Outline Outline 4 Nonlinear Stability Analysis 4 Nonlinear - - PowerPoint PPT Presentation

SLIDE 1

1

• G. Ahmadi

ME 639-Turbulence

• G. Ahmadi

ME 639-Turbulence

Outline Outline

4 4Nonlinear Stability Analysis Nonlinear Stability Analysis 4 4Disturbed Motion Disturbed Motion 4 4Energy Method Energy Method 4 4Uniqueness Theorems Uniqueness Theorems

• G. Ahmadi

ME 639-Turbulence

Basic Motion satisfies the Basic Motion satisfies the Navier Navier-

• Stokes Equation:

Stokes Equation:

V in Re 1 p t

2v

v v v ∇ + −∇ = ∇ ⋅ + ∂ ∂

= ⋅ ∇ v

( )

S

• n

V x v =

Boundary Boundary Conditions Conditions

• G. Ahmadi

ME 639-Turbulence

Disturbed Motion Disturbed Motion

* * p

, v

* 2 * * * *

Re 1 p t v v v v ∇ + −∇ = ∇ ⋅ + ∂ ∂

* =

⋅ ∇ v

S

• n

*

V v =

Boundary Conditions Boundary Conditions

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• G. Ahmadi

ME 639-Turbulence

v v u − =

*

p p* − = π

u u u u v v u u

2

Re 1 t ∇ + π −∇ = ∇ ⋅ + ∇ ⋅ + ∇ ⋅ + ∂ ∂

S

• n

u =

∫

= dV u 2 1 T

2

Energy Stability Energy Stability Analysis Analysis

Boundary Conditions Boundary Conditions

• G. Ahmadi

ME 639-Turbulence

∫ ∫

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⋅ ∇ ⋅ − ⋅ ∇ ⋅ − ⋅ ∇ ⋅ − π ∇ ⋅ − ∇ ⋅ = ∂ ∂ ⋅ = dV Re 1 dV t dt dT

2

u u u u u v u v u u u u u u

( )

u u u

2

∇ − ⋅ ∇ ∇ = × ∇ × ∇

Vector Identities Vector Identities

( ) ( ) ( ) ( )

2

u u u u u × ∇ + × ∇ × ⋅ −∇ = × ∇ × ∇ ⋅

• G. Ahmadi

ME 639-Turbulence

General Energy Stability Equation General Energy Stability Equation

( ) ( )

∫ ∫ ∫

× ∇ − ⋅ × ∇ × = ∇ ⋅ dV dV

2 S 2

u dS u u u u ( ) [ ]

dV dV

S

= ⋅ π = ⋅ ∇ π − π ⋅ ∇ = π ∇ ⋅

∫ ∫ ∫

dS u u u u

( )

∫ ∫

⋅ ∇ ⋅ − × ∇ − = dV dV Re 1 dt dT

2

u v u u

• G. Ahmadi

ME 639-Turbulence

( )

∫ ∫

≥ × ∇ dV u N dV

2 2

u

Korn Korn Inequality Inequality

2

λu ≤ ⋅ ⋅ − u d u T Re N 2 dt dT ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ λ + − ≤

General Energy Stability Equation General Energy Stability Equation

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• G. Ahmadi

ME 639-Turbulence

( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ λ − − ≤ t Re N exp T T

λ ≤ N Re

Stability Stability

∞ → t

T →

u =

v v* =

• G. Ahmadi

ME 639-Turbulence

If for a basic flow of a viscous If for a basic flow of a viscous incompressible fluid in a bounded incompressible fluid in a bounded region of space region of space then the basic flow is stable. then the basic flow is stable.

λ ≤ N Re

• G. Ahmadi

ME 639-Turbulence

If v and v If v and v*

* are two unsteady flows of a

are two unsteady flows of a viscous fluid in a bounded region of space viscous fluid in a bounded region of space having the same velocity distribution at time having the same velocity distribution at time t=0 and on surface boundary S, then they t=0 and on surface boundary S, then they must be identical if must be identical if

λ ≤ N Re

• G. Ahmadi

ME 639-Turbulence

If v and v If v and v*

* are two steady flows of a viscous

are two steady flows of a viscous fluid in a bounded region of space V(t) fluid in a bounded region of space V(t) subject to the same boundary conditions on subject to the same boundary conditions on surface boundary S, then they must be surface boundary S, then they must be identical if identical if

λ ≤ N Re