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Chapter 9. Fluid DynamicsS ll S l A l i Diff i l A l iSmall Scale Analysis: Differential Analysis
Prof. Byoung-Kwon Ahn
bkahn@cnu ac kr http//fincl cnu ac [email protected] http//fincl.cnu.ac.kr
Dept. of Naval Architecture & Ocean EngineeringCollege of Engineering, Chungnam National University
9.1 Introduction
1) Control Volume Analysis:① Chap 5: Control volume (CV)① Chap 5: Control volume (CV)
analysis based on the laws of conservation of mass and energyenergy
② Chap 6: Control volume (CV) analysis based on the law of conservation of momentumconservation of momentum
2) Integral forms of equations are useful for determining overall effectseffects
3) However, we cannot obtain detailed information on the flow field inside the CVfield inside the CV
4) Motivation for Differential Analysis (Chap 9)
Chapter 9: Differential AnalysisShip Hydrodynamics 2
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9.1 Introduction
1) Navier-Stokes equations for incompressible fluid flowi f i i i① Conservation of Mass: Continuity Equation
( ) 0 0V Vt
ρ ρ∂+ ∇ ⋅ = → ∇ ⋅ =
∂② Conservation of Momentum: Momentum Equation
( )t∂
( ) 2VV V p V g
tρ ρ μ ρ∂
+ ⋅∇ = −∇ + ∇ +∂
2) We will learn:① Physical meaning of each term② How to derive ③ How to solve
Chapter 9: Differential AnalysisShip Hydrodynamics 3
9.2 Conservation of Mass - Continuity Equation
1) Conservation of Mass (Chap 5) & Reynolds Transport Theorem (RTT)
Dmd V
Dt t
ρ∂=
∂0n
CV CS
V dAρ+ =∫ ∫
d Vt
ρ∂∂ in outCV
m m= −∑ ∑∫2) Two methods to derive differential form of conservation of mass
①Divergence (Gauss’s) Theorem: Volume Integral ↔ Surface Integral
in outCV
( )f d V∇ ⋅ ( )V A
f n dA= ⋅∫ ∫②Differential CV & Taylor series expansions
V A
( )( ) ( ) ( )
!
nnb a
f b f a f a∞ −
= + ∑Chapter 9: Differential AnalysisShip Hydrodynamics 4
1
( ) ( ) ( )!n
f f fn=
∑
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9.2.1 Continuity Equation: Divergence Theorem
1) Conservation of Mass using Divergence Theorem:
∂ ∂⎡ ⎤0
Dmd V
Dt t
ρ∂= =
∂( )n
CV CS
V dA V d Vt
ρρ ρ∂⎡ ⎤+ = + ∇ ⋅⎢ ⎥∂⎣ ⎦∫ ∫CV∫
2) Integral holds for any CV: General form of the Continuity Equation
( ) 0Vρ ρ∂
+ ∇ =
3) This differential equation is available for all type of flows (Steady, Unsteady,
( ) 0Vt
ρ+ ∇ ⋅ =∂
Viscous, Inviscid, Compressible, Incompressible)① for Steady / Compressible flow: 0Vρ∇ ⋅ =
② for (Steady) / Incompressible flow: density is not f(time, space)
0V∇ ⋅ =
Chapter 9: Differential AnalysisShip Hydrodynamics 5
0V∇
9.2.2 Continuity Equation: Taylor series
1) Consider an infinitesimal control volume dxdydz
2) Consider an infinitesimal control volume dx x dy x dz
3) Approximate the mass flow
2 2
2
( ) 1 ( )( ) ( )
2 2 2!right
u dx dx uu u
x x
ρ ρρ ρ ∂ ∂⎛ ⎞= + + + ⋅⋅ ⋅⎜ ⎟∂ ∂⎝ ⎠3) Approximate the mass flow
rate into or out of each of the 6 faces using Taylor series expansions around pthe center point:
4) Ignore HOTs than O(dx)
5) Conservation of Mass:
0 d Vt
ρ∂=
∂ n
CV CS
V dAρ+∫ ∫
5) Conservation of Mass:
CV CS
d Vt
ρ∂∂ in outCV
m m= −∑ ∑∫
Chapter 9: Differential AnalysisShip Hydrodynamics 6
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9.2.3 Continuity Equation: Taylor series
1) Rate of change of mass within CV:
2) N fl i & f CV
d Vt
ρ∂∂CV
dxdydzt
ρ∂=
∂∫2) Net mass flow rate into & out of CV:
2 2 2
u dx v dy w dzm u dydz v dxdz w dxdy
x y z
ρ ρ ρρ ρ ρ⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞= − + − + −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠
∑ 2 2 2
2 2 2
in x y z
u dx v dy w dzm u dydz v dxdz w dxdy
x y z
ρ ρ ρρ ρ ρ
∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞= + + + + +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠
∑
∑
3) Continuity Equation
2 2 2out x y z∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠
u v wdxdydz dxdydz
t x y z
ρ ρ ρ ρ⎛ ⎞∂ ∂ ∂ ∂= − − −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠t x y z∂ ∂ ∂ ∂⎝ ⎠
0 ( ) 0u v w
Vt x y z t
ρ ρ ρ ρ ρ ρ∂ ∂ ∂ ∂ ∂+ + + = → + ∇ ⋅ =
∂ ∂ ∂ ∂ ∂
Chapter 9: Differential AnalysisShip Hydrodynamics 7
t x y z t∂ ∂ ∂ ∂ ∂
9.2.4 Continuity Equation: Alternative Form
1) Use product rule on divergence term:Material DerivativeMaterial Derivative
( ) 0V V Vt t
ρ ρρ ρ ρ∂ ∂+ ∇ ⋅ = + ⋅∇ + ∇ ⋅ =
∂ ∂
2) Alternative Form of the Continuity Equation:1
0D
VDt
ρρ
+ ∇ ⋅ =
① for Steady / Compressible flow:
Dtρ0Vρ∇ ⋅ =
② for (Steady) / Incompressible flow:
3) Continuity Equation in cylindrical coordinates:
0V∇ ⋅ =
-1i tyθ θ θ
( )1 ( ) 1 ( )0r zur u u
tθρρ ρ ρ
θ∂∂ ∂ ∂
+ + + =∂ ∂ ∂ ∂
1cos y sin tan y
x r rx
θ θ θ= = =
Chapter 9: Differential AnalysisShip Hydrodynamics 8
t r r r zθ∂ ∂ ∂ ∂
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9.3 The Stream Function
1) In general, continuity equation cannot be used by itself to solve for flow field however it can be used to find the missing velocity component if thefield, however it can be used to find the missing velocity component if the flow field is incompressible.
2) Consider the continuity equation for an incompressible 2D flow:
0u v
Vx y
∂ ∂∇ ⋅ = + =
∂ ∂
3) Definition of the Stream Function: one dependent variable (ψ) instead of two dependent variables (u, v)
u vψ ψ∂ ∂
= = −
4) Substitution 3) into 2) yields: which is identically satisfied for any smooth f ti ( )
u vy x∂ ∂
function ψ(x, y).2 2
0x y y x x y y x
ψ ψ ψ ψ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞+ − = − =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠
Chapter 9: Differential AnalysisShip Hydrodynamics 9
9.3 The Stream Function
1) Consider streamlines:
0dy v
vdx udydx u
= → − + =
0dx dyx y
ψ ψ∂ ∂+ =
∂ ∂
2) Total change of ψ:
0d dx dyψ ψψ ∂ ∂
+
① ψ is constant along streamlines
0d dx dyx y
ψ = + =∂ ∂
② Change in ψ along streamlines is zero
③ Curves of constant ψ are streamlines of the flow
Chapter 9: Differential AnalysisShip Hydrodynamics 10
of the flow
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9.3 The Stream Function: Physical Significance
1) Difference in ψ from one streamline to another is equalstreamline to another is equal to the volume flow rate per unit width between the two streamlinesstreamlines
dy dxn i j
d d= −
d V ( )dy dx
V ndA ui vj i j dsd d
⎛ ⎞= ⋅ = + ⋅ −⎜ ⎟⎝ ⎠
jds ds
d V udy vdx dy dx dy x
ψ ψ ψ∂ ∂= − = + =
∂ ∂
ds ds⎜ ⎟⎝ ⎠
y x∂ ∂2
12 1B
B B
V V ndA dV dψ ψ
ψ ψψ ψ ψ
=
== ⋅ = = = −∫ ∫ ∫
Chapter 9: Differential AnalysisShip Hydrodynamics 11
1B B
9.4 Conservation of Momentum – Momentum Eqn.
1) Conservation of Linear Momentum: derivation using the divergence theorem
Body Force Surface Force
F gd Vρ=∑ ( )ij ndA V d Vt
σ ρ∂+ ⋅ =
∂∫ ∫ ( )V V ndAρ+ ⋅∫ ∫CV CS t∂CV CS
Divergence Theorem
ij
CV
dVσ∇ ⋅∫ ( ) CV
VV dVρ∇ ⋅∫
( ) ( ) ijV VV g d Vt
ρ ρ ρ σ∂⎡ ⎤+ ∇ ⋅ − − ∇ ⋅⎢ ⎥∂⎣ ⎦0=∫ t⎢ ⎥∂⎣ ⎦CV
∫This equation holds for any control volume regardless of its size or shape if only if the integrand is identically zero
Chapter 9: Differential AnalysisShip Hydrodynamics 12
if only if the integrand is identically zero.
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9.4.0 Review of the Stress Tensor (Chapter 4)
2) Shear strain rate tensor (Chapter 4)1) Stress tensor
xx xy xzε ε εε ε ε ε
⎛ ⎞⎜ ⎟⎜ ⎟
1 1
ij yx yy yz
zx zy zz
u v u w u
ε ε ε εε ε ε
= ⎜ ⎟⎜ ⎟⎝ ⎠⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞1 1
2 2
1 1
u v u w u
x x y x z
u v v w v
⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞+ +⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂⎜ ⎟⎜ ⎟ ⎜ ⎟
1 1
2 2
1 1
u v v w v
y x y y z
u w v w w
⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂⎜ ⎟= + +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞⎜ ⎟+ +⎜ ⎟ ⎜ ⎟⎜ ⎟2 2z x z y z⎜ ⎟+ +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠
ij ijσ ε∝
Chapter 9: Differential AnalysisShip Hydrodynamics 13
9.4.1 Momentum Equation: Divergence Theorem
1) Cauchy’s Equation:
∂( ) ( ) ijV VV g
tρ ρ σ ρ∂
+ ∇ ⋅ = ∇ ⋅ +∂
2) Alternate form of the Cauchy Equation:
a) ( ) (chain rule)V
V Vt t t
ρρ ρ∂ ∂ ∂= +
∂ ∂ ∂b) ( ) ( ) ( )
t t t
VV V V V Vρ ρ ρ∂ ∂ ∂∇ ⋅ = ∇ ⋅ + ⋅ ∇
0 Conservation of Mass
( ) ( ) ij
VV V V V g
t t
ρρ ρ ρ σ ρ∂ ∂⎡ ⎤+ + ∇ ⋅ + ⋅∇ = ∇ ⋅ +⎢ ⎥∂ ∂⎣ ⎦
0 Conservation of Mass
( ) ij
VV V g
tρ σ ρ
⎡ ⎤∂+ ⋅∇ = ∇ ⋅ +⎢ ⎥∂⎣ ⎦
ij
DVg
Dtρ σ ρ= ∇ ⋅ +
Chapter 9: Differential AnalysisShip Hydrodynamics 14
⎣ ⎦
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9.4.2 Momentum Equation: Taylor series
Inflow and outflow of the xInflow and outflow of the x-component of linear momentum through each face of an infinitesimal control volume (differential CV)
Surface forces acting in the x-direction due to the appropriate t t t hstress tensor component on each
face of the infinitesimal control volume (differential CV)
Chapter 9: Differential AnalysisShip Hydrodynamics 15
9.4.2 Momentum Equation: Taylor series
1) Consider x-momentum:∂
( )x surface x bodyout inCV
F F F u dV mu muρ β β− −
∂= + = + −
∂∑ ∑ ∑ ∑ ∑∫(a) (b)(c) (d)
( ) ( )a u d Vt
ρ∂∂
( )CV
u dxdydzt
ρ∂=
∂
⎡ ⎤∂ ∂ ∂
∫
( ) ( )( ) ( )
((b)
(c)
) ( ) ( ) out in
mu mu uu vu wu dxdydzx y z
F dxdydz
β β ρ ρ ρ
σ σ σ
⎡ ⎤∂ ∂ ∂− = + +⎢ ⎥∂ ∂ ∂⎣ ⎦
⎡ ⎤∂ ∂ ∂+ +⎢ ⎥
∑ ∑
∑ -
-
(c)
d
) (
xx yx zxx surface
xx body
F dxdydzx y z
F g dxdydz
σ σ σ
ρ
= + +⎢ ⎥∂ ∂ ∂⎣ ⎦=
∑
∑
( ) ( ) ( ) ( )xx yx zx x
u uu vu wug
t x y z x y z
ρ ρ ρ ρ σ σ σ ρ⎡ ⎤ ⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂ ∂
+ + + = + + +⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦
Chapter 9: Differential AnalysisShip Hydrodynamics 16
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9.4.2 Momentum Equation: Taylor series
1) Consider x-y-z momentum:
( ) ( ) ( ) ( ): x y zx x x x
u u u uu v wx g
t x y z x y z
ρ ρ ρ ρ σ σ σ ρ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = + + +
∂ ∂ ∂ ∂ ∂ ∂ ∂( ) ( ) ( ) ( )
: x y zy y y y
v v v v
y y
u v wy g
t x y z x y z
ρ ρ ρ ρ σ σ σ ρ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = + + +
∂ ∂ ∂ ∂ ∂ ∂ ∂( ) ( ) ( ) ( )
: x z zzzyz
w w w
y y
u v wz g
t x y z x y z
wρ ρ ρ ρ σ σ σ ρ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = + + +
∂ ∂ ∂ ∂ ∂ ∂ ∂
2) Cauchy’s Equation:
( ) ( ) ijV VV gt
ρ ρ σ ρ∂+ ∇ ⋅ = ∇ ⋅ +
∂
Chapter 9: Differential AnalysisShip Hydrodynamics 17
9.4.2 Momentum Equation: Newton's 2nd Law
1) Consider x-y-z momentum: 2) Cauchy’s Equation
Du ∂ ∂ ∂: xx yx zx x
Dux g
Dt x y z
Dv
ρ σ σ σ ρ∂ ∂ ∂= + + +
∂ ∂ ∂∂ ∂ ∂ ij
DVg
Dtρ σ ρ= ∇ ⋅ +
:
:
xy yy zy yy gDt x y z
Dwz g
ρ σ σ σ ρ
ρ σ σ σ ρ
∂ ∂ ∂= + + +
∂ ∂ ∂∂ ∂ ∂
= + + +
ijDt
3) Closure Problem: unknowns are more than equations①
: xz yz zz zz gDt x y z
ρ σ σ σ ρ= + + +∂ ∂ ∂
① 10 unknownsa. Stress tensor σij : 6 independent componentsb. Density: ρc. Velocity: (u,v,w) 3 independent components
② 4 equations: (1 Continuity Equation + 3 Momentum Equations)③ 6 more equations required to close problem!
Chapter 9: Differential AnalysisShip Hydrodynamics 18
q q p
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9.5.1 Navier-Stokes Equation
1) Separate σij into pressure and viscous stresses
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
σ ij =σ xx σ xy σ xz
σ σ σ
⎛ ⎜ ⎜
⎞ ⎟ ⎟ =
−p 0 0
0 −p 0
⎛ ⎜ ⎜
⎞ ⎟ ⎟ +
τ xx τ xy τ xz
τ τ τ
⎛ ⎜ ⎜
⎞⎟⎟σ ij σ yx σ yy σ yz
σ zx σ zy σ zz⎝ ⎜ ⎜
⎠ ⎟ ⎟
0 p 0
0 0 −p⎝ ⎜ ⎜
⎠ ⎟ ⎟ + τ yx τ yy τ yz
τ zx τ zy τ zz⎝ ⎜ ⎜
⎠⎟⎟
M h i l
2) Mechanical Pressure:
① Incompressible flows: Mean Pressure
Viscous (Deviatoric) Stress Tensor
Mechanical pressure1
( )3m xx yy zzP σ σ σ= − + +
① Incompressible flows: Mean Pressure
② Compressible flows: Thermodynamic Pressure
3) Equations are not improved!
6 unknowns in σij 6 unknowns in τij + P = 7 unknowns
Chapter 9: Differential AnalysisShip Hydrodynamics 19
9.5.2 Navier-Stokes Equation
1) Rheology: the study of the deformation of flowing fluids.
Rheological behavior of fluids – shear stress as a function of shear strain rate.
g2) Newtonian fluids: fluids for which
the shear stress is linearly proportional to the shear strain rate ( i th t t )(air, other gases, water, etc).
3) Reduction in the number of variables is achieved by relating shear stress to strain rate tensorshear stress to strain-rate tensor.
4) Viscous stress tensor for incompressible Newtonian fluid with constant properties
xx xy xz
ij yx yy yz
ε ε εε ε ε ε
⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟with constant properties
2i j i jτ μ ε=1 1
2 2
zx zy zz
u v u w u
x x y x z
ε ε ε⎜ ⎟⎝ ⎠⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞+ +⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠⎜ ⎟⎜ ⎟
5) Newtonian closure is analogousto Hooke’s Law for elastic solids
i j i j1 1
2 2
1 1
u v v w v
y x y y z
u w v w w
⎜ ⎟⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂⎜ ⎟= + +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞⎜ ⎟+ +⎜ ⎟ ⎜ ⎟⎜ ⎟
Chapter 4
Chapter 9: Differential AnalysisShip Hydrodynamics 20
to Hooke s Law for elastic solids2 2z x z y z
⎜ ⎟+ +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠Chapter 4
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9.5.3 Navier-Stokes Equation
1) Stress tensor with Newtonian closure:
⎧2i j i j i jpσ δ μ ε= − + 1 :
0 : ij
i j
i jδ
=⎧⎨ ≠⎩
Kronecker Delta
2) Using the definition of εij2
0 0
u v u w u
x x y x zp
μ μ μ⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞+ +⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠⎜ ⎟⎛ ⎞ ⎜ ⎟0 0
0 0 2
0 0ij
pu v v w v
py x y y z
p
σ μ μ μ−⎛ ⎞ ⎜ ⎟⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂⎜ ⎟ ⎜ ⎟= − + + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ∂ ∂ ∂ ∂ ∂⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟−⎝ ⎠ ⎜ ⎟⎛ ⎞
2
pu w v w w
z x z y zμ μ μ
⎝ ⎠ ⎜ ⎟⎛ ⎞∂ ∂ ∂ ∂ ∂⎛ ⎞⎜ ⎟+ +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠
j ii j i j
V Vp
x xσ δ μ
⎛ ⎞∂ ∂= − + +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠
Chapter 9: Differential AnalysisShip Hydrodynamics 21
i jx x∂ ∂⎝ ⎠
9.5.4 Navier-Stokes Equation
1) Navier-Stokes equations for incompressible & Isothermal flow:
① Continuity Equation:
div( ) 0V V∇ ⋅ = =
② Momentum Equation:
j iV VDV ⎡ ⎤⎛ ⎞∂ ∂
⎢ ⎥⎜ ⎟
( )
j iij ij
i j
VDVg p g
Dt x x
V
ρ σ ρ δ μ ρ∂= ∇ ⋅ + = ∇ ⋅ − + + +⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦
⎛ ⎞∂
2) Closed system of equations:
2( )V
V V p V gt
ρ μ ρ⎛ ⎞∂
+ ⋅∇ = −∇ + ∇ +⎜ ⎟∂⎝ ⎠
2) Closed system of equations:
① 4 equations (continuity and x-y-z momentum equations)
② 4 unknowns (u, v, w, p)
Chapter 9: Differential AnalysisShip Hydrodynamics 22
( p)
Page 12
9.5.5 Navier-Stokes Equation
1) Continuity Equaton:
0u v w
x y z
∂ ∂ ∂+ + =
∂ ∂ ∂
Calculate velocity (U, V, W) and pressure (P) for known geometry with Boundary Conditions (BC), and Initial Conditions (IC)
2) Momentum Equations:2 2 2
:p
x u v w gu u u u u u uρ μ ρ
⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = − + + + +⎜ ⎟⎜ ⎟ 2 2 2
2 2 2
: xx u v w gt x y z x x y z
pv v v v v v v
ρ μ ρ+ + + = + + + +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
+ + + + + + +⎜ ⎟⎜ ⎟ 2 2 2
2 2
: y
py
w w w w w w
u v w gt x y z y x y z
p
ρ μ ρ+ + + = − + + + +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ 2w⎛ ⎞∂
2 2:
w w w w w wpz u v w
t x y z z x yρ μ
⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = − + + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠
2 zgz
w ρ⎛ ⎞∂
+⎜ ⎟∂⎝ ⎠
Chapter 9: Differential AnalysisShip Hydrodynamics 23
9.5.6 Navier-Stokes Equation
1) Continuity Equaton:
① ②① Vector notation:
0iV∂=
∂
② Tensor notation:
0V∇ ⋅ =
2) Momentum Equations:
① Vector notation:
ix∂0V∇
① Vector notation:
2( )V
V V p V gt
ρ μ ρ⎛ ⎞∂
+ ⋅ ∇ = −∇ + ∇ +⎜ ⎟∂⎝ ⎠② Tensor notation:
t∂⎝ ⎠
2
i
i i ij x
j i j j
V V VpV g
t x x x xρ μ ρ
⎛ ⎞ ⎛ ⎞∂ ∂ ∂∂+ = − + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠
Chapter 9: Differential AnalysisShip Hydrodynamics 24
⎝ ⎠ ⎝ ⎠
Page 13
9.5.7 Exact Solutions of the N-S Equation
1) There are about 80 known exact solutionsto the N-S Equation
Convective termto the N-S Equation
2) The can be classified as:① Linear solutions where the convective
term is zero
( )V V⋅∇
term is zero② Nonlinear solutions where convective
term is not zero3) Exact solutions can also be classified by type3) Exact solutions can also be classified by type
or geometry① Shear flows ② Steady channel flows Couette Flow② Steady channel flows③ Unsteady channel flows④ Flows with moving boundaries⑤ Similarity solutions⑤ Similarity solutions⑥ Asymptotic suction flows⑦ Wind-driven Ekman flows
P i ill Fl
Chapter 9: Differential AnalysisShip Hydrodynamics 25
Poiseuill Flow
9.6 How to solve N-S equation
Analytical Fluid Dynamics Computational Fluid DynamicsStep
Analytical Fluid Dynamics(Chapter 9)
Computational Fluid Dynamics
(Chapter 15)
1 Setup Problem and geometry, identify all dimensions and parameters
2 List all assumptions, approximations, simplifications, boundary conditions
3 Simplify PDE’s Build grid / discretize PDE’s
4 Integrate equationsSolve algebraic system of equations including initial conditions and Boundary conditions5
Apply initial conditions and boundary conditions to solve for constants of integrationintegration
6 Verify and plot results Verify and plot results
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9.6.1 Computational Fluid Dynamics (CFD)
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