Covariant Chu-Kovasznay Decomposition: Resolving Thermodynamic Ambiguity in Compressible Flows

Abstract

We establish the Covariant Chu--Kovasznay Decomposition (CCKD), a geometric framework that resolves thermodynamic ambiguity in compressible mode content by formulating the decomposition on the effective acoustic spacetime. Enforcing orthogonality in the covariant Chu energy norm, we show that shock--turbulence interaction, often treated as a scattering source, is, in the idealized linear, inviscid setting, a near-unitary (Chu-isometric) scattering map constrained by conservation of covariant Chu-energy flux. In the canonical Shu-Osher problem, CCKD characterizes the shock as a thermo-acoustic lens, mathematically demonstrating that the transfer of entropy fluctuations into sound follows a geometric blue-shift ($k_{\mathrm{out}}=Λk_{\mathrm{in}}$) analogous to gravitational blue-shift. Thus, while the mean flow produces entropy across the shock, the fluctuation mapping is information-preserving on the retained subspace; practical information loss arises from noise, truncation, and model mismatch, not shock physics.

Figures (10)

• Figure 1: Kinematic validation: Vorticity isolation in a thermal lens. A refractive background bends rays without creating true vorticity. Euclidean post-processing misinterprets curvature as solenoidal content; the covariant projection suppresses this geometric leakage.
• Figure 2: Geometric leakage near an analogue black hole. A horizon geometry amplifies Euclidean projection errors, identifying curvature as vorticity. The covariant projection suppresses this artifact, with residual leakage confined to the regularized interior (inside the horizon).
• Figure 3: Shu--Osher benchmark ($M_1=3$). Top: density profile of the shock--entropy interaction. Bottom: modal Chu-energy fluxes ($J_{\mathcal{S}},J_{\mathcal{P}}$) and total flux $J_{\mathrm{tot}}$. The shock redistributes flux from entropic to acoustic while $J_{\mathrm{tot}}$ remains anchored to the upstream plateau (defect $\varepsilon_J \approx 0.31\%$), consistent with a passive thermo-acoustic lens.
• Figure 4: Shock-wave holography as deterministic scattering. Top: Physical-space view where the shock functions as a thermo-acoustic lens, deterministically encoding upstream entropy into downstream acoustic signatures. Bottom: Fourier-space view illustrating the geometric blue-shift ($k \mapsto \Lambda k$), which establishes an invertible scattering law for exact source reconstruction via transmission and phase factors.
• Figure 5: Spectral signature of near-isometric scattering. Top: Incident entropy mode $|S\rangle$ dominated by low-wavenumber content. Bottom: Scattered acoustic mode $|P\rangle$ exhibiting a distinct spectral blue-shift at the shock interface ($x \approx 0.75$). Here the ket notation denotes the modal coefficient vectors in the chosen spectral basis. This frequency up-conversion indicates the shock functions as a compressive thermo-acoustic lens, mathematically analogous to a gravitational blue-shift.