AOE 5104 Class 8 Online presentations for next class: –Kinematics 2 and 3 Homework 3 (thank you)...

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

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?


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




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


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




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


Flow

ds

VStreamline1. Streamlines 0Vsd

2. StreamsurfacesMake up a function (x,y,z,t) so that surfaces = const. are streamsurfaces. is called a ‘streamfunction’.


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



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).

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.

AOE 5104 Class 8 Online presentations for next class: –Kinematics 2 and 3 Homework 3 (thank you)...

Documents

Transcript of AOE 5104 Class 8 Online presentations for next class: –Kinematics 2 and 3 Homework 3 (thank you)...