Kolmogorov’s Spectral Energy Dynamics in Thermoacoustic Turbulence

11:30AM at LWSN B134
Prof. Carlo Scalo, Purdue University
Kolmogorov’s Spectral Energy Dynamics in Thermoacoustic Turbulence

Periodic heating and cooling of a gas results in the generation of acoustic waves. In thermoacoustically unstable systems, sound waves drive such thermodynamic cycle, resulting in spontaneous generation and exponential self-amplification. In the talk I will discuss what happens when thermoacoustic instability is driven to the limit of shock-wave formation in highly nonlinear broadband acoustic perturbations. Before reaching such extreme conditions, single-frequency harmonic waves grow in amplitude and distort, generating higher frequencies, thereby smaller wavelengths, until the viscous limit is reached: the shock-wave thickness. The same spectral energy dynamics are observed, for example, in a boundary layer undergoing laminar-to-turbulent transition. In this case, Tollmien-Schlichting waves grow exponentially due to hydrodynamic instabilities, generating large eddies, which initiate an energy cascade, feeding progressively smaller eddies. The cascade ends when the eddy size reaches the viscous limit: the Kolmogorov length scale.

Gupta, Lodato and Scalo, AIAA (2017) have revealed for the first time the design of a device capable of generating self-sustaining thermoacoustic shock waves exhibiting spectral energy transfer dynamics identical to the ones predicted by Kolmogorov’s theory for turbulent flows. 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, marking the early and late stages of the energy cascade; (iii) Shock-wave dominated limit cycl}, where energy production is balanced by dissipation occurring at the captured shock-thickness scale. A companion time-domain nonlinear acoustic model has been developed to demonstrate the effect of macrosonic interactions between pressure and heat-flux fluctuations, responsible for saturation and thermodynamic asymmetries. Upon onset of energy cascade, k-th overtone of thermoacoustically amplified mode grows at rate k+1 times the thermoacoustic growth rate. Frequency power spectrum at the shock-dominated limit cycle exhibits -5/2 logarithmic slope, confirmed by a dimensional analysis inspired by Kolmogorov's theory of the hydrodynamic turbulent energy cascade.