Laminar flows have a fatal weakness …

P M V SubbaraoProfessor

Mechanical Engineering DepartmentI I T Delhi

The Stability of Laminar Flows

• 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?

The Philosophy of Instability

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.

Receptivity of Boundary Layer to Disturbances

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.

Sketch of transition process in the boundary layer along a flat Plate

Onset of the unstable two-dimensional Tollmien-Schlichting waves.

Sketch of transition process in the boundary layer along a flat Plate

Development of unstable, three-dimensional waves and the formation of vortex cascades.

Sketch of transition process in the boundary layer along a flat Plate

Bursts of turbulence in places with high vorticity.

Sketch of transition process in the boundary layer along a flat Plate

Intermittent formation of turbulent spots with high vortical core at intense fluctuation.

Sketch of transition process in the boundary layer along a flat Plate

Coalescence of turbulent spots into a fully developed turbulent boundary layer.

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.

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 .

txvxVtxv ,~,

and the pressure field: txpxPtxp ,~,

• 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:

xPxV

&

NS Equations for Flow influenced by Small Disturbances

vVpPvVvVt

vV

~~1~~~2

Performing the differentiation and multiplication, we arrive at: vVpPvvVvvVVVtv

~~1~~~~~22

The small disturbance leading to linear stability theory requires that the nonlinear disturbance terms be neglected. This results in

vpVPVvvVVV

tv

~~11~~~22

vpvvtv

21

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.

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

vpVPVvvVVVtv

~~11~~~22

vpVPVvvVVVtv

~~11~~~22

VPVV

21

Disturbance Conservation Equation

vpVvvV

tv

~~1~~~2

Equation in Cartesian index notation is written as

2

2~~1~~~

j

i

ij

ij

j

ij

i

xv

xp

xVv

xvV

tv

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

• 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.

Two-dimensional Disturbance Equations

2

2

2

2 ~~~1~~~~~

yu

xu

xp

yUv

xUu

yuV

xuU

tu

2

2

2

2 ~~~1~~~~~

yv

xv

yp

yVv

xVu

yvV

xvU

tv

The continuity equation for incompressible flow yields:

0~ vV

0~ v

0 Vas

0~~

yv

xu0

yV

xU

With above equations there are three-equations to solve three unknowns.

Step 4

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

2

2

2

2 ~~~1~~~~~

yu

xu

xp

yUv

xUu

yuV

xuU

tu

2

2

2

2 ~~~1~~~~~

yv

xv

yp

yVv

xVu

yvV

xvU

tv

GDE for Modulation of Disturbance

2

2

2

2 ~~~1~~~

yu

xu

xp

yUv

xuU

tu

2

2

2

2 ~~~1~~

yv

xv

yp

xvU

tv

0~~

yv

xu