[Paper Review] Kadanoff-Baym equations for near-Kolmogorov turbulence

Esteban Calzetta|ArXiv.org|Aug 27, 2009

Fluid Dynamics and Turbulent Flows29 references3 citations

TL;DR

This paper formulates the Kadanoff-Baym equations for near-Kolmogorov turbulence using the two-particle irreducible (2PI) Schwinger-Keldysh effective action, deriving a consistent dynamical framework for velocity and pressure correlations in incompressible, homogeneous turbulent flows. The key result is the derivation of a transport equation describing the approach to Kolmogorov scaling in the inertial range, with explicit verification of the Kolmogorov 4/5 law from the formalism.

ABSTRACT

We use the 2 particle irreducible Schwinger-Keldysh effective action to set up consistent equations for the velocity and pressure correlations of a turbulent flow. We use these equations to derive the Kadanoff-Baym equations describing the relaxation to Kolmogorov turbulence in the absence of mean velocities.

Motivation & Objective

• To develop a consistent field-theoretic framework for turbulence using the 2PI Schwinger-Keldysh effective action.
• To derive dynamical equations for velocity and pressure correlations in incompressible, homogeneous turbulent flows.
• To describe the relaxation dynamics toward Kolmogorov scaling in the inertial range.
• To verify the Kolmogorov 4/5 law within the 2PI CTP formalism as a consistency check on the framework.

Proposed method

• Formulates the 2PI closed-time-path (CTP) effective action for incompressible Navier-Stokes equations with stochastic forcing.
• Derives the Schwinger-Dyson equations for two-point correlation functions of velocity and pressure fields.
• Imposes random Galilean invariance (RGI) on self-energies to ensure physical consistency.
• Applies Fourier transforms to the equations of motion to analyze the inertial range dynamics.
• Uses sum rules and isotropy constraints to determine the structure of the three-point correlation function.
• Verifies the Kolmogorov 4/5 law by computing the third-order velocity structure function from the formalism.

Experimental results

Research questions

• RQ1 How can the Kadanoff-Baym equations be consistently derived for turbulent flows using a variational 2PI effective action formalism?
• RQ2 What is the role of random Galilean invariance in ensuring physical consistency of the self-energies in the absence of mean flow?
• RQ3 How does the formalism describe the relaxation of a nearly homogeneous flow toward Kolmogorov scaling?
• RQ4 Can the Kolmogorov 4/5 law be derived from the 2PI CTP effective action without prior assumptions about the energy spectrum?
• RQ5 What is the structure of the three-point correlation function in the inertial range, and how does it relate to the energy flux?

Key findings

• The three-point correlation function in the inertial range is shown to be proportional to $ \frac{1}{k^2} (\Delta^{rt} k^s + \Delta^{st} k^r) $, with coefficient $ A = \frac{1}{2}(2\pi)^d \epsilon \delta(\mathbf{k}) $, consistent with energy flux conservation.
• The Kolmogorov 4/5 law is derived from the 2PI CTP formalism, with the third-order structure function yielding $ \langle (\mathbf{x} \cdot [\mathbf{u}(\mathbf{x}) - \mathbf{u}(0)])^3 \rangle = -\frac{4}{5} \epsilon r^4 $.
• The two-point correlation function satisfies $ F_1 = -\frac{1}{12} \epsilon r^2 $, which is consistent with the expected $ r^2 $ scaling of the second-order structure function.
• The formalism reproduces the Kármán-Howarth equation and the Kolmogorov 4/5 law without ad hoc assumptions, validating its consistency.
• The sum rule $ i\int d\omega \, \omega \, G^{r+,t+} = 0 $ is used to show that the bare forcing and viscosity are negligible in the inertial range, leading to the vanishing of the three-point function's momentum-space integral.
• The derivation confirms that the energy flux $ \epsilon $ is encoded in the $ \delta(\mathbf{k}) $-singularity of the three-point function, linking the formalism directly to the physical cascade rate.