pith. sign in

arxiv: 2607.01575 · v1 · pith:JFQSAXLNnew · submitted 2026-07-02 · ⚛️ physics.flu-dyn

A second-order diffusive-interface immersed boundary method for incompressible flow with phase change and moving interfaces

Wenyuan Chen , Yantao Yang This is my paper

Pith reviewed 2026-07-03 05:32 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords immersed boundary methoddiffuse interfacephase changesecond-order accuracyincompressible flowmoving interfacesextrapolationautophoresis
0
0 comments X

The pith

Extrapolating the scalar field across the interface restores second-order spatial accuracy in diffuse-interface immersed boundary methods for flows with phase change.

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

The paper identifies local derivative discontinuities as the cause of first-order accuracy loss near phase-changing boundaries in traditional diffuse-interface immersed boundary methods. It introduces a smooth extension that extrapolates scalar fields such as temperature across the interface to enforce derivative continuity. This extension is restricted to scalar transport equations so that the velocity field keeps its standard treatment and the divergence-free condition remains intact. The resulting scheme is tested on one-dimensional evaporation and boiling cases plus autophoretic particle motion, confirming that second-order accuracy is recovered for these multi-physics problems.

Core claim

By extrapolating the scalar field across the interface, the method structurally ensures derivative continuity and thereby restores formal second-order spatial accuracy in diffuse-interface IBMs for phase-changing boundaries, while the velocity field retains the standard diffuse-interface treatment to preserve the divergence-free condition of the incompressible solver.

What carries the argument

The smooth extension strategy, which extrapolates the scalar field across the interface to enforce derivative continuity while leaving the velocity treatment unchanged.

If this is right

  • The method produces second-order spatial accuracy on one-dimensional evaporation and boiling problems.
  • It resolves the spontaneous autophoretic motion of isotropic particles while maintaining incompressibility.
  • Complex multi-physics boundary couplings at moving phase-change interfaces can be captured without order reduction.
  • Derivative continuity is achieved without modifying the velocity solver or its divergence-free enforcement.

Where Pith is reading between the lines

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

  • The scalar-only extension principle may apply to other transport equations in active-matter or reaction-diffusion systems.
  • Implementation in three-dimensional codes could reveal whether the same accuracy gain holds for curved or topologically changing interfaces.
  • The separation of scalar and velocity treatments suggests a modular way to add high-order corrections to existing immersed-boundary codes without re-deriving the projection step.

Load-bearing premise

Local derivative discontinuities are the main source of accuracy loss, and applying the smooth extension only to scalars will not break the divergence-free constraint enforced by the incompressible solver.

What would settle it

A grid-refinement study on the one-dimensional evaporation benchmark that yields only first-order error reduction with the proposed method would falsify the claim of restored second-order accuracy.

Figures

Figures reproduced from arXiv: 2607.01575 by Wenyuan Chen, Yantao Yang.

Figure 1
Figure 1. Figure 1: (a) One-dimensional schematic illustrating the reconstruction of external boundary values [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Configuration of the one-dimensional evaporation (Stefan) problem. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) Interface location versus time for simulations with different grid resolutions, comparing [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Comparison of the temperature distributions obtained by the IB methods with and [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Configuration of the one-dimensional boiling (suction) problem. [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Numerical results for the one-dimensional boiling problem: (a) temporal evolution of the [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Mid-plane temperature contours of the three-dimensional bubble growth process at the [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Numerical results for the three-dimensional bubble growth problem: (a) temporal evo [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Steady-state terminal velocity U∞ as a function of the P´eclet number P e. Our Navier￾Stokes simulations (yellow markers with error bars) are compared against the idealized Stokes-flow theoretical baseline [21] (solid red line) and the fixed Sc = 1 cases (blue squares). (b) Estimated terminal Reynolds number Re as a function of P e under the fixed Schmidt number (Sc = 1) condition [PITH_FULL_IMAGE:figures… view at source ↗
Figure 10
Figure 10. Figure 10: Flow visualization for P e = 10 and Sc = 1 at t = 400. The solute concentration field on the plane x = 10 is shown as a cross-sectional contour map, overlaid with streamlines in the body-fixed frame computed from the relative velocity u − Uc (light-blue curves) 17 [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Transient evolution of the particle swimming velocity over time for P´eclet numbers [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: (a) Steady-state effective squirmer parameter [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
read the original abstract

Accurately resolving interfacial gradients is critical for simulating two-phase flows, particularly those involving phase transitions or active matter. The traditional diffuse-interface immersed boundary methods (IBMs) are highly efficient for such problems, but they typically suffer from a reduction to first-order accuracy near the phase-changing boundaries. We clarify that the main reason is the local derivative discontinuities. Here, we propose a smooth extension strategy to restore formal second-order spatial accuracy. By extrapolating the scalar field across the interface, the method structurally ensures derivative continuity. To preserve the divergence-free condition in incompressible fluid solvers, this smooth extension is applied exclusively to the scalar transport equations. The velocity field retains the standard diffuse-interface treatment. The proposed framework is systematically validated against classical phase-change benchmarks, specifically one-dimensional evaporation and boiling problems. Additionally, the method is applied to the spontaneous autophoretic motion of isotropic particles. The numerical results confirm the capability of our method in resolving the complex multi-physics boundary couplings.

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

1 major / 1 minor

Summary. The manuscript proposes a second-order diffusive-interface immersed boundary method for incompressible flows with phase change and moving interfaces. It identifies local derivative discontinuities as the source of first-order accuracy degradation in standard diffuse-interface IBMs and introduces a smooth extension by extrapolating the scalar field across the interface to enforce derivative continuity. The extension is restricted to scalar transport equations to preserve the divergence-free condition, while the velocity field retains standard diffuse-interface treatment. Validation is reported on one-dimensional evaporation and boiling benchmarks, with an additional application to spontaneous autophoretic motion of isotropic particles.

Significance. If the central accuracy claim holds under the reported coupling, the approach would provide a practical route to second-order spatial accuracy in phase-change IBM simulations without altering the incompressible solver or introducing free parameters, addressing a recurring limitation in diffuse-interface methods for multi-physics interfacial problems.

major comments (1)
  1. [Method and validation sections (Stefan condition implementation)] The core claim that scalar-only extrapolation restores formal second-order accuracy rests on the assumption that derivative discontinuities are the dominant error source and that the Stefan-condition coupling to the (non-extrapolated) velocity field does not degrade the global order. The abstract states that interface velocity is set by the Stefan condition while velocity retains standard treatment; a load-bearing demonstration is therefore required that the computed normal interface speed and subsequent advection remain second-order accurate. A concrete test would be to report observed convergence rates for the interface position and temperature field in the 1D Stefan problems both with and without the scalar extrapolation, including the measured order for the velocity field near the interface.
minor comments (1)
  1. [Abstract and results] The abstract mentions systematic validation on classical benchmarks but does not include quantitative error tables, observed convergence rates, or grid sizes; these should be added to the main text or a dedicated results table for reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and constructive feedback on our manuscript. The major comment raises a valid point about explicitly verifying that the Stefan-condition coupling preserves second-order accuracy for the interface motion and velocity field. We address this directly below and will incorporate the requested additional convergence data in the revised version.

read point-by-point responses

  1. Referee: [Method and validation sections (Stefan condition implementation)] The core claim that scalar-only extrapolation restores formal second-order accuracy rests on the assumption that derivative discontinuities are the dominant error source and that the Stefan-condition coupling to the (non-extrapolated) velocity field does not degrade the global order. The abstract states that interface velocity is set by the Stefan condition while velocity retains standard treatment; a load-bearing demonstration is therefore required that the computed normal interface speed and subsequent advection remain second-order accurate. A concrete test would be to report observed convergence rates for the interface position and temperature field in the 1D Stefan problems both with and without the scalar extrapolation, including the measured order for the velocity field near the interface.

    Authors: We agree that a direct comparison of convergence rates with and without the scalar extrapolation, including the velocity field, would strengthen the validation. Our analysis in the manuscript identifies derivative discontinuities as the primary source of order reduction in standard diffuse-interface IBMs, and the 1D evaporation/boiling benchmarks already show second-order behavior for temperature and interface position under the new method. However, the referee is correct that the manuscript does not explicitly tabulate orders for the non-extrapolated velocity field or provide the with/without comparison. In the revised manuscript we will add these convergence studies for the 1D Stefan problems, reporting observed orders for temperature, interface position, and velocity near the interface in both configurations. This will confirm that the Stefan coupling does not degrade the global order. revision: yes

Circularity Check

0 steps flagged

No circularity: accuracy restoration follows from explicit construction, validated externally

full rationale

The paper states that local derivative discontinuities cause first-order loss and proposes extrapolation on scalars to ensure continuity by construction, applied only to transport equations while velocity retains standard treatment. This is an independent algorithmic modification whose claimed effect on order is asserted from the structural property of the extension, not from any fitted parameter, self-referential definition, or load-bearing self-citation. Validation is performed against external classical benchmarks (1D evaporation/boiling, autophoretic motion), with no equations or results shown to reduce the second-order claim to an input by tautology. The derivation chain therefore remains self-contained against external checks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract alone; no explicit free parameters, invented entities, or non-standard axioms are stated. The work rests on the standard domain assumption of incompressible flow.

axioms (1)
  • domain assumption The velocity field must remain divergence-free to enforce incompressibility.
    Invoked when restricting the smooth extension to scalar transport equations only.

pith-pipeline@v0.9.1-grok · 5699 in / 1256 out tokens · 26439 ms · 2026-07-03T05:32:42.575961+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    C. S. Peskin, Flow patterns around heart valves: a numerical method, Journal of Computa- tional Physics 10 (1972) 252–271

    work page 1972

  2. [2]

    Mittal, G

    R. Mittal, G. Iaccarino, Immersed boundary methods, Annu. Rev. Fluid Mech. 37 (2005) 239–261

    work page 2005

  3. [3]

    L. Wang, G. M. Currao, F. Han, A. J. Neely, J. Young, F.-B. Tian, An immersed bound- ary method for fluid–structure interaction with compressible multiphase flows, Journal of Computational Physics 346 (2017) 131–151

    work page 2017

  4. [4]

    W. Xiao, H. Zhang, K. Luo, C. Mao, J. Fan, Immersed boundary method for multiphase transport phenomena, Reviews in Chemical Engineering 38 (2022) 363–405

    work page 2022

  5. [5]

    P. R. C. Souza, H. R. Neto, M. M. Villar, J. M. Vedovotto, A. A. Cavalini Jr, A. S. Neto, Multi-phase fluid–structure interaction using adaptive mesh refinement and immersed bound- ary method, Journal of the Brazilian Society of Mechanical Sciences and Engineering 44 (2022) 152

    work page 2022

  6. [6]

    H. Yan, G. Zhang, D. Wang, Y. Lu, S. Wang, Improved diffuse interface-immersed bound- ary method for three-dimensional multiphase fluids–structure interaction with moving contact lines, Applied Ocean Research 151 (2024) 104181

    work page 2024

  7. [7]

    Q. Jin, D. Hudson, W. G. Price, A combined volume of fluid and immersed boundary method for modeling of two-phase flows with high density ratio, Journal of Fluids Engineering 144 (2022) 031402

    work page 2022

  8. [8]

    Liu, Z.-X

    X. Liu, Z.-X. Tong, Y.-L. He, Enthalpy-based immersed boundary-lattice boltzmann model for solid-liquid phase change in porous media under local thermal non-equilibrium condition, International Journal of Thermal Sciences 182 (2022) 107786

    work page 2022

  9. [9]

    V. E. Badalassi, H. D. Ceniceros, S. Banerjee, Computation of multiphase systems with phase field models, Journal of computational physics 190 (2003) 371–397

    work page 2003

  10. [10]

    Fadlun, R

    E. Fadlun, R. Verzicco, P. Orlandi, J. Mohd-Yusof, Combined immersed-boundary finite- difference methods for three-dimensional complex flow simulations, J. Comput. Phys. 161 (2000) 35––60

    work page 2000

  11. [11]

    J. H. Seo, R. Mittal, A sharp-interface immersed boundary method with improved mass conservation and reduced spurious pressure oscillations, Journal of computational physics 230 (2011) 7347–7363

    work page 2011

  12. [12]

    Uhlmann, An immersed boundary method with direct forcing for the simulation of partic- ulate flows, Journal of Computational Physics 209 (2005) 448–476

    M. Uhlmann, An immersed boundary method with direct forcing for the simulation of partic- ulate flows, Journal of Computational Physics 209 (2005) 448–476

    work page 2005

  13. [13]

    Kempe, J

    T. Kempe, J. FrHlich, An improved immersed boundary method with direct forcing for the simulation of particle laden flows, Journal of Computational Physics 231 (2012) 3663–3684

    work page 2012

  14. [14]

    A. M. D. Jost, S. Glockner, Direct forcing immersed boundary methods: Improvements to the ghost-cell method, Journal of Computational Physics 438 (2021) 110371. 21

    work page 2021

  15. [15]

    Malan, A

    L. Malan, A. G. Malan, S. Zaleski, P. Rousseau, A geometric vof method for interface resolved phase change and conservative thermal energy advection, Journal of Computational Physics 426 (2021) 109920

    work page 2021

  16. [16]

    Li, The immersed interface method using a finite element formulation, Applied Numerical Mathematics 27 (1998) 253–267

    Z. Li, The immersed interface method using a finite element formulation, Applied Numerical Mathematics 27 (1998) 253–267

    work page 1998

  17. [17]

    M. C. Lai, C. S. Peskin, An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, Journal of Computational Physics 160 (2000) 705–719

    work page 2000

  18. [18]

    B. E. Griffith, C. S. Peskin, On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems, Journal of Computational Physics 208 (2005) 75–105

    work page 2005

  19. [19]

    Treeratanaphitak, N

    T. Treeratanaphitak, N. M. Abukhdeir, Diffuse-interface blended method for imposing physical boundaries in two-fluid flows, ACS omega 8 (2023) 15518–15534

    work page 2023

  20. [20]

    C. Shao, K. Luo, M. Chai, H. Wang, J. Fan, A computational framework for interface-resolved dns of simultaneous atomization, evaporation and combustion, Journal of Computational Physics 371 (2018) 751–778

    work page 2018

  21. [21]

    Michelin, E

    S. Michelin, E. Lauga, D. Bartolo, Spontaneous autophoretic motion of isotropic particles, Physics of Fluids 25 (2013) 061701

    work page 2013

  22. [22]

    Q. Mao, J. E. C. Cabezas, S. Zhao, P. Boivin, J. Favier, An explicit fully one-sided diffuse- interface immersed boundary method for compressible flows, Journal of Computational Physics (2026) 114721

    work page 2026

  23. [23]

    X. Zhu, Y. Chen, K. L. Chong, D. Lohse, R. Verzicco, A boundary condition-enhanced direct- forcing immersed boundary method for simulations of three-dimensional phoretic particles in incompressible flows, Journal of Computational Physics 509 (2024) 113028

    work page 2024

  24. [24]

    R. P. Beyer, R. J. LeVeque, Analysis of a one-dimensional model for the immersed boundary method, SIAM Journal on Numerical Analysis 29 (1992) 332–364

    work page 1992

  25. [25]

    D. B. Stein, R. D. Guy, B. Thomases, Immersed boundary smooth extension: a high-order method for solving pde on arbitrary smooth domains using fourier spectral methods, Journal of Computational Physics 304 (2016) 252–274

    work page 2016

  26. [26]

    W. Chen, S. Zou, Q. Cai, Y. Yang, An explicit and non-iterative moving-least-squares immersed-boundary method with low boundary velocity error, J. Comp. Phys. 474 (2023) 111803

    work page 2023

  27. [27]

    Verzicco, P

    R. Verzicco, P. Orlandi, A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates, Journal of Computational Physics 123 (1996) 402–414

    work page 1996

  28. [28]

    Ostilla-M´ onico, Y

    R. Ostilla-M´ onico, Y. Yang, E. P. Van Der Poel, D. Lohse, R. Verzicco, A multiple-resolution strategy for direct numerical simulation of scalar turbulence, Journal of Computational Physics 301 (2015) 308–321

    work page 2015

  29. [29]

    Vanella, E

    M. Vanella, E. Balaras, A moving-least-squares reconstruction for embedded-boundary formu- lations, Journal of Computational Physics 228 (2009) 6617–6628. 22

    work page 2009

  30. [30]

    Vanella, E

    M. Vanella, E. Balaras, Direct Lagrangian Forcing Methods Based on Moving Least Squares, Springer Singapore, Singapore, 2020, pp. 45–79

    work page 2020

  31. [31]

    S. Wang, X. Zhang, An immersed boundary method based on discrete stream function formu- lation for two- and three-dimensional incompressible flows, Journal of Computational Physics 230 (2011) 3479–3499

    work page 2011

  32. [32]

    Mittal, H

    R. Mittal, H. Dong, M. Bozkurttas, F. Najjar, A. Vargas, A. Von Loebbecke, A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries, Journal of computational physics 227 (2008) 4825–4852

    work page 2008

  33. [33]

    Udaykumar, R

    H. Udaykumar, R. Mittal, P. Rampunggoon, A. Khanna, A sharp interface cartesian grid method for simulating flows with complex moving boundaries, Journal of computational physics 174 (2001) 345–380

    work page 2001

  34. [34]

    J. Kim, D. Kim, H. Choi, An immersed-boundary finite-volume method for simulations of flow in complex geometries, Journal of Computational Physics 171 (2001) 132–150

    work page 2001

  35. [35]

    S. W. Welch, J. Wilson, A volume of fluid based method for fluid flows with phase change, Journal of computational physics 160 (2000) 662–682

    work page 2000

  36. [36]

    Hardt, F

    S. Hardt, F. Wondra, Evaporation model for interfacial flows based on a continuum-field representation of the source terms, Journal of Computational Physics 227 (2008) 5871–5895

    work page 2008

  37. [37]

    Y. Sato, B. Niˇ ceno, A sharp-interface phase change model for a mass-conservative interface tracking method, Journal of Computational Physics 249 (2013) 127–161

    work page 2013

  38. [38]

    Irfan, M

    M. Irfan, M. Muradoglu, A front tracking method for direct numerical simulation of evapora- tion process in a multiphase system, Journal of Computational Physics 337 (2017) 132–153

    work page 2017

  39. [39]

    Scriven, On the dynamics of phase growth, Chemical Engineering Science 10 (1959) 1–18

    L. Scriven, On the dynamics of phase growth, Chemical Engineering Science 10 (1959) 1–18

    work page 1959

  40. [40]

    Kunkelmann, P

    C. Kunkelmann, P. Stephan, CFD simulation of boiling flows using the volume-of-fluid method within openfoam, Numerical Heat Transfer, Part A 56 (2009) 631–646

    work page 2009

  41. [41]

    Michelin, E

    S. Michelin, E. Lauga, Phoretic self-propulsion at finite p´ eclet numbers, Journal of fluid mechanics 747 (2014) 572–604

    work page 2014

  42. [42]

    A. S. Khair, N. G. Chisholm, Expansions at small reynolds numbers for the locomotion of a spherical squirmer, Physics of Fluids 26 (2014)

    work page 2014

  43. [43]

    N. G. Chisholm, D. Legendre, E. Lauga, A. S. Khair, A squirmer across reynolds numbers, Journal of Fluid Mechanics 796 (2016) 233–256

    work page 2016

  44. [44]

    Y. Chen, K. L. Chong, L. Liu, R. Verzicco, D. Lohse, Instabilities driven by diffusiophoretic flow on catalytic surfaces, Journal of Fluid Mechanics 919 (2021) A10

    work page 2021

  45. [45]

    Verzicco, M

    R. Verzicco, M. D. de Tullio, F. Viola, An Introduction to Immersed Boundary Methods, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2025. 23

    work page 2025