Laminar flows have a fatal weakness … P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Evolution & Stability of Laminar Boundary Layer Flows

The Flat Plate Boundary Layer & Experimental Exploration

Measurement of Wall shear Stress

Evolution of Flat Plate Boundary Layer Thickness

Integral View of Flat Plate Boundary Layer

Similarity of An evolving FBL The application of integral balances to the control volume shown in above figure delivers more similarity features. the boundary layer displacement thickness the boundary layer momentum deficiency thickness and the energy deficiency thickness.

Selection of a CV for Engineering BL Analysis The control volume chosen for the analysis is enclosed by dashed lines. This selection is not arbitrary but very clever. Since velocity distributions are known only at the inlet and exit, it is imperative that the other two sides on the control volume be streamlines, where no mass or momentum crosses.

Impermeable Streamlines of A BL The lower side should be the wall itself; hence the drag force will be exposed. The upper side should be a streamline outside the shear layer, so that the viscous drag is zero along this line.

Integral View of A FPBL Integral Method

The partial integration of the second term of the left side of above equation gives: Using continuity equation :

Physical Interpretation of 

The Displacement Thickness Conservation of mass is applied to this Engineering Assuming incompressible flow, this relation simplifies to

 * is the engineering formal definition of the boundary-layer displacement Thickness and holds true for any incompressible flow.

Physical Interpretation of Momentum and displacement thicknesses The ratio of displacement thickness to momentum thickness is called the shape factor, is often used in boundary-layer analyses:

Boundary layer development along a wedge We know that the velocity distribution outside the boundary layer is a simple power function

ODE for Wedge BL Flows Introducing the same relationship for the stream function as This is the so-called Falkner-Skan equation which describes a flow past a wedge.

Effect of Wedge Angle on BL Profile

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 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.

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