Lattice Boltzmann simulations of oscillating-grid turbulence＊

Jin-feng Zhang (张金凤), Qing-he Zhang (张庆河), Jerome P.-Y. Maa, Guang-quan Qiao (乔光全)

1.State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China, E-mail: jfzhang@tju.edu.cn

2.Department of Physical Sciences, Virginia Institute of Marine Science, School of Marine Science, College of William and Mary, Gloucester Point, VA 23062, USA

3.Fourth Harbor Engineering Investigation and Design Institute, Co., Ltd., China Communications Construction Company, Guangzhou 510230, China

(Received January 7, 2015, Revised August 17, 2015)

Lattice Boltzmann simulations of oscillating-grid turbulence＊

Jin-feng Zhang (张金凤)1, Qing-he Zhang (张庆河)1, Jerome P.-Y. Maa2, Guang-quan Qiao (乔光全)3

1.State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China, E-mail: jfzhang@tju.edu.cn

2.Department of Physical Sciences, Virginia Institute of Marine Science, School of Marine Science, College of William and Mary, Gloucester Point, VA 23062, USA

3.Fourth Harbor Engineering Investigation and Design Institute, Co., Ltd., China Communications Construction Company, Guangzhou 510230, China

(Received January 7, 2015, Revised August 17, 2015)

The lattice Boltzmann method is used to simulate the oscillating-grid turbulence directly with the aim to reproduce the experimental results obtained in laboratory. The numerical results compare relatively well with the experimental data through determining the spatial variation of the turbulence characteristics at a distance from the grid. It is shown that the turbulence produced is homogenous quasi-isotropic in case of the negligible mean flow and the absence of secondary circulations near the grid. The direct numerical simulation of the oscillating-grid turbulence based on the lattice Boltzmann method is validated and serves as the foundation for the direct simulation of particle-turbulence interactions.

Oscillating grid, quasi-isotropic homogenous turbulence, mean flow, lattice Boltzmann method

Introduction

The turbulence generated from the oscillatinggrid is characterized by its zero-mean flow, yielding an approximate homogeneity at some distance away from the grid. The intensity of this homogeneous turbulence can be easily controlled, and thus, it is suitable to use it for investigating some phenomena encountered in hydraulic and environmental engineering[1], such as, the free surface fluctuation[2], the particle suspensions and sedimentation[3], and the sediment transport[4].

Many experiments on the oscillating-grid turbulence were conducted for various research purposes.The approximately homogeneous, zero-mean turbulence can be produced by oscillating a symmetrical grid in a water tank[5,6]. The grid is characterized by the diameter of the grid barsdg, the mesh sizeM(defined as the spacing between bars), and the grid solidityσ(defined as the ratio of the area of bars to the total area of the grid). The intensity of the turbulence generated by the oscillating-grid depends on the mesh (dg,Mandσ) as well as the strokeS(the maximum distance of the oscillation) and the frequencyfg. As a rule, to generate a nearly homogenous turbulence of zero-mean flow, the solidity of gridσshould be less than 40%[7], the oscillating frequency[8]should be less than 7 Hz and the measurements should be taken at places 2-3 mesh sizes away[5].

Numerical models were also established for more profound investigations of how to produce homogeneous turbulence by using the oscillating-grid. The direct numerical simulation (DNS)[9], to solve the Navier-Stokes equation numerically, was used to examine the homogeneous turbulence through adding energy continuously and locally into the flow. More models such as those solving the Reynolds Equationsby using thek-εmodels were also used for the oscillating-grid turbulence. However, there was a long debate in the past as to whether it is appropriate to use ak-εmodel to describe the zero-mean-shear turbulence[10]. Therefore, further numerical investigations of the oscillating-grid turbulence are desirable.

The lattice Boltzmann (LB) method[11-13], as a new and effective numerical technique of solving the Navier-Stokes Equation, has been successfully employed in the field of computational fluid dynamics to simulate the turbulent flows, such as the decaying turbulence generated with an initial spectrum and the forced turbulence with a random forcing term[14,15]. For this reason, the LB method can be considered as an alternative of the DNS, if the selected lattice size is small enough. Using the LB method, Djenidi[15]simulated the grid-generated turbulence for a steady mean flow passing through a fixed grid. Although in his study, the turbulence generated by grids is simulated, the use of the mean flow is very different from the use of the oscillating-grid to generate turbulence. This is because the mean flow itself can be a turbulent flow, if the Reynolds number is large. In this study, we simulate the turbulence generated from an oscillating flow through a fixed grid by using the LB method.

1. Numerical methodology

1.1Lattice Boltzmann method

This is a relatively new numerical technique for modeling a physical system response in terms of the dynamics of fictitious particles. In the LB approximation, the fluid is described by the density distribution functionfi(x,t), which describes the number of particles at a lattice sitex, at the timet, with the discrete particle velocityci. The hydrodynamic parameters, such as the mass densityρ, the momentum densityj, and the momentum fluxΠ, can be obtained from this particle distribution as follows[16]

The LB equation describes the time evolution of the particle density distribution functionfi(x,t), and can be expressed as

whereΩi(f)is the collision operator, including the lattice Bhatnagar-Gross-Krook (BGK) model, proposed by Ladd[16]and the multiple-relaxation-time (MRT) model[11]. We use the Ladd’s model,Ωi(f)can be constructed by linearizing the local equilibriumfeq

Here we use the so-called D3Q19 topology, a three-dimensional cubic lattice with 19 particle velocity vectors. A suitable form for the equilibrium distribution of the 19 particle distribution model is

The macro-dynamical behavior can be obtained from the lattice-Boltzmann equation by a multi-scale analysis, i.e., the Chapman-Enskog expansion[18], using an expansion parameterε, defined as the ratio of the lattice spacing to a characteristic macroscopic length, the hydrodynamic limit corresponds toε≤1. It is shown that the lattice-Boltzmann equation reproduces the Navier-Stokes equations with corrections that are of the ordersu2andε2[17].

1.2Model implementation

The computational domain with two grids (Grid 1 and Grid 2) can generate turbulence. Free slip boundary conditions are applied on bothx-andy-boundaries. A bounce-back boundary is imposed at the grid elements to simulate the no-slip conditions. At the inlet and the outlet, an oscillating flow is specified. Ifz-direction is the streamwise direction, the velocity can be expressed asu=0,v=0andw=w0sin(ωt), in whichω=2πfgis the angular frequency and is the characteristic velocity. The oscillating flow is implemented as

This is done by introducing an additional termFi(x,t)[18]in the Boltzmann Eq.(2)

This implementation of a body force is shown to satisfy the continuity and Navier-Stokes equations up to the second order[19].

The motion of the oscillating-grid can be expressed as

whereAis the amplitude,A=S/2andtis the time.

The velocity of the grid is

where the characteristic velocityw0=2πfgAand the angular frequencyω=2πfg.

The characteristic velocityw0and the angular velocityωare specified in Eqs.(5) and (6). Then the oscillating-grid turbulence generated in the lab can be modeled using the LB method. The oscillating-grid turbulence is numerically simulated between the Grid 1 and the Grid 2 (Fig.1).

Fig.1 Sketch (not to scale) of the computational domain with the grid

2. Validation of numerical model by experimental measurement

The homogenous turbulence generated by two oscillating grids is used to validate the numerical model presented in Section 2. The parameters of the numerical model are based on the laboratory experiments conducted by Srdic et al.[8]. In their experiment, a glass box of 0.58 m×0.36 m×0.36 m with grids fitting vertically at either end of the tank (Fig.2) is used. The stainless steel grid is made of square crosssection bars of 0.01 m in size with a mesh sizeMof 0.05 m and a solidityσof 38%. The distance between a grid plane and the closest end wall is 7 cm, and the half-distance between grids (H/2)is 0.22 m. The strokeSis 0.02 m and the oscillation frequency is 3 Hz. Flow velocities are recorded with an acoustic Doppler velocimeter (ADV). The data sampling frequency is 200 Hz and the sampling time is 100 s.

Fig.2 A schematic diagram of the experimental apparatus[8]

In the LB model, the computational domain is 580×360×360 with the grid of a mesh sizeMof 50 and a solidityσof 38%. The characteristic velocity of the oscillating flowu0=0.0188, and the viscosity coefficientν=0.001. The mesh Reynolds numberReM=Mu0/ν=942. The comparison between the numerical results of the normalized root mean squared (rms) velocities(urms,wrms)and the experimental data at the strokeS=0.02 mand the frequencyfg=3Hz shows a quite good agreement (Fig.3).

Fig.3 Comparison of the normalized rms velocitiesurmsand

wrmsbetween the numerical results and the experimental data[8]under the grid condition (S=0.02 m,fg=3Hz)

3. Numerical results and discussions

The computational domain consists of560×400× 400 mesh points (Fig.1). The grids (Grid 1 and Grid 2) to generate turbulence, with a mesh sizeMof 75 and a solidityσof 33%, are placed atz=−200andz=200, respectively. The characteristic velocity of the oscillating flowu0=0.0754, and the viscosity coefficientν=0.001. The mesh Reynolds numberReM=Mu0/ν=5.63× 103. The time and space steps are 10−5s and 10−4m. The Komogrov length scaleη=(ν3/ε)1/4=1.2× 10−4m.

3.1Quasi-isotropic homogenous turbulence

The rms velocities remain nearly constant along the radial directions. The vertical rms velocity is larger than the horizontal rms velocity (Fig.4), which suggests that the effects of the oscillating-grid on the turbulence is significant along the oscillated direction (z-direction).

Fig.4 The rms velocities under the grid condition (S=0.04 m,fg=6 Hz)

Fig.5 The mean horizontal velocities under the grid condition (S=0.04 m,fg=6 Hz)

To determine whether the mean flow is weak in the oscillating-grid turbulence, the time-averaged mean velocity observed is also shown in Fig.5. The mean horizontal velocityuis defined by

F ig.6 (Color online) Model calculated instantaneous 2-D velocity (uandv) fields forReM=5.×103between grids in the(x-y)-plane

In the numerical results, the mean horizontal velocity is close to zero alongz-direction, excluding locations near the lower grid. In the region−0.7＜z/ (H/2)＜0.7, it is apparent that there is no secondary flow and a quasi-isotropic homogenous turbulence occurs. It is difficult to eliminate the secondary flow in the oscillating-grid turbulent system especially near the gird elements, because each bar will generate vorticity due to viscous forces at the edge of the grid.

The velocity vector generated between the grid pair in the(x,y)-plane atz/(H/2)=±0.25and ±0.50is shown in Fig.6. The turbulence has the same structure qualitatively in all parts of the flow field. The figure clearly shows that theuandvvelocity fields are statistically self-similar to validate the usual assumptionu′2=w′2, at least for symmmetrical grids such as used in this study.

Fig.7 Spatial variability of degree of isotropy based on the numerical model, represented by

The coherence of the structures on the (x-z)-plane and the (y-z)-plane remains essentially unchanged and the decay of the turbulence is not significant (Fig.8), which is different from the grid-generated turbulence of the uniform flow with the rapid decay far away from the grid[15]. The oscillating flow passing over the grids, which can generate the isotropic homogenous turbulence based on the LB model, can simulate the oscillating-grid turbulence in the lab and can be used for further studies of the particle-turbulence interactions.

Fig.8 (Color online) Vorticity fields with the dashed vertical white lines indicating the extent of penetration of the grids into the flow field. Blue-green contour indicates negative vorticity and red-yellow contour indicates positive vorticity

3.2Mean flow

A two-dimensional snapshot of the mean velocity fields with the mean vorticity background based on the numerical results is shown in Fig.9. The vorticity scaleωycan be expressed as

The high vorticity distribution near the grid (z/(H/2)=±1.0)shows the existence of the shear stress due to the wake and the jet produced by the grid. The interaction between the jet and the wakes creates a high turbulence near grid. Particularly at the distancez/(H/2)=1.0, the vorticities and the fluid motions are strong due to the secondary flow. It is impossible to completely eliminate the secondary flow in the oscillating-grid turbulent system. However, the time averaging velocity and vorticity fields also indicate a substantial central flow region (between the two grids and away from the column sidewalls), where the mean velocities and the vorticity values are relatively small. It can be seen that for most of the time the secondary mean circulation in the computational domain is weak but not completely absent.

Fig.9 (Color online) Mean velocity fields and vorticity visualization inxzplane

3.3Spectral distribution of velocity fluctuations

For the strokeS=0.04 mand the frequencyfg=6 Hz, the model is used to simulateuandwvelocity time series atz/(H/2)=0 and 0.4 at the center. The energy spectra,Eu(f)andEw(f), are calculated (Fig.10). The inertial subrange exists atz/(H/2)=0within the frequency range of 6-100 for the horizontal component and at the frequency range of 2-8 for the vertical component. There is a wider inertial subrange for the horizontal component than that for the vertical one. The inertial subrange atz/(H/2)=0.4exists within the frequency range of 6-20 for the horizontal component and is not guaranteed to exist for the vertical component, which shows a reasonable large inertial subrange foruvelocity component, but not forwvelocity component.

3.4Integral length scale

The computed integral length scalelis plotted as a function ofzin Fig.11. The integral length scale increases with the increase ofz[20]. A linearl= 0.114zrelationship is obtained by a fitting method. This coefficient 0.114 is within the range reported in previous oscillating-grid turbulence studies[4].

Fig.10 (Color online) Spectrum of energy distributions (S= 0.04 m,fg=6 Hz), respectively at the distance ofz/(H/2)=0andz/(H/2)=0.4

Fig.11 Variation of integral length scalelagainst the distancez

4. Summary and final remarks

A direct numerical simulation of the oscillatinggrid turbulence is carried out. A lattice Boltzmann model for the oscillating-grid turbulence is built with the oscillating flow passing over fixed grids to generate homogenous turbulent flows. The simulation reproduces quite well the experimental results generated by the oscillating-grid mechanism. The quasi-isotropic homogenous turbulence, the spatial variation of the turbulence with distance, the mean flow and the integral length scale are analyzed.

In the range of −0.7＜z/(H/2)＜0.7, the mean velocity is close to zero and no secondary flow occurs. The turbulence produced is quasi-isotropic and homogenous in the range of−0.7＜z/(H/2)＜0.7. The high vorticity distribution near the grid shows that the interaction between the jet and the wakes has created a high turbulence near grid. Furthermore, the inertial subrange exists atz/(H/2)=0within the frequency range of 6-100 for the horizontal component and at the frequency range of 2-8 for the vertical component based on the spectral distribution of the velocity fluctuations. It is also found that the integral length scale increases with the increase of the distancez.

The two-grid configuration where the turbulence in the core region can be considered nearly isotropic is important for many studies, such as the particle suspension and the pollution dispersion in the turbulence. This numerical model can serve as a foundation for the study of the particle-turbulence interactions.

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* Project supported by the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (Grant No. 51621092), the National Natural Science Foundation of China (Grant No. 51579171), the Tianjin Program of Applied Foundation and Advanced-Technology Research (Grant No. 12JCQNJC04100) and TH-1A supercomputer.

Biography:Jin-feng Zhang (1978-), Female, Ph. D., Associate Professor