Central Flow: Unified Phenomena in Science

• Central flow is a multifaceted term describing phenomena in physics, mathematics, and computation characterized by symmetry, hierarchical scaling, and central charge dynamics.
• It plays a critical role in heavy-ion collisions, where ultra-central events reveal collective anisotropies and challenge existing viscous-hydrodynamic models.
• In geometry, fluid dynamics, and astrophysics, central flow underpins integrable curve evolutions, advanced numerical methods, and hierarchical mass accretion processes.

Central flow refers to a diverse set of physical, mathematical, and computational phenomena characterized by flow fields, geometric motion, or renormalization group (RG) behavior with a centralizing or symmetry-constrained structure. This term appears in multiple contexts, notably in the study of heavy-ion collisions (collective flow in central collisions), central-affine curve flows in geometry and integrable systems, multiphysics fluid simulations (central-moment methods), and the flow of central charges in quantum field theory (QFT) and holography.

1. Central Flow in Heavy-Ion Collisions

In relativistic heavy-ion collisions, central flow pertains to collective expansion phenomena—particularly elliptic ($v_2$) and triangular ($v_3$) flow—in events with near-zero impact parameter ("ultra-central"). In this regime, the geometric eccentricity of the initial nuclear overlap vanishes, and the observed collective anisotropies originate almost exclusively from density fluctuations and, in some cases, nuclear deformation (Shen et al., 2015, Giacalone, 2018). The measured flow coefficients $v_n\{2\}$ derived from two-particle cumulants are defined as

$$v_n\{2\} = \sqrt{\left\langle\left\langle e^{i n \Delta\phi}\right\rangle\right\rangle},$$

with the mean taken over all pairs (inner average) and all events (outer average) (Kuroki et al., 2023).

The Ultra-Central Flow Puzzle

Current viscous-hydrodynamic models, evolved from Monte-Carlo Glauber initial conditions and augmented with relativistic fluctuating hydrodynamics, consistently overestimate the ratio $v_2\{2\}/v_3\{2\}$, predicting values $1.3{-}1.4$ versus the experimental result $1.02 \pm 0.05$ (Kuroki et al., 2023). This discrepancy, the "ultra-central flow puzzle," highlights limitations in modeling initial fluctuation spectra and transport properties. Hydrodynamic fluctuations raise both $v_2\{2\}$ and $v_3\{2\}$ but only diminish the $v_2/v_3$ ratio by $\sim 19\%$, insufficient to reconcile with experiment unless further temperature-dependence in $\eta/s$ is included (Kuroki et al., 2023).

2. Central-Affine Curve Flows: Integrability and Geometry

Central-affine curve flows are flows (evolution equations) of curves in $\mathbb{R}^n\setminus 0$ invariant under special linear transformations. For a nondegenerate curve $y(x)$, one imposes the central-affine normalization $\det(y, y_x, \ldots, y_x^{(n-1)})=1$. The central-affine curvature vector $u(x)$ then controls the local geometry, and the "central flow" hierarchy comprises commuting flows for which $u(x, t)$ evolves according to the Gelfand-Dickey (A${}_{n-1}$–KdV) hierarchy: $$L_{t_j} = \left[ (L^{j/n})_+, L \right],\quad L = \partial_x^n + u_1(x)\partial_x^{n-2}+\ldots+u_{n-1}(x) \quad [1411.2725, 1405.4046].$$ For $n=2$ (the plane), Pinkall's third-order central-affine flow induces a KdV evolution for the curvature,

$$y_{xx} = q(x)y,\quad y_t = -q y - \tfrac{1}{2}q_x y_x,\quad q_t = \tfrac{1}{4}(q_{xxx} - 6q q_x) [1405.4046].$$

These flows possess a bi-Hamiltonian structure, explicit Bäcklund transformations, permutability theorems, and admit explicit soliton and finite-gap solutions (Terng et al., 2014, Terng et al., 2014).

3. Central Flow in Multiphysics and Numerical Simulation

In computational fluid dynamics, central-upwind and central-moment methods provide robust, accurate frameworks for simulating compressible flows with shocks and multi-physics effects. The central-upwind schemes are formulated to combine central (low-dissipation) stencils in smooth regions with characteristic-wise reconstruction near discontinuities. For example, the adaptive central-upwind algorithm applies a hybrid of linear central (sixth-order) and upwind (fifth-order) reconstructions with smoothness detectors, significantly reducing computational cost (20–30%) and eliminating spurious vortices in under-resolved shear-layer simulations (Chamarthi, 2024). The key mechanism is the preservation of shear balance by centralizing transverse momentum, which suppresses artificial slip and vortex formation.

In the lattice Boltzmann method (LBM), central-moment relaxation (CM-LBM) employs collision operators in the space of central moments, yielding enhanced Galilean invariance, improved stability at low viscosity, and accuracy for weakly compressible flows. Coarse lattices (D3Q19) with CM-LBM sustain accuracy and stability (1–3% deviation compared to D3Q27) up to moderate Reynolds and Mach numbers, with significant computational savings (Rosis et al., 2020).

4. Central Charge Flow and RG Flows in Field Theory and Holography

The central flow of charges refers to the monotonic RG flow of the central charge (e.g., the $c$-function in 2D CFT or $a$ in 4D SCFT) between UV and IR fixed points. In the context of sine dilaton gravity (sDG), the holographic $c$-function interpolates from $c_{UV}=26$ (the sum of two Liouville CFTs) to $c_{IR}=0$ (pure JT gravity) along the domain-wall coordinate $u$ (Mahapatra et al., 25 Jan 2026). The flow is governed by first-order superpotential (gradient) equations and satisfies monotonicity, $d\dot{c}/du < 0$. Analogous behavior appears in the sine-Gordon model, where the functional RG flow of the scale-dependent $c_k$ function demonstrates $\Delta c = 1$ for the repulsive low-frequency regime upon integrating out to the IR, with the decrease fully accounted for only upon inclusion of wavefunction renormalization beyond the local potential approximation (Bacsó et al., 2015).

In 4D $\mathcal{N}=2$ SCFT, central charge flow induced by RG-relevant deformations (e.g., along the Argyres–Douglas sequence) always decreases monotonically, $a_{\mathrm{UV}} > a_{\mathrm{IR}}$, consistent with the $a$-theorem, with explicit algebraic formulas provided in the Gaiotto–Xie construction (Xie et al., 2013).

5. Central Flow of Mass and Accretion in Astrophysics

In star-forming regions, central flow denotes the hierarchical, organized accretion of gas from parsec to AU scales into compact central cores. Recent high-resolution molecular line and dust continuum mapping of the G351.77–0.54 IRDC reveals a conveyor-belt pattern: large-scale (pc) velocity gradients feed intermediate-scale filaments ($\sim$0.25 pc), which in turn converge on a hub of compact ($\sim$50–10000 AU) protostellar cores (Beuther et al., 19 Feb 2025).

Quantified mass influx rates along filaments are $$\dot{M}_{\rm fil} \sim 10^{-3} M_\odot\,{\rm yr}^{-1}$$, sufficient to sustain high-mass star formation over a free-fall time, consistent with competitive accretion and turbulent core models. On inner (sub-$10^4$ AU) scales, ordered inflow transitions to disordered kinematics due to feedback (outflows, dynamical interactions) from embedded protostars.

6. Central Flow: Unifying Principles and Open Problems

Across these diverse domains, central flow structures reveal universal features:

• Symmetry and Conservation: Central-affine invariance in geometry, Galilean invariance in fluid moments, and conformal symmetry in field-theoretic $c$-functions.
• Hierarchy and Scaling: Flows exhibit hierarchical structure (e.g., in accretion), and in RG flows, monotonic decrease of central charge quantifies loss of degrees of freedom.
• Integrability and Exact Solutions: Central-affine curve flows are integrable, have Lax pairs, and admit multi-soliton solutions via Bäcklund transformations (Terng et al., 2014, Terng et al., 2014).
• Quantitative Constraints: Central flow observables—whether $v_n\{2\}$, central charges, or mass inflow rates—place stringent, multidimensional constraints on underlying microphysics: initial fluctuation spectra, transport coefficients, or nuclear deformation (Kuroki et al., 2023, Giacalone, 2018, Lacey, 2024, Beuther et al., 19 Feb 2025).

Despite these advances, puzzles persist: the resolution of the ultra-central flow puzzle in heavy-ion collisions remains elusive within current hydrodynamic and fluctuating-hydro models, demanding improved initial-state modeling and refined temperature dependence in transport coefficients (Kuroki et al., 2023, Shen et al., 2015, Lacey, 2024).

7. Reference Table: Illustrative Central Flow Applications

Domain	Central Flow Phenomenon	Reference
Heavy-Ion Collisions	Simultaneous $v_2$, $v_3$ response in ultra-central events	(Kuroki et al., 2023)
Integrable Geometry	Central-affine (SL($n$)) curve flows, n-KdV integrables	(Terng et al., 2014)
Computational Fluid Dynamics	Central-upwind/central-moment schemes; vortex suppression	(Chamarthi, 2024, Rosis et al., 2020)
Quantum Field Theory / Holography	RG flow of central charge in sDG, sine-Gordon, 4D SCFT	(Mahapatra et al., 25 Jan 2026, Bacsó et al., 2015, Xie et al., 2013)
Star Formation (Astrophysics)	Hierarchical central accretion from filaments to cores	(Beuther et al., 19 Feb 2025)

Central flow thus refers to flow phenomena, geometric motions, and RG behaviors that are characterized by central symmetry, central charge flow, or centralization in physical space. The continued study of such flows connects deep aspects of dynamics, symmetry, and integrability across mathematics, physics, and astrophysics.