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arxiv: 2604.03281 · v2 · submitted 2026-03-24 · ⚛️ physics.flu-dyn · math.AP· nlin.CD· nlin.PS

A simplified model for coupling Darrieus-Landau and diffusive-thermal instabilities

Prabakaran Rajamanickam This is my paper

Pith reviewed 2026-05-15 00:17 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn math.APnlin.CDnlin.PS
keywords Darrieus-Landau instabilitydiffusive-thermal instabilitypremixed flamesflame-front evolutionhydro-diffusive numberMarkstein numbercellular structuresphenomenological model
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The pith

Simplified model unifies Darrieus-Landau and diffusive-thermal flame instabilities through a cubic coupling term.

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

The paper proposes a minimal phenomenological model that couples the long-wave Darrieus-Landau instability with the short-wave diffusive-thermal instability in premixed flames. A cubic term inserted into the linear dispersion relation encodes the leading interaction between hydrodynamic expansion and diffusive transport, recovering the classical Michelson-Sivashinsky equation when the Markstein number is order unity. In the distinguished crossover regime where the Markstein number scales as the square root of the thermal-expansion parameter, the model produces a generalized evolution equation whose nonlocal stabilising term is controlled by the hydro-diffusive number and remains active even when Markstein stabilisation vanishes. Large-domain simulations of this equation exhibit persistent competition between large DL cusps and fine DT wrinkles, supplying a tractable account of observed accelerated growth rates and fine-scale cellular structures without solving the full conservation equations.

Core claim

A cubic coupling term identified in the linear dispersion relation allows construction of a unified evolution equation for the flame front. In the DL-DT crossover limit the equation contains a nonlocal stabilising operator set by the hydro-diffusive number N = A / delta_L^2, where A is the characteristic area of interaction between hydrodynamic and diffusive transport. This term continues to damp short waves after classical Markstein stabilisation disappears, producing a chaotic regime in which DL cusp structures compete continuously with small-scale DT wrinkles.

What carries the argument

The cubic coupling term in the linear dispersion relation, which encodes the leading-order interaction between hydrodynamic expansion and diffusive transport and generates the nonlocal stabilising operator in the crossover equation.

If this is right

  • For order-unity positive Markstein numbers the model reduces exactly to the Michelson-Sivashinsky equation.
  • When the Markstein number is small the hydro-diffusive number supplies stabilisation independent of Markstein effects.
  • Large-domain integrations produce a chaotic state in which large-scale cusps and small-scale wrinkles remain in persistent competition.
  • The framework accounts for the fine cellular structures and accelerated growth rates seen in experiments without invoking the complete reactive-flow equations.

Where Pith is reading between the lines

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

  • The hydro-diffusive-area concept may generalise to other pairs of long- and short-wave instabilities in reacting flows.
  • The chaotic mixed-scale regime suggests that flame fronts can sustain both large and small wrinkles simultaneously even when one classical mechanism would normally dominate.
  • Extension of the model to three spatial dimensions could expose new cellular patterns arising from the same coupling.
  • Systematic derivation of analogous cubic terms directly from the conservation laws would test whether the phenomenological choice is quantitatively accurate.

Load-bearing premise

The cubic term is assumed to represent the leading-order interaction between hydrodynamic expansion and diffusive transport, chosen phenomenologically from the dispersion relation rather than derived from the full conservation equations.

What would settle it

Full Navier-Stokes simulations with detailed chemistry performed at Markstein numbers of order sqrt(thermal expansion) would reveal whether the predicted persistent competition between cusp and wrinkle scales occurs at the wave-number ratios given by the model.

Figures

Figures reproduced from arXiv: 2604.03281 by Prabakaran Rajamanickam.

Figure 1
Figure 1. Figure 1: Numerical solution in a relatively large domain ( [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical solution in a relatively large domain ( [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Numerical solution in a relatively small domains, [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Numerical solution in a large domain, ν = 0.0005, with λ = +1. Even for positive Markstein numbers, the dynamics becomes chaotic for sufficiently large domain sizes. Remarkably, the chaotic state retains the char￾acteristic single-cusp structure of the DL instability, but the cusp is intermittently destroyed by the emergence of short-wavelength wrinkles and then reforms, giving rise to a complex, recurrent… view at source ↗
Figure 5
Figure 5. Figure 5: Marginal stability curves for different values of ǫ = (r − 1)/2 in the limit β → ∞ where β is the Zeldovich number. The red dashed line corresponds to the formula l = −2 − 8k 2 , applicable when r = 1 (i.e., ǫ = 0), obtained by Sivashinsky [11]. The four solid lines correspond to r = {1.2, 1.5, 2, 5} (i.e., ǫ = {0.1, 0.25, 0.5, 2.0}), obtained by Jackson and Kapila [14]. The green shaded region indicates t… view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of dispersion curves for the Sivashins [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
read the original abstract

A simplified phenomenological model is proposed to couple the long-wave Darrieus--Landau (DL) instability and the short-wave diffusive-thermal (DT) instability in premixed flames. By identifying a cubic coupling term in the linear dispersion relation, representing the leading-order interaction between hydrodynamic expansion and diffusive transport, this framework moves beyond the traditional treatment of these instabilities in isolation. Two distinct asymptotic regimes are identified: the first recovers the classical Michelson--Sivashinsky equation for order-unity positive Markstein numbers $\mathcal M>0$, the second reveals a distinguished DL-DT crossover regime where both instabilities participate at equal order. In this crossover limit, where the Markstein number is small ($\mathcal M \sim \sqrt{\epsilon}$ with $\epsilon$ measuring thermal expansion), a generalized evolution equation is derived featuring a nonlocal stabilising term controlled by the hydro-diffusive number $\mathcal{N} = \mathcal A/\delta_L^2$, where $\mathcal A$ is the hydro-diffusive area -- the characteristic area over which hydrodynamic and diffusive transport processes interact. This term remains active even when Markstein stabilisation vanishes. Numerical solutions in sufficiently large domains based on our model reveal a distinctive chaotic regime in which the characteristic DL cusp structures are in persistent competition with small-scale wrinkles. This minimal unified framework thus captures the essential coupled dynamics governing flame front instability and provides a tractable explanation for the fine-scale cellular structures and accelerated growth rates observed, without recourse to the full complexity of the complete conservation equations.

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

3 major / 2 minor

Summary. The manuscript proposes a phenomenological model to couple the long-wave Darrieus-Landau (DL) and short-wave diffusive-thermal (DT) instabilities in premixed flames. A cubic coupling term is identified in the linear dispersion relation to represent the leading-order interaction between hydrodynamic expansion and diffusive transport. Two regimes are analyzed: recovery of the Michelson-Sivashinsky equation for order-unity positive Markstein numbers, and a distinguished DL-DT crossover where M ~ sqrt(epsilon), yielding a generalized evolution equation with a nonlocal stabilizing term controlled by the hydro-diffusive number N = A / delta_L^2 (A being the hydro-diffusive area). Numerical solutions in large domains exhibit persistent chaotic competition between DL cusps and small-scale DT wrinkles.

Significance. If the central assumptions hold, the model supplies a minimal tractable framework for the coupled dynamics, recovering the known Michelson-Sivashinsky limit as a consistency check and furnishing a plausible mechanism for observed fine-scale cellular structures and accelerated growth rates without solving the full conservation equations. The numerical demonstration of chaotic cusp-wrinkle competition is of potential interest to the combustion and fluid-dynamics communities.

major comments (3)
  1. [Model derivation and crossover regime] The cubic coupling term is introduced by matching the linear dispersion relation rather than derived via matched asymptotics from the full conservation equations. This assumption is load-bearing for the claim that the model captures the essential nonlinear hydro-diffusive interaction in the distinguished limit M ~ sqrt(epsilon).
  2. [Definition of hydro-diffusive number N] The hydro-diffusive area A is defined phenomenologically, with N = A / delta_L^2 then controlling the nonlocal stabilizing term. Because N is constructed directly from this invented quantity and the scaling is chosen to balance the instabilities, the saturation mechanism and reported growth-rate acceleration rest on an unverified parameter choice.
  3. [Numerical results] Numerical solutions are presented for the generalized equation, but no quantitative comparison is made to direct numerical simulations of the complete conservation equations or to experimental data in the M ~ sqrt(epsilon) regime. This leaves the fidelity of the nonlinear coefficients and the predicted chaotic regime untested.
minor comments (2)
  1. [Introduction] The thermal expansion parameter epsilon and the Markstein number script M should be defined explicitly in the introduction before their use in the asymptotic regimes.
  2. [References] Additional references to prior asymptotic and numerical studies of coupled DL-DT instabilities would help situate the novelty of the unified framework.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the positive evaluation of our manuscript and for the detailed comments, which help clarify the scope and limitations of our phenomenological approach. We address each major comment below.

read point-by-point responses

  1. Referee: [Model derivation and crossover regime] The cubic coupling term is introduced by matching the linear dispersion relation rather than derived via matched asymptotics from the full conservation equations. This assumption is load-bearing for the claim that the model captures the essential nonlinear hydro-diffusive interaction in the distinguished limit M ~ sqrt(epsilon).

    Authors: We agree that the cubic term is introduced by matching the linear dispersion relation rather than through a full matched-asymptotic derivation from the conservation equations. This is intentional in a phenomenological model whose goal is to furnish a minimal tractable equation that recovers the Michelson-Sivashinsky limit and exhibits the DL-DT crossover. We will revise the introduction and derivation section to state explicitly that the model is phenomenological, to emphasize the matching procedure, and to note that a rigorous asymptotic derivation remains an open task. revision: partial

  2. Referee: [Definition of hydro-diffusive number N] The hydro-diffusive area A is defined phenomenologically, with N = A / delta_L^2 then controlling the nonlocal stabilizing term. Because N is constructed directly from this invented quantity and the scaling is chosen to balance the instabilities, the saturation mechanism and reported growth-rate acceleration rest on an unverified parameter choice.

    Authors: The hydro-diffusive number N is indeed a phenomenological parameter introduced to quantify the characteristic interaction area in the distinguished limit. Its scaling is chosen so that the nonlocal stabilization balances the DL and DT growth rates when M ~ sqrt(epsilon). We will add a dedicated paragraph discussing the physical motivation for N, possible ways to estimate it from DNS or experiments, and the dependence of the reported dynamics on its value. revision: partial

  3. Referee: [Numerical results] Numerical solutions are presented for the generalized equation, but no quantitative comparison is made to direct numerical simulations of the complete conservation equations or to experimental data in the M ~ sqrt(epsilon) regime. This leaves the fidelity of the nonlinear coefficients and the predicted chaotic regime untested.

    Authors: We concur that quantitative validation against full DNS or experiments in the M ~ sqrt(epsilon) regime would be valuable. Such comparisons lie outside the scope of the present work, which is limited to the derivation and qualitative exploration of the simplified model. The numerical results illustrate the qualitative features (persistent cusp-wrinkle competition) that the model can produce. We will add a statement in the conclusions identifying quantitative validation as an important direction for future research. revision: no

standing simulated objections not resolved
  • Quantitative comparison of the model's nonlinear coefficients and chaotic regime to direct numerical simulations of the full conservation equations or to experiments in the M ~ sqrt(epsilon) regime

Circularity Check

2 steps flagged

Cubic coupling term matched to linear dispersion and hydro-diffusive number defined from invented area reduce nonlinear predictions to model inputs

specific steps
  1. fitted input called prediction [Abstract]
    "By identifying a cubic coupling term in the linear dispersion relation, representing the leading-order interaction between hydrodynamic expansion and diffusive transport, this framework moves beyond the traditional treatment of these instabilities in isolation."

    The cubic term is extracted from linear analysis and inserted into the nonlinear model; the predicted nonlinear regime (persistent chaotic competition between DL cusps and DT wrinkles) is therefore generated by a term whose form was chosen to reproduce the linear behavior rather than derived independently for the nonlinear regime.

  2. self definitional [Abstract]
    "a nonlocal stabilising term controlled by the hydro-diffusive number N = A/δ_L², where A is the hydro-diffusive area -- the characteristic area over which hydrodynamic and diffusive transport processes interact. This term remains active even when Markstein stabilisation vanishes."

    N is defined directly from the hydro-diffusive area A, which is itself introduced as the characteristic interaction area that the stabilizing term is intended to capture; the strength of the central stabilizing mechanism is therefore fixed by the same phenomenological construct it is meant to represent.

full rationale

The paper constructs a phenomenological model by identifying a cubic term from the linear dispersion relation to represent hydro-diffusive interaction and introducing the hydro-diffusive number N via a newly defined area A. These choices are then used to derive the evolution equation and its numerical solutions in the DL-DT crossover. The resulting claims about chaotic competition and accelerated growth rates therefore follow directly from the inserted terms rather than from an independent derivation, producing moderate circularity in the central unified framework. No external benchmarks or first-principles asymptotics are invoked to justify the nonlinear coefficients.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The framework rests on a phenomenological cubic coupling term and the definition of a new hydro-diffusive area A whose scale is not independently measured.

free parameters (1)
  • Markstein-number scaling
    Chosen as M ~ sqrt(epsilon) to place both instabilities at equal order in the crossover regime.
axioms (1)
  • domain assumption Cubic term in the linear dispersion relation represents the leading-order interaction between hydrodynamic expansion and diffusive transport
    Invoked to construct the simplified model beyond isolated DL and DT treatments.
invented entities (1)
  • hydro-diffusive area A no independent evidence
    purpose: Defines the hydro-diffusive number N that controls the nonlocal stabilizing term
    New characteristic area introduced to quantify the interaction scale between hydrodynamic and diffusive processes

pith-pipeline@v0.9.0 · 5583 in / 1358 out tokens · 54302 ms · 2026-05-15T00:17:58.851303+00:00 · methodology

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Reference graph

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