Class 2 - Intro and Vector Algebra

7/27/2019 Class 2 - Intro and Vector Algebra

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AOE 5104 Class 2

Online presentations: Fundamentals

Algebra and Calculus 1

Homework 1, due in class 9/4 Grading Policy

Study Groups

Recitation times (recitations to start week of

9/8) Monday 5-6, 5:30-6:30

Tuesday 5-6, 5:30-6:30


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3a. Ideal Flow

Viscous and compressible effects small (large Re, low M). Flow isa balance between inertia and pressure forces, i.e. acceleration

vector balances the pressure gradient vector

Acceleration vector

Pressure gradient vector

Streamline: Line everywhere tangent to the velocity vector


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http://www.opendx.org


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3b Viscous

FlowViscous region not alwaysconfined to a thin layer

Separation: Large region

of viscous flow produced

when the boundary layerleaves a surface because

of an adverse pressure

gradient, or a sharp

corner


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3c. Compressibility

Incompressible Regime M


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Flow Past a Circular Cylinder

Re = 10,000 and Mach approximately zero

Re = 110,000 and Mach = 0.45 Re = 1.35 M and Mach = 0.64

Pictures are from An Album of Fluid Motionby Van Dyke


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Flow Past a Circular Cylinder

Mach = 0.80 Mach = 0.90 Mach = 0.95 Mach = 0.98



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Flow Past a Sphere

Mach = 1.53 Mach = 4.01



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3c. Compressibility

Some Qualitative Effects

Hypersonic vehicle re-entry

NASA Image Library

Shock wave: Very strong,

thin wave, propagatingsupersonically, producing

almost instantaneous

compression of the flow,

and increase in pressureand temperature.


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3c. Compressibility

Expansion or isentropic

compression wave

Finite wave (often

focused on a corner),

moving at the sound

speed, producing

gradual compression or

expansion of a flow (andraising or lowering of the

temperature and

pressure).

Some Qualitative Effects

Cone-cylinder in supersonic free

flight, Mach = 1.84.

Picture from An Album of Fluid

Motionby Van Dyke.


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Summary

What a fluid is. Its properties. The governinglaws

Reynolds number. Mach number

How Newtons 2nd Law works as a vector

equation Viscous effects: no-slip condition, boundary

layer, separation, wake, turbulence, laminar

Compressibility effects: Regimes, shock waves,isentropic waves.

Initial ideas of concepts such asstreamlines/eddies

Qualitative understanding


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2. Vector Algebra


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Vector basicsVector: A , A

Magnitude: |A |,A

Scalar: p,

Types

Polar vector VelocityV, force F, pressure gradientp

Axial vector

Angular velocity, Vorticity, Area A Unit vector

i,j, k, es, n, A/A

DIR

P

Q


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Vector Algebra

Addition

A + B = C

Dot, or scalar, productA.B=ABcos

E.g. Work=F.s

Flow rate through dA=V.dA orV.ndA

A.B=B.A A.A=A2 A.B=0 if perpendicular

A AB B

C

A

B


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Vector Algebra

Cross, or vector, product

AxB=ABsine

AxB=-BxA AxA=0

AxB=0 ifA andBparallel

A

B

Measured to be


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Vector Algebra Triple Products

1. (A.B)C= (B.A)C

2. Mixed product A.BxC Volume of parallelepiped

bordered by A , B, C May be cyclically permuted

A.BxC=C.AxB=B.CxA

Acyclic permutation changes

sign A.BxC=-B.AxCetc.

3. Vector triple product Ax(BxC)= Vector in plane ofBandC

= (A.C)B (A.B)C

A

B

C

BxC


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PIV of Flow Downstream of a Circular

CylinderChiang Shih , Florida State University


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Cartesian Coordinates

r

ji

k

Coordinatesx, y , z

Unit vectors i,j, k(in

directions of increasing

coordinates) are constant

Position vector

r=xi+ yj+ zk

Vector componentsF= Fxi+Fyj+Fzk

= (F.i)i+ (F.j)j+ (F.k)k

Components same regardless

of location of vector

z

x

y

z

y x

F


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Cylindrical Coordinates

R

er

eez

Coordinates r, , z

Unit vectors er, e, ez(indirections of increasing

coordinates)

Position vector

R= rer+ zez

Vector components

F= Frer+Fe+FzezComponents not constant,

even if vector is constant

r

z

F


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Spherical Coordinates

r

er

e

e

rF

Coordinates r, ,

Unit vectors er, e, e (indirections of increasing

coordinates)

Position vector

r= rer

Vector components

F= Frer+Fe+Fe

Errors on this slide in online presentation


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Vector Algebra in Components

321

321

321

332211.

BBB

AAA

BABABA

eee

BA

BA

works for any orthogonal coordinate system!


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Concept of Differential Change In a

Vector. The Vector Field.

V

-2

-1

0

1

2

y/

L

-2

0

2

-T/ U

L0

1

2

z/L

V+dV

dV

V=V(r,t)

=(r,t)Scalar fieldVector field

Differential change in vector

Change in direction

Change in magnitude


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PP'

er

e

ez

d

r

z

Change in Unit Vectors

Cylindrical System

rdd ee

ee dd r

0zde

e+de

er+de

r

er

ede

der


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Change in Unit Vectors

Spherical System

eee

eee

eee

cossin

cos

sin

ddd

ddd

ddd

r

r

r

r

er

e

e

r

See Formulae for Vector

Algebra and Calculus


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Example

kjir zyx

kjir

Vdt

dz

dt

dy

dt

dx

dt

d

zr zr eer

R=R(t)

Fluid particle

Differentially small

piece of the fluid

material

V=V(t) The position of fluid particle moving in a flowvaries with time. Working in different coordinate

systems write down expressions for the position

and, by differentiation, the velocity vectors.

O

... This is an example of the calculus of vectors with respect to time.

zr

rdtdz

dtdr

dtdr

dtd eeerV

zrdt

dz

dt

dr

dt

dreee

Cartesian

System

Cylindrical

System


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Vector Calculus w.r.t. Time

Since anyvector may be decomposed into

scalar components, calculus w.r.t. time, only

involves scalarcalculus of the components

dtdtdt

ttt

ttt

ttt

BABA

BAB

ABA

BAB

ABA

BABA

.

Documents

Class 2 - Intro and Vector Algebra