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Fluctuation-induced forces in homogeneous isotropic turbulence

Understanding force generation in non-equilibrium systems is a significant challenge in statistical physics. We uncover a surprising fluctuation-induced force between two plates immersed in homogeneous isotropic turbulence using Direct Numerical Simulation. The force is a non-monotonic function of plate separation. The mechanism of force generation reveals an intriguing analogy with fluctuation-induced forces: energy in the fluid is localised in regions of high vorticity, or"worms", which have a characteristic length scale. The magnitude of the force depends on the packing of worms inside the plates, with the maximal force attained when the plate separation is comparable to the characteristic worm length. A key implication of our study is that the length scale-dependent partition of energy in an active or non-equilibrium system determines force generation.

2 non-equilibrium system with coherent spatial structures and well-studied mechanisms of energy dissipation, is hitherto unknown.
In this Letter, we report the first Direct Numerical Simulation of hard plates immersed in Homogeneous Isotropic Turbulence. We demonstrate the existence of an attractive fluctuation-induced "Casimir" force, and reveal that the force is a non-monotonic function of plate separation. By analysing the pressure distribution inside the plates as well as the location of the force maximum, we show that spatially localised "worms" [7], or regions of high vorticity, acts as energy carriers and the ability of the worms to pack inside the plates determines the magnitude of the Casimir-like force.

HIT: A ZERO-MEAN FLUCTUATING TURBULENT FLOW
Homogeneous Isotropic Turbulence (HIT) [8] is a basic model for turbulence, where an idealized turbulent state is simulated by using a triply-periodic computational domain. The domain is randomly forced at the largest wavelengths that fit in the computational box.
The energy then cascades across length-scales, energizing smaller and smaller structures up to scales where viscosity dominates (the Kolmogorov scale), and the energy is dissipated.
If the forcing is continuous in time, statistically stationary states with a continuous energy spectrum can be reached. Intermediate scales are energetic, and play an active role in the cascade. They are known as "inertial" length scales, and are neither artificially forced nor are heavily dissipative. In this inertial range there is a robust relationship between energy and wavenumber known as the "-5/3rds law" [9].
When a plate is submerged in a turbulent fluid, it is subjected to two kinds of loads: viscous and pressure forces. Viscous stresses are mainly tangential, but pressure acts in the normal direction. A difference in pressure on both sides of a plate can lead to a force which attracts or repels plates parallel to each other, generating a Casimir-like effect.
The pressure in a fluid is directly related to the vorticity, i.e. the curl of velocity [10].
Regions of intense vorticity organize themselves into tubular-like structures, also known as worms [7], their radius scales as the Kolmogorov length η K = (ν 3 /ε) 1/4 (a measure of the viscous cut-off length), and their length scales as the integral, or decorrelation length-scale, where ν is the kinematic viscosity of the fluid and ε the energy dissipation in the system.
An estimate for this decorrelation length-scale is provided by the large-eddy length scale, defined as L = k 3/2 /ε, with k the kinetic energy of the flow.
While a single plate in a randomly forced fluid will feel on average symmetric forces on both sides, by having two plates close to each other, we can restrict the worms that "fit" between the plates, thus controlling the pressure fluctuations on one side and generating a net force. This mechanism is analogous to thermal or quantum Casimir forces [1], where a restriction in the fluctuations imposed by the boundaries leads to force generation. Here, as the fluctuations organise themselves into a defined length scale -the "worm" -we would expect the Casimir force to be non-monotonic with the peak force achieved when the plate separation is comparable to the length scale of the worms [4].

A DISJOINING FORCE IN TURBULENT FLOW
To demonstrate Casmir-like force in turbulence, we perform direct numerical simulations of HIT using the Navier-Stokes equations. We use a computational periodic cube of side L, with two square parallel plates of size of l p /L = 0.25, and vary the plate distance d  A Casimir-like force arises as the plates modify the worm structure between the plates by restricting the energy fluctuation modes. Figure 2a shows that the force between the plates is attractive and is a non-monotonic function of plate separation for three different values of turbulent strengths, Re λ . A force can arise as long as the two plates can feel each other, and that happens when the distance between them becomes comparable to the largeeddy length, L. The region between the two plates then has a significantly a lower pressure than the region outside the plates because energetic worms cannot fit in the narrow gap. A non-monotonic force arises because a lower limit is set by viscosity, where the whole flow becomes quiescent between the plates and no worms fit. Comparing Figure 2a and 2b shows that it is the worm length, not diameter, that determines force generation -the location of the force minimum is approximately constant in d/L units but not in d/η K units, suggesting that the important scale is the large-eddy scale, which coincides with worm length. Worms have trouble fitting between the plates when the distance between them is smaller than 0.1L, that is, approximately one-tenth of their length. This worm picture is further confirmed by the fact that the magnitude of the force increases as Re λ increases, as it is known that the vorticity and circulation of worms increases with increasing Re λ [7].

CONFINEMENT OF FLUCTUATIONS AND FORCE GENERATION
A lingering question is how do the fluctuation modes distribute themselves inside the slit. Figure 3a shows the average kinetic energy of the fluid in the area between the two plates for several distances. A narrower plate separation leads to lower mean energy inside the walls, agreeing with our physical picture that the partitioning of energy inside and outside the wall leads to force generation. Moreover, the energy is localised near mid-plane, thus the the worms, which carry the kinetic energy, behave as particles which are repelled by the walls.
Another view of this fluctuation-induced force can be obtained by analysing the pressure fluctuations. Pressure fluctuations in turbulence are known to be asymmetric, and have long tails for negative fluctuations. This is due to both the intimate connection between pressure and vorticity, and the non-Gaussian nature of turbulence. Gaussian velocity fields are known to produce very different pressure distributions [10]. The plates filter the fields and strengthen the intense negative pressure fluctuations. This is shown in Figure 3b

CONCLUSION
In summary, we demonstrate the existence of an attractive turbulent Casimir-like force by direct numerical simulation of two plates immersed in an homogeneous isotropic turbulent flow. The force arises as the energy in a turbulent flow is organised into spatially localised structures that have a finite size, and these structures are extruded from plates that are close together. We show that the force is a non-monotonic function of plate separation, with the peak in the force located when the plate separation is comparable to the characteristic size of those spatial structures. Our study sheds light on the interaction between objects in a turbulent flow. Moreover, we speculate that this mechanism of generating a non-monotonic Casimir force due to non-trivial spatial partition of energy might be a general phenomenology for active and non-equilibrium systems.

METHODS
The incompressible Navier-Stokes (NS) equations (1-2) are solved using an energyconserving second-order centered finite difference scheme in a Cartesian domain with fractional time-stepping. ∂u ∂t u is the fluid velocity vector, ρ f is the density of the fluid and p is the hydrodynamic pressure. F = f tur + f ibm is a force vector which is a sum of two terms; i.e. f tur is a contribution from the forcing needed to generate homogeneous turbulence in the domain, while f ibm is the force needed to enforce the influence of the immersed plates on the flow through the immersed boundary method which is described later. In the NS equations, an explicit Adams-Bashforth scheme is used to discretise the non-linear terms while an implicit Crank-Nicholson scheme is used for the viscous terms [12]. the forcing scheme and its corresponding parameters can be found in Eswaran and Pope [13] and Chouippe and Uhlmann [14]. To spatially discretize the equations, a cubic grid was used, with 240 3 points for Re λ = 65, 360 3 for Re λ = 100, and 480 3 points for Re λ = 140, similar to those used in [14] at the same values of Re λ .
The influence of rigid plates on the surrounding fluid was simulated using an immersed boundary method (IBM) based on the moving least squares (MLS) approximation [11]. In Temporal convergence of the forces was assured by running the simulations until the force from the hydrodynamic pressure on both plates were equal (but oppositely signed) to within 3%. This defined the height of the errorbars in Figure 2. In practice this meant running up to more than 100 large-eddy turnover times defined by T e = u 2 /ε. The shortand long-timescale temporal fluctuations in the force can be seen in Fig. 1.