Post on 03-Jun-2018
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AERO 522 - Viscous Flow
Professor Luis P. Bernal
Introduction
Reading: Whites Viscous Fluid Flow
Chapter 1
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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)
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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,
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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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An Example from Micro Fluidics
Back Cavity
Si
Perforated Electrode
SiSi
Diaphragm
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An Example From Micro Fluidics
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An Example from Micro Fluidics
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
Perforated Electrode
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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An Example from Micro Fluidics
Back Cavity
Si
Perforated Electrode
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
Ref: Laminar Boundary Layers, L. Rosenhead Ed. Section III.13
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
Local Analysis of the Fluid Motion
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
Local Analysis of the Fluid Motion
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
Local Analysis of the Fluid Motion
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
41Local Analysis of the Fluid Motion
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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
Local Analysis of the Fluid Motion
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
42Local Analysis of the Fluid Motion
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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
Local Analysis of the Fluid Motion
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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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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Review of Vector and Tensor Algebra
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
47
Review of Vector and Tensor Algebra
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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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Review of Vector and Tensor Algebra
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Review of Vector and Tensor Algebra
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
Vector and Tensor Operations
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Vector and Tensor Operations
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
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
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
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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
64Conditions at a Free Liquid Surface
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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
67
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
Review of Vector and Tensor Algebra
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