Aerodynamics I
Lecture Notes
Xin Tong
518021910921
F1801002
School of Naval Architecture, Ocean &
Civil Engineering
Shanghai Jiao Tong University
Update: June 12, 2021
1
Introduction
1.2
Classification and practical objectives
1.4
Some fundamental aerodynamic variables
Pressure
dF
p = lim
dA
dA → 0
(1)
Density
ρ = lim
dm
dv
dv → 0
(2)
Temperature
Shear stress
3
KE = kT
2
dFf
τ = lim
dA → 0
dA
where
τ =µ
dV
dy
(3)
(4)
(5)
Figure 1: System International Unites
1.5
Aerodynamic forces and moments
The relation between force systems
L = N cos α − A sin α
D = N sin α + A cos α
2
(6)
The total normal and axial forces per unit span
ˆ TE
ˆ TE
′
N =−
(pu cos θ + τu sin θ) dsu +
(pl cos θ − τl sin θ) dsl
LE
LE
(7)
ˆ TE
ˆ TE
′
A =
(−pu sin θ + τu cos θ) dsu +
(pl sin θ + τl cos θ) dsl
LE
LE
The moment about the leading edge per unit span
ˆ TE
′
MLE =
[(pu cos θ + τu sin θ) x − (pu sin θ − τu cos θ) y] dsu
LE
(8)
ˆ TE
+
[(−pl cos θ + τl sin θ) x + (pl sin θ + τl cos θ) y] dsl
LE
Dynamic pressure
1
q∞ ≡ ρ∞ V∞2
2
Dimensionless forces coefficients
Lift coefficient
(9)
CL ≡
L
q∞ S
(10)
CD ≡
D
q∞ S
(11)
CM ≡
M
q∞ Sl
(12)
Drag coefficient
Moment coefficient
For two-dimensional bodies
L′
cl ≡
q∞ c
1.6
D′
cd ≡
q∞ c
M′
cm ≡
q ∞ c2
(13)
Center of pressure
Center of pressure
xcp = −
′
MLE
N′
(14)
The relation between the moments
c
′
′
= −xcp L′
MLE
= − L′ + Mc/4
4
3
(15)
1.7
Dimensional analysis: The Buckingham PI theorem
From dimensional analysis
CR = f (Re, M∞ )
(16)
Dimensionless force coefficient
CR =
R
(17)
1
2
2 ρ∞ V∞ S
Non-dimensional Reynolds number
Re =
ρ∞ V∞ c
µ∞
(18)
V∞
a∞
(19)
The Mach Number
M∞ =
Flow Assumptions
1
1
µ ∝ T 2,
1.9
a∝T2
(20)
Fluid static dynamics: Buoyance
Hydrostatic equation
dp = −gρdy
(21)
p + ρgh = constant
(22)
expressed as
1.10 Type of flow
1.10.1
Continuum versus free molecule flow
Continuum flow
4
Free molecular flow
Intermediate state
1.10.2
Inviscid versus viscous flow
Viscous flow
Inviscid flow
1.10.3
Incompressible versus compressible flow
Incompressible flow
Compressible flow
1.10.4
Mach number regimes
1.11 Viscous flow: Brief introduction on boundary layer
The shear stress on the wall skin
dV
τw = µ
dy
(23)
y=0
The aero-heating rate
q̇w = −k
dT
dy
(24)
y=0
Where, k is the thermal conductivity of the gas.
The local Reynolds number
Rex =
ρ∞ V∞ x
µ∞
(25)
Compare laminar flow and turbulent flow
(τw )lamin < (τw )turbu
5
(26)
1.12 Applied aerodynamics: Aerodynamic coefficients
For compressible and incompressible flow
CD = f (Re, M∞ ) ,
CD = f (Re)
(27)
The value of drag
D′ = q∞ SCD
(28)
For circle cylinder the Reynolds number
Re =
ρ∞ V∞ d
µ∞
(29)
Nature of drag: Axial force
D′ = pressure drag + skin friction drag
where
ˆ
ˆ
TE
pressure drag =
ˆLETE
skin friction drag =
−pu sin θdsu +
ˆ
τu cos θdsu +
LE
(30)
TE
pl sin θdsl
LE
TE
(31)
τl cos θdsl
LE
The skin friction drag coefficient
Cf =
D′
D′
=
q∞ S
q∞ c(1)
The normal and axial coefficients
ˆ c
ˆ c
1
dyu
dyl
cn =
cf,u
+ cf,l
dx
(Cp,l − Cp,u ) dx +
c 0
dx
dx
0
ˆ c ˆ c
1
dyu
dyl
Cp,u
− Cp,l
dx +
(cf,u + cf,l ) dx
ca =
c 0
dx
dx
0
(32)
(33)
The moment coefficient
ˆ c
ˆ c
1
dyu
dyl
cmLE = 2
cf,u
+ cf,l
xdx
(Cp,u − Cp,l ) xdx −
c
dx
dx
0
0
(34)
ˆ c
ˆ c
dyu
dyl
+
Cp,u
+ cf,u yu dx +
−Cp,l
+ cf,l yl dx
dx
dx
0
0
6
The lift and drag coefficients
cl = cn cos α − ca sin α
(35)
cd = cn sin α + ca cos α
The center of pressure is obtained from
′
′
MLE
MLE
=− ′ ≈− ′
N
L
xcp
2
(36)
Fundamental principles and equations
2.2
Review on vector relations
2.2.1
Introduction on Vector Algebra
Inner product, Dot product
A · B ≡ |A∥B| cos θ
(37)
Vector product, Cross product
A × B ≡ (|A∥B| sin θ)e = G
2.2.2
(38)
Typical Orthogonal Coordinate Systems
Cartesian, Circular cylinder and Spherical.
2.2.3
Scalar and Vector Fields
velocity is a vector quantity
V = Vx i + Vy j + V z k
(39)
where
Vx = Vx (x, y, z, t)
Vy = Vy (x, y, z, t)
Vz = Vz (x, y, z, t)
7
(40)
2.2.4
Scalar and Vector Products
Inner product, Dot product
A = Ax i + Ay j + Az k
B = Bx i + By j + Bz k
(41)
A · B = Ax Bx + Ay By + Az Bz
Vector product, Cross product


i
j k



A×B=
 Ax Ay Az  =i (Ay Bz − Az By ) + j (Az Bx − Ax Bz )
Bx By Bz
+ k (Ax By − Ay Bx )
(42)
2.2.5
Gradient of a Scalar Field
Hamilton operator, Gradient operator
∇≡
∂
∂
∂
i+ j+ k
∂x
∂y
∂z
(43)
Laplacian operator
∆ ≡ ∇2 = ∇ · ∇ =
∂2
∂2
∂2
+
+
∂x2 ∂y 2 ∂z 2
(44)
The directional derivative in the s direction
dp
= ∇p · n
ds
2.2.6
(45)
Divergence of a Vector Field
The time rate of change of the volume is
∇·V =
∂Vx ∂Vy ∂Vz
+
+
∂x
∂y
∂z
8
(46)
2.2.7
Curl of a Vector Field
The rotating angular velocity is one-half of
∇×V =
i
j k
∂ ∂ ∂
∂x ∂y ∂z
V x Vy Vz
∂Vz ∂Vy
=i
−
∂y
∂z
∂Vx ∂Vz
+j
−
∂z
∂x
∂Vy ∂Vx
+k
−
∂x
∂y
(47)
Cylindrical
∇×V =
2.2.11
er reθ ez
∂
∂ ∂
∂r ∂θ ∂z
Vr rVθ Vz
1
r
(48)
Relations Between Line, Surface, and Volume Integrals
˛
Stokes theorem
¨
A · ds =
(∇ × A) · dS
C
S
‹
˚
(49)
divergence theorem
A · dS =
V
S
gradient theorem
‹
(∇ · A)dV
˚
pdS =
V
S
2.4
(50)
∇pdV
(51)
Continuity Equation
By mass conservation
∂
∂t
˚
‹
ρV · dS = 0
ρdV +
V
S
9
(52)
the differential form is
∂ρ
+ ∇ · (ρV) = 0
∂t
2.5
(53)
Momentum Equation
By momentum conservation
‹
‹
˚
˚
∂
(ρV · dS)V = −
pdS +
ρf dV + Fviscous (54)
ρVdV +
∂t
S
S
V
V
differential form in the x component is
∂p
∂(ρu)
+ ∇ · (ρuV) = −
+ ρfx + (Fx )viscous
∂t
∂x
2.6
(55)
An Application of the Momentum Equation
Drag of a two-dimensional body
ˆ b
D′ =
ρ2 u2 (u1 − u2 ) dy
(56)
h
2.7
Energy Equation
The partial differential equation
∂
V2
V2
ρ e+
+∇· ρ e+
V =ρq̇ − ∇ · (pV) + ρ(f · V)
∂t
2
2
′
+ Q̇′viscous + Ẇviscous
(57)
For steady, inviscid and adiabatic flow
V2
∇· ρ e+
V = −∇ · (pV)
2
(58)
where the internal energy
e = cv T
10
(59)
where cv is the specific heat at constant volume
p = ρRT
(60)
which is the perfect gas equation of state.
2.9
Substantial Derivative
The operator of substantial derivative
D
∂
≡
+ (V · ∇)
Dt
∂t
(61)
the substantial derivative in cartesian coordinates
∂
∂
∂
∂
D
≡
+u
+v
+w
Dt
∂t
∂x
∂y
∂z
(62)
2.10 Fundamental Equations in Terms Of the Substantial
Derivative
Continuity equation
Dρ
+ ρ∇ · V = 0
Dt
(63)
Momentum equation in x-component
ρ
Du
∂p
=−
+ ρfx + (Fx )viscous
Dt
∂x
Energy equation
D e + V 2 /2
′
ρ
= ρq̇ − ∇ · (pV) + ρ(f · V) + Q̇′viscous + Ẇviscous
Dt
2.11 Pathlines, Streamlines and Streaklines Of a Flow
11
(64)
(65)
2.12 Angular Velocity, Vorticity, and Strain
The vorticity
ξ = ∇ × V ≡ 2ω
The curl of the velocity is equal to the vorticity
∂w ∂v
∂u ∂w
∂v ∂u
ξ=
−
−
−
i+
j+
k
∂y
∂z
∂z
∂x
∂x ∂y
(66)
(67)
2.13 Circulation
The circulation is
˛
¨
Γ≡−
V · ds = −
C
(∇ × V) · dS
(68)
S
2.14 Stream Function
From mass conservation
∂ ψ̄
∂y
∂ ψ̄
ρv = −
∂x
(69)
1 ∂ ψ̄
r ∂θ
∂ ψ̄
ρVθ = −
∂r
(70)
∆ψ̄
ρ
(71)
ρu =
In terms of polar coordinates
ρVr =
For incompressible flow
∆ψ =
12
2.15 Velocity Potential
In irrotational flow
V = ∇ϕ
(72)
the definition of the gradient in cartesian coordinates
u=
∂ϕ
∂x
v=
∂ϕ
∂y
Vθ =
1 ∂ϕ
r ∂θ
w=
∂ϕ
∂z
(73)
in cylindrical coordinates
Vr =
∂ϕ
∂r
∂ϕ
∂z
(74)
1 ∂ϕ
r sin θ ∂Φ
(75)
Vz =
in spherical coordinates
Vr =
3
∂ϕ
∂r
Vθ =
1 ∂ϕ
r ∂θ
VΦ =
Fundamentals of Inviscid, Incompressible Flow
3.2
Bernoullis Equation
In an inviscid, incompressible flow
1
p + ρV 2 = const
2
(76)
applied along a streamline or throughout the irrotational flow.
3.3
Incompressible Flow In A Duct: The Venturi and Lowspeed Wind Tunnel
The quasi-one-dimensional continuity equation
A1 V1 = A2 V2
13
(77)
The venturi tunnel to measure airspeeds
v
u
2w∆h
u
i
V2 = t h
2
ρ 1 − (A2 /A1 )
3.4
Pitot Tube: Measurement Of Airspeed
Pitot tube measures the total pressure and measure static pressure
s
2 (p0 − p1 )
V1 =
ρ
3.5
(78)
(79)
Pressure Coefficient
The dimensionless pressure
p − p∞
Cp ≡
=1−
q∞
3.7
V
V∞
2
,
1
q∞ = ρ∞ V∞2
2
(80)
Governing Equation For Irrotational, Incompressible Flow:
Laplaces Equation
Incompressible, irrotational flow obeys Laplaces equation
∇2 ϕ = 0
(81)
In Cartesian coordinates
ϕ = ϕ(x, y, z)
∂ 2ϕ ∂ 2ϕ ∂ 2ϕ
∇ ϕ=
+
+
=0
∂x2 ∂y 2 ∂z 2
2
Cylindrical coordinates
1 ∂
∇2 ϕ =
r ∂r
∂ϕ
1 ∂ 2ϕ ∂ 2ϕ
r
+ 2 2 + 2 =0
∂r
r ∂θ
∂z
14
(82)
(83)
For a two-dimensional incompressible flow, a stream function obeys
∂ 2ψ ∂ 2ψ
+ 2 =0
∂x2
∂y
3.7.2
(84)
Wall Boundary Conditions
For inviscid flow, the wall boundary conditions
∂ϕ
= 0,
∂n
3.9
∂ψ
=0
∂s
(85)
Uniform Flow: Our First Elementary Flow
The velocity potential
ϕ = V∞ x + const
(86)
In polar coordinates
ϕ = V∞ r cos θ,
ψ = V∞ r sin θ
(87)
3.10 Source Flow: Our Second Elementary Flow
Defines Λ as the source strength
Vr =
Λ
,
2πr
ϕ=
Λ
ln r,
2π
ψ=
Λ
θ
2π
(88)
3.11 Combination of A Uniform Flow With a Source and Sink
Uniform + Source
ψ = V∞ r sin θ +
15
Λ
θ
2π
(89)
Uniform + Source + Sink
ψ = V∞ r sin θ +
Stagnation point
Λ
(θ1 − θ2 )
2π
r
b2 +
OA = OB =
Λb
πV∞
(90)
(91)
3.12 Doublet Flow: Our Third Elementary Flow
The velocity potential and stream function are
κ cos θ
2π r
κ sin θ
ψ=−
2π r
ϕ=
(92)
where the strength of the doublet is
κ = lΛ
(93)
3.13 Nonlifting Flow Over a Circular Cylinder
Uniform + Doublet
R2
ψ = (V∞ r sin θ) 1 − 2
r
r
where
R=
(94)
κ
2πV∞
(95)
Pressure coefficient
Cp = 1 − 4 sin2 θ
(96)
3.14 Vortex Flow: Our Fourth Elementary Flow
The vortex is
Vθ = −
Γ
,
2πr
ϕ=−
16
Γ
θ,
2π
ψ=
Γ
ln r
2π
(97)
3.15 Lifting Flow Over A Cylinder
Uniform + Doublet + Vortex
R2
ψ = (V∞ r sin θ) 1 − 2
r
+
Γ
r
ln
2π R
(98)
The velocity on the surface of the cylinder is
Vθ = −2V∞ sin θ −
Γ
2πR
(99)
3.16 The Kutta-joukowski Theorem And The Generation Of
Lift
The lift per unit span
L′ = ρ∞ V∞ Γ
(100)
3.17 Nonlifting Flows Over Arbitrary Bodies: The Numerical Source Panel Method
Source panel method
ˆ
n
λi X λj
∂
+
(ln rij ) dsj + V∞ cos βi = 0
2
2π
∂n
i
j
j=1
(101)
(j̸=i)
3.18 Applied Aerodynamics: The Flow Over A Circular
Cylinder
From the figure
CD = f (Re) = f
17
ρ∞ V∞ d
µ∞
(102)
The drag is
1
D = CD q∞ S = CD ρ∞ V∞ 2 dh
2
4
(103)
Incompressible flow over airfoil
4.2
Airfoil Nomenclature
Figure 2: Airfoil nomenclature
4.4
Philosophy of Theoretical Solutions for Low-speed Flow
over Airfoils: The Vortex Sheet
The velocity potential at P due to the entire vortex sheet from a to b is
ˆ b
1
ϕ(x, z) = −
θγds
(104)
2π a
4.5
The Kutta Condition
For the finite-angle trailing edge
γ(TE) = 0
18
(105)
4.6
Kelvins Circulation Theorem And The Starting Vortex
The time rate of change of circulation around a closed curve
DΓ
=0
Dt
4.7
(106)
Classical Thin Airfoil Theory: The Symmetric Airfoil
The fundamental equation of thin airfoil theory
ˆ c
dz
1
γ(ξ)dξ
= V∞ α −
2π 0 x − ξ
dx
(107)
From integration
γ(θ) = 2αV∞
(1 + cos θ)
sin θ
(108)
The lift coefficient
cl = 2πα
(109)
The moment coefficient
πα
cl
=−
2
4
(110)
cl = 2π (α − αL=0 )
(111)
cm,le = −
4.8
The Cambered Airfoil
The lift coefficient
where
αL=0
1
=−
π
ˆ
π
0
dz
(cos θ0 − 1) dθ0
dx
(112)
π
(A2 − A1 )
4
(113)
For 1/4 chord point
cm,c/4 =
19
The center of pressure
xcp
where
4.9
c
π
=
1 + (A1 − A2 )
4
cl
2
An =
π
ˆ
π
0
dz
cos nθ0 dθ0
dx
(114)
(115)
The Aerodynamic Center: Additional Considerations
The aerodynamic center is where moment is independent of AOA
x̄ac = −
where
dcl
= a0 ,
dα
m0
+ 0.25
a0
(116)
dcm,c/4
= m0
dα
(117)
4.10 Lifting Flows Over Arbitrary Bodies: The Vortex Panel
Numerical Method
The governing equation
ˆ
n
X
γj
∂θij
V∞ cos βi −
dsj = 0
2π
∂n
i
j
j=1
(i = 1, 2, . . . , n)
(118)
4.12 Viscous Flow: Airfoil Drag
4.12.1
Estimating Skin-Friction Drag: Laminar Flow
The boundary-layer thickness
5.0x
δ=√
,
Rex
Rex =
20
ρe V∞ x
µ∞
(119)
The skin-Friction drag coefficient
1.328
Cf = √
,
Rec
4.12.2
ρ∞ V∞ c
µ∞
Rec =
(120)
Estimating Skin-Friction Drag: Turbulent Flow
The boundary-layer thickness and skin-Friction drag coefficient
δ=
4.12.3
0.37x
,
Re1/5
x
Cf =
0.074
Re1/5
c
(121)
Transition
The critical Reynolds number for transition
Rexcr =
ρ∞ V∞ xcr
= 5 × 105
µ∞
(122)
The total skin-friction drag coefficient
Cf =
5
x1
x1
(Cf,1 )la + (Cf,c )tu − (Cf,1 )tu
c
c
(123)
Incompressible Flow over Finite Wings
5.1
Introduction: Downwash and Induced
5.2
The Vortex Filament, The Biot-Savart Law, and Helmholtz’s
Theorems
5.2.1
The Biot-Savart Law
The induced velocity
dV =
Γ dl × r
4π r3
21
(124)
for the entire vortex filament
Γ
V(x, y, z) =
4π
˛
dl × r
r3
(125)
let the filament to be infinite long
V=
5.2.2
Γ
eθ
2πh
(126)
The Helmholtz’s Vortex Theorems
1. The strength of a vortex filament is constant along its length.
2. A vortex filament cannot end in a fluid.
5.3
Prandtls Classical Lifting-line Theory
The induced velocity in horseshoe vortex model
w(y) = −
b
Γ
4π (b/2)2 − y 2
The integral of tailing vortex induced velocity
ˆ b/2
(dΓ/dy)dy
1
w (y0 ) = −
4π −b/2 y0 − y
(127)
(128)
The Geometric angle of attack
1
Γ (y0 )
+ αL=0 (y0 ) +
α (y0 ) =
πV∞ c (y0 )
4πV∞
ˆ
b/2
(dΓ/dy)dy
y0 − y
−b/2
(129)
where the effective angle of attack
αeff =
Γ (y0 )
+ αL=0
πV∞ c (y0 )
(130)
and the downwash angle
1
αi (y0 ) =
4πV∞
ˆ
b/2
(dΓ/dy)dy
y0 − y
−b/2
22
(131)
The lift coefficient
CL =
ˆ
2
V∞ S
b/2
Γ(y)dy
(132)
Γ(y)αi (y)dy
(133)
−b/2
The induced drag coefficient
CD,i =
5.3.1
2
ˆ
V∞ S
b/2
−b/2
Elliptical Lift Distribution
Circulation distribution
s
1−
Γ(y) = Γ0
2y
b
2
(134)
Induced velocity distribution
1
w (y0 ) = −
4π
ˆ
b/2
(dΓ/dy)dy
−b/2 (y0 − y)
(135)
Introduce integral transformation
y=
b
cos θ
2
(136)
w
V∞
(137)
Induced angle of attack
αi = −
Aspect Ratio
b2
AR = ,
S
αi =
CL
πAR
(138)
Induced drag coefficient
CD,i =
23
CL2
πAR
(139)
5.3.2
General Lift Distribution
General circulation distribution
Γ(θ) = 2bV∞
N
X
An sin nθ
(140)
1
Induced drag coefficient
CD,i =
CL2
(1 + δ)
πAR
(141)
The span efficiency factor
e=
5.3.3
1
1+δ
(142)
Effect of Aspect Ratio
Lift curve
where
dCL
a0
=a=
a0
dα
1 + πAR
(143)
dCL
= a0
d (α − αi )
(144)
General wing
dCL
=a=
dα
1+
a0
a0
πAR (1
+ τ)
(145)
5.4
A Numerical Nonlinear Lifting-line Method
5.5
The Lifting-surface Theory and the Vortex Lattice Numerical Method
6
Three-Dimensional Incompressible Flow
24
6.2
Three-dimensional Source
If λ is the strength of the source
ϕ=−
6.3
λ
4πr
(146)
Three-dimensional Doublet
Assume µ = λl is the strength of the doublet
µ cos θ
4π r2
(147)
µ = 2πR3 V∞
(148)
ϕ=−
6.4
Flow over a Sphere
Uniform + doublet
Pressure distribution
Cp = 1 −
6.7
9 2
sin θ
4
(149)
Airplane Lift and Drag
For the whole aircraft
CL2
CD = CD,e +
πeAR
(150)
where the parasite drag coefficient
CD,e = CD,o + rCL2
(151)
CL2
πeAR
(152)
Oswald efficiency factor
CD = CD,o +
25
Aircraft Lift to Drag ratio
CL
(πeARCD,o )1/2
=
CD max
2CD,o
15
(153)
Fundamental Principles and Equations of Viscous Flow
15.2 Qualitative Aspects of Viscous Flow
15.4 The Navier-Stokes Equations
The time rate of strain in the xy plane
εxy =
The tangential stresses are
τxy = τyx
∂v ∂u
+
∂x ∂y
(154)
∂v ∂u
=µ
+
∂x ∂y
(155)
The normal stress is
τxx = λ(∇ · V) + 2µ
∂u
∂x
(156)
The x component of the equations
∂u
∂u
∂u
∂u
∂p
∂
∂u
ρ
+ ρu
+ ρv
+ ρw
=−
+
λ∇ · V + 2µ
∂t
∂x
∂y
∂z
∂x ∂x
∂x
∂
∂
∂v ∂u
∂u ∂w
µ
+
+
µ
+
+ ρf x
+
∂y
∂x ∂y
∂z
∂z
∂x
(157)
The y component of the equations
∂v
∂v
∂v
∂v
∂p
∂
∂v ∂u
ρ + ρu
+ ρv
+ ρw
=− +
µ
+
∂t
∂x
∂y
∂z
∂y ∂x
∂x ∂y
∂
∂v
∂
∂w ∂v
+
+
λ∇ · V + 2µ
+
µ
+ ρf y
∂y
∂y
∂z
∂y
∂z
26
(158)
The z component of the equations
∂w
∂w
∂w
∂w
∂p
∂
∂u ∂w
ρ
+ ρu
+ ρv
+ ρw
=− +
µ
+
∂t
∂x
∂y
∂z
∂z ∂x
∂z
∂x
∂w ∂v
∂
∂w
∂
+
µ
+
+
λ∇ · V + 2µ
+ ρf z
∂y
∂y
∂z
∂z
∂z
(159)
15.5 The Viscous Flow Energy Equation
The final form of the energy equation for a viscous flow is
D e + V 2 /2
∂
∂T
∂
∂T
ρ
= ρq̇ +
k
+
k
Dt
∂x
∂x
∂y
∂y
∂
∂T
∂ (uτxx ) ∂ (uτyx )
+
k
− ∇ · pV +
+
∂z
∂z
∂x
∂y
∂ (uτzx ) ∂ (vτxy ) ∂ (vτyy ) ∂ (vτzy )
+
+
+
+
∂z
∂x
∂y
∂z
∂ (wτxz ) ∂ (wτyz ) ∂ (wτzz )
+
+
+
∂x
∂y
∂z
The rate of work done by the surface force in y direction
∂(vp) ∂ (vτxy ) ∂ (vτyy ) ∂ (vτzy )
Ẇy = −
+
+
+
dxdydz
∂y
∂x
∂y
∂z
the rate of work in z direction
∂(wp) ∂ (wτxz ) ∂ (wτyz ) ∂ (wτzz )
Ẇz = −
+
+
+
dxdydz
∂z
∂x
∂y
∂z
(160)
(161)
(162)
15.6 Similarity Parameters
Reynolds number
ρ∞ V∞ D
µ∞
(163)
R
1
2 πD2
2 ρ∞ V∞ 4
(164)
Re =
Force coefficient
CR =
27
Pressure coefficient
Cp =
p
1
2
2 ρ∞ V∞
(165)
V∞
a∞
(166)
γp
ρ
(167)
Mach number
M∞ =
where the speed of sound
r
a=
where the ratio of specific heats
γ=
cp
cv
(168)
where cv and cp are the specific heats at constant volume and constant
pressure.
Prandtl number
Pr =
16
µ∞ cp
frictional dissipation
∝
k
thermal conduction
(169)
Exact Solutions of Viscous Flow
16.2 Couette Flow
Type 1
u = ue
Type 2
y
D
(170)
y
y
u(y)
=P
1−
U
h
h
(171)
h2 dp
P =−
2µU dx
(172)
where the pressure
28
16.3 Poiseuille Flow
Velocity Field
u(y, z) =
1 dp 2
y + z 2 − a2
4µ dx
(173)
Sutherland’s law for the temperature variation of viscosity coefficient
µ
=
µ0
T
T0
3/2
T0 + 110
T + 110
(174)
where the standard sea level values
µ0 = 1.7894 × 10−5 kg/(m · s) T0 = 288.16 K
17
(175)
Introduction to Boundary Layers
17.2 Boundary Layers Properties
Boundary layer thickness
y = δ,
u
= 0.99
ue
Displacement thickness
ˆ y1 ρu
δ∗ ≡
1−
dy ,
ρ
u
e
e
0
Momentum thickness
ˆ
θ=
0
y1
ρu
ρe ue
δ ≤ y1 → ∞
u
1−
dy
ue
(176)
(177)
(178)
It can be stated that
δ > δ∗ > θ
29
(179)
17.3 Boundary Layers Equation
Boundary Layer Equation




∂u ∂v
+
=0
∂x ∂y
∂u
∂u
dU
∂ 2u


+
v
=
U
+
ν
u

∂x
∂y
dx
∂y 2
(180)
Boundary Condition
u(x, 0) = 0 ,
18
v(x, 0) = 0 ,
u(x, ∞) = U
(181)
Laminar Boundary Layers
Transform the independent variables
r
η=y
The stream function
U∞
νx
p
ψ = νxU∞ f (η)
(182)
(183)
Momentum integral equations
dθ
1 dU
τw
+ (2θ + δ ∗ )
=
dx
U dx
ρU 2
(184)
18.2 Blasius Solution
Local Reynolds number
Ux
ν
(185)
δ
5.0
=√
x
Rex
(186)
Rex =
Boundary Layer Thickness
30
Friction coefficient
1.328
Cf = √
Rex
(187)
The integrated skin-friction coefficient
2θx=c
c
(188)
1.721
δ∗
=√
x
Rex
(189)
θ
0.664
=√
x
Rex
(190)
Cf =
Displacement thickness
Momentum thickness
19
Turbulent Boundary Layers
19.2 Reynolds Averaged N-S (RANS) equation
For laminar boundary layer
δ=
5.0x
1/2
Rex
,
Cf =
,
Cf =
1.328
1/2
Rec
(191)
For turbulent boundary layer
δ=
0.37x
Re1/5
x
31
0.074
Re1/5
c
(192)