1. Laminar Flow vs. Turbulent Flow in a Cylindrical Tubeby
Sarat Kunapuli, Sim-Siong Wong, Josh Rexwinkle
GROUP 13

2. Introduction
The study of fluid dynamics involves the explanation and description of the flow of liquids and gases. The are many different kinds of fluid flows, but the one that is most common to us is laminar fluid flow. It is not until now that we have started learning about turbulent flow, which is the kind of flow observed in many technologically significant operations. In the following presentation, we shall discuss and compare both the laminar and turbulent flows.

3. Laminar Flow
In this case, the fluid that is passing through the cylinder is:
- Newtonian (constant viscosity and density)
- flow is isothermal
- is incompressible
- a steady state flow (unchanging in time)
- fully developed flow
Laminar means that the flow is unidirectional and has a single component. In this case, it is v = (0,0,vz ). Laminar flows are simple flows are simple enough to be used for engineering design and are also known as ideal flows. Laminar flow also depends on a dimensionless number called the Reynolds number.

4. The Reynolds number can be expressed as Re = (4Q/D)
The Reynolds number can be expressed as Re = (4Q/D). If Re < 2000, the flow would be considered laminar. An example of a velocity profile of a laminar flow is :
vz = (-CR^2/4)  [ 1 - (r/R)^2]
In a laminar fluid flow, the fluid flows in the shape of a parabola. The following diagram demonstrates a laminar flow in a cylindrical tube.

5. From the diagram from the previous page, we can tell that the velocity
is maximum at the center and zero at the sides (assuming no-slip
boundary conditions).
Turbulent Flow
In a turbulent flow, the component of velocity fluctuates randomly about
mean values. The flow is chaotic at some level. The criterion for
turbulent flow in a cylindrical tube is based on the value of the Reynolds
number. If Re > 4000, the flow is turbulent. The velocity profile for
turbulent flow is uniform with a laminar flow boundary layer.

6. Turbulent flow denotes an unsteady flow condition where streamline
interact causing shear plane collapse and mixing of the fluid.
The seventh power law
Vave = (1 /A)   vz r d dr
Vave = (2 /A)   vz max ((R-r)/R))1/7r d dr
Vave = -(2 /R2)   vz max (Z/R)1/7 (R-Z)dZ let Z = R -r, r =R -Z

7. Vave = (-2vz max / R2)  [(R/R1/7) R0 Z1/7 dZ (-1/R1/7) R0 Z8/7 dZ]
Vave = (-2vz max / R2)  [(R/R1/7) ((7/8)Z8/7)|R0 (-1/R1/7) ((7/15) Z15/7)|R0]
Vave = (-2vz max / R2)  [(-R/R1/7) ((7/8)R8/7) + (1/R1/7) ((7/15) R15/7)]
Vave = (-2vz max / R2)  [ (-7/8)R2 + (7/15)R2]
Vave = (-2vz max)  [(105/120) - (56/120)]
2 (49/120)vz max = (98/120) vave

8. vz max = Vave/0.82
Q = Vave  A
Q = 0.82 vz max  r2
Q = (vz max  r2)