Experimental Investigations and Numerical Simulations of the ViscousFlow around T-Profiles

Mihai SCHIAUA1,a, Andrei DRAGOMIRESCU1,b, and Carmen-Anca SAFTA1,c

University Politehnica of Bucharest, Department of Hydraulics, Hydraulic Machinery andEnvironmental Engineering, 313 Splaiul Independentei, Bucharest, Romania

amschiaua@yahoo.com, bandrei.dragomirescu@upb.ro, csafta.carmenanca@gmail.com

Keywords: viscous flow, T-profile, vortex street, visualization, flow measurement, numerical simulation

Abstract. This paper investigates both experimentally and numerically the flow of a viscous fluidaround T-profiles, i.e. T-shaped bodies, having different geometries. Of main interest it was the vor-tex shedding behind the T-profiles with an eye towards its application to flow measurement in openchannels. Another purpose of the study was to assess to what extent different numerical models canbe used to accurately predict the flow in the vortex street behind T-profiles.

Introduction

Known as Benard-Karman vortex street, the formation of alternate vortices in the wake behind a bluffbody is one of the classical problems of fluid dynamics. The vortices are a result of the flow separationat the back of the body, which occurs from a certain value of the Reynolds number onwards. Withincreasing Reynolds numbers, the vortices start to detach periodically and alternately on either side ofthe body. They form the vortex street as they are carried off by the main flow and eventually disappearas a result of viscous friction.

Birkhoff [1] demonstrated theoretically that in an inviscid fluid the concentration of vortices islimited to circles having diameters of about 25% of the distance between two successive vortices ina row. Schlichting [2] showed that the flow in the wake behind any obstacle becomes turbulent. Also,the velocity profile in the wake at a large distance from the body is independent of the body shape.Roshko [3] observed experimentally that a geometric similarity exists between all vortex streets. Fromhis data, he derived features of a vortex street that are independent of the Reynolds number and hepostulated a universal Strouhal number that is related to wake width. Dumitrescu and Cazacu [4]carried out experimental studies of the laminar viscous flow past a flat plate placed at different attackangles in an open channel. They observed the flow patterns behind the plate at Reynolds numbersin the range between 0.2 and 2 000 for attack angles of 0◦, 45◦, and 90◦. Balan et. al. [5] undertookexperimental and numerical investigations of the planar complex flow of a weak elastic polymersolution around a T-profile placed in an open channel. It was observed that, for the same Reynoldsnumber, the wake in the polymer solution is significantly shorter than in a Newtonian fluid.

The study presented in this paper was focused on experimental and numerical investigations of aNewtonian fluid flowing around T-profiles placed in an open channel. Experimental results obtainedso far and presented in the cited literature show that the Strouhal number calculated with the frequencyof vortex shedding behind a bluff body remains practically constant and independent of the Reynoldsnumber for a certain range of Reynolds numbers. Thus, it becomes possible to determine the averageflow velocity and the flow rate by measuring the vortex shedding frequency. This is basically theworking principle of vortex meters designed for flow measurement in pressure driven flows throughpipes. The purpose of the present study was twofold. On the one hand, it was aimed at investigatingwhether a T-profile could be used as a “shedder” (or vortex generator) for flow measurement in gravitydriven flows through open channels. On the other hand, it was of interest to find a numerical model,either laminar or turbulent, which is best suited to simulate the wall-bounded flow around the T-profilewhile retaining as much as possible of the real features of such a flow. Once found, the model can be

Fig. 1. Geometry of the openchannel used to visualize theflow around T-profiles. Alldimensions are in mm.

used to easily extend the experimental results. The main reason to resort to numerical simulations forthis study is the fact that they are by far less time and resource consuming than the experiments.

In the following, the experimental installation and working procedure are described and experi-mental results are presented and discussed. Next, the numerical simulations are presented in terms ofcomputational domain, equations to be solved, boundary conditions and numerical algorithms. Thenumerical results are compared with the experimental ones. Finally, conclusions are drawn with re-gard to the possibility to use T-profiles for flow measurement in open channels and to the degree towhich different numerical models are able to correctly predict the viscous flow around such profiles.

Experimental

The viscous flow around T-profiles was studied in a closed-circuit open channel. The sketch of thechannel is depicted in Fig. 1. The channel is 3 m long, about 1.2 m wide and 0.35 m high and has fourzones: the test area (1) followed by a 180◦ turn (2), a divergent straight section (3), and a second turn(4) consisting of two 90◦ elbows. The test area has a width lC of 0.23 m and a length of 1.025 m. Thefirst turn accommodates a cross-flow runner which pumps the working fluid. The cross-flow runner isdriven by an electric motor through a reducing gear which offers the possibility to change the speed.To assure a flow free of vortices at the entrance of the test area, the elbows of the second turn areforeseen with flow straighteners and the junction between the last elbow and the test area is made bymeans of a convergent section. The test area is foreseen with a lighting system which provides enoughlight for taking pictures of the flow paths with relatively small shutter speeds.

The T-profiles were made of 3 mm thick sheets of Plexiglas. They were placed with the stemalong the axis of the working area pointing upstream and with the head pointing downstream. Themain dimensions of a profile are the stem length, b, and the head width, Lt . To easily distinguishbetween them, the T-profiles are denoted by a T followed by the length b in mm, a dash, and the widthLt in mm. For example, T500-110 is a T-profile having a stem which is 500 mm long and a headwhich is 110 mm wide.

All experiments were made with water as working fluid. To make the flow paths evident, verysmall particles of Plexiglas were used. They were blown uniformly on the free surface of the flowingwater at the entry of the test area and, after flowing around the T-profile and in its wake, they werecollected with a strainer before entering into the cross-flow runner. The flow paths were photographedwith a bridge camera that provides full manual control over aperture size and shutter speed. For eachcase, the shutter speed was set according to the flow-rate and to the average velocity of the vorticesshed by the T-profile.

On every picture, the length xm of the trace left by a Plexiglas particle along the channel axisupstream of the T-profile was measured. The exposure time texp was read from the EXIF data of the

Table 1. Blockage ratios α for different dimensions of the T-profiles used in the experiments.

Lt [mm] 110 88 66 44 22 10b [mm] (α = 2.1) (α = 2.6) (α = 3.5) (α = 5.2) (α = 10.5) (α = 23)

500 4.54 5.68 7.57 11.36 22.72 50300 2.72 3.41 4.54 6.82 13.64 –100 0.9 1.14 1.51 2.27 4.54 –25 0.22 0.28 0.38 0.57 1.14 –

picture. The maximum velocity in the channel was calculated with the formula

V =xm

texp. (1)

Successive pictures were examined in order to find successive detachments of vortices behind the T-profile. The vortex shedding frequency f was calculated as the inverse of the average time differencebetween two successive detachments. Two significant parameters of the flow are the Reynolds and theStrouhal numbers which were calculated with the formulas

Re =V lC

ν, St =

f Lt

V, (2)

where ν = 10−6 m2/s is the kinematic viscosity of water at the working temperature (about 20◦C).Another importand parameters are the blockage ratio, α = lC/Lt , the aspect ratio, λ = b/Lt , thevelocity of the vortices, VT = dpv/tdcs, and the wake length, LS. The blockage ratio is defined as theratio between channel width and profile width. The vortex velocity is calculated based on the timetdcs between two successive photos and on the distance dpv traveled by the vortex within this timeinterval. The dimensions, corresponding aspect ratios, and blockage ratios of the T-profiles used inthe experiments are summarized in Table 1.

The experiments were carried out in a channel were the lateral walls have a strong influence onthe flow around an obstacle. To account for this influence and to make possible comparisons withprevious studies on the Benard-Karman vortex street, the velocity was corrected with the formula

V ′ =α

α−1V . (3)

With this velocity corrected Reynolds and Strouhal numbers were also calculated.The experiments were performed for Reynolds numbers in the range from 4 000 to 20 000. In this

range, the flow can be considered subcritical according to Rusu [6]. Fig. 2 shows typical flow con-

a) b) c)

Fig. 2. Flow spectra observed behind T-profiles: a) T25-100 at Re= 10976, b) T100-88 at Re= 6380,c) T500-110 at Re = 17400.

a)

0

1

2

3

4

0 5000 10000 15000

St

Re

correction for =a 2.1a = 2.1

correction for =a 2.6a = 2.6

correction for =a 3.5a = 3.5

b)

0

0.5

1

1.5

2

0 10000 20000 30000 40000

St

Re

correction for 2.1a =

correction 2.6for a =

correction 10.5for a =

correction 3.5for a =

a = 2.1

a = 2.6

a = 10.5

a = 3.5

c)

correction 2.1for a =

a = 2.1

correction 2.6for a =

a = 2.6

correction 3.5for a =

a = 3.5

correction 5.2for a =

a = 5.2

correction 10.5for a =

a = 10.5

0

0

0.5

1

1.5

2

2.5

10000 20000 30000

3

40000

Re

St

d)

Fig. 3. Variations of Strouhal number depending on Reynolds number and blockage ratio forT-profiles with a) b = 25 mm, b) b = 100 mm, c) b = 300 mm, and d) b = 500 mm.

figurations observed behind different T-profiles during the experiments. In all cases alternate vorticesformed in the wake of the profiles. With the profiles in the series T25-x, the number of vortices wasbetween 4 and 9 at different flow times. The picture changed for the profiles in the series T100-x,T300-x, and T500-x. As long as the Reynolds number remained below 7 000, up to 4 vortices wereobserved. With increasing Reynolds numbers, the vortices dissipated more rapidly, so that their num-ber decreased. Thus, for Reynolds numbers higher than 11 000 only 2 to 3 vortices were observed.

The variation of the Strouhal number depending on Reynolds number is presented in Fig. 3. Itcan be seen that the Strouhal number remains almost constant at higher values of the blockage ratio,typically for α ≥ 3.5. This is a favorable finding from the point of view of using a T-profile as vortexgenerator in a flow meter for open channels. When the Strouhal number is constant and known be-forehand as a result of experiments, only the vortex shedding frequency f must be measured in orderto determine the flow rate. The reference velocity in the channel can be calculated with the formula

V =f Lt

St. (4)

By multiplying this velocity with the cross-sectional area of the flow and with an appropriate dischargecoefficient found experimentally, the flow rate can be easily determined. It should be noted that, sincewe deal with a free surface flow, the depth of the fluid in the channel must also be measured.

Numerical simulations

The simulations presented in this paper were carried out for the unsteady viscous flow around the pro-file T500-110. The computational domain, presented in Fig. 4, was meshed with an unstructured gridconsisting in 50 838 cells of triangular shape and of boundary layer type. The continuity and momen-

3000

230

500

110

inlet outlet

solid wall

solid wall

Fig. 4. Computational domain for profile T500-110. All dimensions are in mm.

tum equations that describe the flow were integrated numerically in space and time using the FiniteVolume Method and the segregated pressure based solver implemented in the commercial programFluent. The equations were discretized in space with a second order upwind scheme and in time withsecond order accurate backward differences. A first set of simulations were made in the hypothesisthat the flow is laminar. Subsequent simulations were performed considering the flow turbulent andusing the following turbulence models: standard k-ε , SST k-ω (Shear Stress Transport k-ω), and RMS(Reynolds Stress Model). Since no experimental data regarding turbulence was available, the defaultvalues of the turbulence models were kept unchanged. The working fluid was water with the densityρ = 1000 kg/m3 and the dynamic viscosity µ = 0.001 Pas. A mass flow rate of 17.4 kg/s was im-posed as boundary condition at inlet, while the boundary condition at outlet was the evacuation of theentire mass flow rate. To the mass flow rate corresponds an average velocity of 0.07565 m/s that givesa Reynolds number of 17 400 which matches the experiment depicted in Fig. 2c. For all turbulencemodels, a turbulence intensity of 2% was assumed at the inlet. At all solid walls the usual conditionof no-slip was prescribed. The time step used in simulations was of 0.1 s except of the laminar casethat required a time step of 0.01 s.

All simulations were performed in two stages. In the first stage – the flow build up stage –, thenumerical integrations were advanced in time until a stabilization of the drag coefficient of the T-profile around an average value was observed for more than 30 s of the simulation time. In the secondstage, the flow equations were integrated for another 90 s. At each time step, the convergence criterionwas the drop of the residuals of all equations below 10−4.

Some of the results obtained are presented in Fig. 5. These results are to be compared with theexperimental flow depicted in Fig. 2. As it can be seen, the laminar model is able to reproduce thevortex shedding (Fig. 5a). However, in comparison with the vortices observed experimentally, thoseobtained numerically seem to be “squeezed” behind the T-profile. Neither the k-ε model (Fig. 5b), northe SST k-ω model (Fig. 5c) were able to predict vortex detachment. In both cases, the flow remainedsymmetrical, showing two identical stationary vortices behind the profile. The closest to reality wasthe RSM model. When comparing Fig. 2 and Fig. 5d, it can be seen that the RSM model reproduceswith good accuracy in terms of position and size both the vortices shed behind the T-profile and thesmall vortices that appear from time to time at the solid walls.

Conclusions

The viscous flow of a Newtonian fluid around T-profiles placed in an open channel was investigatedboth experimentally and numerically. The study aimed at assessing to what extent T-profiles couldbe used as “shedders” for flow meters and what are the most appropriate numerical models that bestretain the characteristics of wall-bounded free surface flows around T-profiles.

The experiments showed that, for values of the blockage ratio larger than 3.5, the Strouhal numbercalculated with the frequency of the vortex shedding behind the profiles remains almost constant.The small deviations of the Strouhal number from constant values are most probably caused by theinherent measurement errors. Nevertheless, the experimental findings allow us to conclude that T-

Contours of Stream Function (kg/s) (Time=1.1050e+02)T500-110, Re=17400, laminar

FLUENT 6.3 (2d, dp, pbns, lam, unsteady)Feb 12, 2013

3.37e+01

3.20e+01

3.03e+01

2.86e+01

2.69e+01

2.52e+01

2.36e+01

2.19e+01

2.02e+01

1.85e+01

1.68e+01

1.51e+01

1.35e+01

1.18e+01

1.01e+01

8.41e+00

6.73e+00

5.05e+00

3.37e+00

1.68e+00

0.00e+00

a)

Contours of Stream Function (kg/s) (Time=1.2000e+02)T500-110, Re=17400, standard k-epsilon

FLUENT 6.3 (2d, dp, pbns, ske, unsteady)Feb 12, 2013

1.74e+01

1.65e+01

1.57e+01

1.48e+01

1.39e+01

1.31e+01

1.22e+01

1.13e+01

1.04e+01

9.57e+00

8.70e+00

7.83e+00

6.96e+00

6.09e+00

5.22e+00

4.35e+00

3.48e+00

2.61e+00

1.74e+00

8.70e-01

0.00e+00

b)

Contours of Stream Function (kg/s) (Time=1.2000e+02)T500-110, Re=17400, SST k-omega

FLUENT 6.3 (2d, dp, pbns, sstkw, unsteady)Feb 12, 2013

1.74e+01

1.65e+01

1.57e+01

1.48e+01

1.39e+01

1.31e+01

1.22e+01

1.13e+01

1.04e+01

9.57e+00

8.70e+00

7.83e+00

6.96e+00

6.09e+00

5.22e+00

4.35e+00

3.48e+00

2.61e+00

1.74e+00

8.70e-01

0.00e+00

c)

Contours of Stream Function (kg/s) (Time=6.8500e+01)T500-110, Re=17400, RSM

FLUENT 6.3 (2d, dp, pbns, RSM, unsteady)Feb 12, 2013

1.94e+01

1.84e+01

1.75e+01

1.65e+01

1.55e+01

1.45e+01

1.36e+01

1.26e+01

1.16e+01

1.07e+01

9.70e+00

8.73e+00

7.76e+00

6.79e+00

5.82e+00

4.85e+00

3.88e+00

2.91e+00

1.94e+00

9.70e-01

0.00e+00

d)

Fig. 5. Streamlines of the flow behind profile T500-110 ob-tained with different models:a) laminar (at t = 110.5 s),b) standard k-ε (at t = 120 s),c) SST k-ω (at t = 120 s),d) RSM (at t = 68.5 s).

profiles having properly chosen dimensions could be successfully used as vortex generators for flowmeters in gravity driven flows through open channels.

Numerical simulations were carried out only for the flow behind profile T500-110 at Re = 17400.Of the models chosen, the standard k-ε model and the SST k-ω model were unable to reproduce anyvortex shedding. Only the laminar model and the Reynolds Stress Model were able to predict thevortex detachment behind the profile, the latter offering results that are the closest to reality.

References

[1] G. Birkhoff, The formation of vortex street, Journal of Applied Physics 24 (1953) 98-103.

[2] H. Schlichting, Boundary-Layer Theory, McGraw Hill Book Co., Columbus, 1968.

[3] A. Roshko, On the drag and shedding frequency of two-dimensional bluff bodies. TechnicalReport TN 3169, NACA, US Government Printing Office, Washington DC, 1954.

[4] D. Dumitrescu, M.D. Cazacu, Theoretical and experimental considerations on the flow of viscousfluids around a plate at low and average Reynolds numbers, ZAMM 50 (1970) 257-280.

[5] C. Balan, V. Legat, A. Neagoe, D. Nistoran, Experimental investigations and numerical simula-tions for an open channel flow of a weak elastic polymer solution around a T profile, Journal ofExperiments in Fluids 36 (2004) 408-418.

[6] I. Rusu, Phenomena in Hydro-Elasticity, Junimea, Iasi, 1998.