Chapter 9: Differential Analysis Ship Hydrodynamics...

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

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


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=

∑

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∇ ⋅ =


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= −∑ ∑∫


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∂ ∂ ∂ ∂ ∂+ + + = → + ∇ ⋅ =

∂ ∂ ∂ ∂ ∂


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= = =


t r r r zθ∂ ∂ ∂ ∂

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⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞+ − = − =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠


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


of the flow

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

=

== ⋅ = = = −∫ ∫ ∫


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


if only if the integrand is identically zero.

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∝


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= ∇ ⋅ +


⎣ ⎦

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)



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⎡ ⎤ ⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂ ∂

+ + + = + + +⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦



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∂+ ∇ ⋅ = ∇ ⋅ +

∂


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!


q q p

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



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


to Hooke s Law for elastic solids2 2z x z y z

⎜ ⎟+ +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠Chapter 4


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

⎛ ⎞∂ ∂= − + +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠


i jx x∂ ∂⎝ ⎠


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)


( p)


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⎛ ⎞∂

+⎜ ⎟∂⎝ ⎠



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

⎛ ⎞ ⎛ ⎞∂ ∂ ∂∂+ = − + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠


⎝ ⎠ ⎝ ⎠

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


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


9.6.1 Computational Fluid Dynamics (CFD)


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