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arxiv: 2604.26290 · v1 · submitted 2026-04-29 · ⚛️ physics.flu-dyn · physics.comp-ph

Scaling in Supersonic Turbulence: Energy Spectra and Fluxes using High-Fidelity Direct Numerical Simulations

Harshit Tiwari , Dhananjay Singh , Mahendra K. Verma , Rajesh Ranjan This is my paper

Pith reviewed 2026-05-07 12:43 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn physics.comp-ph
keywords supersonic turbulenceenergy spectracompressible turbulencedirect numerical simulationMach numberenergy cascaderotational compressive modesturbulent dissipation
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The pith

In supersonic turbulence, rotational energy spectra steepen toward minus-two scaling with rising Mach number while compressive spectra become shallower.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper presents high-resolution computer simulations of turbulent flows moving faster than sound to track how energy moves from large scales to small ones. The central finding is that higher flow speeds relative to sound speed change the power-law shape of energy spectra differently for two kinds of motion: swirling motions develop a steeper spectrum, while squeezing and expanding motions develop a shallower one. These shifts occur because energy moves across scales from swirling components to compressing ones and because pressure forces perform significant work on the fluid. The results matter for predicting behavior in contexts such as star-forming regions and high-speed vehicle flows, where accurate cascade descriptions affect mixing, heating, and shock formation.

Core claim

High-fidelity direct numerical simulations at 1024 cubed resolution of forced compressible turbulence at turbulent Mach numbers from 0.2 to 3.0 demonstrate that the rotational kinetic energy spectrum steepens from a minus five-thirds power law toward a minus two power law as the Mach number increases. The compressive energy spectrum becomes shallower, deviating from expected minus two scaling. These modifications result from a dominant cross-scale transfer of energy from swirling to compressing flow components in the inertial range, together with significant pressure work contributions. Scaling laws for the root-mean-square compressive velocity and the compressive energy flux follow those of

What carries the argument

The dominant cross-scale transfer of energy from swirling to compressing flow components within the inertial range, together with pressure forces performing work on the fluid.

If this is right

  • Rotational dissipation rates decrease as the turbulent Mach number rises.
  • Compressive dissipation rates and pressure work increase with rising Mach number.
  • Overall energy injection rates depend on the type of forcing applied rather than on Mach number.
  • Root-mean-square compressive velocity and compressive energy flux obey the same scaling relations found in classical shock-dominated turbulence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These mode-specific transfers suggest that large-scale models of sustained supersonic turbulence should track rotational and compressive contributions separately to capture the correct dissipation balance.
  • The Mach-dependent shift may alter mixing and heating rates in other high-speed compressible systems beyond the regimes simulated here.
  • Similar inter-component energy movement could appear in decaying turbulence or in flows with different forcing, providing a testable extension of the cascade picture.

Load-bearing premise

The computer simulations must resolve the smallest eddies and sharp discontinuities without the solution method introducing artificial changes to the measured energy distributions.

What would settle it

An independent higher-resolution simulation or laboratory measurement at high Mach number that finds the rotational energy spectrum staying near the minus five-thirds law instead of approaching minus two would disprove the reported change in scaling.

Figures

Figures reproduced from arXiv: 2604.26290 by Dhananjay Singh, Harshit Tiwari, Mahendra K. Verma, Rajesh Ranjan.

Figure 1
Figure 1. Figure 1: FIG. 1. Time evolution of (a) total kinetic energy view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Flow structures in subsonic and transonic turbulence at view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Flow structures in supersonic turbulence at view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. For view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. For view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. For forced compressible turbulence with (a) view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Plots of normalized energy transfer terms with turbulent Mach number view at source ↗
Figure 8
Figure 8. Figure 8: (b) presents the rotational and compressive RMS velocities (UR, UC ) and the maximum energy fluxes (ΠR, ΠC ). The compressive components align re￾markably well with Burgers equation predictions (solid and dashed black lines). A best-fit analysis yields UC = 0.54(∆V ) 1.08 ≈ ∆V √ 12 , (44) ΠC = 0.038 (∆V ) 3.10 ≈ (∆V ) 3 12L . (45) Interestingly, the rotational component exhibits similar scaling trends with… view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Scalability of view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Case study using supersonic Taylor–Green vortex simulation. (a) Time evolution of the volume-averaged kinetic view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Normalized pressure error along the view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Kelvin-Helmholtz instability diagnostics. (a), (b): Time evolution of volume-integrated kinetic energy view at source ↗
read the original abstract

Supersonic turbulence is vital to astrophysical and high-speed engineering flows, yet its energy transfer mechanisms remain poorly understood. We present high-resolution ($1024^3$) direct numerical simulations (DNS) of forced compressible turbulence across a range of turbulent Mach numbers ($M_t = 0.2$ to $3.0$). Using the GPU-accelerated solver \texttt{DHARA} with a seventh-order, low-dissipation Targeted Essentially Non-Oscillatory (TENO) scheme, we resolve both fine-scale eddies and sharp shock fronts. Our results reveal a fundamental shift in the energy cascade in the supersonic regime. As $M_t$ increases, the rotational kinetic energy spectrum steepens from a Kolmogorov-like $k^{-5/3}$ scaling toward a Burgers-like $k^{-2}$ scaling. Conversely, the compressive energy spectrum becomes shallower, deviating from Burgers scaling. We show that these spectral modifications are driven by a dominant cross-scale transfer of energy from solenoidal to compressive modes within the inertial range, alongside significant contributions from pressure dilatation. Scaling laws for the root-mean-square compressive velocity ($U_C$) and compressive energy flux ($\Pi_C$) are found to mirror classical Burgers turbulence. Finally, we show that while energy injection rates depend on forcing type rather than Mach number, increased $M_t$ leads to decreased rotational dissipation and increased compressive dissipation and pressure dilatation. These findings elucidate intermodal energy cascade mechanisms, advancing our understanding of energy transfers in supersonic turbulence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript reports 1024^3 DNS of forced compressible turbulence at Mt = 0.2–3.0 using the GPU-accelerated DHARA solver with a seventh-order low-dissipation TENO scheme. It claims that rotational kinetic energy spectra steepen from Kolmogorov-like k^{-5/3} toward Burgers-like k^{-2} as Mt increases, while compressive spectra become shallower; these changes are attributed to dominant cross-scale solenoidal-to-compressive energy transfer and pressure dilatation within the inertial range. Additional results include Burgers-like scaling for rms compressive velocity U_C and compressive flux Π_C, Mach-dependent shifts in dissipation (decreased rotational, increased compressive and pressure dilatation), and forcing-type dependence of injection rates.

Significance. If the reported spectral modifications and intermodal transfers are shown to be free of numerical artifacts, the work would provide valuable quantitative insight into energy cascade mechanisms in supersonic turbulence, relevant to astrophysical and high-speed engineering flows. The high-resolution DNS approach with explicit decomposition into rotational/compressive modes and direct computation of cross-scale fluxes represents a clear strength.

major comments (2)
  1. [Abstract and Methods] Abstract and Methods: The claim that the 1024^3 TENO7 simulations 'fully resolve both fine-scale eddies and sharp shock fronts' is not supported by any grid-convergence tests, resolution-sensitivity studies, or explicit checks that inertial-range spectral exponents and flux plateaus remain unchanged under increased resolution or higher scheme order. At Mt=3 the compressive discontinuities are thin; without such evidence the reported steepening of rotational spectra toward k^{-2} cannot be unambiguously attributed to physics rather than residual numerical dissipation.
  2. [Results (spectral analysis)] Results (spectral analysis): No error bars, uncertainty estimates, or details on the selection of inertial-range fitting windows are provided for the reported spectral exponents. This omission makes it impossible to assess the statistical significance of the claimed transition from k^{-5/3} to k^{-2} for rotational modes or the shallowing of compressive spectra.
minor comments (1)
  1. [Abstract] The abstract states that 'scaling laws for U_C and Π_C mirror classical Burgers turbulence' but does not quote the explicit functional forms or the range of Mt over which they hold; these should be stated explicitly in the abstract or early results section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed review of our manuscript. We address each major comment below and have revised the manuscript accordingly to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Abstract and Methods] Abstract and Methods: The claim that the 1024^3 TENO7 simulations 'fully resolve both fine-scale eddies and sharp shock fronts' is not supported by any grid-convergence tests, resolution-sensitivity studies, or explicit checks that inertial-range spectral exponents and flux plateaus remain unchanged under increased resolution or higher scheme order. At Mt=3 the compressive discontinuities are thin; without such evidence the reported steepening of rotational spectra toward k^{-2} cannot be unambiguously attributed to physics rather than residual numerical dissipation.

    Authors: We agree that explicit resolution-sensitivity analysis would strengthen the attribution of the observed spectral changes to physical mechanisms. The 1024^3 resolution and TENO7 scheme were chosen following standard practices for DNS of compressible turbulence at these Mach numbers to capture both eddies and shocks, but the submitted manuscript does not include direct comparisons. In the revised version we will add a dedicated subsection (in Methods or an appendix) presenting spectra and flux plateaus computed at 512^3 and 1024^3 resolutions. This will demonstrate that the inertial-range exponents and the reported steepening of rotational spectra remain unchanged, thereby supporting that the trends are not dominated by residual numerical dissipation. revision: yes

  2. Referee: [Results (spectral analysis)] Results (spectral analysis): No error bars, uncertainty estimates, or details on the selection of inertial-range fitting windows are provided for the reported spectral exponents. This omission makes it impossible to assess the statistical significance of the claimed transition from k^{-5/3} to k^{-2} for rotational modes or the shallowing of compressive spectra.

    Authors: We concur that quantitative uncertainty measures and transparent fitting criteria are necessary for assessing the robustness of the Mach-dependent spectral transitions. In the revised manuscript we will report error bars on all fitted exponents, obtained from the standard deviation across multiple independent time windows in the statistically stationary regime. We will also explicitly state the wavenumber intervals chosen for each fit (identified where the energy flux is approximately constant) and include a brief sensitivity test showing how the exponents vary with modest changes in the fitting window. These additions will allow readers to evaluate the statistical significance of the transition from Kolmogorov-like to Burgers-like scaling. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results emerge from DNS and post-processing

full rationale

The paper's central claims on spectral steepening, intermodal energy transfer, and Burgers-like scalings for compressive quantities are obtained by integrating the compressible Navier-Stokes equations at 1024^3 resolution with the DHARA TENO7 solver, then computing rotational/compressive decompositions, spectra, and fluxes directly from the velocity and pressure fields. No step defines a quantity in terms of a result it is later used to predict, no fitted parameters are renamed as predictions, and no load-bearing uniqueness theorems or ansatzes are imported via self-citation. The observed modifications to the cascade are therefore independent outputs of the numerical experiment rather than tautological restatements of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the compressible Navier-Stokes equations and the numerical properties of the TENO scheme; no new physical entities are introduced and no parameters are fitted to produce the target spectra.

axioms (2)
  • standard math Compressible Navier-Stokes equations govern the evolution of density, velocity, and pressure in the simulated flows
    Invoked implicitly as the basis for all DNS of compressible turbulence
  • domain assumption The seventh-order TENO scheme provides sufficient low-dissipation resolution of both turbulence and shocks at the employed grid resolution
    Required for the claim that observed spectral modifications are physical rather than numerical

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Reference graph

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