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Enstrophy Transfers in Helical Turbulence

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Enstrophy Transfers in Helical Turbulence Enstrophy transfers in helical turbulence Shubhadeep Sadhukhan,1, ∗ Roshan Samuel,2, y Franck Plunian,3, z Rodion Stepanov,4, x Ravi Samtaney,5, { and Mahendra Kumar Verma6, ∗∗ 1Department of Physics, Indian Institute of Technology, Kanpur, Uttar Pradesh 208016, India 2Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 208016, India 3Universit´eGrenoble Alpes, Universit´eSavoie Mont Blanc, CNRS, IRD, IFSTTAR, ISTerre, 38000 Grenoble, France 4Institute of Continuous Media Mechanics UB RAS, Perm, Russian Federation 5Mechanical Engineering, Division of Physical Science and Engineering, King Abdullah University of Science and Technology - Thuwal 23955-6900, Kingdom of Saudi Arabia 6Department of Physics, Indian Institute of Technology, Kanpur, Uttar Pradesh 208016 (Dated: April 23, 2020) In this paper we study the enstrophy transers in helical turbulence using direct numerical simu- lation. We observe that the helicity injection does not have significant effects on the inertial-range energy and helicity spectra (∼ k−5=3) and fluxes (constants). We also calculate the separate con- tributions to enstrophy transfers via velocity to vorticity and vorticity to vorticity channels. There are four different enstrophy fluxes associated with the former channel or vorticity stretching, and one flux associated with the latter channel or vorticity advection. In the inertial range, the fluxes due to vorticity stretching are larger than that due to advection. These transfers too are insensitive to helicity injection. I. INTRODUCTION tensively studied [7{26]. Andr´eand Lesieur [9] studied the effect of helicity on the evolution of isotropic turbu- Turbulence is a classic and still open problem in fluid lence at high Reynolds numbers using a variant of Marko- dynamics. The energetics of homogeneous and isotropic vian eddy-damped quasi-normal (EDQNM) theory. Po- turbulence were explained through the celebrated works lifke and Shtilman [10] showed that the energy decay is of Kolmogorov [1, 2]. Kolmogorov argued that under sta- slowed down by large initial helicity. Waleffe [11] intro- tistically steady state, the rate of energy supplied by an duced helical decomposition of the velocity field and dis- external force is equal to the rate at which energy cas- cussed triadic interactions in helical flows. Yokoi and cades from large to smaller scales which is equal to the Yoshizawa [12] reported the effects of helicity in 3D in- rate of energy dissipation. Obukhov [3] and Corrsin [4] compressible inhomogeneous turbulence with the help of generalized Kolmogorov's theory to isotropic turbulence a two-scale direct-interaction approximation (DIA). Zhou with statistically homogeneous fluctuations of tempera- and Vahala [13] showed that helicity does not alter renor- ture field embedded in a turbulent flow. The theories of malized viscosity. Ditlevsen and Giuliani [14], Chen et al. Kolmogorov, Obukhov, and Corrsin, collectively referred [15], and Lessinnes et al. [16] studied helicity fluxes for to as Kolmogorov-Obukhov theory, predict that the ki- different types of helical modes. Chen et al. [17] stud- 2 −5=3 ied intermittency in helical turbulence. Teitelbaum and netic energy spectrum E(k) = uk=k ∼ k , and the 2 Mininni [18] explored the effect of helicity in rotating spectrum of temperature fluctuations Eζ (k) = ζk =k ∼ k−5=3, where k = 2π=l is the wavenumber with l as the turbulent flows, while Pouquet et al. [19] reported ev- idences for three-dimensionalization recovered at small length scale, and uk and ζk are respectively the veloc- ity and temperature fluctuations at wavenumber k. Re- scales. Stepanov et al. [20] explored the helical bottle- searchers have also studied the spectra of density and neck effect in 3D isotropic and homogeneous turbulence other passive scalars in the same framework. Refer to using a shell model. Maximum helicity states have been Lesieur [5] and Verma [6] for detailed references. It is studied by Biferale et al. [21], Kessar et al. [22], and Sa- important to note that the effect of helicity is absent in hoo and Biferale [23]. Mode-to-mode and shell-to-shell the above-described works. helicity transfers have been studied by Avinash et al. Energy and helicity are two inviscid conserved quanti- [24], and Teimurazov et al. [25, 26]. Helicity may also arXiv:2004.10621v1 [physics.flu-dyn] 22 Apr 2020 ties in three-dimensional (3D) turbulent flows. The effect play a crucial role in magnetohydrodynamics [27, 29, 30] of helicity in fully developed 3D turbulence has been ex- and dynamo action [31{34]. In this paper we derive formulas for mode-to-mode enstrophy transfers and their associated fluxes. These quantities arise due to the nonlinear interactions. In ∗ [email protected] Sec. II we derive the expressions of mode-to-mode en- y [email protected] z [email protected] strophy transfers and the corresponding fluxes. In x [email protected] Sec. III, after introducing the helical forcing that we use, { [email protected] the enstrophy fluxes are calculated from direct simula- ∗∗ [email protected] tions of 3D turbulence using the pseudo-spectral code 2 TARANG [35, 36]. We conclude in Sec. IV. with k0 + p + q = 0. Using the incompressibility condi- tion, we can show that S!!(k0jp; q) + S!!(pjq; k0) + S!!(qjp; k0) = 0; (9) II. ENSTROPHY TRANSFERS AND FLUXES where S!! represents the advection of vortices, thus in- The motion of an incompressible fluid is described by volving enstrophy exchange among the vorticity modes 0 the Navier-Stokes equations !(k ); !(p), and !(q). On the other hand we have !u 0 !u 0 !u 0 @u S (k jp; q) + S (pjq; k ) + S (qjk ; p) 6= 0; (10) = −(u · r)u − rp + νr2u + F; (1) @t where S!u represents the stretching of vortices, thus ei- r · u = 0; (2) ther increasing or decreasing the enstrophy E!. Inci- dentally (10) implies that enstrophy is not a conserved where u and p are the velocity and pressure fields, ν the quantity in 3D hydrodynamics. kinematic viscosity, and F an external force. Now we can split the above transfers into individual The vorticity ! = r × u obeys the following equation: contributions from modes p and q: @! S!!(k0jp; q) = S!!(k0jpjq) + S!!(k0jqjp) (11) = (! · r)u − (u · r)! + νr2! + r × F; (3) @t S!u(k0jp; q) = S!u(k0jpjq) + S!u(k0jqjp); (12) where the first two terms in the right hand side corre- with spond to vorticity stretching and vorticity advection re- S!!(k0jpjq) = −=[(k0 · u(q))(!(p) · !(k0))]; (13) spectively. In Fourier space the vorticity, which is defined !u 0 0 0 as !(k) = ik × u(k), satisfies S (k jpjq) = +=[(k · !(q))(u(p) · !(k ))]: (14) Here S!!(k0jpjq) and S!u(k0jpjq) denote two kinds of d X !(k) = i (k · !(q))u(p) − (k · u(q)) !(p) mode-to-mode enstrophy transfer, both being from p to dt 0 p k with q acting as a mediator. They arise due to advec- − νk2!(k) + ik × F(k); (4) tion and stretching respectively. Here we note that S!!(k0jpjq) = −S!!(pjk0jq) (15) where q = k − p and k = jkj. due to the incompressibility condition, k · u(k) = 0; this result is consistent with (9). Substitution of (13-14) into A. Enstrophy transfers (5) yields d X X Enstrophy, which is defined as W (k) = j!(k)j2=2, sat- W (k0) = S!!(k0jpjq) + S!u(k0jpjq) dt isfies p p − 2νk02W (k0) + k02<fu∗(k0) · F(k0)g; (16) d X W (k) = =f(k · u(q))(!(p) · !∗(k))g dt 0 p and q = −(k + p). − =f(k · !(q))(u(p) · !∗(k))g B. Enstrophy fluxes − 2νk2W (k) + k2<fu∗(k) · F(k)g; (5) Summing the previously discussed mode-to-mode en- where q = k − p. Setting k0 = −k, the above equation strophy transfers over p and k0 and depending on 0 can be rewritten in the form whether p and k belong to the sphere of radius k0 or not, we can define the five following enstrophy fluxes d 0 X !! 0 !u 0 W (k ) = S (k jp; q) + S (k jp; q) !< X X !! 0 dt Π!>(k0) = S (k jpjq) (17) p 0 jp|≤k0 jk j>k0 02 0 02 ∗ 0 0 − 2νk W (k ) + k <fu (k ) · F(k )g; (6) u< X X !u 0 Π!>(k0) = S (k jpjq) (18) !! 0 !u 0 0 where S (k jp; q) and S (k jp; q) are the combined jp|≤k0 jk j>k0 0 transfers of enstrophy from modes p and q to k , defined u< X X !u 0 Π!<(k0) = S (k jpjq) (19) as 0 jp|≤k0 jk |≤k0 S!!(k0jp; q) = −={(k0 · u(q))(!(p) · !(k0))g u> X X !u 0 Π!>(k0) = S (k jpjq) (20) 0 0 0 −={(k · u(p))(!(q) · !(k ))g; (7) jpj>k0 jk j>k0 S!u(k0jp; q) = +=f(k0 · !(q))(u(p) · !(k0))g u> X X !u 0 Π!<(k0) = S (k jpjq); (21) 0 0 0 +=f(k · !(p))(u(q) · !(k ))g (8) jpj>k0 jk |≤k0 3 Performing a sum over all modes inside the sphere of radius k0 yields X d ( + 2νk02)W (k0) − k02<fu∗(k0) · F(k0)g dt 0 < > jk |≤k0 u u !> u< u> = (Π!< + Π!< + Π!<)(k0): (23) Adding (22) and (23) and noticing that (15) implies !< !> Π!> + Π!< = 0, we obtain X d ( + 2νk02)W (k0) − k02<fu∗(k0) · F(k0)g dt k0 u< u> u< u> = (Π!> + Π!> + Π!< + Π!<)(k0): (24) ω< ω> As the left hand side of (24) does not depend on k0, it implies d u< u> u< u> Π!> + Π!> + Π!< + Π!< = 0 (25) dk0 which is valid at all times.
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