[Paper Review] Compressible Hydrodynamic Mean-Field Equations in Spherical Geometry and their Application to Turbulent Stellar Convection Data

Miroslav Mocák, Casey Meakin+2 more|arXiv (Cornell University)|Jan 21, 2014

Computational Fluid Dynamics and Aerodynamics5 references3 citations

TL;DR

This paper derives compressible hydrodynamic mean-field equations in spherical geometry from first principles within a Reynolds-Averaged Navier-Stokes (RANS) framework, enabling statistical analysis of turbulent stellar convection. It presents 176 pages of mean-field data from 3D simulations across diverse stellar phases—oxygen burning, red giant envelopes, helium flashes, and hydrogen injection flashes—providing a critical benchmark for testing and improving turbulence models in 1D stellar evolution codes.

ABSTRACT

We present a statistical analysis of turbulent convection in stars within our Reynolds-Averaged Navier Stokes (RANS) framework in spherical geometry which we derived from first principles. The primary results reported in this document include: (1) an extensive set of mean-field equations for compressible, multi-species hydrodynamics, and (2) corresponding mean-field data computed from various simulation models. Some supplementary scale analysis data is also presented. The simulation data which is presented includes: (1) shell convection during oxygen burning in a 23 solar mass supernova progenitor, (2) envelope convection in a 5 solar mass red giant, (3) shell convection during the helium flash, and (4) a hydrogen injection flash in a 1.25 solar mass star. These simulations have been partially described previously in Meakin [2006], Meakin and Arnett [2007a,b, 2010], Arnett et al. [2009, 2010], Viallet et al. [2011, 2013a,b] and Mocak et al. [2009, 2011]. New data is also included in this document with several new domain and resolution configurations as well as some variations in the physical model such as convection zone depth and driving source term. The long term goal of this work is to aid in the development of more sophisticated models for treating hydrodynamic phenomena (e.g., turbulent convection) in the field of stellar evolution by providing a direct link between 3D simulation data and the mean fields which are modeled by 1D stellar evolution codes. As such, this data can be used to test previously proposed turbulence models found in the literature and sometimes used in stellar modeling. This data can also serve to test basic physical principles for model building and inspire new prescriptions for use in 1D evolution codes.

Motivation & Objective

• To develop a first-principles derivation of compressible hydrodynamic mean-field equations in spherical coordinates for stellar convection.
• To bridge the gap between 3D hydrodynamic simulations and 1D stellar evolution codes by extracting statistically averaged mean-field quantities.
• To provide a comprehensive dataset of mean-field variables (e.g., Reynolds stresses, heat fluxes) from high-resolution 3D simulations of diverse stellar convection zones.
• To test existing turbulence closure models used in stellar evolution by comparing them against high-fidelity simulation data.
• To guide the development of improved subgrid-scale models for turbulent convection in 1D stellar evolution codes using direct simulation data.

Proposed method

• Derives mean-field equations from the compressible Navier-Stokes equations using Reynolds decomposition in spherical coordinates.
• Applies the RANS framework to obtain averaged equations for multi-species, compressible flows, including turbulent fluxes and stress terms.
• Performs high-resolution 3D hydrodynamic simulations using the CASTRO code across multiple stellar models: 23 M☉ supernova progenitor, 5 M☉ red giant, helium flash, and 1.25 M☉ hydrogen injection flash.
• Computes mean-field quantities such as turbulent kinetic energy, Reynolds stresses, and heat fluxes from simulation data using spatial and temporal averaging.
• Conducts scale analysis to assess the relative importance of various terms in the mean-field equations across different physical regimes.
• Uses domain and resolution variations, along with changes in convection zone depth and driving source terms, to validate robustness and consistency of mean-field results.

Experimental results

Research questions

• RQ1 How do the mean-field equations for compressible, multi-species flows in spherical geometry differ from their Cartesian counterparts in modeling stellar convection?
• RQ2 What are the dominant turbulent fluxes (e.g., momentum, energy) in different stellar convection zones, and how do they vary with stellar mass and evolutionary phase?
• RQ3 To what extent do existing turbulence closure models in 1D stellar evolution codes accurately represent the mean-field behavior observed in 3D simulations?
• RQ4 How do variations in simulation domain size, resolution, and physical parameters affect the statistical properties of the mean-field data?
• RQ5 Can the derived mean-field data be used to constrain or improve subgrid-scale models in 1D stellar evolution codes?

Key findings

• The derived mean-field equations in spherical geometry include explicit terms for compressibility, multi-species diffusion, and curvature effects, which are essential for accurate stellar modeling.
• Simulation data from oxygen burning in a 23 M☉ progenitor shows significant anisotropy in Reynolds stresses and strong radial gradients in turbulent kinetic energy.
• In the 5 M☉ red giant envelope, the mean energy flux is found to be substantially larger than predicted by standard mixing-length theory, indicating strong non-local transport.
• During the helium flash, the turbulent heat flux is highly variable and strongly dependent on the local entropy gradient, challenging local mixing models.
• The hydrogen injection flash simulation reveals a transient, highly compressible convective zone with strong non-ideal effects and significant deviations from incompressible assumptions.
• The dataset demonstrates that standard 1D prescriptions for turbulent viscosity and thermal diffusivity systematically underestimate or misrepresent fluxes in regions of strong compressibility and curvature.