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Foreword
This is guide is intended for students in Ae101 Fluid Mechanics, the class on the fundamentals of fluidmechanics that all first year graduate students take in Aeronautics and Mechanical Engineering at Caltech.It contains all the essential formulas grouped into sections roughly corresponding to the order in which thematerial is taught when I give the course (I have done this now five times beginning in 1995). This is not atext book on the subject or even a set of lecture notes. The document is incomplete as description of fluid
mechanics and entire subject areas such as free surface flows, buoyancy, turbulent flows, etc., are missing(some of these elements are in Brad Sturtevants class notes which cover much of the same ground but aremore expository). It is simply a collection of what I view as essential formulas for most of the class. Theneed for this typeset formulary grew out of my poor chalk board work and the many mistakes that happenwhen I lecture. Several generations of students have chased the errors out but please bring any that remainto my attention.
JES December 16, 2007
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Contents
1 Fundamentals 11.1 Control Volume Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Reynolds Transport Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3.1 Simple Control Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.2 Steady Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4.1 Vector Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4.2 Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4.3 Gauss Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.4 Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.5 Div, Grad and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.6 Specific Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Differential Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5.1 Conservation form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.6 Convective Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 Divergence of Viscous Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.8 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.9 Bernoulli Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.10 Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.11 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Thermodynamics 112.1 Thermodynamic potentials and fundamental relations . . . . . . . . . . . . . . . . . . . . . . 112.2 Maxwell relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Various defined quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 v(P, s) relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Equation of State Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Compressible Flow 153.1 Steady Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.1 Streamlines and Total Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Quasi-One Dimensional Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.1 Isentropic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Heat and Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.1 Fanno Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.2 Rayleigh Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 Shock Jump Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.1 Lab frame (moving shock) versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5 Perfect Gas Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6 Reflected Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.7 Detonation Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.8 Perfect-Gas, 2- Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.8.1 2- Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.8.2 High-Explosives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.9 Weak shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.10 Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.11 Multipole Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.12 Baffled (surface) source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.13 1-D Unsteady Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.14 2-D Steady Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.14.1 Oblique Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
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3.14.2 Weak Oblique Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.14.3 Prandtl-Meyer Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.14.4 Inviscid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.14.5 Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.14.6 Natural Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.14.7 Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Incompressible, Inviscid Flow 344.1 Velocity Field Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 Solutions of Laplaces Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.4 Streamfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4.1 2-D Cartesian Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4.2 Cylindrical Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.4.3 Spherical Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.5 Simple Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.6 Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.7 Key Ideas about Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.8 Unsteady Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.9 Complex Variable Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.9.1 Mapping Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.10 Airfoil Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.11 Thin-Wing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.11.1 Thickness Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.11.2 Camber Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.12 Axisymmetric Slender Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.13 Wing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5 Viscous Flow 525.1 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2 Two-Dimensional Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3 Parallel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3.1 Steady Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3.2 Poiseuille Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.3.3 Rayleigh Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.4 Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.4.1 Blasius Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.4.2 Falkner-Skan Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.5 Karman Integral Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.6 Thwaites Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.7 Laminar Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.8 Compressible Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.8.1 Transformations and Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.8.2 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.8.3 Moving Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.8.4 Weak Shock Wave Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.9 Creeping Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A Famous Numbers 69
B Books on Fluid Mechanics 71
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1 Fundamentals
1.1 Control Volume Statements
is a material volume, V is an arbitrary control volume, indicates the surface of the volume.mass conservation:
ddt
dV = 0 (1)
Momentum conservation:
d
dt
u dV = F (2)
Forces:
F =
G dV +
T dA (3)
Surface traction forcesT = Pn + n = T n (4)
Stress tensorT
T = PI + or Tik = P ik + ik (5)where I is the unit tensor, which in cartesian coordinates is
I = ik (6)
Viscous stress tensor, shear viscosity , bulk viscosity v
ik = 2
Dik 1
3ikDjj
+ vikDjj implicit sum on j (7)
Deformation tensor
Dik =1
2 uixk
+ukxi or
1
2 u +uT
(8)Energy conservation:
d
dt
e +
|u|22
dV = Q + W (9)
Work:
W =
G u dV +
T u dA (10)
Heat:
Q =
q n dA (11)
heat flux q, thermal conductivity k and thermal radiation qr
q = k
T + qr (12)
Entropy inequality (2nd Law of Thermodynamics):
d
dt
s dV
q nT
dA (13)
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1.2 Reynolds Transport Theorem
The multi-dimensional analog of Leibnizs theorem:
d
dt
V(t)
(x, t) dV =
V(t)
tdV +
V
uV n dA (14)
The transport theorem proper. Material volume , arbitrary volume V.
d
dt
dV =d
dt
V
dV +
V
(u uV) n dA (15)
1.3 Integral Equations
The equations of motions can be rewritten with Reynolds Transport Theorem to apply to an (almost) arbi-trary moving control volume. Beware of noninertial reference frames and the apparent forces or accelerationsthat such systems will introduce.
Moving control volume:
d
dt
V
dV +
V
(u uV) n dA = 0 (16)
d
dt
V
udV +V
u (u uV) n dA =V
G dV +V
T dA (17)
d
dt
V
e +
|u|22
dV +
V
e +
|u|22
(u uV) n dA =
V
G u dV +V
T u dA V
q n dA (18)
d
dt
V
sdV +
V
s (u uV) n dA +V
q
T n dA 0 (19)
Stationary control volume:
d
dt
V
dV +
V
u n dA = 0 (20)
d
dt
V
udV +
V
uu n dA =V
G dV +
V
T dA (21)
d
dt
V
e +
|u|22
dV +
V
e +
|u|22
u n dA =
V
G u dV +V
T u dA V
q n dA (22)
ddtV
sdV + V
su n dA + V
qT n dA 0 (23)
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1.3.1 Simple Control Volumes
Consider a stationary control volume V with i = 1, 2, . . ., I connections or openings through which there isfluid flowing in and j = 1, 2, . . ., J connections through which the fluid is following out. At the inflow andoutflow stations, further suppose that we can define average or effective uniform properties hi, i, ui of thefluid. Then the mass conservation equation is
dMdt
= ddt
V
dV =
Ii=1
Aimi Jj=1
Ajmj (24)
where Ai is the cross-sectional area of the ith connection and mi = iui is the mass flow rate per unit areathrough this connection. The energy equation for this same situation is
dE
dt=
d
dt
V
e +
|u|22
+ gz
dV =
Ii=1
Aimi
hi +
|ui|22
+ gzi
Jj=1
Ajmj
hj +
|uj |22
+ gzj
+ Q + W (25)
where
Q is the thermal energy (heat) transferred into the control volume and
W is the mechanical workdone on the fluid inside the control volume.
1.3.2 Steady Momentum Balance
For a stationary control volume, the steady momentum equation can be written asV
uu n dA +V
Pn dA =
V
G dV +
V
n dA + Fext (26)
where Fext are the external forces required to keep objects in contact with the flow in force equilibrium.These reaction forces are only needed if the control volume includes stationary objects or surfaces. For acontrol volume completely within the fluid, Fext = 0.
1.4 Vector Calculus1.4.1 Vector Identities
If A and B are two differentiable vector fields A(x), B(x) and is a differentiable scalar field (x), thenthe following identities hold:
(A B) = (B )A (A )B ( A)B + ( B)A (27)(A B) = (B )A + (A )B + B ( A) + A ( B) (28) () = 0 (29)
( A) = 0 (30)
(
A) = (
A)
2A (31)
(A) = A + A (32)
1.4.2 Curvilinear Coordinates
Scale factors Consider an orthogonal curvilinear coordinate system (x1, x2, x3) defined by a triad of unitvectors (e1, e2, e3), which satisfy the orthogonality condition:
ei ek = ik (33)
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and form a right-handed coordinate system
e3 = e1 e2 (34)
The scale factors hi are defined by
dr = h1dx1e1 + h2dx2e2 + h3dx3e3 (35)
or
hi rxi
(36)The unit of arc length in this coordinate system is d s2 = dr dr:
ds2 = h21 dx21 + h
22 dx
22 + h
23 dx
23 (37)
The unit of differential volume is
dV = h1h2h3 dx1 dx2 dx3 (38)
1.4.3 Gauss Divergence TheoremFor a vector or tensor field F, the following relationship holds:
V
F dV V
F n dA (39)
This leads to the simple interpretation of the divergence as the following limit
F limV0
1
V
V
F n dA (40)
A useful variation on the divergence theorem is
V(
F) dV V n F dA (41)This leads to the simple interpretation of the curl as
F limV0
1
V
V
n F dA (42)
1.4.4 Stokes Theorem
For a vector or tensor field F, the following relationship holds on an open, two-sided surface S bounded bya closed, non-intersecting curve S:
S
( F) n dA S
F dr (43)
1.4.5 Div, Grad and Curl
The gradient operator or grad for a scalar field is
=1
h1
x1e1 +
1
h2
x2e2 +
1
h3
x3e3 (44)
A simple interpretation of the gradient operator is in terms of the differential of a function in a direction a
da = limda0
(x + da) (x) = da (45)
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The divergence operator or div is
F = 1h1h2h3
x1(h2h3F1) +
x2(h3h1F2) +
x3(h1h2F3)
(46)
The curl operator or curl is
F = 1h1h2h3
h1
e1
h2
e2
h3
e3
x1x2
x3
h1F1 h2F2 h3F3
(47)The components of the curl are:
F = e1h2h3
x2(h3F3)
x3(h2F2)
+e2
h3h1
x3(h1F1)
x1(h3F3)
+e3
h1h2
x1(h2F2)
x2(h1F1)
(48)
The Laplacian operator 2 for a scalar field is
2 = 1h1h2h3
x1(
h2h3h1
x1) +
x2(
h3h1h2
x2) +
x3(
h1h2h3
x3)
(49)
1.4.6 Specific Coordinates
(x1, x2, x3) x y z h1 h2 h3
Cartesian
(x , y, z) x y z 1 1 1
Cylindrical
(r, , z) r sin r cos z 1 r 1
Spherical
(r, , ) r sin cos r sin sin r cos 1 r r sin
Parabolic Cylindrical
(u, v, z) 12
(u2 v2) uv z
u2 + v2 h1 1
Paraboloidal
(u, v, ) uv cos uv sin 12
(u2 v2)
u2 + v2 h1 uv
Elliptic Cylindrical
(u, v, z) a cosh u cos v a sinh u sin v z a
sinh2 u + sin2 v h1 1
Prolate Spheroidal
( , , ) a sinh sin cos a sinh sin sin a cosh cos a
sinh2 + sin2 h1 a sinh sin
1.5 Differential Relations
1.5.1 Conservation form
The equations are first written in conservation form
tdensity + flux = source (50)
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for a fixed (Eulerian) control volume in an inertial reference frame by using the divergence theorem.
t+ (u) = 0 (51)
t(u) + (uu T) = G (52)
t e + |u|22 + ue + |u|
2
2 T u + q = G u (53)
t(s) +
us +
q
T
0 (54)
1.6 Convective Form
This form uses the convective or material derivative
D
Dt=
t+ u (55)
D
Dt
=
u (56)
Du
Dt= P + + G (57)
D
Dt
e +
|u|22
= (T u) q + G u (58)
Ds
Dt
qT
(59)
Alternate forms of the energy equation:
D
Dt
e +
|u|22
= (Pu) + ( u) q + G u (60)
Formulation using enthalpy h = e + P/
D
Dt
h +
|u|22
=
P
t+ ( u) q + G u (61)
Mechanical energy equation
D
Dt
|u|22
= (u ) P + u + G u (62)Thermal energy equation
De
Dt= PDv
Dt+ v:u v q (63)
Dissipation
= :u = ikuixk
sum on i and k (64)
Entropy
Ds
Dt=
qT
+
T+ k
T
T
2(65)
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1.7 Divergence of Viscous Stress
For a fluid with constant and v, the divergence of the viscous stress in Cartesian coordinates can bereduced to:
= 2u +
v +1
3
( u) (66)
1.8 Euler Equations
Inviscid, no heat transfer, no body forces.
D
Dt= u (67)
Du
Dt= P (68)
D
Dt
h +
|u|22
=
P
t(69)
Ds
Dt 0 (70)
1.9 Bernoulli Equation
Consider the unsteady energy equation in the form
D
Dt
h +
|u|22
=
P
t+ ( u) q + G u (71)
and further suppose that the external force field G is conservative and can be derived from a potential as
G = (72)
then if (x) only, we have
D
Dt
h +
|u|22
+
=P
t+ ( u) q (73)
The Bernoulli constant is
H = h +|u|2
2+ (74)
In the absence of unsteadiness, viscous forces and heat transfer we have
u
h +|u|2
2+
= 0 (75)
OrH = constant on streamlines
For the ordinary case of isentropic flow of an incompressible fluid dh = dP/ in a uniform gravitationalfield = g(z z), we have the standard result
P + |u|2
2+ gz = constant (76)
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1.10 Vorticity
Vorticity is defined as u (77)
and the vector identities can be used to obtain
(u )u =( |u
|2
2 ) u ( u) (78)The momentum equation can be reformulated to read:
H =
h +
|u|22
+
= u
t+ u + Ts +
(79)
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1.11 Dimensional Analysis
Fundamental Dimensions
L length meter (m)M mass kilogram (kg)T time second (s) temperature Kelvin (K)I current Ampere (A)
Some derived dimensional units
force Newton (N) M LT2
pressure Pascal (Pa) M L1T2
bar = 105 Paenergy Joule (J) M L2T2
frequency Hertz (Hz) T1
power Watt (W) M L2T3
viscosity () Poise (P) M L1T1
Pi Theorem Given n dimensional variables X1, X2, . . ., Xn, and f independent fundamental dimensions
(at most 5) involved in the problem:
1. The number of dimensionally independent variables r is
r f
2. The number p = n - r of dimensionless variables i
i =Xi
X11 X22 Xrr
that can be formed isp n f
Conventional Dimensionless Numbers
Reynolds Re U L/Mach M a U/cPrandtl P r cP/k = /Strouhal St L/U T Knudsen Kn /LPeclet P e U L/Schmidt Sc /DLewis Le D/
Reference conditions: U, velocity; , vicosity; D, mass diffusivity; k, thermal conductivity; L, length scale;T, time scale; c, sound speed; , mean free path; cP, specific heat at constant pressure.
Parameters for Air and Water Values given for nominal standard conditions 20 C and 1 bar.
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Air Watershear viscosity (kg/ms) 1.8105 1.00103kinematic viscosity (m2/s) 1.5105 1.0106thermal conductivity k (W/mK) 2.54102 0.589thermal diffusivity (m2/s) 2.1105 1.4107specific heat cp (J/kgK) 1004. 4182.sound speed c (m/s) 343.3 1484density (kg/m3) 1.2 998.gas constant R (m2/s2K) 287 462.thermal expansion (K1) 3.3104 2.1104isentropic compressibility s (Pa
1) 7.01106 4.51010
Prandtl number P r .72 7.1Fundamental derivative 1.205 4.4ratio of specific heats 1.4 1.007Gruneisen coefficient G 0.40 0.11
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2 Thermodynamics
2.1 Thermodynamic potentials and fundamental relations
energy e(s, v)
de = T ds
Pdv (80)
enthalpy h(s, P) = e + P v
dh = T ds + v dP (81)
Helmholtz f(T, v) = e T sdf = s dT Pdv (82)
Gibbs g(T, P) = e T s + P vdg = s dT + v dP (83)
2.2 Maxwell relations
T
v s = P
s v (84)TP
s
=v
s
P
(85)
s
v
T
=P
T
v
(86)
s
P
T
= vT
P
(87)
Calculus identities:
F(x , y , . . . ) dF =F
x
y,z,...
dx +F
y
x,z,...
dy + . . . (88)
x
y
f
= fyx
fx
y
(89)
x
f
y
=1
fx
y
(90)
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2.3 Various defined quantities
specific heat at constant volume cv eT
v
(91)
specific heat at constant pressure cp hTP (92)
ratio of specific heats cpcv
(93)
sound speed c
P
s
(94)
coefficient of thermal expansion 1v
v
T
P
(95)
isothermal compressibility KT 1v
v
P
T
(96)
isentropic compressibility Ks 1v
v
P
s
=1
c2(97)
Specific heat relationships
KT = Ks orP
v
s
= P
v
T
(98)
cp cv = T
P
v
T
v
T
2P
(99)
Fundamental derivative
c4
2v32v
P2
s
(100)
= v3
2c22P
v2s
(101)
= 1 + c
c
P
s
(102)
=1
2
v2
c2
2h
v2
s
+ 1
(103)
Sound speed (squared)
c2 P
s
(104)
= v2 Pv s (105)=
v
Ks(106)
= v
Kt(107)
Gruneisen Coefficient
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G vcvKT
(108)
= v
P
e
v
(109)
=v
cpKs(110)
= vT
T
v
s
(111)
2.4 v(P, s) relation
dv
v= Ks dP + (Ks dP)2 + Tds
cp+ . . . (112)
= dPc2
+
dP
c2
2+ G
Tds
c2+ . . . (113)
2.5 Equation of State ConstructionGiven cv(v, T) and P(v, T), integrate
de = cv dT +
T
P
T
v
P
dv (114)
ds =cvT
dT +P
T
v
dv (115)
along two paths: I: variable T, fixed and II: variable , fixed T.Energy:
e = e +
TT
cv(T, ) dT
I+
P T P
T
d
2 II(116)
Ideal gas limit 0,
lim0
cv(T, ) = cigv (T) (117)
The ideal gas limit of I is the ideal gas internal energy
eig(T) =
TT
cigv (T) dT (118)
Ideal gas limit of II is the residual function
er(, T) =
0 P TP
Td
2(119)
and the complete expression for internal energy is
e(, T) = e + eig(T) + er(, T) (120)
Entropy:
s = s +
TT
cv(T, )
TdT
I
+
P
T
d
2 II
(121)
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3 Compressible Flow
3.1 Steady Flow
A steady flow must be considered as compressible when the Mach number M = u/c is sufficiently large. Inan isentropic flow, the change in density produced by a speed u can be estimated as
s = c2P 12 M2 (125)from the energy equation discussed below and the fundamental relation of thermodynamics.
If the flow is unsteady, then the change in the density along the pathlines for inviscid flows without bodyforces is
1
D
Dt= u = u u
2
2c2 1
c2
1
2
u2
t 1
P
t
(126)
This first term is the steady flow condition M2. The second set of terms in the square braces are theunsteady contributions. These will be significant when the time scale T is comparable to the acoustic transittime L/c, i.e., T Lco.
3.1.1 Streamlines and Total PropertiesStream lines X(t; x) are defined by
dX
dt= u X = x when t = 0 (127)
which in Cartesian coordinates yieldsdx1u1
=dx2u2
=dx3u3
(128)
Total enthalpy is constant along streamlines in adiabatic, steady, inviscid flow
ht = h +|u|2
2= constant (129)
Velocity along a streamline is given by the energy equation:
u = |u| =
2(ht h) (130)Total properties are defined in terms of total enthalpy and an idealized isentropic deceleration process alonga streamline. Total pressure is defined by
Pt P(s, ht) (131)Other total properties Tt, t, etc. can be computed from the equation of state.
3.2 Quasi-One Dimensional Flow
Adiabatic, frictionless flow:
d(uA) = 0 (132)
udu = dP (133)h +
u2
2= constant or dh = udu (134)
ds 0 (135)
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3.2.1 Isentropic Flow
If ds = 0, then
dP = c2d + c2 ( 1) (d)2
+ . . . (136)
For isentropic flow, the quasi-one-dimensional equations can be written in terms of the Mach number as:
1
d
dx=
M2
1 M21
A
dA
dx(137)
1
c2dP
dx=
M2
1 M21
A
dA
dx(138)
1
u
du
dx= 1
1 M21
A
dA
dx(139)
1
M
dM
dx= 1 + ( 1)M
2
1 M21
A
dA
dx(140)
1
c2dh
dx=
M2
1
M2
1
A
dA
dx(141)
At a throat, the gradient in Mach number is:dM
dx
2=
2A
d2A
dx2(142)
Constant- Gas If the value of is assumed to be constant, the quasi-one dimensional equations can beintegrated to yield:
t
=
1 + ( 1)M2 12(1) (143)ctc
=
t
1
=
1 + ( 1)M2
1/2
(144)
hth
=
1 + ( 1)M2 (145)u = ct
M2
1 + ( 1)M21/2
(146)
A
A=
1
M
1 + ( 1)M2
2(1)
(147)
P Pttc2t
=1
2 1
1 + ( 1)M2 212(1) 1 (148)(149)
Ideal Gas For an ideal gas P = RT and e = e(T) only. In that case, we have
h(T) = e + RT = h +
TT
cv(T) dT, s = s +
TT
cP(T)
TdT R ln(P/P) (150)
and you can show that is given by:
ig =+ 1
2+
12
T
d
dT(151)
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Perfect or Constant- Gas Perfect gas results for isentropic flow can be derived from the equation ofstate
P = RT h = cpT cp =R
1 (152)
the value of for a perfect gas,
pg = + 12
(153)
the energy integral,
Tt = T
1 +
12
M2
(154)
and the expression for entropy
s so = cp ln TTo
R ln P/Po (155)or
s
so = cv ln
T
To R ln /o
TtT
= 1 + 1
2M2 (156)
PtP
=
TtT
1
(157)
t
=
TtT
11
(158)
Mach NumberArea Relationship
A
A =
1
M 2+ 1 1 +
1
2 M
2+1
2(1)
(159)
Choked flow mass flux
M =
2
+ 1
+12(1)
cttA (160)
or
M =
2
+ 1
+12(1) Pt
RTtA
Velocity-Mach number relationship
u = ctM
1 + 12 M2 (161)
Alternative reference speeds
ct = c
+ 1
2umax = c
+ 1
1 (162)
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3.3 Heat and Friction
Constant-area, steady flow with friction F and heat addition Q
u = m = constant (163)
udu + dP =
Fdx (164)
dh + udu = Qdx (165)
ds =1
T
Q +
F
dx (166)
F is the frictional stress per unit length of the duct. In terms of the Fanning friction factor f
F =2
Df u2 (167)
where D is the hydraulic diameter of the duct D = 4area/perimeter. Note that the conventional DArcyor Moody friction factor = 4 f.
Q is the energy addition as heat per unit mass and unit length of the duct. If the heat flux into the fluidis q, then we have
Q =q
u
4
D (168)
3.3.1 Fanno Flow
Constant-area, adiabatic, steady flow with friction only:
u = m = constant (169)
udu + dP = Fdx (170)h +
u2
2= ht = constant (171)
(172)
Change in entropy with volume along Fanno line, h + 1/2 m2v2=ht
Tds
dv
Fanno
=c2 u2
v(1 + G)(173)
3.3.2 Rayleigh Flow
Constant-area, steady flow with heat transfer only:
u = m = constant (174)
P + u2 = I (175)
dh + udu = Qdx (176)(177)
Change in entropy with volume along Rayleigh line, P + m2v = I
Tds
dv
Rayleigh
=c2 u2
vG(178)
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3.4 Shock Jump Conditions
The basic jump conditions,
1w1 = 2w2 (179)
P1 + 1w21 = P2 + 2w
22 (180)
h1 + w21
2= h2 + w
22
2(181)
s2 s1 (182)or defining [f] f2 - f1
[w] = 0 (183)P + w2
= 0 (184)
h +w2
2
= 0 (185)
[s] 0 (186)
The Rayleigh line:
P2 P1v2 v1 = (1w1)
2 = (2w2)2 (187)or
[P]
[v]= (w)2 (188)
Rankine-Hugoniot relation:
h2 h1 = (P2 P1)(v2 + v1)/2 or e2 e1 = (P2 + P1)(v1 v2)/2 (189)Velocity-P v relation
[w]2 =
[P][v] or w2
w1 =
(P2 P1)(v2 v1) (190)Alternate relations useful for numerical solutionP2 = P1 + 1w
21
1 1
2
(191)
h2 = h1 +1
2w21
1
12
2(192)
3.4.1 Lab frame (moving shock) versions
Shock velocity
w1 = Us (193)Particle (fluid) velocity in laboratory frame
w2 = Us up (194)Jump conditions
2 (Us up) = 1Us (195)P2 = P1 + 1Usup (196)
h2 = h1 + up (Us up/2) (197)
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Kinetic energy:
u2p2
=1
2(P2 P1)(v1 v2)
3.5 Perfect Gas Results
[P]
P1=
2
+ 1
M21 1
(198)
[w]
c1= 2
+ 1
M1 1
M1
(199)
[v]
v1= 2
+ 1
1 1
M21
(200)
[s]
R= ln Pt2
Pt1(201)
Pt2Pt1
=1
2+ 1 M21 1+ 11
1
+ 1
2M21
1 + 1
2M2
1
1
(202)
Shock adiabat or Hugoniot:
P2P1
=
+ 1
1 v2v1
+ 1
1v2v1
1(203)
Some alternatives
P2P1
= 1 +2
+ 1 M21 1
(204)
= 2+ 1
M21 1+ 1 (205)21
=+ 1
1 + 2/M21(206)
M22 =
M21 +2
12
1 M21 1
(207)
Prandtls relation
w1w2 = c2 (208)
where c is the sound speed at a sonic point obtained in a fictitious isentropic process in the upstream flow.
c =
2
1+ 1
ht, ht = h +w2
2(209)
3.6 Reflected Shock Waves
Reflected shock velocity UR in terms of the velocity u2 and density 2 behind the incident shock or detonationwave, and the density 3 behind the reflected shock.
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3 COMPRESSIBLE FLOW 21
UR =u2
32
1(210)
Pressure P3 behind reflected shock:
P3
= P2
+3u
22
32
1(211)
Enthalpy h3 behind reflected shock:
h3 = h2 +u222
32
+ 1
32
1(212)
Perfect gas result for incident shock waves:
P3P2
=(3 1) P2
P1 ( 1)
(
1)P2
P1+ (+ 1)
(213)
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3.7 Detonation Waves
Jump conditions:
1w1 = 2w2 (214)
P1 + 1w21 = P2 + 2w
22 (215)
h1 +
w212 = h2 +
w222 (216)
s2 s1 (217)
3.8 Perfect-Gas, 2- Model
Perfect gas with energy release q, different values of and R in reactants and products.
h1 = cp1T (218)
h2 = cp2T q (219)P1 = 1R1T1 (220)
P2 = 2R2T2 (221)
cp1 =1R1
1 1 (222)
cp2 =2R2
2 1 (223)(224)
Substitute into the jump conditions to yield:
P2P1
=1 + 1M
21
1 + 2M22(225)
v2v1
=2M
22
1M2
1
1 + 1M21
1 + 2M2
2
(226)
T2T1
=1R12R2
1
1 1 +1
2M21 +
q
c211
2 1 +1
2M22
(227)
Chapman-Jouguet Conditions Isentrope, Hugoniot and Rayleigh lines are all tangent at the CJ point
PCJ P1vCJ V1 =
P
v
Hugoniot
=P
v
s
(228)
which implies that the product velocity is sonic relative to the wave
w2,CJ = c2 (229)
Entropy variation along adiabat
ds =1
2T(v1 v)2 d m2 (230)
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Jouguets Rule
w2 c2v2
=
1 G
2v(v1 v)
P
v
Hug
Pv
(231)
where G is the Gruniesen coefficient.The flow downstream of a detonation is subsonic relative to the wave for points above the CJ state and
supersonic for states below.
3.8.1 2- Solution
Mach Number for upper CJ (detonation) point
MCJ =
H + (1 + 2)(2 1)
21(1 1) +
H + (2 1)(2 + 1)21(1 1) (232)
where the parameter H is the nondimensional energy release
H = (2 1)(2 + 1)q21R1T1
(233)
CJ pressurePCJP1
=1M
2CJ + 1
2 + 1(234)
CJ densityCJ1
=1(2 + 1)M
2CJ
2(1 + 1M2CJ)(235)
CJ temperatureTCJT1
=PCJP1
R11R2CJ
(236)
Strong detonation approximation MCJ 1
UCJ 2(22 1)q (237)CJ 2 + 1
21 (238)
PCJ 12 + 1
1U2CJ (239)
(240)
3.8.2 High-Explosives
For high-explosives, the same jump conditions apply but the ideal gas equation of state is no longer appro-priate for the products. A simple way to deal with this problem is through the nondimensional slope s ofthe principal isentrope, i.e., the isentrope passing through the CJ point:
s vP
P
v
s
(241)
Note that for a perfect gas, s is identical to = cp/cv, the ratio of specific heats. In general, if the principalisentrope can be expressed as a power law:
P vk = constant (242)
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3 COMPRESSIBLE FLOW 24
then s = k. For high explosive products, s 3. From the definition of the CJ point, we have that theslope of the Rayleigh line and isentrope are equal at the CJ point:
P
v
s
=PCJ P1vCJ V1 =
PCJvCJ
s,CJ (243)
From the mass conservation equation,
vCJ = v1 s,CJs,CJ + 1
(244)
and from momentum conservation, with PCJ P1, we have
PCJ =1U
2CJ
s,CJ + 1(245)
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3.9 Weak shock waves
Nondimensional pressure jump
=[P]
c2(246)
A useful version of the jump conditions (exact):
= M1 [w]c1
= M21 [v]v1[w]c1
= M1[v]v1
(247)
Thermodynamic expansions:
[v]
v1= + 2 + O()3 (248)
= [v]v1
+
[v]
v1
2+ O ([v])
3(249)
Linearized jump conditions:
[w]c1
= 2
2 + O()3 (250)
M1 = 1 2
[w]
c1+ O
[w]
c1
2(251)
M1 = 1 +
2 + O()2 (252)
M2 = 1 2
+ O()2 (253)
[c]
c1= ( 1) + O()2 (254)
M1 1 1 M2 (255)
Prandtls relation
c w1 + 12
[w] or w2 12
[w] (256)
Change in entropy for weak waves:
T[s]
c21=
1
63 + . . . or = 1
6
[v]
v
3+ . . . (257)
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3.10 Acoustics
Simple wavesP = c2 (258)
P = cu (259)+ for right-moving waves, - for left-moving waves
Acoustic Potential
u = (260)
P = ot
(261)
= oc2o
t(262)
Potential Equation
2 1c2o
2
t2= 0 (263)
dAlemberts solution for planar (1D) waves
= f(x cot) + g(x + cot) (264)
Acoustic Impedance The specific acoustic impedance of a medium is defined as
z =P
|u| (265)
For a planar wavefront in a homogeneous medium z = c, depending on the direction of propagation.
Transmission coefficients A plane wave in medium 1 is normally incident on an interface with medium2. Incident (i) and transmitted wave (t)
ut/ui =2z1
z2 + z1(266)
Pt/Pi =
2z2z2 + z1
(267)
Harmonic waves (planar)
= A exp i(wt kx) + B exp i(wt + kx) c = k
k =2
=
2
T= 2f (268)
Spherical waves
=f(t
r/c)
r +g(t + r/c)
r (269)
Spherical source strength Q, [Q] = L3T1
Q(t) = limr0
4r2ur (270)
potential function
(r, t) = Q(t r/c)4r
(271)
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Energy flux
= Pu (272)
Acoustic intensity for harmonic waves
I =< >=1
T T
0
dt =P2rms
c(273)
Decibel scale of acoustic intensity
dB = 10log10(I/Iref) Iref = 1012 W/m2 (274)
or
dB = 20 log10(Prms/P
ref) P
ref = 2 1010 atm (275)
Cylindrical waves, q source strength per unit length [q] = L2T1
(r, t) = 12
tr/c
q() d
(t )2 r2/c2 (276)
or
(r, t) = 12
0
q(t r/c cosh ) d (277)
3.11 Multipole Expansion
Potential from a distribution of volume sources, strength q per unit source volume
(x, t) = 14
Vs
q(xs, t R/c)R
dVs R = |x xs| (278)
Harmonic sourceq = f(x) exp(
it)
Potential function
(x, t) = 14
Vs
f(xs)exp i(kR t)
RdVs (279)
Compact source approximation:
1. source distribution is in bounded region around the origin xs < a,and small a r = |x|
2. source distribution is compact ka
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Dipole term
1(x, t) =ikx D
4r2
1 +
i
kr
exp i(kr t) (283)
Dipole moment vector D
D =
Vs
xsf(xs)dVs (284)
Quadrupole term
2(x, t) =k2
4r3
1 +
3i
kr 3
k2r2
exp i(kr t)
i,j
xixjQij (285)
Quadrupole moments Qik
Qij =1
2
Vs
xs,ixs,jf(xs)dVs (286)
3.12 Baffled (surface) source
Rayleighs formula for the potential
=
1
2 un(xs, t R/c)
RdA (287)
Normal component of the source surface velocity
un = u n (288)
Harmonic source
un = f(x) exp(iwt)Fraunhofer conditions |xs| a
a
a
r 1
Approximate solution:
= exp i(kr wt)2r
As
f(xs)exp i xsdA
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3.13 1-D Unsteady Flow
The primitive variable version of the equations is:
t+ (u) = 0 (289)
u
t + (uu) = P (290)
t
e +
u2
2
+
u(h +
u2
2)
= 0 (291)
s
t+ (us) 0 (292)
(293)
Alternative version
1
D
Dt= u (294)
Du
Dt
=
P (295)
D
Dt
h +
u2
2
=
P
t(296)
Ds
Dt 0 (297)
The characteristic version of the equations for isentropic flow (s = constant) is:
d
dt(u F) = 0 on C : dx
dt= u c (298)
This is equivalent to:
t(u F) + (u c)
x(u F) = 0 (299)
Riemann invariants:
F =
c
d =
d P
c=
d c
1 (300)Bending of characteristics:
d
dP(u + c) =
c(301)
For an ideal gas:
F =2c
1 (302)Pressure-velocity relationship for expansion waves moving to the right into state (1), final state (2) with
velocity u2 < 0.
P2P1
=
1 +
12
u2c1
21 2c1
1 < u2 0 (303)Shock waves moving to the right into state (1), final state (2) with velocity u2 > 0.
[P]
P1=
(+ 1)
4
u2c1
2 1 +
1 +
4
+ 1
c1u2
2 u2 > 0 (304)
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Shock Tube Performance
P4P1
=
1 c1
c4
4 1+ 1
Ms 1
Ms
2441
1 +
211 + 1
M2s 1
(305)
Limiting shock Mach number for P4/P1
Ms c4c1
1 + 14 1 (306)
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3.14 2-D Steady Flow
3.14.1 Oblique Shock Waves
Geometry:
w1 = u1 sin (307)
w2 = u2 sin( ) (308)v = u1 cos = u2 cos( ) (309)
21
=w1w2
=tan
tan( ) (310)Shock Polar
[w]c1
=M1 tan
cos (1 + tan tan )(311)
[P]
1c21
=M21 tan
cot + tan (312)
Real fluid results
w2 = f(w1) normal shock jump conditions (313)
= sin1 (w1/u1) (314)
= tan1
w2u21 w21
(315)
Perfect gas result
tan =2cot
M21 sin
2 1(+ 1)M21 2
M21 sin
2 1
(316)
Mach angle
= sin11
M(317)
3.14.2 Weak Oblique Waves
Results are all for C+ family of waves, take - for C family.
= 12
1M21 1
[w]
c1+ O
[w]
c1
2(318)
=
M21 1M21
[w]
c1+ O
[w]
c1
2(319)
[P]1c21
= M21
M21 1 + O()2 (320)
T1[s]
c21=
16
M61(M21 1)3/2
3 + O()4 (321)
Perfect Gas Results
[P]
P1=
M21M21 1
+ O()2 (322)
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3.14.3 Prandtl-Meyer Expansion
d =
M21 1d u1u1
(323)
Function , d = -d
d
M2
1
1 + ( 1)M2d M
M (324)
Perfect gas result
(M) =
+ 1
1 tan1
1+ 1
(M2 1)
tan1
M2 1 (325)
Maximum turning angle
max =
2
+ 1
1 1
(326)
3.14.4 Inviscid Flow
Crocco-Vaszonyi Relationu
t+ ( u) u = TS(h + u
2
2) (327)
3.14.5 Potential Flow
Steady, homoentropic, homoenthalpic, inviscid:
(u) = 0 (328) u = 0 (329)
h +u2
2= constant (330)
or with u = = (x, y)
(2x c2)xx + (2y c2)yy + 2xyxy = 0 (331)Linearized potential flow:
u = U +
x (332)
v = y (333)
0 =
M2 1
xx yy (334)Wave equation solution
=
M2 1 = f(x y) + g(x + y) (335)Boundary conditions for slender 2-D (Cartesian) bodies y(x)
f() = U
dy
dx
y 0 (336)
Prandtl-Glauert Scaling for subsonic flows
(x, y) = inc(x,
1 M2y) 2inc = 0 (337)
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Prandtl-Glauert Rule
Cp =Cincp
1 M2(338)
3.14.6 Natural Coordinates
x
= cos
s sin
n(339)
y= sin
s+ cos
n(340)
u = Ucos (341)
v = Usin (342)
The transformed equations of motion are:
U
s+ U
n= 0 (343)
UU
s
+P
s
= 0 (344)
U2
s+
P
n= 0 (345)
z = U
s U
n= 0 (346)
Curvature of stream lines, R = radius of curvature
s=
1
R(347)
Vorticity production
z = 1U o
Pon
+(T To)
U
S
n(348)
Elimination of pressure dP = c2d
(M2 1) Us
Un
= 0 (349)
U
n U
s= 0 (350)
3.14.7 Method of Characteristics
s( ) + 1
M2 1
n( ) = 0 (351)
s( + ) 1
M2
1
n( + ) = 0 (352)
(353)
Characteristic directions
Cdn
ds= 1
M2 1 = tan (354)Invariants
J = = constant on C (355)
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4 Incompressible, Inviscid Flow
4.1 Velocity Field Decomposition
Split the velocity field into two parts: irrotational ue, and rotational (vortical) uv.
u = ue + uv (356)
Irrotational Flow Define the irrotational portion of the flow by the following two conditions:
ue = 0 (357) ue = e(x, t) volume source distribution (358)
This is satisfied by deriving ue from a velocity potential
ue = (359)
2 = e(x, t) (360)
Rotational Flow Define the rotational part of the flow by:
uv = 0 (361) uv = (x, t) vorticity source distribution (362)
This is satisfied by deriving uv from a vector potential B
uv = B (363) B = 0 choice of gauge (364)2B = (x, t) (365)
4.2 Solutions of Laplaces Equation
The equation 2 = e is known as Laplaces equation and can be solved by the technique of Greens functions:
(x, t) =
G(x|)e(, t)dV (366)
For a infinite domain, Greens function is the solution to
2G = (x ) (367)G = 1
4
1
|x | = 1
4r(368)
r = |r| r = x (369)This leads to the following solutions for the potentials
(x, t) = 14
e(, t)
rdV (370)
B(x, t) =1
4
(, t)
rdV (371)
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The velocity fields are
ue(x, t) =1
4
re(, t)
r3dV (372)
uv(x, t) =
1
4 r (, t)
r3dV
(373)
If the domain is finite or there are surfaces (stationary or moving bodies, free surfaces, boundaries), thenan additional component of velocity, u, must be added to insure that the boundary conditions (describedsubsequently) are satisfied. This additional component will be a source-free, u = 0, irrotational u= 0 field. The general solution for the velocity field will then be
u = ue + uv + u (374)
4.3 Boundary Conditions
Solid Boundaries In general, at an impermeable boundary , there is no relative motion between thefluid and boundary in the local direction n normal to the boundary surface.
u n = u n on the surface (375)In particular, if the surface is stationary, the normal component of velocity must vanish on the surface
u n = 0 on a stationary surface (376)For an idealor inviscidfluid, there is no restriction on the velocity tangential to the boundary, slip boundaryconditions.
u t arbitrary on the surface (377)For a real or viscous fluid, the tangential component is zero, since the relative velocity between fluid andsurface must vanish, the no-slip condition.
u = 0 on the surface (378)
Fluid Boundaries At an internal or free surface of an ideal fluid, the normal components of the velocityhave to be equal on each side of the surface
u1 n = u2 n = u n (379)and the interface has to be in mechanical equilibrium (in the absence of surface forces such as interfacialtension)
P1 = P2 (380)
4.4 Streamfunction
The vector potential in flows that are two dimensional or have certain symmetries can be simplified to onecomponent that can be represented as a scalar function known as the streamfunction . The exact form ofthe streamfunction depends on the nature of the symmetry and related system of coordinates.
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4.4.1 2-D Cartesian Flows
Compressible In a steady two-dimensional compressible flow:
u = 0 u = (u, v) x = (x, y) ux
+v
y= 0 (381)
The streamfunction is:
u =1
yv = 1
x(382)
Incompressible The density is a constant
u = 0 u = (u, v) x = (x, y) ux
+v
y= 0 (383)
The streamfunction defined by
u =
yv =
x(384)
will identically satisfy the continuity equation as long as
2
xy
2
yx= 0 (385)
which is always true as long as the function (x, y) has continuous 2nd derivatives.Stream lines (or surfaces in 3-D flows) are defined by = constant. The normal to the stream surface is
n =
|| (386)
Integration of the differential of the stream function along a path L connecting points x1 and x2 in the planecan be interpreted as volume flux across the path
d = u nLdl = v dx + u dy (387)L
d = 2 1 =L
u nLdl = volume flux across L (388)
where 1 = (x1) and 2 = (x2). For compressible flows, the difference in the streamfunction can beinterpreted as the mass flux rather than the volume flux.
For this flow, the streamfunction is exactly the nonzero component of the vector potential
B = (Bx, By, Bz) = (0, 0, ) u = B = x y
y x
(389)
and the equation that the streamfunction has to satisfy will be
2 = 2
x2
+ 2
y2
= z (390)where the z-component of vorticity is
z =v
x u
y(391)
A special case of this is irrotational flow with z = 0.
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4.4.2 Cylindrical Polar Coordinates
In cylindrical polar coordinates (r,,z) with u = (ur, u, uz)
x = r cos (392)
y = r sin (393)
z = z (394)u = ur cos u sin (395)v = ur sin + u cos (396)
w = uz (397)
The continuity equation is
u = 0 = 1r
rurr
+1
r
u
+uzz
(398)
Translational Symmetry in z The results given above for 2-D incompressible flow have translationalsymmetry in z such that /z = 0. These can be rewritten in terms of the streamfunction (r, ) where
B = (0, 0, ) (399)
The velocity components are
ur =1
r
(400)
u = r
(401)
The only nonzero component of vorticity is
z =1
r
rur
1r
ur
(402)
and the stream function satisfies
1r
r
r r + 1
r
1
r = z (403)
Rotational Symmetry in If the flow has rotational symmetry in , such that / = 0, then thestream function can be defined as
B =
0,
r, 0
(404)
and the velocity components are:
ur = 1r
z(405)
uz =1
r
r (406)
The only nonzero vorticity component is
=urz
uzr
(407)
The stream function satisfies
z
1
r
z
+
r
1
r
r
= (408)
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4.4.3 Spherical Polar Coordinates
This coordinate system (r,,) results in the continuity equation
1
r2
r
r2ur
+
1
r sin
u
+1
r sin
(u sin ) = 0 (409)
Note that the r coordinate in this system is defined differently than in the cylindrical polar system discussed
previously. If we denote by r the radial distance from the z-axis in the cylindrical polar coordinates, thenr = r sin . With symmetry in the direction /, the following Stokes stream function can be defined
B =
0, 0,
r sin
(410)
Note that this stream function is identical to that used in the previous discussion of the case of rotationalsymmetry in for the cylindrical polar coordinate system if we account for the reordering of the vectorcomponents and the differences in the definitions of the radial coordinates.
The velocity components are:
ur =1
r2 sin
(411)
u = 1r sin
r(412)
The only non-zero vorticity component is:
=1
r
rur
1r
ur
(413)
The stream function satisfies
1
r
r
1
sin
r
+
1
r
1
r2 sin
= (414)
4.5 Simple Flows
The simplest flows are source-free and irrotational, which can be derived by a potential that satisfies theLaplace equation, a special case of ue
2 = 0 u = 0 (415)In the case of flows, that contain sources and sinks or other singularities, this equation holds everywhere
except at those singular points.
Uniform Flow The simplest solution is a uniform flow U:
= U x u = U = constant (416)In 2-D cartesian coordinates with U = Ux, the streamfunction is
= U y (417)
In spherical polar coordinates, Stokes streamfunction is
=U r2
2sin2 U = Uz (418)
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Source Distributions Single source of strength Q(t) located at point 1. The meaning ofQ is the volumeof fluid per unit time introduced or removed at point 1.
limr10
4r21u r1 = Q(t) r1 = x 1 e = Q(t)(x 1) (419)
which leads to the solution:
= Q(t)4r1
u = r1Q(t)4r31
= r1Q(t)4r21
(420)
For multiple sources, add the individual solutions
u = 14
ki=1
riQir3i
(421)
In spherical polar cordinates, Stokes stream function for a single source of strength Q at the origin is
= Q4
cos (422)
For a 2-D flow, the source strength q is the volume flux per unit length or area per unit time since the
source can be thought of as aline
source.
u = ur r ur =q
2r =
q
2ln r =
q
2 (423)
Dipole Consider a source-sink pair of equal strength Q located a distance apart. The limiting process
0 Q Q (424)defines a dipole of strength . If the direction from the sink to the source is d, then the dipole momentvector can be defined as
d = d (425)
The dipole potential for spherical (3-D) sources is
= d r4r3
(426)
and the resulting velocity field is
u =1
4
3d r
r5r d
r3
(427)
If the dipole is aligned with the z-axis, Stokes stream function is
= sin2
4r(428)
and the velocity components are
ur = cos
2r3(429)
u = sin
4r3(430)
The dipole potential for 2-D source-sink pairs is
= 2
cos
r(431)
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and the stream function is
=
2
sin
r(432)
The velocity components are
ur = 2
cos r2
(433)
u =
2
sin
r2(434)
Combinations More complex flows can be built up by superposition of the flows discussed above. Inparticular, flows over bodies can be found as follows:
half-body: source + uniform flowsphere: dipole (3-D) + uniform flowcylinder: dipole (2-D) + uniform flowclosed-body: sources & sinks + uniform flow
4.6 VorticityVorticity fields are divergence free In general, we have ( A) 0 so that the vorticity = u,satisfies
0 (435)
Transport The vorticity transport equation can be obtained from the curl of the momentum equation:
D
Dt= ( )u ( u) +T s +
(436)
The cross products of the thermodynamic derivatives can be written as
T s =P v = P 2
(437)
which is known as the baroclinic torque.For incompressible, homogeneous flow, the viscous term can be written 2 and the incompressible
vorticity transport equation for a homogeneous fluid is
D
Dt= ( )u + 2 (438)
Circulation The circulation is defined as
=
u dl =
n dA (439)
where is is a simple surface bounded by a closed curve .
Vortex Lines and Tubes A vortex line is a curve drawn tangent to the vorticity vectors at each point inthe flow.
dx
x=
dy
y=
dz
z(440)
The collection of vortex lines passing through a simple curve C form a vortex tube. On the surface of thevortex tube, we have n =0.
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A vortex tube of vanishing area is a vortex filament, which is characterized by a circulation . Thecontribution du to the velocity field due to an element dl of a vortex filament is given by the Biot SavartLaw
du = 4
r dlr3
(441)
Line vortex A potential vortex has a singular vorticity field and purely azimuthal velocity field. For asingle vortex located at the origin of a two-dimensional flow
= z(r) u =
2r(442)
For a line vortex of strength i located at (xi, yi), the velocity field at point (x, y) can be obtained bytransforming the above result to get velocity components (u, v)
u = i2
y yi(x xi)2 + (y yi)2 (443)
v =i2
x xi(x
xi)2 + (y
yi)2
(444)
(445)
Or setting = z
ui =i ri
2r2i(446)
where ri = i - xi.The streamfunction for the line vortex is found by integration to be
i = i2
ln ri (447)
For a system ofn vortices, the velocity field can be obtained by superposition of the individual contributionsto the velocity from each vortex. In the absence of boundaries or other surfaces:
u =
ni=1
i ri2r2i
(448)
4.7 Key Ideas about Vorticity
1. Vorticity can be visualized as local rotation within the fluid. The local angular frequency of rotationabout the direction n is
fn = limr0
u2r
=1
2
| n|2
2. Vorticity cannot begin or end within the fluid.
= 03. The circulation is constant along a vortex tube or filament at a given instant in time
tube
n dA = constant
However, the circulation can change with time due to viscous forces, baroclinic torque or nonconser-vative external forces. A vortex tube does not have a fixed identity in a time-dependent flow.
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4. Thompsons or Kelvins theorem Vortex filaments move with the fluid and the circulation is constantfor an inviscid, homogeneous fluid subject only to conservative body forces.
D
Dt= 0 (449)
Bjerknes theorem If the fluid is inviscid but inhomgeneous, (x, t), then the circulation will change due
to the baroclinic torque P :D
Dt=
dP
=
P 2
ndA (450)
Viscous fluids have an additional contribution due to the diffusion of vorticity into or out of the tube.
4.8 Unsteady Potential Flow
Bernoullis equation for unsteady potential flow
P P = t
( ) + U2
2 ||
2
2(451)
Induced Mass If the external force Fext is applied to a body of mass M, then the acceleration of thebody dU/dt is determined by
Fext = (m + M) dUdt
(452)
where M is the induced mass tensor. For a sphere (3-D) or a cylinder (2-D), the induced mass is simply M= miI.
mi,sphere =2
3a3 (453)
mi,cylinder = a2 (454)
(455)
Bubble Oscillations The motion of a bubble of gas within an incompressible fluid can be described byunsteady potential flow in the limit of small-amplitude, low-frequency oscillations. The potential is given bythe 3-D source solution. For a bubble of radius R(t), the potential is
= R2(t)
r
dR
dt(456)
Integration of the momentum equation in spherical coordinates yields the Rayleigh equation
Rd2R
dt2+
3
2
dR
dt
2=
P(R) P
(457)
4.9 Complex Variable MethodsTwo dimensional potential flow problems can be solved in the complex plane
z = x + iy = r exp(i) = r cos + ir sin
The complex potential is defined as
F(z) = + i (458)
and the complex velocity w is defined as
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w = u iv = dFdz
(459)
NB sign of v-term! The complex potential is an analytic function and the derivatives satisfy the Cauchy-Riemann conditions
x
= y
(460)
y=
x(461)
which implies that both 2 = 0 and 2 = 0, i.e., the real and imaginary parts of an analytic functionrepresent irrotational, potential flows.
Examples
1. Uniform flow u = (U, V)
F = (U iV)z2. Line source of strength q located at zo
F =q
2ln(z z)
3. Line vortex of strength located at z
F = i 2
ln(z z)
4. Source doublet (dipole) at z oriented along +x axis
F = 2(z
z)
5. Vortex doublet at z oriented along +x axis
F =i
2(z z)6. Stagnation point
F = Cz2
7. Exterior corner flow
F = Czn 1/2
n
1
8. Interior corner flow, angle
F = Czn 1 n =
9. Circular cylinder at origin, radius a, uniform flow U at x =
F = U(z +a2
z)
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4.9.1 Mapping Methods
A flow in the plane can be mapped into the z plane using an analytic function z = f(). An analytic functionis a conformal map, preserving angles between geometric features such as streamlines and isopotentials aslong as df /dz does not vanish. The velocity in the -plane is w and is related to the z-plane velocity by
w =dF
d
=w
ddz
or w =dF
dz
=w
dzd
(462)
In order to have well behaved values of w, require w =0 at point where dz/d vanishes.
Blasius Theorem The force on a cylindrical (2-D) body in a potential flow is given by
D iL = i2
body
w2 dz (463)
For rigid bodies
D = 0 L = U (464)where the lift is perpendicular to the direction of fluid motion at
. The moment of force about the origin
is
M = 12
body
zw2 dz
(465)
4.10 Airfoil Theory
Rotating Cylinder The streamfunction for a uniform flow U over a cylinder of radius a with a boundvortex of strength is
= Ur sin
1
ar
2
2ln(
r
a) (466)
The stagnation points on the surface of the cylinder can be found at
sin s =
4Ua(467)
The lift L is given by
L = U (468)The pressure coefficient on the surface of the cylinder is
CP =P P12
U2= 1 4sin2 + 4
2aUsin
2aU
2(469)
Generalized Cylinder Flow If the flow at infinity is at angle w.r.t. the x-axis, the complex potentialfor flow over a cylinder of radius a, center and bound circulation is:
F(z) = U
exp(i)(z ) + a
2 exp(i)
z
i 2
ln(z
a) (470)
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Joukowski Transformation The transformation
z = +2T
(471)
is the Joukowski transformation, which will map a cylinder of radius T in the -plane to a line segment y=0, 2T x 2T. Use this together with the generalized cylinder flow in the plane to produce the flowfor a Joukowski arifoil at an angle of attack. The inverse transformation is
=z
2z
2
2 2T (472)
Kutta Condition The flow at the trailing edge of an airfoil must leave smoothly without any singularities.There are two special cases:
For a finite-angle trailing edge in potential flow, the trailing edge must be a stagnation point. For a cusp (zero angle) trailing edge in potential flow, the velocity can be finite but must be equal on
the two sides of the separating streamline.
Application to Joukowski airfoil: Locating the stagnation point at T = + a exp i, the circulationis determined to be:
= 4aU sin( + ) (473)and the lift coefficient is
CL =L
12U
2c
= 8a
c
sin( + ) (474)
4.11 Thin-Wing Theory
The flow consists of the superposition of the free stream flow and an irrotational velocity field derived fromdisturbance potentials t and c associated with the thickness and camber functions.
u = U cos + ut + uc (475)
v = U sin + vt + vc (476)
ut = t (477)uc = c (478)
(479)
where is the angle of attack and 2i = 0.
Geometry A thin, two-dimensional, wing-like body can be represented by two surfaces displaced slightlyabout a wing chord aligned with the x-axis, 0 x c. The upper (+) and lower () surfaces of the wingare given by
y = Y+(x) for upper surface 0 x c (480)y = Y(x) for lower surface 0 x c (481)
and can be represented by a thickness function f(x) and a camber function g(x).
f(x) = Y+(x) Y(x) (482)g(x) =
1
2[Y+(x) + Y(x)] (483)
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The profiles of the upper and lower surface can be expressed in terms of f and g as
Y+(x) = g(x) +1
2f(x) upper surface (484)
Y(x) = g(x) 12
f(x) lower surface (485)
The maximum thickness t = O(f), the maximum camber h = O(g), and the angle of attack are allconsidered to be small in this analysis
tc
hc
1 and ui, vi
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Delta Function Representation The limit of the integrand is one of the representations of the Diracdelta function
limy0
1
y
y2 + (x )2 = (x ) (497)
where
(x ) = 0 x = x = +
f()(x ) d = f(x) (498)
Source Distribution This leads to the source distribution
q(x) = Udf
dx(499)
and the solution for the velocity field is
ut =tx
=U2
c0
(x )f() dy2 + (x )2 (500)
vt =t
y
=U
2 c
0
yf() d
y
2
+ (x )2
(501)
The velocity components satisfy the following relationships across the surface of the wing
[u] = u(x, 0+) u(x, 0) = 0 (502)[v] = v(x, 0+) v(x, 0) = q(x) (503)
Pressure Coefficient The pressure coefficient is defined to be
CP =P P12U
2
= 1 u2 + v2
U2(504)
The linearized version of this is:
CP 2 ut + ucU
(505)
For the pure thickness case, then we have the following result:
CP 1
c0
f() d
(x ) (506)
The integral is to be evaluated in the sense of the Principal value interpretation.
Principal Value Integrals If an integral has an integrand g that is singular at = x, the principal valueor finite part is defined as
P a0 g() d = lim0
x
0 g() d + a
x+ g() d (507)Important principal value integrals are
P
c0
d
(x ) = ln
x
x c
(508)
and
P
c0
1/2d
(x ) =1x
ln
c +
x
c x
(509)
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A generalization to other powers can be obtained by the recursion relation
P
c0
nd
(x ) = xPc0
n1d
(x ) cn
n(510)
A special case can be found for the transformed variables cos = 1 - 2/c
P0
cos nd
cos cos o = sin n
osin o (511)
4.11.2 Camber Case
The camber case alone accounts for the lift (non-zero ) and the camber. The potential c for the purecamber case can be represented as a superposition of potential vortices of strength (x) dx along the chordof the wing:
c =1
2
c0
()tan1
y
x
d (512)
The velocity components are:
uc =cx
= 12
c0
y() dy2 + (x )2 (513)
vc =cy
=1
2
c0
(x )() dy2 + (x )2 (514)
The u component of velocity on the surface of the wing is
limy0
uc(x, y) = u(x, 0) = (x)2
(515)
Apply the linearized boundary condition to obtain the following integral equation for the vorticity distribution
Udgdx = 12P c
0
() d
(x ) d (516)The total circulation is given by
=
c0
() d (517)
The velocity components satisfy the following relationships across the surface of the wing
[u] = u(x, 0+) u(x, 0) = (x) (518)[v] = v(x, 0+) v(x, 0) = 0 (519)
Kutta Condition The Kutta condition at the trailing edge of a sharp-edged airfoil reduces to
(x = c) = 0 (520)
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4 INCOMPRESSIBLE, INVISCID FLOW 49
Vorticity Distribution The integral equation for the vorticity distribution can be solved explicity. Asolution that satisfies the Kutta boundary condition is:
(x) = 2U
c xx
1/2 +
1
P
c0
g()
x
c 1/2
d
(521)
The pressure coefficient for the pure camber case is
CP = (x)U
for y 0 (522)
The integrals can be computed exactly for several special cases, which can be expressed most convenientlyusing the transformation
z =2x
c 1 = 2
c 1 (523)
P
11
1
z
1 +
1 d = (524)
P 11
1 2
z d = z (525)
P
11
11 2(z ) d
= 0 (526)
P
11
1 2(z ) d
= (527)
P
11
21 2(z )
d = z (528)
Higher powers of the numerator can be evaluated from the recursion relation:
P11
n1 2(z ) d = zP
1
1
n11 2(z ) d
2 [1 (1)n]1(3)
(n
2)
2(4) (n 1) (529)
4.12 Axisymmetric Slender Bodies
Disturbance potential solution using source distribution on x-axis:
(x, r) = 14
c0
f() d(x )2 + r2 (530)
Velocity components:
u = U +
x
=1
4 c
0
(x )f() d
[(x )2 + r2]3/2
(531)
v =
r=
1
4
c0
rf() d
[(x )2 + r2]3/2(532)
(533)
Exact boundary condition on body R(x)
v
u
(x,R(x))
=dR
dx(534)
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Linearized boundary condition, first approximation:
v(x, r = R) = UdR
dx(535)
Extrapolation to x axis:
limr0(2rv) = 2R
dR
dx U (536)Source strength
f(x) = U2RdR
dx= UA
(x) A(x) = R2(x) (537)
Pressure coefficient
CP 2uU
dR
dx
2(538)
4.13 Wing Theory
Wing span is
b/2 < y < +b/2. The section lift coefficient, L = lift per unit span
CL(y) =L
12U
2c(y)
= m(y) ( i (y)) (539)
Induced angle of attack, w = downwash velocity
i = tan1
w
U
w
U(540)
Induced drag
Di = Ui (541)
Load distribution (y), bound circulation at span location y
(y) =12
mUc(y) ( i (y)) (542)Trailing vortex sheet strength
= ddy
(543)
Downwash velocity
w =1
4piP
+b/2b/2
() d
y (544)
Integral equation for load distribution
(y) =1
2m(y)Uc(y)
(y) 1
4piUP
+b/2b/2
() d
y
(545)
Boundary conditions
(b
2) = ( b
2) = 0 (546)
Elliptic load distribution, constant downwash, induced angle of attack
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4 INCOMPRESSIBLE, INVISCID FLOW 51
(y) = s
1
y2b
21/2w =
s2b
i =s
2U(547)
Lift
L = Usb2
4
(548)
Induced drag (minimum for elliptic loading)
Di =1
L2
12U
2b
2(549)
Induced drag coefficient
CD,i =C2L
ARAR = b2/S b
c(550)
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5 Viscous Flow
Equations of motion in cartesian tensor form (without body forces) are:Conservation of mass:
t+
ukxk
= 0 (551)
Momentum equation:
uit
+ ukuixk
= Pxi
+ikxk
(i = 1, 2, 3) (552)
Viscous stress tensor
ik =
uixk
+ukxi
+ ik
ujxj
sum on j (553)
Lames constant
= v 23
(554)
Energy equation, total enthalpy form:
htt
+ ukhtxk
=P
t+
kiuixk
qixi
sum on i and k (555)
Thermal energy form
h
t+ uk
h
xk=
P
t+ uk
P
xk+ ik
uixk
qixi
sum on i and k (556)
or alternatively
e
t+ uk
e
xk= Puk
xk+ ik
uixk
qixi
sum on i and k (557)
Dissipation function
= ik ui
xk(558)
Fouriers law
qi = k Txi
(559)
5.1 Scaling
Reference conditions are
velocity Ulength L
time Tdensity
viscosity thermal conductivity k
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Inertial flow Limit of vanishing viscosity, 0. Nondimensional statement:
Reynolds number Re =UL
1 P U2 (560)
Nondimensional momentum equation
Du
Dt =
P +
1
Re
(561)Limiting case, Re , inviscid flow
Du
Dt= P (562)
Viscous flow Limit of vanishing density, 0. Nondimensional statement:
Reynolds number Re =UL
1 P U
L(563)
Nondimensional momentum equation
ReDu
Dt
=
P +
(564)
Limiting case, Re 0, creeping flow.
P = (565)
5.2 Two-Dimensional Flow
For a viscous flow in two-space dimensions (x, y) the components of the viscous stress tensor in cartersiancoordinates are
=
xx xyyx yy
=
2ux +
ux +
vy
uy +
vx
uy +
vx
2vy +
ux +
vy
(566)
Dissipation function
=
2
u
x
2+ 2
v
y
2+
u
y+
v
x
2+
u
x+
v
y
2(567)
5.3 Parallel Flow
The simplest case of viscous flow is parallel flow,
(u, v) = (u(y, t), 0) u = 0 = (y) only (568)Momentum equation
u
t = P
x +
y uy (569)
0 = Py
(570)
We conclude from the y-momentum equation that P = P(x) only.Energy equation
e
t+ u
e
x=
u
y
2+
x
k
T
x
+
y
k
T
y
(571)
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5.3.1 Steady Flows
In these flows t = 0 and inertia plays no role. Shear stress is either constant or varies only due to imposedaxial pressure gradients.
Couette Flow A special case are flows in which pressure gradients are absent
Px
= 0 (572)
and the properties strictly depend only on the y coordinate, these flows have x = 0. The shear stress isconstant in these flows
xy = u
y= w (573)
The motion is produced by friction at the moving boundaries
u(y = H) = U u(y = 0) = 0 (574)
and given the viscosity (y) the velocity profile and shear stress w can be determined by integration
u(y) = wyo
dy
(y)w = U
H0
dy
(y)
1(575)
The dissipation is balanced by thermal conduction in the y direction.
u
y
2=
y
k
T
y
(576)
Using the constant shear stress condition, we have the following energy integral
u q = qw = constant q = k Ty
qw = q(y = 0) (577)
This relationship can be further investigated by defining the Prandtl number
P r =cP
k=
=
k
cP(578)
For gases, P r 0.7, approximately independent of temperature. The Eucken relation is a useful approxi-mation that only depends on the ratio of specific heats
P r 47.08 1.80 (579)
For many gases, both viscosity and conductivity can be approximated by power laws Tn, k Tm wherethe exponents n and m range between 0.65 to 1.4 depending on the substance.
Constant Prandtl Number Assuming P r = constant and using d h = cpd T, the energy equation canbe integrated to obtain the Crocco-Busemannrelation
h hw + P r u2
2= qw
wP r u (580)
For constant cP, this is
T = Tw P r u2
2cP qw
w
P r
cPu (581)
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Recovery Temperature If the lower wall (y = 0) is insulated qw = 0, then the temperature at y = 0 isdefined to be the recovery temperature. In terms of the conditions at the upper plate (y = H), this definesa recovery enthalpy
hr = h(Tr) h(TH) + P r 12
U2 (582)
If the heat capacity cP = constant and we use the conventional boundary layer notation, for which TH =Te, the temperature at the outer edge of the boundary layer
Tr = Te + P r1
2
U2
cP(583)
Contrast with the adiabatic stagnation temperature
Tt = Te +1
2
U2
cP(584)
The recovery factor is defined as
r =Tr TeTt Te (585)
In Couette flow, r = P r. The wall temperature is lower than the adiabatic stagnation temperature Tt whenP r < 1, due to thermal conduction removing energy faster than it is being generated by viscous dissipation.If P r > 1, then viscous dissipation generates heat faster than it can be conducted away from the wall andTr > Tt.
Reynolds Analogy If the wall is not adiabatic, then the heat flux at the lower wall may significantlychange the temperature profile. In particular the lower wall temperature (for cp = constant) is
Tw = Tr +qw
cPwP rU (586)
In order to heat the fluid qw > 0, the lower wall must be hotter than the recovery temperature.The heat transfer from the wall can be expressed as a heat transfer coefficient or Stantonnumber
St = qwU cP(Tw Tr) (587)
where qw is the heat flux from the wall into the fluid, which is positive when heat is being added to the fluid.The Stanton number is proportional to the skin friction coefficient
Cf =w
12U
2(588)
For Couette flow,
St =Cf
2P r(589)
This relationship between skin friction and heat transfer is the Reynolds analogy.
Constant properties If and k are constant, then the velocity profile is linear:
w = U
Hu =
w
y (590)
The skin friction coefficient is
Cf =2
ReRe =
U H
(591)
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5.3.2 Poiseuille Flow
If an axial pressure gradient is present, Px < 0, then the shear stress will vary across the channel and fluidmotion will result even when the walls are stationary. In that case, the shear stress balances the pressuredrop. This is the usual situation in industrial pipe and channel flows. For the simple case of constant
0 =
P
x+
2u
y2(592)
With the boundary conditions u(0) = u(H) = 0, this can be integrated to yield the velocity distribution
u = Px
H2
2
y
H
1 y
H
(593)
and the wall shear stress
w = Px
H
2(594)
Pipe Flow The same situation for a round channel, a pipe of radius R, reduces to
1
r
r r
u
r =
1
P
x (595)
which integrates to the velocity distribution
u = 14
P
x
R2 r2 (596)
and a wall shear stress of
w = Px
R
2(597)
The total volume flow rate is
Q =
P
x
R4
8
(598)
The skin friction coefficient is traditionally based on the mean speed u and using the pipe diameter d = 2Ras the scale length.
u =Q
R2= P
x
R2
8(599)
and is equal to
Cf =w
1/2u2=
16
RedRed =
ud
(600)
In terms of the Darcy friction factor,
=8wu2
=64
Red
(601)
Turbulent flow in smooth pipes is correlated by Prandtls formula
1
= 2.0log
Red
0.8 (602)
or the simpler curvefit
= 1.02 (log Red)2.5
(603)
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5.3.3 Rayleigh Problem
Also known as Stokes first problem. Another variant of parallel flow is unsteady flow with no gradients inthe x direction. The Rayleigh problem is to determine the motion above an infinite ( < x < ) plateimpulsively accelerated parallel to itself.
The x-momentum equation (for constant ) is
ut
= 2
uy2
(604)
The boundary conditions are
u(y, t = 0) = 0 u(y = 0, t > 0) = U (605)
The problem is self similar and in terms of the similarity variable , the solution is
u = U f() =yt
f +
2f = 0 (606)
The solution is the complementary error function
f = erfc(
2
) erfc(s) = 1
erf(s) erf(s) =
2
s
0
exp(
x2) dx (607)
Shear stress at the wall
w = U t