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Direct Numerical Simulation of Thermoacoustically Unstable Flames Via Non-Drifting Acoustic Delay Characteristic Boundary Conditions

Benjamin L. Cookman, Jack R. C. King, Raphaël C. Assier, Steven J. Lind

arXiv:2609.05656v1physics.flu-dyncs.NAmath.NAphysics.comp-ph

TL;DR

Thermoacoustic-instability simulations remain scarce because resolving flame dynamics and the surrounding acoustics is computationally demanding. This paper combines delayed-acoustic boundary conditions with drift control to simulate idealized premixed flames in a truncated DNS region. The simulations show primary-instability saturation and, in the 10 cm tube, periodic parametric flame fingering.

  • Problem

    Thermoacoustic-instability DNS remains scarce because it must resolve thin flame layers and acoustics across the combustion system at high computational cost.

    Source: §3 · p. 6

  • Method

    The paper truncates the acoustic domain and models delayed wave reentry with ADCBC, while APILR controls mean inflow and outflow drift.

    Source: §1 · p. 3, §6 · p. 19

  • Results

    In both tube cases primary instability flattened the flames and saturated in growth; the 10 cm case also showed subharmonic cellular parametric fingering.

    Source: §6 · p. 19

  • Takeaways & Limitations

    The boundary methods support DNS of thermoacoustic flame instabilities using a domain limited to the flame and surrounding hydrodynamics.

    Source: §3 · p. 6, §6 · p. 19

    The secondary-instability analysis assumes a standing acoustic field represented by cosine forcing at a specified amplitude and frequency.

    Source: §5.2 · p. 16

Abstract

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Thermoacoustic instability in premixed flames results from the coupling between the flame's heat release, as determined by combustion parameters, and surrounding acoustics, as determined by combustor geometry. A primary instability results in flame flattening as intrinsic flame instability modes are stabilised. Secondary thermoacoustic instability results in a parametric flame instability and drastic growth of acoustic amplitudes. Due to their relative expense, numerical simulations of these phenomena remain scarce. In this work, Direct Numerical Simulations (DNS) of thermoacoustically unstable idealised premixed flames in a tube with acoustically closed upstream and open downstream ends are presented. Results herein demonstrate nonlinear saturation of the primary instability as the flame flattens as well as an oscillating flame fingering characteristic of the unsteady Rayleigh-Taylor effect. To reduce computational cost, we perform DNS only on the region surrounding the flame. Acoustics at in- and outflows are described using the Navier-Stokes Characteristic Boundary Condition (NSCBC) method to model their delayed reentry into the domain in a formulation referred to as the Acoustic Delay Characteristic Boundary Condition (ADCBC) method. A new Averaged Proportional and Integral Linear Relaxation (APILR) method is also introduced, which modifies the Classic Linear Relaxation (CLR) method to maintain time-averaged values of inflow velocity and outflow pressure. Here, an integral control term is used to remove non-zero equilibrium time-averaged inflow velocities which impinge control over flame position. Both new methods demonstrate their capability in inert and counterflow flames test cases. These methods enable the numerical simulation of combustion instabilities at significantly reduced computational expense.

1 Introduction

Thermoacoustic instabilities couple flame heat-release oscillations with chamber acoustics, but their nonlinear behavior remains difficult to predict and costly to simulate. The paper introduces delayed-acoustic boundary conditions and drift control to enable reduced-domain DNS of these instabilities.

1 Introduction

  • Thermoacoustic instability arises when flame heat-release oscillations couple in phase with chamber acoustics; this work focuses on geometry-dependent extrinsic modes.Reflected acoustic waves connect the flame to the combustion chamber’s acoustic modes.Source: §1 · p. 1
  • Primary instability can flatten and stabilize an intrinsically unstable flame, whereas secondary parametric instability can rapidly grow flame curvature and acoustic amplitudes.The secondary regime forms an oscillating cellular flame front and can increase flame surface area and flow speed.Source: §1 · pp. 1–2
  • Existing models capture parts of thermoacoustic behavior, but idealized flame descriptions can exclude complex nonlinear parametric structures observed in experiments.The paper contrasts transverse flame-function models with level-set descriptions and nonlinear flame-response models.Source: §1 · p. 2
  • DNS data remain scarce because simulations must resolve thin reactive and diffusive layers, full-domain acoustics, and restrictive stability-limited time steps.Low-Mach combustion codes also preclude the acoustic waves relevant to these instabilities.Source: §1 · p. 2
  • ADCBC models acoustics beyond the truncated flame region through delayed wave reentry, while APILR controls drift to maintain target mean inflow velocity and outflow pressure.The integral term removes non-zero equilibrium mean inflow drift, helping contain the flame in the DNS region over longer simulations.Source: §1 · p. 3

2 Navier-Stokes Characteristic Boundary Conditions

This section establishes the reacting-flow equations and the NSCBC framework used to separate outgoing waves from boundary conditions for incoming waves. It then describes non-reflecting boundaries and CLR relaxation, including the low-frequency reflection trade-off.

2.1 Governing Equations

  • The model uses two reacting species, reactant R and product P, with constant fluid and reaction properties; several transport and radiation effects are neglected.The paper notes that the setup can be extended to multistep, multispecies chemistry and temperature-varying transport.Source: §2.1 · pp. 3–4

2.2 Characteristic Waves

  • NSCBC applies a locally one-dimensional inviscid approximation to decompose the reacting-flow equations into characteristic waves along the boundary normal.The characteristic fields leaving the domain are evaluated from interior one-sided differences; incoming fields are specified by a physical model.Source: §2.2 · pp. 4–5

2.3 Non-Reflecting Boundaries

  • Non-reflecting NSCBC selects incoming acoustic-wave terms to make the reflection coefficient zero, while the remaining shear and species waves are set separately.The paper assumes no body force or reactions at the boundary for the stated non-reflection terms.Source: §2.3 · p. 5

2.4 Relaxation Terms

  • CLR adds proportional relaxation toward target inflow velocity and outflow pressure, but reflects low-frequency waves nearly fully below its cutoff frequency.Reflection decreases as incident frequency rises relative to the computational domain’s characteristic acoustic frequency.Source: §2.4 · p. 6

3 Acoustic Delay Model

ADCBC models the acoustics outside a flame-focused DNS region as delayed reentry at the in- and outflow boundaries. Tests show accurate reflection and standing-wave behavior, with peak-position error decreasing as the sampling interval decreases.

3 Acoustic Delay Model

  • ADCBC represents each truncated acoustic region by delayed wave reentry, with τU,D ≃ 2LU,D/cU,D at low Mach number.The model assumes constant density and sound speed within each acoustic region and one-dimensional acoustics; it neglects attenuation over the modeled delays.Source: §3 · p. 6

3.1 Acoustic Delay Characteristic Boundary Conditions

  • Uniform boundary averaging makes one one-dimensional acoustic approximation per boundary, valid when computational width W ≪ Ltube.Boundary-velocity sign determines whether the corresponding inflow or outflow conditions are applied.Source: §3.1 · p. 7

3.1.1 Discretisation

  • The solver approximates outgoing characteristics as piecewise constant between samples and stores sample times and values in a memory buffer.The sample interval is bounded under dynamic time stepping; unless otherwise stated, δtsample = 4δt0.Source: §3.1.1 · p. 7

3.2 Preliminary Tests

  • Preliminary tests use one-dimensional, equispaced discretisations, and boundary averaging is considered reasonable when transverse acoustic eigenmodes are negligible.The tests use δx = 30 µm; the same fluid properties are used in all examples.Source: §3.2 · pp. 7–8

3.2.1 Reflected Travelling Acoustic Wave

  • After a single up- and downstream reflection, tests with and without ADCBC show accurate reflection of the pressure disturbance.The ADCBC memory buffers also reconstruct the stored one-dimensional acoustic field from boundary time series.Source: §3.2.1 · p. 8, Figure 2 · p. 8

3.2.2 Convergence over Many Reflections

  • After 20 reflections, ADCBC peak displacements converge to less than 3δx from full DNS, while the acoustic disturbance retains its structure.In the worst case, each upstream reflection imposes an approximate 0.05 mm peak displacement; this is negligible compared with acoustic wavelengths in a 10 cm domain.Source: §3.2.2 · p. 9

3.2.3 Standing Acoustic Wave

  • In the standing-wave test, the desired conditions ua,U = 0 and pa,D = 0 hold, with no noticeable amplitude change or excitation of other modes.The correct standing-wave frequency is reproduced.Source: §3.2.3 · p. 10, Figure 4 · p. 10

4 Drift Control

The section develops APILR to control drift in acoustic boundary conditions while retaining delayed acoustic treatment. Its standing-wave tests show converging inflow drift and a small residual outflow drift.

4 Drift Control

  • ADCBC drift accumulates over time, increases nonlinearly with acoustic amplitude, and is negligible when the truncated boundary lies at an acoustic pressure or velocity node.The drift rate also grows with acoustic frequency; the following model targets drift under persistent acoustics.Source: §4 · pp. 10–11

4.1 A New Linear Relaxation Model

  • For frequencies above the tube frequency, the moving-average gain is less than one and decreases as C increases; it is ≲10^-3 at acoustic DNS frequencies.The moving average acts as a delayed low-pass filter on the signal entering the relaxation terms.Source: §4.1 · p. 11
  • APILR combines integral control with a moving-average-filtered signal to suppress acoustic content and address the nonzero equilibrium error of proportional control.The moving average spans tavg = Cttube, with C > 1; the formulation modifies the relaxation terms used with ADCBC.Source: §4.1 · pp. 11–12

4.2 Non-Drifting Standing Acoustic Waves

  • After t ≳20ttube, APILR brings inflow drift below 0.05pamp; outflow response is similar but retains small equilibrium drift.The residual outflow drift is reported as small enough to have negligible impact on the reacting flows simulated next.Source: Figure 8 · p. 13, §4.2 · p. 13

4.1 A New Linear Relaxation Model

  • Against numerical reflection-coefficient results, the model shows excellent gain agreement across tested frequencies, while phase data are inconsistent in some cases.Figure 7 compares CLR, PILR, and APILR; the authors state the phase discrepancy has little effect because of the overall gain reduction.Source: §4.1 · pp. 12–13, Figure 7 · p. 12

5 Simulation of Thermoacoustically Unstable Counterflow Flames

DNS of idealised premixed flames in closed-open tubes combines a flame-centered domain with acoustic boundary treatment to study thermoacoustic instabilities. Primary growth saturates as flames flatten; the 10 cm case also develops secondary parametric flame fingering, while its onset is not resolution-converged.

5 Simulation of Thermoacoustically Unstable Counterflow Flames

  • The flame-centered 2 cm DNS domain represents a free flame in either a 10 cm or 100 cm closed-open tube, truncating 80% or 98% of tube length, respectively.The modeled flame is not allowed to flash back or advect through the outflow boundary.Source: §5 · pp. 13–14

5.1 Primary Instability; 5.2 Secondary Instability

  • Primary acoustic growth saturates as flames flatten; the 10 cm tube later develops parametric instability with subharmonic cellular flame fingering.The fingering includes oscillation associated with the unsteady Rayleigh-Taylor effect.Source: §5.1 · p. 14, §5.2 · p. 18

5.1 Primary Instability

  • In the 100 cm tube, primary growth reaches quasiperiodic limit-cycle oscillations without secondary instability; in the 10 cm tube, fundamental-mode growth continues after flattening.The origin of the 10 cm case's plateau growth remains unknown.Source: §5.1 · p. 14
  • A 0.01 m s−1 increase in mean inflow velocity would advect the flame 1 cm in 1 s, although oscillation around the 0.20 m s−1 target avoids that drift during primary instability.The boundary conditions remain stable when normal boundary velocity changes sign.Source: §5.1 · p. 15

5.2 Secondary Instability

  • Trial DNS places the intrinsically stable flat-flame width limit at 1.5–2.0 mm, rather than the approximately 0.5 mm predicted from calculated Markstein numbers.The stability analysis treats acoustic amplitude as a standing-wave forcing; parametric instability requires nonzero amplitude and produces subharmonic flame motion.Source: §5.2 · pp. 16–17
  • The 10 cm tube's 1500 Hz fundamental mode excites the predicted k = 3 parametric response, and complex nonlinear flame structure forms after three subharmonic periods.The flame exhibits fingering associated with the unsteady Rayleigh-Taylor effect; turbulence does not develop before the flame reaches the truncated domain's left side.Source: §5.2 · pp. 17–18

5.3 Convergence Results

  • At δx = 36 µm, the DNS does not resolve Darrieus-Landau wrinkling; finer tested resolutions give approximately the same acoustic growth rate, but parametric onset is not converged within one acoustic period.The δx = 18 µm results are retained because their temperature fields avoid the Gibbs phenomenon present at coarser resolutions.Source: §5.3 · p. 18

6 Conclusions

The conclusions present ADCBC and APILR as boundary methods for flame-centered DNS, report their validation and thermoacoustic simulations, and outline broader flame applications.

6 Conclusions

  • ADCBC models truncated upstream and downstream tube acoustics by delaying outgoing waves before their reentry into the flame-centered DNS domain.Travelling-wave tests reproduced signals after repeated reflections, and standing-wave structure persisted over many acoustic periods.Source: §6 · p. 19
  • APILR combines moving-average proportional control with an integral term to suppress velocity and pressure drift and keep a flattened flame stationary.The integral term suppresses non-zero equilibrium drift, including when acoustic amplitude rises after flame flattening.Source: §6 · p. 19
  • In 10 cm and 100 cm closed-open tubes, the flame flattened as primary-instability growth saturated; the 10 cm case also showed periodic fingering at k = 3.The k = 3 response followed the Mathieu-equation prediction when the planar stability limit was calibrated to γc = 2.Source: §6 · p. 19
  • The authors identify more realistic multispecies flames and combined thermoacoustic, turbulent, and thermodiffusive instabilities as directions for applying these boundary conditions.They also propose using stored acoustic values to model frequency-dependent reflection.Source: §6 · p. 19

A Calculation of Acoustic Eigenmodes

The appendix calculates acoustic eigenmodes for a closed-open tube containing a flame interface. It compares the resulting harmonics with equivalent homogeneous-tube modes.

A Calculation of Acoustic Eigenmodes

  • The analysis nondimensionalizes position by tube length, velocity by laminar flame speed, time by acoustic time, and temperature by upstream temperature.Upstream and downstream far-field temperatures are set to 1 and 1 + q, respectively.Source: A Calculation of Acoustic Eigenmodes · p. 21
  • The tube is modeled as nonconvected, with uniform background pressure across the flame and closed-upstream, open-downstream boundary conditions.The flame divides the acoustic fields into upstream and downstream regions.Source: A Calculation of Acoustic Eigenmodes · p. 22
  • The calculation uses low-Mach jump conditions at a passive flame interface, neglecting acoustic radiation from a flame transfer function.The low-Mach assumption yields the velocity jump condition; the model also notes that q = 0 or flame positions at either endpoint recover isothermal acoustics.Source: A Calculation of Acoustic Eigenmodes · p. 22
  • Nontrivial eigenmodes are obtained by satisfying the boundary constraints for complementary solutions on both sides of the flame.The solutions introduce four additional degrees of freedom for each side of the flame for each eigenmode.Source: A Calculation of Acoustic Eigenmodes · p. 22
  • The true harmonics oscillate around the predicted quarter- and three-quarter-wave modes of an equivalent homogeneous tube with the same acoustic period.The eigenmode roots satisfy det(A) = 0 and are found using Newton’s method.Source: A Calculation of Acoustic Eigenmodes · p. 22

B Mathieu Equation Stability Analysis

The appendix analyzes a periodically varying stability system by evolving its matrix solutions over one period. Stability is assessed from the resulting matrix eigenvalues and represented as doubling-rate heatmaps.

B Mathieu Equation Stability Analysis

  • The periodic system is written in matrix form, and its evolution over one period defines the matrix whose eigenvalues determine stability.The matrix of eigenvectors and eigenvalues is used to express solutions after n periods.Source: B Mathieu Equation Stability Analysis · pp. 22–23
  • Stability is determined by the largest eigenvalue magnitude σ of the period matrix: the system is stable for σ < 1 and unstable for σ > 1.The period matrix is computed by integrating from the identity over one period using fourth-order Runge-Kutta with δt = ta/100.Source: B Mathieu Equation Stability Analysis · p. 23
  • Heatmaps report the system’s doubling rate in inverse-time units, defined as ω̄ = f log2(σ).The rate is redimensionalized to units of Hz.Source: B Mathieu Equation Stability Analysis · p. 23
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