Example Turbulent Flow in a Pipe

Next consider flow in a pipe at a flow rate high enough for turbulence to occur. Generally, this occurs whenever the Reynolds number is greater than 2200, where the Reynolds number is defined in terms of the average velocity and the pipe diameter  

You can solve the problem with an inlet velocity that is flat thus you find the entry length it takes to achieve fully developed turbulent flow, and the velocity profile downstream is the fully developed one. When you solve for a kinematic viscosity of 10 m /s (water), diameter of 0.05 m (about 2 inches) and a velocity of 2 m/s (a common optimal velocity), you will obtain a Reynolds number of 10.  

The turbulence model in FEMLAB is in dimensional (SI) units. Turbulence is modeled using the fe-e model. In this model, the turbulent kinetic energy is represented by k, and the rate of dissipation of turbulent kinetic energy is represented by s. Furthermore, the viscosity is augmented by a turbulent eddy viscosity, which is a function of k and s. Special equations have been developed for both variables, and these must be solved along with the momentum equation which has the turbulent eddy viscosity in it as well. All these equations are included in FEMLAB. 

The main caution is that these equations are only an approximation of reality, and the formulas and equations have been chosen to model experimental data. This means that they are only as good as the data they were derived from, and the data is usually found in relatively simple flows. The experiments are definitely not easy to carry out, but they are for idealized situations, such as fully developed flow in a pipe or past a flat plate or in a jet. Thus, the equations should be used with caution. Other methods in turbulence are much more time-consuming and may require banks of computers running for hours. That works for research, but is not suitable for day-to-day engineering work. 

Step 2 Set up the entry problem as was done for laminar flow, except put in a kinematic viscosity of 10 , a radius of 0.025, an inlet velocity of 2 m/s, and a length of 3 m. Note that the momentum equations has been divided by density  

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