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

Outline Outline 4 Nonlinear Stability Analysis 4 Nonlinear Stability Analysis 4 Disturbed Motion 4 Disturbed Motion 4 Energy Method 4 Energy Method 4 Uniqueness Theorems 4 Uniqueness Theorems ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi * p * , Basic Motion satisfies the Basic Motion satisfies the Navier Navier- -Stokes Equation: Stokes Equation: v Disturbed Motion Disturbed Motion ∂ 1 v + ⋅ ∇ = −∇ + ∇ 2 v p in V v v ∂ * 1 ∂ t Re v + ⋅ ∇ = −∇ + ∇ * * * 2 * p v v v ∂ t Re ∇ v ⋅ = 0 * = ∇ v ⋅ 0 ( ) = Boundary on S Boundary v x V v = * Conditions on S Conditions V Boundary Conditions Boundary Conditions ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 1

p * − = − π = * p u v v ⎡ 1 ⎤ ⋅ ∇ 2 − ⋅ ∇ π − ⋅ ∇ ⋅ ∂ dT u u u u u v u ⎢ ⎥ ∫ ∫ = ⋅ = dV dV Re u ⎢ ⎥ ∂ dt t ∂ 1 u − ⋅ ∇ ⋅ − ⋅ ∇ ⋅ ⎣ ⎦ v u u u u u + ⋅ ∇ + ⋅ ∇ + ⋅ ∇ = −∇ π + ∇ 2 u v v u u u u ∂ t Re Vector Identities Vector Identities u = on S ( ) 0 ∇ × ∇ × = ∇ ∇ ⋅ − ∇ 2 Boundary Conditions Boundary Conditions u u u 1 Energy Stability Energy Stability ∫ ( ) ( ( ) ) ( ) = 2 T u dV ⋅ ∇ × ∇ × = −∇ ⋅ × ∇ × + ∇ × 2 u u u u u Analysis Analysis 2 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi ( ) ∫ ∫ ( ) ( ) Korn 2 Korn ∇ × ≥ 2 dV N u dV ∫ ∫ ∫ 2 ⋅ ∇ 2 = × ∇ × ⋅ − ∇ × dV dV u u u u u dS u Inequality Inequality S − ⋅ ⋅ ≤ 2 [ ] ( ) λ u ∫ ∫ ∫ ⋅ ∇ π = ∇ ⋅ π − π ∇ ⋅ = π ⋅ = u d u dV dV 0 u u u u dS S General Energy Stability Equation General Energy Stability Equation General Energy Stability Equation General Energy Stability Equation ⎛ ⎞ dT N dT 1 ≤ − + λ ( ) 2 ⎜ ⎟ T ∫ ∫ 2 = − ∇ × − ⋅ ∇ ⋅ dV dV u u v u dt Re ⎝ ⎠ dt Re ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 2

≤ N If for a basic flow of a viscous If for a basic flow of a viscous Re Stability Stability incompressible fluid in a bounded incompressible fluid in a bounded λ region of space region of space ≤ N ⎧ ⎫ ⎛ ⎞ Re N ( ) ≤ − − λ T T 0 exp ⎜ ⎟ t λ ⎨ ⎬ ⎝ Re ⎠ ⎩ ⎭ then the basic flow is stable. then the basic flow is stable. v * = u = 0 → ∞ T → t 0 v ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi * are two steady flows of a viscous * are two unsteady flows of a If v and v * If v and v are two steady flows of a viscous If v and v * If v and v are two unsteady flows of a fluid in a bounded region of space V(t) fluid in a bounded region of space V(t) viscous fluid in a bounded region of space viscous fluid in a bounded region of space subject to the same boundary conditions on subject to the same boundary conditions on having the same velocity distribution at time having the same velocity distribution at time surface boundary S, then they must be surface boundary S, then they must be t=0 and on surface boundary S, then they t=0 and on surface boundary S, then they identical if identical if must be identical if must be identical if ≤ N ≤ N Re Re λ λ ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 3