[Paper Review] Threshold amplitudes for transition to turbulence in a pipe

Lloyd N. Trefethen, S. Jonathan Chapman+4 more|ArXiv.org|Jul 28, 2000

Fluid Dynamics and Turbulent Flows8 references17 citations

TL;DR

This paper resolves discrepancies in reported threshold amplitudes for transition to turbulence in pipe flow by standardizing on the $L^2$ norm for perturbation amplitude. After reinterpreting experimental data from Draad & Nieuwstadt and Darbyshire & Mullin using this consistent definition, the authors find agreement with theoretical predictions that the minimal amplitude scales as $R^{-3/2 \pm 0.3}$, supporting a critical exponent near $-3/2$ for high Reynolds numbers.

ABSTRACT

Although flow in a circular pipe is stable to infinitesimal perturbations, it can be excited to turbulence by finite perturbations whose minimal amplitude shrinks as R goes to infinity (R = Reynolds number). Laboratory experiments have appeared to disagree with one another and with theoretical predictions about the dependence of this minimal amplitude on $R$, with published results ranging approximately from $R^{-1/4}$ to $R^{-3/2}$. Here it is shown that these discrepancies can be explained by the use of different definitions of amplitude by different authors. An attempt is made to convert the existing results to a uniform definition of amplitude, the nondimensionalized $L^2$ definition common in the theoretical literature. Although subtleties in the physics raise some questions, agreement appears to be reached on a minimal amplitude that scales as $R^{-3/2 \pm 0.3}$.

Motivation & Objective

• To resolve inconsistencies in published threshold amplitude exponents for pipe flow transition, which range from $R^{-1/4}$ to $R^{-3/2}$.
• To standardize amplitude definitions across experimental and theoretical works, which had used incompatible measures, rendering comparisons invalid.
• To convert experimental results from Draad & Nieuwstadt and Darbyshire & Mullin into the $L^2$ amplitude framework commonly used in theoretical fluid dynamics.
• To assess whether experimental data align with theoretical predictions of $\gamma = -3/2$ for the scaling of minimal threshold amplitude with Reynolds number $R$.
• To identify and quantify uncertainties in experimental perturbation injection that affect amplitude scaling estimates.

Proposed method

• Adopt the standard $L^2$ norm for velocity perturbation amplitude, defined using nondimensionalized variables with pipe radius and centerline velocity as scale units.
• Apply a normalization factor to experimental amplitude data to convert from physical to $L^2$-normed amplitudes, using the relation $\epsilon_{L^2} \propto A^{3/4} R^{-1/2}$ for perturbations penetrating $O(1)$ distance into the pipe.
• Account for non-optimality of experimentally injected disturbances by estimating upper bounds on the scaling exponent $\gamma$, acknowledging that actual perturbations may be less efficient at triggering transition.
• Use asymptotic analysis and theoretical predictions from Chapman to compare with adjusted experimental results, particularly for plane Poiseuille and pipe flow.
• Evaluate two possible penetration scenarios (full and partial) to estimate the range of $L^2$ amplitude scaling, leading to adjustments in reported exponents.
• Apply corrections to experimental exponents based on the relationship between injection parameters (e.g., volume, time, velocity) and induced perturbation fields in the pipe.

Experimental results

Research questions

• RQ1 Why do published experimental results on threshold amplitudes for pipe flow transition show such wide variation in scaling exponents, from $R^{-1/4}$ to $R^{-3/2}$?
• RQ2 How can inconsistent amplitude definitions across experimental and theoretical studies be reconciled to enable valid comparison of threshold scaling behavior?
• RQ3 What is the correct scaling exponent $\gamma$ for the minimal amplitude $\epsilon$ that triggers transition to turbulence in pipe flow, when expressed in the standard $L^2$ norm?
• RQ4 To what extent do experimental perturbation injection methods produce disturbances that are optimal for triggering transition, and how does this affect amplitude scaling?
• RQ5 Does the experimental data, when converted to the $L^2$ amplitude framework, converge toward the theoretical prediction of $\gamma = -3/2$?

Key findings

• After conversion to the $L^2$ amplitude definition, the Draad & Nieuwstadt experimental data, originally reported with $\gamma \approx -1$, are adjusted to $-2 \leq \gamma \leq -1$.
• The Darbyshire & Mullin experimental data, originally reported with $\gamma \approx -0.4$ to $-0.2$, are adjusted to $-1.8 \leq \gamma \leq -1.15$ when converted to $L^2$ amplitudes.
• The adjusted experimental results show convergence toward a common scaling exponent of $\gamma = -3/2 \pm 0.3$, consistent with theoretical predictions by Chapman for pipe flow.
• Theoretical asymptotic analysis by Chapman predicts $\gamma = -3/2$ for pipe flow, and the reinterpreted experimental data support this value within the estimated uncertainty range.
• The study identifies significant uncertainties in experimental perturbation injection, particularly the complex relationship between injection parameters and induced velocity fields, which affect amplitude scaling.
• The authors conclude that while no definitive conclusion can be drawn due to multiple uncertainties, the current evidence supports $\gamma \approx -3/2$ as a rough working approximation for the threshold amplitude scaling in pipe flow.