Chapter 7 (Boundary Layer)

Viscous Flow and Boundary Layer

What is viscous Flow?A flow where the effects of the viscosity, thermal

conduction, and mass diffusion are very important.

Influence of ViscosityConsider two solid surface slipping over each other (Ex.

Your book being pushed a cross a table)

There will be a FRICTIONAL FORCE between these objects, which will retard their relative motion.

Viscosity

The quantity μ is a property of the fluid and depends to a great extent on its temperature.

It is a measure of the viscosity of the fluid. The law of friction given by

This law is known as Newton’s law of friction; this law can be regarded as the definition of viscosity.

dyduμτ =

Cont.

The dimensions of viscosity

Dimension of shearing stress :

Dimension of velocity gradient:

From the law of friction, the dimension of the viscosity

2 2.N kg Pam m s

τ → = ≡

1du Secdy−→

.sec.seckg Pa

mμ ⎡ ⎤→ =⎢ ⎥⎣ ⎦

Kinematics viscosity ( )

In all fluid motions in which frictional and inertia force interact

It is important to consider the ratio of the viscosity, μ , to the density, ρ, known as the kinematics viscosity, and denote by

υ

υ

ρμ

=v

Flow of a Fluid over a Solid Surface

The influence of friction between the surface and the fluid adjacent to the surface acts to create a frictional force, which retards the relative motion

Test Cases for Velocity Profile

Laminar BL Velocity Profile

Fully Turbulent BL Velocity ProfileMixing BL Velocity Profile

Cont.

The influence of friction is to create V=0, at the body surface, this is called the NO-SLIP condition which dominant viscous flow.

In any real continuum fluid over a solid surface, the flow velocity is ZERO at the surface. Just above the surface, the Velocity is FINITE.

Development of the Boundary LayerInner and Outer Region

Prandtl (1904): He suggested that fluid motion around object, divided into two regions: Inner Region and Outer Region

Inner Region: A thin region close to the object, where

frictional effects are important

Inner RegionThe inner region is called BOUNDARY LAYER

In this thin boundary layer the velocity of the fluid increases from zero at wall (NO SLIP) to its full value, which corresponds to external frictionless flow.

This B-L will grows from zero thickness at the upstream edge of the body.

Evidently the thickness of the B-L decreases with decreasing Viscosity

Inner region (Cont.)

Even with very small viscosities (Large Reynolds number) the frictional shear stresses, in the B-L are considerable because of the large velocity gradient,

across the flow, whereas outside the B-L they are very small

( )uyτ μ ∂= ∂

τ uy∂∂

Inner region definition

The thickness of the B-L, , grows along a surface.

The flow within B-L begins as laminar flow, but as the layer grows along a surface a transitionregion occurs and the flow in the B-L may become turbulent if the surface is long enough.

Definition: B-L defined as that region where the fluid velocity (parallel to the surface) is less than 99% of the free stream velocity.

Velocity distribution in the B-L

Outer Region

Outer Region: A region, where friction effects may be neglected

Velocity Profile

Lets consider a small element of fluid of unit depth normal to the flow plane, having a unit length in the direction of motion and a thicknessnormal to the flow direction. Where

Shear acting on AB:

Shear stress acting on CD:Streamwise pressure acting on AC: PStreamwise pressure acting on BD:

yδ

yu∂∂

= μτ

yyδττ ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+

xxpp δ⎟⎠⎞

⎜⎝⎛∂∂

+

Cont.

Resulting Force

Assuming u increasing with y, the resulting force can be calculate as

Shear force acting on fluid element:The resulting shearing force in the x-direction:

shear force can be defined as

Net shear force on the element:

yy

yy

δττδττ ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=−⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=yuμτ

yyu δμ ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

2

2

Cont.

Pressure force acting on fluid element

Note: The total thickness of the B-L is very small; the

pressure hardly varies at all normal to the surface. Consequently the net transverse pressure force is zero to be good approximation

xxpx

xppp δδ ⎟

⎠⎞

⎜⎝⎛∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛∂∂

+−

0=∂∂

xp

Velocity Profile (u vs y)

Fact

When becomes very small, the shear stressbecomes negligible, the small gradient exist out to y=∞. Large will result a large shearing stress.

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

yu

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

yu

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=yuμτ

Boundary Layer Thickness and Non-dimensional velocity profile

Definition: B-L thickness (δ) is the region where the fluid velocity is less than 99% of stream velocity.

Non-dimensional velocity profileIn order to compare B-L profile of different

thickness, it is convenient to express the profile shape non-dimensional

Non-dimensional velocity profile

Consider the following

and

The profile shape is given as over the range of y=0 to y=δ

varies from 0 to 0.99

1Uuu = δ

yy =

( )yfu =

u

Types of B-LThere two types of boundary layer: Laminar and turbulent B-L.

Laminar B-L :

In laminar flow the layers of the fluid slide smoothly over one another and there is little interchange of fluid massbetween adjacent layer.

The shearing tractions which develop due the velocity gradients are thus due entirely to the viscosity of the fluid.

Mechanism of Laminar B-L

In laminar B-L, energy from the mainstream is transmitted toward the slower moving fluid near the surface through the medium of viscosity aloneand only a relatively small penetration results.

Consequently an appreciable proportion of the boundary layer flow has a considerably reduce velocity. Throughout the layer, the shearing stress

is given by and the wall shearing stress is

⎟⎠⎞⎜

⎝⎛

∂∂= yuμττ

wyw y

uy

u ⎟⎠⎞⎜

⎝⎛

∂∂=⎟

⎠⎞⎜

⎝⎛

∂∂=

=μμτ

0

Laminar B-L

Laminar B-L (Cont.)

In laminar sub-layer, the shearing action becomes purely viscous and the velocity falls very sharplyand almost linear, with it, to zero at the surface.

Since, at the surface, the wall shearing stress now depends on viscosity only

ww y

u ⎟⎠⎞⎜

⎝⎛

∂∂= μτ

Turbulent B-L

Turbulent flow considerable seemingly random motion exists, in the form of velocity fluctuations both along the mean direction of flow and perpendicular to it.

As a result a of the later there are appreciable transports of mass between adjacent layers. If there is a mean velocity gradient in the flow, then there will be corresponding interchanges of streamwise momentumbetween the adjacent layers which will result in shearing stresses between them.

Turbulent B-L (Cont.)

These shearing stresses may well be of much greater magnitude than those develop as the result of purely viscous action.The velocity profile shape in a turbulent layer is very largely controlled by these Reynolds Stresses, as they are termed. A large Reynoldsstresses are set up due to mass interchanges in a direction perpendicular to the surface, so that energy from the mainstream may easily penetrate to fluid layers quite close to the surface.

Turbulent B-L

This results in these layers having a velocity, which is not much less than that of the mainstream.

However, in layers which are very closed to the surface it is obviously impossible for velocities to exist perpendicular to the surface, so that in a very limited region immediately adjacent to surface, the flow approximates to laminar flow.

Turbulent B-L (Cont.)

It will be clear that the surface friction stress under a turbulent layer will be far greater than under a laminar layer of the same thickness, Since is much greater.

wyu ⎟

⎠⎞⎜

⎝⎛

∂∂

Important Notes

It should be noted, however, that the viscous shear stress relation is only employed in the laminar sub-layer very close to the surface and not throughout the turbulent B-L

The viscous shearing stress at the surface, and thus the surface friction stress, depends only on the slope of the velocity at the surface, whatever the B-L type.

Growth of the B-L along a flat plate

Laminar layer starts to develop from the leading edge, which grows in thickness from zero at the leading edge to some point on the surface where a rapid transition to a turbulent occurs.

Cont.Transition is accompanied by a corresponding rapid thickening of the layer, beyond this transition region, the turbulent layer is exist.

(Transition occurs because of the growth of small disturbance in the B-L)

Turbulent layer continues to thicken steadily as it proceeds towards the trailing edge.

Because of the greater shear stresses within the turbulent layer its thickness much more rapidly than that of the laminar layer, because the momentum defect near the surface is more readily able to influence the mainstream flow at the outer edge of the B-L.

Cont.At trailing edge, the B-L joins with that from the other surface (lower surface of the flat plat) to form a WAKE of retarded velocity also tends to thicken slowly as it flows away downstream.

Surface of Wings or Fuselages

In most aerodynamic problems, the surface is usually that of a streamline from such as a wing or fuselage.

The major difference, affecting the B-L flow in these cases, is that the mainstream velocity and hence the static pressure in a streamwise direction is no longer constant.

Effects of an external pressure gradient

In most aerodynamics applications the mainstream velocity and pressure change in the streamwise direction.

Previously(S:16, 17), the force acting on a small fluid element within B-L is

xxpy

yδδτ

∂∂

−∂∂

Favorable pressure gradient

When the pressure decreases along the surface and velocity along the edge of the B-L increases. the external pressure gradient is said to be favorable. This is because the streamwisepressure forces help to counter the effects of the shearing action and shear stress at the wall

Unfavorable or Adverse Pressure gradient

When the pressure increases and mainstream velocity deceases along the surface, the external pressure gradient is said to be unfavorable or adverse.

This is because the pressure forces reinforce the effects of the shearing action and shear force at the wall

Boundary layer aerodynamic shaped bodies

the flow decelerates more markedly near the walland the B-L grows more rapidly than in the case of the flat plate

The velocity profile is much less full than for a flat plate and develops a point of inflexion.

The adverse pressure gradient is sufficiently strong, the flow near the wall is so greatly decelerated that it begins to reverse direction

Cont.

Flow reversal indicates that the B-L has been separated from the surface

Stall

For airfoil at large angle of attack, the separation may take place not far downstream of the maximum suction point, and a very large wake will develop.

This will cause such a redistribution of the flow over the aerofoil that the large area of low pressure near the upper surface leading edge is seriously reduced, with the result that the lift force is also greatly reduced.

This condition is referred to as the stall

Laminar-Turbulent Transition

The effects of Reynolds number on the transition distance is depends on many parameter such as the following

Pressure gradientSurface roughnessCompressibility effects (related to Mach number)Surface Temperature.Suction or blowing at surfaceFree stream turbulence.

Laminar-Turbulent Transition (Cont.)

Laminar-Turbulent Transition (Cont.)

For incompressible flow past a flat plate, typical transition criterion is

The location for the onset of B-L transitionwould occur at

000,500Re , =trx

μρ e

trxtr u

x ,Re=

The relation between Reynolds number and transition distance

The relation may be summarized as follows:

ForTransition will occur very shortly downstream of the point of minimum

pressure.

If for an airfoil is kept constant, increasing the angle of attack advances the point of minimum pressure to move towards the leading edge on the upper surface, causing transition to move forward The opposite occurs on the lower surface

At constant angle of attack an increase in tends to advance transition

75 10Re10 << L

LRe

LRe

Cont.

For

The transition point may slightly precede the point of minimum pressure

The effect of external pressure gradient on transition also explains how it may be postponed by designing aerofoil with points of minimum pressure further aft.

710Re >L

Transition Once the critical Re is exceeds, The B-L would contains regions with the following

characteristics as it transitioned from the laminar state to a fully turbulent flow:

Stable, laminar flow near leading edge.

Unstable flow containing 2-D Tollmien –Schlichting (T-S) waves.

A region where 3-D unstable waves and hairpin eddies develop.

A region where vortex breakdown produces locally high shear.

Fluctuating, 3-D flow due to cascading vortex breakdown.

A region where turbulent spots form.

Fully turbulent flow.

Cont.

Airfoil at large angle of attack

For airfoil at large angle of attack, the separationmay take place not far downstream of the maximum suction point, and a very large wakewill develop.

This will cause such a redistribution of the flow over the airfoil that the large area of low pressure near the upper surface leading edge is seriously reduced, with the result that the lift force is also greatly reduced. This condition is referred to as the stall.

Cont.

B-L Separation and Vortex Formation

The phenomenon of B-L separation is intimately connected with the pressure distribution in the B-L. In the B-L on a plate NO SEPARATION TAKES PLACE AS NO BACK FLOW OCCURS.

In order to explain the very important phenomenon of B-L separation, let us consider the flow around blunt body (e.g. Circular cylinder).

Separation of the flow around blunt body

Cont.

In frictionless flow

The fluid particles are accelerated on the up stream half from D to E, and decelerated on the downstream half from E to F.

When the flow is started up the motion in the first instant is very nearly frictionless, and remains so as long as the B-L remains thin.

Out side the B-L:

There is a transformation of pressure into kinetic energy along DE, the reversetaking place along EF, so that a particle arrives at F with the same velocity as it had at D.

Cont.

A fluid particle, which moves in the immediate vicinity of the wall in the B-L, remains under the influence of the same pressure field as that that existing outside, because the external pressure is impressed on the B-L.

From E to F, a particle cannot move far into the region of increasing pressure cases them to move in the opposite direction.

Cont.

At a point S on the surface, the profile slopebecomes zero

Downstream of point S, the flow adjacent to the surface may well be in an upstream direction, so that a circulatory movement, in a plane normal to the surface, may take place near the surface.

wyu )( ∂∂

Cont.

Cont.

Cont.

Flow past Cylinders and Spheres(Bluff Bodies )

The flow pattern around a bluff body can change dramatically as the Reynolds number is varied.

An example: flow past a circular cylinder and sphere.

Flow past a circular Cylinder: Very low Reynolds number (less than unity)

The flow behaves as if it were purely viscous with negligible inertia. Such flow is known as CREEPING OR STOKES FLOW.

For such flows there are NO B-L and the effects of viscosity extend an infinite distance from the body.

The streamline is completely symmetrical fore and aft.

The streamline pattern is similar to that potential flow.

Skin friction drag is the only force generated by the fluid flow on the cylinder.

Consequently, the body with the lowest drag for a fixed volume is the sphere.

Flow past a circular Cylinder: 1<Re<5

The streamline pattern remains fairly similar to creeping flow

As Re is increased within this range a more and more pronounced asymmetry develops between the fore and aft direction.

Nevertheless the flow remains attached.

Flow past a circular Cylinder: Re exceeds a value of about 5

A much more profound change in the flow pattern occurs.

The flow separates from the cylinder surface to form a closed wake of recirculating flow.

The wake grows progressively in the length as Re is increased from 5 to 41.

The flow pattern is symmetrical about the horizontal axis and issteady.

The effects of viscosity still extend a considerable distance from the surface, so it is not valid to use the concept of the B-L.

Flow past a circular Cylinder: Re exceeds a value of about 41

The flow becomes unsteady. In the same respects what happen is similar to the early stage of laminar-turbulent transition.

In this case flow, the disturbances develop as vorticesrather than waves, but that small disturbances do not develop into turbulent flow.

In other words, in this range of Reynolds number steady laminar wake develops into an unsteady, but stable laminar wake. In this way a row of vortices are formed, similar to that shown in Fig. C, except that the vortices are formed further downstream in the wake.

Flow past a circular Cylinder: Re rises from 40 to 100

An identifiable thin layer begins to form on the cylinder surface and the disturbance develops increasingly closer to the cylinder.

Flow past a circular Cylinder: Re = 100 and above (<200)

Eddies are shed alternately from the laminar separation points on either side of the cylinder

Thus, vortex will be generated in the region behind the separation point on one side, while a corresponding vortex on the other side will break away from the cylinder and move downstream in the wake.

When the attached vortex reaches a particular strength, it will in turn break away and new vortex will begin to develop again on the second side and so on

Flow past a circular Cylinder: Re = 100 and above (<200)

The wake thus consists of a procession of equal strength vortices, equally spaced but alternating in sign.

This type of wake, which can occur behind all long cylinders of bluff cross section, including flat plates normal to the flow direction, is termed a Von Karmanvortex street or trail (Fig. 6.15a).

In a uniform stream flowing past a cylinder the vortices move downstream at a speed less than the free stream velocity.The reduction in speed being inversely proportional to the streamwise distance separating alternate vortices.

Vortex Shedding

Flow past a circular Cylinder: Vortex Shedding Re = 100 and above


During the formation of any single vortex, an increasing circulation will exist about the cylinder with consequent generation of a lift force.

With the development of vortex, this force will change sign, giving rise to an alternating liftforce on the cylinder at the same frequency as the vortex shedding oscillation of the cylinder.


If the frequency happens to coincide with natural frequency of oscillation of the cylinder, then appreciable vibration may be caused.

This phenomenon can have important consequences in engineering application.

A dramatic example was the failure of the Tacoma Narrows Bridge (Washington state, U.S.A) in 1940. In this case the natural frequency of the bridge deck was close to its shedding frequency in a brisk wind, thereby leading to resonanceresonance and disastrous consequences.

Flow past a circular Cylinder: Shedding frequency Re = 100 and above

A unique relationship is found to exist between Re and Strouhal number:

Definition: Strouhal number is dimensionless parameter involving the shedding frequency.

It is defined by where n is the frequency of vortex shedding (fig 6./15b)

∞= U

nDS

Flow past a circular Cylinder: Shedding frequency

Flow past a circular Cylinder: 200<Re<400 and 3X105<Re<3X106

The regularity of vortex shedding is greatly diminished.

Re≈200: The vortex street persist to great distance downstream. Above this Re, transition to turbulent flow occurs in the wake.

At this Re the vortex street also becomes unstable to 3-D disturbance leading to greater irregularity.


Re≈400:

A further change occurs. Transition to turbulence now occurs close to the separation points on the cylinder. This has a stabilizing effect on the shedding frequency.

This pattern with laminar B-L separation and turbulent vortex wake persists until Re ≈ 3X105 (Fig. 6.14 d).

With laminar separation, the flow separates at points on the front half of the cylinder, thereby forming a large wake and producing a high level of form drag.


When Re reaches value of 3X105:

The laminar B-L undergoes transition to turbulence almost immediately after separation.

The increased mixing re-energize the separated flow casing it to reattach as a turbulent B-L, thereby forming a separation bubble

Cont.At this critical stage, the second and final point of separation, which now takes place in a turbulent layer, moves suddenly downstream, because of the better sticking property of the turbulent layer, and the wake width is very appreciably decreased.

This stage is therefore accompanied by a sudden decrease in the total drag of the cylinder. For this reason , the value of Re at which this transition in flow occurs is called the CRITICAL REYNOLDS NUMBER.

The wake vorticity remains random with no calculable frequency


With further increases in Re (Re>3X105 )The wake width will gradually increase to begin with, as separation points slowly move upstreamround the rear surface.

The total drag continues to increase steadily in this stage due to both pressure and surface friction.

DUspan

dragCD 2

21

∞∞

=ρ

Flow past a circular Cylinder: Drag for Re>1.3X106

CD tends become constant, at about 0.6

Flow past a circular Cylinder: Re ≈3X106

The separation bubble disappears, (Fig f).

This transition has a stabilizing effect on the shedding frequency, which becomes discernable again.

Flow past a circular Cylinder: Re >3X106

Flow behaves similar to Re ≈3X106, but changes little.

Circular Cylinder: Critical Re

The value of critical Re depends on

Smoothness of cylinder surface.

Turbulence level in the incoming free stream.

Increased turbulence, or increased surface roughness, will provoke turbulent reattachment, with its accompanying drag decrease, at lower Re

Flow past a sphere

The behavior of a smooth sphere under similarly varying conditions exhibits the same characteristics as the cylinder, although the Re corresponding to the changes of the flow regime are somewhat different.

The eddying vortex street, typical of bluff cylinder, dose not develops in so regular a fashion behind a sphere. (Fig 16.6, S78)

Application of turbulent B-L (B-L Control )

As we mentioned above that, Increased turbulence, or increased surface roughness, will provoke turbulent reattachment, with its accompanying drag decrease, at lower Re.

This advantage, used in many applications: GOLF BALL AND CRICKET BALLS.

Golf Balls: B-L Control for less drag

The golf balls were manufactured with a dimpled surface to simulate the worn surface (rough). WHY?

The diameter of a golf ball is 42mm, if the ball surface is smooth, Red=3.85x105 is need to start the transition region from laminar to turbulent where the drag is less.

This Re gives a critical velocity in air for smooth ball, just over 135 m/s. This is much higher than average flight path of a driven ball. So, the flow around the ball will belaminar which causing high CD and so experience as high decelerating force throughout, with consequent decrease in range

Golf Balls: B-L Control for less drag

With a rough surface, Precipitating early transition, the critical Re may as low as 105 , which gives a critical speed of a golf ball about 35m/s, which is well below the flight speed.

The ball travels at low CD and so experience as smaller decelerating force throughout, with consequent increase in range.

Cricket Balls: B-L Control to swingthe ball

The art of the seam bowler in cricket is explainable with reference to boundary layer transition and separation. The bowling technique is to align the seam at a small angle to the flight path.

This is done by spinning the ball about an axis perpendicular to the plane of the seam, and using the gyroscopic inertia to stabilize this seam position during the trajectory.

On the side of the front stagnation point where the B-L passes over the seam, it is induced to become turbulent before reaching the point of laminar separation

Cont.On this side, the B-L sticks to a greater angle from stagnation than dose that on the other side where no seam is present to trip the B-L.

The flow past the ball thus becomes asymmetric with larger area of low pressure on the turbulent side, producing a lateral force tending to move the ball in a direction normal to its flight path.

The rang of flight speeds through which phenomenon can be utilized corresponds to those of the medium to medium fast pace bowler.

Cont.

The diameter of cricket ball is about 71 to 72.5mm , If the ball is smooth, the critical speed would be about 75 m/s. The transition to turbulence for the seam-free side occurs at speed in the region of 30 to 35 m/s.

The critical speed for a rough ball with early transition (Re≈105) is about 20 m/s and below this speed the flow asymmetry tends to disappear because laminar separation occurs before the transition, even on the seam side.

Cont.

The speed rang from 20 to 30 m/s, very approximately, the ball may be made to swing by the skilful bowler.

The very fast bowler will produce a flight speed in excess of the upper critical and no swing will be possible.

It is obvious that considerable skill and experience is required to know at just what speed the delivery must be made to do this.

Cont.

Properties of B-LDisplacement thickness (δ*)

Definition: It is the distance that the wall would have to be displaced outward into the free stream in order not to change the flow field, if the fluid were completely inviscid and there were no B-L.

Consider the flow past a flat plate of infinite span held parallel to uniform oncoming stream. Due to the buildup of the B-L on the plate surface a stream tube, which, at the leading edge, is close to the surface, will become entrained in the B-L flow.

Displacement thickness (δ*)


The solid surface had been displaced a small distance into the stream.

The amount by which the surface would be displaced under such conditions is termed the B-L displacement thickness (δ*) and may be calculate as follow,

a.1 -Velocity profile must be known

)(yfu =


a.2 - Mass flow rate in mainstream (mass flux) through the thickness,δ

a.3 - Mass flow rate in mainstream (mass flux) through the thickness δ*

δρδ 11Um =

∗=∗ δρδ

um


a.4 - From a.2 and a.3, the mass flow rate in mainstream through the thickness, δ-δ* is given as

a.5 – From the figure: With the B-L, O‘ABR area is equivalent to area OPQR , equation a.4 can written as

( )∫ −=− ∗

δ

δδ ρρ0 11 dyuUmm

( ) ∗=−∫ δρρρδ

110 11 UdyuU


a.6 - From a.5, the displacement thickness δ* is defined as

OR

dyUu

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−=∗ δ

ρρ

δ0

11

1

ydUu

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

∗ 1

011

1ρρ

δδ


a.7 - For Incompressible flow, density will be constant and Equation a.6 reduces to

( ) yduydUu

∫∫ −=⎟⎟⎠

⎞⎜⎜⎝

⎛−=

∗ 1

0

1

01

11δδ

Properties of B-LMomentum thickness (θ)

Definition: The momentum thickness has same definition as displacement, except that it is based on momentum flux instead of mass flux.

For the typical stream tube within B-L as in Fig. (S 92)

b.1 - Mass flow rate with the stream tube is ρu.

Momentum thickness (θ)b.2 - The rate of momentum defect relative to

mainstream is

b.3 - The rate of momentum defect for thickness θ is given by

b.4 - By using Equivalent area, we can obtain from b.2 and b.3

( ) yuUu δρ −1

θρ 211U

( ) θρρδ 2

1110UdyuUu =−∫

Momentum thickness (θ)

b.5 - From b.4,

Or

For incompressible case, where the density is constant,. Eq. b.5 can be rewritten as

dyUu

Uu

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

δ

ρρ

θ0

111

1

ydUu

Uu

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

1

0111

1ρρ

δθ

( )∫∫ −=⎟⎟⎠

⎞⎜⎜⎝

⎛−=

1

0

1

011

11 yduuydUu

Uu

δθ

Momentum thickness (θ)

b.6 – The momentum thickness concept is conveniently used in transition problem and calculation of skin friction losses.

Properties of B-LKinetic energy thickness (δ**)

Definition: The kinetic energy thickness (δ**) has same definition as momentum thickness and displacement, except that it is based on kinetic energies of the fluid instead of momentum or mass flux.

c.1 - The rate of kinetic energy defect within B-L at any station x is given by the difference between the energy which the element would have at main stream velocity U1 and which it actually has at velocity u being equal to:

( )∫ −δ

ρ0

2212

1 dyuUu

Kinetic energy thickness (δ**)

c.2 - Rate of kinetic energy defect in the thickness δ** is

c.3 - By using equivalent area, Eq. c.1 and Eq. c.2 Can give the following relation

∗∗δρ 3112

1 U

( ) =−∫δρ

0

221 dyuUu ∗∗δρ 3

11U


c.4 - From Eq. c.3, The kinetic energy displacement can be written as,

or

∫ ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=∗∗ δ

ρρ

δ0

2

111

1 dyUu

Uu

∫ ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=

∗∗ 1

0

2

111

1 ydUu

Uu

ρρ

δδ


c.5 - For Incompressible Case (constant density, ρ ), Eq. c.4 can be rewritten as follows:

Note: The integrals on the RHS in Eqs. A.6, b.5 and c. 5

are simply numbers which may be evaluated if the velocity profile is known.

( )∫∫ −=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=

∗∗ 1

0

21

0

2

11

ˆ11 yduuydUu

Uu

δδ

Properties of B-LSurface Friction Drag

Shear stress between adjacent layers of fluid in a laminar flow:

Shear stress at the wall, due to the presence of the B-L:

Where is the wall shear stress or surface friction stress.

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=yuμτ

www y

u⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

= μτ

wτ

Surface Friction Drag

Once the velocity profile of the B-L is known, then the surface or skin friction can be calculated.

Non dimensional local skin friction coefficient can be defined as

2112

1 UC fw ρτ =


For, 2-D flow, the skin friction force per unit width of surface may be evaluated, with reference to following Figure


The skin friction force per unite with on an elemental length (δx) of surface is

Total skin friction force per unit width on length is

xF wδτδ =

∫=l

w dxF0τ


Skin friction force F may be expressed in terms of a non-dimensional coefficient, CFdefined by

Where, SW is the wetted area of the surface.

Total skin frication drag coefficient (CDF) is defined by

WF SUCF 2

21

∞∞= ρ


Where, DF is total skin friction force on both surfaces and S is plan area of plate or airfoil.

For a flat plate or symmetrical aerofoil, at zero incidence (top and bottom surfaces behaves identically), DF=2F and S=SW

SUCDFDF

2

21

∞∞= ρ


For incompressible flow, ρ1=ρ∞=ρ, μ1=μ∞=μand U1=U∞=U

SU

FCFD

2

21

2

∞∞

=ρ

Surface Friction Drag: Flat plate

In flat plate, flows at constant pressure are considered, U1=U∞ are used, but if a general definition is involved, U1 will be used, Subject to above condition, CF can be written as

This equation is applicable to a flat plate only, but on slim aerofoil, for which U1 dose not vary greatly from U∞ over most of the surface, the expression will give a good approximation.

∫ ⎟⎠⎞

⎜⎝⎛=

1

0 LxdCC fF

The Momentum Integral Equation

The accurate evaluation of the quantities such as δ, δ*, δ**, and θ requires the numerical solution of the differential equations of motion.

The required momentum integral equationis derived by considering mass and momentum balances on a thin slice of B-Lof length as shown in the following figure

Slice of BL

Cont.

The quantities vary with x, let take f as quantity acts normal to AB along CD, the resulting force acts on DC can be written as

(1)

Similar By using the same manner, mass and momentum balances on a thin slice of B-Lcan be obtained

xdxdfxfxxf δδ +=+ )()(

Mass Balance

The Momentum Integral EquationMass Balances

Conservative of mass density is assumed to be constant, the mass balance for slice ABCD can be obtained as follow

Volumetric flow rate into AB (Qi)=Volumetric flow rate out across CD ( )+Volumetric flow rate out across AD( ) +

Volumetric flow rate out across BC ( )(2)

xdx

dQQ i

i δ+

xdxdUxVe δδδ 1−

xVsδ

Cont.

Where Qi is defined as(3)

Rearrange Eq. 2 leads to an expression for the as follow

By introducing δ*( ) into above equation, which can be rewritten

(4)

dyuQi ∫=δ

0

∫ −+−=δ δ0 1 Se V

dxdUudy

dxdV

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛−=∗ δ

δ0

1

1 dyUu

( ) se VUdxdV −= ∗

1δ

Y- Momentum Balance

The Momentum Integral EquationY-momentum balance

From Fig. C, the momentum theorem states that the:

Rate at which y-component of momentum crosses( )= The net pressure force in the y-direction acting on the

slice AB ( )

(5)

xVe δρ 2

xpxpw δδ 1−

2 21 1e w e wV x p x p x V p pρ δ δ δ ρ→ = − → = −

Cont.

The net pressure difference across the B-L is negligible (i.e.pw≈p1 ).

For simplicity, The B-L along a flat plate when U1=U∞=Constant and Vs is not considered. Eq. 4 can be rewritten as

(6)2

21 ⎟⎟

⎠

⎞⎜⎜⎝

⎛=−⇒=

∗

∞

∗

∞ dxdUpp

dxdUV w

se

δρδ

Cont.

B-L is very thin compared to the length of the plate thus dδ*/dx<<1, so that its square is negligibly small.

Thus it can be demonstrated that the assumption of a thin B-L implies that the pressure dose not vary appreciably across the B-L. This is one of the major features of B-L Theory. Within the B-L, the pressure is a function of x only. appreciably across the B-L.

X-Momentum Balance

The Momentum Integral EquationX-momentum balance

X-momentum balance for the slice Fig. e:

Rate at which momentum leaves or enters across CD

( ) and AD ( )-Rate at which momentum enters across AB

( ) = Net pressure force in x-direction acting on AB ( ) - acting on CD

( )+ acting on AD ( ) -Surface friction force acting on ABCD ( )

xdx

dMM in

in δ+ xdxdUxUVe δδρδρ 2

11 −

inMδp

( ) xpdxdp δδδ + x

dxdp δδ

XWδτ

Cont.

Let’s write above Eq. Again

Where

( )

21 1

inin e in

W

dM dM x V U x U x Mdx dx

d dp p p x p x Xdx dx

δδ ρ δ ρ δ

δδ δ δ δ δ τ δ

+ + − − =

⎛ ⎞− + + −⎜ ⎟⎝ ⎠

∫=δρ

0

2 dyuM in

dxx →δ

Cont.

Above equation can be simplified as

(7)

∫ −−=+−δ

τρδρρ0 1

21

2we dx

dpVUdxdUdyu

dxd

The Momentum Integral Equation

At the edge of the B-L, the Bernoulli equation can be used as

By using the definition of Ve, δ* and θalong with Bernoulli Eq.. Equation 7 can be rewritten a

(8)

dxdU

UdxdpConstUp 1

12

1 . ρρ −=⇒=+

( )ρτ

ρδθ wsVU

dxdU

UUdxd

=++ ∗1

11

21

Cont.

Eq. 8 is called Von Karman momentum integral equation.

When suitable forms are chosen for the velocity profile the momentum integralequation (Eq. 8) can be solved to provide the variations of δ, δ*, θ and Cf along the surface.

An approximate velocity profile for the laminar boundary layer

In order to use momentum integral equation, an approximate expression is required for the velocity profile ( )

As an example let us choose a third-degree polynomial distribution as

(9)

Where . In order to evaluate the coefficient a, b, c and d, four conditions are required (boundary condition), two at and two at

1Uuu =

32

1

)( ydycybaUuu +++=

δyy =

0=y 1=y

Cont.

At (10.a)

At (10.b)

At (10.c)

To obtain last condition, we need to see Fig. 2 (S15), in that figure the net force acting on ABCD must be zero at y=0, so that

00 =⇒= uy

11 =⇒= uy

01 =∂∂

⇒=yuy

Cont.

At y=0

Since , and from Bernoulli equation , above Eq. Can be rearranged

as

At (10.d)

ydxdpyx

dxdpxy

y ∂∂

=⇒=⎟⎠⎞

⎜⎝⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂∂ τδδδτ 0

δyy =

.21 constUp =+ ρ

0=ydx

dUyu 1

2

2

2

υδ

−=∂∂

Cont.

Appling those B.C (Eq. 10), the coefficientsbecome

(11)

dxdU

c

dcbdcb

a

12

2

0321

0

υδ

−=Λ−=

=++=++

=

Cont.Above equations can be solved for b, c and d to give the following approximate velocity profile:

(12)

The parameter Λ is called the Pohlhausen parameter. It is determines the effect of an external pressure gradient on the shape of the velocity profile.

If Λ>0 is correspond to favourable pressure gradient and Λ<0 is correspond to unfavourable pressure gradient.

( )323 242

123 yyyyyu +−

Λ+−=

Laminar Velocity profile

Velocity profile at various

Λ=-6: Boundary layer separation occurs, the wall shear stress

Λ<-6: flow reversal at the wall develops

0=wτ

Obtain a relation for (δ*/δ), (θ/δ) and Cf

Now, velocity profile in Eq. 12 will be used to obtain (δ*/δ), (θ/δ) and Cf

(13)

(14)

(15)

( )488

311

01Λ

−=−== ∫∗

yduIδδ

( )∫ Λ−Λ

−=−==1

0

2 )61

239(

28011 yduuI

δθ

⎟⎠⎞

⎜⎝⎛ Λ

+=⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

=== 2

322

21 101

21

21

δρμ

δρμ

ρμ

ρ

τUyd

udUy

uUU

Cyw

wf

Cont.

The quantities I and I1 depend only on the shape of the velocity profile, and for this reason they are usually known as shape parameters.

Approximate methods for a B-L on a flat plate with zero pressure gradient

The Momentum Integral Equation (Eq. 8)will be solved to give approximate expressions for the skin friction drag and for the variation of δ, δ*, θ and Cf along a flat plate with laminar, turbulent and mixed laminar/turbulent B-L.

Simplified from the momentum integral equation:

Simplified from the momentum integral equation:

For flat plate: dp/dx=0 and U1=U∞=Constant (dU1/dx=0)Eq. 8 can be reduced to

(16)By introducing shape parameter I =θ/δ into Eq.

16, with assumption that I is independent of x, Eq. Can be rewritten as

(17)

dxdUwθρτ 2

1=

dxdIC fδ2=

Cont.

b- Rate of growth of a laminar B-L on a flat plate:The rate of increase of the B-L thickness δmay be found by integrating Eq. 17, after setting Λ=0 in Eqs. 14 & 15 and substituting for I and Cf, Eq. 17 becomes

δρμδ

∞

==UI

Cdxd f

13140

2

Cont.Therefore

Above Eq,. Leads to

(18)

The other thickness quantities δ* and θ are also evaluated using Eqs 13 and 14 with Λ=0.

∞∞

=⇒=UU

dρμδ

ρμ

δδ13

140213

140 2

( ) 2/1

64.4

xeRx

=δ

Cont.

Thus

(19)

(20)

( ) 2/1

74.1375.0xeR

x==∗ δδ

( ) 2/1

646.0139.0xeR

x== δθ

Drag coefficient

Drag coefficient for a flat plate of streamwise length L with wholly laminar B-L

(21)

Where θ is the value of the momentum thickness at x=L. Thus using Eq. 20 and 21 gives

(22)

( )∫ ∫ ===

L L

fF LL

dxd

LdxC

LC

0 0

221 θθ

( ) ( ) 2/12/1

586.22293.1

xx eFD

eF R

CCR

C ==⇒=

Turbulent Velocity Profile

There is no completely analytical solution for the mean velocity distribution in turbulent flows even for such simple situations as flow over a flat plate or fully developed pipe flow

A commonly employed turbulent B-L profileis the seventh root law profile, which was proposed by Prandtl on the basis of friction loss experiments with turbulent flow in circular pipes correlated by Blasius.


Empirical relationships between the local skin friction coefficient at wells,

Reynolds number, of the flow (based on the average flow velocity in the pipe and the diameter D). Blasius proposed the relationship

(23)

⎟⎠⎞

⎜⎝⎛= 221 UC w

f ρτ

eRU

4/1

0791.0

ef R

C =


Eq. 23 gives good agreement with experiment for values of .

Assuming that the velocity profile in the pipe may be written in the form

(24)Where u is the velocity at distance y from the

wall and a=pipe radius= D/2.

5105.2 ×<eR

nn

ay

Dy

Uu

⎟⎠⎞

⎜⎝⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

21


The value of n can be determined as: Let assume

that where C is a constant to be determined from Eq. 24 as:

(25)

nn

ya

CuU

ayUCu ⎟⎟

⎠

⎞⎜⎜⎝

⎛=⇒⎟

⎠⎞

⎜⎝⎛=

UCU =1


Substituting for surface friction stress at the wall as

(26)From Eq. 25 we know

2 21/ 4

1/ 41/ 42 7 / 4

1/ 4 1/ 4

1 0.0791 12 2

0.0791 1 0.039552

w fe

C U UR

U UD U D

τ ρ ρ

υ υρ ρ

= = =

⎛ ⎞= ⎜ ⎟⎝ ⎠

4/7

4/7

4/74/7

n

ya

CuU ⎟⎟

⎠

⎞⎜⎜⎝

⎛=


Eq. 26 becomes

(27)

It is found that by taking n=1/7, equation (27) expresses the variation of frication coefficient in pipes for .

( ) ( )[ ] 4/14/1

4/7

4/14/74/7

4/7 2103955.0⎟⎠⎞

⎜⎝⎛=

−

υρτ n

n

w yau

C

( ) ( )[ ]4/14/74/7

4/14/7

4/7

0333.0 −= nn a

yu

Cυρ

000,703000 << eR

Turbulent Velocity ProfileSubstituting value of n=1/7 into Eq. 24 gives

This expression thus relates the velocity u at distance yfrom the surface to the velocity at distance a from the surface.

Assuming that this will hold for very large pipes, it can be assumed that the flow at a section along a flat is similar to the flow over flat plate, so that replacing a by will gives the profile for the free B-L on the flat plate as

7/1

1

⎟⎠⎞

⎜⎝⎛=

ay

Uu


On flat plate

(28)

Eq. 28 is Prandtl ‘s seventh root law. In order to find the wall shear stress, Eq. 27 must be used.

The constant C may be evaluated by equating expressions for the total volume flow through the pipe along with Eqs. 25 and 28

7/17/1

1

yuyUu

=⇒⎟⎠⎞

⎜⎝⎛=δ


The constant C:

Substituting for C and n Eq. 27 then gives

Which, on substituting for u from Eq. 28, gives

(29)

( )1/ 7

2 2

0 0

49 602 2 1.22460 49

a a ya U urdr UC a y dy UCa Ca

π π π π⎛ ⎞= = − = ⇒ = =⎜ ⎟⎝ ⎠∫ ∫

4/14/70234.0 ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

yuw

υρτ

4/14/7

10234.0 ⎟⎠⎞

⎜⎝⎛=δυρτ Uw


Finally,(30)

Rate of growth of a turbulent B-L on a flat plate

4/1

4/1

12

1

0468.00468.021

δδυ

ρτ

e

wf RUU

C =⎥⎦

⎤⎢⎣

⎡==

IU

IC

dxd f

2

0468.0

2

4/1

1⎟⎠⎞⎜

⎝⎛

==δυ

δ


Where

Therefore, ( ) ( )

727

725663

97

8711

1

0

1

0

7/97/87/17/11

0=

−=⎥⎦

⎤⎢⎣⎡ −=−=−= ∫∫ yyydyyyduuI

( )

1/ 4

1/ 41

1/ 4 1/ 41/ 4 4/5

1 1

72 0.04682 7

40.241 0.2415

ddx U

d dx xU U

δ υδ

υ υδ δ δ

×=

× ×

⎛ ⎞ ⎛ ⎞⇒ = ⇒ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

5/45/1

1

5/4

4241.05 x

U ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ ×

=⇒υδ


In terms of Rex, this becomes

(31)

In order to estimate the other thickness quantities for the turbulent layer, the following integral must be evaluated

( ) 5/1383.0xeRx

=δ

( ) ( ) 125.0871

8711

1

0

1

0

7/81

0

7/1 =−=⎥⎦⎤

⎢⎣⎡ −=−=−= ∫ ∫∗ yyydyyduδ

δ


( ) ( ) 175.0107

87

107

871

1

0

1

0

7/107/81

0

7/37/12 =−=⎥⎦⎤

⎢⎣⎡ −=−=−= ∫ ∫∗∗ yyydyyyduuδ

δ


We know that I=7/72=0.0973. From the definition of δ*,δ**, and θ with the value of δfrom Eq. 31 along with above integrals, we obtain

(32)

(33)

(34)

( ) 5/1

0479.0125.0xeR

x==∗ δδ

( ) 5/1

0372.00973.0xeR

x== δθ

( ) 5/1

0761.0175.0xeR

x==∗∗ δδ

Boundary layer growths on flat plate


Drag coefficient for a flat plate with wholly turbulent B-L, from Eq. 31 and Eq. 30

(35)

(36)

( )( ) ( ) 5/14/1

20/14/1

1

4/1

1

0595.0383.0

0468.00468.0x

x

e

ef Rx

RUU

C =⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

υδ

υ

( )5/15/9

15/12

15/1

5/11

21 02975.0

20595.0

21 −=== xUU

xUCU fw ρυρυρτ


The total surface friction force and drag coefficient for a wholly turbulent B-L on a flat plate follow as

(37)

∫ ∫ ⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎠⎞

⎜⎝⎛= −1

0

1

0

5/15/1

1

0595.0Lxdx

ULxdCC fF

υ

5/1

1

0

5/45/1

1

0744.0450595.0 −=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛×⎟⎟

⎠

⎞⎜⎜⎝

⎛= eR

Lx

LUυ


Frication Drag

(38)5/11488.02 −== eFD RCC

F


Notes: (B-L on flat plate)

: The B-L will be entirely laminar

The transition from laminar to turbulent is depending on free stream and surface conditions.

For large range of Re the B-L will be partly laminar and partly turbulent.

5103×<eR

Conditions at transition

There are one condition must be satisfiedat transition region. Which is the momentum thickness will remain constant at the transition position, thus

(39)Eq. 39 lead to

tTlTθθ =

( ) ( )∫ ∫ −=−1

0

1

011

L Ttt

yduuyduu TL δδ


The ratio of the turbulent to the laminar B-L is then given by

(40)

Using the values of I previously evaluated, Eq. 40 leads to

(41)

( )

( ) T

L

TL

T

II

yduu

yduuL

t

t =−

−=∫∫

1

0

1

0

1

1

δ

δ

43.10973.0139.0

==t

t

L

T

δ

δ


Eq. 41 indicates that on a flat plate the B-L increases in thickness by about 43% at transition.

It is then assumed that the turbulent layer downstream of transition, will grows as if it had started from zero thickness at same point ahead of transition and developed along the surface so that its thickness reached the value of δT at the length L from the leading edge of the plate.

Mixed B-L flow on a flat plate with zero pressure gradient

At the leading edge, a laminar layer begins to develop, thickening with distance downstream, until transition to turbulence occurs at some Reynolds number


υt

exUR

t1=


At transition the thickness increases suddenly from δLt in the laminar layer to δTtin the turbulent layer, and the later then continues to grow as if it had started from some point on surface distance XTt ahead of transition,

( ) 5/1

383.0

tx

t

t

Te

TT R

x=δ


The total skin friction force coefficient (CF) for one side of the plate of length L= (Skin friction force per unit width for laminar B-L of length Xt +Skin friction force coefficient per unit width for turbulent B-L of length

Where L is wetted surface area per unit widthThe transition position is given by

(42)

txL −

tet RU

x1

υ=

Cont.

The laminar layer θLt at transition is (from Eq. 20 )

On substituting for from Eq. 42, gives

(43)

( )2/1

2/1

12/1 646.0

646.0t

e

tL x

URx

t

t ⎟⎟⎠

⎞⎜⎜⎝

⎛==

υθ

( ) 2/1

1

646.0tt eL R

Uυθ =

Cont

Now for turbulent layer θTt at transition is ( from Eq. 33)

Where, xTt is the equivalent length of turbulent layer to give this thickness in the transition regime that, θLt=θTt , using Eq. 43 & 44 will give

( ) 5/1

0372.0

Tx

t

t

Te

TT R

x=θ

Cont.

Gives

(43)

2/15/4

1

5/45/1

1

2/1

1 0372.0464.0037.0646.0

ttt eTtT

Tt

t RU

xxU

xxU

x ⎟⎟⎠

⎞⎜⎜⎝

⎛=⇒⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ υυυ

8/5

1

5.35tt eT R

Ux υ

=→

Flat plate with zero pressure gradient:

t.The momentum thickness through transition is assumed to be constant, it is clear that the actual surface frication force under the laminar layer of length must be the same as the force which would exist under a turbulent layer of length XTt

)(θfCF =

Cont.

The Total skin frication force for the whole plate may be found simply by calculating the skin frication force under turbulent layer acting over a length from the point at a distance XTt ahead of transition, to the trailing edge.Total effective length of turbulent layer is L-Xt+XTt

From the definition of friction force,

∫ ∫+− +−

== tTt tTtxxL xxL

fw dxCUdxF0 0

212

1 ρτ

Cont.

CF is given in Eq. 35 as

Thus

(46)

( )5/1

5/1

15/1 0595.00595.0 −

⎟⎟⎠

⎞⎜⎜⎝

⎛== x

URC

xef

υ

[ ] ( )L

xxLU

xULU

FC ttTt TtxxLF

5/45/1

1

5/45/1

12

1

0744.0450595.0

2/1+−

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛==

+− υυρ

5/4111

1

0744.0 ⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

υυυυ tTt

xUxULULU

( ) 5/48/55.350744.0tt eee

eF RRR

RC +−=

Cont.where, L is the total chordwise length of the plate, Eq. 46 is not applicable for values of Re less than Ret. Eq. 21 and 22 should be used.

For large values of Re, greater than about 108, the appropriate all turbulent expression should be used.

Eq. 35 & 38 become inaccurate for Re>107 .

At higher , semi-empirical expressions due to Prandtl and Schlichting should be used.

Cont.

Semi-empirical expressions for higher Re(Re>107)

(47)

(48)

( )[ ] 3.210 65.0log2 −−=

xef RC

( ) 58.210log

455.0

eF R

C =

Cont.

For the lower transition of Re of 3x105 , the corresponding value of Re , above which the all turbulent expressions are reasonably accurate, is 107.

Shock wave/ Boundary layer interactions

The shock wave generated by a defected flap will interact with the upstream B-L.

The interaction will generally cause the upstream B-L to separate with locally high heating rates occurring when the flow reattaches.

The extant of the separation, which can cause a loss of control effectiveness, depends on the character of the upstream B-L.

other viscous interaction problems can occur when the shock waves generated by the forebody and other external componentsimpinges on downstream surface of the vehicle.

Again, locally severe heating rates or B-L separation may occur.

Interaction between shock wave and a laminar B-L

Cont.The pressure rise induced by shock wave is propagated upstream through the subsonic portion of the B-L.

Recall that pressure disturbances can affect the upstream flow only if the flow is subsonic.

As a result, the B-L thickness increases and the momentum decreased.

The thickening B-L deflects the external stream and creates a series of compression waves to form a –λ like shock structure.

Cont.If the shock induced adverse pressure gradient is great enough, the skin frication will be reduced to zero and the B-L will separate.

The subsequent behavior of the flow is a strong function of thegeometry.

For a flat plate, the flow reattaches at some distance down stream.

In the case of a convex body, such as an airfoil, the flow may or may not reattach, depending upon the body geometry, the characteristics of the B-L and the strength of the shock wave.

Cont.

If the flow reattaches, a Prandtl-Mayer expansion fan results as the flow turns back toward the surface.

As the flow reattaches and turns parallel to the plate, a second shock wave is formed.

Immediately downstream of reattachment, B-L thickness reaches a minimum. It is in this region that the maximum heating occur.

Interaction between shock wave and a turbulent B-L:

Cont.The length of the interaction is considerably shorter that the interaction length for a laminar B-L.This results because a turbulent B-L has a great momentum than dose a laminar B-L and can therefore overcome a greater adverse pressure gradient.

Furthermore, since the subsonic portion of a turbulent B-L is relatively thin, the region through which the shock induced pressure rise can propagate upstream is limited. As a result, a much greater pressure rise is required to cause a turbulent B-L to separate.

Note

The pressure rise is spread over a much longer distance when B-L is laminar

Boundary Layer Control for the prevention of separation

There are various techniques, which are used to suppress B-L separation.

Some of these required additional power from the propulsion unit (active technique), others do not require additional power (passive technique).

Active techniqueB-L Suction

Cont.The air near the surface at the point of separation is removed through a suction slot. This result in a thinner, more vigorous B-L, which able to progress further along the surface in the adverse pressure gradient without separating.

Repeated suction is necessary if the B-L is to progress far in an adverse pressure gradient. The great disadvantage of this type is that it relies entirely on the necessary engine power being available for suction; an engine failure might well prove catastrophic, since this type of suction will develop little or no lift without suction .

Active technique Control by blowing

Cont.

In this method of preventing separation, which is due to the complete loss of energy of the air flowing immediately adjacent to the surface, is to energize this tired air by means of blowing a thin, high speed jet into it. Care be taken with the design of blowing duct exit. This blowing technique can only applied to the prevention of separation, unlike sucking which may be employed for this purpose as well to delay or avoid transition.

Passive techniqueVortex generating devices

These devices are particularly likely to be used on sweepback wing because, there is a strong tendency for tipward flow to occur over such wings, leading to wing tip stall

Vortex generating devicesA row of vortex generators

The basic principle is to generate an array of small streamwise vortices. These act to encourage increased entrainment, mixing the relatively high speed air more efficient with low speed air, thereby, re-energizing the B-L. Vortex generators promote the reattachment of B-L in separation bubbles.

Vortex generating devices

Wing fences: They also generate a more powerful streamwise vortex

Saw tooth leading edge: Is another common device for generating a powerful streamwise vortex; as is the leading edge strake.

Vortex generating devices

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Chapter 7 (Boundary Layer)

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