1. The Stability of Laminar Flows
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
Laminar flows have a fatal weakness …

2. The Philosophy of Instability
The equations of Fluid dynamics allow some flow patterns.
Given a flow pattern , is it stable?
If the flow is disturbed, will the disturbance gradually die down, or will the disturbance grow such that the flow departs from its initial state and never recovers?

3. Major Classes of Instability in Fluid Dynamics
Wall-bounded flows:
Boundary layers, pipe flows, etc
Any basic flow without inflexion point
viscosity plays a role
sensitive to the form of the basic flow
Free-shear flows:
mixing layers, wakes, jets, etc
less sensitive to the form of the basic flow
Viscosity is not responsible.

4. Receptivity of Boundary Layer to Disturbances

5. Sketch of transition process in the boundary layer along a flat Plate
A stable laminar flow is established that starts from the leading edge and extends to the point of inception of the unstable two-dimensional Tollmien-Schlichting waves.

6. Sketch of transition process in the boundary layer along a flat Plate
Onset of the unstable two-dimensional Tollmien-Schlichting waves.

7. Sketch of transition process in the boundary layer along a flat Plate
Development of unstable, three-dimensional waves and the formation of vortex cascades.

8. Sketch of transition process in the boundary layer along a flat Plate
Bursts of turbulence in places with high vorticity.

9. Sketch of transition process in the boundary layer along a flat Plate
Intermittent formation of turbulent spots with high vortical core at intense fluctuation.

10. Sketch of transition process in the boundary layer along a flat Plate
Coalescence of turbulent spots into a fully developed turbulent boundary layer.

11. Outline of a Typical Stability Analysis
Small disturbances are present in any flow system.
Small-disturbance stability analysis is followed to understand the receptivity of flow.
This analysis is carried-out in seven steps.
1. The flow problem, whose stability is to be studied must have a basic flow solution in terms of Q0, which may be a scalar or vector function.
2. Add a disturbance variable Q' and substitute (Q0 + Q') into the basic equations which govern the problem.

12. Stability of Flow due to Small Disturbances
We consider a steady flow motion, on which a small disturbance is superimposed.
This particular flow is characterized by a constant mean velocity vector field and its corresponding pressure .
We assume that the small disturbances we superimpose on the main flow is inherently unsteady, three dimensional and is described by its vector filed and its pressure disturbance.
The disturbance field is of deterministic nature that is why we denote the disturbances.
Thus, the resulting motion has the velocity vector field:
and the pressure field:

13. NS Equations for Flow influenced by Small Disturbances
Performing the differentiation and multiplication, we arrive at:
The small disturbance leading to linear stability theory requires that the nonlinear disturbance terms be neglected.
This results in

14. Step 3
From the equation(s) resulting from step 2, subtract away the basic terms which Q0, satisfies identically.
What remains is the Governing Equation for evolution of disturbance s.

15. Implementation of Step 3
Above equation is the composition of the main motion flow superimposed by a disturbance.
The velocity vector constitutes the Navier-Stokes solution of the main laminar flow.
Obtain a Disturbance Conservation Equation by taking the difference of above and steady Laminar NS equations

16. Disturbance Conservation Equation
Equation in Cartesian index notation is written as
This equation describes the motion of a three-dimensional disturbance field modulated by a steady three-dimensional laminar main flow field.
A solution to above equation will be studied to determine the stability of main flow.
Two assumptions are made in order to find an analytic solution.
The first assumption implies that the main flow is assumed to be two-dimensional, where the velocity vector in streamwise direction changes only in lateral direction

17. The second assumption concerns the disturbance field.
In this case, we also assume the disturbance field to be two-dimensional too.
The first assumption is considered less controversial, since the experimental verification shows that in an unidirectional flow, the lateral component can be neglected compared with the longitudinal one.
As an example, the boundary layer flow along a flat plate at zero pressure gradient can be regarded as a good approximation.
The second assumption concerning the spatial two dimensionality of the disturbance flow is not quite obvious.
This may raise objections that the disturbances need not be two dimensional at all.

18. Two-dimensional Disturbance Equations
The continuity equation for incompressible flow yields:
With above equations there are three-equations to solve three unknowns.

19. Step 4
Linearize the disturbance equation by assuming small disturbances, that is, Q' << Q0 and neglect terms such as Q’2 and Q’3 ……..

20. GDE for Modulation of Disturbance