# Spectral energy cascade in thermoacoustic shock waves

## TL;DR

This paper investigates the nonlinear development of thermoacoustic waves in a resonator, revealing a transition from harmonic growth to shock-dominated regimes through spectral energy cascade, using theoretical and numerical methods.

## Contribution

It introduces a comprehensive model of thermoacoustic wave regimes based on first principles, highlighting the nonlinear spectral cascade leading to shock formation.

## Key findings

- Identification of three distinct wave regimes
- Observation of nonlinear spectral broadening
- Demonstration of shock-wave limit cycle saturation

## Abstract

We have performed a theoretical and numerical investigation of thermoacoustically amplified quasi-planar nonlinear waves driven to the limit of shock-wave formation in a variable-area 2.58 m long looped resonator designed via eigenvalue analysis to maximize the growth rate of the second harmonic (265 Hz). High-order unstructured fully compressible Navier-Stokes simulations reveal the presence of three regimes: (i) Monochromatic harmonic growth, governed by linear thermoacoustics; (ii) Hierarchical spectral broadening, characterized by nonlinear energy cascade; (iii) Shock-wave dominated limit cycle, where energy production is balanced by dissipation occurring at the captured shock-thickness scale. We have modeled these regimes of thermoacoustic wave amplification based on first-principle governing equations and elucidated the physics of energy density production and saturation to a limit cycle.