On the large-scale vertical velocity intermittency of turbulent wall flows

Abstract: Large-scale intermittency in the vertical velocity (LSI) has received significant attention in studies of coherent structures and their detection using data-driven approaches. However, a theory that predicts the origin of LSI from the Navier-Stokes equations or some approximated version of them at very high Reynolds numbers is yet to be achieved. This letter proposes such a theory for a neutrally stratified wall-bounded turbulent flow based on a dominant balance between inertial and pressure forces. Using multiple flume and wind tunnel experiments, it is shown that the flatness factor ($FF_w$) measuring LSI collapses to a universal trend for all flow configurations within the inertial sublayer (ISL) before reaching a common minimum value above the ISL. A theory that predicts $FF_w$ using second-order statistics and explicitly accommodates large-scale energy anisotropy is tested against a wide range of Reynolds numbers from laboratory to field settings with varied surface roughness conditions. The theory also demonstrates why $FF_w$ cannot be described using down-gradient closure approximations routinely employed in large-scale meteorological and climate models.

• The paper establishes a dominant balance between inertial and pressure forces to derive a universal model for vertical velocity flatness in the inertial sublayer.
• It employs a higher-order Rotta model and a Quasi-Gaussian Approximation to calculate the flatness factor (FF_w) which is validated by extensive laboratory and atmospheric data.
• The study reveals the limitations of gradient-diffusion closures and enhances turbulence modeling for improved predictions in environmental and geophysical applications.

Large-Scale Vertical Velocity Intermittency in Turbulent Wall Flows

Introduction

The study addresses the origin and modeling of large-scale intermittency (LSI) in the vertical velocity component ($w$) of wall-bounded turbulent flows. Quantifying vertical velocity intermittency is crucial for understanding transport mechanisms such as momentum and heat exchange, pollutant dispersion, and sediment suspension. LSI is characterized by infrequent but intense wall-normal velocity events, typically measured by the flatness factor (FF), defined as $FF_w = \overline{w'^4} / (\overline{w'^2})^2$. Despite advances in coherent structure detection and empirical modeling, predictive theoretical frameworks rooted in the Navier-Stokes equations for LSI, particularly for FF's wall-normal variation, remain underdeveloped. The paper closes this gap by establishing a balance between inertial and pressure forces as the dominant mechanism in the inertial sublayer (ISL), elucidating FF's universal behavior across diverse flow configurations.

Theoretical Framework

Limitations of Conventional Closure Schemes

Traditional modeling approaches fall into either gradient-diffusion closures—often assuming a proportionality between FF and the gradient of vertical velocity triple moments—or statistical realizability constraints, which relate higher-order moments via the Cauchy–Schwarz inequality. Empirical data show that gradient closures predict erroneously uniform FF values within the ISL, missing observed weak vertical trends. Realizability models (e.g., $FF_w = \alpha (Sk_w + 1)$ with $\alpha$ empirically tuned between 2.6 and 3.3) capture some correlations between velocity skewness and flatness but are not physically derived from governing equations.

Navier-Stokes Derived Balance

Building upon the averaged Navier-Stokes equations for high-Re flows and neglecting viscous terms, the study establishes a dominant balance between inertial ($\partial_z \overline{w'^5}$) and pressure–velocity interaction ($-\overline{w'^3 \partial_z p'}$) terms in the ISL. The simplified governing equation:

$$\partial_z \overline{w'^5} = -\overline{w'^3 \partial_z p'}$$

necessitates closure schemes for both terms. The pressure–velocity interaction is extended using a higher-order Rotta model, yielding:

$$-\overline{w'^3 \partial_z p'} = \frac{C_R}{2\tau}(\overline{w'^2}q - \overline{w'^4})$$

where $C_R$ is the Rotta constant, $\tau$ is a turbulent relaxation timescale, and $FF_w = \overline{w'^4} / (\overline{w'^2})^2$0 represents twice the turbulent kinetic energy. A Quasi-Gaussian Approximation (QGA) further simplifies the problem by decomposing fourth moments into products of second moments, rendering FF explicitly calculable from normalized second-order statistics.

Modeling the Inertial Term

The inertial term is modeled by assuming a characteristic length scale $FF_w = \overline{w'^4} / (\overline{w'^2})^2$1 proportional to a macro-scale (e.g., integral length or $FF_w = \overline{w'^4} / (\overline{w'^2})^2$2), validating the assumption with experimental data indicating negligible contribution from viscous sublayer scales within ISL, and aligns with asymptotic behavior predicted by the attached eddy model (AEM).

Final Analytical Formulation

The resulting formulation relates FF to normalized variances ($FF_w = \overline{w'^4} / (\overline{w'^2})^2$3, $FF_w = \overline{w'^4} / (\overline{w'^2})^2$4, $FF_w = \overline{w'^4} / (\overline{w'^2})^2$5):

$FF_w = \overline{w'^4} / (\overline{w'^2})^2$6

with a single similarity parameter $FF_w = \overline{w'^4} / (\overline{w'^2})^2$7 reflecting inertial modification. This formulation is consistent across laboratory and atmospheric surface layer data, showing weak dependence on wall-normal distance in the ISL and minimal variation with Reynolds number or surface roughness.

Numerical Results and Model Validation

The universal trend and weak minimum of FF in the ISL were confirmed across a comprehensive corpus of flume and wind tunnel datasets with varying bed types and surface roughness ($FF_w = \overline{w'^4} / (\overline{w'^2})^2$8). The model reliably recapitulates observed FF profiles, demonstrating:

• Collapse to a near-constant FF in the ISL: $FF_w = \overline{w'^4} / (\overline{w'^2})^2$9, independent of $FF_w = \alpha (Sk_w + 1)$0 and flow configuration, for $FF_w = \alpha (Sk_w + 1)$1 and $FF_w = \alpha (Sk_w + 1)$2.
• Minimum FF at ISL/outer layer transition: At $FF_w = \alpha (Sk_w + 1)$3, a local minimum emerges due to the interplay between energy anisotropy and inertial scaling.
• Failure of gradient-diffusion closures: Conventional models cannot reproduce the weak but systematic FF variations observed outside the ISL, especially in buffer and outer layers.
• Robustness for atmospheric surface layer predictions: Field data spanning lakes, grasslands, bare soils, and forested sites validate the proposed model with nearly constant $FF_w = \alpha (Sk_w + 1)$4 and match laboratory-derived trends.

Implications and Future Directions

Practical Implications

This analytical expression for FF enables improved parameterization of vertical velocity intermittency in turbulent wall flows, particularly relevant for geophysical and environmental models. The result highlights that FF cannot be modeled via down-gradient closures, necessitating physics-based closure at the level of balance between inertia and pressure–velocity redistribution. Consequently, applications in meteorology, climate modeling, and environmental engineering—where vertical transport and rare extreme events are critical—should incorporate this universal FF parameterization to account for LSI effects.

Theoretical Implications

The study clarifies the physical origin of large-scale intermittency in wall-bounded turbulence, revealing that energetic anisotropy and inertial–pressure balances drive FF's universal behavior in the ISL. The connection to the attached eddy paradigm further anchors the findings in turbulence theory, supporting the notion that higher-order statistics can be predicted from lower-order variance scalings with a single similarity parameter.

Future Directions

Extending the model to capture FF variations in buffer and outer layers—where the assumptions underlying the ISL fail—remains a challenge. Further theoretical work to address the contribution of triple moment gradients and to refine the closure for highly inhomogeneous, stratified, or transitional conditions is warranted. Additionally, integrating the model into LES and RANS frameworks for predictive simulation of intermittency-driven transport is a promising avenue for advancing turbulence modeling in wall-bounded flows.

Conclusion

The paper provides a rigorous, Navier-Stokes-based theory for large-scale vertical velocity intermittency in turbulent wall flows, validated across laboratory and field datasets. By incorporating inertial and pressure–velocity interaction closures, the resulting expression for FF reveals a universal near-constant trend in the ISL, explains the minimum at the ISL/outer layer boundary, and exposes the deficiencies of traditional gradient-diffusion approximations. The implications for turbulence modeling and environmental prediction are profound, with the single similarity parameter pathway offering practical utility for advancing higher-order turbulence statistics in geophysical flows.