Orr-Sommerfeld Equation

Objective: Obtain an equation whose solutions can be used to predict the wavelengths of the
disturbances which will amplify in a shear layer.
Approach: Start with 2-D incompressible, unsteady flow: Navier-Stokes equations:



Substitute

u = U + u'; v = v'; p = P + p'
The mean flow satisfies the Navier-Stokes equations, so subtract the mean-flow equation to get the equations in terms of the perturbations.  Neglect body-force terms and quadratic terms in the perturbations.  Assume that

Then:

Assume a periodic disturbance. Arbitrary forms of the disturbance can be assumed expanded in a Fourier series, and the solution for each term obtained and summed up (linear assumption, valid for very small amplitudes).
For 2-D flow, the Stream-function can be defined as:
where
, wherel is the wavelength of the disturbance,

, wherebr = 2pf, f being the frequency in cycles per unit time,

and biis the amplification factor.
wherecr is the propagation velocity component in the x-direction. This is known as the "phase velocity".
ci is the degree of damping. If ci is negative, it is amplification.

Thus,


Non-dimensionalize and substitute:



This is the Orr-Sommerfeld equation. Here, all lengths are divided by b, a characteristic length of the mean flow (such as the channel width), or by d, the boundary layer or shear layer thickness. The velocities are divided by Um, the "maximum velocity" of the given flow.
 
 

' denotes differentiation with respect to  or .

The Reynolds number is , or


 

The terms on the l.h.s. are inertia terms: these are of 2nd order

The terms on the r.h.s. are viscous terms: these are 4th order.
 
 

Boundary conditions (wall boundary layer)

y = 0; u' = v' = 0: f = 0; f ' = 0

y = infinity: u' = v' = 0; f = 0; f ' = 0

Obviously, people tried solving the "frictionless stability equation" first ==> led to Rayleigh's theorems on the velocity profile and inflexion point. The result is plotted below. The vertical coordinate is the ratio of shear layer thickness to the wavelength of the  disturbance (see definitions of a  and d above), while the horizontal axis is the Reynolds number based on the maximum (freestream) velocity and the shear layer thickness. The curve below shows the boundaries between regions of stability (where disturbances damp out) and regions of instability (where disturbances amplify). For obvious reasons, this is called a "thumb curve". Below some critical Reynolds number, all sorts of disturbances die out, so the shear layer is stable to any disturbance. For Reynolds numbers greater than this, there are instabilities which can amplify. As Reynolds number becomes very large, most disturbances result in turbulence.

The "inviscid instability" shown requires that the velocity profile have an inflexion point, as given by Rayleigh's 1st theorem. However, under some conditions, there can be instability, despite the absence of an inflexion point in the velocity profile. This is explained as "viscous instability": to model these, we have to include the viscous terms on the right hand side of the Navier-Stokes equations.

Note from the diagram that there is a region of "viscous instability", where oscillations develop despite the absence of an inflexion point in the velocity profile. This region narrows to a point as Reynolds numbner tends to infinity.
 

Effect of pressure gradient

The effect of pressure gradient is equivalent to making velocity profile more stable or  unstable. This is seen below, using values of the parameter L. Negative values of L signify an adverse pressure gradient (pressure increasing downstream), which positive values signify a favorable pressure gradient.