Kiyosi Horiuti, Yohei Takagi and Syouji Abe- Multi modes for the vortex sheet-tube transformation...
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Multi modes for the vortex sheet-tube trans- formation process and viscoelastic effect Kiyosi Horiuti, Yohei Takagi and Syouji Abe Department of Mechano-Aerospace Engineering, Tokyo Institute of Technology, Japan Primary elements which constitute the turbulent flow field are the vortex sheets and the vortex tubes [1, 2]. The vortex tubes are often formed by rolling-up of the vortex sheets. It is generally considered that this rolling-up is attributable to the Kelvin-Helmholtz instability, e.g. [3]. The aim of the present study is to reveal the details of the process for formation of the vortex tube along the vortex sheet, and subsequent formation of hierarchical cluster structure of the tubes. Then, we expl ore the implication of the occurren ce of thi s transformati on process for turbulence energy cascade [2]. In the present study, the eigenvalues were ordered so that the eigenvalue, the eigenvector of which is maximally aligned with the vorticity vector is chosen as the z component, the largest remaining eigenvalue as the + component [1]. The eigenvalues of the strain-rate tensor are denoted as σ z and σ ± . We ha ve utilized the DNS data for the incompress ibl e deca ying/fo rced homogeneous iso trop ic turbulence. In an earl y stage in the deca yin g case , sev eral flat she ets , the lateral extents of whic h we re several times of the inte gral lengt h, emerged. Wit h lapse of time, the sheet-tube transformation abundantly occurred along the sheet and the clusters of tubes were formed. The direction of the axis of the vortex tube, however, was often perpendicular to the vorticity vectors along the vortex sheet. This vorticity configuration is similar to that in Kawahara et al. [4] . Appearance of this vort ici ty confi guratio n is inconsistent wit h the stability analysis for the stagnation-point flow [3], in which it was shown that the vorticity component perpendicular to the direction of the diverging flow decays, and that the parallel component can grow. Detailed analysis revealed that the tube was not formed by the rolling-up of a single sheet, but formed through the interaction of the two different sheets. Initially, these vortex sheets were pla ced perpend icular to eac h othe r. Depending on the alignment of vo rtic it y vector s along the two sheets, three modes of configurations were found. When the vorticit y vector on the two sheets were perpendicular (Mode T), the region with σ z < 0 was formed between the two shee ts. When this compres sio n occurred, the press ure Hessi an terms reacted to relax the occurrence of compression by forming a concentrated low pressure region oriented in the direct ion transverse to the vori cit y directi on along the sheet. Because σ + > 0 and the vortex-st retch ing term in the transvers e (+) direction is positive, the low-press ure region gradually accumulated to form the tube, the axis of which was in the direction transverse (+) to the sheet. In the end, the spiral vo rtex sheet emanating from the tube with transvers e vorticity was formed. In another vorticity configuration, the vorticity vectors along the two sheets were parallel (Mode P), and the axis of the tube formed in mode P was parallel to the vortici ty vec tors on the shee ts . The ci rculation around the tube in Mode P was generall y la rge , and the tube in Mode P per si sted for a rat her long peri od of time. This 1
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Multi modes for the vortex sheet-tube trans-
formation process and viscoelastic effect
Kiyosi Horiuti, Yohei Takagi and Syouji Abe
Department of Mechano-Aerospace Engineering,
Tokyo Institute of Technology, Japan
Primary elements which constitute the turbulent flow field are the vortex sheets and thevortex tubes [1, 2]. The vortex tubes are often formed by rolling-up of the vortex sheets. It isgenerally considered that this rolling-up is attributable to the Kelvin-Helmholtz instability,e.g. [3]. The aim of the present study is to reveal the details of the process for formation of thevortex tube along the vortex sheet, and subsequent formation of hierarchical cluster structure
of the tubes. Then, we explore the implication of the occurrence of this transformationprocess for turbulence energy cascade [2]. In the present study, the eigenvalues were orderedso that the eigenvalue, the eigenvector of which is maximally aligned with the vorticity vectoris chosen as the z component, the largest remaining eigenvalue as the + component [1]. Theeigenvalues of the strain-rate tensor are denoted as σz and σ±.
We have utilized the DNS data for the incompressible decaying/forced homogeneousisotropic turbulence. In an early stage in the decaying case, several flat sheets, the lateralextents of which were several times of the integral length, emerged. With lapse of time,the sheet-tube transformation abundantly occurred along the sheet and the clusters of tubeswere formed. The direction of the axis of the vortex tube, however, was often perpendicular
to the vorticity vectors along the vortex sheet. This vorticity configuration is similar to thatin Kawahara et al. [4]. Appearance of this vorticity configuration is inconsistent with thestability analysis for the stagnation-point flow [3], in which it was shown that the vorticitycomponent perpendicular to the direction of the diverging flow decays, and that the parallelcomponent can grow.
Detailed analysis revealed that the tube was not formed by the rolling-up of a single sheet,but formed through the interaction of the two different sheets. Initially, these vortex sheetswere placed perpendicular to each other. Depending on the alignment of vorticity vectorsalong the two sheets, three modes of configurations were found. When the vorticity vectoron the two sheets were perpendicular (Mode T), the region with σ
z< 0 was formed between
the two sheets. When this compression occurred, the pressure Hessian terms reacted torelax the occurrence of compression by forming a concentrated low pressure region orientedin the direction transverse to the voricity direction along the sheet. Because σ+ > 0 andthe vortex-stretching term in the transverse (+) direction is positive, the low-pressure regiongradually accumulated to form the tube, the axis of which was in the direction transverse(+) to the sheet. In the end, the spiral vortex sheet emanating from the tube with transversevorticity was formed. In another vorticity configuration, the vorticity vectors along the twosheets were parallel (Mode P), and the axis of the tube formed in mode P was parallelto the vorticity vectors on the sheets. The circulation around the tube in Mode P wasgenerally large, and the tube in Mode P persisted for a rather long period of time. This
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configuration was considered in Lundgren [5]. As the frequency of occurrence of this sheet-tube transformation increased, the energy cascade took place, and the energy spectrumshowed a profile close to the −5/3 law. The negative σz and vortex-stretching term in Mode
T played an important role for generation of the energy cascade because the energy transferfrom the z component to the transverse component took place with compression. As theReynolds number was increased, its frequency of occurrence increased. Similar results wereobtained in the forced case.
By terminating an occurrence of this sheet-tube transformation and subsequent energycascade, the turbulence generation may be annihilated. It is known that reduction of tur-bulence generation takes place when the viscoelasticity is introduced into the flow. Weinvestigated the effect of viscoelasticity on the sheet-tube transformation process by DNSusing the Johnson-Segalman constitutive equation [6]. Examination of the enstrophy gen-eration term showed that occurrence of the process with Mode T was substantially reduced
because the vorticity in the stretching (z−
)direction was strengthened along the vortex sheetby the backward transfer of the enstrophy due to the polymer stress into the solvent. Theoccurrence of compression (σ
z< 0) was substantially reduced, and at the center of the
tube-core, the pressure due to the polymer stress took a locally maximum value. Therefore,markedly fewer vortex tubes were generated along the vortex sheet than in the Newtonianflow. The effect of the polymer stress was most dominant in the vortex sheet region becausethe distribution of the polymer energy was primarily concentrated along the sheet.
References
[1] K. Horiuti, 2001, A classification method for vortex sheet and tube structures in turbulent flows, Phys. Fluids 13, 3756–3774.
[2] K. Horiuti, 2003, Roles of nonaligned eigenvectors of strain-rate and subgrid-scale stress
tensors in turbulence generation , J. Fluid Mech. 491, 65–100.
[3] O.S. Kerr, and J.W. Dold, 1994, Periodic steady vortices in a stagnation-point flow , J.Fluid Mech. 491, 307–325.
[4] G. Kawahara, S. Kida, M. Tanaka, & S. Yanase, 1997, Wrap, tilt and stretch of vorticity
lines around a strong thin straight vortex tube in a simple shear flow , J. Fluid Mech. 353,
115–162.[5] T.S. Lundgren, 1982, Strained spiral vortex model for turbulent structures, Phys. Fluids
25, 2193-2203.
[6] M.W. Johnson & D. Segalman, 1977, A model for viscoelastic fluid behavior which allows
non-affine defromation , J. Non-Newt. Fluid Mech. 2, 255–270.
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