Laminar and Turbulent Flows in Pipes
We consider water (at standard 293.2 K temperature) flowing through a long straight pipe with circular cross section of d = 0.1 m (see Figure 1). At the pipe inlet the velocity is uniform and equal to uin. At the pipe outlet the static pressure is equal to 1 atm.
The geometry model used for all the 3D pipe flow calculations is shown in Figure 2 The initial conditions have been specified to coincide with the inlet boundary conditions. The computational domain is reduced to domain (Z ≥ 0, Y ≥ 0) with specifying the flow symmetry planes at Z = 0 and Y = 0.
According to theory (Ref. 21), the pipe flow velocity profile changes along the pipe until it becomes a constant, fully developed profile at a distance of Lin from the pipe inlet. According to Ref. 21, Lin is estimated as:
where Red = ρ·U·d/µ is the Reynolds number based on the pipe diameter d, U is the mass-average flow velocity, ρ is the fluid density, and μ is the fluid dynamic viscosity.
Therefore, to provide a fully developed flow in the pipe at Red under consideration, we will study the cases listed in Table 1. Here, Lpipe is the overall pipe length. All the Flow Simulation predictions concerning the fully developed pipe flow characteristics are referred to the pipe section downstream of the inlet section.
Red |
uin, m/s |
Lin, m |
Lpipe, m |
|---|---|---|---|
0.1 |
10-6 |
0.3 |
0.45 |
100 |
0.001 |
0.3 |
0.45 |
1000 |
0.01 |
3 |
4.5 |
104 |
0.1 |
4 (5)* |
6 (10)* |
105 |
1 |
4 (5)* |
6 (10)* |
106 |
10 |
4 (5)* |
6 (10)* |
*) the lengths in brackets are for the rough pipes.
The flow regime in a pipe can be laminar, turbulent, or transitional, depending on Red. According to Ref. 21, Red = 4000 is approximately the boundary between laminar pipe flow and turbulent one (here, the transitional region is not considered).
Theory (Refs. 21 and 25) states that for laminar fully developed pipe flows (Hagen-Poiseuille flow) the velocity profile u(y) is invariable and given by:
where R is the pipe radius, and dP/dx is the longitudinal pressure gradient along the pipe, which is also invariable and equal to:
The Flow Simulation predictions of dP/dx and u(y) of the laminar fully developed pipe flow at Red = 100 performed at result resolution level 6 are presented in Figure 3 and Figure 4. The presented predictions relate to the smooth pipe, and similar ones not presented here have been obtained for the case of the rough tube with relative sand roughness of k/d = 0.2…0.4 %, that agrees with the theory (Ref. 21).
From Figure 4 one can see that after an entrance length of about 0.15 m the pressure gradient predicted by Flow Simulation coincides with the one predicted by theory. Therefore, the prediction of pipe pressure loss is excellent. As for local flow features, from Figure 3 one can see that the fluid velocity profiles predicted at the pipe exit are rather close to the theoretical profile.
The velocity profile and longitudinal pressure distribution in a smooth pipe at Red = 105, i.e., in a turbulent pipe flow regime, predicted by Flow Simulation at result resolution level 6 are presented in Figure 5 and Figure 6 and compared to theory (Ref. 21, the Blasius law of pressure loss, the 1/7-power velocity profile).
Then, to stand closer to engineering practice, let us consider the Flow Simulation predictions of the pipe friction factor used commonly and defined as:
where L is length of the pipe section with the fully developed flow, along which pressure loss ∆P is measured.
In Figure 7 and Figure 8 (scaled up) you can see the Flow Simulation predictions performed at result resolution level 5 for the smooth pipes in the entire Red range (both laminar and turbulent), and compared with the theoretical and empirical values determined from the following formulas which are valid for fully-developed flows in smooth pipes (Refs. 21, 14, and 25):
It can be seen that the friction factor values predicted for smooth pipes, especially in the laminar region, are fairly close to the theoretical and empirical curve.
As for the friction factor in rough pipes, the Flow Simulation predictions for the pipes having relative wall roughness of k/d = 0.4% (k is the sand roughness) are presented and compared with the empirical curve for such pipes (Refs. 21, 14, and 25) in Figure 8. The underprediction error does not exceed 13%.
Additionally, in the full accordance with theory and experimental data the Flow Simulation predictions show that the wall roughness does not affect the friction factor in laminar pipe flows.
