stabilityandtransitiontoturbulence

stabilityandtransitiontoturbulence - Stability and...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 (1) j ij j j i i x x u u t u = + (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 (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 (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 (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 (10) ( 29 + + + +- = + 2 2 2 2 2 2 1 z v y v x v y p p x v U t v (11) ( 29 + + + +- = + 2 2 2 2 2 2 1 z w y w x w z p p x w U t w (12) For the unperturbed case, we have , 1 2 2 = = +- = z p y p y U x p t U (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 (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 = = = = = = = = 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) +...
View Full Document

Page1 / 8

stabilityandtransitiontoturbulence - Stability and...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online