An Ultimate Combination of Physical Intuition with Experiments… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Boundary Layer Theory for Viscous Fluid Flows

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.

Introduction of Boundary Layer Concept Based on his experimental observations, Prandtl found that effect of the viscosity is confined to a thin viscous layer that he called, the boundary layer.

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

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

General Response of A Second Order System y y

Toward Creeping y y

Response of Flow Field towards Boundary Effects y

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

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:

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

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.

The Art of Asymptotic Thinking This suggests that the term in x-momentum equation can be properly estimated as of order U 2 /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 choiceis to assume that the y-derivatives of u are so large that the balance is with. This is due to the fact that the boundary layer on the plate is observed to be so thin. Thus it makes sense to define A stretched variable Local Reynolds Number

Shape of Boundary Layer In Stretched Coordinates

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

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 .

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

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

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.

Prandtls Viscous Flow Past a General Body Prandtl’s striking insight is clearer when we consider flow past a general smooth body. The boundary layer is taken as thin in the neighborhood of the body. Curvilinear coordinates can be introduced. x the arc length along curves paralleling the body surface and y the coordinate normal to these curves. In the stretched variables, and in the limit for large Re, it turns out that we again get only must be interpreted to mean that the pressure is what would be computed from the inviscid flow past the body.

Bernoulli’s Pressure Prevails in Prandtls Boundary Layer If p  and U  are the free stream values of p and u, then Bernoulli’s theorem for steady flow yields along the body surface It is this p(x) which now applies in the boundary layer. Thus the inviscid flow past the body determines the pressure variation which is then imposed on the boundary layer through the now known function in.

Euler Solution of Wedge Flow

Velocity Outside Boundary Layer for Cylinder

Prandtl’s Boundary Layer Equations We note that the system of equations given below are usually called the Prandtl boundary-layer equations.