A Unified Spectrum for Turbulence in Microfluidic Flow

Abstract

We present a predictive master spectrum describing turbulence-like flows in microfluidic systems. Extending Pao's viscous-range closure, the model introduces (i) an adaptive inertial-range slope dependent on measurable dimensionless numbers and (ii) a physics-specific cutoff that captures entropy-producing sinks such as electrokinetic forcing, compliant walls, active stresses, and interfacial tension. This formulation unifies turbulence regimes -- electrokinetic, active, interfacial, and compressible -- within one compact expression. Comparison with reported data reproduces both spectral slopes and dissipation cutoffs while requiring only global observables (velocity, viscosity, Taylor microscale, and forcing strength). The framework provides a design-level predictive tool for turbulent microflows prior to computationally heavy DNS or CFD.

Figures (4)

• Figure 1: (a) Predicted spectra for electrokinetic turbulence at $V_{\mathrm{pp}}=10,14,20$ V Wang2016_microEKT. (b) Predicted spectra for two electric Rayleigh numbers $Ra_e$Shi2025_quadCascadeEKT, where larger $Ra_e$ activates the electrokinetic term $K$ and reduces the slope toward $-7/5$. Parameters: $\gamma_v=1$, $\alpha_v=4/3$, $C_K=1$.
• Figure 2: Predicted spectrum for interfacial turbulence Padhan2024_interfaceTurbulence. Large $I$ yields $m\simeq4.6$.Parameters: $\gamma_v=1$, $\alpha_v=4/3$, $C_K=1$.
• Figure 3: Predicted spectrum for active bacterial turbulence Wensink2012MesoScaleTurbulence. $m\simeq8/3$. Parameters: $\gamma_v=1$, $\alpha_v=4/3$, $C_K=1$, $\gamma_p=1$, $\alpha_p=1$, $k_\star=1.3\times10^{5}$ m$^{-1}$.
• Figure 4: Slope prediction on distinct physical mechanism: soft-wall elasticity, hydrodynamic oscillation, supersonic plasma turbulence, and active micromachine flow. Parameters: $\gamma_v=1$, $\alpha_v=4/3$, $C_K=1$.