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EML 6930 Computational Fluid Dynamics, Spring, 2020
Please submit answers, all figures and tables in a single pdf or docx format. Submit the required Excel
file as well. For the simulation files, only submit .cas.gz and .dat.gz for each simulation (total 8
Problem 1 (50 pts)
Simulate the steady laminar and turbulent pipe flow using Ansys Fluent as shown in the figure. Choose
proper fluid properties and inlet velocity to simulate flows of Reynolds number (based on the diameter
of the pipe) equaling 655 for laminar flow and 111,569 for turbulent pipe flow, respectively.
Table 1 - Main Dimensions
Radius of Pipe, R
Diameter of Pipe, D
Length of the Pipe, L
Uniform velocity profile is specified at inlet, and the flow will reach the fully developed regions after a
certain distance downstream. No-slip boundary condition will be used on the wall, and zero gauge
pressure for the outlet. You may consider creating a quarter of the pipe to take the advantage of
symmetry of the geometry. This could greatly decrease the number of cells in your mesh.
a) List the material properties and inlet velocities used for your simulations, both laminar and
b) Create three different 3D meshes: coarse, medium, and fine (Note the maximum allowed
number of cells is 512k for the student version). List parameters you changed when creating the
meshes, total number of cells, and minimum Orthogonal Quality. Also use the 2D mesh provided
to perform simulations (When launching Fluent, choose 2D mode and change to Axisymmetric in
Physics>General… tab). Conducting simulations for all meshes (total number of simulations is 8).
Plotting residuals and check the mass balance.
c) Create lines shown in the table below. Here, we assume the x axis is in the axial direction. You
may need to change the coordinates according to the orientation of your geometry.
Use XY Plot to plot velocity profiles at these lines in a single figure for laminar and turbulent
simulations, respectively. Similarly, plot static pressure and velocity magnitude along the
centerline. For all these figures, only show results from the fine 3D mesh and 2D mesh.
d) Extract the Wall Shear Stress values (𝜏 ) along the axial direction on the pipe wall. You can save
it as a .xy file from Fluent and open it with a text editor, and copy the value at x≈7 to the Excel
file. Similarly, extract the pressure value along the centerline to calculate pressure drop from
the inlet to the outlet. Extract the velocity values along the “outlet-line” and copy them to the
Excel. It will be used for plotting the fully-developed velocity profile with analytical or empirical
data (AFD used later is the abbreviation of it). List the y+ value at the end of pipe for all
e) Fill the blanks in the Excel table below for verification and validation. Show proper procedures
for obtaining AFD data. Discuss which mesh solution is closest to the AFD data, and explain why
this is the case? Plot the fully-developed velocity profiles from all meshes with AFD data in a
single figure for laminar and turbulent flows, respectively (You can do it in Excel). Discuss the
developing length for laminar and turbulent pipe flows and compare them with that using the
formulas in the theoretical document.
To know more about analytical or empirical solutions of laminar and turbulent pipe flows, refer to the
theoretical document attached.
In the Excel file, create a table like this and fill the blanks accordingly. U m is the maximum/centerline
velocity at the outlet.
CFD, 𝜏 at
CFD, Um at
Theoretical Fluid Mechanics for Laminar and Turbulent Pipe Flow
For a uniform flow inlet, the flow can be divided into two regimes: developing profile region
and fully-developed flow region. The corresponding velocity profiles and pressure drops are
showing in the figure.
For a laminar flow, the entrance length can be estimated by 𝐿𝑒 ⁄𝐷 ≅ .06𝑅𝑒, where D is the
diameter of the pipe and characteristic length used in the Re number. For a turbulent flow, the
entrance length is 𝐿𝑒 ⁄𝐷 ~4.4𝑅𝑒 1/6.
The skin friction coefficient is a dimensionless skin shear stress which is nondimensionalized by
the dynamic pressure of a free stream, 𝐶𝑓 = 0.5𝜌𝑉
2 , where 𝜏𝑤 is the local wall shear stress and
V is mean velocity at the inlet. For a laminar flow, the skin friction coefficient is related to
Reynolds number as 𝐶𝑓 = 𝑅𝑒. For a turbulent pipe flow, the empirical solution is 𝐶𝑓 =
0.079𝑅𝑒 −0.25. The Darcy friction factor f is also widely used in the literature, which is related to
the skin friction coefficient as 𝑓 = 4𝐶𝑓 .
It is well-known that the fully-developed velocity profile shows a parabolic shape in a laminar
pipe flow from Hagen–Poiseuille equation as, 𝑈 = 1 − (𝑅) , where R is the radius of the pipe,
𝑈𝑚 = 2𝑉 is the maximum velocity and V is the mean velocity.
For a turbulent flow in a circular pipe, 𝑈 = 1 − 2.5 𝑈 𝑙𝑛 𝑅−𝑟, where the maximum velocity
𝑈𝑚 = (1 + 1.326√𝑓)𝑉, and the friction velocity 𝑢 = √𝜏𝑤 ⁄𝜌.