AE 522 Introduction

8/11/2019 AE 522 Introduction

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Aerospace EngineeringUniversity of Michigan

AERO 522 - Viscous Flow

Professor Luis P. Bernal

Introduction

Reading: Whites Viscous Fluid Flow

Chapter 1


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2

AE 522 - Introduct ion

Outline

Course Objectives and Expectations

Review of Thermodynamics

Review of Vector and Tensor Algebra

Kinematics of Flow Fields

Conditions at a Fluid Boundary


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Aero 522: Course Objectives To provide a comprehensive description of the

fundamental flow physics and analysis tools of

viscous effects in fluid flows including internal

flows and external flows (aerodynamics)

3


Physical Understanding

Analysis Engineering Application


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Aero 522: Course Expectations Undergraduate course in fluid mechanics or

aerodynamics

Understanding of thermodynamics concepts andapplication to engineering systems

Physics: good understanding of dimensional

analysis and conservation laws

Math: In this course we will use appliedknowledge/understanding of linear algebra,

ordinary differential equations, partial differential

equations, complex variables,

4



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Review of Thermodynamics

Hypothesis of local thermodynamicequilibrium

Extensive vs. intensive properties 0th Law of Thermodynamics: Temperature Equation of state

Caloric equation of state 1st Law of Thermodynamics: Entropy 2nd Law of Thermodynamics

Transport properties


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Local Thermodynamic Equilibrium

Thermodynamic properties are defined asaverages of molecular properties over the entire

system.

Fluids in motion are not in thermodynamicequilibrium in a strict sense

Hypothesis of Local Thermodynamic Equilibrium Consider the flow around a body of size L.

Thermodynamic variables are defined as molecular

averages over very small regions of the flow.

These regions should be small compared to the size

of the body (> )


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Local Thermodynamic Equilibrium

Knudsen Number: Kn = /L

0.10.01 1 10 102 103 104 105 107 108 109 1010

Free molecular flow

Gas Kinetic TheoryContinuum Fluid Mechanics

and Aerodynamics

1/Kn = L/ In AE 522 we consider only large scale fluid dynamics phenomena.

Therefore theories based on the continuum hypothesis apply.

These theories are not valid for aspects of space flight (e.g. calculationof orbital decay due to aerodynamic drag of satellites).

Consider the result of thermodynamic averages on a number of volumesof size L, spaced a distance of order L

If L < the thermodynamic averages will give different results

If L > the thermodynamic averages will give the same result

is determined by the number of collisions: Mean Free Path


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An Example from Micro Fluidics

Acoustic thrusters use sound to producethrust by generating a synthetic jet


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Back Cavity

Si

Perforated Electrode

SiSi

Diaphragm


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10


An Example From Micro Fluidics


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Question: What is the structural resonant frequencyof the membrane? A: Do a vacuum test Q: At what pressure? A: Mean free path

large compared to

relevant sizeBack Cavity

Si


SiSi

Diaphragm

1 ~n A

Mean free path

n - Number of molecules per unit volume [m-3]. Perfect gas:

A - Molecules cross section area 10-15 cm2 = 10-19 m2

231 38 10

p n k T; k . J /K


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Back Cavity

Si


SiSi

Diaphragm

1 k T~n A p A

For:T = 300 K,

~ 1 mmA = 10-19 m2

Gives

p = 41 Pa = 0.3 Torr

k Tp

A

Better Answer: Consider the effect of the mass of airmoving with the membrane i.e. the apparent mass


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Thermodynamic Properties

Some thermodynamic properties like the pressure and temperature donot depend on the mass of the system. These are called intensiveproperties

Other properties like the volume, internal energy, enthalpy and entropyare proportional to the mass of the system. These are called extensiveproperties

Extensive properties are made intensive by dividing by the mass of thesystem. These are called specific properties, i.e. specific volume, specificinternal energy, specific enthalpy and specific entropy

IMPORTANT: In this course we only use specific properties and becauseof it we drop the word specific

The main thermodynamic properties we use in Fluid Dynamics are Density: [kg/m3] Pressure: p [N/m2] Temperature: T [K] Internal energy or energy: e [m2/s2] Enthalpy: h [m2/s2] Entropy: s [m2/(s2 K)] Viscosity: [(N s)/m2] Thermal conductivity: k [W/(m K)]


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0th Law of Thermodynamics There is a thermodynamic variable, the Temperature, that

characterizes the Thermodynamic state of the system

This variable is always zero or positive: T 0

If T = 0 the system has zero energy IMPORTANT: In thermodynamics it is assumed that two

thermodynamic variables and the composition, uniquely

define the thermodynamic state of the system, i.e.

= (p, T) (equation of state) e = e (p, T) (caloric equation of state) h = h (p, T)

s = s (p, T) = (p, T) k = k (p, T)


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Equation of State

The relation between density, pressure and temperature is the equationof state

Except for a few simple fluids the equation of state is not known

The differential form of the equation of state is always defined

Specific Heat Ratio:

Speed of sound: Bulk modulus:

Coefficient of thermal expansion:Perfect liquid: a ; = 0

T p

d dp dT;p T

(p, T)

1 1

T p

ddp dT;

p T

2d dp dT

a

p vc / c

2

s

pa ;

2

T

p KK ; a

1

pT


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Equation of State: Perfect Gases

A perfect gas is a material formed by molecules that move freely inspace and only interact with each other through collisions

The equation of state for a perfect gas is

Rg is the gas constant:

Ro= 8,313 J/(kg-mol K) is the universal gas constant

Mg is the molecular weight of the gas

For a perfect gas:

g

pR T

og

g

RR

M

2 1

ga R T;T


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Equation of State: Air

Air can be treated as a mixture of perfect gases. The molecular weightis given by the equation

with the mass fraction and the mole fraction

which is also equal to the partial pressure of a component gas

The gas constant for dry air is:

In practice the air density is a function of ambient pressure, p,temperature, T, and relative humidity, r

where pv is the partial pressure of the water vapor which is related tothe saturation vapor pressure, Es, and the relative humidity, r

1mix i i i ii i

M C M X M

287airR J/(kg K)

0 378 v

air

p . p

R T

i i

C i i

X p p

7 5 273 15

35 85610 78 10

. (T . )

T .v s sp r E ; E .


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Equation of State: Air (II) The perfect gas law is not a good approximation near the critical point of a gas This is quantified by the compressibility factor

Z is a function of the reduced pressure, pr, and reduced temperature, Tr, definedas

where pc and Tc are the critical point pressure and temperature respectively

g

pZ

R T

r r

c c

p Tp ; T

p T


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Caloric Equation of State The relation between energy, pressure and temperature is the caloric

equation of state

or

The differential form of this equation is

The specific heats are defined as derivatives of the energy and enthalpy

Specific heat at constant volume

Specific heat at constant pressure

Specific heat ratio

1pdp

dh c dT T

e e(p, T)

ph e h(p, T)

v

ec

T

p

p

hc T

p

v

c

c


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2th Law of Thermodynamics

In an adiabatic closed system the entropy canonly increase (i.e. ds 0)

To meet this requirement the transport propertiesmust satisfy:

where is the viscosity coefficient, is thesecond coefficient of viscosity and k is the

thermal conductivity

20 0 0

3 ; ; k ;


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Transport Properties - Viscosity

Newtons Law of Friction: The friction force acting on a solid surfacedue to the fluid motion is proportional to the velocity gradient.

u(y)y

du

dy

F A d u d y A

The proportionality constant is the viscosity coefficient [N s/m2].

For most fluids (like air, water, fuels, ) the viscosity coefficient is

independent of the fluid motion. These fluids are called newtonian fluids.

For Newtonian fluids the viscosity is a thermodynamic property that canbe modeled using a suitable molecular model and small departures from

thermodynamic equilibrium.

There are many fluids (like blood, ketchup, paints) that do not followNewtons law. These fluids are called non-newtonian fluids.


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Viscosity

Viscosity is a thermodynamic property that depends on temperature andpressure. For single component fluids, it is expressed as a universalfunction of the reduced pressure and temperature

For gases the viscosity increases with temperature For liquids the viscosity decreases with temperature

c r r r c r cF T ,p ; T T T ; p p p


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Viscosity of Gases

For gases at low pressure the viscosity is a function of thetemperature only Gas kinetic theory shows:

Sutherland law:

Power law:

3 2/

o

o o

T ST

T T S

n

o o

T

T

0.67 a


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Fouriers Law of Heat Conduction: The heat transferred per unit timeand area through a surface is proportional to the temperature gradient.

The proportionality constant is the thermal conductivity k [W/(m K)].

The thermal conductivity can also be modeled using a suitable molecularmodel and small departures from thermodynamic equilibrium.

The Prandtl number relates viscous and thermal transport processes

For Gases Pr ~ 1. Molecular transport of heat and momentum occur atapproximately the same rate

For liquids Pr > 1. Molecular transport of momentum is faster than heattransfer

For liquid metals (Mercury) Pr


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Thermal Conductivity

Thermal conductivity is a thermodynamic variable that depends ontemperature and pressure and is a function of the reduced pressure andtemperature

For gases the thermal conductivity increases with temperature

For liquids the thermal conductivity decreases with temperature

c r rk k F T ,p


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Thermal Conductivity of Gases

For gases at low pressure the thermal conductivity is a function ofthe temperature only

Sutherland law:

Power law:

3 2/

o

o o

T Sk T

k T T S

n

o o

k T

k T


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Viscosity and Thermal Conductivity

of Gas Mixtures The viscosity and thermal conductivity of dilute-gas mixtures canbe calculated using the relations (Whites section 1-3.10)

1

1

n

i imix ni

j ijj

x

x 1

1

n

i imix ni

j ijj

x kk

x

ii

i

px molar fraction

p

M molecular weight

21 2 1 4

1 2

1

88

/ /

ji

j i

ij /

i

j

M

M

MM

21 2 1 4

1 2

1

88

/ /

ji

j i

ij /

i

j

Mk

k M

MM


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Kinematics of Flow Fields

Derivative Following the Fluid Element Acceleration of the Fluid Element

Local Analysis of the Fluid Motion Translation Deformation, strain rate Rotation, vorticity

Properties of the strain rate tensor


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Mathematical Description of Fluid Motions

There are two distinct approaches to the mathematical description of fluidmotion: Lagrangian Description and Eulerian Description Lagrangian Description: The position of a fluid element is given as a

function of initial position and time

Here is the vector field describing the position of a fluid element at time t,initially located at

The laws of mechanics apply to material elements and therefore are onlydefined in a lagrangian frame of reference.

Eulerian Description: Fluid properties are given as a function of positionand time

The fluid properties are field variables giving the value of the properties of the

fluid element located at at time t. To derive the conservation laws in fluid mechanics we need to convert from

the eulerian frame of reference to the lagrangian frame of reference Substantive derivative or derivative following the fluid element

r r a, t

r

a.

r, t ; p p r, t ; T T r, t ; V r, t

e e r, t ; h h r, t ; s s r, t

r


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The first term on the right hand side is called the unsteady term orunsteady rate of change

The second term on the right hand side is the advection term or rate ofchange due to advection

Using index notation, the derivative following a fluid element is given by

Derivatives Following the Fluid Element

In the limit dt 0 the rate of change of a fluid property is

t 0

DQ Q Qlim V Q

D t t t

k

k

DQ Q QV

D t t x


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Using index notation, the acceleration is

Acceleration of a Fluid Element

The acceleration of a fluid element is the rate of change of the velocity ofthe fluid particle. Then

DV Va V V

D t t

i i ikik

DV V Va V

D t t x

The last term, the rate of change of the fluid velocity due to advection, isnonlinear. This nonlinearity is the source of a lot interesting fluid flow

phenomena and mathematical difficulty

The advection term can be expressed in terms of the vorticity

2 2V V

V V V V V2 2


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The first term is the strain rate tensorthat characterizes the deformationof the fluid element

The Velocity Gradient Tensor

In general the change of a fluidelement in time t is given by

i i k i k

ik ikk k i k i

V V V V V1 1

ex 2 x x 2 x x

The second term is related to the vorticity and gives the rotation of thefluid element

3 2

i kik 3 1

k i

2 1

0V V1 1

02 x x 2

0

x i k

ik

Vx x t

x

x V x t

The velocity gradient can be written as

x

x x

x

V t

V V x t

In the limit t 0,

it 0i

x d x

i ik

k

d x V xd t x


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Cylindrical Coordinates

Cylindrical coordinates (r, , z): The strain rate tensorin cylindrical coordinates is

The vorticity in cylindrical coordinates is

Ref: Laminar Boundary Layers, L. Rosenhead Ed. Section III.12

r r z r

rr r rz

r r zr z

zr z zz

z r z z

V VV 1 V V V2

r r r r r ze e e

V V V V1 1 V 1 V 1 Ve e e 22 r r r r r z r

e e eVV V 1 V V

2r z z r z

z

r

r z

zr

V1 V

r z

V V

z r

V V1 V

r r r

zVrV

Vz

r0x

x r cos ; y r sin ; z

S h i l C di t


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Spherical Coordinates Spherical coordinates (r, , ):

The strain rate tensorin spherical coordinates is

The vorticity in cylindrical coordinates is


r r r

rr r r

r rr

r

r

VVV 1 V 1 V2 r r

r r r r r r r sine e e

VV V V1 1 V 1 V 1 sine e e r 2

2 r r r r r r sin r sine e e

V V1 V 1r

r r r sin r sin

r

V V V cotsin 1 V2

r sin r sin r r

r

r

r

V1V sin

rsin

V1 1r V

r sin r r V1 1

r Vr r r

0

V

rVV

z

r x

x r sin cos ; y r sin sin ; z r cos

L l A l i f th Fl id M ti


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1

x

2x

1

x

2x

The change of the length of x1 in t is

The rate of chance of the length of x1 is

Local Analysis of the Fluid Motion

Consider the motion during time t of a small fluid element ofsize x1, x2

11 1

1

Vx x t

x

1 11 1 1 1 11 1

V Vx x x t x x t

x x

1 1 111 1

t 0 t 01 1

d x x VVlim lim x t t x

xdt t x

L l A l i f th Fl id M ti


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Then

Similarly, the rate of chance of the length of x2 is

And

The diagonal components of the velocity gradient tensor are the relativerate of change of the length of fluid elements aligned with the coordinate

axes


22 2

2

Vx x t

x

1x

2x

1x

2x

11 1

1

Vx x tx

2 22

2

d x Vx

dt x

1 1

1 1

V 1 d x

x x dt

2 2

2 2

V 1 d x

x x dt

39L l A l i f th Fl id M ti


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The volume of the fluid element is

The rate of change of the volume of the fluid element is

The sum of the diagonal components of the velocity gradient tensor

(the trace of the tensor) gives the relative rate of change of the volumeof the fluid element


31 2

2 3 1 3 1 2

d xd d x d x

x x x x x xdt dt dt dt

1 2 3x x x

31 21 2 3

1 2 3

Vd V Vx x x

dt x x x

31 2

1 2 3

V1 d V V

dt x x x

40Local Analysis of the Fluid Motion


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The rotation of the fluid elementx1 is

Rotation rate of x1 is

Similarly the rotation rate of x2 is

The off-diagonal components of the velocity gradient tensor give therotation rate of fluid elements aligned with the coordinate axes


1 1 2

t 01

d Vlim

dt t x

22 2

2

Vx x t

x

12

2

Vx t

x

1x

2x

1x

2x

11 1

1

Vx x t

x

21

1

Vx t

x

21

1 11

11 1

1

Vx t

xtan

Vx x t

x

12

12 2 12

t 0 t 02 2

2 22

Vx t

d 1 Vxlim lim tan

Vdt t t xx x t

x



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The motion of the fluid elementconsists of:

Translation

Rotation. Consider the diagonal

of the parallelogram formed by x1x2.

The diagonal forms an angle withthe x1 axis

The rotation rate of the fluid element is


1V t

2V t

1x

2x

1x

2x

1

2

1 290

2

1 2d 1 d d

dt 2 dt dt

1 21

902

32 1

1 2

d 1 V V

dt 2 x x 2



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Shear deformation. The shear rate is the rate of change of the angle

There is also a volume change of the fluid element associated with the diagonalelements of the strain rate tensor


1

2

1 2d 1 d dShearRatedt 2 dt dt

1 2

12

2 1

1 V VShearRate e

2 x x

31 2

1 2 3

V1 d V V

dt x x x

43Recap: Velocity Gradient Tensor


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The strain rate tensor eik gives the deformation rate of the fluid element

Recap: Velocity Gradient Tensor The velocity gradient tensor gives the time evolution of fluid elements The change of a fluid element in time

t is given by

i i k i k

ik ikk k i k i

V V V V V1 1

ex 2 x x 2 x x

The second term ik gives the rotation rate of the fluid element with is

also equal to the vorticity3 2

i kik 3 1

k i

2 1

0V V1 1

02 x x 2

0

i ki

k

Vx x t

x

x V x t

The velocity gradient can be written as

x

x x

x

V t

V V x t

In the limit t 0,

it 0i

x d x

i i

k

k

d x V xd t x


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45The Strain Rate Tensor


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Aerospace EngineeringUniversity of Michigan AE 522 - Introduct ion

Another invariant of the strain rate tensor are the principal directions. When the strain rate tensor is expressed in a coordinate system aligned with theprincipal directions it takes the form

The Strain Rate Tensor

1

ik 2

3

e 0 0

e 0 e 0

0 0 e

The principal directions can be determined by solving an eigenvalue problem.The normal strain rates along the principal directions are the eigenvalues. The

principal directions are the eigenvectors

Note that in this particular coordinate system there is no shear deformation

In this case the first scalar invariant I1 takes the form

1 ddiv V 0

d t For incompressible fluid motion and the normal strain rates

can be ordered such that with

i

1 1 2 3

i

V 1 dI e e e div V

x d t

1 2 3e e e 1 3e 0 and e 0

46

R i f V d T Al b


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Scalar fields Vector fields Tensor fields The operator

The gradient The divergence

The curl The velocity gradient and related tensors Strain rate and vorticity

Cylindrical and spherical coordinates

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Review of Vector and Tensor Algebra In Fluid Mechanics we use Scalar, Vectorand Tensorfields Scalar Fields: Scalar fields are used to describe the thermodynamic

state of the fluid and the components of vector and tensor fields

Vector Fields: Vector fields are used to describe properties that havedirectionality like the velocity and position

Vector fields require a coordinate system defined by unit vectors pointingalong three mutually orthogonal directions: the basis. We will use almost

exclusively a cartesian coordinate system for the derivations.A vector field is described by three scalar fields giving the components of the

vector on each coordinate

The component of a vector along a coordinate is given by the scalar product of the

vector with the unit direction vectors. Example - the velocity vector:

The magnitude is:

r, t ; p p r, t ; T T r, t

e e r, t ; h h r, t ; s s r, t

i v j w kV V r, t u r, t r, t r, t

i ; v j; w ku r, t V r, t V r, t V

2 2 2v wV V V u


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Index Notation: We will use index notation frequently in this course.There are two types of indices appearing in equations:

Free indices appear only ones in each term of an equation. They indicate thedirection or component

Repeated indices appear twice in a term. It implies summation over allpossible values of the index

Frequent errors:A free index must appear in all the terms of an equation

An index cannot appear more than two times in a term

Example - The following term appears frequently in this course

In this case the index i is a free index, and the index k is a repeated index.

i i i ik 1 2 3

k 1 2 3

V V V VV V V V

x x x x


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Vector and Tensor Operations


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Vector and Tensor Operations Scalar Product: The scalar products of two vectors is a scalar

given by

Vector Product: The vector product of two vectors is a vector normal to

the plane containing the vectors and magnitude given by

where is a third order tensor called the permutation tensorimk

imk

1 for even permutations 1,2,3 2,3,1 3,1,21 for odd permutations 1,3,2 3,2,1 2,1,3

0 if i m or m k or i k

c a b; c a b sin

a, b

i ia b a b cos a b

1 2 3

1 2 3 i imk m k

1 2 3

e e e

c det a a a ; c a bb b b

52



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The product of a tensor and a vector is another vector

The dyadic product of two vectors is a tensor

ik i kC a b; C a b

i ik kc A b; c A b

53The Operator


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p The is a vector operator given by

Gradient of a Scalar: The gradient of a scalar is a vector given by

Divergence: The divergence of a vector is a scalar given by

Curl: The curl of a vector is a vector given by

i i 2 3i 1 2 3

e e e ex x x x

1 2 3 i

1 2 3 i

p p p pc p e e e ; cx x x x

1 2 3

kimki

1 2 3 m

1 2 3

e e e

aa det ; ax x x x

a a a

31 2 i

1 2 3 i

aa a a

a x x x x

54The Operator


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p Gradient of a Vector: The gradient of a vector is a tensor given by

The 2 operator: The 2 operator is the divergence of gradient. It canbe applied to scalars or vectors. In cartesian coordinates equals the

Laplacian operator

1 1 1

1 2 3

2 2 2 iik

1 2 3 k

3 3 3

1 2 3

a a a

x x x

a a a aB a ; B

x x x x

a a a

x x x

2 2 22

2 2 2

i i 1 2 3

T T T T

T div gradT x x x x x

2 2 kk

i i

aa div grada ; a

x x

55The Velocity Gradient and Related Tensors


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The velocity gradient tensor is

Deformation Rate Tensor: The deformation rate tensor is

Strain Rate Tensor: The strain rate tensor is given by

T

i k

ikk i

V Vdef V V V ; def V

x x

1 1 1

1 2 3

2 2 2 i

ik1 2 3 k

3 3 3

1 2 3

V V V

x x x

V V V VV ; V

x x x x

V V V

x x x

T

i kik

k i

V V V V1 1e def V; e

2 2 2 x x

56Vorticity


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Vorticity: The vorticity is the curl of the velocity

In cartesian coordinates

1 2 3

ki imk

1 2 3 m

1 2 3

e e e

VV det ;x x x x

V V V

x

y

z

w vy z

u w

z xv u

x y

57Cylindrical Coordinates


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Cylindrical coordinates (r, , z):

The strain rate tensorin cylindrical coordinates is

The

vorticityin cylindrical coordinates is


r r z r

rr r rz

r r zr z

zr z zz

z r z z

V VV 1 V V V2

r r r r r ze e e

V V V V1 1 V 1 V 1 Ve e e 22 r r r r r z r

e e eVV V 1 V V

2r z z r z

z

r

r z

z

r

V1 V

r z

V V

z r

V V1 Vr r r

zVrV

Vz

r0

x

x r cos ; y r sin ; z


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59

Boundary Conditions for Fluid Flows


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Boundary Conditions for Fluid Flows

General Considerations

Solid Boundaries

Liquid Surface

Liquid-Vapor & Liquid-Liquid Interfaces

60General Considerations


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Consider a small volume with surfaces parallel to the interface and

very small thickness

As the thickness is reduced and the volume collapses on theinterface the volume and mass go to zero

The temperature and speed of the fluids at the interface must bethe same

If the molecular structure of the two fluids are different there can bea force acting on the interface

If molecules move through the interface (phase change) or if theinterface area changes there is energy change of the system

61Solid Boundaries


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If a liquid or gas is in contact with a solid at the interface the fluid

velocity and temperature satisfy

The condition is called the No Slip boundary condition

Force balance at the interface requires that the stress produced bythe fluid motion be equal to the force acting on the body

In most cases the solid's velocity is known and we are interested incalculating the force acting on the body resulting from the fluid

motion

The heat conduction condition at the wall can have different formsdepending on the thermal conductivity of the solid. The general

expression is

fluid solidV V

fluid solidT T

w fluid solid

fluid solid

T Tq k k

n n

fluid solidV V

w

wall

dV

dn

F A d u d y A

62Solid Boundaries


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If the thermal conductivity is high, a very small temperature changein the solid will support the fluid heat transfer at the wall. Thus, the

solid temperature is uniform . In this case the wall temperature is

known and the wall heat transfer qw is unknown

If the thermal conductivity of the solid is very low compared to thefluid

This is called the adiabatic wall condition

In this case the temperature is not known. The wall temperature isdetermined from the conditions

The wall temperature in this case is called the adiabatic walltemperature (or recovery temperature)

w fluid

fluid

Tq k 0

n

fluid solidV V

fluid

T

0n

63Accommodation Effects


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For gases at low pressure the mean free path could be comparable tothe length scale of the flow

The velocity of the gas at the wall is then given approximately by

This velocity is called the slip velocity. No Slip requires uw = 0

Or in terms of relevant flow properties (c.f. )

where M is the Mach number and is the skin friction

coefficient.

For laminar flow (Re < 5105):

For turbulent flow (Re > 5105):

w

w

duu

dy

2 3 a

w w

f2

u 3 U 20.75 M c

U 4 a U

2

f wc 2 U

1/ 2wf x 1/ 2

x

u 0.4M U xc 0.6Re ; ; ReU Re

1 7

1 7

0 020 027

/ wf x /

x

u . Mc . Re ;

U Re

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At the interface between a liquid and a gas the velocity of the gas

and the liquid must be the same. The interface velocity is thecomponent of the fluid velocity normal to the interface

There is surface tension acting tangent to the interface due todifference molecular force fields in the liquid and gas

Force balance normal to the interface gives

where is the surface tension coefficient and R1, R2 are theprincipal radii of curvature of the surface

R

p

liquid gas

1 2

1 1p p

R R

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Force balance tangent to the liquid surface gives

where are the tangential friction stress on the surface due

to the liquid and gas motion, respectively.

This tangential stress can be caused by temperature changes atthe surface (Marangoni effects)

R

p

liquid gas s

liquid gas,

liquid gas s

s

dT

d T

66Liquid-Liquid & Liquid-Vapor Interfaces


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At a liquid-liquid or liquid-vapor interface the velocity, temperature, shearstress and heat flux must be continuous across the interface

Note that the derivatives of the velocity and temperature are not equal ingeneral since

However for interfaces between a liquid and a vapor when andkk2. The boundary condition for the liquid can be approximated by

1 2 1 2 1 2 1 2V V ; ; T T ; q q

1 2 1 21 1 2 2 1 1 2 2

V V T T; q k q k

n n n n

2 2V T0; 0n n

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Section I Introduction


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Summary Review of Thermodynamics Mean free path and the hypothesis of local

thermodynamic equilibrium

Equation of state. Caloric equation equations. Generaldifferential form of these equations, Perfect gas law

1st and 2nd Laws of Thermodynamics

Transport properties for gases and liquids. Newtonianand non-newtonian fluids


Index notation and tensor algebra Scalar product, vector product

The operator, the gradient, the curl

68Section I Introduction

Summary (Cont )


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Summary (Cont.)

Kinematics of Flow Fields Derivative following the fluid element, acceleration of the

fluid element

The velocity gradient tensor. Strain rate tensor anddeformation of the fluid element. Vorticity and solid bodyrotation of the fluid element

Conditions at a Fluid Boundary Conditions at a solid wall. The No-Slip condition.

Accommodation effects

Heat transfer at a solid wall. Adiabatic wall temperature

(recovery temperature) Conditions at a liquid-gas interface. Surface tension.

Effect of varying surface tension

AE 522 Introduction

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Transcript of AE 522 Introduction