Flow Through a Pipe Elbow

In an engineering context, pipes and fittings are not normally the subject for turbulence modeling. There is a vast amount of tabulated data and correlations that can be used in simplified approaches such as those used in the Pipe Flow Module. But now more and more detailed studies are required, for example, to generate additional correlation data or to understand phenomena caused by the flow. One such example is found in Ref. 1, which models the flow in a 90° pipe elbow in order to understand corrosion and erosion.

This example simulates the same pipe flow as that studied in Ref. 1 using the k-ω turbulence model. The result is compared to experimental correlations.

Model Definition

The model geometry is shown in Figure 1. It includes a 90° pipe elbow of constant diameter, D, equal to 35.5 mm, and coil radius, Rc, equal to 50 mm. The straight inlet and outlet pipe sections are respectively 70 mm and 350 mm long. Only half the pipe is modeled because the xy-plane is a symmetry plane.

Figure 1: Pipe elbow geometry.

The flow conditions are typical to those that can be found in the piping system of a nuclear power plant. The working fluid is water at temperature T=90 °C and the absolute outlet pressure is 20 bar. At this state, the density, ρ, is 965.35 kg/m3 and the dynamic viscosity, μ, is 3.145 × 10−4 Pa·s. The water is approximated as incompressible.

The flow at the inlet is a fully developed turbulent flow with an average inlet velocity of 5 m/s.

Modeling Considerations

Ref. 1 shows separation after the bend which can be due to the fact that the k-ε model has a tendency to perform poorly for flows involving strong pressure gradients, separation and strong streamline curvature (Ref. 2). Therefore the k-ω model is selected model over the k-ε model.

The mesh is constructed to approximately match that used Ref. 1, with the exception that the mesh in the bend itself is unstructured.

Results and Discussion

The resulting streamline pattern is shown in Figure 2. There is a separation zone after the bend which is consistent with the results in Ref. 1. Further downstream, two counterrotating vortices form, caused by the centripetal force (Ref. 3). Observe that only one of the vortices is visible in Figure 2 since the other is located on the other side of the symmetry plane.

Figure 2: Streamlines colored by the velocity magnitude.

A quantitative assessment of the results is given by comparing the so-called diametrical pressure coefficient, ck, to a engineering correlation as is done in Ref. 1. The definition of ck is

(1)

where po and pi are the pressures where the outer and inner radii of the bend intersect the symmetry plane respectively. The diametrical pressure coefficient is commonly measured at half the bend angle, in this case at 45°. Figure 3 shows a surface plot of the pressure in a plane at 45°. po and pi are the maximum and minimum values of the pressure in Figure 3 respectively which gives po − pi ≈ 1.99 × 104 Pa. Since Uavg is 5 m/s, Equation 1 evaluates to 1.6.

Figure 3: Pressure in a plane at 45°.

Ref. 1 compares Equation 1 to the following correlation based on curvatures where Rc/D > 2:

(2)

For the pipe model studied here this evaluates to 1.42. The agreement with the value 1.6 computed above is reasonably good considering that Rc/D in this case is 1.41.

A more common engineering measure is the friction factor, ff, which is related to the head loss, hL, via

(3)

where L is the length of the pipe segment and g is the gravity constant. In the case of a bend, L is the distance along the centerline. The head loss is related to the pressure drop over the pipe segment, Δp, via

(4)

The friction factor can hence be calculated as:

(5)

Equation 5 cannot be evaluated all the way from the start to the end of the bend since there is an exit effect (see Ref. 4 for more details). It is instead evaluated from the start of the bend, 0°, to half way through the bend, that is 45°. This gives L = πRc/4 and Δp evaluates to approximately 400 Pa, which results in a friction factor:

Ref. 4 gives the following correlation for the friction factor through a curved pipe

(6)

where Dc = 2·Rc is the coil diameter. In this case, Equation 6 gives ff = 7.26 × 10−3. The difference of 3% is well within the range of the accuracy of Equation 6, which is 15%.

References

1. G.F. Homicz, “Computational Fluid Dynamic Simulations of Pipe Elbow Flow,” SAND REPORT, SAND2004-3467, Sandia National Laboratories, 2004.

2. F. Menter, “Zonal Two Equation k-ω Turbulence Models for Aerodynamic Flows,” AIAA Paper #93-2906, 24th Fluid Dynamics Conference, July 1993.

3. M.J. Pattison, “Secondary Flows,” Thermopedia,DOI: 10.1615/AtoZ.s.secondary_flows, http://www.thermopedia.com/content/1113/?tid=104&sn=1420

4. R.H. Perry and D.W. Green, Perry’s Chemical Engineers’ Handbook, 7th ed., McGraw-Hill, 1997.

CFD_Module/Verification_Examples/pipe_elbow

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select ω .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
D	35.5[mm]	0.0355 m	Pipe diameter
Lin	70[mm]	0.07 m	Inlet length
Lout	350[mm]	0.35 m	Outlet length
Rc	50[mm]	0.05 m	Coil radius
rhof	965.35[kg/m^3]	965.35 kg/m³	Density
muf	3.145e-4[Pa*s]	3.145E-4 Pa·s	Dynamic viscosity
Uavg	5[m/s]	5 m/s	Average velocity