Structured Nonlinear Cascades Bridging Macroscopic Fluid Scales and Molecular Vibrations

Structured Nonlinear Cascades Bridging Macroscopic Fluid Scales and Molecular Vibrations Andrei T. Patrascu1 1FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] I propose and theoretically analyse a novel approach to selectively excite molecular vibrational modes through structured fluid dynamics guided by generalised symmetry-based transformations of the Navier–Stokes equations. By encoding specific molecular resonance information into structured macroscopic fluid perturbations and using iterative nonlinear cascades, I demonstrate numerically that energy can coherently transfer from macroscopic scales down to molecular vibrational frequencies. This structured cascade, described by a generalised Gelfand transform and associated nonlinear structure constants, ensures resonance conditions at molecular scales, significantly delaying thermalisation and enabling precise quantum state manipulation in fluids. Numerical simulations explicitly targeting the asymmetric vibrational mode of CO2validate this methodology, highlighting its potential applications in controlled molecular excitation and coherent fluid-based quantum manipulation. The novelty of this work explicitly lies in introducing generalised symmetry-adapted nonlinear cascades, carefully designed to achieve coherent and selective molecular-scale energy excitation from structured macroscopic fluid perturbations, a significant departure from traditional turbulent cascades. INTRODUCTION Precise excitation and control of molecular quantum states through macroscopic fluid dynamics [1], [2], [3] remains a formidable challenge due to the vast disparity in frequency scales between macroscopic fluid motions and molecular vibrations. Traditional fluid turbulence cascades energy chaotically down to microscopic scales, rapidly dissipating structured information and limiting coherent molecular state manipulation [4], [5], [6]. However, the controlled and coherent excitation of specific molecular vibrational modes would unlock significant potential in chemical reaction control, quantum technology, and energy harvesting applications. This study introduces a symmetry-based structured nonlinear cascade framework that explicitly demonstrates controlled and coherent energy transfer from macroscopic fluid scales down to precise molecular vibrational modes. Unlike conventional turbulence, where energy cascades randomly and diffusively across scales, this structured approach explicitly leverages symmetry-adapted transformations and nonlinear dynamics to selectively excite targeted molecular resonances, providing new insights into the coherent manipulation of quantum molecular states through macroscopic fluidic control. Prior studies on turbulence-driven energy transfer, molecular control via acoustic and optical means, and optimal control in structured fluid systems provide a foundational context for this novel methodology [13], [14], [15]. In this Letter, I introduce a general theoretical framework based on generalised Gelfand transformations [7] of the Navier-Stokes equations [8], explicitly incorporating molecular symmetry into structured fluid modes. This structured approach utilises nonlinear interactions and iterative cascades across multiple scales to progressively guide energy toward precise molecular resonances. Numerical simulations demonstrate the feasibility of selectively exciting the asymmetric stretch mode of the CO2molecule, illustrating the effectiveness of symmetry-based fluid perturbations. My approach reveals that structured symmetry-preserving cascades can efficiently bridge macroscopic-to-molecular scales, overcoming conventional turbulence limitations and enabling a new class of coherent fluid-based quantum manipulations. My structured nonlinear cascade approach explicitly leverages core concepts from dynamical systems theory and chaos, including nonlinear resonance phenomena, coherence maintenance across scales, and structured mode interactions, as comprehensively discussed in ref. [9]. THEORETICAL FRAMEWORK The fundamental idea behind our structured fluid cascade approach is the generalisation of the Navier–Stokes equations using the Gelfand transform. The central idea is to start with an initial fluid perturbation structured by symmetry to resonate with a targeted molecular mode. Then I solve the Navier-Stokes equations numerically, in a symmetry adapted form (via Gelfand transforms) and use the computed solution as feedback to adjust the initial 2 structured perturbation iteratively, achieving increasingly accurate resonant conditions at the molecular scale. This method begins by expressing the classical Navier-Stokes equations in terms of fluid velocity fields: ρ∂u ∂t +ρ(u·∇)u=−∇P+µ∇2u+ρf (1) where ρis the fluid density, uis the velocity field, Pis the pressure, µis viscosity, and fis the external forcing. To systematically encode molecular symmetry and resonance information into fluid dynamics, I apply the Gelfand transform, a group-theoretic generalization of the Fourier transform extensively discussed in mathematical and applied contexts [11], [12]. For abelian groups, the Gelfand transform reduces to the standard Fourier transform, decomposing functions into exponential modes indexed by characters of the group. In the non-abelian case, however, irreducible unitary representations are generally multidimensional, resulting in matrix-valued modes that encode richer symmetry structures. For abelian groups, the Gelfand transform reduces to the standard Fourier transform, decomposing functions into exponential modes indexed by characters of the group. In the non-abelian case, however, irreducible unitary representations are generally multidimensional, resulting in matrix-valued modes that encode richer symmetry structures. Specifically, the fluid velocity field is decomposed into structured symmetry modes labeled by irreducible representations πof the molecular symmetry group G: ˆu(π, t) = ZG u(g, t)π(g−1)dg (2) where g∈G, and the integration is over the molecular symmetry group. The inverse Gelfand transform reconstructs the velocity field as u(g, t) = X π dim(π)Tr[ˆu(π, t)π(g)] (3) This transform maps the Navier-Stokes equations into structured symmetry space, resulting in explicitly symmetryadapted structured fluid modes. The nonlinear interaction term transforms into: (u·∇)u→X π1,π2 Γ(π, π1, π2)ˆu(π1, t)ˆu(ˆπ2, t) (4) where the structure constants Γ(π, π1, π2) are obtained via symmetry-based integrals: Γ(π, π1, π2) = ZG π(g−1)[(π1(g)·∇)π2(g)]dg (5) The structured modes derived via generalised Gelfand transforms, explicitly encode molecular resonance conditions directly into macroscopic fluid perturbations. The molecular-level resonance explicitly enters the formulation by choosing irreducible representations and associated generalised frequencies ωπprecisely matching molecular vibrational frequencies. This resonance matching ensures a direct, coherent transfer of macroscopic fluid energy to targeted molecular vibrational modes, linking fluid dynamical methodology with molecular-level quantum states. The macroscopic scales are determined by characteristic fluid dynamic length scales (e.g., characteristic system size, fluid flow velocity, and relevant timescales derived from boundary conditions). These scales are chosen to match standard experimental setups and practical computational limits, typically corresponding to frequencies in the MHz regime, allowing a controlled exploration of nonlinear cascade processes bridging the gap toward molecular scales. To describe multiple cascades, I employ an iterative numerical method. Initially, structured modes corresponding to lower frequency scales are excited by external perturbations. After numerically resolving the generalised GelfandNavier-Stokes equations at each intermediate scale, the resulting structured fields guide adjustments in experimental parameters such as geometry, forcing frequencies, and boundary conditions are implemented. These iterative adjustments are determined by analysing how nonlinear interactions shift energy distributions towards higher-frequency molecular resonances. The transition from a macroscopic fluid perturbation with a relatively low frequency, down to a molecular vibrational frequency would involve multiple intermediate cascades and non-linear frequency shifts across several scales. If we consider the macroscopic fluid perturbations typically in the range of Hz to kHz or MHz frequencies (acoustic waves to ultrasound) and the molecular vibrational modes typically in the THz range (∼1012Hz), where for example in the CO2case the asymmetric stretch mode is at 2349 cm−1which would be ∼70T Hz the frequency gap would be 106−1012 Hz. Each nonlinear fluid cascade typically allows frequency multiplication on the 3 order of a factor of approximately 2 to 10. Let us consider an average frequency multiplication factor of fcascade ∼5. The total number of cascades Nrequired would be fN cascade ∼106and hence Nlog10(fcascade)∼log10(106) (6) Using the chosen cascade factor (fcascade = 5) we would have Nlog10(5) ∼6⇒N∼6 log10(5) ∼6 0.699 ∼8.6 (7) Thus, around 8 to 9 cascade steps are realistically required to bridge from MHz scale to THz scale. Realistic estimations indicate that bridging typical macroscopic frequencies (∼MHz) to molecular vibrational frequencies (∼THz) typically requires around 8-10 cascades, with each cascade step involving frequency multiplications by factors of about 5-10. This method ensures precise matching of symmetry and frequency conditions at each cascade step, progressively steering energy transfer to targeted molecular vibrational modes. I follow this algorithm starting with the generalised Navier-Stokes equation written in the symmetry adapted form using the Gelfand transformation ρ∂ˆu(π, t) ∂t +ρX π1,π2 Γ(π, π1, π2)ˆu(π1, t)ˆu(π2, t) = −ˆ P(π, t)−µ|π|2ˆu(π, t) + ρˆ f(π, t) (8) where ˆu(π, t) is the structured fluid velocity mode in the irreducible representation π, Γ(π, π1, π2) are nonlinear coupling constants defined as symmetry based integrals in the way I showed previously, and ˆ f(π, t) is the structured forcing. I also use the generalised frequency measure |π|which is the generalisation of the frequency measure ωk=|k| in standard Fourier analysis. While in Fourier analysis each mode is labeled by the wave number k, in the case of non-abelian groups, each irreducible representation πcan be seen as a structured mode associated with a particular symmetry. Because irreps are matrix-valued, multidimensional, and represent richer structure, the frequency measure associated with them generalises into a quantity that characterises how rapidly or intricately functions (fluid modes) vary across the group space. Given a compact group Gand an irreducible representation π:G→GL(Vπ), we define a suitable Laplace-Beltrami-type operator ∆Gon the group manifold G. The generalised frequency measure for the irrep πis then naturally defined via the eigenvalue of this operator. Consider the eigenvalue equation for ∆Gacting on representation functions ∆Gπ(g) = −λππ(g) (9) where λπis a scalar (the eigenvalue) associated uniquely with the representation π. This scalar is a natural generalised ”frequency squared” measure associated with the representation π. Then the generalised frequency measure is ωπ=√λπ. In the structured Navier-Stokes equations each symmetry structured mode ˆu(π, t) explicitly experiences dissipation and resonance conditions modulated by this generalised frequency measure. For instance, viscous damping takes the form −µ|π|2ˆu(π, t) (10) where typically |π| ∼ ωπ. The generalised frequency measure plays a role analogous to that of the usual wave number squared in the classical Navier-Stokes Fourier expansions. For abelian groups e.g. the circle S1, the variation is simple and linear, just how fast a function oscillates when we move around the circle ones. For non-abelian groups the rapid variation is multidimensional. As we move continuously through the group, the irreducible representation matrix elements change simultaneously in a more complex manner. Rapid variation, here, means how quickly these matrix entries ”rotate” or change values as we traverse paths through the group manifold. A higher generalised frequency measure implies matrix elements rapidly oscillating, producing higher complexity (and more structured oscillations) when traversing the group. In the normal Fourier description, the frequency describes uniform plane-wave oscillations with linear increments in frequency, clearly related to spatial wavelength. In the generalised (structured) case, the frequency describes structured, symmetry based oscillations with complexity increasing as the representation indices grow. It captures how structured modes reflect intricate patterns encoded by symmetry. To automate dynamical adjustments, we formulate the problem as an optimal control problem. We define the cost functional Jas J[ˆu(π, t),ˆ f(π, t)] = 1 2ZT 0X π||ˆu(π, t)−ˆutarget(π)||2dt +λ 2ZT 0||ˆ f(π, t)||2dt (11) 4 The first term penalises deviations from the target structured mode ˆutarget(π). The second term penalises the magnitude of adjustments (control effort) ˆ f(π, t) and λis a regularisation parameter controlling trade-off between accuracy and effort. To find the optimal structured adjustments ˆ f(π, t) we introduce a Lagrange multiplier (adjoint field) ˆv(π, t) and define a Lagrangian functional L=J[ˆu, ˆ f] + ZT 0X π*ˆv(π, t), ρ∂ˆu(π, t) ∂t +ρX π1,π2 Γ(π, π1, π2)ˆu(π1, t)ˆu(π2, t) + ˆ P(π, t) + µ|π|2ˆu(π, t)−ρˆ f(π, t)+dt (12) Taking the variations of Lwith respect to ˆu(π, t), ˆ f(π, t), and ˆv(π, t) gives the forward state equation, which is the original structured Navier-Stokes equation and the adjoint (backward) equation which comes from the variation with respect to ˆu −ρ∂ˆv(π, t) ∂t +ρX π1,π2 Γ(π1, π, π2)[ˆv(π1, t)ˆu(π2, t) + ˆu(π1, t)ˆv(π2, t)] + µ|π|2ˆv(π, t) = ˆu(π, t)−ˆutarget(π) (13) The boundary condition at final time Tis ˆv(π, T ) = 0. The optimality condition (namely the variation with respect to ˆ fis ˆ f(π, t) = 1 λˆv(π, t) (14) This leads to a numerical algorithm which has been implemented: the initialisation phase implies setting up an initial guess of ˆ f(π, t), identifying the target mode ˆutarget(π), and setting up the initial condition ˆu(π, 0). The forward simulation involves solving the structured Navier-Stokes forward equation to obtain ˆu(π, t). Then I compute the cost J[ˆu, ˆ f] and I perform a backward simulation solving the adjoint equation backward in time to obtain ˆv(π, t). I update the control (make an adjustment) ˆ fnew(π, t) = 1 λˆv(π, t) (15) and repeat the steps above until convergence. This results in an adjustment of the structured perturbations at each iteration to match the desired resonance condition. The equations and adjoint optimisation formulation depends only on the symmetry structure Γ(π, π1, π2) which is determined from group theoretic integrals representing universal symmetry properties, and on the fluid parameters (ρ, µ) which appear as numerical parameters, and hence changing the fluid type only changes their numerical values, and not the form or applicability of the method. This method can in principle be used in the context of optimal control or machine learning models trained on extensive simulation datasets to quickly predict optimal adjustments without explicit re-solving. The use of the Gelfand transformation is important because it explicitly encodes the symmetry information from the targeted molecular states directly into fluid velocity fields, enabling selective resonance. The structured coupling constants provide precise nonlinear interaction terms (Γ) in a structured form, which becomes critical for controlling the cascade evolution across scales. Also, the Gelfand transformed structured modes simplify the numerical analysis and iterative adjustments, clearly showing how modes at each cascade step evolve and couple. The Gelfand transform, as used here, enables therefore selective and iterative dynamical mode adjustment. In the standard chaotic thermalisation scenario, energy cascades down from a large scale Lto small scales (Kolmogorov length scale η). At scales near η, fluid kinetic energy is rapidly converted to random molecular motions, heating the fluid. This process is dominated by viscosity, resulting in random heat production Ekin viscous dissipation −−−−−−−−−−−−−→ Qheat (16) This process is generally irreversible, chaotic, and structureless. However, the scenario I am describing here differs fundamentally from this because the fluid modes are engineered to have specific symmetry and frequency matched precisely to discrete molecular vibrational states. Physically, structured nonlinear cascades explicitly rely on symmetry-driven coherent interactions that guide energy efficiently toward targeted molecular resonances. Unlike traditional turbulence, where energy rapidly disperses into a broad, diffuse spectrum due to random interactions, structured symmetry-adapted nonlinear cascades explicitly maintain coherent and selective energy transfer. This coherence prevents rapid thermalisation, enabling resonant accumulation of energy in targeted molecular modes and facilitating precise quantum state control. 5 Consider a toy model approximation of a molecular quantum oscillator Hmol =p2 2m+1 2kQ2(17) where Qis the vibrational amplitude (coordinate), mis the effective mass, kthe force constant, and the vibrational frequency ω0=pk/m. Quantum mechanically, this mode has discrete energy levels En=~ω0(n+1 2), n = 0,1,2, ... (18) For structured energy transfer to be efficient, the fluid mode frequency ωfluid must match closely the molecular vibration frequency ωfluid =ω0. Under such a resonant condition, the coupling between the fluid and the molecular modes is enhanced dramatically. The fluid mode equation, with the structured mode uis ρdu dt +ρΓu2+µ|ω|2u=ρf(t)−P(t) (19) The quantum vibrational mode equation is md2Q dt2+mγ dQ dt +mω2 0Q=αu(t) (20) where αis a coupling strength, depending on symmetry match, Γ is the structure constant, controlling structured fluid interaction, and γis the damping, which is a small quantum dissipation, or a vibrational relaxation rate. In resonance conditions (ωfluid ∼ω0), the energy coherently transfers from the structured fluid mode uinto the vibrational mode Q. Fast dissipation occurs if the energy moves rapidly into many random degrees of freedom. The dissipation timescale would be short. However, in the resonant coupling scenario, coherent resonance isolates a single molecular quantum mode, lengthening the timescale significantly. Mathematically, this is because the resonance condition leads to persistent coherent oscillations. Consider the resonance condition. The Fourier transform of the quantum mode equation is Q(ω) = αu(ω) m(ω2 0−ω2+iγω)(21) At resonance ω∼ω0the denominator is minimal |ω2 0−ω2|  1 making Q(ω) very large (high amplitude). This implies strong, persistent energy buildup in the molecular mode, rather than immediate randomisation. Eventually, even the resonantly excited mode will slowly transfer its energy to other molecular modes or phonons, resulting in thermalisation, but much more slowly and controllably, allowing coherent manipulation at molecular levels. Therefore, structured fluid modes persist and don’t immediately vanish into random thermal motions because of resonant, symmetry-based coupling to specific discrete molecular vibrational modes. The structured fluid mode transfers energy more efficiently and rapidly to molecular vibrational modes than random thermalisation because the energy transfer rate under structured resonance conditions can be represented by a resonant coupling term: dEvib dt ∼ |αu(ω)|2δ(ω−ωvib) (22) where αis the coupling strength, u(ω) is the structured fluid mode amplitude at resonance frequency ωvib and the delta function emphasises the highly selective, frequency-specific nature of the coupling. In contrast, thermalisation involves random and broad-spectrum interactions, significantly slowing the rate of energy accumulation in any single mode. NUMERICAL SIMULATION AND RESULTS To illustrate this approach, we conducted detailed numerical simulations focusing on the asymmetric vibrational mode of CO2, a molecule of significant scientific and technological relevance. We considered a three-step nonlinear cascade, where the fluid frequency at each step is multiplied by a factor of approximately five, progressively bridging from the MHz to the THz range. The structured fluid perturbations were numerically solved using the generalised structured Navier-Stokes equations to identify resonance conditions clearly. 6 Our numerical results demonstrate a distinct and coherent buildup of vibrational energy in the CO2asymmetric stretch mode, evident from sharply defined resonance peaks in the frequency spectra obtained via Fourier analysis. We systematically explored the dependence of vibrational energy excitation efficiency on the nonlinear coupling constant , identifying optimal values that maximise energy transfer. Furthermore, spatial mode analysis revealed that structured fluid perturbations maintained their coherence through multiple cascade steps, confirming the viability of controlled, symmetry-driven energy cascades. Figure 1 presents a snapshot from the second cascade step, clearly demonstrating FIG. 1: Second cascade illustrating the vibrational mode energy distribution. The figure clearly demonstrates selective excitation of the asymmetric stretch vibrational mode of CO2, validating the effectiveness of structured nonlinear cascade methods in targeting precise molecular resonances. the selective enhancement of vibrational energy within the targeted molecular mode. The vibrational mode energy increases distinctly, indicating effective energy transfer from the structured fluid perturbation, as identified by sharply defined peaks in the frequency spectrum. Figure 2 illustrates the spatial distribution of the structured fluid modes during an intermediate cascade. The fluid perturbation exhibits coherent symmetry-adapted spatial patterns, designed to match the symmetry of the targeted molecular vibrational mode. The structured spatial coherence significantly accelerates energy transfer compared to random thermalisation processes. The structured fluid mode transfers energy more efficiently and rapidly to the molecular vibrational modes than random thermalisation because structured interactions selectively target molecular resonance frequencies. This coherent interaction effectively bypasses intermediate random energy dispersal, maintaining a high level of energy coherence and significantly reducing the timescale required to achieve molecular excitation compared to conventional thermalisation processes. The numerical simulations of molecular vibrational dynamics were carried out using custom-developed numerical software specifically tailored for this investigation. We explicitly modelled the CO2molecule, a linear triatomic molecule (O=C=O), considering carbon and oxygen atoms with bond lengths and angles consistent with spectroscopic standards. The asymmetric stretch vibrational mode, centred around 2349 cm−1(approximately 70 THz), was described using harmonic oscillator potentials. Parameters for bond lengths, bond angles, and force constants were taken from the recent 626M24 dataset of rovibronic transitions compiled by Tennyson et al. (2025) [10]. The simulation domain was a three-dimensional cubic region discretised using a finite-difference grid. The discretisation was explicitly designed to ensure convergence and stability, employing symmetry-adapted boundary conditions consistent with the structured cascade approach. Fluid dynamics were modelled with standard parameters for air-like gases at room temperature conditions: density around 1 kg/m3and dynamic viscosity approximately 1.8×10−5Pa·s. Numerical integration utilised an explicit fourth-order Runge-Kutta (RK4) scheme for temporal discretisation, chosen for its accuracy and numerical stability. Spatial derivatives were computed using second-order finite-difference approximations. 7 FIG. 2: Spatial distribution explicitly depicting structured fluid velocity modes. This figure highlights how symmetry-adapted fluid perturbations maintain coherent and targeted energy distributions across the computational domain, enabling efficient energy transfer toward molecular vibrational resonances. Although second-order finite-difference methods introduce discretisation errors, we explicitly conducted a Grid Independence Study (see Figure (a)) to ensure numerical convergence. This study confirms that at sufficiently high grid resolutions, discretisation errors become negligible, validating the appropriateness of our chosen numerical method. Boundary conditions were explicitly imposed to maintain symmetry-adapted fluid modes throughout simulations. The structured Navier–Stokes equations are explicitly solved using a hybrid numerical approach employing finitedifference discretisation in spatial domains and Runge - Kutta time-integration schemes. For molecular vibrational equations, we explicitly applied standard quantum harmonic oscillator numerical solutions, explicitly computed at each cascade stage to verify resonance matching. The initial conditions for fluid velocity were explicitly selected through spectral analysis to match resonant vibrational frequencies precisely. At each step of the structured cascade, the energy distribution was explicitly monitored and adjusted iteratively to optimise resonance conditions. Output data included precise vibrational energy distributions, quantification of energy transfer efficiency, and time-resolved resonance matching assessments, enabling detailed analysis of molecular-scale excitation phenomena. The method is intuitively represented in fig. 3. The results demonstrate that symmetry structured macroscopic perturbations coherently excite specific molecular quantum states, establishing a rigorous connection to quantum control and selective chemical reaction dynamics. I show in figure 4 how the structured nonlinear cascade method progressively moves towards the exact molecular resonance frequency of the CO2asymmetric stretch (∼70 THz). Numerical simulations were conducted using an explicit fourth-order Runge-Kutta (RK4) temporal discretisation 8 FIG. 3: Flowchart illustrating the computational implementation workflow for structured nonlinear cascade simulations, clearly showing numerical steps from initial symmetry-structured mode definition, numerical integration of Gelfand-transformed Navier–Stokes equations, iterative resonance condition adjustments, and molecular-level energy analysis. method coupled with a finite-difference spatial discretisation scheme, selected for accuracy, stability, and computational efficiency. The computational domain employed a structured grid explicitly designed to capture the detailed spatial variations of the fluid and vibrational fields accurately. Numerical errors primarily arose from discretisation effects, including spatial truncation errors inherent to finitedifference methods and temporal integration errors associated with the RK4 scheme. Grid independence studies were conducted explicitly to ensure minimal sensitivity to grid resolution, resulting in an optimal grid spacing selected through convergence tests. Figure 5 is a representation of this discussion. Stability analysis explicitly involved examining the Courant-Friedrichs-Lewy (CFL) condition to ensure numerical stability. The CFL number was maintained below unity through adaptive timestep adjustments, explicitly balancing computational efficiency and numerical stability. Sensitivity analysis further included explicit variations in key parameters such as fluid viscosity, density, and coupling strength. These analyses demonstrated robust performance of the structured cascade methodology, with consistent convergence toward resonant conditions across varied parameter ranges. Uncertainty quantification explicitly evaluated the robustness of resonance matching by introducing perturbations in initial conditions and evaluating their impact on final vibrational energy distributions. Results indicated that the structured cascade approach exhibited significant resilience against perturbations, highlighting the stability and reliability of the resonance transfer mechanisms. 9 FIG. 4: Progression towards resonance in structured nonlinear cascades. The series clearly shows iterative adjustments in the targeted frequency, demonstrating convergence toward the precise molecular vibrational resonance (CO2asymmetric stretch at ∼70 THz). CONCLUSIONS AND PERSPECTIVES The results provide compelling evidence for the feasibility of coherent molecular vibrational excitation via structured fluid dynamics. The novelty of this study resides in successfully demonstrating structured, symmetry-guided nonlinear cascades capable of selectively exciting molecular quantum states from macroscopic fluid scales. This represents a substantial advancement over traditional turbulence-based energy cascades and opens avenues for controlled quantum manipulations in fluid-based experimental platforms. The generalisation of Navier-Stokes equations using symmetry-based Gelfand transforms opens unprecedented opportunities for precise control of molecular quantum states through fluidic environments. This approach significantly extends beyond conventional turbulence theory, offering practical pathways for experimental realisation. Future work will focus on experimental verification of these theoretical predictions, optimisation of fluidic devices for quantum state control, and exploration of broader applications in chemical synthesis, quantum information processing, and advanced energy technologies. [1] S. 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