Averaging the incompressible Navier–Stokes equations over turbulence generates the Reynolds stress tensor, whose six independent components are not determined by the mean fields: the turbulence closure problem, open since Reynolds (1895). Sun (2022, Open Physics) argued that the standard bookkeeping is misleading because the stress tensor is the average of the dyadic ui′uj′u_i' u_j' of the three velocity-fluctuation components, so the primary unknowns are the fluctuation fields u1′u_1', u2′u_2', u3′u_3' and the pressure fluctuation p′p', and proposed that turbulence modeling seek constitutive relations or transport equations directly for these fluctuation fields instead of the six stress components, with the classical goal the source invokes — turbulence predictable by understanding and solving the deterministic Navier–Stokes equations — as the ultimate context. The fluctuation-field formulation has not been developed in the peer-reviewed literature, and the reframed core — a closed model expressed in the fluctuation fields — remains open.

Background

For an incompressible Newtonian fluid, Reynolds (1895) decomposed the velocity and pressure fields into mean and fluctuating parts, ui=ui‾+ui′u_i = \overline{u_i} + u_i' and p=p‾+p′p = \overline{p} + p', where the overbar denotes Reynolds time averaging. Averaging the Navier–Stokes equations then yields the Reynolds-averaged Navier–Stokes (RANS) equations for the mean flow, but the average of the nonlinear convection term produces the Reynolds stress tensor −ρui′uj′‾-\rho \overline{u_i' u_j'} — nine components, six independent by symmetry — which cannot be expressed in terms of the mean fields. Some additional modeled relation is therefore needed before the mean-flow equations can be solved: this is the turbulence closure problem. Classical closures model the stresses themselves. The Boussinesq eddy-viscosity hypothesis represents the stress as a fictitious turbulent diffusivity acting on the mean strain rate; Prandtl's mixing-length models make that diffusivity algebraic; and second-moment (Reynolds-stress transport) closures, exemplified by Launder, Reece and Rodi (1975), derive exact transport equations for the six stress components but must then model third-order moments and the pressure–strain correlation. All contain empirical constants calibrated on limited flow classes, and none is universally valid. Two adjacent strategies sidestep RANS closure: direct numerical simulation (DNS) resolves the fluctuation fields exactly but its cost grows steeply with Reynolds number, ruling out most industrial flows; and probability-density-function (PDF) methods (Pope 1985) model the one-point statistics of the velocity fluctuations through stochastic Lagrangian equations — a mature statistical version of modeling the fluctuations, with its own closure approximations. Data-driven closures trained on DNS and experimental data are currently active (Duraisamy, Iaccarino and Xiao 2019), though learned algebraic stress closures can be ill-conditioned (Wu et al. 2019). Sun (2022, Open Physics), adopting Reynolds' deterministic view of turbulence — under which the RANS equations are deterministic, not statistical — revisited the standard counting of unknowns: since the Reynolds stress tensor is the average of the dyadic ui′uj′u_i' u_j' built from the three fluctuating velocity components rather than a general symmetric tensor, he argued that the literature's six independent stress unknowns reduce to three — functions of u1′u_1', u2′u_2', u3′u_3', together with the pressure fluctuation p′p' — and proposed that modeling concentrate on constitutive relations or transport equations for the fluctuation fields, which have the practical advantage of being experimentally observable, unlike the stress tensor. His conclusions note that the ideas "might be applicable" to compressible turbulent flow, where the Reynolds decompositions of density and temperature introduce additional fluctuating fields, and invoke the classical open-problem context — the deterministic Navier–Stokes equations probably contain all the physics information about turbulence, which can be predicted by understanding and solving them — as the ultimate goal. Two caveats belong to any faithful reading. First, the counting argument is the author's position, not a community consensus: the source adopts Reynolds' deterministic view, under which the fluctuation components are deterministic quantities rather than random fields, but specifying them completely is still equivalent to specifying the full turbulent state, so the reduction from six to three relocates rather than removes the modeling burden. Second, the reframing remains untested in the literature; what it instantiates — the classical closure problem — is nonetheless a precise, long-standing open objective.

Problem Statement

For the turbulent flow of an incompressible Newtonian fluid, construct constitutive relations or transport equations — deterministic in the sense of Reynolds (1895) adopted by Sun (2022), under which the RANS equations are deterministic, not statistical, the overbar denotes Reynolds time averaging, and the fluctuation fields are treated as deterministic quantities rather than random fields — for the three velocity-fluctuation components u1′u_1', u2′u_2', u3′u_3' and the pressure fluctuation p′p' that close the RANS system, so that the Reynolds stresses ui′uj′‾\overline{u_i' u_j'} and hence the mean flow are determined. The relations are sought as an explicit closed system in the fluctuation fields u1′u_1', u2′u_2', u3′u_3' and p′p' alone, with every Reynolds stress and higher-order correlation appearing in the equations replaced by an explicit modeled expression in those fields and the mean flow, so that the mean flow can be predicted without access to the instantaneous velocity or pressure fields; the closure is to be validated by reproducing a named turbulent-flow benchmark, comparing the computed mean velocity and Reynolds stresses against published direct-numerical-simulation or experimental reference data. The closure sought is one genuinely expressed in the fluctuation fields: restatements of the unaveraged or fluctuation Navier–Stokes equations, the definitional identification ui′=ui−ui‾u_i' = u_i - \overline{u_i}, and the Boussinesq eddy-viscosity, mixing-length, and second-moment (Reynolds-stress transport) families do not provide it. This is the classical, still-open turbulence closure problem addressed through the fluctuation fields rather than the six Reynolds-stress components; the classical goal the source invokes — predicting turbulence by understanding and solving the deterministic Navier–Stokes equations — remains the ultimate context, and the source's conclusions note that the ideas might be applicable to compressible turbulent flow, where the Reynolds decompositions of density and temperature introduce additional fluctuating fields.

References

  1. Bohua Sun, 'Revisiting the Reynolds-averaged Navier–Stokes equations', Open Physics 19(1):853–862, published online 2022. DOI: 10.1515/phys-2021-0102, https://doi.org/10.1515/phys-2021-0102
  2. Osborne Reynolds, 'IV. On the dynamical theory of incompressible viscous fluids and the determination of the criterion', Philosophical Transactions of the Royal Society of London A 186:123–164, 1895. DOI: 10.1098/rsta.1895.0004, https://doi.org/10.1098/rsta.1895.0004
  3. B. E. Launder, G. J. Reece, W. Rodi, 'Progress in the development of a Reynolds-stress turbulence closure', Journal of Fluid Mechanics 68(3):537–566, 1975. DOI: 10.1017/S0022112075001814, https://doi.org/10.1017/S0022112075001814
  4. S. B. Pope, 'PDF methods for turbulent reactive flows', Progress in Energy and Combustion Science 11(2):119–192, 1985. DOI: 10.1016/0360-1285(85)90002-4, https://doi.org/10.1016/0360-1285(85)90002-4
  5. Karthik Duraisamy, Gianluca Iaccarino, Heng Xiao, 'Turbulence Modeling in the Age of Data', Annual Review of Fluid Mechanics 51:357–377, 2019. DOI: 10.1146/annurev-fluid-010518-040547, https://doi.org/10.1146/annurev-fluid-010518-040547
  6. Jinlong Wu, Heng Xiao, Rui Sun, Qiqi Wang, 'Reynolds-averaged Navier–Stokes equations with explicit data-driven Reynolds stress closure can be ill-conditioned', Journal of Fluid Mechanics 869:553–586, 2019. DOI: 10.1017/jfm.2019.205, https://doi.org/10.1017/jfm.2019.205
  7. Bo-Hua Sun, 'An equivalent form of the Navier-Stokes equations', Acta Mechanica Sinica 39(2):322241, 2023 (published online October 2022). DOI: 10.1007/s10409-022-22241-x, https://doi.org/10.1007/s10409-022-22241-x