u = t ~ t y ~ t y 0 y I.C. I.C. 1 = a = 2 a 1 / 2 - - PowerPoint PPT Presentation

SLIDE 1

1

ME 637

• G. Ahmadi

ME 637

• G. Ahmadi

Outline Outline 4 4Plate Suddenly Set in Motion Plate Suddenly Set in Motion 4 4Oscillating Plate Oscillating Plate 4 4Unsteady Pipe Flows Unsteady Pipe Flows 4 4Steady Flows in Noncircular Pipes Steady Flows in Noncircular Pipes 4 4Elliptic Cross Section Pipes Elliptic Cross Section Pipes 4 4Triangular Cross Triangular Cross-

• Section Pipes

Section Pipes

ME 637

• G. Ahmadi

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.

ME 637

• G. Ahmadi

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

SLIDE 2

2

ME 637

• G. Ahmadi

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

ME 637

• G. Ahmadi

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.

ME 637

• G. Ahmadi

B.C. B.C. Let Let

2 2

y u t u ∂ ∂ ν = ∂ ∂

x y

Viscous Fluid

Uo cosωt

t cos U u ω =

∞ = y

u =

y =

( )

ay t cos e U u

ky

− ω =

−

ME 637

• G. Ahmadi

( )

ky t cos e U u

ky

− ω =

−

ν ω = 2 k

ν = ν = ω

2

k 2 ak 2

2 2

k a =

( ) ( )

θ − θ − ν = θ ω − sin ak 2 cos a k sin

2 2

( )

ay t sin e U t u

ky

− ω ω − = ∂ ∂

−

( ) ( ) ( )

ay t sin a ay t cos k e U y u

ky

− ω + − ω − = ∂ ∂

−

Navier Navier-

• Stokes Equation

Stokes Equation

Matching Matching Solution Solution

SLIDE 3

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ME 637

• G. Ahmadi

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ ν + ρ − = ∂ ∂ r v r r r 1 dz dP 1 t v

z z

R

z Navier Navier Stokes Stokes

) t , R ( u = ) , r ( u =

ME 637

• G. Ahmadi

R r = ξ

2 2

R t R t ν = ρ µ = τ

( )

ξ ϕ µ − =

2 z

R dz dP 4 1 v

Nondimensional Nondimensional Variables Variables

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ξ ∂ ϕ ∂ ξ ξ ∂ ∂ ξ + = τ ∂ ϕ ∂ 1 4

Navier Navier-

• Stokes

Stokes

ME 637

• G. Ahmadi

Boundary Boundary Conditions Conditions Changing Changing Variable Variable

= ϕ 1 = ξ = ϕ

= τ

ψ − ξ − = φ

2

1

ME 637

• G. Ahmadi

Navier Navier Stokes Stokes

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ξ ∂ ψ ∂ ξ ξ ∂ ∂ ξ = τ ∂ ψ ∂ 1

1 = ξ = ψ

= τ

2

1 ξ − = ψ

Boundary Boundary Conditions Conditions

SLIDE 4

4

ME 637

• G. Ahmadi

Separation of Variables Separation of Variables

( ) ( )

τ ξ = ψ T F

2

d dF d d F 1 T T α − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ξ ξ ξ ξ = & T T

2

= α + & F d dF d F d

2 2 2 2 2

= ξ α + ξ ξ + ξ ξ

τ α −

=

2

Ce T

( ) ( )

αξ + αξ = BY AJ F

Bessel Equation Bessel Equation Bessel Functions Bessel Functions

ME 637

• G. Ahmadi

( )

∞ → Y0

( )

B finite ~ F = ⇒

( ) ( )

J 1 F = α ⇒ =

General General Solution Solution

( )

∑

ξ α = ψ

τ α − n n n

J e A

2 n

Boundary Boundary Conditions Conditions

405 . 2

1 =

α 52 . 5

2 =

α 654 . 8

3 =

α

Eigenvalues Eigenvalues

ME 637

• G. Ahmadi

( )

( ) ( ) ( )

n 2 1 3 n 1 n 2 1 n 2 n

J 8 d J d J 1 A α α = ξ ξ α ξ ξ ξ α ξ ξ − =

∫ ∫ ( )

∑

ξ α = ξ −

n n n 2

J A 1

Initial Initial Condition Condition

Solution Solution

( ) ( )

∑

α α ξ α = ψ

τ α − n n 1 3 n n

J J e 8

2 n

ME 637

• G. Ahmadi

Solution Solution

( ) ( )

∑

τ α −

α α ξ α − ξ − = ϕ

n n 1 3 n n 2

n

e J J 8 1

τ

SLIDE 5

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ME 637

• G. Ahmadi

Navier Navier Stokes Stokes S

const dz dP 1 W

2

= µ = ∇ W =

ME 637

• G. Ahmadi

b

z x y

a

1 b y a x

2 2

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

2 2 2 2

b y a x 1 A w

Ellipse Ellipse

Let Let

ME 637

• G. Ahmadi

( )

dz dP 1 b a b a A 2 b 2 a 2 A w

2 2 2 2 2 2 2

µ = + − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − = ∇ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − + µ − =

2 2 2 2 2 2 2 2

b y a x 1 b a b a dz dP 2 1 A

2 2 3 3

b a b a dz dP 4 Q + µ π − =

NS NS

∫∫

= wdxdy Q

Flow Rate Flow Rate

ME 637

• G. Ahmadi

( ) ( )(

)( )

a 2 y 3 x a 2 y 3 x a x y , x f = + + + − − =

x y a 2a

( )

a 2 y 3 x = + +

( )

a 2 y 3 x = + −

SLIDE 6

6

ME 637

• G. Ahmadi

( )

y , x Af w =

NS NS ( )

dz dP 1 aA 12 y , x f A w

2 2

µ = = ∇ = ∇ dx dP a 12 1 A µ =

Let Let

( )(

)( )

a 2 y 3 x a 2 y 3 x a x dx dP a 12 1 w + + + − − µ =

Solution Solution

ME 637

• G. Ahmadi
• Plate Suddenly Set in Motion

Plate Suddenly Set in Motion

• Oscillating Plate

Oscillating Plate

• Unsteady Pipe Flows

Unsteady Pipe Flows

• Steady Flows in Noncircular

Steady Flows in Noncircular Pipes Pipes

• Elliptic Cross Section Pipes

Elliptic Cross Section Pipes

• Triangular Cross

Triangular Cross-

• Section Pipes

Section Pipes

ME 637

• G. Ahmadi