Boundary Layer Theory

When a real fluid 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 fluid can not slip away from the boundary surface it attains the same velocity as that of the boundary. This is called no slip condition.

Boundary LayersAs a fluid flows over a body, the no-slip condition ensures that the fluid next to the boundary is subject to large shear. A pipe is enclosed, so the fluid is fully bounded, but in an open flow at what distance away from the boundary can we begin to ignore this shear?

There are three main definitions of boundary layer thickness:1. 99% thickness2. Displacement thickness3. Momentum thickness

Definition of a Fluida fluid, such as water or air, deforms continuously when acted on by shearing stresses of any magnitude. WaterOilAirWhy isnt steel a fluid?

Viscosity The viscosity is the property of fluid by virtue of which a fluid it offers resistance to deformation under the influence of a shear force.It offers resistance to the movement of one layer of fluid over another adjacent layer of fluid.

Fluid Deformation between Parallel PlatesSide viewForce F causes the top plate to have velocity U.What other parameters control how much force is required to get a desired velocity?Distance between plates (b)Area of plates (A)Viscosity!

Shear Stresschange in velocity with respect to distancedimension ofTangential force per unit areaRate of angular deformationrate of shear

Fluid classification by response to shear stressNewtonianIdeal FluidIdeal plasticShear stress Rate of deformation1

Newtonian and Non-Newtonian FluidThe viscosity of the non-Newtonian fluid is dependent on the velocity gradient as well as the condition of the fluid.

Newtonian Fluidsa linear relationship between shear stress and the velocity gradient (rate of shear),the slope is constantthe viscosity is constant

non-Newtonian fluidsslope of the curves for non-Newtonian fluids varies

If the gradient m is constant, the fluid is termed as Newtonian fluid. Otherwise, it is known as non-Newtonian fluid. Fig.1.5 shows several Newtonian and non-Newtonian fluids.

Fluid ViscosityExamples of highly viscous fluids______________________Fundamental mechanismsGases - transfer of molecular momentumViscosity __________ as temperature increases.Viscosity __________ as pressure increases.Liquids - cohesion and momentum transferViscosity decreases as temperature increases.Relatively independent of pressure (incompressible)molasses, tar, 20w-50 oilincreases_______increases

Role of ViscosityStaticsFluids at rest have no relative motion between layers of fluid and thus du/dy = 0Therefore the shear stress is _____ and is independent of the fluid viscosityFlowsFluid viscosity is very important when the fluid is moving zero

Dynamic and Kinematic ViscosityKinematic viscosity (__) is a fluid property obtained by dividing the dynamic viscosity (__) by the fluid density[m2/s]Connection to Reynolds number!

Thickness of Boundary LayerVelocity within the boundary layer increases from zero at the boundary surface to the velocity of the main stream asymptoticallyThat distance from the boundary surface in which the velocity reaches 99% of the velocity of the main stream.

Displacement Thickness The distance by which the boundary surface would have to be displaced outwards so that the actual discharge deficit would be same as that of an ideal (or frictionless) fluid past the displaced boundary.

Displacement thicknessThere is a reduction in the flow rate due to the presence of the boundary layerThis is equivalent to having a theoretical boundary layer with zero flowUUd

Displacement thicknessThe areas under each curve are defined as being equal:andEquating these gives the equation for the displacementthickness:

Momentum Thickness The distance from the actual boundary surface such that the momentum flux corresponding to the main stream velocity V through this distance is equal to the deficiency or loss in momentum due to the boundary layer formation.

Momentum thicknessIn the boundary layer, the fluid loses momentum, so imagining an equivalent layer of lost momentum:andEquating these gives the equation for the momentumthickness:

99% ThicknessUU is the free-stream velocity(x) is the boundary layer thickness when u(y) ==0.99U

Flat Plate: Parallel to FlowxyWhy is shear maximum at the leading edge of the plate?boundary layer thicknessshear

Factors Affecting the Thickness of BLBLT increases as the distance from the leading edge increasesBLT decreases with the increase in the velocity of flow of approaching stream fluidGreater is the kinematic viscosity, greater is the BLTConsiderably affected by the pressure gradient in the direction of flowIf pressure gradient is ve in the case of converging flow, boundary layer growth is retarded because the resulting pressure force acts in the direction of flow and it accelerates the retarded flow in the BL.

Boundary Layer ConceptsTwo flow regimesLaminar boundary layerTurbulent boundary layerwith laminar sub-layerCalculations ofboundary layer thicknessShear (as a function of location on the surface)Drag (by integrating the shear over the entire surface)

Force and momentum in fluid mechanics - refresherNewtons laws still apply. Consider a stream tube:u1,A1q1=u1A1u2,A2q2=u2A2mass entering in time, t, is u1A1tmomentum entering in time, t, is m1 = (u1A1t)u1momentum leaving in time, t, is m2 = (u2A2t)u2Impulse = momentum change, F = (m2 m1)/ t = (u22A2-u12A1)

The von Karman Integral Equation (VKI)Boundary LayerABCDFlow enters on AB and BC, and leaves on CD122 - 1xUu1(y)u2(y)

VKIThe momentum change between entering and leaving the control volume is equal to the shear force on the surface:(CD)(AB)(BC)By conservation of fluid mass, any fluid entering the control volume must also leave, therefore Force on fluid

VKIAs x 0, the two integrals on the right become closer andthe equation may be written as a differential:The integral is the definition of the momentum thickness, soif V(x)

Laminar boundary layer growth + d dyBoundary layer => Inertia is of the same magnitude as Viscositya) Inertia Force: a particle entering the b.l. will be slowed from a velocity U to near zero in time, t. giving force FI V/t. But u=x/t => t l/V where V is the characteristic velocity and l the characteristic length in the x direction.Hence FI V2/l b) Viscous force: F /y 2u/y2 V/2since V is the characteristic velocity and the characteristic length in the y direction(x)

Laminar boundary layer growthComparing these gives:So the boundary layer grows according to Alternatively, dividing through by l, the non-dimensionalisedboundary layer growth is given by:Note the new Reynolds numbercharacteristic velocity and characteristic lengthV2/l VU/2

Boundary layer growth

Length Reynolds Number

Flow at a pipe entrylUdIf the b.l. meet while the flow is still laminar the flow in the pipe will be laminarIf the b.l. goes turbulent before they meet, then the flow in the pipe will be turbulent

Length Reynolds number and Pipe Reynolds numberThe critical Reynolds number for flow along a surface is Rl=3.2*105In a pipe, the Reynolds number is given byConsidering a pipe as two boundary layers meeting, d=2a=2and from above The mean velocity in the pipe, um, is comparable to the free-stream velocity, UIf Rl=3.2*105 then Re=5657

Boundary layer equations for laminar flowThese may be derived by solving the Navier-Stokes equationsin 2d.ContinuityMomentumUAssume:1. The b.l. is very thin compared to the length2. Steady state

Boundary layer equations for laminar flowThis gives Prandtls b.l. equation:rate of change of u with x is small compared to yBlasius produced a perfect solution of these equations validfor 0

Blasius Solution

Chart2

0

1

2

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7

y'

u/U

y'

Sheet1

f' (u/U)y'

00

0.329791

0.629772

0.846053

0.955524

0.991555

0.998986

0.999927

Sheet2

Sheet3

Chart2

0

1

2

3

4

5

6

7

y'

u/U

y'

Sheet1

f' (u/U)y'f' (or u/U)f''

0000.332

0.3297910.3300.332

0.6297720.6300.323

0.8460530.8460.267

0.9555240.9560.161

0.9915550.9920.064

0.9989860.9990.002

0.9999271.0000.000

(after Howarth)

Sheet2

Sheet3

Laminar skin frictionThe shear stress at the surface can be found by evaluatingthe velocity gradient at the surfaceThe friction drag force along the surface is then found byintegrating over the lengthwhere b is the breadth of the surface

Laminar skin frictionFrom the Balsius solution, the gradient of the velocityprofile at y=0 yields the result:The shear force can be obtained by integration along the surfaceThe frictional drag coefficient can then be calculated

Turbulent boundary layersTurbulent Boundary layers are usually thicker than laminar ones.Velocity distribution in a turbulent boundary layer is much more uniform than that in a laminar boundary layerLarge velocity change occur in a relatively small vertical distanceVelocity gradient is steeper in a turbulent boundary layer

Turbulent boundary layersThe assumption is made that the flat plate approximates to the behaviour in a pipe. The free stream velocity, V, corresponds to the velocity at the centre, and the boundary layer thickness, , corresponds to the radius, R.1/7 Power LawFrom experiments, one possibility for the shape of the boundary layer profile is and measurements of the shear profile give

Turbulent boundary layersPutting the expression for the 1/7 power law into the equations for displacement and momentum thickness=99%dm

Turbulent boundary layersbecomesEquating this to the experimental value of shear stress:Integrating gives:The turbulent boundary grows as x4/5, faster than the laminar boundary layer.

Turbulent boundary layersMomentum thicknessTo find the total force, first find the shear stressthen integrate over the plate lengthFor a plate of length, l, and width b,

Logarithmic boundary layerFrom the mixing length hypothesis it can be shown thatthe profile is logarithmic, but the experimental valuesare different from those in a pipeand the friction coefficient(A is a correction constant if part of the b.l. is laminar)

Problem 1A plate 0.50m X 0.20m has been placed longitudinal in a stream of crude oil which flows with undisturbed velocity of 6.0 m/sec. Given that oil has a specific gravity of 0.9 and kinematic viscosity of 1 stoke, calculate the boundary layer thickness and shear stress at the middle of plate. Also calculate friction drag on one side of the plate.

Problem 3A rectangular plate of length a and width b is towed lengthwise through water with velocity Va and subsequently widthwise with velocity Vb. The boundary layer is laminar and the plate experiences equal drag in both the cases. Determine the ratio of velocities Va and Vb in terms of dimensions of the plate.

Problem 4A smooth rectangular plate 1.25 m wide x 25 m long moves through water in the direction of its length. The drag force on the two sides of the plate is estimated to be 8500 N. Workout (i) velocity with which the plate moves, (ii) boundary layer at the trailing edge of the plate and (iii) whether boundary layer changes from laminar to turbulent? If so determine distance at which the laminar boundary layer existing at the leading edge transforms into turbulent boundary layer. Assume Cf = 0.0018

Problem 5Water flows down a smooth wide concrete apron into a river. Assuming that a turbulent boundary layer forms, estimate shear stress and the boundary layer thickness 50 m downstream of the entrance to the apron. Use following data:Main stream velocity V = 7.0 m/secDynamic viscosity

Velocity distribution is given by

Shear stress at the wall

Problem 6Air moves over a flat plate with a uniform free stream velocity of 10 m/sec. At a position 15 cm away from the front edge of the plate, what is the boundary layer thickness? Use a parabolic profile in the boundary layer. For air, take kinematic viscosity 1.5 x 10-5 m2/sec and density of air 1.23 kg/m3.

Problem 7A wind tunnel has a cross section at its inlet of 1.0 m by 1.0 m and a length of 10.0 m. Wind at uniform velocity of 15 m/sec enters the tunnel at 20 oC. Determine the cross sectional dimensions at the end of the test section which will yield zero pressure gradient along its length. Assume velocity distribution in turbulent boundary layer to follow the law and m2/sec

Problem 8A plate 3.0 m X 1.5 m is held in water moving at 1.25 m/sec parallel to its length. If the flow in the boundary layer is laminar at the leading edge of the plate, determineWhether the flow changes from laminar to turbulent? If so, calculate the distance from the leading edge where the boundary layer flow changes from laminar to turbulent flow.The thickness of the boundary layer at this section, and The frictional drag on the plate considering both its sides. (Assume that constant A is not known).

Problem 9A small low-speed wind tunnel is being designed for calibration of hot wires. The air is at 190C. The test section of the wind tunnel is 30 cm in diameter and 30 cm in length. The flow through the test section must be as uniform as possible. The wind tunnel speed ranges from 1 to 8 m/s, and the design is to be optimized for an air speed of V = 4.0 m/s through the test section. (a) For the case of nearly uniform flow at 4.0 m/s at the test section inlet, by how much will the centerline air speed accelerate by the end of the test section? (b) Recommend a design that will lead to a more uniform test section flow.

ProblemBelow are given the wind tunnel measurement of velocity at different elevations from the boundary. What is the free stream velocity? Determine the thickness of nominal boundary. Also determine the exponent n in the equation.

Problem 8If the velocity in laminar boundary layer over a flat plate is assumed to be given by the second order polynomial, determine its form using the necessary boundary conditions. Hence calculate displacement and momentum thickness.

Quadratic approximation to the laminar boundary layer

Chart2

0

1

2

3

4

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7

y'

u/U

y'

Chart1

00

0.20.2

0.40.4

0.60.6

0.80.8

11

1.21.2

1.41.4

Blasius (exact)

Quadratic

u/U

y/d

Sheet1

f' (u/U)y'f' (or u/U)f'' (or du/dy)

00000

0.3297910.3300.332

0.6297720.6300.323

0.8460530.8460.267

0.9555240.9560.161

0.9915550.9920.064

0.9989860.9990.002

0.9999271.0000.000

(after Howarth)

F(quadratic)u/U (Blasius)y/d

000

0.360.329790.2

0.640.629770.4

0.840.846050.6

0.960.955520.8

10.991551

0.960.998981.2

0.840.999921.4

Sheet2

Sheet3

Chart1 (2)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Quadratic

Chart1 (3)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Blasius (exact)

u/U

y/d

Chart2

0

1

2

3

4

5

6

7

y'

u/U

y'

Chart1

00

0.20.2

0.40.4

0.60.6

0.80.8

11

1.21.2

1.41.4

Blasius (exact)

Quadratic

u/U

y/d

Sheet1

f' (u/U)y'f' (or u/U)f'' (or du/dy)

00000

0.3297910.3300.332

0.6297720.6300.323

0.8460530.8460.267

0.9555240.9560.161

0.9915550.9920.064

0.9989860.9990.002

0.9999271.0000.000

(after Howarth)

F(quadratic)u/U (Blasius)y/d

000

0.360.329790.2

0.640.629770.4

0.840.846050.6

0.960.955520.8

10.991551

0.960.998981.2

0.840.999921.4

Sheet2

Sheet3

Chart1 (2)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Quadratic

Chart1 (3)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Blasius (exact)

u/U

y/d

Quadratic approximation to the laminar boundary layerRemember - boundary layer theory is only applicable insidethe boundary layer.This is sometimes written with =y/ and F()=u/U asIt provides a good approximation to the shape of the laminar boundary layer and to the shear stress at the surface

Turbulent Boundary Layer

Laminar Sub-Layer

what is this?