Boundary Layer Theory.

Introduction

Let’s start understanding the overview of the topic by a simple example. Now, imagine that you are watching a moving crowd from bird’s eye view. And you will observe sometimes that apart of crowd is moving fast and a part of crowd is moving slowly with this reference you can imaginarily draw boundary differentiating crowd on basis of their relative velocity this theory of boundary’s in flowing fluid is said as boundary layer theory.

 

Fig :(a)

                                          Image reference : https://www.nuclear-power.net/nuclear-engineering/fluid-dynamics/boundary-layer/

 

Let’s take a look at above diagram (Fig :(a))

In above diagram a flow of fluid on a flat plate is shown. Here in this diagram the fluid is considered to be flowing with free stream velocity (U0) and this fluid in our discussion will be called Free stream and is considered to have a Uniform velocity in Y-Direction along X-direction. Now, when the free stream comes in contact with the solid plate the lower layer of fluid gets attracted to the solid plate due to Vander-walls forces and adhesive forces. If in case the lower layer of fluid gets stuck on the solid plate then this condition is known as no slip condition. Now in this condition fixing a certain point on X-axis and moving in upper Y-Direction. We can see an increase in velocity and eventually at a certain point where fluid velocity will be equal to  99.99% of free stream velocity. And on tracing this point along X-axis we can plot a curve or a layer (in 3-D view) this layer is called Boundary layer.

Now, Let’s review a few basic concepts to understand what does Boundary layer separates.

A). Laminar Flow : Flow is described as laminar when all the fluid (gas or liquid) flows in layers.

B). Turbulent flow :Flow is described as Turbulent flow, when type of fluid (gas or liquid) flow in which the fluid undergoes irregular fluctuations, or mixing, in contrast to laminar flow.

                                                                                                                                                       Reference : https://www.britannica.com/science/turbulent-flow

What does boundary layer depict ?

A boundary layer can show a contour of three zones namely:

1). Laminar Zone.

2). Transition Zone.

3). Turbulent Zone.

The deciding factor for the type of zone under the layer is decided by the Reynolds Number. (Re)

1). If Re < 5 x 105 then the zone is said to be Laminar.

2). If Re > 5 x 105 then the zone is said to be Turbulent.

3). If Re = 5 x 105 then the zone is said to be Transition.

 

Boundary Layer thickness (𝛿)

                                          Image reference : https://gradeup.co/concept-of-boundary-layer-and-its-growth-i-d633ef01-75b7-11e6-99b9-f5ba37a77793

It is defined as the distance from the boundary of the solid body measured in the Y- direction to the point, where the velocity of the fluid is approximately equal to 0.99 times the free stream velocity (U0) of the fluid. It is denoted by the symbol 𝛿. For laminar and turbulent zone, it is denoted as:

1.      𝛿lam  = Thickness of laminar boundary layer.

2.      𝛿tur   = Thickness of turbulent boundary  layer.

3.      𝛿’      = Thickness of transition boundary layer.

                                                                                         

Displacement Thickness (𝛿*)

Image reference : https://gradeup.co/concept-of-boundary-layer-and-its-growth-i-d633ef01-75b7-11e6-99b9-f5ba37a77793

It is defined as the distance, measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for the reduction in flow rate on account of boundary layer formation . It is denoted by 𝛿* .It is also defined as :

"The distance perpendicular to the boundary , by which the free-stream is displaced due to the formation of boundary layer".

 

Expression for Displacement Thickness (𝛿*) is

Here,

u = velocity of fluid at element strip.

U = free stream velocity.

𝛿*= Displacement thickness.

Momentum Thickness (ϴ)

Momentum thickness is defined as the distance, measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for the reduction in momentum of the flowing fluid on account of boundary layer formation. It is denoted by ϴ.

 

Expression for Momentum Thickness (ϴ) is

Here,

u = velocity of fluid at element strip.

U = free stream velocity.

ϴ = Momentum thickness.                                                                              

 

Energy Thickness(𝛿**)

It is defined as the distance measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for the reduction in kinetic energy of the flowing fluid on account of boundary layer formation. It is denoted by 𝛿 **.

Expression for Momentum Thickness (𝛿 **.) is

Here,

u = velocity of fluid at element strip.

U = free stream velocity.

ϴ = Momentum thickness.                                                                              Reference : R.K Bansal – A Textbook of Fluid Mechanics and Hydraulic Machines

DRAG FORCE ON A FLAT PLATE DUE TO BOUNDARY LAYER

Consider the flow of a fluid having free-stream velocity equal to U , over a thin plate as shown in fig. The drag force on the plate can be determined if he velocity profile near the plate is known .  consider a small length of the plate is shown in fig.

 

 

 

 

Equation is known as van karman momentum integral equation for boundary layer flows.

This is applied to :

1.      Laminar boundary layers

2.      Transition boundary layers , and

3.      Turbulent boundary layer flows.

1.Local co-efficient of Drag[cd]-It is define as the ratio of the shear stress 𝛕0  to  the  quantity ½ pU2 . It is denoted by CD.

Hence,

 

2. Average cofficent of drag[CD]-It is define as the ratio of the total drag force to the quantity1/2 ϼAU2.It is called co-efficient  of drag and is denoted by

 Hence,

 

 Where, A=Area of the surface (or plate)

          U= Free stream velocity

         ϼ = Mass density of fluid.

 

3.Boundary conditions for the velocity profiles-The followings are the boundary conditions which must be satisfied by any velocity profile , whether it is laminar boundary layer zone ,or in turbulent boundary layer zone :

 

                           I.          At y=0 , u= 0 and du/dy has some final value.

                         II.          At δ =0 , u= U

               III.      At y=  δ , du/dy =0 

Reference : R.K Bansal – A Textbook of Fluid Mechanics and Hydraulic Machines

Turbulent boundary layer on flate plate

The Turbulent layer is the 3rd region of the boundary layer. If the length of the plate is more than the distance x, the thickness of boundary layer will go on increasing in the downstream direction. Then the laminar boundary layer becomes unstable and motion of fluid within it, is disturbed and irregular which leads to a transition from laminar to turbulent boundary layer. This short length over which the boundary layer flow changes from laminar to turbulent is called transition zone. Further downstream the transition zone, the boundary layer is turbulent and continues to grow in thickness. This layer is called turbulent boundary layer. In the Turbulent boundary layer zone there are three sublayers, firstly is Viscous sublayer. The Viscous sublayer is very thin layer just above the surface of the plate. It starts rom transition zone. The next is Buffer sublayer which is just above the viscous layer and the last is the main turbulent region which covers the most portion of the turbulent boundary layer. The thickness of the boundary layer, drag force on one side of the plate and co-efficient of drag due to turbulent boundary layer provided the velocity profile known. 

Reference : https://www.youtube.com/watch?v=jRjOTXPhPuE&feature=youtu.be

 Analysis of turbulent flow

If Reynolds number is more than 5* 105   and less than 107  the thickness of boundary layer and drag coefficient is given by

Where x= distance from leading edge

Rex = Reynolds number of length x

Rel = Reynolds number at end of plate

And if in case Reynolds number is greater than 107  but less than 109  Schlichling has given a empirical equation

Total drag on plate

L = total length of plate , b= width of plate

A= length of laminar boundary layer

If the length of transition region is considered negligible then,

L-A =lenght of tubulent boundary layer

We know the drag on flate platefor laminar as well as turbulent boundary layer based on our assumptions that turbulent boundary layer starts from the leading edge .this is valid only when length of laminar boundary layer is negligible. But if not so, then the following process is used,

1.      Fing the length of leading edge upto ehich laminar boundary layer exists. This is done by equating  5*105 =Ux/v. The value of x gives the length of laminar boundary layer. Let this be equal to A.

2.      Find drag using blasius solution of laminar boundary layer fot length A.

3.      Find drag due to turbulent boundary layer for whole length of plate.

4.      Find the drag due to turbulent boundary layer for length A only

Then total drag on plate = drag given by 2+ drag given by 3 - drag given b

    = drag due to laminar boundary layer for length A+ drag due to turbulent boundary layer of total

        length L - drag due to turbulent boundary layer of length A    

               

Reference : R K Bansal (book of fluid enginnering )

Link:  www.researchgate.net

Separation in Boundary Layer 

Whenever there is relative movement between a fluid and a solid surface, whether externally round a body or internally in enclosed passage a boundary layer exists with the viscous force present in the layer of fluid close to the surface. In this thin layer of fluid, the velocity varies from zero to free stream velocity in the direction normal to the solid body. As the fluid flows along the length of surface the thickness of the boundary layer also increases. The fluid layer adjacent to the solid surface must do work against the surface friction at the expense of its kinetic energy. This loss of energy is recovered the adjacent fluid layer through the momentum exchange process; thus, velocity of layer goes on decreasing. Along the length of the solid surface a stage may come when the boundary may not be able to keep sticking to solid surface, if it cannot provide the kinetic energy to overcome the resistance offered by the solid surface. Hence the boundary layer will be separated from the surface. This is known as Boundary layer Separation. 

Effect of Adverse Pressure Gradient 

More dramatic effect, of boundary layer separation in aircraft wings is aerodynamic stall. At relatively low angles of attack, for example during cruise, the adverse pressure gradient acting on the top surface of the wing is benign, and the boundary layer remains attached over the entire surface. As the angle of attack is increased, however, so does the pressure gradient. At some point the boundary layer will start to separate near the trailing edge of the wing, and this separation point will move further upstream as the angle of attack is increased. If an aero foil is positioned at a sufficiently large angle of attack, separation will occur very close to the point of maximum thickness of the aero foil and a large wake will develop behind the point of separation. This wake redistributes the flow over the rest of the aero foil and thereby significantly impairs the lift generated by the wing. As a result, the lift produced is seriously reduced in a condition known as aerodynamic stall. Due to the high pressure drag induced by the wake, the aircraft can further lose airspeed, pushing the separation point further upstream and creating a deleterious feedback loop where the aircraft literally starts to fall out of the sky in an uncontrolled spiral. 

 Boundary Layer Separation Over a Cylinder

 

 

Influencing Parameters

The tendency of a boundary layer to separate primarily depends on the distribution of the adverse or negative edge velocity gradient duo / ds(s) < 0  along the surface, which in turn is directly related to the pressure and its gradient by the differential form of the Bernoulli relation, which is the same as the momentum equation for the outer inviscid flow.

But the general magnitudes of duo / ds  required for separation are much greater for turbulent than for laminar flow, the former being able to tolerate nearly an order of magnitude stronger flow deceleration. A secondary influence is the Reynolds number. For a given adverse duo / ds  distribution, the separation resistance of a turbulent boundary layer increases slightly with increasing Reynolds number. In contrast, the separation resistance of a laminar boundary layer is independent of Reynolds number — a somewhat counterintuitive fact.

 

Effect of Boundary Layer Separation

More dramatic effect, of boundary layer separation in aircraft wings is aerodynamic stall. At relatively low angles of attack, for example during cruise, the adverse pressure gradient acting on the top surface of the wing is benign, and the boundary layer remains attached over the entire surface. As the angle of attack is increased, however, so does the pressure gradient. At some point the boundary layer will start to separate near the trailing edge of the wing, and this separation point will move further upstream as the angle of attack is increased. If an aero foil is positioned at a sufficiently large angle of attack, separation will occur very close to the point of maximum thickness of the aero foil and a large wake will develop behind the point of separation. This wake redistributes the flow over the rest of the aero foil and thereby significantly impairs the lift generated by the wing. As a result, the lift produced is seriously reduced in a condition known as aerodynamic stall. Due to the high pressure drag induced by the wake, the aircraft can further lose airspeed, pushing the separation point further upstream and creating a deleterious feedback loop where the aircraft literally starts to fall out of the sky in an uncontrolled spiral. 

Boundary layer separation over the top surface of wing

 

Another effect of boundary layer separation is regular shedding vortices, known as a Kármán vortex street. Vortices shed from the bluff downstream surface of a structure at a frequency depending on the speed of the flow. Vortex shedding produces an alternating force which can lead to vibrations in the structure. If the shedding frequency coincides with a resonance frequency of the structure, it can cause structural failure. These vibrations could be established and reflected at different frequencies based on their origin in adjacent solid or fluid bodies and could either damp or amplify the resonance.

 

Methods of Prevention

Due to such dramatic effects of boundary layer separation, attempts should be made to avoid separation by various methods. Such as,

1.      Suction of slow-moving fluid by a suction slot.

2.      Supplying additional energy from a blower.

3.      Providing a bypass in the slotted wings.

4.      Rotating boundary in the direction of flow.

5.      Providing small divergence in a diffuser.

6.      Providing guide blades in a bend.

7.      Providing a trip-wire ring in the laminar region for the flow over sphere.

References:

1. R.K Bansal – A Textbook of Fluid Mechanics and Hydraulic Machines.

2. https://www.britannica.com/science/turbulent-flow

3.www.gradeup.com

4. www.researchgate.net