[Paper Review] The turbulence velocity gradient tensor formed additively by normal and non-normal tensors

Christopher J. Keylock|White Rose Research Online (University of Leeds, The University of Sheffield, University of York)|Aug 3, 2016

Fluid Dynamics and Turbulent Flows3 citations

TL;DR

This paper proposes a novel additive decomposition of the turbulence velocity gradient tensor into normal and non-normal components using Schur decomposition, revealing that non-normality captures critical dynamics beyond eigenvalue-based topology. The method introduces new topological invariants (κ₁, κ₂, κ₃) that reveal strain alignment structures and non-local pressure Hessian effects, offering a richer framework for modeling dissipation in high-Reynolds-number flows.

ABSTRACT

We decompose the velocity gradient tensor for turbulence into normal and non-normal parts, and condition our analysis on the strain eigenvector alignments between these tensors. We identify states that always enhance, and always counteract the axisymmetric expansion state, and give a rationale for decomposing the production balance term into its constituents: complex behavior arises when the dominant strain alignments involve the non-normal tensor. Finally, we develop a topological analysis framework where mathematical bounds on two of the three variables leads to an analysis in two planes.

Motivation & Objective

• To develop a new mathematical framework for analyzing turbulence by decomposing the velocity gradient tensor into normal and non-normal components.
• To overcome limitations of traditional Q–R topology by incorporating non-normal dynamics absent in eigenvalue representations.
• To link non-normality to physical mechanisms such as strain alignment and pressure Hessian effects in homogeneous isotropic turbulence.
• To provide a new topological representation space (κ₁–κ₂, κ₁–κ₃) that captures Lagrangian dynamics and intermittency in dissipation.
• To enable improved closure models for high-Reynolds-number flows by revealing hidden dynamical structures in the velocity gradient tensor.

Proposed method

• Uses complex Schur decomposition to split the velocity gradient tensor A into B (eigenvalue matrix) and C (non-normal residual matrix), such that A = B + C.
• Defines non-normality via the Frobenius norm of the off-diagonal Schur matrix N, with ||A||_F² = ||B||_F² + ||C||_F².
• Introduces normalized invariants: κ₁ = ||S_B||_F / ||A||_F, κ₂ = ||S_B||_F / ||S_A||_F, and κ₃ = ||S_B||_F / ||Ω_A||_F to quantify relative strain and enstrophy contributions.
• Analyzes joint probability distributions of κ₁–κ₂ and κ₁–κ₃ planes, conditioned on discriminant D = Q³ + (27/4)R² and sign of R.
• Uses strain alignment between eigenvectors of A and B (eigenvectors of the normal part) to interpret dominant flow topologies.
• Applies the framework to homogeneous isotropic turbulence data, linking non-normal dynamics to pressure Hessian and production terms.

Experimental results

Research questions

• RQ1 How does decomposing the velocity gradient tensor into normal and non-normal components reveal dynamics not captured by eigenvalue-based topology?
• RQ2 What is the role of non-normality in shaping strain alignment and coherent structure formation in turbulence?
• RQ3 How do the new invariants (κ₁, κ₂, κ₃) reflect the relative contributions of strain and enstrophy production in different topological regimes?
• RQ4 Can the κ₁–κ₂ and κ₁–κ₃ planes reveal non-local effects such as pressure Hessian influence on dissipation?
• RQ5 How do Lagrangian dynamics in the new topological space differ from those in standard Q–R space?

Key findings

• The non-normal component C captures dynamics independent of the eigenvalue spectrum, with ||C||_F² = ||A||_F² - ||B||_F² quantifying non-normality.
• In the D < 0 region (real eigenvalues), κ₁ is typically positive, indicating dominant axisymmetric expansive straining in the normal part B.
• When D > 0, κ₁ > 0 for R < 0 and Q > 0, showing alignment between A and B eigenvectors under specific topological conditions.
• For [e_i^(A), e_i^(C)] alignments, κ₁ is strongly negative, indicating non-normal contributions dominate in certain regions.
• The κ₁–κ₃ plane reveals that R > 0 regions are associated with κ₃ > 0, indicating enhanced strain in the normal component.
• The Vieillefosse tail region shows anomalous behavior in [e₂^(A), e₂^(C)] alignment, indicating unique non-normal dynamics not visible in production terms.