Laminar flows have a fatal weakness …

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

The Stability of Laminar Flows

Wake flow of a Flat Plate

Laminar Jet Flows

Mixing Layer Flow

Gotthilf Heinrich Ludwig Hagen

• Hagen in studied Königsberg, East Prussia, having among his teachers the famous mathematician Bessel.

• He became an engineer, teacher, and writer and published a handbook on hydraulic engineering in 1841.

• He is best known for his study in 1839 of pipe-flow resistance.

• Water flow at heads of 0.7 to 40 cm, diameters of 2.5 to 6 mm, and lengths of 47 to 110 cm.

• The measurements indicated that the pressure drop was proportional to Q at low heads and proportional (approximately) to Q2 at higher heads.

• He also showed that Δp was approximately proportional to D−4.

The Overshadowed Ultimate Truth

• In an 1854 paper, Hagen noted that the difference between laminar and turbulent flow was clearly visible in the efflux jet.

• This jet was either “smooth or fluctuating,” and in glass tubes, where sawdust particles either “moved axially” or, at higher Q, “came into whirling motion.”

• Thus Hagen was a true pioneer in fluid mechanics experimentation.

• Unfortunately, his achievements were somewhat overshadowed by the more widely publicized 1840 tube-flow studies of J. L. M. Poiseuille, the French physician.

Laminar-Turbulent Transition

Transition process along a flat plate

Can a given physical state withstand a disturbance and still return to its original state?

Stability of a Physical State

Stable Unstable neutral stability

Stable for small disturbances but unstable for large ones

Outline of a Typical Stability Analysis

• All small-disturbance stability analyses follow the same general line of attack, which may be listed in seven steps.

1. We seek to examine the stability of a basic solution to the physical problem, 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.

3. From the equation(s) resulting from step 2, subtract away the basic terms which Q0, satisfies identically.

• What remains is the disturbance equation.

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

• If the linearized disturbance equation is complicated and multidimensional, it can be simplified by assuming a form for the disturbances, such as a traveling wave or a perturbation in one direction only.

6. The linearized disturbance equation should be homogeneous and have homogeneous boundary conditions.

• It can thus be solved only for certain specific values of the equation's parameters.

• In other words, it is an eigenvalue problem.

7. The eigenvalues found in step 6 are examined to determine when they grow (are unstable), decay (are stable), or remain constant (neutrally stable).

• Typically the analysis ends with a chart showing regions of stability separated from unstable regions by the neutral curves.

Stability of Small Disturbances

• We consider a statistically 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:

txvxVtxv ,~,

and the pressure field: txpxPtxp ,~,

Small Disturbances

vVpPvVvVt

vV

~~1~~~

2

Performing the differentiation and multiplication, we arrive at:

vVpPvvVvvVVVt

v

~~1~~~~~

22

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

vpVPVvvVVV

t

v

~~11~~~

22

• 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 Laminar NS equations

vpVPVvvVVV

t

v

~~11~~~

22

vpVPVvvVVV

t

v

~~11~~~

22

VPVV

21