Analysis of Perceivable (Non-Creeping) Fluid flows

1. 17th December 1903 Analysis of Perceivable (Non-Creeping) Fluid flows P M V Subbarao Professor Mechanical Engineering Department I I T Delhi An Ultimate Combination of Physical Intuition with Experiments…

2. Degree of Creeping • An almost imperceptible flow field (creeping flow field) completely respects the presence of a solid boundary. • How to define the degree of creeping? • What if the fluid particle can move much faster than the speed at which the effect of solid boundary propagates into the flow field? • How deep the presence of a boundary can propagate into the flow field? • No effect of Wall at all or something else?

3. Introduction of Boundary Layer Concept Based on his experimental observations, A Genius Intuitively concluded that effect of the viscosity is confined to a thin viscous layer that he called, the boundary layer.

4. An Ingenious Lecture • A29 year old professor in Hanover, Germany delivered in a 10 minutes address in 1904 on this topic. • This concept is a classic example of an applied science greatly influencing the development of mathematical methods of wide applicability. • Prof. Ludwig Prandtl. • Prandtl had done experiments in the flow of water over bodies, and sought to understand the effect of the small viscosity on the flow. • Realizing that the no-slip condition had to apply at the surface of the body, his observations led him to the conclusion that the flow was brought to rest in a thin layer adjacent to the rigid surface. • The boundary layer.

5. The Boundary Layer Effect : The Leader of Asymptoticity • Prandtl reasoning suggested that the Navier-Stokes equations should have a somewhat simpler form owing to the thinness of this layer. • This led to the equations of the viscous boundary layer. • Boundary-layer methods now occupy a fundamental place in many asymptotic problems for partial differential equations. • Ludwig Prandtl, with his fundamental contributions to hydrodynamics, aerodynamics, and gas dynamics, greatly influenced the development of fluid mechanics as a whole. • His pioneering research in the first half of the 20th century that founded modern fluid mechanics.

6. Analytical Proof for Prandtls Intuition & Experiments Consider non-dimensional NS Equations for incompressible flow Steady State non-dimensional NS Equations for incompressible flow Steady State Incompressible non-dimensional NS Equations

7. Equivalent ODE to NS A selected property of any fluid flow field can be approximated as:

8. General Response of A Second Order System y y

9. Toward Creeping y y

10. Response of Flow Field towards Boundary Effects y

11. Applications of the limit of Very large Re Flow over a Wedge

12. Prandtls Large Reynolds Number 2-D Incompressible Flow The free-stream velocity will accelerate for non-zero values of β: where L is a characteristic length and m is a dimensionless constant that depends on β:

13. The Measure of Wedge Angle The condition m = 0 gives zero flow acceleration corresponding to the Blausius solution for flat-plate flow. The boundary layer is seen to grow in thickness as x moves from 0 to L.

14. Two-dimensional Boundary Layer Flows In dimensionless variables the steady incompressible Navier-Stokes equations in two dimensions may be written: The boundary layer is seen to grow in thickness as x moves from 0 to L.

15. The boundary layer is seen to grow in thickness as x moves from 0 to L. The Art of Asymptotic Thinking - I This suggests that the term in x-momentum equation can be properly estimated as of order U2/L In the dimensionless formulation, should be taken as O(1) at large Re. If this term is to balance the viscous stress term, then the natural choice is to assume that the y-derivatives of u are so large that the balance is with .

16. The Art of Asymptotic Thinking - II An invaluable fact intuitively confirmed by Prandtl is that the boundary layer on the plate is observed to be so thin. If this inertial term is to balance the viscous stress term, then the natural choice is to assume that the y-derivatives of u are so large that the balance is with . Viscous Term Thus it makes sense to define a local non-dimensional number A stretched variable Local Reynolds Number

17. Shape of Boundary Layer In Stretched Coordinates

18. The stretched N-S Equations 2-D incompressible continuity equations In order to keep this essential equation intact and as of order unity: Stretched coordinate: The stretched variable must be compensated by a stretched form of the y-velocity component: 2-D incompressible continuity equations in stretched coordinates:

19. Prandtls Intuition Prandtl would have been comfortably guessed this definition. The boundary layer on the plate was so thin that there could have been only a small velocity component normal to its surface. Thus the continuity equation will survive our limit Re .

20. X - Momentum Equation in Stretched Coordinates Returning now to consideration of x-momentum equation, retain the pressure term as O(1). X-momentum equation in stretched coordinates: In the limit Re , with stretched variables, this amounts to dropping the term

21. y-Momentum Equation in 2-D Boundary Layer Flows Use these stretched variables in y-momentum equation Thus in the limit Re  the vertical momentum equation reduces to

22. The Conclusions from Intuitive Mathematics • The pressure does not change as we move vertically through the thin boundary layer. • That is, the pressure throughout the boundary layer at a station x must be the pressure outside the layer. • At this point a crucial contact is made with inviscid fluid theory. • The “pressure outside the boundary layer” should be determined by the inviscid theory. • Since the boundary layer is thin and will presumably not disturb the inviscid flow very much. • In particular for a flat plate the Euler flow is the uniform stream- the plate has no effect and so the pressure has its constant free-stream value.