Influence of a gap on flow around horizontal cylinder

 

Zdenek Chara¹, Pavel Vlasak¹, Jaroslav Pollert jun. ²

 

¹ Institute of Hydrodynamics, ASCR, Pod Patankou 5, Prague 6, Czech Republic

² CTU - LERMO, Prague 6, Thakurova 7, Czech Republic

¹ Pod Patankou 5, Prague 6, 166 12, Czech Republic

Tel.:+420-2-24310671, Fax: +420-2-322181, E-mail: chara@ih.cas.cz, ih@ih.cas.cz

 

 

Abstract

The paper presents results of an experimental investigation of an open channel flow around horizontal circular cylinder placed at different positions above smooth bed for cylinder Reynolds numbers in the range 2.10³-10⁴. The main attention was paid to the detail investigation of velocity distribution along a vertical going through the center of the cylinder. The experiments were conducted for two flow regimes - shallow flow (when the cylinder behaves like a weir) and backwater flow (when the water surface is practically unaffected by the presence of the cylinder). Based on measured velocity profiles the ratios of mean and maximal velocities through the gaps to the velocities over the cylinder were determined as well as the shear velocities on the channel bed. The computational software Fluent was used to simulate both types of flow over the cylinder and the results were compared with the experimental results.

 

Keywords: LDA measurements, numerical simulation, open channel, flow around cylinder

 

Introduction

A lot of research work has been carried out in the field of an open channel flow around horizontal circular cylinder placed perpendicularly to the flow direction. Such arrangement can simulate many engineering applications - subaqueous pipelines or cables, flow around elements consolidating channel beds or flow around natural obstacles (fallen trees in rivers).

Bearman and Zdravkovitch (1978) investigated the influence of gap between the cylinder and bed on vortex shedding and pressure distribution on the bed and on the cylinder. Fredsoe and Hansen (1987) used a propeller to measure the velocity profiles in the gap and between the top of the cylinder and water surface. They reported that the gap flow velocity was nearly equal to the velocity of flow above the cylinder in the contrary to the potential theory that predicts increase the gap flow velocity when the gap itself is decreasing. Chiew (1991) measured the velocity distribution over a horizontal cylinder and pressure distribution around the cylinder placed in the shallow flow where the high blockage ratio of the cylinder leaded to the choking of the flow. He presented some function describing a gap flow rate in the dependency of undisturbed approach flow depth, gap size and diameter of the cylinder.

The present study is focused on the LDA measurements of longitudinal velocity distribution in the gap and also in the part over the cylinder. From the velocities measured near the bed the shear velocities were estimated in the dependency of the gap opening. Second part of the paper shows comparison of experimental results with the simulation of the flow by the commercial software Fluent (V4.4).

 

Experimental arrangements

The experiments were performed in a horizontal laboratory flume 5.6 m long, 0.25 m wide and 0.25 m deep. Both bed and walls were made from glass tables to ensure hydraulic smooth conditions. To control the water level downstream end of the channel was equipped by the controlled weir. Two flow regimes were investigated - normal (shallow) flow and backwater flow. To justify the shallow flow condition the controlled weir was not used. The depth was relatively low and the cylinder itself acted as some kind of a weir and hence upstream the cylinder the depth was always higher than the original value of undisturbed flow.

In the second case (backwater flow) the controlled weir increased the water depth, mean velocity decreased and the cylinder did practically not affect the water surface.

Three Plexiglas cylinders of diameters 12, 20 and 30 mm were used for the backwater flow tests and the same cylinders plus a cylinder of diameter 6 mm were used for the shallow flow tests.

Reynolds numbers based on diameter of the cylinder and approaching velocity were in the range 2.10³-10⁴. The cylinders were placed 3.8m downstream the entrance section and fixed directly on side walls. The undisturbed flow depths h₀ (depths of flow without the cylinder measured in the location where the cylinders were fixed) varied from 0.074 to 0.11 m in the backwater flow, in the shallow flow the depth h₀ was 0.0425 m. The unit flow rate was kept on constant value q=19.1l/s/m. Main hydraulic parameters of flows without the cylinder are summarized in Table 1. Shear velocities were estimated from velocity gradient measured near the channel bed.

 

Table 1.

 

The velocity profiles were measured by the differential LDA system in the middle of the channel width, where quasi two-dimensional flow was observed. The system consisted of the argon-ion laser Coherent Innova 70 and the LDA components (including beam expender) assembled from the Dantec optical system 55X. The Dantec Burst Spectrum Analyzer (BSA-enhanced) was used to process velocity data collected in backscatter mode. All optical parts (transmitting section, receiving photomultiplier as well as the argon-ion laser) were placed on a remote controlled xyz support that enabled precise location of a measuring volume.

 

Results and discussion

Fig. 1 shows several non-dimensional velocity profiles in the backwater flow measured in the gap for different cylinder diameters and different gap openings. The velocities and the vertical coordinates are non-dimensioned by the average gap velocity and by the height of the gap, respectively. Similar shapes of the gap velocity profiles were also observed in the shallow flow.

 

 

Fig. 1 Velocity profiles in the gap for backwater flow

 

Fig. 2 shows the influence of the gap opening on the ratio of the maximal velocities just below the cylinders to those over the cylinders in both flow regimes. The gap opening is normalized by critical gap height G_crit. The critical gap height is gap opening for which all flow is going only through the gap. This value depends on the cylinder diameter and the undisturbed flow depth h₀ and from the experimental data the following relations were found

 

(1)

 

The second relation (for G/D>2.15) is very close to the equation published by Chiew (1991). As can be seen in the Fig. 2 the ratio of maximal velocities is increasing with decreasing gap height that is in accordance with potential theory, but the values are much smaller (0.9-1.1) than values predicted by potential theory.

 

 

Fig. 2 Ratio of maximal velocities under and over the cylinder

 

Since the flow in the gap is accelerating the streamwise negative pressure gradient (favourable pressure gradient- FPG) is increasing. The effect of FPG on flow properties may be among other described by the accelerating parameter K. Due to the circular shape of the obstacle the parameter K is just zero in the gap under the cylinder, but very close upstream of the measuring position the parameter K achieved values greater than 3.10^-6. For such high values of parameter K a departure of velocity profile from standard log-law profile could be expected, Warnack and Fernholz (1998). This was confirmed by the experiments as can be seen in Fig. 3, where an example of semi-logarithmic plot of U⁺ versus y⁺ is shown for backwater flows under the cylinder of diameter 12 mm. For comparison, a theoretical log-law profile according Coleman and Alonso (1983) is attached, too. This theoretical curve was in very good agreement with our experimental data measured in the flows without cylinders. The gap velocity data lie partly above the theoretical log-law profile mainly in the viscous sublayer where follow linear relation (U⁺=y⁺) much longer than for normal flow.

 

 

Fig. 3 Non-dimensional velocity profiles in the backwater flow for cylinder of diameter 12 mm

 

Based on the measured velocities in the laminar sublayer the shear velocities on the channel bed in vertical passing through cylinder axis were estimated supposing that the linear relation (U⁺=y⁺) is satisfied. The influence of gap openings on shear velocities is shown in Fig. 4. The shear velocities were normalized by those for flows without cylinders. For gap opening G/D greater than 2 the shear velocities are about 20% higher than for undisturbed flow, for minimal gap sizes (G/D<0.1) the shear velocities are more than 2 times higher.

 

 

Fig. 4 Variation in shear velocities at channel bed with gap opening

 

Comparison of calculated and measured velocity profiles

To simulate flow around circular obstacles the program Fluent (V4.4) was used. Two flow patterns were simulated -the cylinder of diameter 30 mm was placed on the bed (G=0) both in backwater and shallow flow. Implemented model was applied with time step 0.0002 sec. The results are presented in Fig. 5 and 6. Fig. 5 shows non-dimensional plot of measured and calculated velocity profiles over the cylinder both in the backwater flow and the shallow flow. The velocities are normalized by average velocity. The measured velocities in backwater flow are practically constant but the calculated values significantly differ mostly near the cylinder surface. In the case of the shallow flow the measured and calculated data are in better coincidence. In this case the program predicts smaller velocities near the cylinder surface. Fig.6 presents measured and calculated shape of water surface over the cylinder in the case of shallow flow. The calculated curve fits the experimental data quite good mainly downstream the cylinder. In the upstream part the calculated water surface is slightly bellow the measured data.

 

 

Fig. 5 Comparison of measured and calculated velocity profiles over the cylinder

 

 

Fig. 6 Comparison of measured and calculated water surface over the cylinder

 

 

Conclusion

From the presented paper several conclusion can be drawn:

·        According to the potential theory the ratio of maximal velocities under and over the cylinder increases with decreases gap opening, but the values vary only in the range 0.9-1.1.

·        The velocity profiles in the gap significantly differ from theoretical log-law profile due to the strong favourable pressure gradient

·        The simulation program Fluent gives relatively good results for shallow flow, but for backwater flow it greatly overestimates velocities near the cylinder surface.

 

Acknowledgement

This research was supported in part under Grant No. A2060603 of the Grant Agency of the ASCR, Project No. VS988037 of the Ministry of Education and Grant No. 103/97/0860 of the Czech Grant Agency.

 

NOTATION

D = diameter of cylinder

G = gap size

G_crit = gap size for which all flow is going through the gap

K = accelerating parameter (K=n/U²(dU/dx))

h = flow depth

h_o = undisturbed flow depth

h_top = flow depth above the cylinder

q = unit flow rate

Re = Reynolds number based on hydraulic radius

Fr = Froude number

U = longitudinal velocity component

U_m = mean velocity

U_mo = undisturbed mean velocity

U_{m bot} = mean velocity in the gap

U_top = maximal velocity above the cylinder

U_bot = maximal velocity bellow the cylinder

U_* = shear velocity at bed

U_*o = undisturbed shear velocity at bed

U⁺ = normalized (dimensionless) local longitudinal velocity (U⁺=U/U_*)

x = streamwise direction

y = vertical direction with the origin at the channel bed

y⁺ = dimensionless y-coordinate (y⁺= y U_* /n)

 

REFERENCES

[1] Bearman, P.W., and Zdravkouvich, M.M. (1978), J. Fluid. Mech., 89 (1),33-47.

[2] Chiew, Y.M. (1991), J. Wtrway., Port, Coast. and Oc. Engrg., ASCE, 117 (2), 120-135

[3] Coleman, N.L., and Alonso, C.V. (1983), J. Hydr. Engrg., ASCE, 109 (2), 175-188.

[4] Fredsoe, J., and Hansen, E.A. (1987), J. Wtrway., Port, Coast., and Oc. Engrg., ASCE, 113 (2); 139-155.

[5] Warnack, D., and Fernholz, H.H. (1998), J. Fluid Mech., 359, 357-381