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The Effect of Heat Loss During the Early Stages of Flame Propagation and Tulip Flame Formation

This study utilizes direct numerical simulations of fully compressible reactive Navier-Stokes equations with detailed chemistry to demonstrate how wall heat losses and geometric confinement jointly influence early-stage flame propagation and the formation of tulip flames in hydrogen-air combustion within confined channels.

Original authors: Mikhail A. Liberman, Chengeng Qian

Published 2026-07-17

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CC0 1.0

Original authors: Mikhail A. Liberman, Chengeng Qian

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a campfire. The flames dance, flicker, and spread, but they don't just move randomly; they follow invisible rules of physics that dictate how fast they grow and what shape they take. This is the world of combustion science, a field that helps us understand everything from how car engines run to how we can prevent dangerous explosions. At the heart of this study is a specific, fascinating dance called the "tulip flame." When a fire starts at one end of a long tube and rushes toward the other, it doesn't stay a simple, round ball. Instead, it stretches out like a finger, then suddenly flips inside out, curling backward to look exactly like a blooming tulip flower. Scientists have known about this shape for over a century, but figuring out exactly why it happens and what factors change its behavior has been a tricky puzzle. One major question has been: does the heat escaping into the walls of the tube change the story, or is the flame's shape determined purely by the air and gas moving around?

This paper dives deep into that question using powerful computer simulations to act as a virtual laboratory. The researchers, Mikhail A. Liberman and Chengeng Qian, built a digital model of a flame racing through a narrow channel, but with a twist: they didn't just pretend the walls were perfect insulators. Instead, they programmed the walls to actually "suck" heat out of the fire, letting it travel through the solid material and escape into the air outside. They tested this with different types of walls (like quartz and copper) and different channel lengths to see how much the "leaking" heat changed the tulip's shape.

Here is what they found in their virtual experiments. First, they confirmed that the tulip flame is indeed a hydrodynamic phenomenon—meaning it's caused by the movement of the gas and pressure waves, not by the heat leaking out. In fact, the paper argues against older ideas that suggested the flame's shape was caused by chemical instabilities or the way heat diffuses through the gas. The simulations show that the "inversion" (the moment the flame flips backward) happens because of rarefaction waves, which are like sound waves that pull the flame back, and this happens so fast that heat loss doesn't have time to stop it.

However, the heat loss does play a subtle role. The researchers discovered that while the heat escaping through the walls doesn't stop the tulip from forming, it does act like a gentle brake. In their simulations, the flame moved slightly slower when the walls were allowed to cool it down compared to when the walls were perfectly insulated. This effect was more noticeable in cylindrical tubes (like a real pipe) than in flat, 2D channels because a pipe has more wall surface area relative to the volume of hot gas, allowing more heat to escape.

The study also tackled a common shortcut scientists used to take: assuming the walls stay at a constant, cold temperature (like an ice block) or that they are perfectly insulated. The simulations showed that neither of these simple models is quite right. In reality, the inner surface of the wall heats up quickly as the fire passes, but the heat doesn't have enough time to travel all the way through a thick wall (like a 2 mm thick quartz or copper wall) to the outside during the early stages of the flame's journey. This means the outside of the wall stays cool, but the inside gets hot, creating a complex temperature gradient that simple models miss.

Interestingly, the paper suggests that if the walls were extremely thin (less than 0.1 mm for copper), the heat would escape much faster, potentially changing the flame's behavior more dramatically. But for the thicker, more realistic walls used in the study, the heat loss was a minor player in the early stages, only becoming a bigger factor if the flame were to travel for a much longer time or if the mixture were less reactive. Ultimately, the paper concludes that while heat loss is a real physical process that slows the flame down slightly, the spectacular "tulip" shape is a robust feature of fluid dynamics that forms regardless of whether the walls are hot, cold, or somewhere in between. The flame flips because of the physics of the gas itself, not because the walls are cooling it down.

Technical Summary: The Effect of Heat Loss During the Early Stages of Flame Propagation and Tulip Flame Formation

Problem Statement
The propagation of premixed flames in confined channels is a fundamental process relevant to explosion safety, internal combustion engines, and pollutant emissions. A critical phenomenon in this process is the formation of the "tulip flame" (TF), where a flame front initially convex toward the direction of propagation inverts to become concave. While recent studies have identified the tulip flame formation as a purely hydrodynamic phenomenon driven by flame-generated rarefaction waves, the influence of wall heat losses on these early stages remains a subject of investigation. Previous numerical studies often relied on simplified approaches, such as single-step chemical kinetics and isothermal wall boundary conditions, which fail to capture the complex thermal interactions between high-temperature combustion products and channel walls. This work addresses the need for a comprehensive numerical investigation of heat-loss effects using a more realistic physical model that accounts for convective, conductive, and radiative heat transfer mechanisms.

Methodology
The authors employed direct numerical simulations (DNS) of the fully compressible, multicomponent reactive Navier–Stokes equations coupled with conductive heat transfer within the channel walls. The study utilized a high-order numerical method (fifth-order WENO finite-difference scheme) and adaptive mesh refinement (AMR) to resolve flame structures, pressure waves, and their interactions.

Key methodological features include:

Chemical Model: A detailed multistep chemical kinetic model (21 reactions, 9 species) for stoichiometric hydrogen–air combustion, replacing the single-step models used in prior heat-loss studies.
Heat Transfer Modeling: Unlike previous works using isothermal boundaries, this study models heat loss through a three-stage process:
1. Convective heat transfer from hot combustion products to the inner wall surface (modeled via Newton's cooling law with a calculated Nusselt number).
2. Thermal conduction through the solid wall (solving the thermal conduction equation in the wall material).
3. Convective and radiative heat losses from the outer wall surface to the surroundings.
Geometries: Simulations were conducted in two-dimensional (2D) channels and cylindrical tubes with varying aspect ratios ( $L/D_{in} = 6$ and $12$). Wall materials included quartz and copper with varying thicknesses (primarily 2 mm).
Boundary Conditions: Symmetry conditions were applied at the centerline, no-slip conditions at the walls, and adiabatic conditions at the closed ignition end. Gravitational effects were neglected as their timescale exceeds that of tulip flame formation.

Key Results
The simulations compared flame dynamics under adiabatic wall conditions against those with realistic heat losses:

Impact on Early Flame Dynamics: During the initial stages of propagation (approximately the first 1–2 ms), heat loss has a relatively minor effect on the overall flame acceleration and the timing of flame front inversion. The time at which the flame front begins to invert (the onset of tulip flame formation) remains largely consistent between adiabatic and heat-loss cases for the geometries studied.
Flame Velocity and Pressure: While heat loss causes a slight reduction in flame tip velocity immediately before inversion, the effect is subtle in the early stages. However, in cylindrical tubes, the surface-to-volume ratio is higher, leading to more significant heat losses that suppress pressure rise more effectively than in 2D channels.
Wall Thermal Conductivity and Thickness:
- For thick walls (2 mm), thermal penetration depth is insufficient to reach the outer wall surface during the early propagation phase, even for highly conductive copper. Consequently, outer surface heat losses (convection/radiation) are negligible in this timeframe.
- The inner wall temperature varies significantly over time, contradicting the assumption of isothermal walls often used in simplified models.
- Increasing the convective heat transfer coefficient (simulating higher Reynolds numbers) accelerates flame front inversion by reducing the axial flame velocity more rapidly.
Dimensionality Effects: Flame front inversion occurs faster in cylindrical (axisymmetric) and 3D geometries compared to 2D channels due to geometric differences in flame surface area and heat release rates. However, the relative impact of heat loss is more pronounced in cylindrical tubes than in 2D channels due to the larger surface-to-volume ratio, despite the shorter propagation time.
Distorted Tulip Flames: The formation of distorted tulip flames (DTF) and the subsequent evolution are influenced by the intensity of pressure waves. Heat loss slightly alters the timing and intensity of these interactions but does not fundamentally change the hydrodynamic mechanism of inversion.

Significance and Claims
The paper claims that the commonly used simplified models—specifically those employing single-step chemistry and isothermal wall boundary conditions—do not adequately describe the actual thermal effects during early flame propagation. The study demonstrates that:

Realistic Modeling is Necessary: The inner wall temperature changes considerably during the early stages, making the isothermal assumption invalid for capturing transient thermal interactions.
Hydrodynamic Dominance: The mechanism of tulip flame formation remains a purely hydrodynamic phenomenon driven by rarefaction waves, even in the presence of heat loss. This distinguishes freely propagating flames from those in heat-recirculating microtubes, where flame inversion can be a purely thermal phenomenon.
Limitations of Boundary Conditions: The authors note that adiabatic conditions suppress necessary heat loss (overestimating acceleration), while isothermal conditions assume walls do not heat up (underestimating acceleration). Real experimental scenarios lie between these extremes.
Future Relevance: While heat loss has a minor influence on the early stages of flame propagation in the studied configurations, the authors suggest its effects may become much more pronounced at later stages, such as during deflagration-to-detonation transition (DDT), where longer time scales allow for greater heat accumulation.

The study concludes that for thick-walled channels, outer surface heat losses are negligible during the early stages, and the primary thermal interaction is the transient heating of the inner wall surface, which must be resolved dynamically rather than assumed constant.

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