versión impresa ISSN 0327-0793
Lat. Am. appl. res. vol.40 no.3 Bahía Blanca jul. 2010
Polar profile of the wall pressure on cylindrical bars in yawed gas flow
R. Marino†, V. Herrero‡, N. Silin†, J. Converti† and A. Clausse§, ‡
† CNEA-EA-CONICET, 8400 Bariloche, Argentina.
‡ Universidad Austral, 1063 Buenos Aires, Argentina.
§ CNEA-CONICET and Universidad Nacional del Centro, 7000 Tandil, Argentina.
Abstract - The polar distribution of wall pressures in cross flow through a cell of four cylindrical tubes inclined at different angles 30°≤α≤90° was experimentally studied using flow air at near atmospheric pressure flowing at a maximum velocity of 30.8 m/sec (2200≤R ≤6100). The experiments show that the pressure coefficient is strongly influenced by the inclination angle. For perpendicular cross flow, the experiments were compared with those obtained from potential flow theory showing poor agreement. A model based on the curvature of the stream lines in the gap between bars agrees very well with the pressure coefficient at the gap.
Keywords - Gas Flow. Yawed Flow. Cylindrical Tubes. Pressure Drop. Boundary Layer.
Any solid obstacle immersed in a flow generates a resistance, which cannot be correctly predicted unless the separation of the boundary layer on the wall in the rear of the obstacle is considered. Boundary layer theory predicts accurately the point of separation, but fails in calculating the wall pressure profile, which is responsible for the drag force. The problem of separation of boundary layer in gas flow received the attention of researchers for many years (Prandtl, 1952), mostly in aerodynamic research related to the development of aviation industry (Wu and Chen, 2000). Turbine gas generators have introduced several interesting issues, such as flow around tubes and cross flow through tube bundles amongst others. Flow around different tube configurations has also applications in nuclear industry, especially in advanced gas cooled reactors. A careless fluid dynamic design of fuel elements might lead to flow induced vibrations affecting normal behavior of the core structure (Kishor et al., 2006). Therefore the understanding of gas dynamics around fuel bundles and the generation of experimental data to support design are important issues (Fahmi et al., 1989).
The general problem of flow across circular cylinders is a well known problem of fluid dynamics. A good review can be found in Zdravkovich (2003). The issue was extensively studied in surface flow for its applications in support columns of river bridges. For internal flow, the studies were aimed mainly to heat exchangers. Fornberg (1991) analyzed the incompressible cross-flow past a row of circular cylinders. Williamson (1988, 1992) investigated the three-dimensional transition of the flow behind a circular cylinder. Schewe (1983) found that the drag coefficient and the vortex shedding frequency are not sensitive to the Reynolds number within the subcritical regime (300<Re<3x105). A number of numerical calculations of the cross-flow around cylinders were presented in 2D and 3D geometries. An updated review of these studies can be found in Zhao et al. (2009).
In numerous applications, such as the flow past cables, subsea pipelines and heat exchangers, the direction of the flow is not strictly perpendicular to the cylinder axis. This kind of flows can be ideally represented by a wake flow downstream of a yawed cylinder. Flows past a yawed cylinder have been studied by a number of investigators both experimentally and numerically (Zhao et al., 2009; King, 1977; Thakur et al., 2004; Chiba and Horikawa, 1987; Marshall, 2003; Lucor and Karniadakis, 2003; Kim et al., 2006; Fowler and Bejan, 1994). Experimental results showed that, for an isolated long cylinder, the drag coefficient normalized by the velocity component perpendicular to the cylinder, are approximately independent on the yaw angle. In case of ﬂow past a yawed cylinder of ﬁnite length, it was shown that the wake vortices far from the upstream end of the cylinder are approximately parallel to the cylinder. The vortices near the upstream end of the cylinder are aligned at an angle larger than the cylinder inclination.
In this article, the distribution of wall pressure around a bundle cell three tubes with different inclinations to the flow was experimentally and theoretically studied. Gas flow around rod bundles similar configurations were studied in the past (Peybernès, 2005; Zdravkovich, 2003), although pressure profiles on the wall tube were not reported in the open literature for such bundles.
II. EXPERIMENTAL SETUP AND METHOD
The experimental setup consists of a rectangular channel. Air is forced, by an axial compressor, through a set of cylindrical bars at variable controlled inclination. The flow is homogenized by means of three metallic grids in the inlet section (Fig. 1).
Figure 1: Inlet section (Lengths in mm)
The uniformity of the velocity profile was verified measuring its local value in the cross section by means of a hot-wire anemometer.
The flow in the test section (Figs. 2a, 2b) is forced through two metallic bars of circular cross section (OD 10.86 mm) in the center and two lateral half bars. The gap between bars is 2 mm, and the inclination of the array is measured by an attached protractor (0.5° accuracy). The outlet section (Fig. 3) prevents downstream interference on the test section flow and provides a convenient location for the measurement of bulk exit conditions.
Figure 2a: Test section (Lengths in mm)
Figure 2b: Cross view of the Test section (Lengths in mm)
Figure 3: Outlet section (Lengths in mm).
The flow rate is measured by means of an elliptical Pitot tube (Preso Ellipse) located in a stainless steel pipe SCH 30, ∅ = 2", 1.35 m long, which ensures the flow development at the point of measurement. The Pitot tube is connected to a DP Cell Honeywell ST 300. A RTD (thermal sensor) is used to measure fluid temperature (Fig. 4).
Figure 4: Mass flow measurement system. All lengths are expressed in meters.
A 0.5 mm diameter hole (at the vertical midpoint position) in one of the central bars of the test section for pressure measurement was designed to rotate on its axis. The difference between the internal pressure of the bar and the bulk pressure in the inlet section was measured with another DP Cell Honeywell ST 300 for Δp>60 mbar and a DP Cell Siemens Sitrans PDS-III for Δp≤60 mbar (Fig. 5).
Figure 5: Definition of the polar angle.
Measurements were performed for several steady gas flow rates. Once the steady state temperature was reached, the differential pressure between the bar wall and the inlet was recorded for different angular positions θ (θ = 0 being the flow direction) (Fig. 5). Measurements were also done adjusting the array of bars to different inclination angles 30°≤α≤90° respect to the flow direction.
III. EXPERIMENTAL RESULTS
Figures 6 to 12 show the θ dependence of the wall pressure coefficient CD defined as:
where ρ is the gas density, pw(θ) is the wall pressure at the polar angle θ, and p∞ and v∞ are the inlet pressure and characteristic velocity (being the mass flow rate, and ρ∞ and A∞ are the inlet air density and cross section area. Each graphic corresponds to measurements performed at constant inclination α, and the curves are parameterized with v∞. Since the measurements were symmetric for 180°≤ θ ≤360°, only the range 0°≤ θ ≤180° is shown.
Figure 6: Polar profile of the pressure coefficient for a flow perpendicular to the array of bars (α = 90°). Experimental results (symbols). Inlet flow velocity: 12.06 m/sec (), 18.66 m/sec (), 24.9 m/sec (), 30.81 m/sec ().
Figure 7: Polar profile of the pressure coefficient for a yawed flow across an array of bars inclined α = 80°. Inlet flow velocity: 12.06 m/sec (), 18.66 m/sec (), 24.9 m/sec (), 30.81 m/sec ().
Figure 8: Polar profile of the pressure coefficient for a yawed flow across an array of bars inclined α = 70°. Inlet flow velocity: 12.06 m/sec (), 18.66 m/sec (), 24.9 m/sec (), 30.81 m/sec ().
Figure 9: Polar profile of the pressure coefficient for a yawed flow across an array of bars inclined α = 60°. Inlet flow velocity: 12.06 m/sec (), 18.66 m/sec (), 24.9 m/sec (), 30.81 m/sec ().
Figure 10: Polar profile of the pressure coefficient for a yawed flow across an array of bars inclined α = 50°. Inlet flow velocity: 12.06 m/sec (), 18.66 m/sec (), 24.9 m/sec (), 30.81 m/sec ().
Figure 11: Polar profile of the pressure coefficient for a yawed flow across an array of bars inclined α = 40°. Inlet flow velocity: 12.06 m/sec (), 18.66 m/sec (), 24.9 m/sec (), 30.81 m/sec ().
Figure 12: Polar profile of the pressure coefficient for a yawed flow across an array of bars inclined α = 30°. Inlet flow velocity: 12.06 m/sec (), 18.66 m/sec (), 24.9 m/sec (), 30.81 m/sec ().
The hydraulic diameter of the system is calculated as Dh = 2/(h-1 + ε-1), where h is the channel height and ε is the gap between bars. The resulting Reynolds number, Re = v∞Dh/v ranges 2200 ≤ Re ≤ 6100.
It can be observed that CD is almost independent of the flow rate within the experimental errors. Figure 13 shows the values of CD at θ = 0 (stagnation point) for each experimental condition. It can be seen that the values are around CD = 1, although they somehow differ specially at low velocities, probably due to the larger uncertainties in this region. The lowest pressure is measured at θ = 90°. For θ > 90° there is a slight pressure recovery and then the pressure remains constant until θ = 180°, which is an indication of the detachment of the boundary layer. At α = 90° the pressure recovery occurs around θ = 110°, closer to θ = 90° than in a flow passing around an isolated bar (White, 1996); this suggests that the presence of neighbor bars favors the detachment of the boundary layer.
Figure 13: Pressure coefficient at the stagnation point (absolute errors go from 0.9 at lower flows to 0.3 at higher flows).
As the inclination of the array increases, i.e. α decreases, the minimum pressure at 90° and the back pressure at 180° increases. Also, the detachment point moves progressively to the rear, reaching θ = 140° for α = 30°. These effects are reasonable since the planar cross-section shapes tend to ellipses, which are aerodynamically better the longer is the axis parallel to the flow. Fig. 14 shows the dimensionless rear wall pressure averaged between 45 g/sec and 120 g/sec for different inclination angles. The corresponding correlation is:
Figure 14: Dependence of the rear pressure coefficient on the inclination angle.
IV. THEORETICAL ANALYSIS
A theoretical analysis was performed to check the consistency of the experimental results for flow perpendicular to the bars α = 90°. In such case, the flow is two-dimensional and some simplifications can be considered to assess the pressure coefficient.
A. Potential flow
The easiest approach that can be applied to calculate the wall pressure produced by a fluid flow around tube bundles is the incompressible two-dimensional potential flow passing through an infinite row of cylinders of radii a and distance b between centers. The stream function of a single test bar located at x = 0 and y = 0 is given by:
Generalizing for an infinite row gives:
The corresponding velocity at the wall is given by:
where y = yn + nb.
Following the current line attached to the wall of the central cylinder, the Bernoulli equation gives:
Combining Eqs. (1), (5) and (6), the dimensionless wall coefficient results:
Figure 15 compares the polar profile of the wall pressure as predicted by the incompressible potential theory. The graphic variable was changed in order to visualize differences close to CD = 1. It can be seen that the potential theory disagrees with the experiments for θ > 60°. The experimental curve corresponding to a single bar in turbulent flow is also shown for comparison. It can be seen that the influence of neighbor bars is not neglectable.
Figure 15. Comparison of polar profiles of the pressure coefficient for a flow perpendicular to the array of bars (α = 90°). Experimental results (symbols), potential flow theory (dashed curve), isolated bar in turbulent flow (solid curve) (White, 1996).
B. Wall pressure at (α = 90°)
The wall pressure at α = 90° for the case of an infinite row of cylinders perpendicular to the flow can be analyzed by considering the balance of forces on a small control volume. The pressure gradient normal to curvilinear stream lines in an invicid two-dimensional flow is given by (Shapiro, 1953):
where R is the radius of curvature of the stream lines.
Let us locate, for simplicity, the y-coordinate origin at the midpoint between two bars. From Eq. (8), the pressure along coordinate y passing through the center of the cylinders and normal to the streamlines varies according to:
The curvature radius of the stream lines R(y) is infinite at the center of the gap (y = 0), and a at the wall (y = ε/2). Assuming a linear variation of R-1(y) and a constant average velocity across the gap, Eq. (9) can be integrated to calculate the pressure difference between the midpoint (y = 0) and the wall (y = ε/2). This yields:
The average velocity in the gap, , satisfies:
where A∞ and Ag are the cross flow areas of the channel free from obstacles and the minimum cross flow area between bars. In the present case, A∞/Ag = 6.43.
From Bernoulli equation:
Combining Eq. (1) with Eqs. (10) to (12), the pressure coefficient at θ = 90° results in:
which gives a value CD(90°)= -48 very close to the experimental result (-51.0±1.2).
The polar distribution of wall pressures around a circular tube of a cell bundle in yawed air cross flow was measured for different tube inclinations and flow rates. The experiments showed that the pressure coefficient is strongly influenced by the inclination angle and almost not affected by the flow rate (within the experimental range). The rear pressure coefficient can be linearly correlated with the inclination angle.
For perpendicular flow, the experiments were compared with the potential flow theory, showing that it greatly underestimate the absolute value of the pressure coefficient. This is unfortunate since recent analytical treatments using this theory were proposed to calculate general tubes configurations (Crowdy, 2006), which would have been a useful tool in designing tube bundles for gas cross flows. However, a model based in the curvature of the stream lines in the gap between bars was proposed to assess the pressure coefficient at the gap, giving excellent results.
1. Chiba, K. and A. Horikawa, "Numerical solution for the ﬂow of viscoelastic ﬂuids around an inclined circular cylinder. II. The hydrodynamic force on a circular cylinder," Rheologica Acta, 26, 255-265 (1987). [ Links ]
2. Crowdy, D., "Analytical solutions for uniform potential flow past multiple cylinders," European Journal of Mechanics, B 25, 459-470 (2006). [ Links ]
3. Fahmi, M., M. Hussein, S. Mohamed and S. El-Shobokshy, "Experimental Investigation of the effects of extended surfaces on the performance of tube banks in cross flow," J. King Saud. Univ. Eng. Sci. Conf., Riyadh, 1, 213-228 (1989). [ Links ]
4. Fornberg, B., "Steady incompressible flow past a row of circular cylinders," J. Fluid Mech., 225, 655-671 (1991). [ Links ]
5. Fowler, A. and A. Bejan, "Forced convection in banks of inclined cylinders at low Reynolds numbers," Int. J. Heat Fluid Flow, 15, 90-99 (1994). [ Links ]
6. Kim, T., H. Hodson and T. Lu, "On the prediction of pressure drop across banks of inclined cylinders," Int. J. Heat Fluid Flow , 27, 311-318 (2006). [ Links ]
7. King, R., "Vortex excited oscillations of yawed circular cylinders," ASME J. Fluids Eng., 99, 495-502 (1977). [ Links ]
8. Kishor, K., A. Meher and R. Rama, "Optimization of the number of spacers in a nuclear fuel bundle with respect to flow-induced vibration," Nuclear Engineering and Design, 236, 2348-2355 (2006). [ Links ]
9. Lucor, D. and G. Karniadakis, "Effects of oblique inﬂow in vortex-induced vibrations," Flow, Turbulence and Combustion, 71, 375-389 (2003). [ Links ]
10. Marshall, J., "Wake dynamics of a yawed cylinder," ASME J. Fluids Eng. 125, 97-103 (2003). [ Links ]
11. Peybernès, J., "Evaluation of the forces generated by cross-flow on PWR fuel assembly," IAEA-TECDOC, 1454, 13 (2005). [ Links ]
12. Prandtl, L., Essentials of Fluid Dynamics, London, Blackie & Son Ltd (1952). [ Links ]
13. Schewe, G., "On the force ﬂuctuations acting on a circular cylinder in cross-ﬂow from subcritical up to transcritical Reynolds numbers," J. Fluid Mech., 133, 265-285 (1983). [ Links ]
14. Shapiro, A., The dynamics and thermodynamics of compressible flow, 1, 281, The Ronald Press Co., New York (1953). [ Links ]
15. Thakur, A., X. Liu and J. Marshall, "Wake ﬂow of single and multiple yawed cylinders," ASME J. Fluids Eng., 126, 861-870 (2004). [ Links ]
16. White, F.M., Fluid mechanics, MGraw-Hill, 3rd Ed., Mexico (1996). [ Links ]
17. Williamson, C., "The existence of 2 stages in the transition to three-dimensionality of a cylinder wake," Phys. Fluids, 31, 3165-3168 (1988). [ Links ]
18. Williamson, C., "The Natural and forced formation of spot-like vortex dislocations in the transition of a wake," J. Fluid Mech., 243, 393-441 (1992). [ Links ]
19. Wu, T. and C. Chen, "Laminar boundary-layer separation over a circular cylinder in uniform shear flow," Acta Mechanica, 144, 71-82 (2000). [ Links ]
20. Zhao, M., L. Cheng and T. Zhou, "Direct numerical simulation of three-dimensional ﬂow past a yawed circular cylinder of inﬁnite length," J. Fluids Structures, 25, 831-847 (2009). [ Links ]
21. Zdravkovich, M., Flow Around Circular Cylinders: 1 Fundamentals, 2 Applications, Oxford University Press, London (2003). [ Links ]
Received: June 8, 2009
Accepted: September 10, 2009
Recommended by Subject Editor: Walter Ambrosini