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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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1 FUNDAMENTALS 4

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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1 FUNDAMENTALS 5

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 FUNDAMENTALS 7

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 FUNDAMENTALS 8

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 FUNDAMENTALS 9

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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1 FUNDAMENTALS 10

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 11

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 THERMODYNAMICS 12

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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2 THERMODYNAMICS 13

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 15

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:


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:


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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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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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 34

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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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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(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 52

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


or alternatively

e

t+ uk

e

xk= Puk

xk+ ik

uixk

qixi


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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5 VISCOUS FLOW 53

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 VISCOUS FLOW 54

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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5 VISCOUS FLOW 55

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 VISCOUS FLOW 56

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 VISCOUS FLOW 57

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

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