Laminar pipe flow problem
Problem statement :
Consider fluid flowing through a circular pipe of constant radius as illustrated below. The figure is not to scale. The pipe diameter D = 0.2 m and length L = 3 m. Consider the inlet velocity to be constant over the cross-section and equal to 1 m/s. The pressure at the pipe outlet is 1 atm. Take density ρ = 1 kg/ m3 and coefficient of viscosity µ = 2 x 10 -3 kg/(m*s). These parameters have been chosen to get a desired Reynolds number of 100 and don't correspond to any real fluid.
To find :
- Velocity vectors
- Velocity magnitude contours
- Pressure contours
- Velocity profile at the outlet
We'll verify the results by following a systematic process which includes comparing the results with the analytical solution in the fully-developed region.
Hand Calculations :
1. Reynolds Number = ρ*v*D/µ = 1*1*0.2/ 2 x 10 -3 = 100
2. U(velocity at outlet )(fully developed flow at centreline) = 2*V = 2m/s
3. 
Velocity of pipe at outlet (from central line to pipe wall)along diameter
| Y [ m ] | analytical final values |
| 0.00E+00 | 2.00E+00 |
| 1.01E-03 | 2.00E+00 |
| 2.02E-03 | 2.00E+00 |
| 3.03E-03 | 2.00E+00 |
| 4.04E-03 | 2.00E+00 |
| 5.05E-03 | 1.99E+00 |
| 6.06E-03 | 1.99E+00 |
| 7.07E-03 | 1.99E+00 |
| 8.08E-03 | 1.99E+00 |
| 9.09E-03 | 1.98E+00 |
| 1.01E-02 | 1.98E+00 |
| 1.11E-02 | 1.98E+00 |
| 1.21E-02 | 1.97E+00 |
| 1.31E-02 | 1.97E+00 |
| 1.41E-02 | 1.96E+00 |
| 1.52E-02 | 1.95E+00 |
| 1.62E-02 | 1.95E+00 |
| 1.72E-02 | 1.94E+00 |
| 1.82E-02 | 1.93E+00 |
| 1.92E-02 | 1.93E+00 |
| 2.02E-02 | 1.92E+00 |
| 2.12E-02 | 1.91E+00 |
| 2.22E-02 | 1.90E+00 |
| 2.32E-02 | 1.89E+00 |
| 2.42E-02 | 1.88E+00 |
| 2.53E-02 | 1.87E+00 |
| 2.63E-02 | 1.86E+00 |
| 2.73E-02 | 1.85E+00 |
| 2.83E-02 | 1.84E+00 |
| 2.93E-02 | 1.83E+00 |
| 3.03E-02 | 1.82E+00 |
| 3.13E-02 | 1.80E+00 |
| 3.23E-02 | 1.79E+00 |
| 3.33E-02 | 1.78E+00 |
| 3.43E-02 | 1.76E+00 |
| 3.54E-02 | 1.75E+00 |
| 3.64E-02 | 1.74E+00 |
| 3.74E-02 | 1.72E+00 |
| 3.84E-02 | 1.71E+00 |
| 3.94E-02 | 1.69E+00 |
| 4.04E-02 | 1.67E+00 |
| 4.14E-02 | 1.66E+00 |
| 4.24E-02 | 1.64E+00 |
| 4.34E-02 | 1.62E+00 |
| 4.44E-02 | 1.60E+00 |
| 4.55E-02 | 1.59E+00 |
| 4.65E-02 | 1.57E+00 |
| 4.75E-02 | 1.55E+00 |
| 4.85E-02 | 1.53E+00 |
| 4.95E-02 | 1.51E+00 |
| 5.05E-02 | 1.49E+00 |
| 5.15E-02 | 1.47E+00 |
| 5.25E-02 | 1.45E+00 |
| 5.35E-02 | 1.43E+00 |
| 5.45E-02 | 1.40E+00 |
| 5.56E-02 | 1.38E+00 |
| 5.66E-02 | 1.36E+00 |
| 5.76E-02 | 1.34E+00 |
| 5.86E-02 | 1.31E+00 |
| 5.96E-02 | 1.29E+00 |
| 6.06E-02 | 1.27E+00 |
| 6.16E-02 | 1.24E+00 |
| 6.26E-02 | 1.22E+00 |
| 6.36E-02 | 1.19E+00 |
| 6.46E-02 | 1.16E+00 |
| 6.57E-02 | 1.14E+00 |
| 6.67E-02 | 1.11E+00 |
| 6.77E-02 | 1.08E+00 |
| 6.87E-02 | 1.06E+00 |
| 6.97E-02 | 1.03E+00 |
| 7.07E-02 | 1.00E+00 |
| 7.17E-02 | 9.71E-01 |
| 7.27E-02 | 9.42E-01 |
| 7.37E-02 | 9.13E-01 |
| 7.47E-02 | 8.83E-01 |
| 7.58E-02 | 8.52E-01 |
| 7.68E-02 | 8.21E-01 |
| 7.78E-02 | 7.90E-01 |
| 7.88E-02 | 7.58E-01 |
| 7.98E-02 | 7.26E-01 |
| 8.08E-02 | 6.94E-01 |
| 8.18E-02 | 6.61E-01 |
| 8.28E-02 | 6.28E-01 |
| 8.38E-02 | 5.94E-01 |
| 8.48E-02 | 5.60E-01 |
| 8.59E-02 | 5.26E-01 |
| 8.69E-02 | 4.91E-01 |
| 8.79E-02 | 4.55E-01 |
| 8.89E-02 | 4.20E-01 |
| 8.99E-02 | 3.84E-01 |
| 9.09E-02 | 3.47E-01 |
| 9.19E-02 | 3.10E-01 |
| 9.29E-02 | 2.73E-01 |
| 9.39E-02 | 2.35E-01 |
| 9.49E-02 | 1.97E-01 |
| 9.60E-02 | 1.58E-01 |
| 9.70E-02 | 1.19E-01 |
| 9.80E-02 | 8.00E-02 |
| 9.90E-02 | 4.02E-02 |
| 1.00E-01 | 4.71E-06 |
Mesh :
Nodes : 4221 ; Elements: 4000
Boundary Conditions :
Mathematical model :
Velocity at oulet in ansys fluent
| Velocity u [ m s^-1 ] | Y [ m ] |
| 2.00E+00 | 0.00E+00 |
| 1.99E+00 | 1.01E-03 |
| 1.99E+00 | 2.02E-03 |
| 1.99E+00 | 3.03E-03 |
| 1.99E+00 | 4.04E-03 |
| 1.99E+00 | 5.05E-03 |
| 1.99E+00 | 6.06E-03 |
| 1.98E+00 | 7.07E-03 |
| 1.98E+00 | 8.08E-03 |
| 1.98E+00 | 9.09E-03 |
| 1.97E+00 | 1.01E-02 |
| 1.97E+00 | 1.11E-02 |
| 1.96E+00 | 1.21E-02 |
| 1.96E+00 | 1.31E-02 |
| 1.95E+00 | 1.41E-02 |
| 1.95E+00 | 1.52E-02 |
| 1.94E+00 | 1.62E-02 |
| 1.93E+00 | 1.72E-02 |
| 1.93E+00 | 1.82E-02 |
| 1.92E+00 | 1.92E-02 |
| 1.91E+00 | 2.02E-02 |
| 1.90E+00 | 2.12E-02 |
| 1.89E+00 | 2.22E-02 |
| 1.89E+00 | 2.32E-02 |
| 1.88E+00 | 2.42E-02 |
| 1.87E+00 | 2.53E-02 |
| 1.86E+00 | 2.63E-02 |
| 1.84E+00 | 2.73E-02 |
| 1.83E+00 | 2.83E-02 |
| 1.82E+00 | 2.93E-02 |
| 1.81E+00 | 3.03E-02 |
| 1.80E+00 | 3.13E-02 |
| 1.78E+00 | 3.23E-02 |
| 1.77E+00 | 3.33E-02 |
| 1.76E+00 | 3.43E-02 |
| 1.74E+00 | 3.54E-02 |
| 1.73E+00 | 3.64E-02 |
| 1.71E+00 | 3.74E-02 |
| 1.70E+00 | 3.84E-02 |
| 1.68E+00 | 3.94E-02 |
| 1.67E+00 | 4.04E-02 |
| 1.65E+00 | 4.14E-02 |
| 1.63E+00 | 4.24E-02 |
| 1.62E+00 | 4.34E-02 |
| 1.60E+00 | 4.44E-02 |
| 1.58E+00 | 4.55E-02 |
| 1.56E+00 | 4.65E-02 |
| 1.54E+00 | 4.75E-02 |
| 1.52E+00 | 4.85E-02 |
| 1.50E+00 | 4.95E-02 |
| 1.48E+00 | 5.05E-02 |
| 1.46E+00 | 5.15E-02 |
| 1.44E+00 | 5.25E-02 |
| 1.42E+00 | 5.35E-02 |
| 1.40E+00 | 5.45E-02 |
| 1.38E+00 | 5.56E-02 |
| 1.35E+00 | 5.66E-02 |
| 1.33E+00 | 5.76E-02 |
| 1.31E+00 | 5.86E-02 |
| 1.28E+00 | 5.96E-02 |
| 1.26E+00 | 6.06E-02 |
| 1.23E+00 | 6.16E-02 |
| 1.21E+00 | 6.26E-02 |
| 1.18E+00 | 6.36E-02 |
| 1.16E+00 | 6.46E-02 |
| 1.13E+00 | 6.57E-02 |
| 1.10E+00 | 6.67E-02 |
| 1.08E+00 | 6.77E-02 |
| 1.05E+00 | 6.87E-02 |
| 1.02E+00 | 6.97E-02 |
| 9.93E-01 | 7.07E-02 |
| 9.63E-01 | 7.17E-02 |
| 9.34E-01 | 7.27E-02 |
| 9.05E-01 | 7.37E-02 |
| 8.75E-01 | 7.47E-02 |
| 8.44E-01 | 7.58E-02 |
| 8.13E-01 | 7.68E-02 |
| 7.82E-01 | 7.78E-02 |
| 7.50E-01 | 7.88E-02 |
| 7.19E-01 | 7.98E-02 |
| 6.86E-01 | 8.08E-02 |
| 6.52E-01 | 8.18E-02 |
| 6.19E-01 | 8.28E-02 |
| 5.86E-01 | 8.38E-02 |
| 5.52E-01 | 8.48E-02 |
| 5.17E-01 | 8.59E-02 |
| 4.82E-01 | 8.69E-02 |
| 4.46E-01 | 8.79E-02 |
| 4.11E-01 | 8.89E-02 |
| 3.76E-01 | 8.99E-02 |
| 3.38E-01 | 9.09E-02 |
| 3.01E-01 | 9.19E-02 |
| 2.64E-01 | 9.29E-02 |
| 2.26E-01 | 9.39E-02 |
| 1.89E-01 | 9.49E-02 |
| 1.49E-01 | 9.60E-02 |
| 1.10E-01 | 9.70E-02 |
| 7.04E-02 | 9.80E-02 |
| 3.10E-02 | 9.90E-02 |
| 0.00E+00 | 1.00E-01 |
comparison of analytical values with ansys fluent values
Results :
1. Velocity contour
2. Pressure contour
3. Velocity vectors
