TY - JOUR

T1 - Finite-thickness effect on speed of a counter-rotating vortex pair at high Reynolds numbers

AU - Habibah, Ummu

AU - Nakagawa, Hironori

AU - Fukumoto, Yasuhide

N1 - Funding Information:
We are grateful to Yuji Hattori and Stefan Llewellyn Smith for stimulating discussions and invaluable comments. UH was supported by Indonesia DIKTI’s scholarship for doctoral students and young researchers. YF was supported in part by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (Grant No. 16K05476).
Publisher Copyright:
© 2018 The Japan Society of Fluid Mechanics and IOP Publishing Ltd.

PY - 2018/3/29

Y1 - 2018/3/29

N2 - We establish a general formula for the translational speed of a counter-rotating vortex pair, valid for thick cores, moving in an incompressible fluid with and without viscosity. We extend to higher order the method of matched asymptotic expansions developed by Ting and Tung (1965 Phys. Fluids 8 1039-51). The solution of the Euler or the Navier-Stokes equations is constructed in the form of a power series in a small parameter, the ratio of the core radius to the distance between the core centers. For a viscous vortex pair, the small parameter should be where ν is the kinematic viscosity of the fluid and Γ is the circulation of each vortex. A correction due to the effect of finite thickness of the vortices to the traveling speed makes its appearance at fifth order. A drastic simplification is achieved of expressing it solely in terms of the strength of the second-order quadrupole field associated with the elliptical deformation of the core. For a viscous vortex pair, we exploit the conservation law for the hydrodynamic impulse to derive the growth of the distance between the vortices, which is cubic in time.

AB - We establish a general formula for the translational speed of a counter-rotating vortex pair, valid for thick cores, moving in an incompressible fluid with and without viscosity. We extend to higher order the method of matched asymptotic expansions developed by Ting and Tung (1965 Phys. Fluids 8 1039-51). The solution of the Euler or the Navier-Stokes equations is constructed in the form of a power series in a small parameter, the ratio of the core radius to the distance between the core centers. For a viscous vortex pair, the small parameter should be where ν is the kinematic viscosity of the fluid and Γ is the circulation of each vortex. A correction due to the effect of finite thickness of the vortices to the traveling speed makes its appearance at fifth order. A drastic simplification is achieved of expressing it solely in terms of the strength of the second-order quadrupole field associated with the elliptical deformation of the core. For a viscous vortex pair, we exploit the conservation law for the hydrodynamic impulse to derive the growth of the distance between the vortices, which is cubic in time.

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U2 - 10.1088/1873-7005/aaa5c8

DO - 10.1088/1873-7005/aaa5c8

M3 - Article

AN - SCOPUS:85047274240

VL - 50

JO - Fluid Dynamics Research

JF - Fluid Dynamics Research

SN - 0169-5983

IS - 3

M1 - 031401

ER -