A gauge identity for interscale transfer in inhomogeneous turbulence

Abstract

The local definition of interscale energy transfer is missing in inhomogeneous turbulence research. This manifests as a discrepancy between the subgrid-scale production $Π^{\mathrm{SGS}}$ and the increment-based transfer density $Π^{\mathrm{KHMH}}$. Here, this missing definition is found by identifying a gauge freedom in the spatial transport of energy, yielding the identity: $Π^{\mathrm{SGS}} = \int G_\ell Π^{\mathrm{KHMH}} \, d\boldsymbol{r} + \nabla \cdot \boldsymbol{J}_{\mathrm{gauge}}$. The formulations are proven to differ strictly by the divergence of the current $\boldsymbol{J}_{\mathrm{gauge}}$. Validation against the analytical Womersley solution confirms the identity to within machine precision ($<10^{-14}$). The current $\boldsymbol{J}_{\mathrm{gauge}}$ is identified as the mechanism for redistribution toward compliant boundaries. Both measures are shown to converge to the unique Duchon--Robert dissipation $D(u)$, unifying the theoretical framework for non-stationary turbulence.

Figures (4)

• Figure 1: Phase-space visualization of the gauge transformation. In homogeneous regions (right), energy cascades vertically ($\Pi^{\mathrm{SGS}} \approx \Pi^{\mathrm{KHMH}}$). In inhomogeneous near-wall regions (left), the SGS flux deviates from the vertical cascade. This deviation is the gauge current $\bm{J}_{gauge}$. The divergence of this current at the compliant boundary ($x=0$) reconciles the energy budget.
• Figure 2: Divergence of Interscale Diagnostics in Womersley Flow. (a) Radial profiles of velocity amplitude $|\hat{u}^*|$ for various Womersley numbers ($\alpha$), illustrating the kinematic inhomogeneity near the wall ($y^* \to 1$). (b) The "Gauge Gap" distribution, defined as $\Pi^{\mathrm{SGS}} - \Pi^{\mathrm{KHMH}}$. This gap is non-zero and concentrated in the near-wall shear layer, demonstrating that standard interscale diagnostics diverge in inhomogeneous regions.
• Figure 3: Validation of the Gauge Identity (Theorem 2). (a) Energy flux density budget for $\alpha=10$. The red shaded region represents the divergence of the gauge current ($\nabla \cdot \bm{J}_{gauge}$), which perfectly accounts for the difference between the SGS production ($\Pi^{\mathrm{SGS}}$) and the local interscale transfer ($\Pi^{\mathrm{KHMH}}$). (b) The equation residual for the gauge identity. The residual remains within machine precision ($< 10^{-14}$) across the entire domain, confirming the identity is exact.
• Figure 4: Physics of the Gauge Current. (a) Radial distribution of the gauge flux magnitude $|J_{gauge}|$, showing the spatial drift of subgrid energy toward the compliant boundary ($y^*=1$). (b) Scaling of the peak gauge flux with the Womersley number $\alpha$. The linear increase demonstrates that the gauge mechanism is dominant in high-frequency, pulsatile flows such as arterial hemodynamics.

Theorems & Definitions (14)

• Definition 1: Admissible averaging
• Theorem 1: Gauge Uniqueness
• proof
• Remark 1: Boundary conditions and Physicality
• Lemma 1: Germano-type Identity
• proof
• Theorem 2
• proof
• Corollary 1: The Boundary Gauge Theorem
• proof