** Senthil Seliyan Elango **

** Title:** Fluid Dynamics

**Area: **

**Country:**

**Program: **

** Available for Download: ** Yes

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** INTRODUCTION **

**Real fluids **

The flow of real fluids exhibits viscous effect, which are they tend
to "stick" to solid

surfaces and have stresses within their body.

You might remember from earlier in the course Newton's law of
viscosity:

This tells us that the shear stress, , in a fluid is proportional to
the velocity gradient - the

rate of change of velocity across the fluid path. For a "Newtonian"
fluid we can write:

where the constant of proportionality, is known as the coefficient
of viscosity (or

simply viscosity). We saw that for some fluids - sometimes known as
exotic fluids - the

value of changes with stress or velocity gradient. We shall only
deal with Newtonian

fluids.

In his lecture we shall look at how the forces due to momentum
changes on the fluid and

viscous forces compare and what changes take place.

**DESCRIPTION **

**Laminar and turbulent flow **

If we were to take a pipe of free flowing water and inject a dye
into the middle of the

stream, what would we expect to happen?

This

this

or this

Actually both would happen - but for different flow rates. The top occurs when the fluid

is flowing fast and the lower when it is flowing slowly.

The top situation is known as

In laminar flow the motion of the particles of fluid is very orderly with all particles

moving in straight lines parallel to the pipe walls.

But what is fast or slow? And at what speed does the flow pattern change? And why

might we want to know this?

The phenomenon was first investigated in the 1880s by Osbourne Reynolds in an

experiment which has become a classic in fluid mechanics.

He used a tank arranged as above with a pipe taking water from the centre into which he

injected a dye through a needle. After many experiments he saw that this expression

where = density,

would help predict the change in flow type. If the value is less than about 2000 then flow

is laminar, if greater than 4000 then turbulent and in between these then in the transition

zone.

This value is known as the Reynolds number, Re:

Laminar flow: Re < 2000

Transitional flow: 2000 < Re < 4000

Turbulent flow: Re > 4000

What are the units of this Reynolds number? We can fill in the equation with SI units:

i.e. it has

dimensionless) quantity. Thus the Reynolds number, Re, is a non-dimensional number.

We can go through an example to discover at what velocity the flow in a pipe stops being

laminar.

If the pipe and the fluid have the following properties:

water density = 1000 kg/m3

pipe diameter

(dynamic) viscosity, = 0.55x103 Ns/m2

We want to know the maximum velocity when the Re is 2000.

If this were a pipe in a house central heating system, where the pipe diameter is typically

0.015m, the limiting velocity for laminar flow would be, 0.0733 m/s.

Both of these are very slow. In practice it very rarely occurs in a piped water system - the

velocities of flow are much greater. Laminar flow does occur in situations with fluids of

greater viscosity - e.g. in bearing with oil as the lubricant.

At small values of Re above 2000 the flow exhibits small instabilities. At values of about

4000 we can say that the flow is truly turbulent. Over the past 100 years since this

experiment, numerous more experiments have shown this phenomenon of limits of Re for

many different Newtonian fluids - including gasses.

What does this abstract number mean?

We can say that the number has a physical meaning, by doing so it helps to understand

some of the reasons for the changes from laminar to turbulent flow.

It can be interpreted that when the inertial forces dominate over the viscous forces (when

the fluid is flowing faster and Re is larger) then the flow is turbulent. When the viscous

forces are dominant (slow flow, low Re) they are sufficient enough to keep all the fluid

particles in line, then the flow is laminar.

In summary:

Laminar flow

Re < 2000

'low' velocity

Dye does not mix with water

Fluid particles move in straight lines

Simple mathematical analysis possible

Rare in practice in water systems.

Transitional flow

2000 > Re < 4000

'medium' velocity

Dye stream wavers in water - mixes slightly.

Turbulent flow

Re > 4000

'high' velocity

Dye mixes rapidly and completely

Particle paths completely irregular

Average motion is in the direction of the flow

Cannot be seen by the naked eye

Changes/fluctuations are very difficult to detect. Must use laser.

Mathematical analysis very difficult - so experimental measures are used

Most common type of flow.

Up to this point on the course we have considered ideal fluids where there have been no

losses due to friction or any other factors. In reality, because fluids are viscous, energy is

lost by flowing fluids due to friction which must be taken into account. The effect of the

friction shows itself as a pressure (or head) loss.

In a pipe with a real fluid flowing, at the wall there is a shearing stress retarding the flow,

as shown below.

If a manometer is attached as the pressure (head) difference due to the energy lost by the

fluid overcoming the shear stress can be easily seen.

The pressure at 1 (upstream) is higher than the pressure at 2.

We can do some analysis to express this loss in pressure in terms of the forces acting on

the fluid.

Consider a cylindrical element of incompressible fluid flowing in the pipe, as shown

The pressure at the upstream end is

by p to (p-p).

The driving force due to pressure (F = Pressure x Area) can then be written

driving force = Pressure force at 1 - pressure force at 2

The retarding force is that due to the shear stress by the walls

As the flow is in equilibrium,

driving force = retarding force

Giving an expression for pressure loss in a pipe in terms of the pipe diameter and the

shear stress at the wall on the pipe.

The shear stress will vary with velocity of flow and hence with Re. Many experiments

have been done with various fluids measuring the pressure loss at various Reynolds

numbers. These results plotted to show a graph of the relationship between pressure loss

and Re look similar to the figure below:

This graph shows that the relationship between pressure loss and Re can be expressed as

As these are empirical relationships, they help in determining the pressure loss but not in

finding the magnitude of the shear stress at the wall

In general the shear stress

possible to calculate a theoretical value for a given velocity, fluid and pipe dimension.

In laminar flow the paths of individual particles of fluid do not cross, so the flow may be

considered as a series of concentric cylinders sliding over each other - rather like the

cylinders of a collapsible pocket telescope.

As before, consider a cylinder of fluid, length L, radius

a pipe.

We are in equilibrium, so the shearing forces on the cylinder equal the pressure forces.

By Newtons law of viscosity we have

, where y is the distance from the wall.

As we are measuring from the pipe centre then we change the sign and replace y with r

distance from the centre, giving

Which can be combined with the equation above to give

In an integral form this gives an expression for velocity,

Integrating gives the value of velocity at a point distance r from the centre

At

so, an expression for velocity at a point r from the pipe centre when the flow is laminar is

Note how this is a parabolic profile (of the form y = ax2 + b ) so the velocity profile in the

pipe looks similar to the figure below

What is the discharge in the pipe?

So the discharge can be written

This is the Hagen-Poiseuille equation for laminar flow in a pipe. It expresses the

discharge

), diameter of the pipe and the

viscosity of the fluid.

We are interested in the pressure loss (head loss) and want to relate this to the velocity of

the flow. Writing pressure loss in terms of head loss

This shows that pressure loss is directly proportional to the velocity when flow is

laminar.

It has been validated many time by experiment.

It justifies two assumptions:

1. fluid does not slip past a solid boundary

2. Newton's hypothesis.

When a fluid flows over a stationary surface, e.g. the bed of a river or the wall of a pipe,

the fluid touching the surface is brought to rest by the shear stress o at the wall. The

velocity increases from the wall 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, = distance from wall to point where u = 0.99 umainstream

The value of 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 oat 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.

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.

Above we noted that the boundary layer grows from zero when a fluid starts to flow over

a solid surface. As is passes over a greater length more fluid is slowed by friction

between the fluid layers close to the boundary. Hence the thickness of the slower layer

increases.

The fluid near the top of the boundary layer is dragging the fluid nearer to the solid

surface along. The mechanism for this dragging may be one of two types:

The first type occurs when the normal viscous forces (the forces which hold the fluid

together) are large enough to exert drag effects on the slower moving fluid close to the

solid boundary. If the boundary layer is thin then the velocity gradient normal to the

surface, (

also large. The corresponding force may then be large enough to exert drag on the fluid

close to the surface.

As the boundary layer thickness becomes greater, so the velocity gradient become

smaller and the shear stress decreases until it is no longer enough to drag the slow fluid

near the surface along. If this viscous force was the only action then the fluid would come

to a rest.

It, of course, does not come to rest but the second mechanism comes into play. Up to this

point the flow has been laminar and Newton's law of viscosity has applied. This part of

the boundary layer is known as the laminar boundary layer

The viscous shear stresses have held the fluid particles in a constant motion within layers.

They become small as the boundary layer increases in thickness and the velocity gradient

gets smaller. Eventually they are no longer able to hold the flow in layers and the fluid

starts to rotate.

This causes the fluid motion to rapidly become turbulent. Fluid from the fast moving

region moves to the slower zone transferring momentum and thus maintaining the fluid

by the wall in motion. Conversely, slow moving fluid moves to the faster moving region

slowing it down. The net effect is an increase in momentum in the boundary layer. We

call the part of the boundary layer the turbulent boundary layer.

At points very close to the boundary the velocity gradients become very large and the

velocity gradients become very large with the viscous shear forces again becoming large

enough to maintain the fluid in laminar motion. This region is known as the laminar sub-

layer. This layer occurs within the turbulent zone and is next to the wall and very thin - a

few hundredths of a mm.

Despite its thinness, the laminar sub-layer can play a vital role in the friction

characteristics of the surface.

This is particularly relevant when defining pipe friction - as will be seen in more detail in

the level 2 module. In turbulent

than the thickness of the laminar sub-layer then this increases the amount of turbulence

and energy losses in the flow. If the height of roughness is less than the thickness of the

laminar sub-layer the pipe is said to be smooth and it has little effect on the boundary

layer.

In laminar flow the height of roughness has very little effect

As flow enters a pipe the boundary layer will initially be of the laminar form. This will

change depending on the ration of inertial and viscous forces; i.e. whether we have

laminar (viscous forces high) or turbulent flow (inertial forces high).

From earlier we saw how we could calculate whether a particular flow in a pipe is

laminar or turbulent using the Reynolds number.

Laminar flow: Re < 2000

Transitional flow: 2000 < Re < 4000

Turbulent flow: Re > 4000

If we only have laminar flow the profile is parabolic - as proved in earlier lectures - as

only the first part of the boundary layer growth diagram is used. So we get the top

diagram in the above figure.

If turbulent (or transitional), both the laminar and the turbulent (transitional) zones of the

boundary layer growth diagram are used. The growth of the velocity profile is thus like

the bottom diagram in the above figure.

Once the boundary layer has reached the centre of the pipe the flow is said to be fully

developed. (Note that at this point the whole of the fluid is now affected by the boundary

friction.)

The length of pipe before fully developed flow is achieved is different for the two types

of flow. The length is known as the entry length.

Laminar flow entry length 120 diameter

Turbulent flow entry length 60 diameter

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

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.

When the pressure increases in the direction of flow the situation is very different. Fluid

outside the boundary 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 surface as 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.

The increasing area of flow causes a velocity drop (according to continuity) and hence a

pressure rises (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 provides the optimum

balance between length of meter and danger of boundary layer separation which would

cause unacceptable pressure energy losses.

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.

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.

Two separation zones occur in bends as shown above. The pressure at b must be greater

than at a as it must provide the required radial acceleration for the fluid to get round the

bend. There is thus an adverse pressure gradient between a and b so separation may occur

here.

Pressure at c is less than at the entrance to the bend but pressure at d has returned to near

the entrance value - again this adverse pressure gradient may cause boundary layer

separation.

Flow past a cylinder

The pattern of flow around a cylinder varies with the velocity of flow. If flow is very

slow with the Reynolds number ( v diameter/ less than 0.5, then there is no separation

of the boundary layers as the pressure difference around the cylinder is very small. The

pattern is something like that in the figure below.

If 2 < Re < 70 then the boundary layers separate symmetrically on either side of the

cylinder. The ends of these separated zones remain attached to the cylinder, as shown

below.

Above a Re of 70 the ends of the separated zones curl up into vortices and detach

alternately from each side forming a trail of vortices on the down stream side of the

cylinder. This trial in known as a Karman vortex trail or street. This vortex trail can easily

be seen in a river by looking over a bridge where there is a pier to see the line of vortices

flowing away from the bridge. The phenomenon is responsible for the whistling of

hanging telephone or power cables. A more significant event was the famous failure of

the Tacoma narrows bridge. Here the frequency of the alternate vortex shedding matched

the natural frequency of the bridge deck and resonance amplified the vibrations until the

bridge collapsed.

Normal flow over a aerofoil (a wing cross-section) is shown in the figure below with the

boundary layers greatly exaggerated.

The velocity increases as air it flows over the wing. The pressure distribution is similar to

that shown below so transverse lift force occurs.

If the angle of the wing becomes too great and boundary layer separation occurs on the

top of the aerofoil the pressure pattern will change dramatically. This phenomenon is

known as stalling.

When stalling occurs, all, or most, of the 'suction' pressure is lost, and the plane will

suddenly drop from the sky! The only solution to this is to put the plane into a dive to

regain the boundary layer. A transverse lift force is then exerted on the wing which gives

the pilot some control and allows the plane to be pulled out of the dive.

Fortunately there are some mechanisms for preventing stalling. They all rely on

preventing the boundary layer from separating in the first place.

1. Arranging the engine intakes so that they draw slow air from the boundary layer

at the rear of the wing though small holes helps to keep the boundary layer close

to the wing. Greater pressure gradients can be maintained before separation take

place.

2. Slower moving air on the upper surface can be increased in speed by bringing air

from the high pressure area on the bottom of the wing through slots. Pressure will

decrease on the top so the adverse pressure gradient which would cause the

boundary layer separation reduces.

3. Putting a flap on the end of the wing and tilting it before separation occurs

increases the velocity over the top of the wing, again reducing the pressure and

chance of separation occurring.

Pipe Network simulates steady flow of liquids or gases under pressure. It can simulate

city water systems, car exhaust manifolds, long pipelines with different diameter pipes in

series, parallel pipes, groundwater flow into a slotted well screen, soil vapor extraction

well design, and more. Enter flows at nodes as positive for inflows and negative for

outflows. Inflows plus outflows must sum to 0. Enter one pressure in the system and all

other pressures are computed. All fields must have a number, but the number can be 0.

You do not need to use all the pipes or nodes. Enter a diameter of 0.0 if a pipe does not

exist. If a node is surrounded on all sides by non-existent pipes, the node's flow must be

entered as 0.0. The program allows a wide variety of units. After clicking Calculate, the

arrows "<--, -->, v, ^" indicate the direction of flow through each pipe (to the left, right,

down, or up).

Losses can be computed by either the Darcy-Weisbach or Hazen-Williams (HW) method,

selectable by clicking on the "Roughness, e" drop-down menu. If HW is used, then the

fluid must be selected as "Water, 20C (68F)".

The H, V, Re output field is scrollable using the left and right arrow keys on your

keyboard. Velocity is in m/s if metric units are selected for flow rate Q, and ft/s if

English units are selected for Q.

The pipe network calculation uses the steady state energy equation, Darcy Weisbach or

Hazen Williams friction losses, and the Hardy Cross method to determine the flow rate in

each pipe, loss in each pipe, and node pressures. Minor losses (due to valves, pipe bends,

etc.) can be accounted for by using the equivalent length of pipe method.

The Hardy Cross method is also known as the single path adjustment method and is a

relaxation method. The flow rate in each pipe is adjusted iteratively until all equations

are satisfied. The method is based on two primary physical laws:

1. The sum of pipe flows into and out of a node equals the flow entering or leaving the

system through the node.

2. Hydraulic head (i.e. elevation head + pressure head, Z+P/S) is single-valued. This

means that the hydraulic head at a node is the same whether it is computed from upstream

or downstream directions.

Pipe flows are adjusted iteratively using the following equation,

until the change in flow in each pipe is less than the convergence criteria.

n=2.0 for Darcy Weisbach losses or 1.85 for Hazen Williams losses.

Our calculation gives you a choice of computing friction losses

Weisbach (DW) or the Hazen-Williams (HW) method. The DW method can be used for

any liquid or gas while the HW method can only be used for water at temperatures typical

of municipal water supply systems. HW losses can be selected with the menu that says

"Roughness, e (m):". The following equations are used:

Hazen Williams's equation (Mays, 1999; Streeter et al., 1998; Viessman and Hammer,

1993) where k=0.85 for meter and seconds units or 1.318 for feet and seconds units:

Where "log" is base 10 logarithm and "ln" is natural logarithm.

After computing flow rate Q in each pipe and loss H in each pipe and using the input

node elevations Z and known pressure at one node, pressure P at each node is computed

around the network:

density [F/L3].

Minor losses such as pipe elbows, bends, and valves may be included by using the

equivalent length of pipe method (Mays, 1999). Equivalent length (Leq) may be

computed using the following calculator which uses the formula Leq=KD/f. f is the

Darcy-Weisbach friction factor for the pipe containing the fitting, and cannot be known

with certainty until

know f ahead of time, a reasonable value to use is f=0.02, which is the default value. We

also recommend using f=0.02 even if you select Hazen-Williams losses in the pipe

network calculation. K values are from Mays (1999).

For example, there is a 100-m long 10-cm diameter (inside diameter) pipe with one fully

open gate valve and three regular 90o elbows. Using the minor loss calculator, Leq is 1.0

m and 1.25 m for the fully open gate valve and each elbow, respectively. The pipe length

you should enter into the pipe network calculator is 100 + 1.0 + 3(1.25) = 104.75 m. The

calculator allows a variety of units such as m, cm, inch, and ft for diameter; and m, km,

ft, and miles for equivalent length. If a fitting is not listed, select "User enters K" and

enter the K value for the fitting.

The pipe network calculation has many applications. Two examples will be provided.

1. Municipal water supply system. A water tower is located at node D. The other nodes

could represent industries or homes. Enter the water withdrawals at all the nodes as

negative numbers, then enter the inflow to the network from the water tower at node D as

a positive number equal to the sum of the withdrawals from the other nodes. Usually,

cities require a certain minimum pressure everywhere in the system, often 40 psi. Use

the drop-down menu to select the node that you expect will have the lowest pressure -

possibly the node furthest from D or the one at the highest elevation; we'll use node I.

Enter the pressure at node I as 40 psi. Enter all the pipe lengths, diameters and node

elevations. Then click "Calculate". You can use your right and left arrow keys to scroll

to the left and right to see the velocity in each pipe. Typically, you want pipe velocities

to be around 2 ft/s. If you are designing a system (as opposed to analyzing a system that

is already in place), vary the pipe diameters until the pipe velocities are reasonable and

pressure at node D is as low as possible to minimize the height of the water tower. There

will be a trade-off between pressure at D and pipe diameters. Smaller diameter pipes

will save money on pipes but will require a taller water tower. The water tower height is

proportional to the pressure at D according to h=P/S, where P is the pressure at D. S is

the weight density of the water, and h is the water tower height required.

2. Manifold. A manifold has multiple inflows at various positions along the same

pipeline, and one outflow. Let node I be the outflow and use all other nodes A-H as

inflow locations; so flow is from node A through pipes 1, 2, 5, 7, 6, 8, 11, and 12 and out

node I. Enter the diameters and lengths of these pipes and the desired inflows at nodes

A-H. Enter the outflow at node I as a positive number equal to the sum of the inflows at

nodes A-H. Enter the diameters of pipes 3, 4, 9, and 10 as 0.0 since they are non-existent

pipes. Enter the elevations of all nodes. For a horizontal pipe, set all the elevations to

the same value or just to 0.0 to keep it simple. From the drop-down menu, select the

node where you know the pressure and enter its pressure. Clicking "Calculate" will give

the flow rate in all pipes and the pressure at all the nodes.

The discussion thus far has been rather general and has introduced many important ideas

and principles. Fluid flow behavior has been demonstrated. Numerous references to

airfoil or streamline shapes have been made. Viscous flow of the boundary layer and

unsteady flow in the turbulent wake have been examined. The flow is two-dimensional

since velocity and other flow parameters vary normal to the free-stream direction as well

as parallel to it. With these ideas in mind, one may now study aircraft operating in a

subsonic flow.

Fluid Mechanics by Dr.Andrew Sleigh

Fluid Mechanics with engineering Application-

J.Franzini/E.Finnemore, McGraw Hill

800-993-0066 (Toll Free in US)

808-924-9567 (Internationally)

808-947-2488 (Fax)