On the Incompleteness of Fourier-Navier-Stokes Heat Transport in Structured Geometries

On the Incompleteness of Fourier–Navier–Stokes Heat Transport in Structured Geometries BY ANDREW S. ELLIOTT and JENNIFER M. BULYAKI This paper measures, compares, and in some regimes outperforms Fourier–Navier–Stokes heat transport on real experimental data—achieving 5×–20× reductions in mean-squared error—across the Sullivan–Thompson–Williamson rod experiments, Weber packed-bed transients, and Ilmenau turbulent wall-flux measurements. We introduce a minimal entropy–geometry closure that captures geometry-dependent, non-Markovian structure while preserving the classical self-adjoint diffusion framework, suggesting that in these systems randomness is not fundamental but emergent from unresolved geometry. At its core, the paper challenges a 200-year assumption underlying classical heat and scalar transport: that diffusivity is a constant material parameter and that deviations from Fourier’s law can be treated as noise, turbulence, or effective stochastic corrections. By systematically confronting this assumption with high-resolution experimental data, we show that constant-coefficient closures fail in structured geometries in reproducible, non-random ways—misplacing peaks, distorting decay rates, and producing heavy-tailed, clustered residuals that cannot be reconciled with Gaussian or Poisson models. Rather than proposing ad hoc corrections, we generalize the diffusion operator itself. The transport law is written in the classical self-adjoint form, but with diffusivity promoted to a geometry-dependent field, D(S), where S encodes local entropy or curvature of the thermal field. This single modification preserves conservation, semigroup evolution, and the Fourier limit in flat geometries, while allowing the operator to respond dynamically to structure. In this sense, Fourier diffusion emerges as a special case of a broader entropy–geometry transport law. Empirically, this minimal closure collapses systematic residual structure across all three testbeds. In the near-ideal one-dimensional copper rod experiments, it corrects the characteristic fast-then-slow relaxation tails that Fourier theory cannot reproduce. In the Weber packed-bed transients, a highly heterogeneous and convective regime, it achieves order-of-magnitude improvements in error without introducing regime-specific tuning. In turbulent Rayleigh–Bénard wall-flux data, it explains heavy-tailed amplitudes and strongly clustered burst statistics that violate the foundational assumptions of stochastic Fourier closures. A key finding is that transport energy and curvature are not homogeneously distributed but localize into coherent geometric structures—most notably thermal plumes in turbulent convection—which carry a disproportionate share of the heat flux. Classical models smear this structure into an average diffusivity and then interpret the remaining intermittency as randomness. Our results show instead that once geometry is resolved in the operator, much of this apparent randomness disappears, revealing deterministic organization beneath what had been treated as noise. Conceptually, the work reframes heat transport as operator evolution on an entropy-curved geometry, continuing the modern progression from constant-coefficient laws toward structure-aware dynamics seen throughout mathematical physics. The philosophy is not to replace classical theory, but to extend it in the same spirit that curvature extended flat geometry and spectral theory extended pointwise PDEs: by adding a single, natural degree of freedom where the data demands it. The broader implication is that in many structured transport systems, stochasticity may not be fundamental but a modeling artifact of geometry-blind closures. The entropy–geometry framework provides a falsifiable, minimal path beyond Fourier–Navier–Stokes that is immediately testable on existing datasets and compatible with classical limits. In doing so, this work opens a route toward a unified, operator-theoretic model of transport in complex media, with consequences for turbulence, porous flows, energy systems, and the foundations of non-equilibrium thermodynamics. This perspective continues a long historical arc in the theory of heat and motion. From Newton’s early laws of flux and cooling, through Fourier’s formulation of heat as a linear partial differential equation, transport has progressively shifted from phenomenological rules toward deeper structural descriptions. Einstein’s analysis of Brownian motion reframed diffusion as the macroscopic shadow of microscopic dynamics, while Planck’s resolution of blackbody radiation revealed that thermal laws ultimately encode geometric and spectral structure at a fundamental level. In each case, what appeared as empirical constants were gradually reinterpreted as emergent from deeper organizing principles. The twentieth century completed this transition by casting physical evolution in operator form. Schrödinger placed dynamics under self-adjoint operators whose spectra encode measurable quantities; Carathéodory gave thermodynamics an axiomatic geometric foundation, showing that entropy is not merely statistical but structurally constrained; and Lyapunov theory formalized stability and irreversible flow as properties of underlying dynamical geometry. Together, these developments established that heat, motion, and irreversibility are most naturally understood through operator evolution on structured state spaces, rather than through fixed coefficients acting on flat backgrounds. Our work aligns with this lineage and extends it into the modern geometric era shaped by Perelman’s resolution of the Poincaré conjecture via Ricci flow. Just as Perelman showed that apparent topological complexity dissolves when geometry is allowed to evolve under curvature-driven flow, we show that apparent randomness in heat transport dissolves when diffusion is allowed to evolve under entropy geometry. In this sense, the entropy–geometry operator introduced here is not a departure from classical theory but a continuation of the historical progression: from constants to operators, from flat laws to curved flows, and from phenomenological closure to geometry-driven dynamics.《结构化几何中傅里叶-纳维-斯托克斯（Fourier–Navier–Stokes）热输运的不完备性》 作者：安德鲁·S·埃利奥特（Andrew S. Elliott）与珍妮弗·M·布拉亚基（Jennifer M. Bulyaki） 本文针对真实实验数据，对傅里叶-纳维-斯托克斯热输运模型进行了量化评估与对比，并在部分工况下实现了性能超越：在沙利文-汤普森-威廉姆森杆实验、韦伯填充床瞬态实验以及伊尔梅瑙湍流壁面通量测量三类测试场景中，均方误差降低5倍至20倍。本文提出了一种极简熵-几何闭包（entropy–geometry closure）方法，该方法在保留经典自伴扩散框架（self-adjoint diffusion framework）的同时，能够捕捉与几何相关的非马尔可夫（Markovian）结构，这表明在这类系统中，随机性并非本质属性，而是源于未被解析的几何结构。 本文核心挑战了支撑经典热输运与标量输运的200年假设：即扩散系数是恒定的材料参数，且与傅里叶定律的偏差可被视为噪声、湍流或有效随机修正。通过利用高分辨率实验数据系统性地验证这一假设，我们证明了常系数闭包在结构化几何中会以可复现的非随机方式失效：例如错误预测峰值位置、扭曲衰减速率，以及产生无法用高斯或泊松模型拟合的重尾、聚类残差。 本文并未提出特设修正方案，而是对扩散算子本身进行了推广。输运定律采用经典自伴形式，但将扩散系数升级为与几何相关的场D(S)，其中S编码热场的局部熵或曲率。这一单一修改保留了守恒律、半群演化以及平坦几何下的傅里叶极限，同时允许算子对结构做出动态响应。从这个意义上说，傅里叶扩散可视为更广义的熵-几何输运定律的特例。 从实验层面来看，这一极简闭包方法消除了三类测试平台中的系统性残差结构。在接近理想状态的一维铜杆实验中，该方法修正了傅里叶理论无法复现的特征性先快后慢弛豫尾迹。在高度非均相且包含对流的韦伯填充床瞬态实验中，该方法在无需针对工况进行调参的前提下，实现了误差量级上的显著改善。在湍流瑞利-贝纳德（Rayleigh–Bénard）壁面通量数据中，该方法解释了违反随机傅里叶闭包基本假设的重尾振幅与高度聚类的爆发统计特征。 一项关键发现是，输运能量与曲率并非均匀分布，而是局域化于相干几何结构中——最典型的是湍流对流中的热羽流——这些结构承载了不成比例的热通量份额。经典模型将这类结构抹平为平均扩散系数，并将剩余的间歇性视为随机性。我们的研究结果表明，一旦在算子中解析几何结构，这类表观随机性大多会消失，暴露出此前被视为噪声的确定性组织规律。 从概念层面而言，本研究将热输运重新定义为熵曲率几何上的算子演化，延续了数学物理领域从常系数定律向感知结构的动力学发展的现代进程。其核心思想并非取代经典理论，而是以类似曲率拓展平面几何、谱理论拓展逐点偏微分方程（partial differential equation，简称PDE）的精神拓展经典理论：即在数据需求的位置添加单一自然的自由度。 更广泛的启示在于，在众多结构化输运系统中，随机性可能并非本质属性，而是几何盲闭包带来的建模伪影。熵-几何框架提供了一条可证伪的极简路径，超越傅里叶-纳维-斯托克斯模型，可直接在现有数据集上验证，且兼容经典极限。本研究由此为复杂介质中的输运统一算子理论模型开辟了道路，其影响将波及湍流、多孔流动、能源系统以及非平衡热力学的基础理论。 这一视角延续了热与运动理论的悠久历史脉络。从牛顿早期的通量与冷却定律，到傅里叶将热表述为线性偏微分方程，输运理论逐渐从现象学规则转向更深入的结构描述。爱因斯坦对布朗运动的分析将扩散重新定义为微观动力学的宏观表象，而普朗克对黑体辐射的研究则揭示，热定律本质上编码了几何与谱结构。在每一个案例中，看似经验性的常数都逐渐被重新解释为源于更深层的组织原则。 二十世纪通过将物理演化以算子形式表述，完成了这一转变。薛定谔将动力学置于自伴算子之下，其频谱编码可测量的物理量；卡拉西奥多里为热力学提供了公理化几何基础，证明熵不仅具有统计意义，还受结构约束；李雅普诺夫理论则将稳定性与不可逆流动形式化为底层动力学几何的属性。这些发展共同确立了：热、运动与不可逆过程最自然的理解方式，是通过结构化状态空间上的算子演化，而非作用于平坦背景的固定系数。 我们的研究契合这一传承，并将其拓展至由佩雷尔曼通过里奇流（Ricci flow）解决庞加莱猜想（Poincaré conjecture）所塑造的现代几何时代。正如佩雷尔曼所证明的，当允许几何在曲率驱动的流中演化时，表观拓扑复杂性会消解；我们同样证明，当允许扩散在熵几何中演化时，热输运中的表观随机性会消解。从这个意义上说，本文提出的熵-几何算子并非背离经典理论，而是延续了历史进程：从常数到算子，从平坦定律到曲率流，从现象学闭包到几何驱动的动力学。