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
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid3 Dr. Sameer Shadeed
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
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid4 Dr. Sameer Shadeed
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
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid5 Dr. Sameer Shadeed
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
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid6 Dr. Sameer Shadeed
Schematic diagram of Reynolds apparatus
Types of Flow Reynolds Experiment
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
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid7 Dr. Sameer Shadeed
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
Types of Flow Reynolds Experiment
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid8 Dr. Sameer Shadeed
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
Types of Flow Reynolds Experiment
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid9 Dr. Sameer Shadeed
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
Types of Flow Reynolds Experiment
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid10 Dr. Sameer Shadeed
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 Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid11 Dr. Sameer Shadeed
• 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
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid12 Dr. Sameer Shadeed
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
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid13 Dr. Sameer Shadeed
)FlowLaminar(21001032
103205.0
8601.06.0
R
VdR
Shear Stress in Laminar Flow
dy
dv
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid14 Dr. Sameer Shadeed
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
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid15 Dr. Sameer Shadeed
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
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid16 Dr. Sameer Shadeed
Example 3
Solution:
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid17 Dr. Sameer Shadeed
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.
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid18 Dr. Sameer Shadeed
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)
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid19 Dr. Sameer Shadeed
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
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid20 Dr. Sameer Shadeed
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.
Development of Boundary Layer
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
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid21 Dr. Sameer Shadeed
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
Development of Boundary Layer
As the fluid passes along the plate, the zonein which shearing action occurs tend tospread further outwards. This zone is knownas ‘boundaryboundary layerlayer’.
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid22 Dr. Sameer Shadeed
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
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid23 Dr. Sameer Shadeed
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
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid24 Dr. Sameer Shadeed
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
Development of Boundary Layer
Below these values, laminar boundary layersare to be expected, whereas ReynoldsReynoldsNumbersNumbers above these values suggest that theflow is transitioning to turbulent layers
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid25 Dr. Sameer Shadeed
Shear Stress and Head Loss
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid26 Dr. Sameer Shadeed
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
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid28 Dr. Sameer Shadeed
rr is any distance measured from the centre line
Example 4
Solution:
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid29 Dr. Sameer Shadeed
Example 5
Solution:
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid30 Dr. Sameer Shadeed
Example 6
Solution:
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid31 Dr. Sameer Shadeed
Solution:
Example 6
Solution (cont):
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid32 Dr. Sameer Shadeed
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.
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid33 Dr. Sameer Shadeed
In many practical problems, these alternations are sosmall that they may be neglected
Velocity Distribution and Its Significance
Correction factors
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid34 Dr. Sameer Shadeed
Velocity Distribution and Its Significance
Taking A and C as typical streamlines, it is noted thateach stream line is associated with different energy line
Fluid Mechanics-2nd Semester 2010- [7] Flow of a Real Fluid35 Dr. Sameer Shadeed