v = 0 Continuity Continuity Navier- -Stokes Stokes Navier - - PowerPoint PPT Presentation

SLIDE 1

1

• G. Ahmadi

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• G. Ahmadi

ME 437/537

• Navier

Navier-

• Stokes Equation

Stokes Equation

• Steady Parallel Flows

Steady Parallel Flows

• Boundary Conditions

Boundary Conditions

• Unsteady Parallel Flow

Unsteady Parallel Flow

• Boundary Layer Flows

Boundary Layer Flows

• Flow over a Flat Plate

Flow over a Flat Plate

• Blasius

Blasius Solution Solution

• Laminar Boundary Layer

Laminar Boundary Layer

• G. Ahmadi

ME 437/537

Continuity Continuity Navier Navier-

• Stokes

Stokes

= ⋅ ∇ v

v v v v

2

P 1 t ∇ ν + ∇ ρ − = ∇ ⋅ + ∂ ∂

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Navier Navier Stokes Stokes

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) z w y w x w ( z p g ) z w w y w v x w u t w ( ) z v y v x v ( y p g ) z v w y v v x v u t v ( ) z u y u x u ( x p g ) z u w y u v x u u t u (

2 2 2 2 2 2 z 2 2 2 2 2 2 y 2 2 2 2 2 2 x

∂ ∂ + ∂ ∂ + ∂ ∂ µ + ∂ ∂ − ρ = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ ρ ∂ ∂ + ∂ ∂ + ∂ ∂ µ + ∂ ∂ − ρ = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ ρ ∂ ∂ + ∂ ∂ + ∂ ∂ µ + ∂ ∂ − ρ = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ ρ

z w y v x u = ∂ ∂ + ∂ ∂ + ∂ ∂

Incompressible Fluid Incompressible Fluid

4 Equations for 4 4 Equations for 4 unknowns u,v,w and p unknowns u,v,w and p

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y h u =

w , v ), y ( u = =

Cartesian Coordinates Cartesian Coordinates

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dy u d x p g

2 2 x

= µ + ∂ ∂ − ρ B Ay y ) x p g ( 2 1 u

2 x

+ + ∂ ∂ − ρ µ − =

General Solution General Solution

Cartesian Coordinates Cartesian Coordinates

• G. Ahmadi

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A viscous fluid sticks to its boundary, A viscous fluid sticks to its boundary, u ufluid

fluid=

=u uwall

wall

Shear Stress=0 Shear Stress=0

Solid Walls Solid Walls Free Surface Free Surface

uwall

Fluid

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u u1

1= u

= u2

2

Between Two Fluids Between Two Fluids

τ τ1

1=

= τ τ2

2

Velocity and shear stress are Velocity and shear stress are the same at the interface. the same at the interface.

uwall

Fluid 1 Fluid 2

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dy u d x p g

2 2 x

= µ + ∂ ∂ − ρ

B Ay y ) x p g ( 2 1 u

2 x

+ + ∂ ∂ − ρ µ − =

Cartesian Cartesian

) dr dv r ( dr d r 1 z p g

z z

= µ + ∂ ∂ − ρ

B r ln A r ) z p g ( 4 1 v

2 x z

+ + ∂ ∂ − ρ µ − =

Cylindrical Axial Cylindrical Axial

r B Ar v + =

θ

r v dr dv r 1 dr v d

2 2 2

= − +

θ θ θ

Rotating Rotating

r v dr dp

2 θ

ρ =

• G. Ahmadi

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x y

Viscous Fluid

Uo

2 2

y u t u ∂ ∂ ν = ∂ ∂

y =

U u =

∞ = y

u =

t =

u =

B.C. B.C. I.C. I.C.

• G. Ahmadi

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Similarity Solution Similarity Solution Let Let

1

t ~ t

a

t ~ y a 2 1 = 2 / 1 a = t 2 y ν = η

( )

η = f U u

Navier Navier-

• Stokes

Stokes Similarity Similarity Variables Variables

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B.C. B.C.

f 2 f = ′ η + ′ ′ 1 ) ( f = ) ( f = ∞

2

ce f

η −

= ′

( )

η − = η π − =

∫

η η −

erf 1 d e 2 1 f

1

2 1

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ν = t 2 y erfc U u

NS NS Solution Solution

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ME 437/537

0.0 0.2 0.4 0.6 0.8 1.0

u/Uo

0.5 1 1.5 2 2.5 3

y

tν=0.0025 tν=0.062 tν =0.25 tν=1 tν=4

Variation of velocity profile with time. Variation of velocity profile with time.

• G. Ahmadi

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• G. Ahmadi

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l

• U
• U

δ

Laminar Boundary Layer Laminar Boundary Layer

1 / << δ l

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Displacement Displacement Thickness Thickness

Shape Shape Factor Factor

99 . U u =

∫

∞

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = δ

*

dy U u 1

∫

∞

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = θ dy U u 1 U u

θ δ =

*

H

Momentum Momentum Thickness Thickness Boundary Layer Boundary Layer Thickness, Thickness, δ δ

Distance at which

• G. Ahmadi

ME 437/537

) y v x v ( y p 1 y v v x v u ) y u x u ( x p 1 y u v x u u

2 2 2 2 2 2 2 2

∂ ∂ + ∂ ∂ ν + ∂ ∂ ρ − = ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ν + ∂ ∂ ρ − = ∂ ∂ + ∂ ∂

y v x u = ∂ ∂ + ∂ ∂ Steady Two Steady Two-

• D Flows

D Flows

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2 2

y u dx dp 1 y u v x u u ∂ ∂ ν + ρ − = ∂ ∂ + ∂ ∂

y v x u = ∂ ∂ + ∂ ∂

Boundary Boundary Conditions Conditions

• U

u y at v , u y at = ∞ = = = =

Ludwig Ludwig Prandtl Prandtl

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2 2

y u y u v x u u ∂ ∂ ν = ∂ ∂ + ∂ ∂

y v x u = ∂ ∂ + ∂ ∂

Boundary Boundary Conditions Conditions

• U

u y at v , u y at = ∞ = = = =

x

• U
• U

δ

y

atm

P P =

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• U

x

) x ( δ

y

• U
• U

x U

• ν

= η y

• U

/ u

1

) ( ' f U u

• η

=

Blasius

• G. Ahmadi

ME 437/537

x U

• ν

= η y

) ( ' f U u

• η

=

x U ) ( " f y u

• ν

η = ∂ ∂

' " f 2 " ff = +

Blasius Blasius Equation Equation Boundary Boundary Conditions Conditions 1 ' f at ' f , f at = ∞ = η = = = η Boundary Boundary Layer Layer Eq Eq. .

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Numerical Numerical Solution Solution

f”(0)=0.332 5

• U

x 5 ν = δ

2 1 x

Re 5 x

−

= δ Boundary Layer Boundary Layer Thickness, Thickness, δ δ

x U ) ( " f U dy du

• y

ν µ = µ = τ

=

1

η

f ’=u/Uo f f ”

Experimental data

SLIDE 7

7

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• F

R 664 . R ) ( " f 2 U 2 1 C = = ρ τ =

Friction Coefficient Friction Coefficient

l l

l

e e 2

• D

R 328 . 1 R ) ( " f 4 U 2 1 D C = = ρ =

Drag Coefficient Drag Coefficient

• G. Ahmadi

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U x 721 . 1 dy U U 1 ν = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = δ

∫

∞

U x 664 . dy U U 1 U U ν = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = θ ∫

∞

51 . 2 H

*

= θ δ =

Displacement Displacement Thickness Thickness Shape Shape Factor Factor Momentum Momentum Thickness Thickness

• G. Ahmadi

ME 437/537

• Navier

Navier-

• Stokes Equation

Stokes Equation

• Steady Parallel Flows

Steady Parallel Flows

• Boundary Conditions

Boundary Conditions

• Unsteady Parallel Flow

Unsteady Parallel Flow

• Boundary Layer Flows

Boundary Layer Flows

• Flow over a Flat Plate

Flow over a Flat Plate

• Blasius

Blasius Solution Solution

• Laminar Boundary Layer

Laminar Boundary Layer

• G. Ahmadi

ME 437/537