Selective Energy and Enstrophy Modification of Two-Dimensional

Selective energy and enstrophy modification of two-dimensional decaying turbulence Aditya G. Nair∗, James Hanna & Matteo Aureli Department of Mechanical Engineering, University of Nevada, Reno Abstract In two-dimensional decaying homogeneous isotropic turbulence, kinetic energy and enstrophy are respectively transferred to larger and smaller scales. In such spatiotemporally complex dynamics, it is challenging to identify the important flow structures that govern this behavior. We propose and numerically employ two flow modification strategies that leverage the inviscid global conservation of energy and enstrophy to design external forcing inputs which change these quantities selectively and simultaneously, and drive the system towards steady-state or other late-stage behavior. One strategy employs only local flow-field information, while the other is global. We observe various flow structures excited by these inputs and compare with recent literature. Energy modification is characterized by excitation of smaller wavenumber structures in the flow than enstrophy modification. 1 Introduction Turbulent flows exhibit nonlinear interactions over a wide range of spatiotemporal scales. In two-dimensional (2D) decaying turbulence, the rate of energy dissipation is considerably slowed by kinetic energy transfer to large-scale coherent vortex cores through the inverse energy flux mechanism [2,3,5, 13, 14], while the enstrophy dissipation rate is enhanced by enstrophy transfer to small scale eddies [21]. The identification and modification of collective structures in the flow that accelerate or decelerate the inverse energy flux mechanism or alter the enstrophy cascade is a fundamental question [9, 10]. This is unlikely to be addressed by flow modification strategies based on linearization of the Navier–Stokes equations and reduced-order/surrogate representations that lack strict adherence to conservation laws. The objective of the current work is to tie a flow modification strategy directly to the governing equations and their ensuing conservation laws. A comparison of the behavior of decaying 2D turbulence with either well-developed vortices or phase- scrambled initial conditions revealed the crucial role of coherent vortex structures in suppressing the cascade rate [14]. More recent studies by Jiménez [11, 12] have shed light on the connections between dipoles (counter-rotating vortices) and streams (formed by concatenation of dipoles) on the kinetic energy of the flow. These findings have been corroborated by network-theoretic approaches using induced velocity and flow perturbations [22]. However, these recent data-intensive approaches either require machine learning to extract templates of dynamical significance, or some explicit knowledge of vortical interactions in the flow. In the present work, we provide explicit physical definitions of forcing terms to be added to the Navier- Stokes equations, capable of independently altering conserved quantities, and automatically revealing the flow structures of significance. Several previous efforts have focused on exclusive modification of single conserved quantities in fluid arXiv:2108.01137v1 [physics.flu-dyn] 2 Aug 2021 flows. Vallis et al. [19] and Shepherd [17] modified the Euler equations to alter energy while preserving topological invariants associated with the vorticity field. This enabled the discovery of isolated energy extrema and stable steady equilibria. Sadourny & Basdevant [16] designed, and Vallis & Hua [20] later employed, a subgrid-scale closure scheme that dissipates enstrophy while preserving energy. Another approach was developed by Gay-Balmaz & Holm [6] and applied to dissipate the squared helicity while preserving energy in 3D incompressible flow. A relevant early work with a more general context is that of Morrison [15]. Our approach is inspired by the recently introduced framework of exterior dissipation [1,7], ∗ Corresponding author ([email protected]). 1 which enables selective modification of multiple conserved quantities. We present local and global flow modification approaches with three objectives: (i) selectively and simultaneously alter any integral conserved quantities; (ii) identify characteristic flow structures that accelerate or decelerate the inverse energy flux and enstrophy cascade; (iii) discover and efficiently approach stable steady or slowly-varying states. In certain limits our local approach leads to similar behavior as those of Vallis et al. [19] or Sadourny & Basdevant [16]. 2 Approach We consider the 2D flow of homogenous and incompressible fluid within a fixed bi-periodic square domain D. The governing dynamics is given by the forced Navier-Stokes equations, mu = −u · ru − r?/d ¸ ar2u ¸f , r · u = 0, (1) mC u | {z } |{z} eu du where u = u¹x,Cº is the velocity, ? the pressure, d the density, a the kinematic viscosity, and fu the external forcing to be designed. In the inviscid, unforced case, the flow admits the integral quadratic invariants &8 of kinetic energy 퐸 and enstrophy Ω, defined as ¹ ¹ ≡ = 1 · ≡ Ω = 1 · &1 퐸 2 u u dx,&2 2 ! ! dx, (2) D D where ! = r ×u = lkˆ is the vorticity field. We may express the Eulerian time derivative of these conserved quantities as d ¹ mu ¹ ¹ ¹ &8 = b8 · dx = b8 · eu dx ¸ b8 · du dx ¸ b8 · fu dx. (3) dC D mC D D D ¯ · The contribution of the Euler term in Eq. (3) vanishes [4,8], i.e. D b8 eu dx = 0. The evolution of kinetic energy and enstrophy are governed by b1 = u and b2 = r × !, respectively; using a vector calculus identity, ! · ! = ! · r × u = u · ¹r × !º ¸ r · ¹u × !º, but due to the bi-periodic boundary conditions the divergence term vanishes, and we have simply ! · ! = u · ¹r × !º. Along with Eq. (1), we also consider the forced vorticity transport equation in the form ml ml mk ml mk 2 = − ¸ ar l ¸ 5l, (4) mC mH mG mG mH |{z} | {z } 3l 퐽 ¹l,kº ˆ where u = r × kk defines the streamfunction k , and 5l is the external forcing to be designed. The integral kinetic energy and enstrophy invariants [2] and their Eulerian time derivatives may be expressed as ¹ ¹ d ¹ ml ≡ = 1 ≡ Ω = 1 2 = &1 퐸 2 lk dx,&2 2 l dx, &8 18 dx, (5) D D dC D mC where the energy has been rewritten by eliminating a divergence term due to the bi-periodic boundary ¯ ¹ º conditions. Similarly to before, D 18 퐽 l, k dx = 0. Note that the 18 in Eq. (5) are distinct from the b8 = m 1 · = r × ˆ · mu in Eq. (3). We find that 11 k, by a manipulation of mC 2 u u kk mC and elimination of a divergence term, while clearly 12 = l. The vector and scalar equations (1) and (4) will be the respective starting points for local and global approaches to systematically modify the inviscid conservation or viscous nonconservation of the integral quantities &8. Each approach generates a unique forcing term, as described below. 2 2.1 Local formulation We first construct an external forcing term fu = fu ¹x,Cº in Eq. (1) using only local flow field information. Inspired by Hanna [7] and Aureli & Hanna [1], we define fu = −n1 ¹b1 ^ b2º · b2 − n2 ¹b2 ^ b1º · b1, (6) where b1 = u and b2 = ¹r × !º as discussed above, and n1 and n2 are constant coefficients. The wedge product ^ of two vectors in three dimensions is 2b1 ^b2 = b1b2 −b2b1, and the terms in (6) can be rearranged into double cross products, n1 2 n2 2 fu = − jjr × !jj u − »u · ¹r × !º¼ r × ! − jjujj r × ! − »u · ¹r × !º¼ u . 2 2 (7) | {z } | {z } b2×¹b1×b2º b1×¹b2×b1º The first term exclusively alters the integral flow energy while the second exclusively alters the integral flow enstrophy, allowing for independent manipulation of these two quantities1. If n1 = 0, energy is conserved while enstrophy is increased (decreased) if n2 < 0 (n2 ¡ 0). Similarly, if n2 = 0, enstrophy is conserved while energy is increased (decreased) if n1 < 0 (n1 ¡ 0). However, for 2D turbulence, enstrophy cannot be increased as it is bounded by the initial value of flow palinstrophy [2], so in what follows n2 ≥ 0. In the inviscid limit, the rates of change of the integrals &8 can be rearranged into the simple forms ¹ ¹ ¹ n8 2 n8 2 b8 · fu dx = − »u × ¹r × !º¼ dx = − »u · r!¼ dx, (8) D 2 D 2 D where we have used the fact that ¹r!º · u vanishes for 2D flow. These two rates are proportional, and thus the quantity 퐸/n1 − Ω/n2 is conserved [7], which seems to ensure that the resulting dynamics are not simply driven to a trivial equilibrium when both quantities are dissipated. The rates vanish when the two b8 align. Remarkably, when n2 = 0 the rate of change of energy is the same as that generated by the method of Vallis et al. [19], and when n1 = 0 the rate of change of enstrophy is the same as that generated by the method of Sadourny & Basdevant [16]. 2.2 Global formulation We next construct an alternative external forcing term 5l = 5l ¹x,Cº in Eq. (4) incorporating global flow field information. Interpreting the integrals of Eq. (5) as inner products of square-integrable functions on D allows for a formal analogy with the framework of Aureli & Hanna [1], in which scalar fields are now regarded as infinite-dimensional vectors. We mimic the algebraic construction of exterior dissipation in Aureli & Hanna [1] by introducing an array B = ¹11, 12º = ¹k, lº produced by stacking the vectors 18, and the Gram matrix G of the inner products of 11 and 12, ¹ k2 kl G = 2 dx. (9) D kl l The global forcing is constructed as 1 5l = − B adj¹Gº, (10) 2 1 The construction of b2 involved the discarding of a divergence term. Thus, the first term in fu does actually change the local vorticity magnitude and enstrophy, but its effects are a pure divergence that does not affect the global conservation. 3 where adj¹•) denotes the adjugate (transpose of the cofactor matrix) and = ¹n1, n2º is an array containing the constant coefficients exclusively modulating the rates of 퐸 and Ω, respectively.

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