stabilityandtransitiontoturbulence

# stabilityandtransitiontoturbulence - Stability and...

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Unformatted text preview: Stability and Transition to Turbulence Linear estimates for neutral stability To estimate conditions under which parallel flows begin to be unstable, look at conditions under which an initially infinitesimal disturbance will grow exponentially. Begin with Navier-Stokes equations: = + i i x u t ∂ ∂ρ ∂ ∂ρ (1) j ij j j i i x x u u t u ∂ ∂τ ∂ ∂ρ ∂ ∂ρ = + (2) To simplify the problem, let us consider incompressible flow, z w y v x u x u i i ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + = = (3) + + +- = + + + 2 2 2 2 2 2 1 z u y u x u x p z u w y u v x u u t u ∂ ∂ ∂ ∂ ∂ ∂ ν ∂ ∂ ρ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ (4) + + +- = + + + 2 2 2 2 2 2 1 z v y v x v y p z v w y v v x v u t v ∂ ∂ ∂ ∂ ∂ ∂ ν ∂ ∂ ρ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ (5) + + +- = + + + 2 2 2 2 2 2 1 z w y w x w z p z w w y w v x w u t w ∂ ∂ ∂ ∂ ∂ ∂ ν ∂ ∂ ρ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ (6) Since we are focusing on parallel or fully developed flows, let us assume that u(y) only. A small velocity perturbation can be applied in each direction such that p x p p w w v v u y U u ′ + → ′ → ′ → ′ + → ) ( ) ( (7) where ) ( ) ( , , x p p y U w v u << ′ << ′ ′ ′ (8) Installing these into eqn. 4-6, results in = ′ + ′ + ′ z w y v x u ∂ ∂ ∂ ∂ ∂ ∂ (9) ′ + ′ + + ′ + ′ +- = ′ + ′ + ′ + 2 2 2 2 2 2 ) ( ) ( 1 ) ( z u y u U x u x p p y u v x u U t u u ∂ ∂ ∂ ∂ ∂ ∂ ν ∂ ∂ ρ ∂ ∂ ∂ ∂ ∂ ∂ (10) ( 29 ′ + ′ + ′ + ′ +- = ′ + ′ 2 2 2 2 2 2 1 z v y v x v y p p x v U t v ∂ ∂ ∂ ∂ ∂ ∂ ν ∂ ∂ ρ ∂ ∂ ∂ ∂ (11) ( 29 ′ + ′ + ′ + ′ +- = ′ + ′ 2 2 2 2 2 2 1 z w y w x w z p p x w U t w ∂ ∂ ∂ ∂ ∂ ∂ ν ∂ ∂ ρ ∂ ∂ ∂ ∂ (12) For the unperturbed case, we have , 1 2 2 = = +- = z p y p y U x p t U ∂ ∂ ∂ ∂ ∂ ∂ ν ∂ ∂ ρ ∂ ∂ (13) Subtracting eqn. 13 from 10 for example, gives ′ + ′ + ′ + ′- = ′ + ′ + ′ 2 2 2 2 2 2 1 z u y u x u x p y U v x u U t u ∂ ∂ ∂ ∂ ∂ ∂ ν ∂ ∂ ρ ∂ ∂ ∂ ∂ ∂ ∂ (14) If we nondimensionalize eqns. 9, 14, 11, and 12 using 2 , , , , , , U p p L z z L x y L x x L tU t U w w U v v U u u ρ = = = = = = = = gives = ′ + ′ + ′ z w y v x u ∂ ∂ ∂ ∂ ∂ ∂ (15) ′ + ′ + ′ + ′- = ′ + ′ + ′ 2 2 2 2 2 2 Re 1 z u y u x u x p y U v x u U t u ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ (16) ′ +...
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stabilityandtransitiontoturbulence - Stability and...

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