler equation
Ignoring the viscous terms in Eq (C.89) yields the Euler equations which are given below in the vector form.
UEFG + + + = 0 where, t x y z
ρ ρU ρV ρW 2 ρU p+ρU ρWU ρUW U=ρV ,E= ρUV ,F= p+ρV2 ,G= ρVW (C.95) 2 ρ W ρUW ρ V W p+ρW ρE ρE+pU ρE+pV t t t ρEt +p W
It may be noted that the continuity equation is same in Eqs. (C.89) and (C.95).
C.9.2.1 Various form of Bernoulli equation
Using Eq.(C.14a) the momentum equation for inviscid flow can be written as :
V 2 1 VVV/ 2 - =F - p (C.96) t body ρ
2 2 2 2 where V = U +V +W and Fbody = Xi + Yj + Zk The x-component of Eq.(C.96) is : U U U U 1 p +U +V +W = X- (C.97) t x y z ρx Similar equations can be written for the Y and Z- components of momentum equation. Assuming the flow to be steady and ignoring the body force, Eq. (C.96) reduces to:
2 1 VVV/ 2 - = p (C.97a) ρ Equation (C.97a) can be integrated in the following two special cases.
(i) Along a streamline where V V .d r is zero. In this case integration of Dept. of Aerospace Engg., Indian Institute of Technology, Madras 40
Turbulence Prof. E.G. Tulapurkara Chapter-3 Eq. (C.97a) gives :
2 V dp + = const. (C.98) 2 ρ (ii) If the flow is irrotational then ×V is zero every where and integration of Eq. (C.97a) gives :
2 V dp +=constant every where and not just along a streamline. (C.99) 2 ρ If the fluid is barotropic i.e. the density is a function only of the pressure then the second term in Eq. (C.99) can be integrated. The two special cases are as follows. a) In a steady incompressible flow, ρ , is constant, this yields: p + 1/ 2ρ V 2 =constant. (C.100) This is called Bernoulli equation for steady incompressible, inviscid flow. b) In an isentropic flow, ρ = (constant) p1/. Then, Equation (C.99) gives :
2 V p +=constant. (C.101) 2 -1 ρ This is called Bernoulli equation for steady compressible inviscid flow.
Remarks:
(i) For an unsteady, irrotational flow for which the body force field is conservative, the generalised Bernoulli equation is :
2 V dp + + + = constant everywhere. (C.102) t2 ρ
Where is velocity potential and is the body force potential such that
X = - / x ; Y = - / y ; Z = - / z ;
(ii) From equation of state , p/ρ = RT .
Further, = C /R -1 p
p Hence, = C T = h. Dept. of Aerospace -1 ρ p Engg., Indian Institute of Technology, Madras 41
Turbulence Prof. E.G. Tulapurkara Chapter-3
Substituting this in Eq.(C.101) gives : h+ V 2 / 2 = constant for inviscid, steady compressibile flow. (C.103)
C.9.3 Crocco’s equation
For an inviscid fluid the equation of motion, in the absence of body forces, is : V 1 2 VVV = - p - / 2 (C.104) t ρ
The following relationship is obtained from thermodynamics :
1 T s = h- p where, s is entropy. ρ Noting, that the stagnation enthalpy h is h+V 2 / 2 , Eq. (C.104) can be rewritten o as : V -V ×ω = T s- h ; ω = × V (C.105) t o Equation (C.105) is called Crocco‟s equation. For incompressible flow it reduces to
V 1 2 -V xω = - p , where p = p+ 1/ 2ρ V is the stagnation pressure (C.106) t ρ o o The following conclusions can be drawn from Eqs. (C.105) and (C.106).
(i) In a steady flow, where entropy (s) and stagnation enthalpy (ho) are the same in the flow field, the vorticity is zero everywhere. Same is true in an incompressible flow with constant stagnation pressure. However, vorticity can be non-zero in an unsteady flow. (ii) The r.h.s of Eq.(C.105) indicates that the vorticity can be produced in a steady flow when gradients of entropy and/or stagnation enthalpy are present. This can happen in the case of boundary layers, curve shocks and combustion.
C.9.4 Velocity potential equation
For an irrotational flow a velocity potential exists such that
U=- ,V=- ,W=- (C.107) x y z
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Turbulence Prof. E.G. Tulapurkara Chapter-3
Substituting these in continuity equations yields :
ρx + ρ y + ρ z = 0 (C.108) x y z
From Eq. (C.99) , dp = -ρdV 2 /2=ρd222 + + /2 (C.109) x y z
Further, the speed of sound (a) is given by : a = dp dρ = p ρ = RT . (C.110)
Hence, Eq.(C.109) can be rewritten as : ρ dρ = - d222 + + / 2 , (C.111) a2 x y z This expression can be used to find the derivatives of ρ in different directions. Substituting these in Eq. (C.108) yields :
22 2 1-xz + 1-y + 1- - 2x y - 2xz - 2 y z = 0 (C.112) 2 xx 2 yy 2 zz 2xy 2 xz 2 yz a a a a a a Equation (C.112) is the velocity potential equation for a compressible flow. For an incompressible flow; a and Eq.(C.112) reduces to :
xx+ yy + zz = 0 (C.113) In a two-dimensional flow Eq.(C.113) reduces to :
xx+ yy = 0 (C.113a) C.9.5 Boundary layer equations From Eq. (C.52) it is seen that when the kinematic viscosity () is taken as zero the terms containing the second derivatives of U and V are eliminated and the equations reduce to Euler equations. This process reduces the order of the differential equation and as such some of the boundary conditions would not be satisfied by the solution. The Euler equations, being a first order system, require only one boundary condition viz. the no penetration condition at the body surface. However, N-S equations require two boundary conditions e.g. both normal and tangential velocities are to be zero on the surface of the body, when the body is stationary. In 1904 L. Prandtl proposed the boundary layer theory and showed that the Navier-Stokes equations can be simplified to
Dept. of Aerospace Engg., Indian Institute of Technology, Madras 43
Turbulence Prof. E.G. Tulapurkara Chapter-3 yield approximate solutions for flow at high Reynolds numbers. The basic argument underlying the theory is as follows. Under conditions of normal temperature and pressure a fluid satisfies the no-slip condition i.e. on the surface of a solid body the relative velocity between the fluid and the solid wall is zero. However, a velocity of the order of the freestream velocity is reached in a very thin layer called boundary layer (Fig.C.9). The velocity gradient normal to the surcface (U/y) is very high in the boundary layer. Hence, even if is small, the shear stress (U/y) in the boundary layer may be large or may be comparable to other terms in the N-S equations. Outside the boundary layer the gradient U/y is very small and the viscous terms can be ignored.
Fig.C.9 Schematic of boundary layer
The simplification of N-S equations is based on the analysis of the „order of magnitude‟ of various terms in these equations.
It may be noted that O ( ) means „of the order of‟. It is generally understood that : (a) A quantity is of order one or O (1) when it lies approximately between 1/ 10 and
10 . (b) A quantity is of order 0.1 or O (0.1) when it lies approximately between 1/10 10 and 1/ 10 .
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Turbulence Prof. E.G. Tulapurkara Chapter-3 (c) A quantity is of order 0.01 or O (0.01) when it lies approximately between 1/10 10 and 1/100 10. (d) A quantity is of order 10 or O (10) when it lies approximately between 10 and 10 10 (e) A quantity is of order 100 or O (100) when it lies approximately between 10 10 and 100 10 And so on. When the order of magnitude changes by one, the value of the quantity changes by a factor of 10. For the order of magnitude analysis of the terms in the NS equations, it is convenient to consider the N-S equations in the non-dimensional form (Eq.C. 92). For simplicity, the steady, incompressible, two-dimensional case is considered here. For this case, the equations in the non-dimensional form are as follows.
U*V* + = 0 (C.114) x * y *
U* U* p* 1 22 U* U* U* + V * = - +22 + (C.115) x* y* x*ReL x* y* V* V* p*1 22 V* V* U* + V * = - +22 + (C.116) x* y* y*ReL x* y*
Note :
ρ V L VL x* = x/L, y* =y/L, U* = U/ V, V* = V/ V, ReL = = ; V is the free stream μ velocity and L is the characteristic length which in this case can be the length of the plate on which the boundary layer develops.
To carry out the order of magnitude analysis, it is assumed that U = O (V) ; x = O(L) and y/L = O(); << 1 i.e. = O (0.01). In other words : x* is of O (1) , y* is of O (), U*
Dept. of Aerospace Engg., Indian Institute of Technology, Madras 45
Turbulence Prof. E.G. Tulapurkara Chapter-3 is of O(1) The order of magnitude of the other quantities will be decided by the order of magnitude analysis of Eq.(C.114) to (C.116) which is presented below. (I) Since, U* and x* are of O (1), the term U*/x* is also of order 1. Further, in Eq.(C.114) the r.h.s. equals zero. As a consequence, the two terms on the l.h.s. namely U*V* U* and are of the same order ; though they may have different sign. Since, x * y * x* is of the order 1 the quantity .V*/y* should also be of order one. Now, the assumption that the boundary layer is thin is taken into consideraion by taking y*(= y/L) as of the V* order of . Since, is of the order of 1, and y* is of order , as a consequence, the y* quantity V* is of order . (II) As noted in the previous paragraph, (a) U* and x* are of order one, (b) V* and y* are of order . Consequently, V*/x* and 2V*/x*2 are of the order of and 2U*/x*2 is of the order of unity. The orders of magnitude of the various terms are given below the equations in Eqs.(C.117) and (C.118).
U* U* p* 1 22 U* U* U* + V * = - +22 + (C.117) x* y* x*R x* y* eL 1 1 1/ 1 1/2
V* V* p*1 22 V* V* U* + V * = - + + (C.118) x* y* y*R x*22 y* eL
1 1 ( /2)
In the above analysis the order of magnitudes of p * / x * and p * / y * are not known a
2U* priori. In Equation (C.117) the second term inside the bracket on r.h.s. ( ) is of the y*2
2U* order of 1/2 which is very large. Hence, the first term in the bracket ( ) which is of x*2 the O(1) can be ignored. Secondly, in order that the terms on the r.h.s. and l.h.s. of this equation are of the same order of magnitude, the Reynolds number (ReL) must be of the
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Turbulence Prof. E.G. Tulapurkara Chapter-3 order of 1/2. Thus, for the above order of magnitude analysis to be valid, the Reynolds number should be high i.e. of the order of 1/2. It may be pointed out that when the term 2U* ( ) is ignored, the nature of the equations changes from elliptic to parabolic. x*2
(iii) Equation (C.118) shows that the terms on the l.h.s. are of order . Noting from 2 previous paragraph that (ReL) is of the order 1/ , the second term on the r.h.s. is also seen to be of the order . Hence, p*/y* in Eq.(C118) is also of order . In other words, the change of static pressure (p) across the height of the boundary layer is very small. As an approximation, the static pressure can be taken as constant across the boundary layer and equal to the outside pressure. This conclusion has very significant implications. (a) The pressure inside the boundary layer is not a variable. (b) Outside the boundary layer the viscous effects are negligible and one can apply the following Bernoulli equation. 11 p+ ρV22 = p+ ρU (C.119) 22 e
Thus, if Ue, the velocity outside the boundary layer, is known then p can be calculated. With these simplifications the equation for two-dimensional steady boundary layer can now be written in the dimensional form as: U U 1 dp 2 U U + V = - +v 2 (C.120a) xy ρ dx y UU + = 0 (C.120b) xy
Remarks: (i) In the set of Eq. (C.120 a & b), U and V are the only unknowns, dp/dx has to be prescribed. The usual way to solve these equations is to initially obtain the velocity distribution past the body assuming fluid to be inviscid. Then, this velocity distribution is used to get dp/dx using Eq. (C.119). Using this dp/dx or Ue (dUe/dx), Eqs. (C.120 a & b) are solved. The boundary conditions are: at y = 0 , U = 0 and V = 0 ; at y = or since is not known initially, one can take that at y = , U = Ue(x). (ii) Equations (C.114) to (C.116) are elliptic in nature whereas Eqs. (C.120 a & b) are parabolic in nature. Numerical solution of parabolic equations is much easier than that
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Turbulence Prof. E.G. Tulapurkara Chapter-3 of elliptic equations. Appendix D briefly presents the steps to obtain the boundary layer over a flat plate developing in a uniform external stream. Example D.1 shows that ReL 5 in a typical case, is 3.57 x 10 which appears to confirm the requirement that ReL should be of order (1//2) . The length of the plate (L), in the same example is 500 mm and the height of the boundary layer is obtained as 4.19 mm. This also shows that the ratio of the height of the boundary layer to the plate length is of the order of 0.01 or 1/. It is also found experimentally that the pressure inside a laminar boundary layer is almost constant across the height of the boundary layer. These confirmations of the assumptions made at the beginning of the formulation of the boundary layer theory, are the hallmark of an appropriate theory and the work of a genius like Prandtl. Several tributes were paid, in 2004, on the occasion of the centenary of boundary layer theory. Reference C.12 is one such tribute. (iii) Reference C.6 chapter 5 presents, boundary layer equations in various cases. (iv) Boundary layer form of the energy equation for steady, incompressible flow, with constant and , is:
2 2 T T2 T U p U U +V = α2 + +μ / ρcp (C.121) x y y ρCp x y where, = thermal diffusivity = / Cp. The equation can be used to find temperature distribution when Mach number is low i.e. less than 0.3. (v) The boundary layer equations are applicable when the gradient along y-axis is much larger as compared to that along x-direction. The flow has a preferred direction and diffusion is much larger in the cross-stream direction. Thus, these equations can also be used to predict thin shear flows like wakes, jets and mixing layer. Hence, boundary layer equations are also called thin shear flow equations. (vi) Boundary layer equations should not be used when the flow field has regions of separated flow.
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Turbulence Prof. E.G. Tulapurkara Chapter-3 C.10 Momentum integral equation This equation is obtained by integrating the boundary layer equations, derived in section C. 9.5, across the boundary layer. This equation is used (1) as a check on the numerical computations carried out using differential equations (Ch.6), and (ii) in approximate methods to predict boundary layers which are called integral methods. The derivation of momentum integral equation for a 2-D incompressible flow is as follows.
Fig.C.10 Velocity profile in boundary layer The equations of motion in this case are : UV + = 0 xy (C.122)
U U 1 dp U U +V = - + xY ρ dx y y
The boundary condition are : (a) at y = 0, U = 0, and (b) at y = , U =Ue (x) where, = boundary layer thickness. 1 dp U Noting that , =U e and integrating the momentum equation gives: ρ dxe x
δδUUU U U +V -Ue dy =v dy e (C.123) 00x y x y y
UU 0 Now, r.h.s. of Eq.(C.123) equals, 0 = -; Note that a equals zero at y = δ and y 0 ρy
U 0 (b) equals at y = 0 y ρ
y U From continuity V = - dy 0 x Dept. of Aerospace Engg., Indian Institute of Technology, Madras 49
Turbulence Prof. E.G. Tulapurkara Chapter-3 Hence, Eq.(C.123) becomes :
δ UUU y dU U - dy - Ue0 dy = - (C.124) x y xe dx ρ y=0 0 Integrating by parts yields :
y δ UUUU δ dydy=Ue dy-U dy y x x x 0 00 Thus, Eq.(C.124) becomes δ UUdU 2U -U -Ue dy = - 0 ee 0 x x dx ρ δδ dU UU-U dy+e U-Udy=- o Or ee (C.125) 00x dx ρ δ U The displacement thickness (δ or δ** ) is defined as : (δ or δ ) = 1- dy 11 (C.126) 0 Ue δ UU The momentum thickness δ or θ is definedas: δ or θ = (1- )dy 22 (C.127) 0 UUee Substituting from Eqs.(C.126) and (C.127) into Eq.(C.125) gives: d dU 0e= (U2*θ)+δ U (C.128) ρ dxee dx
0 Further, skin friction drag coefficient (Cf ) is defined as : Cf = 2 (C.129) 1/ 2ρUe δ* The shape parameter (H) is defined as : H = (C.130) θ
Consequently, Eq.(C.128) becomes dθθ dU C + (H+2)e = f (C.131) dx Ue dx 2 This equation is also know as Von Karman momentum integral equation. Referring to Cebeci and Smith (Ref.C.8, chapter 3) the momentum integral equation for two- dimensional compressible flow is : dθθ 2 dUe Cf + (H+2-Me ) = (C.132) dx Ue dx 2 Reference C.8, chapter 3 may be referred to for integral equation for mean energy .
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Turbulence Prof. E.G. Tulapurkara Chapter-3 C.11 Vorticity transport equation
While solving the Navier-Stokes equations for incompressible flow, one encounters the difficulty that there is no separate equation for pressure. This difficulty is overcome in several ways. In 2-D flows it is possible to eliminate it (pressure) and work in terms of vorticity component (ς ) and stream function (). An equation for vorticity, derived from the Navier-Stokes equations, is called vorticity transport equation. It can be derived, in 2-D cases, using the following steps. The 2-D N-S equations are: UV + = 0 (C.133) xy
U U U 1 p 22 U U +U + V = - + 22 + (C.134) t x y ρ x x y
V V V 1 p 22 V V +U + V = - + 22 + (C.135) t x y ρ y x y
Differentiating Eq.(C.134) with y and Eq.(C.135) with x yields the following two equations.
UUUUVUU 22 12 p 3 U 3 U = - +ν+ + +U + + 2 23 (C.136) y t y x x y y y y ρ x y y x y
VUVVVVV 22 12 p 3 V 3 V + +U + + = - + 32 + (C.137) x t y x x y x y x y ρ x y x x y Subtracting Eq.(C.136) from (C.137) yields:
VUUUVVUVVUVU -- +- ++U -+V - txyyxyxxy xxy yxy
22 VUVU = 22 (C.138) x x y y x y
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Turbulence Prof. E.G. Tulapurkara Chapter-3
UV VU From Eq.(C.133), + = 0 , and noting that ς = - , Eq.(C.138) can be xy xy rewritten as :
ς ς ς 22 ς ς +U + V =22 + (C.139) t x y x y Equation (C.139) is called vorticity transport or vorticity transfer equation. ψψ Now,from definition of stream function, U = , V = - (C.140) yx
VU 22ψψ Hence, ς = - = - - = -2 ψ (C.141) x y x22 y Substituting for from Eq.(C.141) in Eq.(C.139) yields: ψ 2 ψ ψ 2ψ+ - 2 ψ = 4 ψ (C.142) t y x x y Remarks: (i) Equation (C.142) involves only one dependent variable viz. ψ . It can be solved numerically, with appropriate boundary conditions. This gives . Subsequently, the velocity components can be obtained using Eq. (C.140). The pressure can be obtained in terms of velocity components as (see section C.12 for derivation) : 2 2 2 UVUV p = - + +2 (C.143) x y y x
(ii) Taking curl of Eq.(C.52) , the following vorticity transport equation in three- dimensional flow is obtained. Dω In vector form : =ω . V +2 ω ; ω = ξ i +η j +ς k C.144 Dt 2 ωi ω i U i ω i In tensor form : +Ujj = ω + C.145 t xj x j x j x j Equation (C.139) is the vorticity transport equation in 2-D case. Equation (C.144) is the vorticity transport equation in 3-D case. Comparing these equations, it is noticed that the first term on the right hand sides of Eq.(C.144) is absent in the two dimensional
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Turbulence Prof. E.G. Tulapurkara Chapter-3 case (Eq. C.139). This term {ωV. } represents stretching of vortex lines by mean flow gradients. Absence of this term has implications for turbulent flows.
C.12 Poisson’s equation for pressure in incompressible flow
Consider the unsteady N-S equation for incompressible flow (Eq.C.51) and assume that the body force terms are negligible. Now, (a) differentiate the first equation of Eq.(C.51) by „x‟, (b) differentiate second of Eq.(C.51) by „y „ , (c) differentiate third of Eq.(C.51) by „z „ and (d) add the resulting three equations. The following equation is obtained after UVW noting that in an incompressible flow is zero. x y z 2p 2 p 2 p + + = xyz2 2 2
2UVW 22UV 2 UW 2 UV 2 2 2 VW 2 UW 2 VW 2 2 -ρ + + + + + + + + x2 xy xz xy y 2 yz xz yz z 2 (C.146) This equation can be written in tensor form as :
2 2 p UUij =-ρ (C.147) xj x j x i x j
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Turbulence Prof. E.G. Tulapurkara Chapter-3 Remarks: (i) Equation (C.147) does not involve time. Thus, at each instant of time the pressure field is determined by corresponding velocity field. In other words pressure is a global variable and approximating pressure in terms of local velocity field may not be accurate. This has implications while formulating the models of turbulence. (ii) Equation (C.147) has the form of a Poisson‟s equation. Hence, this equation is called Poisson‟s equation for pressure. (iii) If the body force field is not negligible or the coordinate system is not Cartesian, then additional terms would appear in Eq.(C.147). Reference C.9, chapter 2 and Ref. C.10, chapter 2 be referred to for further details.
General remark : Reference C.11, chapter 13 be referred to for derivation of equations for the following additional quantities.
1 U U Rate of strain : S =i + j ij 2 xji x Vorticity : Ω=×V 1 Enstrophy : 2 ii
References
C.1 Schlichting,H. and Gersten,K. “Boundary layer theory” 8th edition, Springer Verlag, Berlin, 2000. C.2 Duncan, W.J., Thom, A.S. and Young, A.D. “An elementary treatise on the mechanics of fluids”, ELBS. London, 1975. C.3 Shapiro. A.H. “Basic equations of fluid flow” In Hand Book of Fluid Dynamics edited by V.L.Streeter, McGraw Hill, New York 1961. C.4 Hirsch, C. “Numerical computation of internal and external flows, Vol.I” John Wiley, New York 1989. C.5 Muralidhar K, and Sundararajan T. “Computational fluid flow and heat transfer” 2nd edition, Narosa Publishing House, New Delhi, 2003. C.6 Pletcher R.H, Tannehill, J.C. and Anderson, D.A. “Computational fluid mechanics and heat transfer”, Taylor & Francis, CRC Press, Boca Raton, 2013. Dept. of Aerospace Engg., Indian Institute of Technology, Madras 54
Turbulence Prof. E.G. Tulapurkara Chapter-3 C.7 Bertin, J.J. and Smith, M.L. “Aerodynamics for engineers”, Prentice-Hall, 2nd Edition, Upper Saddle River, NJ, 1989. C.8 Cebecci, T. and Smith, A.M.O. “Analysis of turbulent boundary layers” Academic, New York, 1974. C.9 Pope, S.B. “Turbulent flows” Cambridge University Press, Cambridge, U.K 2000. C.10 Libby, P.A. “Introduction to turbulence” Taylor and Francis, Washington, D.C 1996. C.11 Tsinober, A., “An informal conceptual introduction to turbulence”, 2nd Edition, Springer, Dordrecht, 2009. C.12 Tulapurkara, E.G. “Hundred years of boundary layer – Some aspects “ Sadhana, Vol. 30, pp. 499 – 512, 2005. This article can be downloaded from internet (www.google.com)
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