## Fluid MechanicsProperties of Fluids Fluid Statics Control Volume Analysis, Integral Methods Applications of Integral Methods Potential Flow Theory Examples of Potential Flow Dimensional Analysis Introduction to Boundary Layers Viscous Flow in Pipes |
## Viscous Flow in PipesPipes are closely associated with fluid flow. Many examples of pipe flow are encounted in every-day life - for water and gas supply at houses and offices, in fluid machinery, for transporting oil over long distances, etc. A flow can be an Figure 1: Flow Classification
## Classification of Flows, Laminar and Turbulent Flows
A flow can be Laminar,
Turbulent or Transitional in nature. Very different results are obtained
depending on which type of flow is encountered.
This was demonstrated in the
experiments conducted by Figure 2: Reynolds ExperimentTo measure in detail what is happening an experiment may
be conducted using modern day electronic equipement, Hot Wire Anemometer. This instrument
measures instantaneous velocities at a point. The traces of velocity
at the three regimes of flow are shown in Figure 3: Hot Wire
Signals for Turbulent flow (top), Transitional flow (middle) and
Laminar Flow (bottom)
Flow in a pipe is laminar if the Reynolds Number (based on diameter
of the pipe) is less than Figure 4: Flow at the entrance to a pipe
Consider a flow entering
a pipe. The entering flow is assumed to be uniform, so inviscid.
As soon as the flow encounters the walls of the pipe velocity changes take place.
Viscosity imposes a "
Consequently the velocity components are each zero on the wall, ie., $$L_e/D≈4.4({Re}_D)^{1/6}\text" , for a Turbulent Flow"$$ where
At critical condition,
i.e., L for a laminar flow is _{e}/D138.
Under turbulent conditions it ranges from 18 (at Re) to _{D} = 400095 (at Re)_{D}=108## Pressure along the pipe
The forces acting along the pipe
are inertial, viscous force
due to shear and the pressure forces. Gravity can be ignored if the pipe is horizontal.
When the flow is fully developed the pressure gradient and shear forces balance each other and the flow
continues with a constant velocity profile. The pressure gradient remains constant.
In the entrance region the fluid is decelerating.
A balance is achieved with inertia, pressure and shear forces. The
pressure gradient is not constant in this part of the flow and decreases as shown in Figure 5 : Pressure
distribution along the flow in a pipe.
## Fully Developed Laminar Flow in a PipeConsidering a fully developed laminar flow in a pipe it is possible to derive an expression for the velocity profile and then calculate useful practical results. This derivation can be carried out in a number of ways - (1) by a Control Volume Analysis, (2) from Navier-Stokes Equation or (3) by Dimensional Analysis. ## Volumetric Flow Rate
A quantity of interest in pipe flows is the volumetric flow rate, which
is obtained by integrating the velocity profile. Considering a disc
of thickness integrating leading to
If an average velocity, The volumetric flow rate written in terms of pressure gradient becomes, ## Correction for non-horizontal pipes
If the pipe considered in
the previous analysis was not horizontal, then
gravity effects should be included when calculating velocity and volumetric flow rates.
Referring to Figure 8: Flow through an inclined pipe.
Accordingly, the force balance becomes The pressure difference term in other equations needs to be replaced as well. Accordingly, and ## Energy Considerations, Friction factorFigure 9: Energy balance for a pipe flow.Again considering the pipe flow as
in the previous section, an energy analysis can be carried out. With reference
to Since the flow is incompressible and the pipe cross-section area is constant, Now applying the energy equation for a steady flow, Note that every term in the above equation has the dimension of length. As a fully developed flow is being considered
, then
A force balance in the Note that, $L\sin(θ)=z_2-z_1=Δz$ Dividing by $πγR^2$ gives, An inspection of these equations shows that Again this result is valid for both laminar and turbulent flows. ## Dimensional AnalysisThrough a dimensional analysis it is possible to derive an expression for "loss of head" in a pipe flow. Assuming that pressure drop is proportional to pipe length it can be shown that Thus pressure drop now becomes, with
The non-dimensional
quantity It is possible to show that ## Turbulent Flow through PipesTurbulent Flows can be calculated using the Navier-Stokes equation along with a model for turbulence. Several models have been developed for the purpose. They start from the simple algebraic models to the most involved Reynolds stress modelling. In addition, Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) are among the recent computational methods to handle turbulent flows. A full discussion of these numerical methods is beyond the scope of this section on Fluid Mechanics.
Based on experimental findings a velocity profile can be arrived at called
the ## Logarithmic Overlap LawFigure 10 : Logarithmic Overlap LawFigure 11 : Velocity and Shear Stress profiles for a Turbulent Boundary
Layer
A typical boundary layer velocity profile for a turbulent flow about a flat plate is shown in
## Wall Layer
The wall layer is the one closest to the wall. In this layer the flow is
dominated by viscous shear force. From It is established that for this layer, where
Since, by definition, $ν=μ/ρ$, the wall layer extends from the wall to a ## Overlap LayerOverlap layer is in between the wall layer and the outer layer. As the name indicates, both laminar and turbulent shear stresses prevail in this region. The velocity profile is given by the logarithmic law,
where the Karman constant, ## Outer Layer Outer Layer is next to
the Overlap Layer. It is found that in this layer ## Power Law Velocity ProfileIn many engineering calculations, it is possible to simplify this complex velocity profile model by using a simple power law approximation,
The exponent " For a power law it can be shown that the ratio of average velocity to the centreline velocity is given by, |