Fluid Mechanics 2 - Formula Sheet

Conservation Equation
∂ρ
Continuity: + ∇ · (ρV) = 0
∂t
Incompressible continuity: ∇ · V = 0

Momentum:
Du ∂p ∂τxx ∂τxy ∂τxz
x: ρ =− + ρBx + + +
Dt ∂x ∂x ∂y ∂z
Dv ∂p ∂τyx ∂τyy ∂τyz
y: ρ =− + ρBy + + +
Dt ∂y ∂x ∂y ∂z
Dw ∂p ∂τzx ∂τzy ∂τzz
z: ρ =− + ρBz + + +
Dt ∂z ∂x ∂y ∂z

DV →
−
Navier-Stokes: ρ = ρ B − ∇p + µ∇2 V
Dt
→
−
where V = (u , v , w ), B is body force.

Cylindrical coordinates:

Incompressible continuity: 1 ∂
r ∂r (rVr ) + 1 ∂
r ∂θ (Vθ ) + ∂
∂z (Vz ) = 0

Navier-Stokes:
∂Vr Vθ ∂Vr Vθ2 1 ∂p 2 ∂Vθ
+ Vr ∂V + Vz ∂V Vr
2


r: ∂t ∂r +
r
r ∂θ − r ∂z = − ρ ∂r + gr + ν ∇ Vr −
r
r2 − r 2 ∂θ

∂Vθ 1 ∂p
+ Vr ∂V Vθ ∂Vθ Vr Vθ
+ Vz ∂V 2 ∂Vr Vθ
2


θ: ∂r + + ∂z = − ρr ∂θ + gθ + ν ∇ Vθ − −
θ θ
∂t r ∂θ r r 2 ∂θ r2

∂Vz Vθ ∂Vz 1 ∂p
z: ∂t + Vr ∂V
∂r +
z
r ∂θ + Vz ∂V z 2
∂z = − ρ ∂z + gz + ν∇ Vz

Mathematical operators:

Gradient: ∇V = ∂V
∂r ir + 1 ∂V
r ∂θ iθ + ∂V
∂z iz
→
−
Divergence: ∇ · V = 1 ∂
r ∂r (rVr ) + 1 ∂Vθ
r ∂θ + ∂Vz
∂z

Laplacian: ∇2 = 1 ∂ ∂ 1 ∂2 ∂2


r ∂r r ∂r + r 2 ∂θ 2 + ∂z 2

1
Spherical coordinates:

Mathematical operators:

Gradient: ∇V = ∂V
∂r ir + 1 ∂V
r ∂θ iθ + 1 ∂V
r sin θ ∂ϕ iϕ
→
−
Divergence: ∇ · V = 1 ∂ 1 ∂Vθ 1 1 ∂Vϕ


r 2 ∂r r 2 Vr + r ∂θ + r tan θ Vθ + r sin θ ∂ϕ

Laplacian: ∇2 V = 2 ∂V
r ∂r + ∂2V
∂r 2 + 1 ∂V
r 2 tan θ ∂θ + 1 ∂2V
r 2 ∂θ 2 + 1 ∂2V
r 2 sin2 θ ∂ϕ

High and Low Reynolds Hydrodynamics

Pipe head loss: hf = f Ld V2g
2

Laminar friction factor: flam = 64

Red
 
Turbulent friction factor for smooth and fully rough wall: √1 = −2 log ε
(3.7d) + 2.51
√
f (Red f )

Logarithmic Velocity Proles for Turbulent Flow

q
Shear velocity: u∗ = τw
ρ
yu∗
 
u (y) 1
Logarithmic velocity prole: ∗
= ln +B ; α = 0.4
u α ν
eu∗
Smooth pipes: B = 5.5 <5
ν
eu∗
 
1 eu∗
Completely rough zone: B = 3.4 − ln ν > 70
α ν
∗
Transition zone: u(y) 1 y
5 < euν < 70


u∗ = α ln e + B

where B for the transition zone is given by an approximate curve, e is the roughness.

Figure 1: Plot of B for transition zone

2
Stress and Strain Relations
Stokes' viscosity law:
 
2
τxx = −p + µ 2˙xx − (˙xx + ˙yy + ˙zz ) ; τxy = 2µ˙xy
3
 
2
τyy = −p + µ 2˙yy − (˙xx + ˙yy + ˙zz ) ; τyz = 2µ˙yz
3
 
2
τzz = −p + µ 2˙zz − (˙xx + ˙yy + ˙zz ) ; τxz = 2µ˙xz
3

The strain rate:

∂Vx ∂Vy ∂Vz

˙xx = ; ˙yy = ; ˙zz =
∂x ∂y ∂z
     
1 ∂Vy ∂Vx 1 ∂Vx ∂Vz 1 ∂Vy ∂Vz
˙xy = + ; ˙xz = + ; ˙zy = +
2 ∂x ∂y 2 ∂z ∂x 2 ∂z ∂y

Newtonian and non-Newtonian

 
uids
Newtonian uid: τij = µ ∂Vi
∂xj +
∂Vj
∂xi

Bingham plastic: τ = τ0 + µ du
dy for τ ≥ τ0
 n
Non-Newtonian uid - power law: τ = K du
dy

Euler's equation
→
− → →
−
DV − →− ∂V 1
Euler's equation: = V ·∇ V + = − ∇p − g∇→ −z
Dt ∂t ρ
 2
V →
− →
− 1
Euler's equation for steady ow: ∇ − V × curl V = − ∇p − g∇→ −
z
2 ρ
V2 p
Euler's equation for steady and irrotational ow - Brenoulli's equation: + − gz = const
2 ρ

Potential velocity, stream function and circulation

∂ψ ∂φ ∂ψ ∂φ ∂φ
u=− =− ; v= =− ; w=−
∂y ∂x ∂x ∂y ∂z
1 ∂ψ ∂φ ∂ψ 1 ∂φ
Vr = − =− ; Vθ = =−
r ∂θ ∂r ∂r r ∂θ
z→ −
Circulation: Γ = V · ds

3
Elementary ow solutions
Uniform ow in positive x direction: φ = −U0 x ; ψ = −U0 y ; u = U0 ; v=0
Λ Λ Λ
Two dimensional source: φ = − ln (r) ; ψ=−
θ ; Vr = ; Vθ = 0
2π 2π 2πr
χ cos θ χ sin θ χ cos θ χ sin θ
Doublet: φ = − ; ψ= ; Vr = − ; Vθ = − 2
r r r2 r
Λ Λ Λ
Simple Vortex (counter clock wise): φ = θ ; ψ = ln (r) ; Vr = 0 ; Vθ =
2π 2π 2πr
Cylinder:
χ cos θ χ sin θ
φ = −U0 r cos θ − ; ψ = −U0 r sin θ + ;
r r r
χ cos θ χ sin θ χ
Vr = U0 cos θ − ; Vθ = −U0 sin θ − ; R=
r2 r2 U0

Cylinder with rotation:

χ cos θ Λ χ sin θ Λ
φ = −U0 r cos θ − + θ ; ψ = −U0 r sin θ + − ln (r) ;
r 2π r 2π r
χ cos θ χ sin θ Λ χ
Vr = U0 cos θ − 2
; Vθ = −U0 sin θ − 2
− ; R=
r r 2πr U0

Boundary layer ow

Laminar at plate ow:
δ 4.96 0.664 1.328
=√ ; cf = √ ; CD = √ Rex < 105
x Rex Rex ReL

Turbulent at plate ow - Smooth plates:

δ 0.37 0.0577 0.074  ν 1/4  y 1/7
= 1/5
; cf = 1/5
; CD = 1/7
; τw = 0.0225ρU 2 ; u = U0 5·105 < Rex < 107
x Rex Rex ReL Uδ δ

Turbulent at plate ow:

δ 0.16 0.027 0.031
= 1/7
; cf = 1/7
; CD = 1/7
107 < Rex
x Rex Rex ReL

von-Karman: !
Z δ
dp d 2


−δ − τw = ρ u − um u dy
dx dx 0

where um is the local main-stream velocity.

4
Various
Lift and drag coecients: CL = 1
L
2 ; CD = 1
D
2
2 ρU∞ A 2 ρU∞ A
 2
Pressure Coecient: Cp = p−p∞
1 2 =1− u
U∞
2 ρU∞
Z η
y 2 02
Stokes' rst problem - sudden acceleration problem : u (y , t ) = U (1 − erf (η)) ; η = √ ; erf (η) = √ e−η dη 0
2 νt π 0

Stokes' second problem - oscillating plate problem: u (y , t ) = U e−ky cos (ωt − ky) ; k = 2νω
p