Nonlinear responses of the premixed V-flame subjected to dual-frequency disturbances

The two-way interaction between the unsteady flame heat release rate (HRR) and acoustic waves can lead to combustion instability within combustors. Previous studies have typically characterised premixed flame responses to pure harmonic forcing, assuming dynamically linear or weakly nonlinear behaviour, to quantify flame–acoustic interactions. By combining third-order asymptotic analysis with numerical simulations of the upper G <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>G</mml:mi> </mml:math> $G$ -equation, this study investigates the nonlinear response of laminar premixed V-flames subjected to dual-frequency velocity perturbations ( upper S t 1 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>S</mml:mi> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> $St_1$ and upper S t 2 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>S</mml:mi> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> $St_2$ , dimensionless frequencies). The positive correlation between disturbance propagation speed u Subscript c <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> $u_c$ and frequency upper S t <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>S</mml:mi> <mml:mi>t</mml:mi> </mml:math> $St$ is captured by integrating a velocity-potential model with calibration against existing experimental data. The mechanism by which the disturbance at one forcing frequency, say upper S t 2 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>S</mml:mi> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> $St_2$ , affects the flame dynamic response at the other forcing frequency, upper S t 1 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>S</mml:mi> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> $St_1$ , is studied in detail. The perturbation at upper S t 2 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>S</mml:mi> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> $St_2$ </jats:inline-