Boundary layer, Fluid mechanics – civilengineering4u

Introduction

Boundary Layers

When a fluid flows over a stationary surface, e.g. the bed of a river, or the wall of a pipe, the fluidtouching the surface is brought to rest by the shear stress t_o at the wall. The velocity increases from thewall to a maximum in the main stream of the flow.

Looking at this two-dimensionally we get the above velocity profile from the wall to the centre of the flow.

This profile doesn’t just exit, it must build up gradually from the point where the fluid starts to flow past the surface – e.g. when it enters a pipe.

If we consider a flat plate in the middle of a fluid, we will look at the build up of the velocity profile as the fluid moves over the plate.

Upstream the velocity profile is uniform, (free stream flow) a long way downstream we have the velocity profile we have talked about above. This is the known as fully developed flow. But how do we get to that state?

This region, where there is a velocity profile in the flow due to the shear stress at the wall, we call the boundary layer. The stages of the formation of the boundary layer are shown in the figure below:

We define the thickness of this boundary layer as the distance from the wall to the point where the velocity is 99% of the ‘free stream’ velocity, the velocity in the middle of the pipe or river.

boundary layer thickness, d = distance from wall to point where u = 0.99 u_mainstream

The value of d will increase with distance from the point where the fluid first starts to pass over the boundary – the flat plate in our example. It increases to a maximum in fully developed flow.

Correspondingly, the drag force D on the fluid due to shear stress t_o at the wall increases from zero at the start of the plate to a maximum in the fully developed flow region where it remains constant. We can calculate the magnitude of the drag force by using the momentum equation. But this complex and not necessary for this course.

Our interest in the boundary layer is that its presence greatly affects the flow through or round an object. So here we will examine some of the phenomena associated with the boundary layer and discuss why these occur.

Hydraulic gradient line.

It is defined as the line which gives the sum of pressure head ( P/?g) and datum head (z) of a flowing fluid in a pipe with respect to some reference line or is the line which is obtained by joining the top of all vertical ordinates, showing the pressure head ( P/?g) of a pipe from the center of the pipe. It is briefly written as H.G.L

Major energy loss and minor energy loss in pipe

The loss of head or energy due to friction in pipe is known as major loss while the loss of energy due to change of velocity of the flowing fluid in magnitude or direction is called minor loss of energy.

Total Energy line

It is defined as the line, which gives sum of pressure head, datum head and kinetic head of a flowing fluid in a pipe with respect to some reference line.

Equivalent pipeline

An Equivalent pipe is defined as the pipe of uniform diameter having loss of head and discharge of a compound pipe consisting of several pipes of different lengths and diameters.

Water Hammer in pipes.

In a long pipe, when the flowing water is suddenly brought to rest by closing the valve or by any similar cause, there will be a sudden rise in pressure due to the momentum of water being destroyed. A pressure wave is transmitted along the pipe. A sudden rise in pressure has the effect of hammering action on the walls of the pipe. This phenomenon of rise in pressure is known as water hammer or hammer blow.

The boundary layer is called laminar, if the Renolds number of the flow is defined as Re = U x X / v is less than 3X10⁵

If the Renolds number is more than 5X10⁵, the boundary layer is called turbulent boundary.

Where, U = Free stream velocity of flow X = Distance from leading edge v = Kinematic viscosity of fluid

Chezy’s formula.

Chezy’s formula is generally used for the flow through open channel.

V =C.Rt(mi)

Where , C = chezy’s constant, m = hydraulic mean depth and i = hf/ L.

Problem 1

A crude of oil of kinematic viscosity of 0.4 stoke is flowing through a pipe of diameter 300mm at the rate of 300 litres/sec. find the head lost due to friction for a length of 50m of the pipe.

Problem 2

Find the type of flow of an oil of relative density 0.9 and dynamic viscosity 20 poise, flowing through a pipe of diameter 20 cm and giving a discharge of 10 lps. Solution :

s = relative density = Specific gravity = 0.9 m = Dynamic viscosity = 20 poise = 2 Ns/m².

Dia of pipe D = 0.2 m; Discharge Q = 10 lps = (10 / 1000) m³/s; Q = AV. So V = Q / A = [10 / (1000 X ( ^p₄ ( 0.2)²) )] = 0.3183 m/s.

Kinematic viscosity = v = m / ? = [ 2 / (0.9X1000)] = 2.222X10^-3 m² / s. Reynolds number Re = VD / v

Re = [0.3183 X 0.2 / 2.222X10^-3] = 28.647; Since Re ( 28.647) < 2000,

It is Laminar flow.

Formula for finding the loss of head due to entrance of pipe hi

hi = 0.5 ( V² / 2g)

Formula to find the Efficiency of power transmission through pipes

n = ( H – hf) / H

where, H = total head at inlet of pipe. hf = head lost due to friction

Problem 3

Hydro dynamically smooth pipe carries water at the rate of 300 lit/s at 20^oC (r = 1000 kg/m³, n = 10^-6 m²/s) with a head loss of 3m in 100m length of pipe.

Determine the pipe diameter. Use f = 0.0032 + (0.221)/ (Re)^0.237 equation for f where hf = ( fXLXV²)/ 2gd and Re = (rVD/m)

Boundary layer separation

(i) Convergent flows: Negative pressure gradients

If flow over a boundary occurs when there is a pressure decrease in the direction of flow, the fluid will accelerate and the boundary layer will become thinner.

This is the case for convergent flows.

The accelerating fluid maintains the fluid close to the wall in motion. Hence the flow remains stable and turbulence reduces. Boundary layer separation does not occur.

(ii) Divergent flows: Positive pressure gradients

When the pressure increases in the direction of flow the situation is very different. Fluid outside theboundary layer has enough momentum to overcome this pressure which is trying to push it backwards. The fluid within the boundary layer has so little momentum that it will very quickly be brought to rest,and possibly reversed in direction. If this reversal occurs it lifts the boundary layer away from the surfaceas shown below.

This phenomenon is known as boundary layer separation.

At the edge of the separated boundary layer, where the velocities change direction, a line of vortices occur (known as a vortex sheet). This happens because fluid to either side is moving in the opposite direction.

This boundary layer separation and increase in the turbulence because of the vortices results in very large energy losses in the flow.

These separating / divergent flows are inherently unstable and far more energy is lost than in parallel or convergent flow.

(iii) A divergent duct or diffuser

The increasing area of flow causes a velocity drop (according to continuity) and hence a pressure rise(according to the Bernoulli equation).

Increasing the angle of the diffuser increases the probability of boundary layer separation. In a Venturi meter it has been found that an angle of about 6 o provides the optimum balance between length of meter and danger of boundary layer separation which would cause unacceptable pressure energy losses.

(iv) Tee-Junctions

Assuming equal sized pipes, as fluid is removed, the velocities at 2 and 3 are smaller than at 1, the entrance to the tee. Thus the pressure at 2 and 3 are higher than at 1. These two adverse pressure gradients can cause the two separations shown in the diagram above.

(v) Y-Junctions

Tee junctions are special cases of the Y-junction with similar separation zones occurring. See the diagram below.

Downstream, away from the junction, the boundary layer reattaches and normal flow occurs i.e. the effect of the boundary layer separation is only local. Nevertheless fluid downstream of the junction will have lost energy.