[Paper Review] Variation on the Kolmogorov Forcing: Asymptotic Dissipation Rate Driven by Harmonic Forcing

Bertrand Rollin, Yves Dubief+1 more|arXiv (Cornell University)|Mar 11, 2009

Fluid Dynamics and Turbulent Flows22 references15 citations

TL;DR

This study investigates how the shape of harmonic body forces affects the asymptotic energy dissipation rate in 3D periodic turbulent flows. Using direct numerical simulations and variational optimization, it shows that the dimensionless dissipation factor β depends strongly on forcing shape—qualitatively matching theoretical bounds at infinite Reynolds numbers, though quantitative predictions remain limited.

ABSTRACT

The relation between the shape of the force driving a turbulent flow and the upper bound on the dimensionless dissipation factor $β$ is presented. We are interested in non-trivial (more than two wave numbers) forcing functions in a three dimensional domain periodic in all directions. A comparative analysis between results given by the optimization problem and the results of Direct Numerical Simulations is performed. We report that the bound on the dissipation factor in the case of infinite Reynolds numbers have the same qualitative behavior as for the dissipation factor at finite Reynolds number. As predicted by the analysis, the dissipation factor depends strongly on the force shape. However, the optimization problem does not predict accurately the quantitative behavior. We complete our study by analyzing the mean flow profile in relation to the Stokes flow profile and the optimal multiplier profile shape for different force-shapes. We observe that in our 3D-periodic domain, the mean velocity profile and the Stokes flow profile reproduce all the characteristic features of the force-shape. The optimal multiplier proves to be linked to the intensity of the wave numbers of the forcing function.

Motivation & Objective

• To examine the influence of non-trivial, multi-wave-number forcing functions on the upper bound of the dimensionless dissipation factor β in 3D periodic turbulence.
• To compare analytical bounds derived from variational optimization with results from direct numerical simulations (DNS) at moderate Reynolds numbers.
• To investigate the relationship between the mean velocity profile, Stokes flow profile, and the optimal multiplier profile across different forcing shapes.
• To assess whether the optimization framework of Doering et al. (2003) accurately predicts quantitative dissipation behavior in finite-Reynolds-number flows.

Proposed method

• Formulates a body-force-driven incompressible Navier-Stokes system in a 3D periodic domain with harmonic forcing functions of the form f(y) = sin(ky) + A·sin(3ky).
• Applies variational optimization to derive upper bounds on the dimensionless dissipation factor β = εl/U³ using the method of Doering & Constantin.
• Performs direct numerical simulations (DNS) of the Navier-Stokes equations with periodic boundary conditions to compute β and mean velocity profiles.
• Compares DNS results with analytical bounds, focusing on the dependence of β and velocity profiles on the amplitude and wave number composition of the forcing.
• Analyzes the optimal multiplier profile ψₘ to understand its relation to the forcing function and mean velocity field.
• Uses normalized force, mean velocity, and optimal multiplier profiles to visualize structural correlations across different forcing amplitudes and wave numbers.

Experimental results

Research questions

• RQ1 How does the shape of a multi-harmonic body force influence the upper bound of the dimensionless dissipation factor β in 3D periodic turbulence?
• RQ2 To what extent do the analytical bounds on β derived at infinite Reynolds number hold qualitatively and quantitatively in finite-Reynolds-number DNS?
• RQ3 How do the mean velocity profiles in turbulent flows reflect the structural features of the forcing function in a 3D periodic domain?
• RQ4 What is the relationship between the optimal multiplier profile ψₘ and the forcing function or mean velocity profile?
• RQ5 Does the addition of higher wave number components to the forcing function lead to a predictable, non-linear increase in β?

Key findings

• The dimensionless dissipation factor β depends strongly on the shape of the forcing function, with higher wave number components contributing less to dissipation unless their amplitude is sufficiently large.
• The analytical upper bound on β derived via variational optimization captures the qualitative dependence on forcing shape at infinite Reynolds number, and this behavior persists qualitatively at moderate Reynolds numbers in DNS.
• The mean velocity profile in turbulent flow reproduces all characteristic features of the forcing function, such as changes in slope and local extrema, especially when the forcing includes multiple wave numbers.
• The optimal multiplier profile ψₘ is primarily sensitive to the dominant wave number in the forcing function and becomes insensitive to secondary components when their amplitude is small.
• As the amplitude of the secondary wave number component increases, ψₘ evolves to reflect the combined wave number structure, and its amplitude decreases slowly, stabilizing when the secondary term dominates.
• The relation between ψₘ and the mean velocity profile is unclear—ψₘ does not reflect secondary features in the mean velocity profile even when they are present, indicating a decoupling between the multiplier and the flow response.