concise fluid mechanics

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Concise Fluid Mechanics

A.V.Smirnov

c Draft date September 12, 2004


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Contents

Contents i

Preface v

Nomenclature vii

1 Properties and Variables 1

1.1 Kinematic variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Substantial derivative . . . . . . . . . . . . . . . . . . . . . 3

1.1.3 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.4 Strain rate and vorticity . . . . . . . . . . . . . . . . . . . . 4

1.2 Thermodynamic Variables . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Equations of state . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Fluid Proper ties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Thermodynamic properties . . . . . . . . . . . . . . . . . . 8

1.3.2 Transport Properties . . . . . . . . . . . . . . . . . . . . . . 9

1.3.3 Other properties . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Fundamental Laws 15

2.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . 15

i


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ii CONTENTS

2.1.1 General formulation . . . . . . . . . . . . . . . . . . . . . . 15

2.1.2 Constant density flow . . . . . . . . . . . . . . . . . . . . . 16

2.1.3 Stream function . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 General formulation . . . . . . . . . . . . . . . . . . . . . . 182.2.2 Constant density flow . . . . . . . . . . . . . . . . . . . . . 21

2.2.3 Vorticity formulation . . . . . . . . . . . . . . . . . . . . . . 21

2.2.4 Potential flow . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.5 2D limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.6 Viscous limit . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.7 Inviscid limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.8 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 29

2.3 Pressure Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32


2.3.2 Constant density flow . . . . . . . . . . . . . . . . . . . . . 32

2.3.3 Viscous limit . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 33

2.4 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


2.4.2 Constant density flow . . . . . . . . . . . . . . . . . . . . . 372.4.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 39

2.5 Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 39

2.5.1 Invariant forms . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5.2 Non-inertial coordinate systems . . . . . . . . . . . . . . . 40

2.6 The Law of Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.6.1 PI-Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.6.2 Non-dimensional formulations . . . . . . . . . . . . . . . . . 46

2.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Laminar flows 53


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CONTENTS iii

3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Confined flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.1 Flow between parallel plates . . . . . . . . . . . . . . . . . 54

3.2.2 Axially moving concentric cylinders . . . . . . . . . . . . . . 56

3.2.3 Rotating concentric cylinders . . . . . . . . . . . . . . . . . 57

3.2.4 Poiseuille flow through ducts . . . . . . . . . . . . . . . . . 59

3.2.5 Combined Couette-Poiseuille flows . . . . . . . . . . . . . 63

3.2.6 Non-circular ducts . . . . . . . . . . . . . . . . . . . . . . . 64

3.3 Unsteady flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3.1 Fluid oscillation above infinite plate . . . . . . . . . . . . . . 66

3.3.2 Unsteady flow between infinite plates . . . . . . . . . . . . 68

3.4 Creeping flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.4.1 Stokes flow around a sphere . . . . . . . . . . . . . . . . . 72

3.4.2 2D Creeping flows . . . . . . . . . . . . . . . . . . . . . . . 76

3.4.3 Lubrication theory . . . . . . . . . . . . . . . . . . . . . . . 76

3.5 Boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.5.1 Flat plate integral analysis . . . . . . . . . . . . . . . . . . 81

3.5.2 Laminar boundary layer equations . . . . . . . . . . . . . . 85

3.5.3 Blasius solution . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.5.4 Reynolds analogy . . . . . . . . . . . . . . . . . . . . . . . 92

3.5.5 Free shear flows . . . . . . . . . . . . . . . . . . . . . . . . 94

3.6 Integral methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4 Turbulent flows 105

4.1 Transition to turbulence . . . . . . . . . . . . . . . . . . . . . . . . 105

4.2 Turbulence Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 1054.2.1 LES models . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.2.2 RANS models . . . . . . . . . . . . . . . . . . . . . . . . . 107


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iv CONTENTS

Bibliography 113

A Introduction to Tensor Calculus 115

A.1 Coordinates and Tensors . . . . . . . . . . . . . . . . . . . . . . . 116

A.2 Cartesian Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

A.2.1 Tensor Notation . . . . . . . . . . . . . . . . . . . . . . . . 120

A.2.2 Tensor Derivatives . . . . . . . . . . . . . . . . . . . . . . . 126

A.3 Curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . 128

A.3.1 Tensor invariance . . . . . . . . . . . . . . . . . . . . . . . 128

A.3.2 Covariant differentiation . . . . . . . . . . . . . . . . . . . . 132

A.3.3 Orthogonal coordinates . . . . . . . . . . . . . . . . . . . . 134

A.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

B Curvilinear coordinate systems 143

C Solutions to problems 147

D Midterm Exam Topics: Laminar Flow Solutions 193

E Final Exam Topics 197

E.1 Fundamental Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

E.2 Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 198

E.3 Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

E.4 Turbulence Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Index 203


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Preface

The idea behind this book is to provide a formal but concise introduction to theo-retical fluid mechanics.

The book covers the traditional topics of fluid mechanics, such as the funda-mental equations of motion, compressible and incompressible forms, and invari-ant formulations, special analytical solutions, laminar flows, elements of boundary

layer theory, main aspects of turbulence modeling and numerical methods. The

emphasis is on viscous flow phenomena.

The author tried to put more emphasis on mathematical rigor rather thanon lengthy narrative. Tensor notation is used extensively throughout the book.

However, the knowledge of tensor calculus is not required of the reader, sinceenough introductory material is provided in the appendix.

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vi


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Nomenclature

is defined as , or is equivalent to

. Note:

partial derivative over time:

partial derivative over :

control volume

time-th component of a coordinate ( =0,1,2), or

fluid velocity:

strain tensorany variable of coordinates and timestress tensor

viscosity: kinematic viscosity

RHS Right-hand-sideLHS Left-hand-side

NS Navier-Stokes equation

PDE Partial differential equation.. Continued list of items

vii


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Chapter 1

Properties and Variables

Consider a dimensional space of real numbers representing physicalspace of dimension ( =2,3) and time. The state of the fluid will be represented

by real functions of coordinates and time ( , ), continuous with theirderivatives up to the second order. This is an Eulerian description in which both

and represent a set of independent variables. In an alternative Lagrangian

descriptionthe fluid is specified by a set of moving particles. These fluid particlesform a continuum and their coordinates are themselves functions of time. Conse-

quently, time becomes the only independent variable in this case. Our objectivewill be to formulate the laws of fluid motion in Eulerian, fixed-space coordinates.

The set of independent variables can be extended beyond space coordi-nates and time by introducing properties of the fluid. These properties describedifferent physical processes and are classified accordingly as thermodynamic

properties, transport properties, etc.

Dependent variables are functions of the independent variables implicitly

expressed in a physical law. Dependent variables can also be classified accordingto the physical process they describe, and we shall consider only two types:kinematic variables and thermodynamic variables.

1.1 Kinematic variables

1.1.1 Velocity

Lets define a fluid particle as an infinitely small element of the fluid.

1


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2 CHAPTER 1. PROPERTIES AND VARIABLES

Definition 1.1.1 Fluid velocity

Fluid velocity, , is equal to the velocity of a fluid particleat a given point in space and time, :

(1.1)

The definition above can be inverted to define a particle trajectory.

Definition 1.1.2 Particle trajectory

For a given vector field of fluid velocities, , particle trajectory, , is

a solution to the following problem:

From these definitions it follows that the vector of velocity is tangential toparticle trajectory at each point in the fluid. Particle trajectory is also called astreamline.

In what follows we shall drop the super-index , and also adopt the followingdot-notation for particle velocity:

(1.2)

meaning that a time derivative of fluid particle position is taken along the particles

trajectory at a space point . This notation should not create a confusion sincespace coordinates do not depend on time, while the fluid-particle coordinatesdo. Hence when using time derivatives of coordinates, as in (1.2), we will mean

fluid particle coordinates.

We shall use the same dot notation for partial time derivatives of other fluid

variables at a fixed point in space. For example, for any variable,

(1.3)


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1.1. KINEMATIC VARIABLES 3

1.1.2 Substantial derivative

Definition 1.1.3 Fluid element

Byfluid element we shall understand a finite volume of a fluid, which is small

enough that the velocity of all its points can be approximated by the velocity of a

single fluid particle inside the element.

Partial derivatives (1.3) describe changes in a variable at a fixed space pointattributed to the explicit time dependence of the variable. There are also changes

brought about solely by the motion of the fluid, i.e. due to the fact that differentfluid elements cross the given space point, changing the fluid variable at that

point. To account for all the changes we introduce a substantial derivative. It isequal to the rate of change of fluid variable inside a fluid element moving with thevelocity of the fluid.

Consider a change of a fluid variable inside a fluid element as the

fluid element moved to a nearby position :

(1.4)

where we used a Taylor expansion up to the first order. We can rewrite it in amore compact tensor notation (Sec.A):

(1.5)

Considering that the displacement follows the fluid element, it should bea product of velocity and time: . Then we can rewrite the expression

above in terms of variable change, , as follows:

(1.6)

And dividing both sides by , and taking the limit of we have:

(1.7)


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this expression represents a substantial derivative of a fluid variable, which de-

scribes the change of that variable in the coordinate system moving with the fluidelement. The last term , in (1.7) is also called a convective derivative.

1.1.3 Acceleration

Applying (1.7) to velocity itself, we have the relation for flow acceleration:

(1.8)

where we used the dummy index rule (A.2.16).

1.1.4 Strain rate and vorticity

We can formally represent a velocity derivative as a sum of a symmetric andan asymmetric parts:

where these parts becomes the newly introduced strain rate ( ) and vorticity

( ) tensors.

Definition 1.1.4 Strain rate tensor

Thestrain rate tensor, , is defined as:

(1.9)

Definition 1.1.5 Vorticity tensor

Thevorticity tensor, is defined as

(1.10)


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1.2. THERMODYNAMIC VARIABLES 5

Definition 1.1.6 Vorticity vector

Thevorticity vector, , is defined as

(1.11)

From this definition, and the definition of vorticity tensor (1.10), we have:

(1.12)

Using the definition (A.23) of the permutation tensor we can write thecomponents of (1.11) explicitly as:

1.2 Thermodynamic Variables

Classical thermodynamics was formulated for equilibrium states. Even though

fluid flow is not generally in equilibrium, we can apply thermodynamical conceptsusing a quasi-equilibrium approximation, which assumes that the flow changes

slowly enough, so that at each point a local thermodynamic equilibrium is reached.

1.2.1 Equations of state

The equation of state relates important thermodynamic variables, such as pres-

sure, , temperature, , and density, :

(1.13)


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such as an ideal gas law:

(1.14)

where is the gas constant. Depending on the number of parameters in the

system, there can be several equations of state so as to keep the number ofindependent variables, to comply to the Gibbs rule [1]:

where is the number of components and is the number of phases in the

system. For example, for the water-vapor mixture we have and, then . Thus the state of water-vapor mixture can be completely

described by the pressure or temperature only.

1.2.2 Energy

The first law of thermodynamics expresses the principle of energy conservation

related to the thermodynamic variables: internal energy1, , heat, , and work,:

(1.15)

that is, the change of energy of the fluid element is equal to the heat inflow plus

the work done on that element. Rewriting this in terms of specific values, i.e.values related to the unit of mass ( ), we have:

(1.16)

Considering only the mechanical work, we have

where is the specific volume: , and the minus sign signifies

that the work done on the system is positive when it is compressed and negativewhen it is inflated. From the definition of entropy[1] we have:

1In thermodynamics books it is usually denoted by , but we reserve this symbol for fluid velocity


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1.2. THERMODYNAMIC VARIABLES 7

Thus (1.16) becomes:

(1.17)

which can be expressed in terms of density:

(1.18)

This relation implies that is a function of and only:

(1.19)

Relation (1.19) constitutes the equation of state as dictated by the energy con-servation law.

Another form of energy identified in thermodynamics is enthalpy. It is definedas:

(1.20)

and analogously to (1.17), we obtain:

(1.21)

Expressing the entropy, , from relations (1.17) and (1.21) we obtain the

so-called relations, well known in thermodynamics:


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1.3 Fluid Properties

Fluid properties are given as free parameters in a physical law. Along with thespace coordinates and time they form the set of independent variables of thesystem.

1.3.1 Thermodynamic properties

Specific heat at constant volume determines the amount of thermal energythat is needed to be transfered to the substance to heat it up by one degree,while keeping it at a constant volume:

(1.22)

For ideal gases the internal energy, depends only on temperature, and the

relation above can be rewritten as:

(1.23)

Specific heat at constant pressure, , is defined as the amount of energy

needed to be supplied to the substance to heat it up by one degree at constantpressure:

(1.24)

where the enthalpy, , is used instead of internal energy, , since it accounts forthe work done by the pressure forces to extend/compress the substance. For an

ideal gas this translates to:

(1.25)


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1.3. FLUID PROPERTIES 9

1.3.2 Transport Properties

A general form for the transport law is presumed to obey the gradient approxima-

tion in a form:

(1.26)

where is the kinematic or thermodynamic property, which represents a depen-dent variable of the problem, flux is the amount of property passed through aunit area per unit time and gradient is the direction of maximum change in prop-

erty. Both flux and gradient are a vectors. The coefficient of proportionality,represents the transport property controlling the transport process (1.26).

The coefficient of viscosity

The coefficient of viscosity is introduced to quantify the process of momentum

transport.

Newtonian fluid: In elasticity theory the resistance force is proportional to defor-

mation:

Similarly in fluid mechanics the resistance to fluid motion is proportional to

the velocity change in the direction normal to the fluid motion (strain).

Since generally the stresses and strains are tensors (Sec.1.1.4, 2.2.1), the

relation above is usually written as:

(1.27)

Definition 1.3.1 Newtonian fluid

A fluid with the linear relationship between stresses and strains, like (1.27)is called aNewtonian fluid.


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Definition 1.3.2 Coefficient of viscosity

The coefficient of viscosity is introduced as a proportionality constant be-tween the shear stress and strain. Considering one component of stress tensor

at: i=1, j=2, we have:

The kinematic viscosity is defined as

Using a more common notation for vector components:we have:

(1.28)

Viscosity usually decreases with temperature for liquids and increases forrarefied gases [2].

Non-newtonian fluid: For non-Newtonian fluids the relation between the stressand the strain is non-linear, for example

Thermal Conductivity

Following the gradient approximation (1.26), we presume the heat transport toobey the relation:

which in quantitative terms becomes:

(1.29)

where is the wall heat flux per unit of time ( ). This is the ex-pression of the Fouriers law, where is a heat conduction coefficient, or heat


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(1.32)

and relating the mass flux to the fluid velocity across the boundary: ,

we have:

There are two non-dimensional numbers relating the momentum-to-massand mass-to-heat transport processes:

Schmidt number: momentum transport / mass transort

Lewis number: heat transport / mass transport

1.3.3 Other properties

Speed of sound

The speed of sound for compressible flow is defined as the rate of propagation ofsmall pressure perturbations, and it is found to be equal to [3]:

where


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1.4. PROBLEMS 13

Bulk modulus

The bulk modulus expresses the change of density with increasing pressure at a

constant temperature:

and is used in acoustic problems.

Coefficient of thermal expansion

The Coefficient of thermal expansion relates density to temperature changes:

(1.33)

and is used in the problems of natural convection.

1.4 Problems

Problem 1.4.1 Mass diffusivity in terms of concentration

Show how to obtain (1.31) from (1.32).

Problem 1.4.2 Lubrication

A plate of mass with an area slides down a long

incline at angle , on which there is a film of oil of thickness ,with viscosity . Assuming the plate does not deform the oil

film estimate (1) the terminal sliding velocity , and (2) the time required for the

plate to accelerate from rest to of the terminal velocity.


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Chapter 2

Fundamental Laws

2.1 Conservation of Mass

2.1.1 General formulation

The conservation of mass dictates that:

which also means that

The total change of mass inside the control volume will consist of changes

of mass inside the volume because of density changes that may occur at eachpoint of the flow, and the influx or out-flux of mass through the boundary. This can

be expressed as:

where is the unit normal vector to the boundary, is the surface area element,and the last integral spans all the boundary of the control volume.

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16 CHAPTER 2. FUNDAMENTAL LAWS

Lets define the surface area vector, as:

(2.1)

By Gauss theorem we can convert the last integral to the volume integral:

and finally:

Considering the arbitrary nature of the control volume selection, we con-clude:

(2.2)

This is a general relation of mass conservation valid for both compressibleand incompressible flows. Differentiating the second term by parts, and using therelation of substantial differentiation (1.7) the latter can be rewritten as:

(2.3)

2.1.2 Constant density flow

For a constant density flow , and from (2.2) it follows:

(2.4)

which is also called the continuity equation or incompressibility condition. Vector

field satisfying (2.4) is also called solenoidalor divergence-free.

Another form of this relation can be obtained by combining (2.3) and (2.4):

(2.5)


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2.1. CONSERVATION OF MASS 17

2.1.3 Stream function

Lets introduce the stream function, which is closely related to the mass flow rate.

The stream function , which is a vector in 3D, also called the streamline-

vorticity function is defined such as to satisfy the relation

(2.6)

or, using the nabla operator (A.32):

(2.7)

which analogously to (1.13) is

(2.8)

In two dimensions the streamline function is defined as

i.e. a vector normal to the plane, and therefore (2.8) are reduced to:

(2.9)

The following relation between the stream function and the mass flow ratecan be shown for a two dimensional case:

(2.10)

where is the element of the surface normal to the velocity . This relation canalso be proved more rigorously for a 3D space.


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Remark 2.1.1 Existence of stream-function

It should be noted that sometimes, instead of definition (2.6), the stream-function is defined from a simpler relation:

(2.11)

where the density is omitted. In this case the stream function can only be used todescribe incompressible flow. This can be shown by computing the divergence ofvelocity vector :

which is true due to the symmetric identity (A.27) and the symmetry of with

respect to the order of differentiation. This becomes especially obvious in a 2D

case:

Thus, in terms of definition (2.11) the stream function can only exist for in-

compressible flows.

2.2 Conservation of Momentum


According to Newtons law a particle of mass is accelerated by the action of aforce as:

which will apply to a particle of both constant and a variable mass. Applying this

to a fluid particle of density and velocity in a small control volume, we have

(2.12)


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2.2. CONSERVATION OF MOMENTUM 19

The forces acting on a fluid element come from the possible external forces,

like gravity, electromagnetic fields, etc. (body forces), and forces caused by theinteraction of this fluid element with neighboring fluid elements or boundaries

(surface forces). Body forces relate to the unit of volume and surface forces relate

to the unit of area.

(2.13)

where is the volumetric density of the body force . It corresponds to a forcefield like electromagnetic, gravity, etc. Generally it can serve as a source term

connecting this equation to other equations.

Definition 2.2.1 Stress tensor

The surface force term in (2.13) is called the stress tensor.

Using this definition and applying Gauss theorem to the last term in (2.13),

we have:

(2.14)

Now, comparing (2.14) with (2.12), we have

Using the fact the control volume was chosen arbitrarily, the integral sign

could be dropped, and we have:

(2.15)

Using the definition of substantial derivative (1.7), we have:

(2.16)


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Using the hypothesis of Newtonian fluid (1.27), and general considerations

of symmetry for the case of isotropic and homogeneous fluid, a general relationbetween the stress tensor and the strain tensor can be written as [2, p.66]:

(2.17)

where is the pressure, is the coefficient of bulk viscosity, which is only impor-tant for compressible flows.

Sometimes it is convenient to separate the stress tensor (2.17) into thepressure-related and viscous parts:

(2.18)

where is the viscous stress tensor defined as

(2.19)

Parameter is the coefficient of bulk viscosity, which can only be important for

variable density flows [2]. Thus, for incompressible flows (2.19) becomes:

(2.20)

where we used the definition of strain rate tensor (1.9). Using the definition of theviscous stress (2.19) we can write (2.16) as:

(2.21)

When represents the gravity forces: we can rewrite equation

(2.21) as:

(2.22)

Equations (2.2), (2.22) and (2.19) represent a general case of compressible

unsteady flow. The dependent variables are two scalars: density, and pressure,, and a vector - velocity, . Considering that equation (2.21) is a vector equation,

this leaves one more equation to close the system. The appropriate candidate forthis is the equation of state (1.13).


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The viscous term in equation (2.22) can be further simplified for constant densityflows. Using (2.20) we can rewrite it as:

(2.23)

where we used the continuity relation: for incompressible flow (2.4).

Substituting this into (2.22), we have:

(2.24)

which is an incompressible form of momentum equation, also referred to as theNavier-Stokes equation (NS).

2.2.3 Vorticity formulation

Our objective will be to replace the velocity vector in a constant density NS equa-

tion (2.24) with a vorticity vector (1.11). For this purpose consider a cross productbetween nabla operator (A.32) and vorticity vector:

(2.25)

Using (1.12) the cross product (2.25) can be rewritten as:

(2.26)

Using the tensor identity (A.29):


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which can be rewritten as

to match the indexes, we can simplify the cross product (2.25):

(2.27)

And finally (2.26) becomes:

(2.28)

For constant density flows it follows from (2.4) that , and

(2.29)

Thus we can replace the diffusive term in (2.24) with the cross-product

above.

Now lets consider the convective term in (2.24). Using the constantdensity assumption ( ) and now considering the cross product of the type

, we can repeate the steps as in (2.26) and obtain1:

(2.30)

1See Problem 2.7.1


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Thus

(2.31)

And substituting (2.29) and (2.31) into the momentum equation (2.24) wehave:

(2.32)

Rearranging the terms, and using the relation (A.34), we have:

(2.33)

This equation can also be rewritten as:

(2.34)

This is a NS equation in vorticity formulation for the incompressible flow2.

2.2.4 Potential flow

Lets consider the irrotational flowwhere the vorticity vector is zero ( ). Thisflow is also called potential flow, since the velocity vector can be replaced by a

gradient of a scalar function, , also called a velocity potential function:

This is possible, because the gradient of a scalar function also satisfies the condi-tion of zero vorticity, which follows from the definition of the vorticity vector (1.12)

and the symmetry of the second derivative of with respect to order of differenti-ation, :

2Note that we achieved only a partial success in our objective to replacing the velocity vector with the

vorticity vector, but thats the best we can do.


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In the case of steady state incompressible potential flow the continuity con-

dition (2.4) translates into the Laplace equation for the velocity potential:

(2.35)

Thus, the solution of the problem in this case is reduced to finding a single scalar

function from equation (2.35). Another important relation in this case can be ob-tained from equation (2.34), which, after eliminating time derivatives and vorticityterms reduces to:

(2.36)

which is a weak formulation of the Bernoullis Equation3.

2.2.5 2D limit

Lets rewrite (2.33) as

(2.37)

Where we denoted the term in parentheses by . The equality above is a firstrank tensor equality with terms of type . Lets now form a cross product betweenthis equality and of the type (see also (A.32)):

(2.38)

Using the definition of (A.32), we get:

3See also (2.78)


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(2.39)

where we used abbreviations for the last two terms on the RHS. Sinceis a symmetric tensor then by (A.2.22), we have .

Noticing that by the definition of vorticity vector: , we can reduce

the equation (2.39) to the form:

(2.40)

Using the permutation property (A.24): , we have:

where we used the incompressibility condition (2.4): .

In a 2D limit we have and . Lets consider only

the 3-rd component of the equation above, i.e.

(2.41)

Then the term can be simplified as:

Applying the same manipulations to the term, we have:

(2.42)

And in the 2D limit:


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After substituting , into (2.41), it becomes:

(2.43)

which is the 2D limit of (2.37).

Stream-function formulation

By the definition of the stream-function (2.6) we have:

(2.44)

Substituting (2.44) into (1.12) we can find relation between the vorticity vec-tor and the stream-function:

(2.45)

In the 2D limit: , , , and ,

the first of the last two terms in (2.45) will vanish: , and we have:

(2.46)


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Now (2.43) can be rewritten in terms of the stream-function only:

(2.47)

Thus, the flow is now completely defined by a scalar field , which

is obtained as a solution of (2.47). Note that since (2.47) contains 4th orderderivatives of , it involves more complex boundary conditions, and poses

higher differentiability requirements on .

In the case of irrotational flow ( ), equation (2.46) reduces to the

Laplace equation for the stream-function:

(2.48)

with the boundary conditions derived from the relation between and the velocity

field (2.9). Solving the equation for stream function is usually preferred over solv-ing an equation for the vorticity, since the velocity field can be obtained from thestream function by a simple differentiation of type (2.6) or (2.9), whereas obtain-ing the velocity field from the vorticity as in (1.12) would require a more laborious

integration.

2.2.6 Viscous limit

Consider the incompressible NS equation (2.24). In the viscous limit we shall

assume the viscous term to be much larger than the convective term. Thus in theviscous formulation we shall simply neglect the convective terms:

(2.49)

Using the expression of vorticity vector (1.12), we can express the aboveequation in terms of vorticity vector only:

(2.50)

which is an incompressible viscous limit of the NS equation in vorticity formulation(see Problem.2.7.3).


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In the case of steady state flow this equation simplifies to a Laplace equation

for the vorticity vector:

2.2.7 Inviscid limit

The fluid with a zero viscosity is called an ideal fluid, and the flow of such a fluidis called inviscid. Consider equation (2.22) in the limit of inviscid flow, when the

viscous tensor, , vanishes:

(2.51)

In the incompressible limit it will simplify to:

(2.52)

This is Euler equation for inviscid incompressible flow. It can also be rewrit-ten in terms of substantial derivative (1.8):

(2.53)

Conservation of vorticity

It can be shown that the inviscid flow preserves vorticity. For this purpose lets

form a vector product between the nabla operator and equation (2.53):

where the last equality is due to the symmetry of and identity (A.27). We can

transform the term according to:

(2.54)


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where we used the skew-symmetric property of (A.24). Finally, using the

definition of vorticity vector (1.12), we obtain:

(2.55)

which means that the vorticity is conserved ( ). In particular, this means

that if the flow was irrotational ( ), it will remain so4. In this case the problemof inviscid flow can be solved using velocity potential function (Sec.2.2.4).

The momentum flux

Since for incompressible flow (2.4), we can rewrite (2.51) as:

And introducing the momentum fluxas

(2.56)

we have the Euler equation in momentum-flux formulation:

(2.57)

2.2.8 Boundary conditions

Equation system (2.2), (2.21), (2.19) and (1.13) may not have a unique solutionfor any boundary conditions. Generally, the character of the equation system,

i.e. hyperbolic, parabolicor eliptic[4, 5], may change depending on the boundaryconditions and the region of space and time. However, there are several types

of boundaries that are typically considered, and that usually lead to well posedproblems.

4See Problem 2.7.4


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Inlet

At the inlet boundary the value of the velocity is usually specified . This boundarycondition is known as Dirichlet boundary condition. Note, that this is not always

the case, since a pressure can be prescribed as the inlet condition instead, whenthe Poisson equation for pressure (2.60) is used.

Outlet

Depending on the character of the equation system the boundary conditions mayor may not need be specified at the outlet boundary. The most common outlet

boundary condition is the condition of the zero boundary-normal velocity deriva-

tive (Neuman boundary).

In more complex flow situations there may not be a clear distinction between

the inlet and the outlet, since the flow may reverse. These types of situations are

hard to solve in a consistent manner and should be avoided by repositioning theinlet/outlet of the domain so as to comply to either Dirichlet or Neuman boundary

conditions.

Fluid-solid interface (Wall)

In the case of a fluid-solid boundary the flow velocity is set equal to the velocity of

the wall, which covers the cases of both stationary and moving boundaries. Thisboundary condition is called a no-slip boundary condition.

In some cases a finite velocity jump may be imposed at the boundary, inwhich case this is called a slip boundary condition.

The specification of the velocity alone at the boundary may not be enough,since the momentum equation (2.21) contains second order velocity derivatives

in the viscous term ( ), which means that the first order derivatives should be

given at the boundary. However, with the no-slip condition at the wall, the velocityderivatives at the wall can be considered zero. This follows from the continuity

relationship (2.4). For example, if we consider velocity components and asbeing parallel to the wall, and normal to the wall, then from the no-slip conditionwe have:

and consequently:


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Thus from continuity (2.4) we must have:

In cases when the forces on the wall need to be estimated they can berelated to the boundary normal forces due to pressures, and shear forces thatcan be related to the stress tensor via (2.14).

It should be noted that if the boundary is moving with acceleration additional

non-inertial terms should be introduced into the boundary conditions (Sec.2.5.2).

Fluid-fluid interface (Free surface)

The gas-liquid or liquid-liquid boundary conditions are also called free-boundary

conditions. They consist of the requirements that the pressure, velocities and

the fluxes of mass and momentum be continuous functions across the interface.

This means that these quantities should have the same values on both sides ofthe interface. The position of the interface surface will then be determined as asolution to the flow equations subjected to the free surface boundary conditions.

In cases where surface tension effects are important, they should enter intothe pressure boundary condition, namely the extra boundary pressure should be

added on both sides of the interface. This pressure should be inversely pro-

portional to the local surface curvature. Since the surface curvature generally

depends on the direction selected on the surface to measure the curvature, oneform of its estimate may be to set it proportional to the sum of inverse curvatureradii in two orthogonal directions:

(2.58)

where is the coefficient of surface tension. Note, that the forces resulting fromthe pressure terms should always act normal to the surface. Shear forces at theboundary are usually considered to be zero.


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2.3 Pressure Equation


In the solution of the equation system (2.2), (2.21), (2.19), and (1.13), is compli-

cated by the fact that the equation of mass conservation (2.2) does not containpressure, and the equation of momentum conservation (2.21) contains both ve-

locity and pressure [6]. In the case of compressible flow the equation of state(1.13) can be used as an additional relation between pressure and density. How-ever, for the incompressible flow the continuity equation (2.4) has velocity only,

and can not be effectively used in combination with the momentum equation.

To make the equation system better conditioned, the continuity equation (2.4) isusually replaced by the Poisson equation for pressure. To obtain the equation

for pressure, lets apply the divergence operator, , (A.2.32) to themomentum equation (2.22):

(2.59)

After substituting the first term on the LHS from (2.2), considering that isa constant, and rearranging terms, we have:

(2.60)

which is a Poisson equation for pressure. It has to be solved together with themomentum equation (2.21) and the relation of the state law (1.13).


As it was pointed out the pressure equation is mainly used for incompressibleflows where it replaces the continuity equation (2.4). In this case we can simplifythe pressure equation (2.60) by applying the continuity condition to (2.60):


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2.3. PRESSURE EQUATION 33

Using the incompressible form of the viscous stress tensor , (2.20), and thecontinuity relation, , it can be shown that the last term will be zero:

and finally we obtain:

(2.61)

This is the incompressible form of the pressure equation, also called the Pois-

son equation for pressure. It should be considered together with the momentumequation (2.24).

2.3.3 Viscous limit

Considering the viscous limit (2.49), and taking divergence of this equation, we

have:

(2.62)

which is the Laplace equation for pressure.


Pressure equation is an elliptic second order PDE. As such it requires the spec-ification of two sets of boundary conditions, which usually are the values of thepressure and its boundary normal derivatives.

At the solid walls the boundary-normal derivative of pressure is usually setto zero, which is a Neuman boundary condition. The value of pressure at the wall


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comes out as the solution which satisfies this condition. However, because the

Poisson equation is a second order equation, the Neuman boundary conditionalone will result in an indeterminate solution, when adding any constant to the

pressure will still satisfy the equation. Fixing the value for the pressure in at least

one point will remove this uncertainty. Thus, other conditions at the inlet/outletboundaries are usually applied. At the inlet/outlet pressure values are given and

the boundary normal derivatives are usually set to zero.

In the case when the velocity is also specified at the inlet/outlet, both pres-

sure and velocity specifications should be consistent so as not to create a over-

defined problem. Specifying either pressure or velocity alone will be enough in

many cases. However, this will depend on the character of the flow and the dis-cretization scheme used to solve the equations [4].

2.4 Energy Equation


Usually the flow field is a carrier for the transport of other variables of the contin-uum media. One important variable is energy.

The balance of energy in a control volume can be written as

(2.63)

where the LHS is the rate of energy change inside the control volume, and thetwo terms on the RHS represent total heat inflow into the control volume and work

done on it. This relation is also known as the second law of thermodynamics.Note that both heat and work are transported into the control volume through itsboundary, so they can be represented by flux-vectors.

Energy:

(2.64)


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2.4. ENERGY EQUATION 35

where the minus sign in front of the gravity term means the potential energy in-

creases as we move against the gravity force.

Heat: The total heat change inside the volume can be related to the heat fluxthrough the boundary of the volume:

(2.65)

where is the heat flux, which is the rate of heat inflow through a unit area perunit of time, and the vector-element of the boundary introduced in (2.1). Minussign occurs because of the convention of surface normal vectors to point outside

of the volume, meaning that is actually the out-flux of heat whereas wasdefined as an incoming heat by the virtue of (2.63). Applying the Gauss theorem

to (2.65) we have:

(2.66)

It is postulated that the heat flux is proportional to temperature gradient:

where the minus sign signifies that the heat flows from higher to lower tempera-

tures. This relation is known as the Fouriers law. Substituting it into (2.66), we

have:

(2.67)

Work: In analogy to (2.65) we introduce the flux of external work through thefluid element:

(2.68)

The work flux vector through the area can be computed as


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We saw in (2.2.1) that the surface forces are described by the stress tensor

. The time derivative of the displacement of fluid element is given by velocity(1.1). Thus5:

(2.69)

Applying Gauss theorem to (2.68) and combining it with (2.69) we obtain:

(2.70)

Differentiating by parts, we have:

(2.71)

We can eliminate the derivative of : , by expressing it from (2.15) withreplaced by the gravity force :

where we used the symmetry of : . Now we can rewrite (2.71) as:

Substituting the above into (2.70), we have:

(2.72)

Combining (2.64), (2.67) and (2.72), we have

5Note that the positive signs in (2.68) and (2.69) follow the convention that the work done on the system

is positive when the direction of external force coincides with the direction of displacement.


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2.4. ENERGY EQUATION 37

which after rearranging terms and dropping the integration sign due to the arbi-trariness of our choice of the control volume becomes:

And after canceling the same terms on both sides, we have:

(2.73)

which is the equation for the rate of change of energy density valid for both com-pressible and incompressible fluid.

In an ideal gas approximation we can express the LHS in terms of tempera-ture using the thermodynamic relation (1.23):

(2.74)

It is useful to express (2.73) in terms of enthalpy (1.20). For this purpose wecan add to both sides of (2.73) and obtain:

(2.75)

which is the equation for the rate of change of enthalpy valid for both compressible

and incompressible fluid.


Lets substitute from (2.17) into (2.73), and consider that :

(2.76)


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Using the definition of viscous stress (2.19), we can rewrite this as:

which after applying the continuity relation (2.4) this reduces to:

(2.77)

where we should use the incompressible form of viscous stress (2.20):

Equation (2.77) can be used in combination with the momentum equation

(2.21) to derive a strong form of the Bernoullis equation6:

(2.78)

We can also rewrite (2.77) in terms of temperature change, and velocity. Us-

ing the definition of specific heat (1.25), substituting the velocity from the relationof viscous stress (2.20), and assuming , we have:

(2.79)

This is an incompressible heat conduction equation.

Heat dominated flow

In the cases with small pressure and velocity gradients, or when fluid viscosityis small compared to the heat conductivity, which corresponds to small Prandtl

number (1.30), equation (2.79) simplifies to:

(2.80)

6See Problem 2.7.5


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2.5. CURVILINEAR COORDINATES 39

where we also presumed that the heat conduction coefficient, , is a constant.

This is a limit case of (2.79) for the case of heat dominated flow. Note that sincethe substantial derivative is used for T, we still have the convective terms present:

(2.81)

which is also called a heat convection equation.


Generally temperature may experience a jump at the boundary [2]. Usually thisjump is small and the temperature of the fluid at the wall is considered to be equalto the temperature of the wall.

Since the equation (2.81) is a second order differential equation, we wouldneed to specify temperature derivatives in addition to specifying temperature val-

ues at the boundary. These conditions can be obtained from the considerationof energy conservation. In particular, heat flux across the boundary should be

conserved. Thus from (2.81) we obtain

(2.82)

where is a boundary-normal unit vector, and is the heat flux across the bound-ary.

2.5 Curvilinear Coordinates

Physical laws should not depend on the choice of a coordinate system. This is

expressed in the terminology of tensor calculus as coordinate invariance. Tensorsare designed to be invariant under coordinate transformations (Remark A.3.3).Therefore, tensor relations provide a consistent way of writing physical laws.

There are two aspects of expressing physical laws in tensor forms: identify-ing , physical components, and forming invariant expressions.


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2.5.1 Invariant forms

The scalar product (Definition A.3.4) was constructed to be invariant. By virtue

of its invariance it represents a physical entity. Using the invariant forms of thescalar product (Corollary A.3.5), we can rewrite the expression for the substantialderivative(1.7) in invariant form:

(2.83)

Correspondingly, the mass conservation equation (2.2) will be expressed as

(2.84)

and the momentum equation (2.22) becomes:

(2.85)

where the covariant and contravariant velocities and stress tensors are linked by

the conjugate tensor relations (A.42), (A.43):

where and is the metric tensor (A.40) and its conjugate (A.38). A moredetailed description of curvilinear coordinate systems including an important caseof an orthogonal coordinate system can be found in (App.A.3.3). A derivation of

Laplace operator in cylindrical coordinates using chain differentiation is given in(Sec. B).

2.5.2 Non-inertial coordinate systems

Generally, if we consider space and time as a single continuum represented bya single set of coordinates, we have no distinction between the curvilinear and


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have , and (2.87) becomes

(2.89)

Assuming that the rotation is constant:

we can differentiate (2.89) further to obtain

(2.90)

where

(2.91)

and using 2.88 we have

(2.92)

Substituting (2.88), (2.91) and (2.92) to (2.90) we get

(2.93)

The extra acceleration terms involving arise due to rotation and are interpreted

as being the result of the Coriolis force.

An important special case of a non-inertial coordinate systems is a coordi-nate system undergoing a pure rotation with a constant angular velocity. Assumethat the moving coordinate system is undergoing a rotation with and

. Then we can align the origins of the two coordinate system to


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2.6. THE LAW OF SIMILARITY 43

eliminate altogether. Lets also assume that the axis of rotation is normal to the

plane , that is . With these assumptions (2.93) becomes

(2.94)

Where and are the additional acceleration vectors (Problem 2.7.7). Thecorresponding Coriolis forces are introduced into the equations of motion in a ro-

tating coordinate system:

with being the displaced mass. In computations of continuum media dy-

namics is replaced with mass element :

where is the density of the fluid and is the face-normal velocity across theface of area of a control volume (see Problem 2.7.8).

2.6 The Law of Similarity

The law of similarity[7, 8] enables in some situations to use a single solution to the

equations of fluid motion to represent a whole family of different cases. Consideran example of a steady flow past a solid body, where the flow velocity upstream

of the body is . Consider also several cases of such flows when the body has

the same shape but different sizes, . Now, if the only fluid property affectingthis process is the kinematic viscosity, , then in all these cases the distributionof velocity should be a function of space coordinates , and of at least

three additional parameters, ( ):

(2.95)

The number of parameters8 can be reduced by considering the dimensions

of physical units in which they are measured:8A parameter can be looked at as just another independent variable, like space coordinate or time.

However, we treat them separately, since parameters are specific for each physical law, whereas are

not.


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We can multiply the equality (2.102) by and get:

(2.103)

This is an extra relation imposed in addition to our physical law (2.97) by virtue

of scale invariance or homogeneity of our law with respect to length-scaling [9].In a similar manner we can arrive at two more relations imposed because ofhomogeneity with respect to other two primary dimensions: time and mass:

(2.104)

(2.105)

Thus we have three more relations in addition to our physical law (2.97),

which means that the number of variables can be reduced from to .

If we use a more complex law that involves an additional primary dimension,

such as temperature, then we can reduce the number of variables of the problemby 4. Generally, if we have primary dimensions and independent variables in

the problem, then the independent variables can be reduced to non-dimensional parameters. These parameters can be different depending on thechoice of scaling factors used in transformation (2.100). Generally, normalization

(2.100) does not have to be done by primary dimensions, but can be used withrespect to any group of variables that do not form a so called PI-group, i.e. theirproducts of the type (2.98) can not be reduced to a non-dimensional number, no

matter what powers are used [10, 11]. This constitutes the essence of the PI-theorem [11]. It lays a more rigorous foundation for the law of similarity [7, 8],

which means that the same solution can be reused by rescaling the variables.

2.6.2 Non-dimensional formulations

To formulate a physical law in dimensionless variables we should introduce di-mensional scales for each variable. The scale, representing the variable, will be

denoted with the same symbol, but with the subscript 0. Scales can be introduced

for both scalar, vector, and general tensor variables. Thus, the scale for a vectorvariable will be denoted as10 . In some situations there can be different scales

10Subscript 0 shouldnt be confused with the vector component, since vector components are numbered

with one


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for different components, in which case we shall use a different notation.

Space and time derivatives should also be scaled. Thus, if we considerspace and time as independent variables: , and as dependent variables ve-

locity, density, and pressure we can introduce the following non-dimensional vari-ables:

(2.106)

where we use the Nabla operator to denote the space derivative.

After the dimensional variables are replaced withe the dimensionless onesby means of (2.106), one should look into the physics of the problem and see if

some extra relations between the scales can be applied. For example, in some

problems the characteristic velocity scale can be related to length and time scales

as: . This can be the case in the problem of a steady flow around a fixedobject. However, a steady flow around a rotating object will have an independenttime scale related to the period of rotation.

After all possible eliminations of scales were done, one should try to con-struct dimensionless combinations of scales, or non-dimensional parameters.

There can be several different ways in which these parameters can be selected.

This process can be formalized somewhat [10], but there is still a room for subjec-tive judgment on which dimensionless combinations of scales are most appropri-

ate as parameters for the problem at hand. No matter how these parameters areselected the PI-theorem states that their minimum number can be as low as ,

where is the number of dimensional scales and is the number of primary di-

mensions of the problem. If all the dimensionless parameters have been correctlyidentified, it should be possible to replace all the dimensional scales with these

parameters, thereby rendering the physical law in a dimensionless formulationwith the minimum set of independent parameters.

Lets consider several cases of non-dimensional formulations and of appli-cation of the PI-theorem.

Mass conservation law

Lets write a non-dimensional formulation of the mass conservation law (2.2):

(2.107)


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where we used the nabla operator (A.32) to simplify further analysis. There are

three primary dimensions in this case ( ): [length], [time] and [mass]. Usingthe scaling transformations (2.106), the non-dimensional form of (2.107) is:

As can be seen, all the dimensional parameters canceled out from the equa-

tion.

Momentum equation

Consider the steady-state limit of the incompressible momentum equation givenby the Navier-Stokes equation (2.24):

(2.108)

Since this is a constant density formulation, density becomes a parameterof the problem: . Considering this, and transforming to the non-dimensional

variables according to (2.106) we obtain:

(2.109)

To simplify things, lets select for the pressure scale the dynamic pressure:. By doing this we state that pressure scale, , is not an independent

parameter of our problem, but is related to the density and velocity scales. Now,lets make each term of (2.109) dimensionless, by multiplying the whole equationby :

The equation includes four dimensional parameters: . Since thisis a steady-state formulation, time is no longer a dimension of the problem, andthere are only two primary dimensions left ( ): [length], and [mass]. Thus,


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the number of independent parameters can be as low as . If we

introduce two non-dimensional numbers: Reynolds number:

(2.110)

and Froude number, relating the forces of inertia to gravity:

(2.111)

then we obtain the non-dimensional form of the momentum equation:

(2.112)

with the four non-dimensional variables: and , and two non-dimensional

parameters: (see also Problem 2.7.9).

Boundary conditions

Some non-dimensional parameters arise from the boundary conditions. For ex-

ample, non-dimensionalizing the boundary condition of the energy equation (2.82)

leads to:

(2.113)

where is the Nusselt number:

(2.114)

where is the wall heat flux ( ), the characteristic length-scale,the heat conduction coefficient, (1.29), and the characteristic temperature

difference between the wall and the fluid.


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Boundary conditions at the free surface give rise to additional parameters,

such as Froude number, relating inertia forces to gravity, (2.111), Weber number,relating inertia to surface tension:

(2.115)

where is the coefficient of surface tension entering the boundary condition(2.58).

Other non-dimensional parameters may appear as new phenomena areadded into the physical law [2].

2.7 Problems

Problem 2.7.1 Derivation of the vorticity equation

Obtain the result outlined in (2.30).

Problem 2.7.2 2D vorticity limit

Perform the missing steps in (2.42).

Problem 2.7.3 Incompressible viscous limit

Derive (2.50) from (2.49).

Problem 2.7.4 Conservation of circulation

Thevelocity circulation is defined as

(2.116)

where the integration is over any closed loop inside the fluid.

Show that for irrotational flow ( ): .

Problem 2.7.5 Bernoullis equation

Using the energy equation (2.77):


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2.7. PROBLEMS 51

and momentum equation (2.21):

derive thestrong formulation of theBernoullis equation:

and formulate its applicability limits.

Problem 2.7.6 Volume change inside a moving boundary

Suppose that a region of space is enclosed by a moving boundary. Thevelocity of motion of the boundary, , is given at each point on the boundary.Show that the rate of change of the volume, , of that region will be equal to:

(2.117)

where is the unit normal vector to the boundary and surface area element,

and find the coefficient .

Problem 2.7.7 Rotating coordinates

Obtain explicit relations for the components of acceleration vectors inin (2.94) in terms of .

Problem 2.7.8 Rotation with separated coordinate origins

Consider a simple rotation with as in (2.94), but now let theorigin of the rotating coordinate system rotate with the same around the

origin of :

Derive the expression for in this case.


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Problem 2.7.9 Nondimesionalizing energy equation

Write a non-dimensional form of the heat convection equation (2.79):

selecting for the pressure scale. Determine the minimum number ofdimensionless parameters. Write the equation using the Eckert number (3.114)as one of the parameters:

(2.118)


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Chapter 3

Laminar flows

3.1 Assumptions

Flow equations discussed in Chapter 2 provide analytical solutions only in somespecial cases. In this chapter we shall consider the equations for incompressibleflow: (2.4), (2.24) and (2.74), assuming that all the coefficients are constant:

(3.1)

(3.2)

(3.3)

where is the Hydrostatic pressure:

(3.4)

For Newtonian incompressible fluids the viscous stress term, , has the form(2.23):

(3.5)

Definition 3.1.1 Laminar flow

Lets make an assumption of laminar flow which states that the time scaleof changes in the flow can not be lower than the time-scale of the motion of the

53


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54 CHAPTER 3. LAMINAR FLOWS

boundary or any external sources. In other words, if there is any repeatability in

the motion of the boundary or in the external forces then the frequencies associ-ated with either factors can not be lower than the frequencies of the flow motion.

It means that neither the boundaries not external forces can induce any addi-

tional frequencies in the flow. In the limit case of non-moving boundaries andnon-changing forces the flow should not depend on time, which means that all

the dependent variables should become functions of spatial coordinates only.

The conditions of laminar flow defined by (3.1.1) are realized when the con-tribution of the non-linear term in the momentum equation (3.2) is small, or when

the contribution of the viscous term dominates. This is usually the casewhen non-dimensional Reynolds number (2.110):

(3.6)

is small. In practical situations the smallness of corresponds to .

Navier-Stokes equation, (3.2) is known to have very few analytical solutions.

This is mainly due to the non-linear convective term , which is the maincause for the rich dynamical features of fluid flow. For this reason, most of thecases that provide analytical solution do not include the convective term. Below

we shall consider several such cases.

3.2 Confined flows

Probably the simplest of confined flows are the flows between moving surfaces,which belong to the category of Couette flows[2].

3.2.1 Flow between parallel plates

Lets consider a flow between two parallel plates, one of which is moving relativeto the other with a constant velocity (Fig. 3.1).

We are looking for a two-dimensional solution, since by the assumption of

laminar flow (3.1.1) and from the symmetry of the problem we do not expectany changes in the transverse direction. We are also looking for a steady-statesolution, thus all the variables will be the functions of axial and vertical coordinatesonly: , and only two velocity components need to be considered


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3.2. CONFINED FLOWS 55

Figure 3.1: Flow between parallel plates: the lower plate is at rest, the upper plate is

moving with velocity .

. Since the plates are considered to be infinite no variable shouldchange in direction either. Thus, the only independent variable of the problembecomes the vertical coordinate , which we shall denote as . Likewise, from

the symmetry of the problem the only non-zero component of velocity is , whichwe shall denote by . In addition to this we can also assume the pressure to be

constant. This can be explained by the absence of normal stresses in this flow.With these assumptions the momentum and energy equations (3.2), (3.3) reduce

to:

(3.7)

(3.8)

Equations (3.7) and (3.8) can be solved with the boundary conditions u(0)

= 0 and u(H) = U, and and , with the solution (See Prob-lem 3.7.2):

(3.9)

and the shear stress:

(3.10)

The non-dimensional friction coefficient, , becomes inversely proportionalto the Reynolds number:

(3.11)


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and the Poiseuille number:

(3.12)

Solution to the temperature equation (3.8) produces a quadratic depen-dence on (See Problem 3.7.2):

The dimensionless Brinkman number is introduced as a relative measure of vis-cous forces to thermal fluxes:

(3.13)

In the momentum equation (3.7) the effect gravity was neglected under theassumption that the gravity force acts normal to the direction of the flow. Generallyit may not be the case, but the solution procedure remains essentially the same

(see Problem 3.7.3).

3.2.2 Axially moving concentric cylinders

In this case only the axial component of velocity vector is non-zero:

, and it only depends on : . The appearance of anyother velocity component, or a dependence on other coordinates will lead to the

violation of the assumption of a laminar flow (Definition 3.1.1). Substituting thisform of the solution into the momentum equations in cylindrical coordinates, we

find that only the an axial momentum equation takes a non-trivial form:

(3.14)

A solution that satisfies this equation is:

(3.15)


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where are the constants, which can be determined from the boundary con-

ditions:

Substituting this into (3.15) we have:

and the solution becomes:

(3.16)

Remark 3.2.1 Pulling an infinite rod

Consider the problem above with the boundary conditions:

Then, applying this conditions to (3.15), we have

from which it follows that . Thus the problem of pulling an infinite roddoes not have a steady-state solution.

3.2.3 Rotating concentric cylinders

In this case we have: , and .

Continuity:

-momentum:

(3.17)


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-momentum (Problem 3.7.5):

(3.18)

Boundary conditions:

The equation that determines the velocity is (3.18). A solution that willsatisfy this equation has the form:

(3.19)

Substituting it into the boundary conditions, we find the constants , ,and the final solution becomes:

(3.20)

Remark 3.2.2 Flow inside a rotating cylinder

If we set the solution above will become:

which is asolid body rotation. Thus the steady-state solution for the flow inside arotating cylinder is a solid body rotation.

Remark 3.2.3 Rotating an infinite rod

If we solve the problem above with the boundary conditions:


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we can obtain the following system for the coefficients :

(3.21)

from which we have: , and the solution becomes:

(3.22)

Substituting it into (3.17) and integrating it, we can obtain the pressure dis-

tribution:

and the pressure distribution is

where the constant is found from the boundary condition: . Thus,

the pressure distribution is:

(3.23)

It is useful to compute the rotational momentum (torque) that arises in a

system of two rotating cylinders (Problem 3.7.6).

3.2.4 Poiseuille flow through ducts

Lets consider a case of a straight duct with a constant cross-sectional area, .Since the area does not change its form, the length-scale of the problem will be

the characteristic size of the duct1: .1See Sec.3.2.6 for another measure of duct diameter


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Compared to the case of infinite plates, a duct has an entrance and it has a

finite cross-sectional area. The existence of an entrance causes entrance effects,such that the flow is three dimensional over some distance from the entrance,

that is all three components of velocity are non-zero: , and each

component also depends on all three spatial coordinates: .

However, we assume that this transition region will end at some distanceafter the entrance and be replaced by a fully developed flow region, where axialvelocity does no longer depend on the axial coordinate, i.e.

(3.24)

It should be noted that the existence of a fully developed laminar regimeis only an assumption, but it happens so, that there is a solution satisfying thisassumption. However, it also happens that there are other solutions, which do not

satisfy this assumption, i.e. unsteady turbulent regime. Which solution is realized

in reality depends on the magnitude of the Reynolds number. At the low Reynoldsnumbers the fully developed regime occurs in circular ducts at distances between30 and 100 duct diameters from the entrance.

In a fully developed flow in addition to (3.24), the velocity components nor-mal to the axis should be zero: . This is because the ap-pearance of any velocity component normal to the axis can not be sustained for a

long period of time since there is no pressure gradient imposed in that direction.

On the other hand, any short time appearance of such components will violate

the assumption of a steady-state laminar nature of the flow in a fully developedregion. Thus, we can use only the axial component, which we shall denote as

. With these assumptions, in a fully developed flow the axial momentumequation (3.2) can be simplified to:

(3.25)

The other two momentum equations simplify to . Thus, dependsonly on : , and we can write:

Since the second term on the RHS of (3.25) does not depend on, and then it follows from (3.25) that should not depend on either. But


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since depends only on , it follows that is a constant. Thus, dividing

equation (3.25) by , and introducing dimensionless variables:

where is the appropriate measure of the duct cross-sectional size, we obtain athe following boundary value problem for the dimensionless velocity :

(3.26)

where subscript stands for wall and index spans only the variables that

depends on: . This is a Poisson equation in a confined 2D domain with theno-slip velocity at the boundary (Dirichlet boundary conditions).

Remark 3.2.4 Reynolds number independence

Note that the Reynolds number does not enter the equation (3.26), and

therefore, should not affect the solution. This is because we excluded non-steadysolutions and turbulence from our consideration.

The circular pipe

Consider a fully developed flow region in a circular pipe of radius . The natural

coordinate system for this case is cylindrical: , with being the axial coor-dinate. Following the discussion of the previous section, only the axial velocitycomponent, will be non zero, which we shall denote for simplicity as . As itwas shown, it can only depend on and . However, because of the symmetry of

the duct, the dependence on will inevitably lead to time-dependent solution andviolate the assumption of fully developed flow. Thus, we should have ,

and using the Laplacian operator in cylindrical coordinates, (3.26) reduces to:

(3.27)

which gives the parabolic solution:


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and in dimensional units:

(3.28)

This solution is called the Poiseuille flow.

The Volumetric flow rate through the pipe can be computed as

(3.29)

The mean duct velocity is defined as:

The wall shear stress:

(3.30)

Darcy friction factor:

Skin-friction coefficient

(3.31)

where the Reynolds number is based on duct diameter: .

Poiseuille number

(3.32)


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We can find the constant after substituting this solution into (3.34):

(3.37)

From (3.36) we can see, that when the velocity changes sign at the lowerwall ( ). This is called flow separation point, and according to (3.37) it corre-sponds to the pressure gradient:

(3.38)

A pressure gradient greater than this will cause the flow at the lower wall toreverse2.

3.2.6 Non-circular ducts

Because the problem of the fully developed duct flow was reduced to a well posedand well studied boundary value problem based on the Poisson equation thereare variety of analytical solutions obtained for ducts of various shapes [2].

There are several convenient measures that are introduced for ducts of ar-bitrary shapes.

The cross-sectional length-scale of the duct is called the Hydraulic diame-ter, which is introduced as a generalization of relation for a diameter of a circle.

For a circle of diameter the relation between its area and a perimeter

is: . Thus, for any non-circular duct of perimeter and area the

hydraulic diameter is defined as:

(3.39)

The mean wall shear stress is defined as:

(3.40)

2See Problem 3.7.7


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3.3. UNSTEADY FLOWS 65

where the integration is done over the perimeter of the duct.

In a fully developed flow each element of the fluid between cross-sectionalplanes at and moves with a constant velocity. Thus the sum of forces on

that element should be zero. This means that the friction force at the wall shouldexactly balance the axial force pushing the fluid element due to the pressure drop

in the duct. Thus, we have the following relation between the mean wall shearstress and the pressure gradient:

and using the definition of the wall mean shear stress (3.40), we have:

3.3 Unsteady flows

Some unsteady flows in the ducts can still be solved analytically. To introduceunsteadiness we formally use the same assumptions that lead to (3.26), but now

we shall reinstall the time derivative from (3.2), which with these assumptionsbecomes:

(3.41)

where as in Sec.3.2.4, is the axial component of velocity in the duct,which is the only non-zero velocity component. Following the same reasoningas in Sec.3.2.4, we conclude that the last term can not depend on any spatial

coordinate. Since we consider unsteady flows, this term can still depend on time.However, in this problem we can combine velocity and pressure into a single joint

variable:

And the final equation becomes:

(3.42)


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This equation has a paraboliccharacter, which means that one independent

variable - time - is asymmetric with respect to direction. Specifically, at any pointin time the solution will depend only on the previous points on the time axis, but

not on the subsequent points. This difference in the directions of time: the future

and the past, is the result of the first order time derivative in the equation (3.42).In contrast, the second order Laplacian space derivative, makes all space

directions equivalent. It should be noted that since , the Laplacianoperator in (3.42) includes only two components: .

Equation (3.42) is still simple enough to provide analytical solutions in sev-

eral cases. The important cases include:

1. Starting flow in a duct.

2. Pipe flow due to oscillating pressure gradient.

3. Fluid oscillating above an infinite plate.

4. Unsteady flow between infinite plates.

3.3.1 Fluid oscillation above infinite plate

Suppose that the plate is oscillating in direction , which we shall denote as

, and the velocity of oscillation is:

(3.43)

In this case the solution for the fluid velocity will depends on only one direction :. Then equation (3.42) can be rewritten as:

(3.44)

We can note that in this case the motion in time is periodic, while the changes

in space are aperiodic. Correspondingly, we may look for a solution form whichis a periodic function in time and a decaying function of space. One form of the

solution that will satisfy this equation is:

(3.45)


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where is an unknown function of . If we substitute the latter into (3.44), well

get the equation for :

(3.46)

where , and the parameter was introduced for brevity. This equation

has a solution:

where is a constant. Thus, the solution for , (3.45), is:

Using the definition of , (3.46), and the identity: , we can write:

and for we obtain:

where . Considering only the real solution, we have:

Constant and can be found from the initial conditions. In case of a moving

plate and a stagnant flow-field at infinity (3.43), we have:

Thus , , and the final solution is:

(3.47)


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In case of a stationary plate and a flow-field oscillating as ,

we can obtain the solution by subtracting the equation above from (t):

(3.48)

It can be checked by a direct substitution that the above equation satisfies (3.44)and the boundary conditions: , and .

3.3.2 Unsteady flow between infinite plates

Consider two parallel plates separated by a distance , and a fluid with viscos-

ity initially at rest is filling up the space between the plates. Suppose that theupper plate starts moving with velocity . We are looking for the solution for the

unsteady flow-field between the plates. This case is similar to the one describedabove, but now the solution should be aperiodic in time, and in fact it can be peri-odic in space, since any periodic function with a spatial period equal to the plate

separation, , will be suitable. The equation (3.44) is still valid in this case. How-ever, now we have an explicit length-scale, , and using dimensionless variablesbecomes more attractive. Lets define the non-dimensional variables as:

(3.49)

Then equation (3.44) can be rewritten in the non-dimensional variables as:

And selecting the time scale as , we have:

This equation should be solved for the unknown function with the boundaryconditions:


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(3.50)

(3.51)

(3.52)

(3.53)

where (3.50) describes the fixed lower plate, (3.51) describes the moving upperplate, (3.52) is the initial and (3.52) the final velocity distributions. Note, that the

final velocity distribution was obtained before as the steady state solution for this

case (Sec.3.2.1). As can be seen, boundary condition (3.51) is non-zero. Tosimplify our search for the right solution, it would be nice if we could look for a

function which is zero at the boundaries. To make the boundary conditions bothzero lets look for a solution that is represented by the difference between the

steady-state solution , (3.53), and , since this function will be zeroat both plates:

Now, if we replace with the new unknown function: , we obtain the

following boundary value problem:

(3.54)

(3.55)

(3.56)(3.57)

where we have all the boundary values zero. We can look for a solution in formof separated variables:

(3.58)

Substituting this into (3.54) we obtain relationship between and :


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where as in (3.46) we introduced a constant . This time, however, is not known

in advance, and should be determined from the boundary conditions. The relationabove is identically satisfied by:

(3.59)

(3.60)

with , and unknown constants in addition to . Out of these the constantcan be absorbed into and , since and enter as a product in (3.58). So,

without loss of generality we can set . The boundary conditions on

(3.55) - (3.58) can be translated to and as:

(3.61)

(3.62)(3.63)

(3.64)

where the last equality (3.64) is satisfied identically by virtue of (3.60). From (3.61)and (3.59) if follows that . Similarly, from (3.62) it follows that , where

is an integer number. The only way to reconcile boundary condition (3.63):, with boundary conditions (3.61) and (3.62) and the analytical form of

given by (3.59), is to express as a Fourier series:

Coefficients can be found as:

Computing the integral: , we obtain:

Thus for we have:


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3.4. CREEPING FLOWS 71

And the final function becomes:

(3.65)

3.4 Creeping flows

If we combine equations (3.2) and (3.5), and use the expression for substantialderivative (1.7), we obtain yet another form of Navier-Stokes equation (2.24):

The LHS of this equation represents the inertial forces. The assumption of creep-

ing flow or Stokes flow states that the inertial forces are negligible. With thisassumption the last equation becomes:

(3.66)

Differentiating over , we get:

from which we obtain a Laplace equation for pressure:

(3.67)

Forming a product with and using the symmetric identity (A.27), we ob-tain:


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Swapping indexes and in this equation and subtracting one from another we

can express it in terms of vorticity vector (1.12):

(3.68)

Thus both the vorticity and pressure satisfy Laplace equation in a creeping flow.

Important cases of creeping flow include:

1. Fully developed duct flow. Re-number independent.

2. Flow about immersed bodies (Stokes solution or the sphere).

3. Flow in narrow but variable passages. (Lubrication theory).

4. Flow through porous media.

3.4.1 Stokes flow around a sphere

Consider a laminar viscous flow around a sphere of radius , with the velocity atinfinity . The solution to this problem will be two-dimensional, since by symmetry

nothing should depend on the azimuthal direction. It was shown in Sec.2.2.5 thatin a 2D limit the vorticity vector has only one component and it is related to t

concise fluid mechanics

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