Fluctuation-induced force in homogeneous isotropic turbulence

DISCUSSION

We now discuss the nature and origin of the computed attractive force. Any surface (plate here) immersed in a flow field experiences two kinds of stresses: viscous stresses and pressure, which act in the tangential and normal direction, respectively. A difference in local pressure on both sides of a rigid plate leads to a force along the normal direction. The pressure in a fluid is directly related to the velocity gradients in the flow, i.e., the curl of velocity, also known as vorticity (ω), and the rate of strain in the fluid σ by the relation ∇²(p/ρ_F) = ω²/2 − σ² = 2Q (5), where Q acts like a source in the pressure evolution equation. A series of classical studies in the past have looked at the consequences of the above equation and the nature of distribution of pressure in the flow (10–12). In a homogeneous and isotropic turbulent flow field, large-scale structures and strain-dominated regions are markers for positive pressure fluctuations, while thin and tubular high-intensity vorticity structures lead to negative pressure fluctuations. Another interesting observation is that the pressure distribution is highly skewed toward negative pressure with exponential tails as opposed to more Gaussian-like behavior toward positive pressure fluctuations. This is primarily a consequence of the interaction between strain-dominated and vorticity-dominated regions in the flow where it is clearly observed that high-strain regions are typically associated with high vorticity, while the converse does not necessarily hold (10). This leads to a bias in the source term (Q) for the pressure Poisson equation and consequently extremely low-pressure fluctuations within regions of high-intensity vorticity structure. The interaction between structures with different characteristics and length scales leads to a nontrivial pressure distribution in the flow.

The forces acting on the plates are directly governed by the local pressure, which, in turn, is governed by the organization of the underlying energy field and vorticity. While energy is contained in the largest wavelengths, regions of intense vorticity organize themselves into tubular-like structures, referred to as worms in literature (13). The radius of these tubular structures scales as the Kolmogorov length η_K = (ν³/ε)^1/4 (a measure of the viscous cutoff length), where ε is the energy dissipation in the system. Their length scales as the integral, or decorrelation, length scale. An estimate for this decorrelation length scale is provided by the large-eddy length scale, defined as L = k^3/2/ε, with k = 3/2ρ_Fu′² the kinetic energy of the flow and u′ the root mean squared velocity fluctuation in one direction (in our simulations, L is approximately constant). The complex organization of energy and vorticity into spatial structures in hydrodynamic turbulence, which affects the nature of positive and negative pressure fluctuations combined with the geometrical confinement induced by the plates, leads to the Casimir-like force.

A single plate in a randomly forced fluid will feel, on average, symmetric forces on both sides. However, two rigid plates placed next to each other restrict the structures that “fit” between the plates. Such a geometrical confinement completely excludes high-energy structures except when the plates are far apart. The geometrical confinement also affects the magnitude and direction of intense vorticity structures that fit between the plates (see the Supplementary Materials for a quantification of anisotropy). Both effects modify the pressure fluctuations and generate a net force. The mechanism of force generation 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 energy and vorticity fluctuations organize themselves into defined and distinct length scales, we would expect the Casimir force to be nonmonotonic, with the peak force achieved when the plate separation is comparable to the length scale of the excluded structures (4). As Re_λ increases, the magnitude of the force increases, possibly because of increase in the intensity of thin vortex structures, which are markers for negative pressures. The plate distance at which optimal force is achieved varies between d/ℒ=0.05 and 0.1 as we vary the Reynolds number between Re_λ = 65 and 140. To fully understand how the plate distance for optimal force generation varies with Reynolds number, a denser sweep of the parameter space is required, and this is anticipated in future studies.

To quantitatively understand the above arguments, we examine how the fluctuation modes distribute themselves inside the slit. Figure 2 shows an instantaneous snapshot of the flow for two plate distances. The left panel clearly shows that the energetic structures have lengths comparable to or larger than the plates. They can only penetrate between the plates for the large plate distances. Meanwhile, the right panels clearly show the tubular vorticity structures. By comparing both figures, we can see how for close distances, the plates organize the tubular structures within, while for large distances, they are more disordered.

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The time-averaged kinetic energy and the marker for intense vortex structures Q in the planes normal to the plates (Fig. 2, C and D) further corroborate the relationship between energy, vorticity structures, and force generation. A narrower plate separation leads to lower mean energy inside the walls, agreeing with our physical picture that the plates filter the fluctuations in between and outside the plates. However, no qualitative change is seen in the profiles of kinetic energy close to the maximum, and the nonmonotonicity of the force cannot be explained from energy statistics alone. For the Q profiles, there is a qualitative change of behavior around the plate separation at which the maximum force is attained (green and yellow curves). The mean value of Q increases within the gap for plate separations at which the maximum force is found, indicating the significance of structures where vorticity dominates strain rate (14). At very close distances between the plates, viscous effects damps the evolution of the vortical structures in the gap. The whole flow becomes quiescent between the plates, and no structures fit. As the plate separation increases, the region between the plates becomes populated, first by intense vorticity structures and later by energy-containing structures.

By immersing the plates into the flow, we selectively filter the flow fields. In Fig. 3, we plot the pressure within and outside the immersed plates. We note that while the sharp edges of the plates are clearly visible in this figure, this is not the source of the force. To ensure this, in Fig. 1, we reported that an attractive force was also found between two spheres. Instead, we can understand the force from the fact that when we place two plates in the flow, we exclude the energy-containing structures from between the plates, which generates high pressures at the plate centers. At the same time, we force the intense vorticity structures to interact in close quarters within the confinement, which enhances vortex stretching and produces even more intense vortex structures, which decreases the pressure (13). The pressure on the outside section of the plate can be seen to increase greatly between the left and middle pictures, as structures are excluded from a larger volume. The pressure in the inside remains relatively low in comparison to the outside, generating the attractive force. As the plates separate further, the prominent peak in the pressure fluctuations at the midgap fades, and the pressure fluctuations profile between the plates becomes horizontal; the interaction between intense vortical structures is now less localized. This coincides with a sharp decrease in the force magnitude between the plates.

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Because of the intimate link between vorticity and pressure, one can expect this effect to show up in the pressure fluctuations, shown in Fig. 3B. Around d/ℒ = 0.075, a sharp peak of pressure fluctuations develops at the midgap. The peak is present for the plate separations that cause the highest attractive forces. The profiles of Q (in Fig. 2B) and the pressure fluctuations indicate the enhancement of intense vortex structures (source of low-pressure regions), which also contributes to optimal force generation. To reassure the reader that the force in not a numerical artifact, we further characterize the nature of force generation in the Supplementary Materials.

In summary, we demonstrate the existence of an attractive turbulent Casimir-like force by direct numerical simulation of two plates immersed in a homogeneous isotropic turbulent flow. Our study sheds light on the interaction between objects in a turbulent flow at very close distances. We anticipate that this force can be experimentally realized in setups previously used for studying single-phase and multiphase HIT (15). Further research to understand the Re_λ dependence is needed. Moreover, we speculate that this mechanism of generating a nonmonotonic Casimir-like force due to nontrivial spatial redistribution of energy containing structures and intense vortex filaments might be a general phenomenology for active and nonequilibrium systems and could appear in large classes of biological fluids comprising microbial suspensions that exhibit notable analogies with turbulent flows (16–19).