Two-point Turbulence Closures in Physical Space

This paper introduces a predictive two-point statistical closure framework for turbulence formulated in physical space, which avoids Fourier transforms by using discrete evolution equations and EDQNM-inspired nonlinear closures to accurately model homogeneous isotropic turbulence while offering advantages for inhomogeneous and anisotropic flows where spectral methods are ill-conditioned.

The Gist

Imagine you are trying to predict how a pot of boiling water moves, or how smoke swirls from a chimney. This is the world of turbulence. It's chaotic, messy, and incredibly hard to predict because every tiny swirl of water or air affects every other swirl around it.

For decades, scientists have tried to build "rulebooks" (mathematical models) to predict this chaos. The most successful rulebooks so far have been written in a special language called Spectral Space. Think of Spectral Space like looking at a complex painting through a prism: instead of seeing the paint strokes, you see the specific colors (frequencies) that make up the image. It's great for smooth, uniform things, but if the painting has sharp edges, cracks, or sudden changes (like a shockwave in a supersonic jet), the prism breaks, and the picture gets blurry.

This paper introduces a new way to write the rulebook. Instead of using the prism (Spectral Space), the authors write the rules directly in Physical Space—the actual, real-world view where you can see the water and the air.

Here is a breakdown of their approach using simple analogies:

1. The Problem: The "Too Many Variables" Puzzle

In turbulence, to predict how a specific swirl will move next, you need to know how it interacts with all its neighbors.

• Old Way (Single-Point): Scientists used to look at just one tiny drop of water and guess what its neighbors were doing based on average rules. This is like trying to predict traffic in a city by only looking at one car and guessing the behavior of the whole highway. It often fails because it misses the big picture.
• The Two-Point Solution: The authors decided to look at two points at once. Imagine holding two hands out; you can feel the tension and distance between them. By studying the relationship between two points in the fluid, they can capture how energy moves from one swirl to another much more accurately.

2. The Innovation: Walking Instead of Flying

Most advanced turbulence models (like the famous EDQNM model) rely on the "prism" method (Fourier transforms) to do their math. It's fast and elegant for smooth, uniform flows.

• The Paper's Trick: The authors realized that if you stay in Physical Space (the real world), you don't need the prism. Instead of flying over the city to see the whole map, they decided to walk the streets.
• How they did it: They used a method called Finite Differences. Imagine you want to know how steep a hill is. Instead of using a magic telescope, you just measure the height of the ground at your feet and the height of the ground a few steps away. By doing this repeatedly across a grid, they can calculate how the fluid moves without ever leaving "physical space."

3. The "Eddy Damping" (The Shock Absorber)

Turbulence is full of energy that needs to be dissipated (lost as heat). In the old models, they used a "shock absorber" (called eddy damping) to stop the math from going crazy.

• The authors had to invent a new kind of shock absorber that works in Physical Space. They created a "smart viscosity" that acts like a sponge, soaking up the chaotic energy exactly where it needs to be, based on the local conditions of the flow.

4. The Pressure Problem: The "Ghost" Force

In fluids, pressure acts instantly everywhere. If you push water here, the pressure changes there immediately. This is called a "non-local" effect.

• In the old "prism" models, this was easy to handle. In the new "walking" model, it's hard. The authors had to solve a complex math puzzle involving triple integrals (imagine calculating the total weight of a cloud by summing up every single raindrop in a 3D sphere). They managed to write this out in their new language, showing that even though it's computationally heavy, it's possible.

5. Did it Work? (The Test Drive)

The authors tested their new "Physical Space" rulebook against two things:

1. The Old Rulebook: They compared it to the best spectral models for smooth, decaying turbulence (like smoke slowly fading away). Result: It matched perfectly.
2. Real Data: They compared it to super-computer simulations (Direct Numerical Simulations) of forced turbulence (like a fan blowing air). Result: It captured the energy transfer and the "swirliness" of the flow very accurately.

Why Does This Matter? (According to the Paper)

The paper claims this is a proof of concept. It proves you can build a high-accuracy turbulence model without using the "prism" (Fourier transforms).

The authors suggest this is a crucial first step for tackling harder problems where the prism breaks down, such as:

• Compressible flows: Air moving so fast it creates shockwaves (like a supersonic jet).
• Discontinuities: Flows with sudden jumps or breaks.

In Summary:
The authors built a new, robust way to predict how turbulent fluids move by staying in the "real world" (Physical Space) rather than translating the problem into a different language (Spectral Space). They showed that by using a grid-based approach and clever math tricks to handle pressure and energy loss, they can predict turbulence just as well as the old methods, but with a framework that is ready to handle the messy, sharp-edged problems of the real world.

Technical Summary

1. Problem Statement

Traditional turbulence modeling often relies on single-point closures (e.g., RANS), which struggle to accurately capture non-local interactions, anisotropy, and scale-to-scale energy transfer. While two-point closures (e.g., EDQNM, DIA) address these limitations, they have historically been formulated almost exclusively in spectral space (Fourier space).

The reliance on spectral methods presents significant barriers for complex engineering applications:

• Discontinuities: Spectral methods are ill-posed for flows with shocks or discontinuities (common in compressible flows).
• Inhomogeneity: Extending spectral closures to inhomogeneous flows is mathematically difficult and computationally expensive.
• Boundary Conditions: Spectral methods struggle with complex boundary conditions compared to physical space methods.

The authors aim to develop a predictive two-point statistical closure framework formulated entirely in physical space. The goal is to retain the physical fidelity of two-point statistics (capturing non-locality and energy transfer) while avoiding the mathematical constraints of Fourier transforms, thereby enabling application to complex, discontinuous, and inhomogeneous flows.

2. Methodology

The authors derive a closure model for Incompressible Homogeneous Isotropic Turbulence (HIT) as a proof of concept, with the intent to generalize to anisotropic and inhomogeneous flows.

A. Governing Equations and Discretization

• Physical Space Formulation: Instead of transforming Navier-Stokes equations into spectral space, the authors work directly with the evolution equations for two-point velocity correlation tensors ($R_{ij}$) and third-order moments.
• Discrete Derivatives: Spatial derivatives are approximated using finite differences (specifically Radial Basis Function-Finite Difference, RBF-FD, for high accuracy near $r=0$). This converts the Partial Differential Equations (PDEs) of the correlation functions into a system of Ordinary Differential Equations (ODEs) in matrix-vector form.
  • Linear terms (viscous diffusion) are represented by a radial Laplacian matrix ($\mathbf{L}$).
  • The evolution of the longitudinal correlation function $f(r)$ is governed by a matrix ODE involving differentiation matrices ($\mathbf{D}$) and identity matrices ($\mathbf{I}$).

B. Closure Strategy (EDQNM Adaptation)

The closure follows the phenomenological principles of the Eddy-Damped Quasi-Normal Markovian (EDQNM) model but adapts it to physical space:

1. Quasi-Normality: Fourth-order moments are approximated as products of second-order moments (Gaussian assumption).
2. Markovian Approximation: The time history of the third-order moments is approximated by evaluating spectra at the current time, allowing for analytical time integration.
3. Eddy Damping: To correct the unphysical behavior of the quasi-normal approximation (lack of reversibility and incorrect relaxation), an eddy-viscosity term is introduced.
   • In physical space, this is implemented by modifying the molecular viscosity in the matrix exponential solution.
   • The eddy viscosity is modeled as a function of the local velocity increment (via the second-order structure function), ensuring correct damping rates at small scales.

C. Handling Non-Local Pressure

A unique challenge in physical space is the pressure-Poisson equation, which introduces non-locality via a triple integral over the domain.

• The authors derive the pressure contribution to the third-order moment evolution.
• They demonstrate that while the pressure term involves complex triple integrals (requiring numerical quadrature), the net contribution of pressure to the energy transfer (spectral energy evolution) is zero due to the solenoidal nature of the flow. This simplifies the closure significantly, allowing the focus to remain on advection terms.

D. Numerical Implementation

• Grid: A geometrically spaced grid is used for the correlation distance $r$ to resolve sharp gradients near $r=0$ (where derivatives are critical) while covering large scales.
• Time Integration: An IMEX (Implicit-Explicit) scheme is used. The stiff viscous terms are treated implicitly (using matrix exponentials or backward Euler), while nonlinear terms are treated explicitly.
• Matrix Exponential: The linear solution involves computing the matrix exponential of the Laplacian operator, which is solved efficiently using Krylov subspace methods.

3. Key Contributions

1. First Physical-Space EDQNM Framework: The paper presents the first systematic derivation of an EDQNM-style closure entirely in physical space, bypassing the need for Fourier transforms.
2. Matrix-Vector Formulation: The evolution of turbulence statistics is cast as a linear algebra problem ($\dot{\mathbf{f}} = \mathbf{A}\mathbf{f} + \mathbf{b}$), where spatial derivatives are handled via differentiation matrices. This naturally incorporates non-local length-scale information.
3. Analytical Treatment of Pressure: The authors provide a rigorous derivation showing how the non-local pressure term can be handled in physical space and prove its cancellation in the energy budget, simplifying the closure.
4. Eddy Damping in Physical Space: A novel formulation for eddy damping is introduced using the Laplacian matrix and structure functions, replacing the spectral wavenumber-dependent viscosity.

4. Results and Verification

The model was verified against two benchmarks:

1. Decaying HIT (Comparison with Spectral EDQNM):
   • The physical space model successfully replicated the decay of the longitudinal ($f(r)$) and lateral ($g(r)$) correlation functions.
   • It correctly captured the Kolmogorov inertial scaling ($E(\kappa) \sim \kappa^{-5/3}$) and the decay laws for turbulent kinetic energy and integral length scales (Saffman and Batchelor spectra).
   • Minor discrepancies were observed at very large correlation lengths due to grid truncation and Hankel transform oscillations, but the inertial range was accurately resolved.
2. Forced HIT (Comparison with DNS):
   • The model was tested against Direct Numerical Simulation (DNS) data for statistically stationary forced turbulence ($Re_\lambda \approx 433$).
   • The physical space model accurately predicted the time-averaged energy spectrum and longitudinal structure functions.
   • It correctly captured the Kolmogorov $-5/3$ scaling and the correct magnitude of the third-order moment (energy transfer), validating the nonlinear closure.

5. Significance and Future Outlook

• Bridging the Gap: This work bridges the gap between the physical fidelity of two-point closures and the applicability of physical space methods. It opens the door for applying advanced turbulence closures to compressible flows with shocks, reacting flows, and complex geometries where spectral methods fail.
• Computational Trade-offs: While the physical space approach avoids Fourier transforms, it introduces computational costs associated with evaluating triple integrals (for pressure) and large matrix operations. The authors propose using Fast Multipole Methods (FMM) and sparse matrix techniques to mitigate these costs for general inhomogeneous/anisotropic flows.
• Extensibility: The framework is designed to be extensible. The authors outline how the method can be generalized to anisotropic flows (using SO(3) decomposition and spherical harmonics) and inhomogeneous flows (using local coarse grids with embedded two-point stencils).

In conclusion, this paper establishes a viable, predictive framework for two-point turbulence closure in physical space, demonstrating that high-fidelity statistical modeling can be achieved without spectral transformations, thus paving the way for more robust turbulence modeling in complex, real-world engineering flows.