Post on 22-Dec-2015
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