AOE 5104 Class 9

• Online presentations for next class:– Kinematics 2 and 3

• Homework 4 (6 questions, 2 graded, 2 recitations, worth double, due 10/2)

• No office hours this week

Kinematics

Kinematics of Velocity

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

v

p

Dt

Dff

V

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

www.lavision.de

Kinematic Concepts - Velocity3. 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. Infinite number of stream surfaces that contain a given streamline. A streamline must appear at the intersection of two stream surfaces.

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

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 ;;

Flow

1 = const.2 = const.

Mathematical Description

21 V

)definition(By 0.. 21 V

0. V

21DirDir V

where = (x,y,z,t) and scalar

To find we take

So,

Steady flow: = ,Incompressible flow: = 1, Unsteady flow: streamlines largely meaningless

0. V0. V

1

2

V

Example: 2D – Flow Over An Airfoil

Take z2

100

11121 zyx

kji

V

xv

yu

11 ,

y

x

z

Find consistent relations for the steamfuncitons (implicit or in terms of the velocity field).

VV

k 2

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

eree

r

Find a set of streamfunctions.

Kinematics of Vorticity

Hermann Ludwig Ferdinand von Helmholtz (1821-1894)

Vorticity =V• 2 circumferentially averaged

angular velocity of the fluid particles • Sum of rotation rates of 2

perpendicular fluid lines• Non-zero vorticity doesn’t imply

spin .=0. Incompressible?• Direction of ?

U

y

Always true!

Can be anything compared to V that the curl produces

No spin, but a net rotation rate

Circulation

•

• Macroscopic rotation of the fluid around loop C

• Non-zero circulation doesn’t imply spin

• Connected to vorticity flux through Stokes’ theorem

• Stokes’ for a closed surface?

C

dsV.

U

y

c

CS

ddS sVnΩ ..

Open Surface Swith Perimeter CndS

0. dSS

nΩ

Net outflow of vorticity is zero

Flow Past a ‘Cookie-Tin’

Pictures are from “An Album of Fluid Motion” by Van Dyke

Side view

Top view

‘Horseshoe vortex’

Re ~ 4,000

Large Eddy Simulation Re=5000

George Constantinescu IIHR, U. Iowa

Kinematic Concepts - VorticityBoundary layer growing on flat plate

Cylinder projecting from plate

Vortex Line: A line everywhere tangent to the vorticity vector. Vortex lines may not cross. Rarely are they streamlines. Thread together axes of spin of fluid particles. Given by ds=0.

Vortex sheet: Surface formed by all the vortex lines passing through the same curve in space. No vorticity flux through a vortex sheet, i.e. .ndS=0

Vortex tube: Vortex sheet rolled so as to form a tube.

ndS

Vortex line

Vortex tube

Vortex sheet

Vortex Tube

0. S

dSnΩ

dSdS nΩnΩ ..1sec2sec

Section 1

Section 2

ndS

c

CS

ddS sVnΩ ..

Since

12

So, we call

The Vortex Tube Strength

Implications (Helmholtz’ Vortex Theorems, Part 1)

• The strength of a vortex tube (defined as the circulation around it) is constant along the tube.

• The tube, and the vortex lines from which it is composed, can therefore never end. They must extend to infinity or form loops.

• The average vorticity magnitude inside a vortex tube is inversely proportional to the cross-sectional area of the tube

But, does the vortex tube travel along with the fluid, or does it have a life of it’s own?

Fluid loop C at time t

Same fluid loop at time t+dt

Vdt

(V+dV)dt

If it moves with the fluid, then the circulation around the fluid loop shown should stay the same.

CCC Dt

Ddd

Dt

Dd

Dt

D

Dt

D sVs

VsV ...

ds

dtDt

Ddd

ss

0

sVd

Dt

D

Dt

D

C

.

Fluid loop at time t

Same fluid loop at time t+dt

Vdt

(V+dV)dt

v

p

Dt

Dff

V

C

dDt

D

Dt

Ds

V.So the rate of change of around the fluid loop is

Now, the momentum eq. tell us that

Body force per unit mass

Pressure force per unit mass

Viscous force per unit mass, say fv

CCC

ddp

dDt

Dsfssf v ...

‘Body force torque’

‘Pressure force torque’

Viscous force ‘torque’

So, in general

Body Force Torque

kf g

SC

dSd nfsf ..Stokes Theorem

For gravity 0 f

So, body force torque is zero for gravity and for any irrotational body force field

Therefore, body force torque is zero for most practical situations

Pressure Force Torque0.

1.

1

SC

dSpdp ns

If density is constant

So, pressure force torque is zero. Also true as long as = (p).

Pressure torques generated by

• Curved shocks

• Free surface / stratificationEarth ScienceandEngineeringImperialCollege UK

Shock in a CD Nozzle

Bourgoing & Benay (2005), ONERA, France Schlieren visualizationSensitive to in-plane index of ref. gradient

Viscous Force Torque• Viscous force torques are non-zero where viscous forces

are present ( e.g. Boundary layer, wakes)

• Can be really small, even in viscous regions at high Reynolds numbers since viscous force is small in that case

• The viscous force torques can then often be ignored over short time periods or distances

Implications

In the absence of body-force torques, pressure torques and viscous torques…

• the circulation around a fluid loop stays constant: Kelvin’s Circulation Theorem

• a vortex tube travels with the fluid material (as though it were part of it), or– a vortex line will remain coincident with the same fluid line– the vorticity convects with the fluid material, and doesn’t diffuse– fluid with vorticity will always have it– fluid that has no vorticity will never get it

CCC

ddp

dDt

Dsfssf v ...

‘Body force torque’

‘Pressure force torque’

Viscous force ‘torque’

Helmholtz’ Vortex Theorems, Part 2

0Dt

D

Lord William Thompson Kelvin (1824-1907)

Vorticity Transport Equation

• The kinematic condition for convection of vortex lines with fluid lines is found as follows

0dsΩDt

D

VΩΩ

.Dt

D

After a lot of math we get....