OUTLINE FOR Chapter 2
Transcript of OUTLINE FOR Chapter 2
OUTLINE FOR Chapter 2
REVIEW OF VECTOR RELATIONS (I)
AERODYNAMICS (W1-2-2)
REVIEW OF VECTOR RELATIONS (II)• Scalar Fields (pressure, density, temperature..)
• Vector Fields (velocity..)
• Scalar and Vector Products
• Gradients of Scalar Fields
1. Its magnitude is the maximum rate of change of p per unitlength of the coordinate space at the given point.2. Its direction is that of the maximum rate of change of p atthe given point.
• Directional derivatives:
AERODYNAMICS (W1-2-3)
Cartesian:
Cylinerical: Spherical:
REVIEW OF VECTOR RELATIONS (III)• Divergence of a Vector Field => Scalar
The physical meaning of divergence is the rate of change of the volume of a moving fluid element, per unit volume. <== see Chapter 2.3 for detail!
• Curl of a Vector Field => Vector
The physical meaning of curl is twice of the angular velocity vector of a fluid element.
AERODYNAMICS (W1_2_5)
REVIEW OF VECTOR RELATIONS (IV)
• Line integral
• Surface integral
• Volume integral
Relations between Line, Surface and Volume integrals:
Stokes theorem
Consider a vector field
Divergence theorem
Gradient theorem
• Line integral to Surface integral
• Surface integral to Volume integral
METHODS OF ANALYSIS
• System method In mechanics courses.Dealing with an easily identifiable rigid body.
• Control volume method In fluid mechanics course.Difficult to focus attention on a fixed identifiable
quantity of mass.Dealing with the flow of fluids.
System Method• A system is defined as a fixed, identifiable quantity of
mass.• The boundaries separate the system from the surrounding.• The boundaries of the system may be fixed or movable. No
mass crosses the system boundaries.
Piston-cylinder assembly: The gas in the cylinder is the system. If the gas is heated, the piston will lift the weight;The boundary of the system thus move.Heat and work may cross the boundaries, but the quantity of matter remain fixed.
Control Volume Method• Control Volume (CV) is an arbitrary volume in space
through which the fluid flows.• The geometric boundary of the control volume is called the
Control Surface (CS).
• The CS may be real or imaginary.• The CV may be at rest or in motion.
METHODS OF DESCRIPTION
• Lagrangian description => System• Eulerian description => Control volume
Lagrangian Description
• The fluid particle is colored, tagged or identified.• Determining how the fluid properties associated with the
particle change as a function of time. Example: one attaches the temperature-measuring device
to a particular fluid particle A and record that particle’s temperature as it moves about. TA = TA(t)=T (xo,yo,zo, t)where particle A passed through coordinate (xo,yo,zo) at to
The use of may such measuring devices moving with various fluid particles would provide the temperature of these fluid particles as a function of time.
• Attention is focused on a material volume (MV) and follow individual fluid particle as it move.
Eulerian Description• Attention is focused on the fluid passing through a
control volume (CV) fixed in the space.• Obtaining information about the flow in terms of what
happens at the fixed points in space as the fluid flows past those points.
• The fluid motion is given by completely prescribing the necessary properties as a functions of space and time.
Example: one attaches the temperature-measuring device to a particular point (x,y,z) and record the temperature at that point as a function of time. T = T ( x , y , z , t ) => field concept.The independent variables are the spatial coordinates ( x , y , z) and time t
Field Representation of flow• At a given instant in time, any fluid property ( such as density,
pressure, velocity, and acceleration) can be described as a functions of the fluid’s location.This representation of fluid parameters as functions of the spatial coordinates is termed a field representation of flow.
• The specific field representation may be different at different times, so that to describe a fluid flow we must determine the various parameter not only as functions of the spatial coordinates but also as a function of time.
• EXAMPLE: Temperature field T = T ( x , y , z , t )• EXAMPLE: Velocity field
ktzyxwjtzyxvitzyxuV ),,,( ),,,( ),,,( ++=
Ball, with mass m, released from rest at a height y= 0. Air resistance is neglected FD=0. (a) Find the Lagrangian description of this velocity field. (b) Find the Eulerian description of this velocity field.
y
Basic Laws
• Conservation of mass – Continuity Equation• Conservation of (angular) Momentum
- Newton’s second law of motion.• Conservation of Energy
The first law of thermodynamics
Analysis of any problem in fluid mechanics necessarily includes statement of the basic laws governing the fluid motion. The basic laws, which applicable to any fluid, are:
BASIC LAWS FOR A SYSTEM- Conservation of Mass
• Conservation of Mass Requiring that the mass, M, of the system be constant.
Where the mass of the system
0)(
=== ∫ systemV
system
dVDtD
DtDM
dtdM ρ
BASIC LAWS FOR A SYSTEM- Conservation of Momentum
• Newton’s Second Law Stating that the sum of all external force acting on the
system is equal to the time rate of change of linear momentum of the system.
Where P is the linear momentum of the system
∫==
=
)(systemVsystem
dVVDtD
DtPD
dtPdF ρ
BASIC LAWS FOR A SYSTEM- Conservation of Energy
• The First Law of Thermodynamics Requiring that the energy of system be constant.
dEWQ =−δδ
Where E is the total energy of the system and et is the total energy of the system per unit mass
e is specific internal energy, V the speed, and z the height of a particle having mass dm.
)()(∫==
=−
systemV tsystem
dVeDtD
DtDE
dtdEWQ ρ
gzVeet ++=2
2
How To Derive Control Volume Formulation
BASIC LAWS
System Method Control Volume Method
Governing EquationSystem Formulation
Governing EquationControl Volume Formulation
Reynolds Transport TheoremIntegral (large) control volume
Differential (differential) control volume) Total (material) derivative
Transformation between Lagrangianand Eulerian Description
• It is more nature to apply conservation laws by using Lagrangian description (ie. Material Volume).
• However, the Eulerian description (ie. Control Volume) is preferred for solving most of problem in fluid mechanics.
• The two descriptions are related and there are a transformation formula called Reynolds transport theorem and material derivative between Lagrangian and Eulerian descriptions.
Reynolds Transport Theorem
∫∫∫∫∫∫∫∫ •+∂∂
=CSVCsystem
SdVVdt
VdDtD
ααα
By converting the surface integral to volume integral by use of Gauss theorem
∫∫∫∫∫ •∇=•VCCS
VdVSdV )(
αα
Langragian derivative of a volume integral of a given property
This is the fundamental relation between the rate of change of any arbitrary extensive property, α, of a system and the variations of this property associated with a control volume.
∫∫∫∫∫∫∫∫∫∫∫∫ •∇+∂∂
=•∇+∂∂
=VCVCVCsystem
VdVt
VdVVdt
VdDtD ))(()(
ααααα
Conservation of Mass• Basic Law for Conservation of Mass• The system and integral (large) control volume
formulation ---- Reynolds Transport Theorem
α=ρ
ααα∇•+
∂∂
= utDt
D
Material derivativeContinuity equation
A partial differential equation => velocity is continuous
0)( =•∇+∂∂ V
t
ρρ
0)()()( =•∇+=•∇+∇•+∂∂
=•∇+∂∂ V
DtDVV
tV
t
ρρρρρρρ
0=∫∫∫system
VdDtD ρ
∫∫∫∫∫∫ •∇+∂∂
=VCsystem
VdVt
VdDtD ))((
ααα
0))(( =•∇+∂∂
= ∫∫∫∫∫∫VCsystem
VdVt
VdDtD
ρρρ
How To Derive Control Volume Formulation
BASIC LAWS
System Method Control Volume Method
Governing EquationSystem Formulation
Governing EquationControl Volume Formulation
Reynolds Transport TheoremIntegral (large) control volume
Differential (differential) control volume) Total (material) derivative
Material Derivative (I)• Let α(x,y,z,t) be any field variable, e.g., ρ, T, V=(u,v,w), etc.
(Eulerian description)• Observe a fluid particle for a time period δt as it flows
(Langrangian description)• During the time period, the position of the fluid particle will
change by amounts δx , δy , δz, while its vale of α will change by an amount
• As one follow the fluid particle,So
which is called the material, total, or substantial derivative.
zz
yy
xx
tt
δαδαδαδαδα∂∂
+∂∂
+∂∂
+∂∂
=
),,(),,( wvutz
ty
tx
=δδ
δδ
δδ
zw
yv
xu
ttz
zty
ytx
xttDtD
t ∂∂
+∂∂
+∂∂
+∂∂
=∂∂
+∂∂
+∂∂
+∂∂
==→
ααααδδα
δδα
δδαα
δδαα
δ 0lim
Material Derivative (II)• Use the notation D/Dt to emphasize that the
material derivative is the rate of change seen by an observer “following the fluid.”
• The material derivative express a Langrangianderivative in terms of Eulerian derivatives.
• In vector form,
ααααααα )( ∇•+∂∂
=∂∂
+∂∂
+∂∂
+∂∂
= Vtz
wy
vx
utDt
D
EXAMPLE OF SUBSTANTIAL DERIVATIVE
The velocity flow field of a steady state flow is given by the equations: u=-x ; v=y
The temperature of the field is described by the following expression: T(x,y,t)=xt+3xy
Determine the time rate of change of temperature of a fluid element as it passes through the point (1, -2) at time t=6.
the time rate of change of temperature of a fluid element
local derivative
convective derivative
yTv
xTu
tT
DtDT
∂∂
+∂∂
+∂∂
=
5)3)(2()66)(1(1)3()3)(( −=−+−−+=++−+=∂∂
+∂∂
+∂∂
= xyytxxyTv
xTu
tT
DtDT
x
y
Ball, with mass m, released from rest at a height y= 0. Air resistance is neglected FD=0. (a) From the Lagrangian description of this velocity field, find
the acceleration of the ball.(b) From the Eulerian description of this velocity field, , find the acceleration of the ball.
y
0=∇•+∂∂
= Mut
MDt
DM
Conservation of Mass• Basic Law for Conservation of Mass• The system and differential control volume
formulation ---- Material derivative
0=Dt
DM
α=M
0)()(=∇•+∇•+
∂∂
+∂∂
=∇•+∂
∂ ρρρρρρ uVVutVV
tVu
tV
0)()( =∇•+∂∂
+∇•+∂∂
=∇•+∇•+∂∂
+∂∂ Vu
tVu
tVuVVu
tVV
t ρρρρρρρ
0)()1( =•∇+=+=+ VDtD
DtVD
VDtD
DtVD
VDtD
ρρρρρρ
VM ρ=
Volume dilation = divergence of velocity field
ααα∇•+
∂∂
= utDt
D
Physical Meaning of V
•∇
CONSERVATION OF MASSRectangular Coordinate System
• The differential equation for conservation of mass:The continuity equation
By “Del” operator
The continuity equation becomes
0)()( =•∇+∂∂
=•∇+∇•+∂∂
=•∇+ Vt
VVt
VDtD
ρρρρρρρ
Description and Classification of Fluid Motions
Continuity Equation for Incompressible FluidDefinition of Incompressible fluid:As a given fluid is followed, not only will its mass be observed to remain constant, but its volume, and hence its density, will be observed to remain constant.
0)( =ρDtD
Continuity eq.
Continuity equation
Material derivative
Follow a fluid particle
01==•∇
DtVD
VV
?cosntant =ρ
0)( =•∇+∂∂
=•∇+ Vt
VDtD
ρρρρ
Stratified-Fluid Flow• A fluid particle along the line ρ1 or ρ2 will have its
density remain fixed at ρ=ρ1 or ρ=ρ2
0)( =ρDtD
Follow a fluid particleStratified-fluid flow is considered to be incompressible, but ρ is not constant (ρ≠constant ) everywhere ie. ə ρ/ ə x ≠0, ə ρ/ əy≠0,
Stratified-fluid flow may occurs in the ocean (owing to salinity variation) or in the atmosphere (owing to temperature variations). For 2D steady state stratified flow, the continuity equation should be
0)()(=
∂∂
+∂
∂yv
xu ρρ
Since the control volume is fixed in space, the time derivative can be placed inside the volume integral
Apply divergence theorem:
Integral form of continuity equation becomes:
Integral form continuity equation:
Differential form of continuity equation:
Steady and Unsteady flows
Unsteady:
Steady:
Compressible and Incompressible flows
Compressible:
Incompressible: constant
Summary of Continuity Equation
0)( =ρDtD
OUTLINE FOR Chapter 2-2
AERODYNAMICS (W1-2-1)
Basic Laws
• Conservation of mass – Continuity Equation• Conservation of (angular) Momentum
- Newton’s second law of motion.• Conservation of Energy
The first law of thermodynamics
Analysis of any problem in fluid mechanics necessarily includes statement of the basic laws governing the fluid motion. The basic laws, which applicable to any fluid, are:
BASIC LAWS FOR A SYSTEM- Conservation of Momentum
• Newton’s Second Law Stating that the sum of all external force acting on the
system is equal to the time rate of change of linear momentum of the system.
Where P is the linear momentum of the system
∫∫∫==
=
systemsystem
VdVDtD
DtPD
dtPdF ρ
• The forces act on fluid particles:– Body forces ( gravity, electromagnetic ).
– Surface forces ( pressure, viscous ).
Total viscous forces =
Review of Momentum Equation (I)
AERODYNAMICS (W1-2-8)
Momentum Equation
Physical principle:
Force = time rate of change of momentum
Body farces: gravity, electromagnetic forces, or anyother forces which “act at a distance on the fluidinside V.
Force:
Surface forces: pressure and shear stress acting onthe control surface S.
Total viscous forces =
Time rate of change of momentum
The integral form of momentum equations
AERODYNAMICS (W1-2-9)
Divergence theorem
The differential form of momentum equations (Navier-Stokes equations)
For unsteady 3D flow, compressible or incompressible, viscous or inviscid
For
Euler equations for steady inviscid flow
Inviscid, Incompressible: constant
xPuVVu∂∂
−=∇•+•∇ρ1)(
yPvVVv∂∂
−=∇•+•∇ρ1)(
zPwVVw∂∂
−=∇•+•∇ρ1)(
ρ
ρ
ρ-1
-1
-1
Review of Momentum Equation (II)
)()( 2
2
2
2
yu
xuf
xP
yuv
xuu
tu
x ∂∂
+∂∂
++∂∂
−=∂∂
+∂∂
+∂∂ µρρ
xx Fdxdzyudydz
xudxdydzfdPdydz
DtDudxdydzma =
∂∂
+∂∂
++−== )()(ρµρρ
dx
dzdy
12 PPdP −=
2P1P
)(
orce
dxdzyudydz
xuF
dxdzdydzfshear
visoucs
yxxx
∂∂
+∂∂
=
+=
µ
ττ
)()()( 2
2
2
2
yu
xuf
dxdP
dxdydzdxdz
yu
dxdydzdydz
xuf
dxdydzdPdydz
DtDu
xx ∂∂
+∂∂
++−=∂∂
+∂∂
++−= µρµρρ
)()()( 2
2
2
2
yu
xuf
dxdP
yuv
xuu
tu
DtDu
x ∂∂
+∂∂
++−=∂∂
+∂∂
+∂∂
= µρρρ
ααα∇•+
∂∂
= utDt
D Material derivative
SUMMARY
AERODYNAMICS (W1-2-10)
For steady, incompressible, inviscid flow, no body force
Continuity equation:
Momentum equation:
xP
zuw
yuv
xuuuV
∂∂
−=∂∂
+∂∂
+∂∂
=∇•ρ1
yP
zvw
yvv
xvuvV
∂∂
−=∂∂
+∂∂
+∂∂
=∇•ρ1
zP
zww
ywv
xwuwV
∂∂
−=∂∂
+∂∂
+∂∂
=∇•ρ1
0=∂∂
+∂∂
+∂∂
=•∇zw
yv
xuV
kwjviuzyxV
++=),,(
Vorticity• Defining Vorticity ζ which is a measurement of the
rotation of a fluid element as it moves in the flow field:
• In cylindrical coordinates system:
Vyu
xvk
xw
zuj
zv
ywi
×∇=
∂∂
−∂∂
+
∂∂
−∂∂
+
∂∂
−∂∂
== ωζ 2
∂∂
∂∂
∂∂
=×∇===
wvuzyx
kji
VVcurl
ωζ 2
∂∂
−∂∂
+
∂∂
−∂∂
+
∂∂
−∂∂
=×∇θθ
θθ
θ rz
zrzr
Vrr
rVr
er
Vz
Vez
VVr
eV 111
Fluid Rotation
∂∂
−∂∂
=
∂∂
−∂∂
=
∂∂
−∂∂
=
yu
xv
xw
zu
zv
yw
Z
y
x
212121
ω
ω
ω
( ) ( ) ( )[ ]
∂∂
−∂∂
+
∂∂
−∂∂
+
∂∂
−∂∂
=++==×∇=yu
xvk
xw
zuj
zv
ywikjiV zyx
((22 ωωωωζ
= 0 Irrotational≠ 0 rotational
Rotational flow Irrotational flow
0=×∇= VVcurl
0≠×∇= VVcurl
Irrotational flow rotational flow
Irrotational and Rotational flows
W3 8
EXAMPLE OF VORTICITY
( )2 ( )2
CIRCULATION
From Stokes’ theorem
Definition:
Irrotational flow
C
Vds
V∞
Circulation Lift
Example: Circulation in a uniform flow
u = V∞
v = 0
Irrotational flow
For arbitrary close curve C
(irrotational)
Stokes’ theorem
AERODYNAMICS (W1-3-6)
STREAMFUNCTION AND VELOCITY POTENTIAL
Streamfunction Ψ: definition
streamfunction Ψ properties:
automatically satisfy continuity equation
Velocity Potential φ: definition
Velocity potential φ properties:
automatically satisfy irrotational conditon
1.
2. Relationship between streamfunction and velocity potential
= 0
SUMMARY
substantial derivative
local derivative
convective derivative
= 0 Irrotational≠ 0 rotational
1. Substantial derivative
2. Streamline
3. Vorticity
4. Circulation
Irrotational flow
5. Streamfunction Ψ:
6. Velocity Potential φ: