Turbulent Solutions in Fluid Dynamics

• Turbulent solutions are mathematically exact characterizations of nonlinear hydrodynamic and kinetic PDEs that reveal the dynamic skeleton and statistical structure of turbulence.
• They combine advanced numerical methods and analytical techniques to capture invariant structures, coherent flow patterns, and non-uniqueness in turbulent regimes.
• Their study provides practical insights for turbulence control, model reduction, and bridging fluid dynamics with broader mathematical and physical theories.

Turbulent solutions are mathematically exact or analytically characterized solutions—often unstable or exhibiting non-uniqueness—of nonlinear hydrodynamic, kinetic, and related PDEs that reproduce or underpin the emergent features and statistical properties observed in fully developed turbulence. These solutions arise in diverse settings: as time-periodic or relative-periodic orbits of the Navier–Stokes equations, as mean-field or ensemble solutions of kinetic or generalized hydrodynamic equations, as traveling waves of effective models for turbulent transport, or as statistical ensembles in path-dependent limits of vanishing dissipation. Their study reveals both the geometrical and dynamical skeleton of turbulence, its statistical and intermittency structure, and the limitations or extensions required beyond classical phenomenology.

1. Dynamical Systems Paradigm: Invariant Solutions and Exact Coherent Structures

The modern dynamical systems approach to turbulence interprets observed complexity as organized around a web of invariant solutions—equilibria, traveling waves, periodic and relative-periodic orbits—of the Navier–Stokes equations. These are computed numerically in high-dimensional discretizations using Newton–Krylov solvers, often initialized from filtered near-recurrences in DNS or experimental data (Kawahara et al., 2011, Suri et al., 2016, Avila et al., 2012).

• In pipe and channel flows, streamwise-localized relative periodic orbits are found to bifurcate from laminar flow and act as organizing centers for the onset and maintenance of turbulence, with their stability spectra and manifold structure delineating laminar–turbulent boundaries, intermittent puffs, and chaotic saddles (Avila et al., 2012).
• In weakly turbulent quasi-2D flows, laboratory and numerical data confirm that turbulent trajectories visit neighborhoods of unstable periodic orbits (UPOs), with statistics matching those of turbulence, and that the departure from such UPOs follows their unstable manifolds for multiple correlation times, enabling local flow forecasting (Suri et al., 2016, Suri et al., 2020).
• For lower Reynolds numbers, these invariant solutions reproduce coherent near-wall structures, buffer-layer cycles, and even secondary flows in complex geometries. Weighted averages over their networks recover time-averaged turbulence statistics such as the Prandtl log law and the dissipation-range scaling (Kawahara et al., 2011). This finite-dimensional skeleton persists up to moderate Re, though the catalog must be extended to capture high-Reynolds-number turbulence.

2. Analytical and Kinetic Theory: Superposition, Similarity, and the Generalized Hydrodynamics Approach

Analytical solutions for turbulent mean flows, structured fluctuations, and boundary layers have been derived in frameworks extending beyond classical Navier–Stokes hydrodynamics.

• The Generalized Boltzmann Equation (GBE) and its hydrodynamic reduction (AHE/GHE) admit both parabolic (laminar) and superexponential (turbulent) stationary solutions. The turbulent mean profile is accurately represented as a linear superposition of these two modes, parametrized by a weighting factor γ and a boundary-layer thickness parameter δ, the latter coinciding with the Kolmogorov microscale and universal across a wide Reynolds-number range (Fedoseyev, 2023, Fedoseyev, 17 Dec 2025). A viscous-dissipation minimization principle rigorously determines the superposition coefficients, yielding profiles in quantitative agreement (within 5%) with canonical turbulent channel and pipe flow data up to $Re\sim 10^7$, outperforming both the classical log law and the laminar prediction (Fedoseyev, 11 Apr 2024).
• In this analytic formalism, turbulence emerges as oscillations or weighted mixtures between the laminar and turbulent states. The turbulence source is identified as the mean cross-flow (transverse velocity), and suppression of such disturbances via wall control or inlet noise management can eliminate turbulence entirely, consistent with experimental observations (Fedoseyev, 17 Dec 2025).

3. Non-Uniqueness, Pathwise Limits, and Mathematical Constructions of Turbulent Solutions

Rigorous analysis of the incompressible Navier–Stokes and Euler equations reveals infinitely many smooth (non-weak) turbulent-type solutions in both periodic and bounded domains, challenging deterministic uniqueness even at the level of classical analysis.

• Regular steady solutions of the static Euler equations exist in infinite families parameterized by spectral data (eigenmodes) under given boundary conditions (Han, 17 Feb 2025, Han, 2023).
• Associated Navier–Stokes solutions decay exponentially in time but, as viscosity $\nu\rightarrow 0$ and $t\rightarrow\infty$ along specific "random" paths (with $\nu t = \text{const}$), their limits persist as nontrivial steady Euler flows—random in the sense that different paths select different limiting states.
• The double limit $\lim_{\nu\to 0} \lim_{t\to\infty}$ or $\lim_{t\to\infty} \lim_{\nu\to 0}$ returns the trivial state, exhibiting non-commutation, and thus the “randomness of turbulence” can arise as a purely regular, pathwise, deterministic phenomenon (Han, 17 Feb 2025, Han, 2023).
• The implications are profound: even for smooth initial data, turbulence is mathematically realized as a failure of uniqueness in the inviscid limit and as a fundamentally path-dependent process.

4. Model Reductions and Traveling-Wave Turbulent Solutions in Effective Media

Turbulent phenomena in complex transport, such as filtration and scalar mixing, can manifest as traveling or stationary solutions with novel features in degenerate, nonlinear, or statistically averaged equations.

• For nonlinear double-degenerate parabolic equations modeling turbulent filtration, existence theorems guarantee monotonic, compactly supported traveling-wave solutions for a wide class of exponents and absorption laws (Prinkey, 2019). These “turbulent solutions” exhibit explicit power-law asymptotics and compact support, corresponding to finite-speed propagation and sharp interfaces absent in classical diffusion.
• In decaying turbulence, the advection–diffusion of a passive scalar exhibits a unique, quantized solution: a shell structure formed by nested, piecewise-parabolic profiles as explicit analytic solutions of the loop calculus representation of the Navier–Stokes equations in the ideal limit. These quantized shells, physically corresponding to coherent scalar sheets, are robust to finite Schmidt numbers and external forcing (Migdal, 14 Apr 2025).

5. Turbulent Solutions in Advanced Theoretical Frameworks

Contemporary theoretical advances have recast turbulence as a geometric or stochastic ensemble phenomenon, linking hydrodynamics to electrodynamics, string theory, and stochastic PDEs.

• The loop-space calculus framework represents turbulence via linear diffusion equations in function space, admitting analytic solutions (Euler ensemble) and unifying the spatial and temporal scaling laws of turbulence. Critical exponents controlling intermittency and decay emerge as spectra determined by the zeros of the Riemann zeta function. The theory predicts new, physically observable phenomena including log-periodic oscillations and phase transitions in MHD turbulence, and connects hydrodynamic turbulence to solvable regimes in Yang–Mills gradient flow and string theory (Migdal, 4 Nov 2025).
• The FLRW-toroidal, KAM spin-network approach constructs explicit turbulent solutions as geodesics on intersecting de Sitter tori in high-dimensional configuration space, with unique energy-spectrum exponents predicted and singularity/intermittency phenomena suppressed by embedded geometric symmetries (Scott, 12 Mar 2024).
• In completely integrable dispersive PDEs, such as the one-dimensional cubic Schrödinger and binormal flow, turbulent solutions constructed at or beyond critical function space regularity exhibit all hallmarks of turbulence: energy cascade in Fourier shells, Talbot effect revivals, and multifractal filament trajectories (Banica et al., 18 Dec 2024).

6. Physical Interpretation, Control, and Relevance to Modelling

The existence and structure of turbulent solutions inform turbulence modeling, prediction, and control across engineering and fundamental physics.

• Their presence as organizing centers, “attractors,” or “repellers” in phase space underpins the statistical behavior and recurrence properties of turbulent flows. Weighted averages over these solutions can reproduce ensemble statistics without reliance on ad hoc closure models (Kawahara et al., 2011, Suri et al., 2020).
• Turbulence control strategies—such as enforcing specific wall-normal mean flows or manipulating inlet boundary conditions—can suppress or restore turbulence, corresponding precisely to the structure of analytical solutions (e.g., setting the cross-flow component to zero returns the flow to laminar profiles) (Fedoseyev, 17 Dec 2025).
• In advanced reduced-order or parametric modeling (e.g., using proper generalized decomposition with RANS equations), the structure of the solution manifold and its low-dimensional representation are informed by the nature of turbulent solutions and their dominant manifolds, facilitating rapid parametric control and uncertainty quantification (Tsiolakis et al., 2020).

7. Extensions, Open Problems, and Future Directions

The study and application of turbulent solutions continue to develop along several fronts:

• Extension to higher Reynolds numbers and the detailed structure of invariant solutions' networks remains computationally challenging and theoretically significant for universal turbulence modeling (Kawahara et al., 2011).
• The role of random solutions, non-uniqueness, and path-dependent limits is an area of active mathematical exploration, particularly regarding their implications for predictability, selection principles, and turbulence closure (Han, 17 Feb 2025, Han, 2023).
• Generalization to compressible flows, non-Newtonian fluids (notably, the emergence of elasto-inertial turbulence and modified spectral slopes in polymeric flows), and to more complex multiphysics systems is an ongoing effort informed by rigorous and semi-analytical constructions (Mitishita et al., 2022).
• Connections to statistical mechanics, quantum turbulence, geometric measure theory, and integrable systems continue to deepen, revealing turbulence as a multifaceted phenomenon at the intersection of analysis, geometry, and physics (Banica et al., 18 Dec 2024, Scott, 12 Mar 2024, Migdal, 4 Nov 2025).

Turbulent solutions therefore provide both a foundational framework for understanding the dynamics of turbulence and a concrete bridge between mathematical analysis, physical phenomenology, and engineering modeling across a broad class of PDE systems.