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Advances in Mechanical Engineering

Volume 2013 (2013), Article ID 592940, 10 pages

http://dx.doi.org/10.1155/2013/592940

## Comparisons of LES and RANS Computations with PIV Experiments on a Cylindrical Cavity Flow

^{1}School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China^{2}State Key Laboratory of Hydro-Power Equipment, Harbin 150040, China^{3}School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China^{4}TSI Incorporated, Minneapolis, MN 55126, USA

Received 13 January 2013; Revised 6 March 2013; Accepted 20 March 2013

Academic Editor: Tomoaki Kunugi

Copyright © 2013 Wen-Tao Su et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A comparison study on the numerical computations by large eddy simulation (LES) and Reynolds-averaged Navier-Stokes (RANS) methods with experiment on a cylindrical cavity flow was conducted in this paper. Numerical simulations and particle image velocimetry (PIV) measurement were performed for two Reynolds numbers of the flow at a constant aspect ratio of ( is the radius of the cylindrical cavity, and is liquid level). The three components of velocity were extracted from 100 sequential PIV measured velocity frames with averaging, in order to illustrate the axial jet flow evolution and circulation distribution in the radial direction. The results show that LES can reproduce well the fine structure inside the swirling motions in both the meridional and the horizontal planes, as well as the distributions of velocity components and the circulation, in good agreement with experimental results, while the RANS method only provided a rough trend of inside vortex structure. Based on the analysis of velocity profiles at various locations, it indicates that LES is more suitable for predicting the complex flow characteristics inside complicated three-dimensional geometries.

#### 1. Introduction

Turbulence is a universal flow phenomenon in the nature world and engineering application systems. Research on turbulence has long been a hot topic, which directly influences the prediction of environment and engineering application. There are currently three main approaches for numerically simulating turbulent flows, that is, Reynolds-averaged Navier-Stokes (RANS), large eddy simulation (LES), and direct numerical simulation (DNS). For RANS method, its governing equation (Reynolds time- and space-averaged equation) “flattens” the local flow parameters, and so it only predicts the overall characteristics of turbulent flows. Since it computes the flow parameters by dividing them into the mean part (average), which represents the main flow, and the fluctuating part, which is calculated by turbulence models, it greatly reduces the computation burden. However, the results are strongly influenced by the turbulence models adopted. For DNS method, it directly solves the Navier-Stokes (N-S) equation without introducing any turbulence models. Thus, DNS needs very fine computational meshes and large computer resources in order to resolve the smallest turbulence scale (Kolmogorov scale). So, currently it cannot handle complex flow at a relatively large flow Reynolds number () and is still far from the needs for real engineering applications.

Recently, LES was used widely in the engineering applications, due to a balance of great development of computer capacity and computational cost. The basic idea of LES is to decompose the flow variables into large-scale parameters and small-scale parameters by using a “cutoff” filter function. Thus, it explicitly simulates the large eddies but parameterizes the small eddies using a subgrid scale (SGS) model, by which it can provide accurate results at different length-scales above the cutoff threshold in a complex fluid flow.

LES was initially proposed for global weather forecasting by meteorologist Smagorinsky [1]. After this pioneering work, LES method has been greatly developed so far in respect to the SGS model development, improvement of numerical simulation efficiency for LES, robustness of LES for turbulent flow with complex flow passage geometry, and application area extension (e.g., [2–10] among a large amount of others). Many studies show that the LES method can simulate complex flow phenomena such as vortex shedding, buoyancy influence, curved streamlines, and rotation flow, where the RANS method cannot capture the delicate flow structure in such flow conditions.

Among the complex flow phenomena, swirling flow is a fine example to test the numerical computation with different turbulent models. Swirling flows in a cylindrical cavity have attracted much attention in the past decades due to such kind of flows having close associations with frequently encountered natural phenomena like tornadoes, complex engineering applications such as centrifugal machinery and cylindrical cyclone separator, and classical studies on fluid mechanics such as flow bifurcation, symmetry breaking, and vortex break-down phenomena. The vortex structures in the rotated cavity flow show abundant flow patterns, which crucially depend on , the height to radius aspect ratio, , and also the fluid properties. Lopez et al. [11, 12] have conducted numerical simulations and experiments on both deep and shallow cylinder flows, showing that the rotating waves were the main pattern of vorticity distribution. Li et al. [13] investigated the flow patterns in a cylindrical cavity flow of viscoelastic fluid by using a two-dimensional (2D) particle image velocimetry (PIV) system, indicating that the viscoelasticity of fluids influenced both meridional and planar flow characteristics and then the flow transition to symmetry breaking.

It is conjectured that flow characteristics in swirling flow should be well captured by the LES methods. Due to the rapid development in computer performance and numerical simulation techniques, computational fluid dynamics (CFD) developed greatly in the past decades. The commercial CFD software greatly facilitates the research on the turbulent flows in complicated flow geometries. We have carried out a comparative study on the performances of LES and RANS methods in predicting complicated three-dimensional flow characteristics in a hydraulic power machinery model [14], showing that LES is much superior to RANS for this complex flow, and only the LES results were well coinciding with the experiments. This paper aims at making further comparative study on LES and RANS computations focusing on the characteristics of the cylindrical cavity flow, especially on the vorticity distribution and velocity profiles. A carefully regulated 2D PIV experiment is performed to obtain database of velocity distribution and vortex structures for comparisons.

#### 2. Experimental Procedures

The experimental setup is shown in Figure 1. The cylindrical cavity (inner diameter of mm) and its end-wall (driven by a precise stepper motor via the timing belt) are made of transparent acrylic resin. The cylindrical cavity flow is driven by the rotation of the end-wall, and the liquid flow in the cavity has a free surface in contact with the air. The hollow cylinder is bathed in a transparent acrylic resin rectangular container with the same working fluid. For detailed information of this experimental apparatus, one refers to [13].

For a swirling flow in a cylindrical container, the curved wall may bring about serious quantitative deviation to the PIV-measured velocity field in the meridional plane due to optical refraction if the difference of refractive index between the working fluid and the transparent wall material is obviously large. Therefore, tuning of working fluid refractive index has been performed in order to match with the transparent acrylic resin wall. The sodium iodide (NaI) is used as a refractive index tuning material. Eventually, an NaI solution of 61 wt% concentration is finalized so as to assure the refractive index of 1.49 in both the working fluid and the bath fluid, which are similar with that of acrylic resin. It is known that this solution is chemically stable, and its refractive index is thermally stable [15]. For the measurement of refractive index, a refractive-index meter is used at the temperature of 295 K. By using this refractive-index-matched NaI solution, image distortions have been essentially eliminated, as seen from Figure 2 showing the comparison between images of light passing through the water bath and that passing through the prepared NaI solution bath.

A standard 2D two-component PIV system (TSI Inc., Shoreview, MN, USA) is utilized to measure the velocity fields in two planes, that is, the meridional plane and lateral plan as indicated in Figure 1 as sheet 1 and sheet 2, respectively, in the cylindrical cavity swirling flow. For the illumination system of PIV, the double-pulsed YAG200-NWL (New Wave) laser generator has a power output of 200 mJ/pulse, maximum repetition rate of 15 Hz, and the emission wavelength of 532 nm. The thickness of light sheet is 0.5 mm. Sets of digital images are captured by 12-bit CCD cameras (TSI Powerview Plus with pixel resolution). A TSI synchronizer box controls the strobing and timing of the camera and laser. The dual frame acquisition rate is 8 Hz, and so the sampling frequency of velocity frames is 4 Hz. For the resolution of measurement, all the picture frames in the - plane (the Cartesian coordinates is shown in Figure 3) cover the full height of liquid, that is, an area of mm^{2} with the spacing between adjacent vectors mm. And in the - plane, the images cover a planar disc area of mm^{2}, with the spacing between adjacent vectors mm and mm at mm and 30.0 mm planes, respectively. The interrogation area is set as pixels (with 50% overlap in each direction) for velocity vector analyses.

Teflon tracer particles with an average diameter of 10 *μ*m are used to seed the flow. Although the density of tracer particle is somewhat smaller than that of working fluid, for the measurement within 30 minutes, the tracer particles follow the working fluid very well. Carefully adjusting the position (both horizontal and perpendicular) of laser sheets and the focal distance, the tracer particles can be uniformly illuminated and so clear images can be sampled. To meet the demand of this measurement, a Nikkon 50 mm/F1.8 camera lens is used with camera to capture the flow field in the meridional plane (sheet 1 and camera 1 as indicated in Figure 1), and a Sigma Micro 105 mm/F2.8 lens is for the horizontal plane measurement (sheet 2 and camera 2). The illuminated image in the horizontal plane (sheet 2) is taken from the end-wall bottom through a mirror set at 45° angle.

The measured cylindrical cavity flow has a constant aspect ratio of . PIV images are acquired for 100 dual frames (200 PIV photographs) for each run of measurements. For the image post processing, flow vectors are obtained by TSI INSIGHT 3G software (Version 3.3), then the bad vectors are filtered by using a homemade FORTRAN program based on the “three-sigma” principle.

#### 3. Numerical Simulation Procedures

To study the fine structures of the swirling flow with free surface, numerical simulations are conducted based on RANS method and LES method, respectively. The solvers in commercial CFD code FLUENT (version 6.3) are utilized to solve the governing equations defined for the problem.

The computational model adopted in the numerical simulation is shown in Figure 3. The height of liquid level is 0.06 m under stationary state, which is the same as experiment. In order to capture the gas-liquid interface, an air domain with 0.02 m height is added over the water domain. The computational domains are meshed with structured grids. Near the cylinder wall and the bottom end-wall around which the high shear stresses exist, the grids are well refined. For the refinement, the viscous layer grid is set less than 8, where is the wall friction velocity and is the kinematic viscosity.

The generated mesh system contains over 1.4 million cells in the whole domain and is displayed in Figure 3(b) and Figure 3(c). Though the liquid phase is water in numerical simulations, the results are still comparable with experiments since Res of the simulated cylindrical cavity flow are set to be the same as that of corresponding experimental cases.

Numerical simulations are performed for an incompressible fluid flow. The LES and RANS methods are introduced briefly as follows.

LES method is realized in a way of reducing the number of freedom degrees in the simulated turbulent flow. This is done by calculating only the low-frequency modes (corresponding to turbulent structures larger than the cutoff threshold) but modeling the high frequency (smaller than the cutoff threshold) ones. The scale separation is performed by filtering in space the N-S equations. Small eddies can be modeled by the additional term in the LES governing equations resulting from the filtered N-S equations. The additional term is referred to as the SGS model which is derived from the interaction of unresolved scales and resolved scales. Under the assumption of constant fluid property, the governing equations of LES are as follows: where represents the subfilter tensor comprising the terms that are not expressed directly from resolved scales, . In order to obtain the correct subfilter tensor, Smagorinsky-Lilly model is used in this study [16].

The most widely used approach in industrial applications for the modeling of turbulent flow is RANS. From the viewpoint of engineering applications, the focus is the change of time-averaged flow field induced by the turbulence. Therefore, solving the time-averaged N-S equations is taken into consideration. This approach assumes that nonconvective transport in a turbulent flow is governed by random three-dimensional turbulence possessing a broadband spectrum with no distinct frequencies. The model represents a very wide range of scales with the smallest scales being influenced by the fluid viscosity. In this study, the standard - model and high-accuracy RNG - model in RANS are adopted. The RNG - model improves the accuracy of the prediction on the turbulent vortices and near-wall flow, compared with the standard one.

To consider the air-water interface at the free surface of swirling flow, the volume of fluid (VOF) model for two-phase flow is adopted to conduct numerical simulation. In VOF model, there is no interpenetration for parameters in two or more fluids (or phases). A new variable for each additional phase is added into the model, that is, the volume fraction of the phase in the computational cell. And the sum of the volume fraction of all phases in each cell reaches unity. The fields for all variables and properties are shared by the phases and represent volume-averaged values, as long as the volume fraction of each phase at each location is known. Based on the volume fraction of the th fluid in the cell , the tracking of the interface between the phases is completed by solving the continuity equation for volume fraction of one or more phases. For the th phase, the equation is defined as [16] where is mass transfer from th phase to th phase, and is mass transfer from th phase to th phase. in (3) is hereby the same velocity vector in N-S (2).

The bottom end-wall of the geometry is set as the slip wall (the absolute velocity near the wall is nonzero) with constant rotational velocity which ensures the same as the one in the corresponding experimental case. The sidewall surface of the cylinder is set as nonslip wall condition, and the top plane of liquid is defined as pressure outlet with a standard atmospheric pressure. Fluid media in the simulation are air and water. In VOF model, air is set as the primary phase, while water is set as the secondary phase.

The SIMPLE scheme is used for the computation of velocity-pressure coupling. PRESTO! and the first-order upwind scheme are, respectively, adopted to discretize the pressure equation and the volume fraction equation. For the momentum equation discretization in LES, bounded central differencing scheme is utilized. For standard and RNG - models, second-order upwind scheme is used to discretize the momentum equation, turbulent kinetic energy () equation, and turbulent dissipation rate () equation. For the time marching, it is set as the implicit second-order method.

#### 4. Results and Discussions

For convenience, a cylindrical coordinates (, , ) (correspondingly, the velocity field is (, , )) is defined instead of the Cartesian coordinates (, , ), with the origin locating at the center of the bottom end-wall. Therefore, horizontal and meridional planes refer to (-) plane and (-) plane, respectively. The Reynolds number of the investigated cylindrical cavity flow is defined as , where is angular velocity. Two flow cases are tested at and 6541, respectively. The Froude number is defined as , where is the gravitational accelerator, and it reads 0.0186 and 0.1165 for the two investigated cases, respectively. For a free-surface rotating flow at number less than 1, the surface perturbation in an open container is trivial [17]. In this case, therefore, the deformation of air-water interface is considered to be negligible, and the surface tension does not influence the flow structures.

The flow structures in the meridional plane and two parallel planes are captured in both experiments and numerical simulations. Distributions of the radial and axial velocity components, and , and the circulation , respectively, along the radius are also obtained from the meridional and horizontal planes. Comparisons of these results obtained from LES and RANS with that of the experiment are presented as follows.

##### 4.1. Flow Structures in the Meridional Plane

The 2D velocity fields in the meridional plane (-) obtained from PIV measurement and numerical simulations of the cylindrical cavity flow are shown in Figure 4. The vectors are normalized velocities , , and colored contours represent the normalized circumferential vorticity calculated by and distributions in the (-) plane. Note that all the PIV measurement results are treated by a 100-frames-averaging, due to the steady flow structures at the given flow conditions. As gets higher, the flow should be more irregular, so the RNG - model is introduced to make a comparison within the RANS category and to check how it performs comparing with others.

Figure 4 depicts the velocity field and vorticity obtained by LES and RANS (RNG - model is used additionally for ) and PIV measurement for the same configuration and setup. The swirling flow driven by the bottom end-wall displays an axisymmetric pattern, which radially flows outwards in the Ekman layer near the bottom and is restricted into the vertically axial direction near the cylinder sidewall, forming a jet-like shear layer flow. Finally, in the axis of the cylinder, the fluid overturns near the interface then diverts down into the bulk. This “swirling nature,” in both the azimuthal and vertical directions, is much similar to the tealeaf motion with the spoon stirring in a cup. Compared with the RANS method (for both - models), the vector profile and vorticity distribution by LES method in the (-) plane are much sharper and more reasonable as a symmetric swirling flow and agree with the PIV measurement better. Note that the result by RNG - model proves to be better than that by standard - model on the vorticity structures, and velocity distribution in the shear layer is also better. That should be due to the good treatment of RNG - model near the wall. However, the two RANS models both do not predict well the flow near the free surface and in the bulk where the fluid changes direction. And in comparison, LES shows much better results which resemble the experimental measurement. It implies that the high-accuracy - model still cannot reproduce as fine detailed information as LES does. Therefore, in the following, we just show the results of RANS by using standard - model, along with the LES and PIV for .

At least, the result by RANS with a - model shows an overall prediction of the axisymmetric structures in the (-) plane; however, it encompasses a much larger zone of overturning flow in the bulk as compared with LES and PIV measurement results. Such difference probably stems from the numerical simulation approach of RANS, which has the “smearing effect” for the turbulence modeling [18].

The LES method reproduces better the PIV measurement result. Nevertheless, more discrepancies appear when gets high. When the flow gets more chaotic, the fluid motions under the small scales will be more robust to transport the injected energy. Therefore, the flow information at the high frequency needs refined girds and numerical method to predict, which is just limited by SGS models in the LES. The bulk flow by LES is more irregular than experimental result, which should be due to the limitation of SGS model adopted in this study. Anyway, for both numerical methods (LES and RANS), the vertical Ekman layers near the sidewall of cylinder coincide well with that of experimental result, indicating the local mesh for simulation is good enough. And more detailed differences come from the essence of the numeric treatment and turbulence models.

##### 4.2. Flow Structures in the Horizontal Plane

For the flow pattern in the parallel plane (-), the motions show an axisymmetric pattern as well. Figures 5 and 6 display the simulated and measured velocity field together with the normalized axial vorticity at depths of and 1.2 for two different Res, respectively. As depicted in the experimental setup, the lower horizontal plane is physically mm, near to the bottom end-wall, the flow is strongly influenced by the end-wall rotating motion. So the velocity in this (-) plane should be composed of a large azimuthal component and a small radial component. The larger the rotation speed is, the more chaotic the flow in the bulk becomes. At the upper horizontal plane mm, the flow in the bulk shows axisymmetric pattern, which is consistent with that in the (-) plane. Again, the result by LES method is shaper than that by RANS method, and for the vectors both agree well with experimental result, but for the vorticity distribution, compared with RANS method LES result shows more similar pattern as that of experiment at the two measured Res. However, the PIV images at the parallel planes are truly rough in the rim, which should be due to the possible contamination of particle tracers closed to the wall during the long-time measurement.

To compare the - models within RANS method, the RNG - model performs clearer prediction near the bottom wall (Figure 5), whereas LES method outperforms their both near the cylinder wall (Figure 6). This indicates that LES is more suitable for simulating the swirling flow, with high shearing effect existing near the wall.

##### 4.3. Radial Distributions of Velocity Components and Circulation

To further compare the simulated results with experiments, the radial distributions of two velocity components, and , and the circulation at two liquid levels ( and 1.2), and two Res ( and 6541) are calculated and shown in Figures 7 and 8, respectively. The distributions of and are extracted from velocity vectors in the (-) plane as plotted in Figure 4. The distribution of is calculated using the velocity field in the two (, ) planes as plotted in Figures 5 and 6. Note that the RNG - model is used for high to make better illustration, and it shows much similar result with that by standard - model.

For the radial distribution of , the peaky structure moves towards the bulk when the observed plane varies from low to high (from to 1.2, as seen from the comparison between Figures 7(a) and 7(b)). The width of peak gets narrow as rises (as seen from the comparison between the upper and lower ones in Figure 7), which means the jet flow near the sidewall of cylinder becomes stronger, and the shear layer correspondingly becomes thinner. More importantly, it is seen from Figure 7 that, for all the runs, the overall variation trend of along the radial direction obtained by LES matches with the PIV measurements much better than that by RANS, although there are still some quantitative differences between LES and PIV measurement results.

For the radial distribution of , the magnitude is apparently smaller than that of . Again, the matching between LES method and PIV result is fairly good. As for the result by RANS method, the evolution of both and flattens in the radial direction, especially in the core region of the cylinder. It indicates that the LES method for such a swirling flow can capture the detailed characteristics of the vortex motions but RANS can only capture overall ones.

For a cylindrical cavity flow, the circulation is an important parameter to illustrate the influence of swirling motion on the structure of the vortex flow by showing its radial distributions in different horizontal cross-sections [13]. As shown in Figure 8, the circulation continuously increases from the core (about zero magnitude) to a maximum near the sidewall and then decreases to zero on the sidewall of the cylinder. With the increase of , the maximum decreases in its amplitude, and its location moves away from the sidewall. It can be seen from Figure 8 that the overall variation trend of in the radial direction can be captured by LES method quite well, as compared with PIV measurements. Particularly in the bulk flow, the matches between LES and PIV measurement are good enough that the plateau zones are successfully predicted. The RANS method fails to capture this feature in the bulk flow though it also depicts good circulation distribution near the wall.

From the comparisons of flow structures in the cylindrical cavity flow obtained by LES, RANS and PIV measurement, it is confirmed that LES can reproduce the overall information of swirling flow, and probably most of engineering flows. As speculated, RANS method loses precision in the prediction of detailed information of vortex structures, but can only provide the gross patterns of swirling flow.

#### 5. Conclusions

Comprehensive comparisons between LES, RANS simulations, and PIV measurement for a swirling flow in a cylinder with bottom end-wall rotating have been carried out in this paper, through presenting the important parameters including velocity, vorticity distributions in both the meridional and parallel planes, and the radial distributions of different velocity components and circulation, at two Reynolds numbers, respectively.

The comparison with PIV measurement shows that LES method can capture more detailed information in such a swirling motion, with good matches in flow structures, overall characteristics of the radial distributions of the investigated velocity components and the circulation distribution. However, RANS method fails to regenerate the fine structures in the bulk. Together with our previous study, it is again verified that LES has great potential in serving as a powerful tool for the prediction of engineering flows inside complicated geometries.

#### Acknowledgments

This work is supported by National Natural Science Foundation of China (51276046, 51076036), Specialized Research Fund for the Doctoral Program of Higher Education of China (20112302110020), National Key Technology R&D Program (2011BAF03B01), and Heilongjiang Scientific Funds for Distinguished Young Scientist (JC201115).

#### References

- J. Smagorinsky, “General circulation experiments with the primitive equation,”
*Monthly Weather Review*, vol. 91, no. 3, pp. 99–164, 1963. - J. W. Deardorff, “Convective velocity and temperature scales for the unstable planetary boundary layer and Rayleigh convection,”
*Journal of Atmospheric Sciences*, vol. 27, pp. 1211–1213, 1970. - U. Schumann, “Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli,”
*Journal of Computational Physics*, vol. 18, no. 4, pp. 376–404, 1975. View at Scopus - N. N. Mansour, P. Moin, W. C. Reynolds, and J. H. Ferziger, “Improved methods for large-eddy simulations of turbulence,” in
*Symposium on Turbulent Shear Flows*, pp. 14.21–14.29, Pennsylvania State University, 1977. - U. Schumann, “Subgrid length-scales for large-eddy simulation of stratified turbulence,”
*Theoretical and Computational Fluid Dynamics*, vol. 2, no. 5-6, pp. 279–290, 1991. View at Publisher · View at Google Scholar · View at Scopus - M. Germano, U. Piomelli, P. Moin, and W. H. Cabot, “A dynamic subgrid-scale eddy viscosity model,”
*Physics of Fluids A*, vol. 3, no. 7, pp. 1760–1765, 1991. View at Scopus - N. Y. Liu, X. Y. Lu, and L. X. Zhang, “A dynamic SGS model for large eddy simulation of stratified shear-flow,”
*Scientia Sinica Mathematica*, no. 2, pp. 145–153, 2000. - X. L. Tang, Z. D. Qian, and Y. L. Wu, “Second-order dynamic sub-grid-scale stress model with double coefficients,”
*Advances in Water Science*, vol. 15, no. 1, pp. 50–55, 2004. View at Scopus - X. X. Li, R. E. Britter, T. Y. Koh et al., “Large-eddy simulation of flow and pollutant transport in urban street canyons with ground heating,”
*Boundary-Layer Meteorology*, vol. 137, no. 2, pp. 187–204, 2010. View at Publisher · View at Google Scholar · View at Scopus - X. X. Li, R. E. Britter, L. K. Norford, T. Y. Koh, and D. Entekhabi, “Flow and pollutant transport in urban street canyons of different aspect ratios with ground heating: large-eddy simulation,”
*Boundary-Layer Meteorology*, vol. 142, no. 2, pp. 289–304, 2012. - A. H. Hirsa, J. M. Lopez, and R. Miraghaie, “Symmetry breaking to a rotating wave in a lid-driven cylinder with a free surface: experimental observation,”
*Physics of Fluids*, vol. 14, no. 6, pp. L29–L32, 2002. View at Publisher · View at Google Scholar · View at Scopus - J. M. Lopez, F. Marques, A. H. Hirsa, and R. Miraghaie, “Symmetry breaking in free-surface cylinder flows,”
*Journal of Fluid Mechanics*, vol. 502, pp. 99–126, 2004. View at Publisher · View at Google Scholar · View at Scopus - F. C. Li, M. Oishi, Y. Kawaguchi, N. Oshima, and M. Oshima, “Experimental study on symmetry breaking in a swirling free-surface cylinder flow influenced by viscoelasticity,”
*Experimental Thermal and Fluid Science*, vol. 31, no. 3, pp. 237–248, 2007. View at Publisher · View at Google Scholar · View at Scopus - W. T. Su, F. C. Li, X. B. Li, X. Z. Wei, and Y. Zhao, “Assessment of LES performance in simulating complex 3D flows in turbo-machines,”
*Engineering Applications of Computational Fluid Mechanics*, vol. 6, no. 3, pp. 355–364, 2012. - K. Yuki, “Visualization of complex flow structures by matched refractive-index PIV method,” in
*Aeronautics and Astronautics*, M. Mulder, Ed., 2011. *Fluent User Manual*, Version 6.3.2., 2006.- B. Freeze, S. Smolentsev, N. Morley, and M. Abdou, “Characterization of the effect of Froude number on surface waves and heat transfer in inclined turbulent open channel water flows,”
*International Journal of Heat and Mass Transfer*, vol. 46, no. 20, pp. 3765–3775, 2003. View at Publisher · View at Google Scholar · View at Scopus - N. C. Nguyen, P. O. Persson, and J. Peraire, “RANS solutions using high order discontinuous Galerkin methods,” in
*Proceedings of the 45th AIAA Aerospace Sciences Meeting and Exhibit*, pp. 11080–11095, Reno, Nev, USA, January 2007. View at Scopus