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…

3a. Ideal Flow Viscous and compressible effects small (large Re, low M). Flow is a 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

http://www.opendx.org

3b Viscous FlowViscous region not always confined to a thin layerSeparation: Large region of viscous flow produced when the boundary layer leaves a surface because of an adverse pressure gradient, or a sharp corner

3c. Compressibility

• Incompressible Regime M<0.3– Negligible compressibility effects

• Subsonic Regime 0.3<M<0.7– Quantitative effects, no qualitative effects

• Transonic Regime 0.7<M<1.3– Large regions of subsonic and supersonic flow. Large

qualitative effects.• Supersonic Regime M>1.3

– Almost entirely supersonic flow. Large qualitative effects

Importance of compressibility effects governed byc

VM

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 Motion” by Van Dyke

Flow Past a Circular Cylinder

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

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

Flow Past a Sphere

Mach = 1.53 Mach = 4.01

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

3c. CompressibilitySome Qualitative Effects

Hypersonic vehicle re-entryNASA Image Library

Shock wave: Very strong, thin wave, propagating supersonically, producing almost instantaneous compression of the flow, and increase in pressure and temperature.

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 (and raising 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 Motion” by Van Dyke.

Summary• What a fluid is. Its properties. The governing laws• Reynolds number. Mach number• How Newton’s 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 as streamlines/eddies• Qualitative understanding

2. Vector Algebra

Vector basics Vector: A, A

Magnitude: |A|, A

Scalar: p, Types– Polar vector

• Velocity V, force F, pressure gradient p

– Axial vector • Angular velocity , Vorticity Ω, Area A

– Unit vector• i, j, k, es, n, A/A …

MAG

DIRP

Q

Vector Algebra

• Addition

A + B = C

• Dot, or scalar, product

A.B = ABcos• E.g. Work=F.s

• Flow rate through dA=V.dA or V.ndA

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

AAB B

C

A

B

Vector Algebra

• Cross, or vector, product

AxB=ABsine

• AxB=-BxA• AxA=0• AxB=0 if A and B parallel

A

B

Measured to be <180o

Perpendicular to A and B in direction given by RH rule rotation from A to B

Parallelogram area is |AxB|

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 permutedA.BxC=C.AxB=B.CxA• Acyclic permutation changessign A.BxC=-B.AxC etc.

3. Vector triple product• Ax(BxC) = Vector in plane of B and C

= (A.C)B – (A.B)C

A

B

C

BxC

PIV of Flow Downstream of a Circular Cylinder

Chiang Shih , Florida State University

Cartesian Coordinates

r

ji

k

• Coordinates x, y , z

• Unit vectors i, j, k (in directions of increasing coordinates) are constant

• Position vector r = x i + y j + z k

• Vector components F = Fx i+Fy j+Fz k = (F.i) i+ (F.j) j+ (F.k) k

Components same regardless of location of vector

z

x

y

z

y x

F

Cylindrical Coordinates

R

er

e

ez

• Coordinates r, , z

• Unit vectors er, e, ez (in directions of increasing coordinates)

• Position vector R = r er + z ez

• Vector components F = Fr er+F e+Fz ez

Components not constant, even if vector is constant

r

z

F

Spherical Coordinates

r

er

e

e

rF

• Coordinates r, ,

• Unit vectors er, e, e (in directions of increasing coordinates)

• Position vector r = r er

• Vector components F = Fr er+F e+F e

Errors on this slide in online presentation

Vector Algebra in Components

321

321

321

332211.

BBB

AAA

BABABA

eee

BA

BA

…works for any orthogonal coordinate system!

Concept of Differential Change In a Vector. The Vector Field.

V

-2

-1

0

1

2y/ L

-2

0

2-T /U L

0

1

2

z / L

V+dV

dV

V=V(r,t)

=(r,t)Scalar field

Vector field

Differential change in vector• Change in direction• Change in magnitude

PP'

er

e

ez

d

r

z

Change in Unit Vectors – Cylindrical System

rdd ee ee dd r

0zde

e+de

er+der

er

e

de

der

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”

Example

kjir zyx

kjir

Vdt

dz

dt

dy

dt

dx

dt

d

zr zr eer

R=R(t)

Fluid particleDifferentially small piece of the fluid material

V=V(t) The position of fluid particle moving in a flow varies 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

r dt

dz

dt

dr

dt

dr

dt

de

ee

rV

zr dt

dz

dt

dr

dt

dreee

Cartesian System

Cylindrical System

Vector Calculus w.r.t. Time

• Since any vector may be decomposed into scalar components, calculus w.r.t. time, only involves scalar calculus of the components

dtdtdt

ttt

ttt

ttt

BABA

BAB

ABA

BAB

ABA

BABA

.