boundary conditions -...

Equation of motion 1

Equation of MotionNavier-Stokes Equation

Boundary conditions

Types of boundaries① Rigid wall② Permeable wall③ Free surface④ Infinite medium

Geometry

Motion of walls

⇒ Rigid impermeable wall



Boundary conditions

Impermeable wall

y x

u U

n̂nn ˆˆ ⋅=⋅ Uu

0ˆ =⋅nuwhen U=0

x,u

y,v

v=0



Boundary conditions

No-slip condition

No relative motion between a wall and fluid immediately next to it.

nUnu ˆˆ ×=×

0or

0ˆ

==

=×

wu

nuwhen U=0



Boundary conditions

No-slip on the impermeable wall

Uu =

Wall at rest 0=u



Boundary conditions – consideration on stress

∂∂

+∂∂

∂∂

+

∂∂

+∂∂

∂∂

+

⋅∇+

∂∂

∂∂

zu

xw

zxv

yu

yu

xu

xµµλµ2

Viscous force

0 ≠0 at y=0 0



Boundary conditions – consideration on stress

0=

∂∂

+∂∂

yxv

yuµ

u=v=w=0 at y=0 for all x and z

=0

Same consideration on y- and z- components

00

;0;==

∂∂

∂∂

yy yw

yu µµ



Boundary conditions – infinite medium

∞→→ ruu as0


Scope of this course

1. Laws of classical mechanics apply.

2. Length scale of the flow >> molecular mean free path

3. Incompressible flow 非圧縮性流れ

4. Newtonian fluids

5. No free surface

6. Exclude electromagnetic effects



Incompressibility – if ρ=constant

Continuity equation

( ) 0=⋅∇+∂∂ ut

ρρ 0=⋅∇ u

Momentum equation

Fuu+∇+−∇= 2µρ p

DtD



Condition for Incompressibility

( ) constant, =≡ Tpρρ Equation of state

Tp δδδρ &for0≈

Liquid density is very little changed.Gas density changes by changing pressure.




10

<<∆ρρ

ρρρ ∆+= 0

p∆←∆ρ

FupDt

uD+∇+−∇= 2µρ




u u u pt

ρ ∂ + ⋅∇ −∇ ∂ ∼

2p uµ∇ ∇∼u u p⋅∇ −∇∼

Steady state without body force

20 p µ−∇ + ∇∼ u




u u p⋅∇ −∇∼21

2p u uux x x

ρ ρ∂ ∂ ∂=

∂ ∂ ∂∼

( )2∆ U∆P ρL L∼

x

p, u

L

∆P, ∆U

Order of magnitude estimate

( )2∆P ∆ Uρ∼




Order of magnitude estimate

( )2∆P ∆ Uρ∼Selecting a reference frame such that a reference velocity is zero.

∆U ~ U ; ∆(U2) ~ U2




( )2∆P ∆ Uρ∼

2∆P Uρ∼

Pρ βρ∆

∆∼

2 1paρ ρβ

∂= ∂

∼

22

2 2

∆ρ ∆P U Maρ ρa a

=∼ ∼

1<<= 2

22

aUMa

Ma : Mach number




0

0.1

0.2

0.3

0.4

0 50 100 150 200 250

AirWater

Velocity　(m/s)

Ma2




2p uµ∇ ∇∼

Ma2 << Re




2p uµ∇ ∇∼

2

2

2

p uµx x∆P ∆UµL L

U∆P µL

∂ ∂∂ ∂∼

∼

∼

1

2

2

2 2 2

2

∆ρ ∆Pβ∆Pρ ρaµU νU νUρa L a L a LUMaRe

= =

= <<

∼ ∼

∼

Ma2 << Re


Equation of Motion

Equation of Motion : Summary

Continuity equation

( ) 0=⋅∇+∂∂ ut

ρρ

Equation of motion

Fuu+∇+−∇= 2µρ p

DtD

∇⋅+∂∂

= utDt

D


Equation of Motion


Continuity equationConstant density flow

0=⋅∇ u( ) 0=⋅∇+∂∂ ut

ρρ

Equation of motion

FupDt

uDρ

νρ

11 2 +∇+∇−= ∇⋅+∂∂

= utDt

D


Equation of Motion


Fupuutu

ρν

ρ11)( 2 +∇+∇−=∇⋅+

∂∂

=0 : inviscid (no viscosity)=0 : very low speed

=0 : steady flow =0 : no body force


Equation of Motion


Fuρ

νρ

11 2 +∇+∇−= pDtDu

Nondimensionalization

Fu +∇+−∇= 2

Re1p

DtDu

νUL

=Re

boundary conditions -...

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