[Paper Review] A multi-layer model for turbulent kinetic energy in pipe flows

Xi Chen, Fazle Hussain+1 more|arXiv (Cornell University)|Dec 30, 2011

Water Systems and Optimization3 citations

TL;DR

This paper proposes a multi-layer model for turbulent kinetic energy in pipe flows using a novel energy length function, accurately predicting the mean streamwise turbulent kinetic energy profile across a wide Reynolds number range. It identifies a critical Reynolds number ($Re_\tau$) beyond which a scaling anomaly emerges, leading to a second peak in the kinetic energy profile.

ABSTRACT

A multi-layer model of an energy length function is developed by employing recent results of the authors. The theory predicts the complete, mean streamwise turbulent kinetic-energy profile (MKP), in good agreement with empirical data for a wide range of Reynolds numbers (Re). In particular, a critical $Re_τ$ is predicted, beyond which a scaling anomaly appears and MKP develops a second peak.

Motivation & Objective

• To develop a comprehensive multi-layer model for turbulent kinetic energy in pipe flows using recent theoretical advances.
• To accurately predict the mean streamwise turbulent kinetic-energy profile (MKP) across a broad range of Reynolds numbers.
• To identify and explain the emergence of a scaling anomaly in turbulent kinetic energy profiles at high Reynolds numbers.
• To predict the existence of a second peak in the MKP beyond a critical $Re_\tau$, challenging conventional scaling assumptions.

Proposed method

• The model employs a multi-layer framework based on a newly derived energy length function to describe turbulent energy distribution across the pipe cross-section.
• It integrates recent theoretical results to ensure consistency with known turbulent flow physics and empirical data.
• The energy length function is calibrated to reproduce observed mean streamwise turbulent kinetic energy profiles from experimental data.
• The model uses asymptotic analysis to identify critical transitions in scaling behavior at high Reynolds numbers.
• It applies a layered decomposition of the flow field to capture distinct behaviors in the core, buffer, and wall-adjacent regions.
• Theoretical predictions are validated against empirical data across a wide range of Reynolds numbers.

Experimental results

Research questions

• RQ1 At what critical Reynolds number does the turbulent kinetic energy profile exhibit a scaling anomaly in pipe flows?
• RQ2 Does the mean streamwise turbulent kinetic-energy profile develop a second peak at high Reynolds numbers, and if so, under what conditions?
• RQ3 How does the proposed multi-layer model with an energy length function improve prediction accuracy across varying Reynolds numbers?
• RQ4 What physical mechanisms underlie the emergence of a second peak in the turbulent kinetic energy profile?
• RQ5 Can the model reconcile discrepancies between classical scaling laws and empirical observations in high-Reynolds-number pipe flows?

Key findings

• The model accurately predicts the complete mean streamwise turbulent kinetic-energy profile across a wide range of Reynolds numbers, showing strong agreement with empirical data.
• A critical $Re_\tau$ is identified beyond which a scaling anomaly emerges in the turbulent kinetic energy profile.
• The model predicts the formation of a second peak in the mean streamwise turbulent kinetic-energy profile at high Reynolds numbers, a feature not captured by classical models.
• The emergence of the second peak is linked to a breakdown in traditional scaling assumptions, indicating a transition in turbulent energy distribution.
• The energy length function formulation successfully captures the multi-scale nature of turbulence, particularly in the near-wall and core regions of pipe flows.