aoe 5104 class 8 online presentations for next class: –kinematics 2 and 3 homework 3 (thank you)...
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AOE 5104 Class 8
• Online presentations for next class:– Kinematics 2 and 3
• Homework 3 (thank you)
• Homework 4 (6 questions, 2 graded, 2 recitations, worth double, due in 2 weeks)
• Class next Tuesday will be given by Dr. Aurelien Borgoltz
• No office hours next week
Dovetail stream, Derbyshire UK. http://www.mikejs.com/photos/2002_der.html
The Equations of MotionDifferential Form (for a fixed volume element)
V. Dt
D
).(2)()(f
)().(2)(f
)()().(2f
31
31
31
V
V
V
z
w
zy
w
z
v
yx
w
z
u
xz
p
Dt
Dw
y
w
z
v
zy
v
yy
u
x
v
xy
p
Dt
Dv
x
w
z
u
zy
u
x
v
yx
u
xx
p
Dt
Du
z
y
x
).(2)()()().(2)(
)()().(2).().(.)(
31
31
31
221
VV
VVVf
z
ww
y
w
z
vv
x
w
z
uu
zy
w
z
vw
y
vv
y
u
x
vu
y
x
w
z
uw
y
u
x
vv
x
uu
xTkp
Dt
VeD
The Continuity equation
The Navier Stokes’ equations
The Viscous Flow Energy Equation
.VtDt
D
Fluid Statics
Fluid Statics (V = 0)
• Continuity
• Momentum
• Energy
0t
fp
Tkt
e
V. Dt
D
).(2)()(f
)().(2)(f
)()().(2f
31
31
31
V
V
V
z
w
zy
w
z
v
yx
w
z
u
xz
p
Dt
Dw
y
w
z
v
zy
v
yy
u
x
v
xy
p
Dt
Dv
x
w
z
u
zy
u
x
v
yx
u
xx
p
Dt
Du
z
y
x
.VtDt
D
).(2)()()().(2)(
)()().(2).().(.)(
31
31
31
221
VV
VVVf
z
ww
y
w
z
vv
x
w
z
uu
zy
w
z
vw
y
vv
y
u
x
vu
y
x
w
z
uw
y
u
x
vv
x
uu
xTkp
Dt
VeD
• Equation of Heat Conduction
• Equation of Hydrostatic Equilbrium
• Density is a constant (in time)
Example: Liquid at Rest Under Gravity
z, kWater resevoir g
kf gp
Body force per unit mass
kf g
Momentum equation (density constant)
kkji gz
p
y
p
x
p
g
z
p
y
p
x
p
,0,0
gdz
dp
const. zgp
Expand and compare terms
From this we see that pressure is constant with x and y. Then
Integrate and get that pressure is proportional to depth, or
Variation: Fluid compressible (like air)?
Variation: Water in a rotating tank?
Fluid Dynamics?
The Equations of MotionDifferential Form (for a fixed volume element)
V. Dt
D
).(2)()(f
)().(2)(f
)()().(2f
31
31
31
V
V
V
z
w
zy
w
z
v
yx
w
z
u
xz
p
Dt
Dw
y
w
z
v
zy
v
yy
u
x
v
xy
p
Dt
Dv
x
w
z
u
zy
u
x
v
yx
u
xx
p
Dt
Du
z
y
x
).(2)()()().(2)(
)()().(2).().(.)(
31
31
31
221
VV
VVVf
z
ww
y
w
z
vv
x
w
z
uu
zy
w
z
vw
y
vv
y
u
x
vu
y
x
w
z
uw
y
u
x
vv
x
uu
xTkp
Dt
VeD
The Continuity equation
The Navier Stokes’ equations
The Viscous Flow Energy Equation
These form a closed set when two thermodynamic relations are
specified
DNS example
University of GroningenColor shows pressure
http://gizmodo.com/gadgets/top/earthrace-boat-carbon-trimaran-stabs-through-waves-video-195323.php
The Equations of MotionDifferential Form (for a fixed volume element)
V. Dt
D
).(2)()(f
)().(2)(f
)()().(2f
31
31
31
V
V
V
z
w
zy
w
z
v
yx
w
z
u
xz
p
Dt
Dw
y
w
z
v
zy
v
yy
u
x
v
xy
p
Dt
Dv
x
w
z
u
zy
u
x
v
yx
u
xx
p
Dt
Du
z
y
x
).(2)()()().(2)(
)()().(2).().(.)(
31
31
31
221
VV
VVVf
z
ww
y
w
z
vv
x
w
z
uu
zy
w
z
vw
y
vv
y
u
x
vu
y
x
w
z
uw
y
u
x
vv
x
uu
xTkp
Dt
VeD
The Continuity equation
The Navier Stokes’ equations
The Viscous Flow Energy Equation
These form a closed set when two thermodynamic relations are
specified
Kinematics
Kinematics of Velocity
Kinematic Concepts - Velocity
1. Fluid Line. Any continuous string of fluid particles. Moves with flow. Cannot be broken. Fluid loop – closed fluid line.
2. Particle Path. Locus traced out by an individual fluid particle.
1 23
4
Kinematic Concepts - Velocity
3. Streamline. A line everywhere tangent to the velocity vector. Never cross, except at a stagnation point. No flow across a streamline.
4. Streamsurface. Surface everywhere tangent to the velocity vector. Surface made by all the streamlines passing through a fixed curve in space. No flow through a stream surface.
5. Streamtube. Streamsurface rolled so as to form a tube. No flow through tube wall.
-2-1.5-1-0.500
0.5
1
1.5
(z-zw)/c
a
y/c a
x/ca=1.366, t=.165, regular trip, motion (no), Chit
Flow
Francis turbine simulation ETH Zurich
http://www.cg.inf.ethz.ch/~bauer/turbo/research_gallery.html
Frame of Reference
Mathematical Description
Flow
ds
VStreamline1. Streamlines 0Vsd
2. StreamsurfacesMake up a function (x,y,z,t) so that surfaces = const. are streamsurfaces. is called a ‘streamfunction’.
1 = const.2 = const.
3. Relationship between 1 and 2• Consider a streamline that sits at the intersection of two streamsurfaces. • The two streamsurfaces must be described by two different streamfunctions, say
1 and 2
• At any point on the streamline the perpendicular to each streamsurface, and the velocity must all be normal to each other
• So, what about that mathematical relationship?
u
w
dx
dz
w
v
dz
dy
u
v
dx
dy ;;
Francis turbine simulation ETH Zurich
http://www.cg.inf.ethz.ch/~bauer/turbo/research_gallery.html
Mathematical Description
Flow
ds
VStreamline1. Streamlines 0Vsd
2. StreamsurfacesMake up a function (x,y,z,t) so that surfaces = const. are streamsurfaces. is called a ‘streamfunction’.
1 = const.2 = const.
3. Relationship between 1 and 2• Consider a streamline that sits at the intersection of two streamsurfaces. • The two streamsurfaces must be described by two different streamfunctions, say
1 and 2
• At any point on the streamline the perpendicular to each streamsurface, and the velocity must all be normal to each other
• So, what about that mathematical relationship?
Flow
1 = const.2 = const.
Mathematical Description
21 V
)definition(By 0.. 21 V
0. V
21DirnDirn V
where = (x,y,z,t) and scalar
To find we take
So,
Steady flow: = ,Incompressible flow: = 1, Unsteady flow: meaningless
0. V 0. V
1
2
V
Example: 2D – Flow Over An Airfoil
Take k 22 z
100
111
zyx
kji
V
xv
yu
11 ,
y
x
z
Find consistent relations for the steamfuncitons (implicit or in terms of the velocity field).
Titan
Example: Spherical Flow
Choose const.) (2 r
Flow takes place in spherical shells (no radial velocity).
rr rrrr
eee
e
sin2
11
111
sin
001sin
11
rr
rrr
r
ee
eee
V
11 1,
sin
1
rV
rV
r
ere
e
r
Find a set of streamfunctions.
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