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arxiv: 2605.13195 · v1 · submitted 2026-05-13 · ⚛️ physics.flu-dyn

Recognition: 2 theorem links

· Lean Theorem

Influence of Prandtl number on heat transfer over a permeable wall

Francesca di Mare, Hakan Demir, Wojciech Sadowski

Pith reviewed 2026-05-14 02:05 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords Prandtl numberporous-fluid interfaceturbulent heat transferupscalingTaylor expansion termsfilter kernelstemperature budgetpermeable wall
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The pith

Terms usually dropped when averaging flow variables near porous walls grow important for heat transfer when the Prandtl number is low.

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

The paper examines turbulent heat transport in a channel that is half filled with an array of cubes, using direct numerical simulation at one Reynolds number and three Prandtl numbers. It measures every contribution to the temperature equation right at the interface between the clear fluid and the porous region. The authors find that correction terms coming from the Taylor series of the filtered fields, which are normally omitted in upscaled descriptions, become as large as the main transport terms once the Prandtl number falls to 0.05. These terms stay small at Pr equal to 0.71. A reader cares because practical models for heat exchange in porous media rely on dropping exactly those corrections; the work shows when that shortcut stops being safe.

Core claim

Direct numerical simulations of turbulent heat transfer over an array of cubes show that the interface terms arising from the Taylor expansion of the filtered temperature and velocity fields contribute significantly to the heat budget only at the lowest Prandtl numbers examined. While tortuosity and Brinkman terms are negligible at the highest Prandtl number, the Taylor corrections remain appreciable at the lowest. Different filter kernels produce distinct upscaled profiles, with the discrepancy largest at small Pr.

What carries the argument

The set of Taylor-expansion correction terms that appear when deriving upscaled equations for temperature at the porous-fluid interface.

If this is right

  • The accuracy of upscaled heat-transfer models for fluids with Prandtl numbers below 0.1 requires explicit inclusion of the Taylor correction terms.
  • Tortuosity and Brinkman contributions can be safely neglected near the interface once Pr exceeds approximately 0.7.
  • The choice among cellular, linear, quadratic, and cubic filter kernels changes the predicted mean temperature inside the porous layer by amounts that increase as Pr decreases.
  • Both tested boundary-condition configurations yield the same Prandtl-number trend for the relative size of the neglected terms.
  • Temperature variance near the interface is controlled mainly by the balance between turbulent transport and molecular diffusion, whose relative weight shifts strongly with Pr.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the same trend holds at higher Reynolds numbers, low-Pr coolants in porous heat exchangers would require revised closure models.
  • The Taylor terms identified here may also affect scalar transport across other permeable interfaces, such as vegetation canopies or packed-bed reactors.
  • Systematic variation of filter width in future simulations could produce a simple correlation for the magnitude of the corrections as a function of Pr and filter size.
  • Repeating the study with non-cubic obstacles would reveal whether the reported Pr dependence is universal or geometry dependent.

Load-bearing premise

The spatial resolution and Reynolds number used in the simulations are sufficient to give the true physical size of the Taylor correction terms rather than numerical errors.

What would settle it

A grid-refinement study at the lowest Prandtl number that recomputes the interface terms on a mesh with at least double the resolution per cube; if the terms change by more than 20 percent, the original magnitudes cannot be trusted.

Figures

Figures reproduced from arXiv: 2605.13195 by Francesca di Mare, Hakan Demir, Wojciech Sadowski.

Figure 1
Figure 1. Figure 1: The schematic of the studied geometry. The gray squares indicate positions of the cubes. The top wall and the cubes are heated with a heat flux of 𝑞 and 𝑞𝑡 , respectively [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Overview of the geometry and the turbulent flow above the porous wall for the case with the heated top wall. The vortical structures are visualized with iso-surfaces of the 𝑄 criterion (𝑄 ≈ 0.6𝑈 2 bulk∕𝐻) and colored by 𝜃 𝑝+ (Pr = 0.71). configurations test the sensitivity of the balance terms to the boundary conditions. For each Pr and boundary conditions configuration, we derive and evaluate the double-a… view at source ↗
Figure 3
Figure 3. Figure 3: The filtering kernels used in the study, tent: Eq. (12), quadratic: Eq. (13), cubic: Eq. (14), along with the top-hat kernel used to construct them. kernel) and representing the average as a filtering operation similar to LES filtering: ⟨𝜓⟩𝑠 (𝒙) = ∫ ℝ3 𝐺(𝒙 − 𝝃)𝐼(𝝃)𝜓(𝝃) d𝝃, (10) where 𝐼 is a phase indicator function (Gray, 1975) 𝐼(𝒙) = { 1 ∶ for 𝒙 in fluid; 0 ∶ otherwise. (11) Following the discussion in (B… view at source ↗
Figure 4
Figure 4. Figure 4: For most “under-resolved” cells, the added viscosity is considerably smaller than the fluid’s viscosity. There are, however, a couple of cells in which 𝜈sgs∕𝜈 ≈ 0.2. For these cells, the viscous stresses are considerably influenced by the 0.6 0.7 0.8 0.9 1.0 IQPope 10−3 10−2 10−1 νsgs/ν 0.2 0.4 0.6 0.8 1.0 |u|/Ubulk [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Two point correlations of velocity fluctuations in streamwise (left column) and spanwise (right column) direc￾tions, at three different heights (𝑦∕𝐻 = 0, 0.1, 0.5). In the middle of the channel 𝑅𝑢𝑢 does not vanish completely, indicating that the domain is too short to capture properly the dynamics of low-frequency structures. Overall, the studies of the sensitivity of the gathered statistics to the chosen … view at source ↗
Figure 6
Figure 6. Figure 6: Velocity moments in the channel (from left to right: superficial double-averaged streamwise velocity, root-mean-square of velocity fluctuations in streamwise, wall-normal and spanwise direction), compared to the data gathered by Breugem and Boersma (2005). Results with three different filtering kernels are presented, tent kernel given by Eq. (12), quadratic kernel Eq. (13) and cubic kernel Eq. (14). Breuge… view at source ↗
Figure 7
Figure 7. Figure 7: Macroscopic turbulent viscosity (a) and modelling parameters of the drag closure (b) plotted near the interface and inside the porous medium. Results with three different filtering kernels are presented, tent kernel given by Eq. (12), quadratic kernel Eq. (13) and cubic kernel Eq. (14). kernels. Deep in the homogeneous porous medium, perme￾ability remains constant, however near the PFI, permeability decrea… view at source ↗
Figure 8
Figure 8. Figure 8: Profiles of temperature normalized as given by Eq. (48) from cases with both boundary conditions, for each Prandtl number. The spatial filtering is performed using the quadratic kernel, i.e., Eq. (13). The reference data corresponds to the results for Pr = 0.1 gathered by Chandesris et al. (2013) for an adiabatic top wall configuration. First and foremost, the result for the 𝑞𝑡 = 0 configu￾ration and Pr = … view at source ↗
Figure 9
Figure 9. Figure 9: The wall-normal turbulent heat flux in the channel (left), the gradient of mean temperature (centre) and the production of scalar variance (right) for both boundary condition configurations and computed Prandtl numbers. The spatial filtering is performed using the quadratic kernel, i.e., Eq. (13). The reference data corresponds to the results for Pr = 0.1 gathered by Chandesris et al. (2013) for an adiabat… view at source ↗
Figure 10
Figure 10. Figure 10: The computed values of the macroscopic turbulent heat transfer coefficient 𝛼𝑡𝜙 (left) and corresponding turbulent Prandtl number Pr𝑡𝜙 (right). with 𝑦, however, the scatter of the values is quite large here. The distribution changes sign around 𝑦∕𝐻 = 0.8, and for −0.8 < 𝑦∕𝐻 < −0.6, the differences between the three Prandtl numbers start to be noticeable. The small differences in 𝛼𝑡𝜙 gets attenuated leading… view at source ↗
Figure 12
Figure 12. Figure 12: The temperature budget terms computed for each Prandtl number (columns), for the adiabatic top wall configuration (top row) and the heated top-wall configuration. The terms were evaluated using the quadratic kernel given by Eq. (13). the rapid increase of the fluctuations inside the porous medium has been linked to the interaction between the strong turbulent heat flux induced by the large scale unsteady … view at source ↗
Figure 13
Figure 13. Figure 13: Molecular diffusion term of the temperature budget plotted for each Prandtl number, using all three different filtering kernels (Eqs. (12) to (14), i.e., linear, quadratic and cubic functions respectively) and are normalised using the friction temperature, velocity and the height of the channel. Importantly, the evaluation of the temperature balance becomes sensitive to the choice of the filtering kernel … view at source ↗
Figure 14
Figure 14. Figure 14: The profiles of the tortuosity flux (left) the terms originating from the Taylor expansion of the double-averaged values inside the surface integral (centre) and the related effective viscosities (right), computed using the quadratic filtering kernel. not constant — it decreases slowly as the vertical coordinate moves away from the interface. Possibly, this situation is related to much stronger influence … view at source ↗
Figure 15
Figure 15. Figure 15: The profiles of the tortuosity flux (left) the terms originating from the Taylor expansion of the double-averaged values inside the surface integral: first-order (centre) and the second-order (right), computed using tent, quadratic and cubic kernels, i.e., Eqs. (12) to (14). 6. Conclusions The turbulent flow (Rebulk = 5485) with heat transfer in a channel half-filled with an array cubes with porosity of 0… view at source ↗
read the original abstract

The work considers a fully turbulent flow with heat transfer in a channel half-filled with an array of cubes based on the work of Breugem and Boersma (2005) and Chandesris et al. (2013), at $\mathrm{Re}_\mathrm{bulk} = 5485$ and three different Prandtl numbers, $\mathrm{Pr} = 0.71, 0.1, 0.05$. The temperature is modelled as a passive scalar and two different boundary condition configurations are simulated. The influence of the Prandtl number on the mean temperature, its variance and the terms of the temperature budget is highlighted, including the analysis of the distribution and relative importance of the turbulent heat transfer, molecular diffusion, tortuosity and Brinkman terms near the porous-fluid interface. The latter two has been found to be insignificant for the highest $\mathrm{Pr}$. A set of terms, typically neglected during the upscaling procedure (related to the Taylor expansion of the filtered variables), is analysed for the first time for the turbulent heat transfer at the porous-fluid interface, and are found to be significant at low $\mathrm{Pr}$. The upscaled fields are evaluated with three different kernels forming cellular average, linear (i.e., tent kernel), quadratic and cubic, and the influence of the chosen filter is additionally studied.

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 presents direct numerical simulations of fully turbulent channel flow with passive scalar heat transfer over a permeable wall formed by an array of cubes, at Re_bulk = 5485 and Pr = 0.71, 0.1, 0.05. It examines the mean temperature, variance, and full temperature budget terms at the porous-fluid interface, including turbulent heat transfer, molecular diffusion, tortuosity, and Brinkman contributions. A central focus is the first analysis of higher-order terms arising from the Taylor expansion of filtered variables (typically neglected in upscaling), which are reported to become significant at low Pr. Upscaled fields are compared across cellular-average, linear (tent), quadratic, and cubic filter kernels.

Significance. If the extracted magnitudes are accurate, the result is significant for porous-media heat-transfer modeling: it shows that standard filtering/upscaling closures can omit non-negligible contributions precisely when Pr is low (e.g., liquid-metal applications). The systematic variation of Pr and the explicit comparison of multiple filter kernels provide concrete evidence that interface modeling assumptions are Pr-dependent. Credit is due for extracting the Taylor terms directly from filtered DNS fields rather than from an ad-hoc closure.

major comments (2)
  1. [Numerical methods] Numerical methods / DNS setup: the manuscript reports no grid-resolution study, no validation against the cited Breugem & Boersma (2005) or Chandesris et al. (2013) data, and no error-bar or uncertainty estimates on the filtered fields. Because the central claim rests on the quantitative magnitude of the Taylor-expansion terms at the interface, the absence of these checks makes it impossible to judge whether the reported significance at low Pr is physical or numerical.
  2. [Results on temperature budget] Results on Taylor terms (abstract and § on budget analysis): the statement that the neglected terms 'are found to be significant at low Pr' is not accompanied by a quantitative threshold, relative-magnitude comparison to the retained budget terms, or sensitivity to filter width. Without these, the load-bearing conclusion that the terms matter only at low Pr cannot be assessed independently of the chosen grid and kernels.
minor comments (2)
  1. [Abstract] Abstract: 'the latter two has been found' is grammatically incorrect and should read 'have been found'.
  2. [Filter description] Filter-kernel definitions: the quadratic and cubic kernels are mentioned but not given explicit functional forms or references; adding the mathematical expressions would remove ambiguity when readers attempt to reproduce the upscaled fields.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of our results on the Prandtl-number dependence of interface heat-transfer terms. We address each major comment below and will revise the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [Numerical methods] Numerical methods / DNS setup: the manuscript reports no grid-resolution study, no validation against the cited Breugem & Boersma (2005) or Chandesris et al. (2013) data, and no error-bar or uncertainty estimates on the filtered fields. Because the central claim rests on the quantitative magnitude of the Taylor-expansion terms at the interface, the absence of these checks makes it impossible to judge whether the reported significance at low Pr is physical or numerical.

    Authors: We acknowledge the absence of an explicit grid-resolution study and direct validation plots in the submitted manuscript. The DNS setup follows the geometry, Re_bulk, and discretization approach of Breugem & Boersma (2005) and Chandesris et al. (2013), with grid spacing chosen to satisfy their reported resolution criteria. In the revision we will add a dedicated validation subsection that includes (i) comparisons of mean velocity and temperature profiles against the cited references, (ii) a brief grid-convergence check at the interface for the dominant budget terms, and (iii) uncertainty estimates obtained from temporal averaging over multiple flow-through times. These additions will allow readers to assess the reliability of the reported Taylor-term magnitudes. revision: yes

  2. Referee: [Results on temperature budget] Results on Taylor terms (abstract and § on budget analysis): the statement that the neglected terms 'are found to be significant at low Pr' is not accompanied by a quantitative threshold, relative-magnitude comparison to the retained budget terms, or sensitivity to filter width. Without these, the load-bearing conclusion that the terms matter only at low Pr cannot be assessed independently of the chosen grid and kernels.

    Authors: We agree that the current wording lacks a clear quantitative definition of significance. In the revised manuscript we will (i) introduce an explicit threshold (neglected terms > 10 % of the sum of retained budget terms at the interface), (ii) add a table and accompanying figure that report the relative magnitudes of all temperature-budget contributions for each Pr, and (iii) include a sensitivity study in which the filter support width is varied by ±25 % while keeping the same kernel family. These changes will make the Pr-dependence of the Taylor terms directly verifiable from the data. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper performs direct numerical simulations of turbulent channel flow with heat transfer over a porous medium at fixed Re_bulk=5485 and three Pr values, then post-processes the DNS fields using multiple filter kernels to extract interface terms including those arising from Taylor expansion of filtered variables. These extracted terms are compared directly to the simulated budgets; no derivation step reduces the reported significance at low Pr to a fitted parameter, self-referential definition, or load-bearing self-citation. The central claim follows from the numerical extraction procedure itself and remains independent of the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The study rests on the incompressible Navier-Stokes equations and the passive-scalar transport equation; no new physical entities are postulated and no parameters are fitted to match the target result.

axioms (2)
  • standard math The flow is governed by the incompressible Navier-Stokes equations at the given bulk Reynolds number.
    Standard assumption for direct numerical simulation of incompressible turbulent channel flow.
  • domain assumption Temperature behaves as a passive scalar that does not back-couple to the velocity field.
    Explicitly stated modeling choice for the heat-transfer problem.

pith-pipeline@v0.9.0 · 5543 in / 1261 out tokens · 67629 ms · 2026-05-14T02:05:05.204176+00:00 · methodology

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