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arxiv: 1907.10092 · v1 · pith:BWKFYJKLnew · submitted 2019-07-23 · 🧮 math.NA · cs.NA

On URANS Congruity with Time Averaging: Analytical laws suggest improved models

William Layton , Michael McLaughlin This is my paper

Pith reviewed 2026-05-24 16:51 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords URANSturbulence modelingone-equation modeltime averaginglength scaleNavier-StokesPrandtl modelconsistency conditions
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The pith

A kinematic choice of turbulence lengthscale makes the one-equation URANS model satisfy four consistency conditions without adjustments.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves that defining the turbulence lengthscale in the standard one-equation model as l equals square root of two times square root of k times the time filter window tau leads to automatic satisfaction of four key requirements. These include recovering the Navier-Stokes solution as the window shrinks to zero, vanishing lengthscale at walls, bounded total energy, and appropriate long-time statistical behavior of dissipation. The result holds by direct derivation from the commutation properties of time averaging without needing extra damping or tuning parameters. Readers interested in turbulence simulation would note that this removes common empirical fixes from the model.

Core claim

The report proves that a kinematic specification of the model's turbulence lengthscale by l(x,t)=√2 k^{1/2}(x,t) τ , where τ is the time filter window, results in a 1-equation model satisfying Conditions 1,2,3,4 without model tweaks, adjustments or wall damping multipliers.

What carries the argument

The kinematic turbulence lengthscale defined as l(x,t) = √2 √k(x,t) ⋅ τ that directly incorporates the time-averaging window into the Prandtl one-equation model.

Load-bearing premise

The commutation and limit properties of the time-averaging operator applied to the Navier-Stokes equations hold exactly as used in the derivation of the model.

What would settle it

A numerical simulation of a simple flow, such as decaying turbulence, where the total energy grows unbounded or the wall condition is violated despite the lengthscale formula.

Figures

Figures reproduced from arXiv: 1907.10092 by Michael McLaughlin, William Layton.

Figure 4.1
Figure 4.1. Figure 4.1: Discretization of Ω 4.1. Test 1: Flow between 2d offset circles. For the first test, we con￾sider a two-dimensional rotational flow obstructed by a circular obstacle with no-slip boundary conditions. Let Ω1 ⊂ R 2 , where Ω1 = {(x, y) ∈ R 2 : x 2 + y 2 < 1} \ {(x, y) ∈ R 2 : (x − .5)2 + y 2 ≤ .01}. The domain Ω1 is discretized via a Delaunay triangulation with a maximal mesh width of .01; a plot is given … view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: 2d Flow statistics for both models. 4.2. Test 2: Flow between 3d offset cylinders. The second test is a 3d analogue of the first. It shows similar differences in the two models. Taking Ω1 to be the domain given in the first test, we define Ω = Ω1 × (0, 1), a cylinder of radius and height one with a cylindrical obstacle removed. The domain Ω was discretized with Delaunay tetrahedrons with a maximal mesh w… view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Average mixing length comparison before, we start the flow from rest (v0 = (0, 0, 0)T ) and let the kinematic viscosity ν = 0.0001. The flow evolves via the body force f(x, y, z;t) = min{t, 1}(−4y(1 − x 2 − y 2 ), 4x(1 − x 2 − y 2 ), 0)T , and is observed over the time interval (0, 10], with ∆t = .05 and the initial conditions for k being set in the same way as the first test. Below, we present the evolu… view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Kinematic mixing length model velocity and vorticity. [PITH_FULL_IMAGE:figures/full_fig_p016_4_4.png] view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: Flow statistics for the 3d offset cylinder problem. [PITH_FULL_IMAGE:figures/full_fig_p017_4_5.png] view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Streamlines for the 3d offset cylinder problem. [PITH_FULL_IMAGE:figures/full_fig_p018_4_6.png] view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: Velocity magnitude for the 3d offset cylinder problem. [PITH_FULL_IMAGE:figures/full_fig_p019_4_7.png] view at source ↗
read the original abstract

The standard $1-$equation model \ of turbulence was first derived by Prandtl and has evolved to be a common method for practical flow simulations. Five fundamental laws that any URANS model should satisfy are \[ \begin{array} [c]{ccc} \textbf{1.} & \text{Time window:} & \begin{array} [c]{c} \tau\downarrow 0\text{ implies }v_{\text{{\small URANS}}}\rightarrow u_{\text{{\small NSE}}}\text{ \&}\\ \text{ }\tau\uparrow\text{implies }\nu_{T}\uparrow \end{array} \\ \textbf{2.} & l(x)=0\ \text{at walls:} & l(x)\rightarrow 0\text{ as }x\rightarrow walls,\\ \textbf{3.} & \text{ Bounded energy:} & \sup_{t}\int\frac{1} {2}|v(x,t)|^{2}+k(x,t)dx<\infty\\ \textbf{4.} & \begin{array} [c]{c} \text{Statistical }\\ \text{equilibrium:} \end{array} & \lim\sup_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\varepsilon _{\text{model}}(t)dt=\mathcal{O}\left( \frac{U^{3}}{L}\right) \\ \textbf{5.} & \begin{array} [c]{c} \text{Backscatter}\\ \text{possible:} \end{array} & \text{(without negative viscosities)} \end{array} \] This report proves that a \textit{kinematic} specification of the model's turbulence lengthscale by \[ l(x,t)=\sqrt{2}k^{1/2}(x,t)\tau\text{ }, \] where $\tau$\ is the time filter window, results in a $1-$equation model satisfying Conditions 1,2,3,4 without model tweaks, adjustments or wall damping multipliers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to prove that the kinematic choice l(x,t)=√2 k^{1/2}(x,t) τ in the standard Prandtl one-equation URANS model satisfies four listed conditions (time-window limits, wall vanishing, bounded energy, statistical equilibrium) without additional damping functions or parameter adjustments, by exploiting commutation and limit properties of a time-averaging filter applied to the Navier-Stokes equations.

Significance. If the derivation is complete and the operator assumptions hold, the result supplies an explicit, parameter-free link between a one-equation closure and time-filtered NSE that automatically recovers the correct limits as τ→0 and τ→∞. This would be a concrete analytical contribution to URANS modeling. The manuscript does not supply machine-checked proofs or reproducible code, so the strength rests entirely on the transparency of the algebraic steps.

major comments (2)
  1. [Abstract] Abstract and the central derivation: the claim that l=√2 k^{1/2} τ satisfies Condition 1 (τ↓0 ⇒ ν_T→0 and τ↑ ⇒ ν_T↑) is asserted but the explicit algebraic steps that produce the √2 prefactor from the filtered convective term and the energy balance are not shown; without those steps it is impossible to verify that the factor is derived rather than selected to enforce the desired scaling ν_T ~ k τ.
  2. [Abstract, Conditions 3 and 4] The proof that the model satisfies Condition 3 (sup_t ∫ |v|^2 + k dx < ∞) and Condition 4 (long-time average of ε_model = O(U^3/L)) relies on the time filter commuting exactly with the nonlinear term and on the standard Prandtl production/dissipation form remaining unaltered by the time-dependent l; if either assumption fails, the boundedness and equilibrium statements do not follow from the given l alone.
minor comments (2)
  1. [Abstract] The five conditions are presented as a numbered array but Condition 5 (backscatter) is listed without any statement of whether the proposed model satisfies it; either remove the condition or add an explicit remark.
  2. [Abstract] Notation: the symbol τ is used both for the filter window and implicitly in the length-scale definition; a short clarifying sentence would prevent confusion with the usual turbulence time scale.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and the recognition that a complete derivation would constitute a concrete analytical contribution. We address the two major comments below with clarifications and commit to revisions that improve transparency while preserving the manuscript's core claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the central derivation: the claim that l=√2 k^{1/2} τ satisfies Condition 1 (τ↓0 ⇒ ν_T→0 and τ↑ ⇒ ν_T↑) is asserted but the explicit algebraic steps that produce the √2 prefactor from the filtered convective term and the energy balance are not shown; without those steps it is impossible to verify that the factor is derived rather than selected to enforce the desired scaling ν_T ~ k τ.

    Authors: We agree that the algebraic origin of the √2 prefactor requires explicit display. It arises by substituting the kinematic length scale into the modeled eddy viscosity, equating the resulting dissipation term to the filtered convective contribution in the integrated energy balance, and imposing exact recovery of the inviscid limit as τ→0 together with linear growth of ν_T as τ→∞. In the revised manuscript we will add a dedicated subsection (immediately after the statement of the five conditions) that carries out these steps from the time-filtered NSE through the commutation identity to the final expression l=√2 k^{1/2} τ. This will make clear that the prefactor is fixed by the scaling requirements rather than chosen ad hoc. revision: yes

  2. Referee: [Abstract, Conditions 3 and 4] The proof that the model satisfies Condition 3 (sup_t ∫ |v|^2 + k dx < ∞) and Condition 4 (long-time average of ε_model = O(U^3/L)) relies on the time filter commuting exactly with the nonlinear term and on the standard Prandtl production/dissipation form remaining unaltered by the time-dependent l; if either assumption fails, the boundedness and equilibrium statements do not follow from the given l alone.

    Authors: The proofs for Conditions 3 and 4 do rest on two standard operator assumptions: (i) exact commutation of the fixed-window time average with the convective term, which holds for the continuous averaging operator employed in the derivation, and (ii) preservation of the classical Prandtl production/dissipation structure when l is inserted only through ν_T. We will revise the manuscript to state these assumptions explicitly in the introduction and in the paragraphs preceding the proofs of Conditions 3 and 4, together with a short justification based on the properties of the averaging filter. Under these assumptions the bounded-energy and statistical-equilibrium statements follow directly from the kinematic choice of l; we will also add a remark noting that any numerical implementation in which commutation is only approximate would constitute a separate discretization issue outside the scope of the continuous analysis. revision: yes

Circularity Check

1 steps flagged

Kinematic lengthscale l=√2 k^{1/2} τ chosen to satisfy conditions by construction

specific steps
  1. self definitional [Abstract]
    "This report proves that a kinematic specification of the model's turbulence lengthscale by l(x,t)=√2 k^{1/2}(x,t) τ , where τ is the time filter window, results in a 1-equation model satisfying Conditions 1,2,3,4 without model tweaks, adjustments or wall damping multipliers."

    The lengthscale is defined using the time window τ and √2 prefactor specifically to enforce the required ν_T scaling with τ (Condition 1) and wall behavior (Condition 2) when inserted into the standard Prandtl 1-equation form. Satisfaction of the conditions is therefore a direct algebraic consequence of the chosen definition rather than an independent prediction.

full rationale

The paper's central claim is that specifying l kinematically as √2 k^{1/2} τ produces a model satisfying Conditions 1-4. This reduces directly to the input choice: the form is selected so ν_T scales with τ (ensuring Condition 1), l→0 at walls when k→0 (Condition 2), and the energy/statistical bounds follow from the standard Prandtl structure under the assumed commutation properties. The √2 prefactor is required to produce the exact scaling limits rather than emerging from an independent derivation. While the paper invokes external operator assumptions, the satisfaction of the target conditions is equivalent to the kinematic definition itself.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard 1-equation model form, the mathematical properties of the time-averaging operator, and the kinematic definition of l; no new physical entities are postulated.

free parameters (1)
  • sqrt(2) prefactor = √2
    Chosen so that the kinematic l produces the required limits and energy bounds in the derivation.
axioms (2)
  • domain assumption Standard Prandtl 1-equation model equations
    The paper begins from this established model form.
  • standard math Commutation and limit properties of the time-averaging operator on NSE
    Invoked to establish the congruity with the five conditions.

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Works this paper leans on

37 extracted references · 37 canonical work pages · 1 internal anchor

  1. [1]

    Brossier and R

    F. Brossier and R. Lewandowski , Impact of the variations of the mixing length in a first order turbulent closure system,ESAIM: Mathematical Modelling and Numerical Analysis 36.2 (2002): 345-372

    work page 2002

  2. [2]

    Bulicek and J

    M. Bulicek and J. Malek , Large data analysis for Kolmogorov’s 2 equation model of turbu- lence, Nonlinear Analysis 50(2018) 104-143

    work page 2018

  3. [3]

    Bulíček, R., Lewandowski and J

    M. Bulíček, R., Lewandowski and J. Malek , On evolutionary Navier-Stokes-Fourier type systems in three spatial dimensions, Commentationes Mathematicae Universitatis Caroli- nae 52.1 (2011) 89-114

    work page 2011

  4. [4]

    Boussinesq, Essai sur la théorie des eaux courantes, Mémoires présentés par divers savants à l’Académie des Sciences 23 (1877): 1-680

    J. Boussinesq, Essai sur la théorie des eaux courantes, Mémoires présentés par divers savants à l’Académie des Sciences 23 (1877): 1-680

    work page

  5. [5]

    Chacon-Rebollo and R

    T. Chacon-Rebollo and R. Lewandowski , Mathematical and numerical foundations of tur- bulence models and applications, Springer, New-York, 2014

    work page 2014

  6. [6]

    Davidson, Turbulence: an introduction for scientists and engineers

    P. Davidson, Turbulence: an introduction for scientists and engineers. Oxford Univ. Press, 2015

    work page 2015

  7. [7]

    Darrigol, Worlds of flow, Oxford, 2005

    O. Darrigol, Worlds of flow, Oxford, 2005

    work page 2005

  8. [8]

    Doering and C

    C. Doering and C. Foias , Energy dissipation in body-forced turbulence, J. Fluid Mech., 467 (2002), 289-306

    work page 2002

  9. [9]

    Doering and P

    C.R. Doering and P. Constantin , Energy dissipation in shear driven turbulence, Physical review letters 69.11 (1992): 1648

    work page 1992

  10. [10]

    Doering and J.D

    C. Doering and J.D. Gibbon , Applied Analysis of the Navier-Stokes Equations, Cambridge Univ. Press, Cambridge, 1995

    work page 1995

  11. [11]

    Durbin and B.A

    P.A. Durbin and B.A. Pettersson Reif , Statistical theory and modeling for turbulent flows, Second Edition, Wiley, Chichester, 2011

    work page 2011

  12. [12]

    Eckert, The dawn of fluid dynamics, Wiley-VCH, Weinheim, 2006

    M. Eckert, The dawn of fluid dynamics, Wiley-VCH, Weinheim, 2006

    work page 2006

  13. [13]

    Frisch, Turbulence, Cambridge Univ

    U. Frisch, Turbulence, Cambridge Univ. Press, Cambridge, 1995

    work page 1995

  14. [14]

    Layton , Numerical Analysis of two Ensemble Eddy Viscosity Mod- els of Fluid Motion, accepted: NMPDEs, 2014, published online : 15 JUL 2014, DOI: 10.1002/num.21908

    Nan Jiang and W. Layton , Numerical Analysis of two Ensemble Eddy Viscosity Mod- els of Fluid Motion, accepted: NMPDEs, 2014, published online : 15 JUL 2014, DOI: 10.1002/num.21908

    work page doi:10.1002/num.21908 2014

  15. [15]

    Layton , Algorithms and models for turbulence not at statistical equilib- rium, Computers & Mathematics with Applications, 71 (2016) 2352-2372

    Nan Jiang and W. Layton , Algorithms and models for turbulence not at statistical equilib- rium, Computers & Mathematics with Applications, 71 (2016) 2352-2372

    work page 2016

  16. [16]

    Johnson, E.N

    F.T. Johnson, E.N. Tinoco and N.J. Yu ,Thirty years of development and application of CFD at Boeing Commercial Airplanes, Seattle, Computers & Fluids, 34(10):1115–1151,2005

    work page 2005

  17. [17]

    Layton, The 1877 Boussinesq conjecture: Turbulent fluctuations are dissipative on the mean flow, TR 14-07, www.mathematics.pitt.edu/research/technical-reports, 2014

    W. Layton, The 1877 Boussinesq conjecture: Turbulent fluctuations are dissipative on the mean flow, TR 14-07, www.mathematics.pitt.edu/research/technical-reports, 2014

    work page 2014

  18. [18]

    Layton, L.G

    W. Layton, L.G. Rebholz, C. Trenchea , Modular nonlinear filter stabilization of methods for higher Reynolds numbers flow, Journal of Mathematical Fluid Mechanics 14 (2012), 325-354

    work page 2012

  19. [19]

    R. Lewandowski, The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity, Nonlinear Analysis, 28 (1997), 393-417

    work page 1997

  20. [20]

    Lewandowski and B

    R. Lewandowski and B. Mohammadi, Existence and positivity results for theφ− θ model and a modifiedk− ε model, Math. Model Methods Appl. Sci.3(1993) 195-215

    work page 1993

  21. [21]

    Mathieu and J

    J. Mathieu and J. Scott , An introduction to turbulent flows, Cambridge, 2000

    work page 2000

  22. [22]

    Mohammadi and O

    B. Mohammadi and O. Pironneau , Analysis of the K-Epsilon Turbulence Model, Masson, Paris, 1994

    work page 1994

  23. [23]

    Pakzad, Damping Functions correct over-dissipation of the Smagorinsky Model,Mathe- matical Methods in the Applied Sciences 40 (2017), no

    A. Pakzad, Damping Functions correct over-dissipation of the Smagorinsky Model,Mathe- matical Methods in the Applied Sciences 40 (2017), no. 16, DOI 10.1002/mma.4444

    work page doi:10.1002/mma.4444 2017

  24. [24]

    Pope, Turbulent Flows,Cambridge Univ

    S. Pope, Turbulent Flows,Cambridge Univ. Press, Cambridge, 2000

    work page 2000

  25. [25]

    Prandtl, Über ein nenes Formelsystem für die ausgebildete Turbulenz, Nacr

    L. Prandtl, Über ein nenes Formelsystem für die ausgebildete Turbulenz, Nacr. Akad. Wiss. Göttingen, Math-Phys. Kl., (1945) 6-16

    work page 1945

  26. [26]

    Prandtl, On fully developed turbulence, in: Proceedings of the 2nd International Congress 20 of Applied Mechanics, Zurich (1926) 62-74

    L. Prandtl, On fully developed turbulence, in: Proceedings of the 2nd International Congress 20 of Applied Mechanics, Zurich (1926) 62-74

    work page 1926

  27. [27]

    Saint-Venant (Barré) , Note à joindre au Mémoire sur la dynamique des fluides, CRAS 17(1843), 1240-1243

    A.J.C. Saint-Venant (Barré) , Note à joindre au Mémoire sur la dynamique des fluides, CRAS 17(1843), 1240-1243

    work page

  28. [28]

    Spalart, Philosophies and fallacies in turbulence modeling, Progress in Aerospace Sci- ences, 74:1–15, 2015

    P.R. Spalart, Philosophies and fallacies in turbulence modeling, Progress in Aerospace Sci- ences, 74:1–15, 2015

    work page 2015

  29. [29]

    Starr, Physics of Negative Viscosity Phenomena, McGraw Hill, NY, 1968

    V.P. Starr, Physics of Negative Viscosity Phenomena, McGraw Hill, NY, 1968

    work page 1968

  30. [30]

    Taylor, Eddy motion in the atmosphere,Phil

    G.I. Taylor, Eddy motion in the atmosphere,Phil. Trans. of Royal Soc. Series A 215 (1915) 1-26

    work page 1915

  31. [31]

    V assilicos, Dissipation in turbulent flows, Ann

    J.C. V assilicos, Dissipation in turbulent flows, Ann. Rev. Fluid Mech. 47 (2015) 95-114

    work page 2015

  32. [32]

    Vergassola, S

    M. Vergassola, S. Gama and U. Frisch , Proving the existence of negative, isotropic eddy viscosity, pp. 321-328 in: Solar and Planetary Dynamics (eds.: M. Proctor, D. Mattheus and A. Rucklidge) Cambridge U. Press, Cambridge, 1994

    work page 1994

  33. [33]

    Vreman, An eddy-viscosity subgrid-scale model for turbulent shear flow: algebraic theory and applications,Phys

    A.W. Vreman, An eddy-viscosity subgrid-scale model for turbulent shear flow: algebraic theory and applications,Phys. Fluids 16 (2004), 3670-3681

    work page 2004

  34. [34]

    W ang, The time averaged energy dissipation rates for shear flows, Physica D, 99 (1997) 555-563

    X. W ang, The time averaged energy dissipation rates for shear flows, Physica D, 99 (1997) 555-563. 2004

    work page 1997

  35. [35]

    Wilcox, Turbulence Modeling for CFD, DCW Industries, La Canada, 2006

    D.C. Wilcox, Turbulence Modeling for CFD, DCW Industries, La Canada, 2006

    work page 2006

  36. [36]

    Zi-Niu Wu and Song Fu , Positivity of k-epsilon turbulence models for incompressible flow, Mathematical Models and Methods in Applied Sciences 12 (2002): 393-406

    work page 2002

  37. [37]

    Quantification of Model Uncertainty in RANS Simulations: A Review

    H. Xiao and P. Cinnella , Quantification of Model Uncertainty in RANS: A Review , arxiv.org/pdf/1806.10434.pdf , 2018. 21

    work page internal anchor Pith review Pith/arXiv arXiv 2018