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Normalizing Flows as a Novel PDF Turbulence Model
Fluid Dynamics · Submitted 10 Jan 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The NF-PDF closure model allows for direct sampling from the underlying probability density functions of fluctuating turbulent quantities, such that ensemble-averaged quantities can then be computed.
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The abstract, sentence by sentence
In this paper, we propose normalizing flows (NF) as a novel probability density function (PDF) turbulence model (NF-PDF model) for the Reynolds-averaged Navier-Stokes (RANS) equations. We propose to use normalizing flows in two different ways: firstly, as a direct model for the Reynolds stress tensor, and secondly as a second-moment closure model, i.e. for modeling the pressure-strain and dissipation tensor in the Reynolds stress transport equation. In classical PDF closure models, a stochastic differential equation has to be modeled and solved to obtain samples of turbulent quantities. The NF-PDF closure model allows for direct sampling from the underlying probability density functions of fluctuating turbulent quantities, such that ensemble-averaged quantities can then be computed. To illustrate this approach we demonstrate an application for the canonical case of homogeneous shear turbulence. Training data are extracted from high-fidelity direct numerical simulations (DNS).
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, we propose normalizing flows (NF) as a novel probability density function (PDF) turbulence model (NF-PDF model) for the Reynolds-averaged Navier-Stokes (RANS) equations. | contribution 1.00 | 0.83 | 0.28 | asserted 0.97 |
| s1 | We propose to use normalizing flows in two different ways: firstly, as a direct model for the Reynolds stress tensor, and secondly as a second-moment closure model, i.e. for modeling the pressure-strain and dissipation tensor in the Reynolds stress transport equation. | method 0.56 | 0.80 | 0.10 | asserted 0.91 |
| s2 | In classical PDF closure models, a stochastic differential equation has to be modeled and solved to obtain samples of turbulent quantities. | background 1.00 | 0.04 | 0.03 | asserted 1.00 |
| s3 | The NF-PDF closure model allows for direct sampling from the underlying probability density functions of fluctuating turbulent quantities, such that ensemble-averaged quantities can then be computed. | method 0.69 | 0.46 | 0.21 | asserted 0.96 |
| s4 | To illustrate this approach we demonstrate an application for the canonical case of homogeneous shear turbulence. | method 0.61 | 0.89 | 0.04 | asserted 0.99 |
| s5 | Training data are extracted from high-fidelity direct numerical simulations (DNS). | method 1.00 | 0.82 | 0.02 | asserted 1.00 |
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The paper
Original abstract
In this paper, we propose normalizing flows (NF) as a novel probability density function (PDF) turbulence model (NF-PDF model) for the Reynolds-averaged Navier-Stokes (RANS) equations. We propose to use normalizing flows in two different ways: firstly, as a direct model for the Reynolds stress tensor, and secondly as a second-moment closure model, i.e. for modeling the pressure-strain and dissipation tensor in the Reynolds stress transport equation. In classical PDF closure models, a stochastic differential equation has to be modeled and solved to obtain samples of turbulent quantities. The NF-PDF closure model allows for direct sampling from the underlying probability density functions of fluctuating turbulent quantities, such that ensemble-averaged quantities can then be computed. To illustrate this approach we demonstrate an application for the canonical case of homogeneous shear turbulence. Training data are extracted from high-fidelity direct numerical simulations (DNS).
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How the literature uses it
1 arXiv paper cite this in 1 sentence
What it cites · 2 sentences without a confident label
Uses the method or tool of 2 papers
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In this work, we parameterize our normalizing flow with a sequence of real-valued non-volume preserving (real NVP) transformations [ref] .
Density estimation using Real NVP · Normalizing Flows · 1.00
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A conditional version of the affine coupling layer was established in [ref] which allows for conditional sample creation.
Guided Image Generation with Conditional Invertible Neural Networks · Normalizing Flows · 0.92
Cites as background 1 paper
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Beck et al. [ref] used Deep Neural Networks for LES turbulence modelling, while Novati & Koumoutsakos [ref] used reinforcement learning to adjust the Smagorinsky constant in LES simulations.
Automating Turbulence Modeling by Multi-Agent Reinforcement Learning · Introduction · 0.95
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