Experimental and numerical study on the flapping motion of ...

SCIENCE CHINA Technological Sciences Progress of Projects Supported by NSFC

October 2013 Vol.56 No.10: 2391–2397 doi: 10.1007/s11431-013-5333-z

Experimental and numerical study on the flapping motion of submerged turbulent plane jet SUN JianHong1*, ZHAO LiQing1 & HSU ChinTsau2 1

College of Aerospace Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China; Department of Mechanical Engineering, Hong Kong University of Science & Technology, Hong Kong, China

2

Received June 28, 2013; accepted August 15, 2013; published online September 9, 2013

A submerged, vertical turbulent plane water jet impinging onto a free surface will be self-excited into a flapping oscillation when the jet velocity, leaving the jet orifice, exceeds a critical value. The flapping phenomenon was verified simultaneously in this paper by laser Doppler velocimeter measurement and numerical analyses with volume of fluid approach coupled with a large eddy simulation turbulent model. The general agreement of mean velocities between numerical predictions and experimental results in self-similar region is good for two cases: Reynolds numbers 2090 and 2970, which correspond to the stable impinging jet and flapping jet. Results show that the flapping jet is a new flow pattern for submerged turbulent plane jets with characteristic flapping frequency, and that the decay of the mean velocity along the jet centerline is considerably faster than that of the stable impinging state. turbulent plane jet, flapping motion, laser Doppler velocimeter, volume of fluid, large eddy simulation Citation:

Sun J H, Zhao L Q, Hsu C T. Experimental and numerical study on the flapping motion of submerged turbulent plane jet. Sci China Tech Sci, 2013, 56: 23912397, doi: 10.1007/s11431-013-5333-z

1 Introduction Jets have been studied extensively due to their wide engineering application. One such example is the disposal of wastewater through diffusers at the bottom of rivers, lakes or ocean. For a submerged turbulent plane jet in water of finite depth impinging vertically onto free surface, the jet will be self-excited to flap horizontally when the jet velocity, leaving the jet orifice, exceeds a critical value. One important characteristic of the flapping jet is that it can enhance mixing by flapping-induced Reynolds stresses [1]. In some cases the instability of jet can be utilized to enhance the dilution. There is therefore a considerable interest in the design and control of diffusers mentioned above. The flapping motion in a turbulent plane jet was first *Corresponding author (email: [email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2013

studied by Goldschmidt and Bradshaw [2]. They investigated the “puffing” and “flapping” modes of oscillation in two-dimensional (2D) jets discharging into an infinite stagnant ambient and showed the existence of the “flapping” mode. The flapping behavior was associated with the coherent structure in the turbulent plane jet. The coherent structure in a turbulent plane jet was visualized by Goldschmidt et al. [3] with a smoke wire method. However, Wygnanski and Gutmark [4] and Moum et al. [5] did not detect the flapping mode but showed only the existence of large eddy structures. Thereafter, the flapping motion of a turbulent plane jet was confirmed by Hsu et al. [6] and Sun [7] using a laser induced florescence (LIF) technique. Hsu et al. [6] suggested that the flapping behavior was attributed to the self-excited jet instability, associated with the pressure restoring force, due to gravity by surface displacement. Recently, Espa et al. [8] conducted experiments and smoothed tech.scichina.com

www.springerlink.com

2392

Sun J H, et al.

Sci China Tech Sci

particle hydrodynamics simulations to study the instability of a vertical plane jet introduced from the bottom of a finite depth laterally-confined water environment. It should be noted that the flow region in Espa’s study was laterally confined and the ratio of water depth to orifice width H/d was too small to reach the self-similar regime [9] for a submerged plane jet. On the other hand, a few studies have been done to determine the critical condition for the plane jet instability. Madarame et al. [10] observed the self- induced oscillations of a vertical jet and established a relation between the flapping frequency and the water depth. Wu et al. [11] delineated the critical jet exit velocity and the jet flapping frequency in a dimensional form. Hsu et al. [6] provided more general dimensionless parametric relations for the onset of flapping instability. Till now, detailed flow field characteristics of such flapping plane jets have received little attention in literature. Meanwhile, main research work was carried out by experimental study, no clear numerical results about the flow structures of flapping plane jet have been reported. In this study, to investigate the flow field characteristics of flapping jet, refined experiments were conducted at two Reynolds numbers Re=2090 and 2970, which corresponded to stable impinging jet and flapping jet. The ratio of water depth to orifice width H/d was kept at 50. The jet velocities were measured with laser Doppler velocimeter (LDV). In addition to the results of detailed measurements, the flow field will be visualized numerically using the large eddy simulation (LES) approach. The numerical study is undertaken to well understand the flow structure of the flapping motion.

2 Experimental set-up As shown in Figure 1, the flow facility consisted of a water tank, a jet generator and water recirculation system. The jet was generated by pumping water through a jet generator located at the middle of the water tank, which is 6 m long, 0.4 m wide and 0.4 m high. The jet generator has a conver-

October (2013) Vol.56 No.10

gent bell-shaped jet orifice of 395 mm long and 2 mm wide (d=2 mm). With the water depth of H=100 mm, we have the ratio of water depth to orifice width H/d=50. To maintain the water level at constant, two outlets located at the far ends of the tank bottom were connected to the pump and then to the jet generator, to form a water recirculation system [7]. The flow field was visualized by a laser induced fluorescence (LIF) system, using a dye of water solution of Rhodamine 6G. The images were recorded by a 3-CCD digital video camera. The velocities of the turbulent jet were measured with a LDV system mounted on a three-dimensional traverse. To increase the data rate of the LDV system, the flow field was seeded with titanium oxide (TiO2) particles of 1−5 m in diameter. As shown in Figure 1, the coordinates (x, z) were chosen such that the jet was symmetric to the plane x=0 and z was measured upward from the jet orifice.

3 Numerical procedure To predict the highly oscillatory nature of the flapping jet, volume of fluid approach coupled with a large eddy simulation (LES) turbulent model is applied in this study. The threedimensional time dependent space filtered Navier-Stokes (N-S) equations can be written as [12]  u   u u   p   g     u   T u   t 





 FST  C     SGS ,

 (1)

  u  0,

(2)

C  u C  0, t

(3)

where an overbar denotes the spatial filter and a tilde represents the Favre filter, i.e.     . The variable  is the volume mass, u velocity, p pressure, g the gravitational acceleration,  the dynamic viscosity, FST the surface tension force and C the volume fraction. When the filtered volume fraction C is not constant, the filtered density  and viscosity  can be prescribed by

  C  1  1  C   0 ,   C  1  1  C   0 .

(4)

The variables with superscripts 1 and 0 refer to variables in the phase of water and air, respectively. Sub-grid closure term in eq. (1) is defined as .  SGS   uu   uu Figure 1 Schematic of the experimental set-up: water tank, jet generator and water recirculation system.

(5)

In LES, the Smagorinsky-Lilly model was used to model

Sun J H, et al.

Sci China Tech Sci

the sub-grid scale (SGS) stress tensor, shown as,

and time step 0.1d Wo .

 SGS   t u   T u   2t S , 1  u u j Sij   i  2  x j xi

(6)

4

  , 

(7)

where t   L2s S is the SGS viscosity, and S  2 Sij Sij .



Ls is the length scale and is defined as Ls  min  d , Cs 



with k=0.42. The variable d is the distance to the closest wall. The value of Smagorinsky constant is chosen as Cs= 0.1, which has been proven to yield good results for a wide range of flow conditions. The governing equations are numerically solved by SIMPLE method [13]. In eq. (1), a continuum surface model proposed by Brackbill et al. [14] is implemented for the treatment of the surface tension effects. Spatial discretization is achieved using a second-order-accurate central scheme. Temporal integration of the transient terms is performed by a second-order-accurate implicit scheme. This numerical strategy has already been applied with success to a wide range of turbulent shear flows such as jets in cross-flow [15, 16] and the temporal mixing layer [17]. We have carefully examined the physical model and numerical approach used in this study and verified that the calculated results are reliable. As shown in Figure 2, water enters vertically into the computational domain of 30H×10d×5H with uniform jet exit velocity Wo. For the purpose of maintaining water level at constant in simulation, two uniform velocity outlets are applied at the two ends of the tank bottom. The upper boundary condition for air in the calculation is set to inlet condition with the same pressure. Periodic boundary conditions are applied in the spanwise (y) direction. Other boundaries are set to solid wall with no-slip boundary condition. The three-dimensional computer model of the jet was constructed using Gridgen software, and hexahedral computational grids were generated for the geometry. After careful test, grid number is 273×81×200 in the transverse (x), spanwise (y) and longitudinal (z) directions, respectively,

Results and discussion

In the present study, the water depth and the jet orifice width are maintained at H=100 mm and d=2 mm. Based on Hsu’s empirical formula [6], the corresponding critical jet exit velocity Wcr is about 1.31 m s-1. In order to reproduce the stable impinging jet and flapping impinging jet, the jet exit velocities Wo are chosen as 1.21 and 1.72 m s1, which give Reynolds numbers of Re=Wod/v=2090 and 2970, respectively, where v is the fluid viscosity. The flow fields are visualized by LIF and measured by LDV. 4.1

Visualization of two types of flow pattern

Figure 3 shows the LIF image of stable impinging jet at Re= 2090 without flapping motion. As shown in Figure 3, the surface deforms to form a humped ridge at the free surface due to the impingement of the vertical plane jet. When the jet exit velocity Wo is smaller than the critical jet flapping velocity Wcr, the hump does not oscillate horizontally, but only fluctuates very weakly due to the impingement of the coherent large eddies. When the jet exit velocity is larger than the critical velocity (Wo>Wcr), surface deformation (SD) becomes to oscillate horizontally. Figure 4 shows the LIF images of the flapping jet in one period T. Note that the starting time in Figure 4 is arbitrary. The oscillation of SD was observed to synchronize with the jet flapping motion. As shown in Figure 4(a), the jet flaps to the left maximum displacement location at t=0. At t=T/4, the jet oscillates backward to the jet center plane above the jet orifice as shown in Figure 4(b). Figure 4(c) shows that the plane jet continues to flap towards the right of the jet orifice and reaches the right maximum displacement at t=2T/4. The plane jet then oscillates to the center plane position again at t=3T/4 as shown in Figure 4(d). Finally the jet flaps back to the left maximum displacement position at t=T as shown in Figure 4(e) to complete one cycle of oscillation. The dynamic characteristics of a jet are closely associated with the vortical structures in the flow field. To clearly demonstrate the vortex structures, the numerical results in Figure 5 show instantaneous snapshots of the flow field depicted by isosurface of the Q criterion [18] Q

Figure 2

Computational domain.

2393

October (2013) Vol.56 No.10



1  2  S   2

2

,

(8)

 and   denote the strain and the rotation tensor, where S respectively. A positive value of Q presents the regions in which the rotation exceeds the strain. After the surface impingement, the vertical turbulent jet will transform into two horizontal surface jets. For the flapping situation, as sketched

2394

Sun J H, et al.

Sci China Tech Sci

October (2013) Vol.56 No.10

the free surface region for the stable impinging situation as sketched in Figure 5(a). On the other hand, the horizontal oscillation phenomenon for the flapping jet is also captured by LES simulation. As shown in Figure 5(b), the jet flaps away the center plane (x=0) and towards the positive x direction. 4.2

Figure 3

Image of LIF visualization for turbulent plane jets at Re=2090.

in Figure 5(b), the horizontal surface jet is turned downward and a large part of its fluid is re-entrained, forming a strong recirculation cell in the whole vertical jet region. In contrast to the flapping situation, recirculation is only formed near

Figure 4

Flapping frequency

To obtain the frequency of the flapping motion, the signals of centerline velocities, u and w at z/d=28 are obtained by LDV measurement and LES simulation. The power spectral densities (PSDs) based on LES simulation are shown in Figure 6. As shown in Figure 6(a), the characteristic frequencies, St (=f0d/Wo), for velocities u and w are about 0.0019 and 0.0038, respectively. For comparison, Table 1 lists the frequency of those signals mentioned above. As shown in Table 1, the flapping frequency obtained by LES simulation

Image of LIF visualization of a flapping motion for turbulent plane jets at Re=2970.

Figure 5

Instantaneous vortical structures by isosurface of the Q-criterion (Q=0.001).

Sun J H, et al.

Figure 6

Table 1 z/d=28

Sci China Tech Sci

Non-dimensional frequency for centerline velocities u and w at Stw

LDV measurement

0.002

0.004

LES simulation

0.0019

0.0038

is in good agreement with the experimental measurement. On the other hand, the frequency of longitudinal velocity (w) oscillation at the jet centerline is twice of the transverse velocity (u) oscillation. This is merely a consequence of large amplitude jet flapping motion. For the measurement volume of LDV located at the jet centerline, the maximum of the longitudinal velocity passes the measurement volume twice in one flapping cycle. Mean velocity profiles

The theoretical distributions of mean velocities in self-similar region for stable impinging jet without flapping motion were studied by Kuang et al. [9], shown as, w Wc u Wc

Figure 7

    sec h 2  ,  2 

    sec h 2   2

2395

Power spectral densities of centerline velocities at z/d=28: (a) u; (b) w.

Stu

4.3

October (2013) Vol.56 No.10

      tanh    2

(9)  . 

(10)

Here,



denotes the time-averaged component,

Wc

the centerline mean velocity,   x ( z  zo ) , zo the jet virtual origin,   0.067 [9] for stable impinging jet. To quantitatively characterize the flapping jet in self-similar region, the distributions of longitudinal and transverse mean velocities by LDV measurement are shown in Figure 7 at z/d=10, 15, 20 and 25. For comparison, the theoretical predictions for stable impinging jet are also shown in Figure 7. For the longitudinal mean velocities in Figure 7(a), the theory for stable impinging jet has predicted the basic feature of longitudinal velocity distribution for flapping jet. However, with the increase of z/d, the deviation of longitudinal velocity distribution in Figure 7(a) from theoretical prediction of stable impinging jet becomes larger. This is mainly due to the large amplitude oscillation of flapping motion, which greatly enhances the spreading of the jet. For transverse mean velocities in Figure 7(b), an entrainment of fluid into the jet flow can be found. Again, the theory for stable impinging jet has predicted the basic feature of transverse mean velocity distribution for flapping jet, although there may be some uncertainty due to higher entrainment in shallow water and the experiment set-up. The difference between theory and experiment as shown in Figure 7(b) may be due to the existence of stronger recirculation cells than stable impinging jet as observed in Figure 5. To quantitatively validate the reliability of the present LES simulation, the mean longitudinal velocities and transverse

Distribution of longitudinal mean velocities (a) and transverse mean velocities (b) by LDV measurement for flapping jet when Re=2970.

2396

Sun J H, et al.

Figure 8

Sci China Tech Sci

October (2013) Vol.56 No.10

The longitudinal and transverse mean velocities of flapping jet obtained numerically and experimentally.

velocities for flapping jet are presented and compared with experimental data in Figure 8 at z/d=10, 15, 20 and 25. The lines in Figure 8 represent computational results and the symbol denotes experimental data. As shown in Figure 8, the mean velocities agree well with experimental data. In Figure 9, the variation of centerline longitudinal mean velocity  Wc Wo  obtained by LDV measurement and LES simulation for flapping jet is reported. For comparison, the result of stable impinging jet without flapping motion is also plotted in Figure 9. Good agreement between numerical prediction and experimental data can be found in Figure 9. On the other hand, as shown in Figure 9, the decay rule for flapping jet is about 0.8, which is apparently faster than that without flapping motion.

5 Conclusions Submerged turbulent plane jets impinging onto free surface were investigated experimentally and numerically at two Reynolds numbers 2090 and 2970, which corresponded to stable impinging jet and flapping jet. When H/d=50 and

Figure 9 Variation of centerline longitudinal mean velocity when Re= 2090 and 2970.

Re=2970, both LIF visualization and LES simulation results show that the plane jet would flap periodically. The results of LDV measurement show that the theoretical predictions for stable impinging jet can predict the basic feature of longitudinal and transverse velocity distributions for flapping jet, although there are some deviations due to the existence of flapping motion. The mean velocity along the jet centerline

Sun J H, et al.

Sci China Tech Sci

for the flapping jet is found to decay considerably faster than that of the stable impinging state. On the other hand, the general agreement between numerical predictions and experimental results for velocity profiles in self-similar region is good, which validates the reliability of the present LES simulation.

October (2013) Vol.56 No.10

7

8

9 10

This work was supported by the National Natural Science Foundation of China (Grant No. 10472046), the Priority Academic Program Development of Jiangsu Higher Education Institutions, grants from the Postgraduate Research and Innovation Project of Jiangsu Province (Grant No.CX08B_035Z) and PhD Thesis Innovation and Excellence Fund of Nanjing University of Aeronautics & Astronautics (Grant No. BCXJ08-01).

1

2 3 4 5 6

Sun J H, Zhao L Q, Hsu C T. Theoretical analyses on flapping motion of submerged turbulent plane jets. Mod Phys Lett B, 2005, 19: 1471–1474 Goldschmidt V W, Bradshaw P. Flapping of a plane jet. Phys Fluids, 1973, 16: 354–355 Goldschmidt V M, Moallemi M K, Oler J W. Structures and flow reversal in turbulent plane jets. Phys Fluids, 1983, 26: 428−432 Wygnanski I, Gutmark E. Lateral motion of the two-dimensional jet boundaries. Phys Fluids, 1971, 14: 1309−1311 Moum J N, Kawall J G, Keffer J F. Structure features of the plane turbulent jet. Phys Fluids, 1979, 22: 1240−1244 Hsu C T, Kuang J, Sun J H. Flapping instability of vertically impinging turbulent plane jets in shallow water. J Eng Mech, 2001, 127: 411−420

11 12

13

14 15 16 17

18

2397

Sun J H. Flapping turbulent plane jets in shallow water and interacting with surface waves. Dissertation for Doctoral Degree. Hong Kong: The Hong Kong University of Science and Technology, 2001 Espa P, Sibilla S, Gallati M. SPH simulations of a vertical 2-D liquid jet introduced from the bottom of a free surface rectangular tank. Adv Appl Fluid Mech, 2008, 3: 105−140 Kuang J, Hsu C T, Qiu H H. Experiments on vertical turbulent plane jets in water of finite depth. J Eng Mech, 2001, 127: 18−26 Madarame H, Iida M, Okamoto K, et al. Jet-flutter: self-induced oscillation of upward plane jet impinging on free surface. In: Proceedings of the Asia-Pacific Vibration Conference ’93, Kitakyushu: the Dynamics, Measurement and Control Division of the Japan Society of Mechanical Engineers, 1993. 265−270 Wu S, Rajaratnam N, Katopodis C. Oscillating vertical plane turbulent jet in shallow water. J Hydraul Res, 1998, 36: 229−234 Larocque J, Riviere N, Vincent S, et al. Macroscopic analysis of a turbulent round liquid jet impinging on an air/water interface in a confined medium. Phys Fluids, 2009, 21: 065110 Patankar S V, Spalding D B. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int J Heat Mass Tran, 1972, 15: 1787−1806 Brackbill J U, Kothe B D, Zemach C. A continuum method for modeling surface tension. J Comput Phys, 1992, 100: 335−354 Majander P, Siikonen T. Large-eddy simulation of a round jet in a cross-flow. Int J Heat Fluid Fl, 2006, 27: 402−415 Jones W P, Wille M. Large-eddy simulation of a plane jet in a cross-flow. Int J Heat Fluid Fl, 1996, 17: 296−306 Vreman B, Geurts B, Kuerten H. Comparison of numerical schemes in large-eddy simulation of the temporal mixing layer. Int J Numer Meth Fl, 1996, 22: 297−311 Jeong J, Hussain F. On the identification of a vortex. J Fluid Mech, 1995, 285: 69−94