fluid mechanics-61341 - an-najah videos · 7 fluid mechanics-2nd semester 2010- [7] ... no mixing,...

Fluid Mechanics-61341

An-Najah National UniversityCollege of Engineering

Chapter [7]

Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid Dr. Sameer Shadeed1

Dr. Sameer Shadeed

Chapter [7]

Flow of a Real Fluid

Introduction In the earlier chapters, the basic equations of

continuity, energy and momentum were introducedand applied to fluid flow cases where theassumption of frictionless flow (or idealideal fluidfluid flow)was made

Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid2 Dr. Sameer Shadeed

It is now necessary to introduce concepts whichenable the extension of the previous work to realrealfluidsfluids in which viscosity is accepted and frictionaleffects cannot be ignored

The concept of ReynoldsReynolds NumberNumber as an indicationof flow type will be used extensively

Real Fluid In aa realreal fluidfluid viscosity produces resistance to

motion by causing shear or friction forcesbetween fluid particles and between these andboundary walls

Due to this viscousviscous effectseffects, fluid tends to


Due to this viscousviscous effectseffects, fluid tends to‘stick’ to solid surfaces and have stresses withintheir body

The inclusion of viscosity allows the existenceof two physically distinct flow regimes, known aslaminarlaminar and turbulentturbulent flow

Types of Flow

Theoretically the physical nature of fluidflow can be categorized into three types,i.e. laminarlaminar, transitiontransition and turbulentturbulent flow


To predict whether the flow will belaminarlaminar, transitiontransition or turbulentturbulent, it isnecessary to explore the characteristics offlow in each of these region

Types of Flow

Laminar Flow Turbulent Flow


Types of Flow Reynolds Experiment

The physical nature of fluid flow has been studied indetailed by OsborneOsborne ReynoldsReynolds (1883) using theapparatus shown in the figure


Schematic diagram of Reynolds apparatus


In this experiment, A filament of dye was injected to the flow of water The discharge was carefully controlled, and

passed through a glass tube so that observationscould be made Reynolds discovered that the dye filament would


Reynolds discovered that the dye filament wouldflow smoothly along the tube as long as thedischarge is low By gradually increased the discharge, a point is

reached where the filament became wavy A small further increase in discharge will cause

vigorous eddying motion, and the dye mixedcompletely with water

Three distinct patterns of flow were recognized:

ViscousViscous oror LaminarLaminar:: In which the fluid particlesappear to move in definite smooth parallel path withno mixing, and the velocity only in the direction of flow

TransitionalTransitional:: In which some unsteadiness



TransitionalTransitional:: In which some unsteadinessbecomes apparent (the wavy filament)

TurbulentTurbulent:: In which the flow incorporates aneddying or mixing action. The motion of a fluidparticle within a turbulent flow is complex andirregular, involving fluctuations in velocity anddirections



Flow Patterns

ReynoldsReynolds experiment also revealed that theinitiation of turbulence was a function of fluidvelocity, viscosity, and a typical dimension. This ledto the formation of the dimensionless ReynoldsReynoldsNumberNumber (R)(R)

Vdor

VdR



Where = density = dynamic viscosityV = mean velocityd = pipe diameterν = kinematic viscosity

Vd

orVd

R

Critical Reynolds Numbers

Laminar flow Transitional flow Turbulent flow • R < 2100;

• 'low' velocity;

• Dye does not mix with water;

• Fluid particles

• 2100 < R < 4000;

• 'medium' velocity; and

• Dye stream wavers in water -

• R > 4000;

• 'high' velocity;

• Dye mixes rapidly and completely;

• Particle paths completely


• Fluid particles move in straight lines;

• Simple mathematical analysis possible; and

• Rare in practice in water systems.

wavers in water -mixes slightly.

• Particle paths completely irregular;

• Average motion is in the direction of the flow;

• Changes/fluctuations are very difficult to detect;

• Mathematical analysis very difficult - so experimental measures are used; and

• Most common type of flow.

If the pipe and the fluid have the following properties, atwhat velocity the flow in a pipe stops being laminar?

water density, = 1000 kg/m3

pipe diameter, d = 0.5mdynamic viscosity, = 0.55x10-3 N.s/m2 (kg/m.s)

Solution: We want to know the maximum velocity for

Example 1


Solution: We want to know the maximum velocity forlaminar flow, i.e. when R =2100

smV

dVor

dVR

/0023.0

5.01000

1055.021002100

2100

3

Oil of viscosity 0.05 kg/m.s and density 860 kg/m3

flows in a 0.1 m diameter pipe with a velocity of 0.6 m/s.Determine the type of flow.

Solution: first, we want to calculate the ReynoldsReynoldsNumberNumber

Example 2


)FlowLaminar(21001032

103205.0

8601.06.0

R

VdR

Shear Stress in Laminar Flow

dy

dv


The first attempt to express turbulent shear stressin mathematical form was made by BoussinesqBoussinesq(1877), who followed the pattern of the laminar flowequation and wrote

Shear Stress in Turbulent Flow

dy

dv

Where εε is the eddyeddy viscosityviscosity which is a property ofthe flow (note the fluid alone) which dependedprimarily on the structure of the turbulence

From its definition the eddyeddy viscosityviscosity can be seen tohave the disadvantageous feature of varying frompoint to point throughout the flow

dy


Shear stress for any flow (laminar and/or turbulent)can be expressed as

Shear Stress in any Flow

dy

dv)(

This equation can cover the combined situationwhere both viscous action and turbulent action arepresent in a flow

The first factor μμ represents effects of viscousaction, and the second factor εε account for effects ofturbulent action

dy


Example 3

Solution:


Boundary LayersWhether a flow is in general laminar or turbulent,

the effects of the viscosity of the fluid are greatest inregions close to solid boundaries.

It is a characteristic of all real fluid, where theviscous effect causes it to "stick" to solid surfacesand have stresses within their body.


and have stresses within their body.

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 shearstress o at the wall.

The velocity increases from zero at the wall to amaximum in the main stream of the flow.

Boundary Layers Looking at this two-dimensionally we get

the velocity profile from the wall to thecentre of the flow as shown in the figure

This profile doesn't just exit, it must buildup gradually from the point where the fluidstarts to flow past the surface (e.g. whenit enters a pipe)


it enters a pipe)

If we consider a flat plate in the middle of a fluid, we will lookat the build up of the velocity profile as the fluid moves overthe plate

This lead to the development of a thin layer of fluid adjacentto the surface in which the fluid velocity varies from zero at thesurface to the free stream value a short distance out from thesurface. This layer has been called the boundaryboundary layerlayer forthe surface

Development of Boundary Layer

To develop the boundary layer concept, let’s considera flow pass over a stationary which lies parallel to theflow.

The flow just upstream the plate has a uniform freestream velocity, VVss.

As the flow comes into contact with the plate, the


As the flow comes into contact with the plate, thelayer of fluid immediately adjacent to the platedecelerates due to viscous friction and comes to rest.

The fluid in contact with the plate surface has zerovelocity, known as ‘no slip’ condition.

A velocity gradient exists between the fluid in the freestream and the plate surface.


Newton’s law of viscosity tells us that the shearstress, , in a fluid is proportional to the velocitygradient (the rate of change of velocity across the fluidpath).

dv


Shearing action between the layer of fluid in contactwith the plate and second layer of the fluid forced thesecond layer to decelerate.

This creates a shearing action between the third layerof fluid and so on.

dy


As the fluid passes along the plate, the zonein which shearing action occurs tend tospread further outwards. This zone is knownas ‘boundaryboundary layerlayer’.


Outside the boundary layer, the flowremains effectively free of shear, so the fluidis not subjected to viscosity related forces.Thus, the fluid outside the boundary layermay be assumed to act like an idealideal fluidfluid.



Development of boundary layer along a flat plate

Development of Boundary Layer The boundary layer, which starts at the leading edge of the

surface, is a laminar layer which grows in thickness alongthe surface. Eventually, at some distance downstream, thislaminar layer become unstable, transforming into a turbulentlayer.

As with any other sheared flow, flow in boundary layer canbe laminarlaminar or turbulentturbulent, depending on the local ReynoldsReynolds


be laminarlaminar or turbulentturbulent, depending on the local ReynoldsReynoldsNumbersNumbers

TwoTwo ReynoldsReynolds NumbersNumbers are widely used to define thecharacter of the boundary layer; they are defined in terms ofthe undistributed velocity VVoo, boundary layer thickness ,and distance from the leading edge of the plate xx

For smooth polished plate, translation toturbulent flow occurs when the values of theseReynoldsReynolds NumbersNumbers are 500000 and 3900,respectively


Below these values, laminar boundary layersare to be expected, whereas ReynoldsReynoldsNumbersNumbers above these values suggest that theflow is transitioning to turbulent layers


Shear Stress and Head Loss


The drop in the energy line between points 1 and 2(the head loss hL1-2) is given by

Shear Stress and Head Loss

The effect of shear stress on the control volume is tocause a loss of energy in the flow that is proportional tothe stress and the length of the section over which thestress acts and inversely proportional to the effectiveradius of the control volume. For this simple case ofincompressible flow

Where Rh is the hydraulic radius (Rh = A/P)A = the cross sectional dataP = wetted perimeter

For full pipe flow with a circular conduit (Rh = r/2 = d/4)Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid27 Dr. Sameer Shadeed

incompressible flow

Shear Stress and Head Loss in Circular Conduits

rr is any distance

The relationships of Eq. 7.36 and 7.37 have beendeveloped without regard to the flow regime; thus, itfollows that they are applicable to both laminar andturbulent flow in pipes


rr is any distance measured from the centre line

Example 4

Solution:


Example 5

Solution:


Example 6

Solution:


Solution:

Example 6

Solution (cont):


These deviation from theuniform velocity distributionof ideal fluid flow

Velocity Distribution and Its Significance

The shearing stresses of laminar and turbulent flowproduce velocity distributions characterized by reducedvelocities near boundary surfaces

of ideal fluid flownecessitate alternations inthe methods for calculatingof velocity head andmomentum flux.


In many practical problems, these alternations are sosmall that they may be neglected


Correction factors



Taking A and C as typical streamlines, it is noted thateach stream line is associated with different energy line



The exact head loss


fluid mechanics-61341 - an-najah videos · 7 fluid mechanics-2nd semester 2010- [7] ... no mixing,...

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