Residual-based closure model for density-stratified incompressible turbulent flows

Lixing Zhu, Arif Masud

Research output: Contribution to journalArticlepeer-review


This paper presents a locally and dynamically adaptive residual-based closure model for density stratified incompressible flows. The method is based on the three-level form of the Variational Multiscale (VMS) modeling paradigm applied to the system of incompressible Navier–Stokes equations and an energy conservation equation for the relative temperature field. The velocity, pressure, and relative temperature fields are additively decomposed into overlapping scales which leads to a set of coupled mixed-field sub-problems for the coarse- and the fine-scales. In the hierarchical application of the VMS method, the fine-scale velocity and relative temperature fields are further decomposed, leading to a nested system of two-way coupled fine-scale level-I and level-II variational subproblems. A direct application of bubble functions approach to the fine-scale variational equations helps derive fine-scale models that are nonlinear and time dependent. Embedding the derived model from the level-II variational equation in the level-I variational equation helps stabilize the convection-dominated mixed-field thermodynamic subproblem. Locally resolving the unconstrained level-I variational equation yields the residual-based turbulence model which is a function of the residual of the Euler–Lagrange equations of the conservation of momentum, mass, and energy. The derived model accommodates forward- and back-scatter of energy and entropy and embeds sub-grid scale physics in the computable scales of the problem. The steps of the derivation show that it is essential to apply the concept of scale separation systematically to the coupled system of equations and it is critical to preserve the coupling between flow and thermal phases in the fine-scale variational equations. The method has been implemented with hexahedral and tetrahedral elements with equal order interpolations for the velocity, pressure, and temperature fields. Several canonical flow cases are presented that include Rayleigh–Bénard instability, Rayleigh–Taylor instability, and turbulent plane Couette flow with stable stratification.

Original languageEnglish (US)
Article number113931
JournalComputer Methods in Applied Mechanics and Engineering
StatePublished - Dec 2021


  • Boussinesq approximation
  • Density stratification
  • Hierarchical methods
  • Incompressible turbulent flows
  • Variational multiscale method

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Physics and Astronomy(all)
  • Computer Science Applications


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