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Transcript of AOE 5104 Class 2 Online presentations: –Fundamentals –Algebra and Calculus 1 Homework 1, due in...
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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 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
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http://www.opendx.org
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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
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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
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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 Motion” by Van Dyke
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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
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Flow Past a Sphere
Mach = 1.53 Mach = 4.01
Pictures are from “An Album of Fluid Motion” by Van Dyke
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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.
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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 (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.
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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
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2. Vector Algebra
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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
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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
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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|
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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 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
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PIV of Flow Downstream of a Circular Cylinder
Chiang Shih , Florida State University
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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
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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
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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
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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
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
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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+der
er
e
de
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 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
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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
.