class 2 - intro and vector algebra
TRANSCRIPT
-
7/27/2019 Class 2 - Intro and Vector Algebra
1/26
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
-
7/27/2019 Class 2 - Intro and Vector Algebra
2/26
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
-
7/27/2019 Class 2 - Intro and Vector Algebra
3/26
http://www.opendx.org
-
7/27/2019 Class 2 - Intro and Vector Algebra
4/26
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
-
7/27/2019 Class 2 - Intro and Vector Algebra
5/26
3c. Compressibility
Incompressible Regime M
-
7/27/2019 Class 2 - Intro and Vector Algebra
6/26
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
-
7/27/2019 Class 2 - Intro and Vector Algebra
7/26
Flow Past a Circular Cylinder
Mach = 0.80 Mach = 0.90 Mach = 0.95 Mach = 0.98
Pictures are from An Album of Fluid Motionby Van Dyke
-
7/27/2019 Class 2 - Intro and Vector Algebra
8/26
Flow Past a Sphere
Mach = 1.53 Mach = 4.01
Pictures are from An Album of Fluid Motionby Van Dyke
-
7/27/2019 Class 2 - Intro and Vector Algebra
9/26
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.
-
7/27/2019 Class 2 - Intro and Vector Algebra
10/26
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.
-
7/27/2019 Class 2 - Intro and Vector Algebra
11/26
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
-
7/27/2019 Class 2 - Intro and Vector Algebra
12/26
2. Vector Algebra
-
7/27/2019 Class 2 - Intro and Vector Algebra
13/26
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
-
7/27/2019 Class 2 - Intro and Vector Algebra
14/26
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
-
7/27/2019 Class 2 - Intro and Vector Algebra
15/26
Vector Algebra
Cross, or vector, product
AxB=ABsine
AxB=-BxA AxA=0
AxB=0 ifA andBparallel
A
B
Measured to be
-
7/27/2019 Class 2 - Intro and Vector Algebra
16/26
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
-
7/27/2019 Class 2 - Intro and Vector Algebra
17/26
PIV of Flow Downstream of a Circular
CylinderChiang Shih , Florida State University
-
7/27/2019 Class 2 - Intro and Vector Algebra
18/26
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
-
7/27/2019 Class 2 - Intro and Vector Algebra
19/26
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
-
7/27/2019 Class 2 - Intro and Vector Algebra
20/26
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
-
7/27/2019 Class 2 - Intro and Vector Algebra
21/26
Vector Algebra in Components
321
321
321
332211.
BBB
AAA
BABABA
eee
BA
BA
works for any orthogonal coordinate system!
-
7/27/2019 Class 2 - Intro and Vector Algebra
22/26
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
-
7/27/2019 Class 2 - Intro and Vector Algebra
23/26
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
-
7/27/2019 Class 2 - Intro and Vector Algebra
24/26
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
-
7/27/2019 Class 2 - Intro and Vector Algebra
25/26
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
-
7/27/2019 Class 2 - Intro and Vector Algebra
26/26
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
.