Module-2 Boundary Layer€¦ · Module-5 Dimensional Analysis and Similitude 3 Darshan Institute of Engineering & Technology, Rajkot DEFINITIONS: 1. Laminar Boundary Layer: Consider

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Module-2

Boundary Layer

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98795 10743

Applied Fluid Mechanics (2160602)

13-02-2020

Prof. Mehul Pujara

2Module-5 Dimensional Analysis and Similitude Darshan Institute of Engineering & Technology, Rajkot

INTRODUCTION:

According to boundary layer theory,the flow of fluid in the neighborhood ofthe solid boundary may be divided intotwo regions.

1. A very thin layer of the fluid, calledthe boundary layer, in the immediateneighborhood of the solid boundary,where the variation of velocity fromzero at the solid boundary to free-stream velocity in the directionnormal to the boundary takes place.

2. The remaining fluid, which is outsidethe boundary layer. The velocityoutside the boundary layer isconstant and equal to free-streamvelocity.


DEFINITIONS:

1. Laminar Boundary Layer:

Consider the flow of a fluid, havingfree-stream velocity (U), over a smooththin plate which is flat and placedparallel to the direction for free streamof fluid as shown in Fig.

Plate is stationary and hence velocityof fluid on the surface of the plate iszero.

A velocity gradient is set up in the fluidnear the surface of the plate. Thisvelocity gradient develops shearresistance, which retards the fluid.

The boundary layer region begins atthe sharp leading edge.

At subsequent pointsdownstream the leading edge,the boundary layer regionincreases because the retardedfluid is further retarded. This isalso referred as the growth ofboundary layer.


DEFINITIONS:

1. Laminar Boundary Layer:

Near the leading edge of the surfaceof the plate, where the thickness issmall, the flow in the boundary layer islaminar though the main flow isturbulent. This layer of the fluid is saidto be laminar boundary layer. This isshown by AE in Fig.

The Reynold number is given by

x = Distance from leading edge,

U = Free-stream velocity of fluid,

v = Kinematic viscosity of fluid,

Reynold number equal to 5 x 105 for aplate.


DEFINITIONS:

2. Turbulent Boundary Layer:

The thickness of boundary layer willgo on increasing in the downstreamdirection.

Then the laminar boundary layerbecomes unstable and motion of fluidwithin it, is disturbed and irregularwhich leads to a transition fromlaminar to turbulent boundary layer.

This short length over which theboundary layer flow changes fromlaminar to turbulent is called transitionzone.


DEFINITIONS:

3. Laminar Sub-layer:

This is the region in the turbulentboundary layer zone, adjacent to thesolid surface of the plate as shown inFig.

In this zone, the velocity variation isinfluenced only by viscous effects.

The velocity gradient can beconsidered constant. Therefore, theshear stress in the laminar sub-layerwould be constant and equal to theboundary shear stress τ0.


DEFINITIONS:

4. Boundary Layer Thickness (δ):

It is defined as the distance from theboundary of the solid body measuredin the y-direction to the point, wherethe velocity of the fluid isapproximately equal to 0.99 times thefree stream velocity (U) of the fluid.

For laminar and turbulent zone it isdenoted as :

1. δlam = Thickness of laminar boundarylayer,

2. δtur = Thickness of turbulentboundary layer,

3. δ’ = Thickness of laminar sub-layer,


DEFINITIONS:

5. Displacement Thickness (δ *):

It is defined as the distance,measured perpendicular to theboundary of the solid body, by whichthe boundary should be displaced tocompensate for the reduction in flowrate on account of boundary layerformation.

At a distance x from the leading edgeconsider a section 1-1.

The velocity of fluid at B is zero and atC, which lies on the boundary layer, isU.

Thus velocity varies from zero at B toU at C, where BC is equal to thethickness of boundary layer DistanceBC = δ

Let

y = distance of elemental strip fromthe plate,

dy=thickness of the elemental strip,

u=velocity of fluid at the elementalstrip,

b = width of plate.


DEFINITIONS:


Then area of elemental strip,

dA = b x dy

Mass of fluid per second flowingthrough elemental strip

= p x Velocity x Area of elemental strip

= p x u x dA

=px u x b x dy ...(i)

Then mass of fluid per second flowingthrough elemental strip

= p x Velocity x Area

= p x U x b x dy ...(ii)

This reduction in mass/secflowing through elemental strip

= mass/sec given by equation (ii) -mass/sec given by equation (i)

= pUbdy – pubdy

= pb(U - u)dy.


DEFINITIONS:


Total reduction in mass of fluid/sflowing through BC due to plate

=

= ...(iii)

Loss of the mass of the fluid/secflowing through the distance δ *

= p x Velocity x Area

= p x U x δ * x b ...(iv)

Equating equation (iii) and (iv)


DEFINITIONS:

6. Momentum Thickness (θ):

Momentum thickness is defined as thedistance, measured perpendicular tothe boundary of the solid body, bywhich the boundary should bedisplaced to compensate for thereduction in momentum of the flowingfluid on account of boundary layerformation.

Consider the section 1-1 at a distancex from leading edge. Take anelemental strip at a distance y from theplate having thickness (dy).

Momentum of this fluid

= Mass x Velocity

= (pubdy)u

Momentum of this fluid in theabsence of boundary layer

= (pubdy)U

Loss of momentum throughelemental strip

= (pubdy)U - (pubdy) x u

= pbu(U - u)dy


DEFINITIONS:

6. Momentum Thickness (θ):

Total loss of momentum/sec throughBC

…(i)

Let θ = distance by which plate isdisplaced when the fluid is flowingwith a constant velocity U.

Loss of momentum/sec of fluidflowing through distance θ with avelocity U

= Mass of fluid through θ x velocity

= (p x area x velocity) x velocity

= [p x θ x b x U] x U

= p θ bU2 …(ii)

Equating equ. (i) and (ii)


Von Karman Momentum Integral Equation:

Consider the flow of a fluid having free-stream velocity equal to U, over athin plate as shown in Fig.

The drag force on the plate can be determined if the velocity profile near theplate is known.

Consider a small length ∆x of the plate at a distance of x from the leadingedge as shown in Fig. (a). The enlarged view of the small length of the plateis shown in Fig. (b).



The shear stress τ0 is given by,

Where is the velocity distribution near the

plate at y = 0.

Then drag force or shear force on a small distance∆x is given by,

∆ Fd = shear stress x area

= τ0x ∆x x b

where ∆ Fd = drag force on distance ∆x

Let u = velocity at any point within the boundary layer

b = width of plate



Then mass rate of flow entering through the side AD,

Mass rate of flow leaving the side BC,

+



Mass rate of flow entering DC

= mass rate of flow through BC - mass rate of flowthrough AD,

Now let us calculate momentum flux throughcontrol volume.

Momentum flux entering through AD,



Momentum flux entering through AD,

Momentum flux leaving the side BC,

Momentum flux entering the side DC ,

= mass rate through DC x velocity



Rate of change of momentum of the control volume,

= Momentum flux through BC - Momentum fluxthrough AD - momentum flux through DC

…(i)



The only external force acting on the control volumeis the shear force acting on the side AB in thedirection from B to A as shown in Fig. (b).

The value of this force is given by equation as,

∆ Fd = τ0x ∆x x b

Total external force in the direction of rate of changeof momentum,

= - τ0x ∆x x b …(ii)

Equating equation (i) and (ii)



Equating equation (i) and (ii)



The equation is equal to momentumthickness θ.

The above Equation is known as Von Karmanmomentum integral equation for boundary layer flows.


References:

1. Fluid Mechanics and Fluid Power Engineering by D.S. Kumar, S.K.Kataria & Sons

2. Fluid Mechanics and Hydraulic Machines by R.K. Bansal, LaxmiPublications

3. Fluid Mechanics and Hydraulic Machines by R.K. Rajput, S.Chand & Co

4. Fluid Mechanics; Fundamentals and Applications by John. M. CimbalaYunus A. Cengel, McGraw-Hill Publication

Module-2 Boundary Layer€¦ · Module-5 Dimensional Analysis and Similitude 3 Darshan Institute of Engineering & Technology, Rajkot DEFINITIONS: 1. Laminar Boundary Layer: Consider

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