AOE 5104 Class 2 - PowerPoint PPT Presentation

1 / 26
About This Presentation
Title:

AOE 5104 Class 2

Description:

Its properties. The governing laws. Reynolds number. Mach number ... F = Fr er F e F e. Errors on this in online presentation. Vector Algebra in Components ... – PowerPoint PPT presentation

Number of Views:22
Avg rating:3.0/5.0
Slides: 27
Provided by: williamd56
Learn more at: http://www2.esm.vt.edu
Category:
Tags: aoe | class | property

less

Transcript and Presenter's Notes

Title: AOE 5104 Class 2


1
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, 530-630
  • Tuesday 5-6, 530-630

2
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
3
http//www.opendx.org
4
3b. Viscous Flow
Importance of viscous effects governed by
Boundary layer Thin region adjacent to a solid
surface where friction slows the flow.
There is no pressure gradient across a boundary
layer
No-slip condition Fluid immediately adjacent to
a solid surface does not move relative to it
5
3b Viscous Flow
Viscous region not always confined to a thin
layer
Separation Large region of viscous flow produced
when the boundary layer leaves a surface because
of an adverse pressure gradient, or a sharp corner
6
3c. Compressibility
Importance of compressibility effects governed by
  • Incompressible Regime Mlt0.3
  • Negligible compressibility effects
  • Subsonic Regime 0.3ltMlt0.7
  • Quantitative effects, no qualitative effects
  • Transonic Regime 0.7ltMlt1.3
  • Large regions of subsonic and supersonic flow.
    Large qualitative effects.
  • Supersonic Regime Mgt1.3
  • Almost entirely supersonic flow. Large
    qualitative effects

7
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
8
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
9
Flow Past a Sphere
Mach 1.53
Mach 4.01
Pictures are from An Album of Fluid Motion by
Van Dyke
10
3c. Compressibility
Some Qualitative Effects

Shock wave Very strong, thin wave, propagating
supersonically, producing almost instantaneous
compression of the flow, and increase in pressure
and temperature.
Hypersonic vehicle re-entry NASA Image Library
11
3c. Compressibility
Some Qualitative Effects
  • 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).

Cone-cylinder in supersonic free flight, Mach
1.84. Picture from An Album of Fluid Motion by
Van Dyke.
12
Summary
  • What a fluid is. Its properties. The governing
    laws
  • 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 as
    streamlines/eddies
  • Qualitative understanding

13
2. Vector Algebra
14
Vector basics
Q
  • Vector A, A
  • Magnitude A, A
  • Scalar p, ?
  • Types
  • Polar vector
  • Velocity V, force F, pressure gradient ?p
  • Axial vector
  • Angular velocity ?, Vorticity ?, Area A
  • Unit vector
  • i, j, k, es, n, A/A

MAG
DIR
P
15
Vector Algebra
B
A
B
A
  • Addition
  • A B C
  • Dot, or scalar, product
  • A.B ABcos?
  • E.g. WorkF.s
  • Flow rate through dAV.dA or V.ndA
  • A.BB.A A.AA2 A.B0 if perpendicular

C
A
?
B
16
Vector Algebra
  • Cross, or vector, product
  • AxBABsin?e
  • AxB-BxA
  • AxA0
  • AxB0 if A and B parallel

A
?
B
Parallelogram area is AxB
Measured to be lt180o
Perpendicular to A and B in direction given by RH
rule rotation from A to B
17
Vector Algebra Triple Products
  • (A.B)C (B.A)C
  • Mixed product A.BxC
  • Volume of parallelepiped
  • bordered by A, B, C
  • May be cyclically permuted
  • A.BxCC.AxBB.CxA
  • Acyclic permutation changes
  • sign A.BxC-B.AxC etc.
  • Vector triple product
  • Ax(BxC) Vector in plane of B and C
  • (A.C)B (A.B)C

B
C
A
BxC
18
PIV of Flow Downstream of a Circular Cylinder
Chiang Shih , Florida State University
19
Cartesian Coordinates
  • 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 iFy jFz k
  • (F.i) i (F.j) j (F.k) k
  • Components same regardless of location of vector

z
F
k
j
i
r
z
x
y
y
x
20
Cylindrical Coordinates
  • Coordinates r, ? , z
  • Unit vectors er, e?, ez (in directions of
    increasing coordinates)
  • Position vector
  • R r er z ez
  • Vector components
  • F Fr erF? e?Fz ez
  • Components not constant, even if vector is
    constant

F
ez
e?
er
R
z
r
?
21
Spherical Coordinates
Errors on this slide in online presentation
  • Coordinates r, ? , ?
  • Unit vectors er, e?, e? (in directions of
    increasing coordinates)
  • Position vector
  • r r er
  • Vector components
  • F Fr erF? e?F? e?

er
e?
F
r
e?
?
r
?
22
Vector Algebra in Components
works for any orthogonal coordinate system!
23
CFD of Flow Around a Fighter
FinFlo Ltd
24
Concept of Differential Change In a Vector. The
Vector Field.
???(r,t)
Scalar field
VV(r,t)
Vector field
V
dV
VdV
  • Differential change in vector
  • Change in direction
  • Change in magnitude

25
Change in Unit Vectors Cylindrical System
de?
erder
e?
ez
e?de?
der
er
P'
e?
P
z
er
r
d?
?
26
Change in Unit Vectors Spherical System
er
e?
r
e?
?
r
See Formulae for Vector Algebra and Calculus
?
27
Example
Fluid particle Differentially small piece of the
fluid material
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.
VV(t)
RR(t)
Cartesian System
O
Cylindrical System
... This is an example of the calculus of vectors
with respect to time.
28
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
Write a Comment
User Comments (0)
About PowerShow.com