Proceedings of the 44th National Conference on Fluid Mechanics and Fluid Power December 14-16, 2017, Amrita University, Amritapuri Campus, Kollam, Kerala, India
FMFP2017–PAPER NO. 32
Vortex Identification and Energy Quantification for Flow past a Fixed and Transversely Rotating Sphere at Re 300 Shashank S. Tiwari
Ashwin W. Patwardhan
Jyeshtharaj B. Joshi
Department of Chemical Engineering, Institute of Chemical Technology, Mumbai-019
Department of Chemical Engineering, Institute of Chemical Technology, Mumbai-019 Email:
[email protected]
Homi Bhabha National Institute, Mumbai-094 Email:
[email protected]
Email:
[email protected]
Abstract
= von Kármán constant
The unsteady planar symmetric regime which occurs at Re=300 for a flow past stationary sphere and for a transversely rotating sphere is investigated using LES and DES turbulence models. The objective of this work is to quantify the energy content, analyse the periodicity behaviour of the vortex shedding phenomenon and identify the vortex structures. Rigorous comparisons have been made with previously available DNS and experimental results from literature. It was found that eddies of the length of inertial subrange could be well captured by the LES simulations, and to a lesser extent by the DES simulations. Other time-averaged statistics of parameters such as drag coefficient, lift coefficient, Strouhal (Sr) number, separation angle, pressure coefficients and transverse velocities were also in good agreement. It was concluded that it is advisable to perform simulations using comparatively less computationally intensive turbulence models like LES or DES if the solutions are to be obtained for an engineering scale equipment where well predicted time averaged parameters solve the purpose. However it is essential to go for DNS if the purpose is to study the intricate physics involved behind transitioning of the flow from laminar to fully turbulent.
d = Distance to the closest wall C S = Smagorinsky constant
V = Volume of the computational cell t = Turbulent viscosity 1.0
INTRODUCTION
Flow past bluff bodies has been a fascinating problem in the area of physics, mathematics and engineering due to the applicability of the problem in various fields such as building/stack/bridge vibrations, chemical reactors, astronomical bodies to mention a few. Also, such type of flow exhibits a complicated three dimensional behaviour in the wake region as the Reynolds number increases which results in further convoluted phenomenon like generation of hairpin vortices, breaking of the axisymmetric wake, drag crisis at Re of about 3 x 105, vortex induced vibrations [1] [2]. One of the most widely studied Re in this regards is Re of 300 at which the two threaded wake originating from sphere head takes the shape of a hairpin and the vortex shedding takes place [4]. Most of these studies have been either on developing new turbulence models which could help in deciphering the complicated physics behind this 3D flow or to statistically determine the effect of Re on various time averaged parameters [7][8]. Some studies in the past have surely been focussed on quantifying the energy content of these eddies, but none of these have explicitly compared DNS, LES and DES turbulence models for predicting the energy content in the resolved length and time scales thereby making a critical observation on the quality of resolution obtained using these models.
Keywords: LES, DES, Turbulence, Energy Spectra, Hairpin vortex
Nomenclature C D = Time averaged drag coefficient CL = Time averaged lift coefficient
Ls = Mixing length for sub-grid scales
1
The objective of this work is to visualise the vortex at a Re of 300 for both stationary and rotating sphere, identify the shedding frequency using Fast Fourier Transform (FFT) and obtain the energy content in the resolved eddies, further comparing them with the available DNS and experimental results in literature.
The turbulent eddy viscosity involved in the last term of the above equation is further given as follows
2.0
formula
t L2s S
(3)
Here, S 2 S ij S ij and Ls is calculated using the following
METHODOLOGY
Ls min d ,CsV 1/ 3
The computational domain along with the boundary conditions which were implemented for both the LES and DES models have been illustrated in Fig.1. The spatial sensitivity analysis performed have been tabulated in Table 1. Mesh was generated using the SnappyHexMesh tool in OpenFoam®. For using this tool the following methodology was adopted: (1) Exporting the sphere geometry (which was created in Salome) to OpenFoam in STL format (2) Creating a blockMesh of a cubical enclosure around the sphere (3) Creating a castellatedMesh by first dividing all cells cut by the sphere STL, refining cells and discarding cells outside of STL borders. (4) Cells were refined into much smaller ones near the sphere where boundary layer separation occurs, three boxes were created at increasing distances from sphere so that sudden transition from very fine grids to coarse grid could be avoided. The mesh inside the refinement boxes have been shown in Fig.2. (5) Snapping the boundary cells on to the geometry (6) Adding more layers at the refinement box zones (7) Exporting the generated mesh to MSH format (so that the mesh can be read by Fluent) using the FoamMeshToFluent utility tool of OpenFOAM®.
(4)
The advantage of using dynamic Smagorinsky model over Smagorinsky-Lily model is that is eliminates the necessity to specify the model constant C S as it calculates C S , dynamically from solutions obtained for the different scales of motion.
Fig. 1: Computational domain and boundary conditions
Transient simulations were run for a total of 10 sec so as to obtain steady results. For both meshing and solution purpose Intel® Core™ i7-5820K CPU, 3.3 GHz, x64-based processor with 64 GB DDR4 RAM. It took around 10 days for both DES and LES simulations of stationary and rotating sphere cases. LES and DES simulations have been performed in ANSYS Fluent® Academic Research, Release 17. The dynamic Smagorinsky model was used for modelling the sub-grid scale stresses in the LES model, while a combination of the k-ω SST RANS model and LES (for near wall) was used for simulating the flow using DES. The Navier-Stokes equation along with the SGS equation used for LES are enlisted below [9]:
ui 0 t xi
Table 1: Sensitivity analysis for stationary sphere at Re 300 Domain Size (L x W x H)
(1)
p ij ui ui u j t x j xi x j
Fig. 2: Refinement boxes in the near wall region of sphere for a plane cut in the middle of the domain in XZ plane for 2.1 million mesh size
(2)
2
LES
8D x 3D x 3D
DES
8D x 3D x 3D
Number of cells (million)
CD
1.5 1.8 2.1 2.5 1 1.2 1.5 1.8
0.693 0.685 0.672 0.672 0.687 0.685 0.666 0.667
Based on the spatial sensitivity analysis carried out for different mesh sizes using drag coefficients as the criteria results in using a mesh containing 2.1 million cells for LES calculations and 1.5 million cells for DES calculations for both stationary and transversely rotating sphere.
respect to time. The comparison of the time averaged values of both the overall drag coefficients and Strouhal number were found to be in fair agreement with those reported in literature and the comparison have been shown in Table 2. Table 2: Comparison of drag coefficient, lift coefficient, Strouhal number at Re = 300 for stationary and rotating sphere
3.0
RESULTS AND DISCUSSION Vortex visualization using the Q-Criterion method obtained for the stationary and the rotating sphere at Re of 300 have been shown in Fig. 3(a) and 3(b) respectively. A layer of fluid is adhered over the sphere for both the cases. For the stationary sphere case, a hairpin vortex can be clearly seen to be form at some distance downstream from the sphere. Another similar vortex begins to form at the head of the previously formed vortex and this phenomenon further continuous in the subsequent downstream region. Whereas, in case of rotating sphere the two legs of the wake extending from the base of the sphere further seem to extend to further downstream distance as compared to the stationary case. Also, unlike stationary sphere case there is no as such hairpin like structure formed in the rotating sphere case and just a slight twist of two legs is visualised from the Q-criterion visualization.
Stationary LES DES [4] [6] Transversely Rotating LES DES [5] [3]
CD
Sr
0.672 0.684 0.656 0.668
0.138 0.138 0.137 0.133
0.667 0.670 0.657 0.658
0.133 0.136 0.134 0.134
The vorticity plots for the stationary sphere and rotating sphere are shown in Fig. 4(a) and 4(b) respectively followed by plots of velocity contours in Fig. 5(a) and 5(b) and pressure coefficient contour plot in Fig. 6(a) and 6(b).
(a)
(a)
(b) Fig. 3: Isosurfaces plot of wake for (a) stationary sphere and (b) rotating sphere at Re=300 using Q-Criteria for LES simulations
(b)
As identifiable from the above vortex visualization plots there exists a certain frequency with which the shedding of the vortices take place. It is important to determine the value of this frequency as this shedding phenomenon is closely related to breaking of the axisymmetric wake which takes place at Re 300. To find the shedding frequency in terms of non-dimensional Strouhal number ( Sr fD / U ) Fast
Fig. 4: Vorticity plots of (a) stationary sphere and (b) rotating sphere at Re=300 for LES simulations
A detailed power spectrum analysis of the velocity components helped in determining the capability of LES and DES models for resolving eddies of various length scales. A comparison of power spectrum analysis with that in published literature shows that due to the isotropic assumption of small scale eddies as done by LES model, the smallest scale eddies are not exactly captured in LES and DES and thus it is essential to go for DNS simulations for obtaining fully resolved temporal and spatial scales. It is only after performing statistical analysis on fully resolved
Fourier Transform was performed over the transverse direction velocities monitored over a point in the wake region during the simulations. Besides, Sr, the overall drag coefficient at the wall of the sphere was also monitored with
3
4.0
DNS data of velocity components that a possible theoretical explanation of the complicated physics behind the yet unexplained phenomenon associated with flow past bluff bodies can be thoroughly addressed and a reliable conclusion can be drawn from it.
CONCLUSIONS
LES and DES simulations were performed for fluid flowing through a stationary and a transversely rotating sphere at a Re of 300. The vortex visualization done using the Q-criterion method clearly shows the difference in orientation of the vortex originating from the base of the sphere for both the cases which is in well agreement with previous reported results in literature. The values of Strouhal number and drag coefficients agree well with those in previously reported DNS and experimental results in literature. The results from energy spectra illustrated that DES and LES turbulence model are proficient enough to resolve eddies of length scales of inertial subrange to some extent and for getting accurate time averaged statistics, however if the purpose of simulations is to perform a detailed structure analysis of the vortex shedding which takes place and then to understand the intricate physics involved in the flow then it is essential to solve the smallest eddies which are of the Kolmogorov length scales, which is only possible using DNS.
(a)
Acknowledgements We would like to acknowledge ICT-DAE Centre for Chemical Engineering Education and Research for providing the computational facilities for simulations. REFERENCES
(b) Fig. 5: Velocity plots of (a) stationary sphere and (b) rotating sphere at Re=300 for LES simulations
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(a)
(b) Fig.6: Pressure coefficient plots of (a) stationary sphere and (b) rotating sphere at Re=300 for LES simulations
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