Page 253 - Computational Fluid Dynamics for Engineers

Stability and Transition

8.1 Introduction

The current methods for predicting transition from laminar to turbulent flow
are based on the solution of the unsteady Navier-Stokes equations (DNS) dis-
cussed in [1], on the solutions of the parabolized stability equations (PSE) dis-
cussed in [2, 3] and on the solutions of the linear stability equations discussed
in Section 2.5. While DNS approach offers exciting possibilities, it is currently
limited to some simple flows. Its computer requirements are large for transition
calculations on complex bodies. The only engineering calculation method for
n
predicting transition at this time is the e -method based on the solutions of the
Orr-Sommerfeld (OS) equation given by Eq. (2.5.13)

2
2
<f> iv - 2a (j)" + a V = iR(au - uj)((j)" - a (/)) - iRau"<f> (2.5.13)
subject to boundary conditions, which for wall boundary-layer flows are given
by Eqs. (P2.10.1) and (P2.10.2), or

y = , </> = </>' = Q (8.1.1a)
0
2
y = 6, (D - £)<!> + (ft + &)(D + 6 ) 0 = 0 (8.1.1b)
2
(£ + &)(0 -£?)0 = O
Before we discuss the solution procedure it is useful to review the properties
of the OS equation by rewriting Eq. (2.5.13) as
2
4
(j) iv - 2a V + a (f) = iaR(u - c)(</>" - a 0) - iRau"</> (8.1.2)
where c is the dimensionless complex propagation velocity of the disturbance
related to its dimensional value c* by