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

3. Wake flow of a Flat Plate

4. Laminar Jet Flows

5. Mixing Layer Flow

6. 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 Q 2 at higher heads. He also showed that Δp was approximately proportional to D−4.

7. 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.

8. Laminar-Turbulent Transition

9. Transition process along a flat plate

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

11. Stability of a Physical State Stable Unstable neutral stability Stable for small disturbances but unstable for large ones

12. 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, Q 0, which may be a scalar or vector function. 2. Add a disturbance variable Q' and substitute (Q 0 + Q') into the basic equations which govern the problem. 3. From the equation(s) resulting from step 2, subtract away the basic terms which Q 0, satisfies identically. What remains is the disturbance equation. 4. Linearize the disturbance equation by assuming small disturbances, that is, Q' << Q 0 and neglect terms such as Q’ 2 and Q’ 3 ……..

13. 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.

14. 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: and the pressure field:

15. 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

16. 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