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arxiv: 1907.00049 · v1 · pith:M5JBIJ7Qnew · submitted 2019-06-22 · ❄️ cond-mat.soft · physics.flu-dyn

Statistical properties of thermally expandable particles in soft Rayleigh-Benard convection

Kim M. J. Alards , Rudie P. J. Kunnen , Herman J. H. Clercx , Federico Toschi This is my paper

Pith reviewed 2026-05-25 18:22 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.flu-dyn
keywords Rayleigh-Bénard convectioninertial particlesthermal expansionboundary layersdirect numerical simulationbuoyancy
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The pith

Particles with greater thermal expansion than the fluid spend a time in boundary layers that stays constant for fast response but grows for slower response.

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

The paper examines inertial particles in Rayleigh-Bénard convection where the particles expand thermally more than the fluid. This mismatch produces a buoyancy force that drives particles out of the thermal boundary layers at the plates toward the bulk. Direct numerical simulations show that the residence time inside those layers remains constant for small thermal response times but increases once the response time exceeds a threshold value, and the time also lengthens as the particles' expansion coefficient relative to the fluid decreases. A stripped-down one-dimensional model that lets particle motion depend only on the buoyancy force from thermal inertia reproduces the same two regimes of behavior.

Core claim

Because particles expand more than the fluid, they become lighter than the fluid near the hot bottom plate and heavier near the cold top plate, so the net Archimedes force pushes them toward the bulk. The characteristic time particles spend inside the thermal boundary layers is independent of the thermal response time τ_T when τ_T ≲ 1 and rises with τ_T when τ_T ≳ 1; the residence time also increases as the relative expansion coefficient K = α_p / α_f decreases. These trends appear in the DNS data and are recovered qualitatively by a one-dimensional model whose only ingredient is buoyancy arising from thermal inertia.

What carries the argument

Buoyancy force generated by the thermal-expansion mismatch between particles and fluid, parameterized by thermal response time τ_T and expansion ratio K.

If this is right

  • Residence time inside the thermal boundary layers is independent of τ_T for small response times.
  • Residence time increases with τ_T once the response time exceeds the threshold.
  • Residence time increases as the relative expansion coefficient K decreases.
  • The one-dimensional buoyancy model reproduces the same two regimes observed in the full DNS.

Where Pith is reading between the lines

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

  • The same expulsion mechanism may alter steady-state particle concentration profiles inside convective cells.
  • The dependence on τ_T and K could be checked in laboratory experiments that use particles with controlled thermal properties.
  • The effect may appear in other thermally driven flows that contain expandable particles.

Load-bearing premise

The one-dimensional model that lets particle motion depend only on buoyancy from thermal inertia is assumed to capture the essential trends seen in the three-dimensional simulations.

What would settle it

A set of simulations or experiments in which the boundary-layer residence time varies continuously with τ_T even for values below 1, or fails to increase with τ_T above 1.

Figures

Figures reproduced from arXiv: 1907.00049 by Federico Toschi, Herman J. H. Clercx, Kim M. J. Alards, Rudie P. J. Kunnen.

Figure 1
Figure 1. Figure 1: The vertical distribution, ni , of tracers (gray crosses) and thermally responsive particles (different colors) in the Rayleigh–B´enard cell. Results are shown for three different values of K: (a) K = 1.1, (b) K = 2 and (c) K = 10 and for different τT as reported in the legend of panel (a) (see also table 2). The solid vertical line shows the thermal boundary layer thickness, δT = 0.022H. Because of symmet… view at source ↗
Figure 2
Figure 2. Figure 2: (a) The fluid temperature, Tf , in a horizontal plane at z = 0.012H in the Rayleigh– B´enard cell. (b-g) The same temperature field, together with particles with vertical position zp < 0.015H for different values of K and τT : (b) K = 1.1, τT = 0.1, (c) K = 1.1, τT = 4, (d) K = 2, τT = 0.1, (e) K = 2, τT = 4, (f) K = 10, τT = 0.1 and (g) K = 10, τT = 4. Axes and colorbars are as in panel (a) and the color … view at source ↗
Figure 3
Figure 3. Figure 3: (a) The temperature difference hTp −Tf (xp)izi , averaged horizontally and in time within horizontal slabs at central vertical position zi for fluid tracers (gray lines with crosses) and ther￾mally responsive particles (different colors) in Rayleigh–B´enard convection for K = 1.1. (b) PDFs of Tp − Tf (xp), measured in the thermal boundary layer (BL) at the bottom plate for K = 1.1. (c) and (d) show similar… view at source ↗
Figure 4
Figure 4. Figure 4: PDFs of the particle residence time, tδT , inside the thermal boundary layer (BL) at the plates for different values of K: (a) K = 1.1, (b) K = 2 and (c) K = 10 and various τT as reported in the legend of panel (a) (see also table 2). The thermal BLs have a thickness of δT = 0.022H. Errors are estimated as the deviation between the PDFs measured in the thermal BL at the top and bottom plates. Error bars fa… view at source ↗
Figure 5
Figure 5. Figure 5: The average residence time, htδT i of thermally responsive particles in the thermal bound￾ary layers at the plates as a function of τT , for K = 1.1 (blue), K = 2 (red) and K = 10 (green) for DNS (symbols). The dashed lines show fits of the function y = a(K) + b(K)x, where the fitting coefficients a(K) and b(K) are as reported in the legend. thermal BL residence time as tδT = a(K) + b(K)τT , (14) where bot… view at source ↗
Figure 6
Figure 6. Figure 6: (a) The average residence time, htδT i, for the DNS (symbols) and the residence time, tδT , for the 1D model (lines) of thermally responsive particles in the thermal boundary layers at the plates as a function of τT , for K = 1.1 (blue), K = 2 (red) and K = 10 (green). (b) The same data, but now the axes are re-scaled for the 1D model data where the vertical axis is multiplied by 3 and the horizontal axis … view at source ↗
read the original abstract

The dynamics of inertial particles in Rayleigh-B\'{e}nard convection, where both particles and fluid exhibit thermal expansion, is studied using direct numerical simulations (DNS). We consider the effect of particles with a thermal expansion coefficient larger than that of the fluid, causing particles to become lighter than the fluid near the hot bottom plate and heavier than the fluid near the cold top plate. Because of the opposite directions of the net Archimedes' force on particles and fluid, particles deposited at the plate now experience a relative force towards the bulk. The characteristic time for this motion towards the bulk to happen, quantified as the time particles spend inside the thermal boundary layers (BLs) at the plates, is shown to depend on the thermal response time, $\tau_T$, and the thermal expansion coefficient of particles relative to that of the fluid, $K = \alpha_p / \alpha_f$. In particular, the residence time is constant for small thermal response times, $\tau_T \lesssim 1$, and increasing with $\tau_T$ for larger thermal response times, $\tau_T \gtrsim 1$. Also, the thermal BL residence time is increasing with decreasing $K$. A one-dimensional (1D) model is developed, where particles experience thermal inertia and their motion is purely dependent on the buoyancy force. Although the values do not match one-to-one, this highly simplified 1D model does predict a regime of a constant thermal BL residence time for smaller thermal response times and a regime of increasing residence time with $\tau_T$ for larger response times, thus explaining the trends in the DNS data well.

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 / 1 minor

Summary. The manuscript uses DNS to study inertial particles with thermal expansion coefficient larger than the fluid in Rayleigh-Bénard convection. Particles experience a net force toward the bulk due to opposing Archimedes forces. The central result is that the residence time inside the thermal boundary layers is constant for τ_T ≲ 1 and increases with τ_T for τ_T ≳ 1; the time also increases with decreasing K = α_p / α_f. A 1D model in which particle motion depends only on buoyancy arising from thermal inertia reproduces the same qualitative regimes (constant then increasing) even though numerical values do not match one-to-one.

Significance. If the reported regimes are robust, the work isolates the effect of thermal inertia on particle residence times near walls in buoyancy-driven flows. The combination of 3D DNS data with an independent 1D reduction is a positive feature; the model supplies a falsifiable mechanistic hypothesis even if quantitative agreement is imperfect.

major comments (2)
  1. [Abstract] Abstract: the claim that the 1D model 'explains the trends in the DNS data well' rests on qualitative regime agreement, yet the manuscript acknowledges that values 'do not match one-to-one' without reporting a quantitative discrepancy measure (e.g., relative L2 error or ratio of slopes) or demonstrating that forces omitted from the 1D model (lift, shear-induced drag modulation, or particle-induced flow alteration) remain negligible across the τ_T range.
  2. [Abstract] Abstract and DNS results section: no error bars, bootstrap uncertainties, or grid/convergence checks are supplied for the residence-time curves that define the constant (τ_T ≲ 1) versus increasing (τ_T ≳ 1) regimes; without these, the sharpness of the transition at τ_T ≈ 1 cannot be assessed.
minor comments (1)
  1. Notation: K is defined as α_p / α_f but the symbol is introduced only in the abstract; a brief reminder in the main text would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and the positive evaluation of the work's significance. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the 1D model 'explains the trends in the DNS data well' rests on qualitative regime agreement, yet the manuscript acknowledges that values 'do not match one-to-one' without reporting a quantitative discrepancy measure (e.g., relative L2 error or ratio of slopes) or demonstrating that forces omitted from the 1D model (lift, shear-induced drag modulation, or particle-induced flow alteration) remain negligible across the τ_T range.

    Authors: The 1D model is intentionally reduced to isolate the buoyancy mechanism arising from thermal inertia; its purpose is to test whether this single effect is sufficient to produce the observed transition between constant and increasing residence-time regimes at τ_T ≈ 1. We agree that the presentation would be strengthened by a quantitative measure of agreement (e.g., the ratio of the slopes in the τ_T ≳ 1 regime) and by a short discussion of why lift, shear-induced drag modulation, and particle-induced flow alteration remain sub-dominant for the Stokes numbers and particle Reynolds numbers realized in the DNS. These additions will be included in the revised manuscript. revision: yes

  2. Referee: [Abstract] Abstract and DNS results section: no error bars, bootstrap uncertainties, or grid/convergence checks are supplied for the residence-time curves that define the constant (τ_T ≲ 1) versus increasing (τ_T ≳ 1) regimes; without these, the sharpness of the transition at τ_T ≈ 1 cannot be assessed.

    Authors: The residence times are ensemble averages over several thousand particle trajectories per parameter combination. While the trends are reproducible across independent runs, we acknowledge that explicit uncertainty quantification would allow a clearer assessment of the transition sharpness. In the revision we will add bootstrap-derived error bars on the residence-time curves and a brief statement on the grid resolution and convergence checks already performed for the underlying DNS. revision: yes

Circularity Check

0 steps flagged

No circularity: DNS trends and independent 1D model are separate evidence

full rationale

The paper reports residence-time trends directly from 3D DNS as a function of τ_T and K. It then introduces a separate 1D model whose only force is buoyancy from thermal inertia; the model is not fitted to the DNS data and is explicitly stated to mismatch quantitative values while reproducing the same qualitative regimes (constant for τ_T ≲ 1, increasing for τ_T ≳ 1). No equation reduces a prediction to a fitted parameter, no self-citation chain is load-bearing, and no ansatz or uniqueness theorem is smuggled in. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard Boussinesq fluid assumptions plus two study parameters (τ_T, K) that are varied rather than fitted to the residence-time data itself.

free parameters (2)
  • K = α_p / α_f
    Ratio of particle to fluid thermal expansion coefficients; varied parametrically in the DNS.
  • τ_T
    Particle thermal response time; varied parametrically in the DNS and 1D model.
axioms (2)
  • domain assumption Both particles and fluid obey thermal expansion with constant coefficients α_p and α_f
    Invoked in the setup to produce differential buoyancy inside the thermal boundary layers.
  • standard math Boussinesq approximation remains valid for the density variations induced by temperature
    Standard assumption in Rayleigh-Bénard convection simulations.

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Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    A. S. Ackerman, M. P. Kirkpatrick, D. E. Stevens, and O. B. Toon. The impact of humidity above stratiform clouds on indirect aerosol climate forcing. Nature, 432(7020):1014–1017, 2004

    work page 2004

  2. [2]

    Ahlers, S

    G. Ahlers, S. Grossmann, and D. Lohse. Heat transfer and large scale dynamics in turbulent Rayleigh–B´ enard convection.Rev. Mod. Phys. , 81:503–537, 2009

    work page 2009

  3. [3]

    Arcen, A

    B. Arcen, A. Tani` ere, and M. Khalij. Heat transfer in a turbulent particle-laden channel flow. Int. J. Heat Mass Transfer , 55(23):6519–6529, 2012

    work page 2012

  4. [4]

    Armenio and V

    V. Armenio and V. Fiorotto. The importance of the forces acting on particles in turbulent flows. Phys. Fluids , 13(8):2437–2440, 2001

    work page 2001

  5. [5]

    Calzavarini, M

    E. Calzavarini, M. Cencini, D. Lohse, and F. Toschi. Quantifying Turbulence- Induced Segregation of Inertial Particles. Phys. Rev. Lett. , 101:084504, 2008

    work page 2008

  6. [6]

    Douady, Y

    S. Douady, Y. Couder, and M. E. Brachet. Direct observation of the intermit- tency of intense vorticity filaments in turbulence. Phys. Rev. Lett. , 67:983–986, 1991

    work page 1991

  7. [7]

    Engelmann

    S. Engelmann. Advanced Thermoforming. John Wiley & Sons, New Jersey, 2012. 20

    work page 2012

  8. [8]

    G.M. Faeth. Mixing, transport and combustion in sprays. Prog. Energy Com- bust. Science, 13(4):293–345, 1987

    work page 1987

  9. [9]

    Joshi, H

    P. Joshi, H. Rajaei, R. P. J. Kunnen, and H. J. H. Clercx. Effect of particle injection on heat transfer in rotating Rayleigh–B´ enard convection. Phys. Rev. Fluids, 1:084301, Dec 2016

    work page 2016

  10. [10]

    R. M. Kerr. Rayleigh number scaling in numerical convection. Journal of Fluid Mechanics, 310:139–179, 1996

    work page 1996

  11. [11]

    A. B. Kostinski and R. A. Shaw. Fluctuations and Luck in Droplet Growth by Coalescence. Bull. Am. Meteorol. Soc. , 86(2):235–244, 2005

    work page 2005

  12. [12]

    J. G. M. Kuerten, C. W. M. van der Geld, and B. J. Geurts. Turbulence modification and heat transfer enhancement by inertial particles in turbulent channel flow. Phys. Fluids , 23(12):123301, 2011

    work page 2011

  13. [13]

    J. G. M. Kuerten and A. W. Vreman. Collision frequency and radial distribution function in particle-laden turbulent channel flow.Int. J. Multiphase Flow , 87:66– 79, 2016

    work page 2016

  14. [14]

    Lakkaraju, L

    R. Lakkaraju, L. E. Schmidt, P. Oresta, F. Toschi, R. Verzicco, D. Lohse, and A. Prosperetti. Effect of vapor bubbles on velocity fluctuations and dissipation rates in bubbly Rayleigh–B´ enard convection.Phys. Rev. E , 84:036312, 2011

    work page 2011

  15. [15]

    Lakkaraju, R

    R. Lakkaraju, R. J. A. M. Stevens, P. Oresta, R. Verzicco, D. Lohse, and A. Pros- peretti. Heat transport in bubbling turbulent convection. Proc. Natl. Acad. Sci., 110(23):9237–9242, 2013

    work page 2013

  16. [16]

    Lavezzo, H.J.H

    V. Lavezzo, H.J.H. Clercx, and F. Toschi. Role of thermal plumes on particle dispersion in a turbulent Rayleigh–B´ enard cell.Journal of Physics: Conference Series, 318(5):052011, 2011

    work page 2011

  17. [17]

    Lessani and M

    B. Lessani and M. H. Nakhaei. Large-eddy simulation of particle-laden turbulent flow with heat transfer. Int. J. Heat Mass Transfer , 67:974–983, 2013

    work page 2013

  18. [18]

    M. R. Maxey and J. J. Riley. Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids , 26(4):883–889, 1983

    work page 1983

  19. [19]

    E. E. Michaelides and Z. Feng. Heat transfer from a rigid sphere in a nonuniform flow and temperature field. Int. J. Heat Mass Transfer , 37(14):2069–2076, 1994

    work page 2069

  20. [20]

    M. H. Nakhaei and B. Lessani. Effects of solid inertial particles on the velocity and temperature statistics of wall bounded turbulent flow. Int. J. Heat Mass Transfer, 106:1014–1024, 2017

    work page 2017

  21. [21]

    Okada and T

    M. Okada and T. Suzuki. Natural convection of water-fine particle suspension in a rectangular cell. Int. J. Heat Mass Transfer , 40(13):3201–3208, 1997

    work page 1997

  22. [22]

    Oresta and A

    P. Oresta and A. Prosperetti. Effects of particle settling on Rayleigh–B´ enard convection. Phys. Rev. E , 87:063014, 2013

    work page 2013

  23. [23]

    Oresta, R

    P. Oresta, R. Verzicco, D. Lohse, and A. Prosperetti. Heat transfer mechanisms in bubbly Rayleigh–B´ enard convection.Phys. Rev. E , 80:026304, 2009

    work page 2009

  24. [24]

    S. L. Post and J. Abraham. Modeling the outcome of drop–drop collisions in Diesel sprays. Int. J. Multiphase Flow , 28(6):997–1019, 2002. 21

    work page 2002

  25. [25]

    W. E. Ranz and W. R. Marshall. Evaporation from droplets, parts i and ii. Chem. Eng. Prog. , 48(4):173–180, 1952

    work page 1952

  26. [26]

    Reigada, R

    R. Reigada, R. M. Hillary, M. A. Bees, J. M. Sancho, and F. Sagu´ es. Plankton Blooms Induced by Turbulent Flows.Proc.: Biol. Sci. , 270(1517):875–880, 2003

    work page 2003

  27. [27]

    E. W. Saw, R. A. Shaw, S. Ayyalasomayajula, P. Y. Chuang, and ´A. Gylfason. Inertial Clustering of Particles in High-Reynolds-Number Turbulence. Phys. Rev. Lett., 100:214501, 2008

    work page 2008

  28. [28]

    R. A. Shaw. Particle-Turbulence Interactions in Atmospheric Clouds. Ann. Rev. Fluid Mech., 35(1):183–227, 2003

    work page 2003

  29. [29]

    K. D. Squires and Hidekatsu Y. Preferential concentration of marine particles in isotropic turbulence. Deep Sea Res. Part I: Oceanogr. Res. Pap. , 42(11):1989– 2004, 1995

    work page 1989

  30. [30]

    van Aartrijk and H

    M. van Aartrijk and H. J. H. Clercx. The dynamics of small inertial particles in weakly stratified turbulence. J. Hydro Environ. Res. , 4(2):103–114, 2010

    work page 2010

  31. [31]

    van Aartrijk and H

    M. van Aartrijk and H. J. H. Clercx. Vertical dispersion of light inertial particles in stably stratified turbulence: The influence of the Basset force. Phys. Fluids , 22(1):013301, 2010

    work page 2010

  32. [32]

    E. P. van der Poel, R. Ostilla-M´ onico, J. Donners, and R. Verzicco. A pen- cil distributed finite difference code for strongly turbulent wall-bounded flows. Computers & Fluids , 116:10–16, 2015

    work page 2015

  33. [33]

    Verzicco and R

    R. Verzicco and R. Camussi. Numerical experiments on strongly turbulent ther- mal convection in a slender cylindrical cell. J. Fluid Mech. , 477:19–49, 2003

    work page 2003

  34. [34]

    Z. Wang, V. Mathai, and Sun C. Self-sustained biphasic catalytic particle tur- bulence. Nature Communications, page In press, 2019

    work page 2019

  35. [35]

    Wetchagarun and J

    S. Wetchagarun and J. J. Riley. Dispersion and temperature statistics of inertial particles in isotropic turbulence. Phys. Fluids , 22(6):063301, 2010. 22

    work page 2010