Theory Of Boundary Layer

Introduction

When a real fluid flows past a solid boundary, a layer of fluid which comes in contact with the boundary surface adheres to it on account of viscosity. Since this layer of the fluid cannot slip away from the boundary surface it attains the same velocity as that of the boundary. If the boundary is stationary, the fluid velocity at the boundary surface will be zero. Thus at the boundary surface the layer of the fluid undergoes retardation.

Therefore in the immediate vicinity of the boundary surface, the velocity of the fluid increases gradually from zero at boundary surface to the velocity of the mainstream. This region is known as BOUNDARY LAYER.

Causes Of Its Formation

Large velocity gradient (Larger variation of velocity in relatively smaller distance) leading to appreciable shear stress.

Consists of two layers:

CLOSE TO BOUNDARY : large velocity gradient, appreciable viscous forces.

OUTSIDE BOUNDARY LAYER: viscous forces are negligible, flow may be treated as non-viscous or inviscid.

Development Of Boundary Layer

The boundary layer starts at the leading edge of a solid surface and the boundary layer thickness increases with the distance x along the surface.

Near the leading edge of the solid surface, where thickness is small, the flow is laminar (LAMINAR BOUNDARY LAYER UPTO Re 3 x 105 ~ 3.2 x 105

LAMINAR BOUNDARY LAYER PROFILE – PARABOLIC

As the thickness of the layer increases in the downstream direction, the laminar layer becomes unstable, leading to transition from laminar to turbulent boundary layer. Re ~ 5.5 x 105 ( Onset of turbulent BL)

TURBULENT BOUNDARY LAYER – PROFILE BECOMES LOGARITHMIC

Turbulent flow is characterized by greater interchange of mass momentum and energy within the fluid particles.

The velocity profile is more uniform in turbulent BL.

Velocity gradient is higher in turbulent BL, hence shear stresses are higher.

BL depends on Reynold’s number & also on the surface roughness. Roughness of the surface adds to the disturbance in the flow & hastens the transition from laminar to turbulent

Parameters Of Boundary Layer

BOUNDARY LAYER THICKNESS (δ)

The velocity within a boundary layer approaches the free stream velocity value asymptotically, and so the limit of boundary layer is not easily defined.

A distance δ is prescribed at the velocity lies within 1 % of the asymptotic value u = 0.99 U0

δ becomes the measure of the thickness of a region in which major portion of the velocity distribution takes place

Parameters Of Boundary Layer

DISPLACEMENT THICKNESS (δ*)

Consider an elementary strip of thickness dy & at a distance y from the plate surface.

Area of elementary strip dA = b x dy, where b is the width of the plate of this page.

Mass flow rate through this strip = ρ x flow velocity x area = ρub x dy

In absence of the plate, the fluid would have moved with a constant velocity equal to free stream velocity U0

Corresponding mass flow rate = ρU0b x dy

Loss in mass flow rate through the elemental strip = ρU0b x dy – ρub x dy = ρ(U0 – u) b x dy

Where δ is the value of y at which u = U0

Hence we define the displacement thickness as the thickness of flow moving at the free stream velocity and having the flow rate equal to the loss in flow rate on account of boundary layer formation.

Momentum Thickness (Θ)

Loss in mass flow rate due to velocity defect = ρ(U0 – u) b dy

Loss in momentum = ρ(U0 – u) b dy x u

The momentum thickness (θ) is defined as the thickness of flow moving at free stream velocity and having the same momentum flux equal to the deficiency of the momentum flux in the region of boundary layer.

Momentum thickness can be conceived as the transverse distance by which the boundary layer should be displaced to compensate for the reduction in the momentum of the flowing fluid on account of the boundary layer formation

Energy Thickness (Δ**)

Mass of fluid = ρubdy

K.E = ½ * ρu x b x dy x u2

K.E. in absence of boundary layer = ½ * ρu x b x dy x U02

loss in K.E through the elemental strip = ½ * ρu x b x dy x (U02 – u2)

Total loss of K.E.

The energy thickness δ** is defined as the thickness of the flow moving at the free stream velocity & having the energy equal to the deficiency of energy in the boundary layer region

K.E . Through distance δ** = ½ * (ρbδ**U0)xU02 = ½ * (ρbδ**)xU03

Energy thickness may be conceived as the transverse distance by which the boundary layer should be displaced to compensate for the reduction in energy of the flowing fluid on account of the boundary layer formation

2-D Boundary Layer Eqn.

PRANDTL’S BOUNDARY LAYER EQUATIONS

Consider steady two dimensional incompressible viscous flow in x-direction along the wall & y normal to the wall

BASIC EQUATIONS (NAVIER STOKES EQUATION & CONTINUITY EQUATIONS)

BL On Flat Plate

BLASIUS SOLUTION

He showed that the boundary equations can be solved exactly for u & v assuming free stream velocity U as constant or ∂U/∂x =0

With an ingenious coordinate transformation, Blasius showed that the dimensionless velocity profile u/U is a function only of a single composite dimensionless variable

Momentum Integral Eqn.

A complete description of the boundary layer with the aid of non-linear differential equation is very cumbersome.

Therefore approximate solution is required.

Since boundary layer is satisfied in a stratum near the wall & near the region of transition. In the remaining region of the fluid, the mean over the differential equation is satisfied.

The mean is taken over the whole thickness of the boundary layer. Such a mean is obtained from the momentum equation, by integrating over the boundary layer thickness

Karman Pohlhausen Method For Flow Over Flat Plate

Outcome Of The Results Derived So Far

BOUNDARY LAYER THICKNESS INCREASES AS THE SQUARE ROOT OF THE DISTANCE x FROM THE LEADING EDGE AND INVERSELY AS SQUARE ROOT OF FREE STREAM VELOCITY.

WALL SHEAR STRESS IS INVERSELY PROPORTIONAL TO THE SQUARE ROOT OF x AND DIRECTLY PROPORTIONAL TO 3/2 POWER OF U

LOCAL & AVERAGE SKIN FRICTION VARY INVERSELY AS SQUARE ROOT OF BOTH x & U

Boundary Layer Separation

In adverse gradient, the second derivative of velocity is positive at the wall, yet it must be negative at the outer layer (y=δ) to merge smoothly with the mainstream flow U(x).

It follows that the second derivative must pass through zero somewhere in between, at the point of inflexion, and any boundary layer profile in an adverse must exhibit a characteristic S –shape.