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A face-centred finite volume method for laminar and turbulent incompressible flows

Luan M. Vieira, Matteo Giacomini, Ruben Sevilla, Antonio Huerta

Abstract

This work develops, for the first time, a face-centred finite volume (FCFV) solver for the simulation of laminar and turbulent viscous incompressible flows. The formulation relies on the Reynolds-averaged Navier-Stokes (RANS) equations coupled with the negative Spalart-Allmaras (SA) model and three novel convective stabilisations, inspired by Riemann solvers, are derived and compared numerically. The resulting method achieves first-order convergence of the velocity, the velocity-gradient tensor and the pressure. FCFV accurately predicts engineering quantities of interest, such as drag and lift, on unstructured meshes and, by avoiding gradient reconstruction, the method is less sensitive to mesh quality than other FV methods, even in the presence of highly distorted and stretched cells. A monolithic and a staggered solution strategies for the RANS-SA system are derived and compared numerically. Numerical benchmarks, involving laminar and turbulent, steady and transient cases are used to assess the performance, accuracy and robustness of the proposed FCFV method.

A face-centred finite volume method for laminar and turbulent incompressible flows

Abstract

This work develops, for the first time, a face-centred finite volume (FCFV) solver for the simulation of laminar and turbulent viscous incompressible flows. The formulation relies on the Reynolds-averaged Navier-Stokes (RANS) equations coupled with the negative Spalart-Allmaras (SA) model and three novel convective stabilisations, inspired by Riemann solvers, are derived and compared numerically. The resulting method achieves first-order convergence of the velocity, the velocity-gradient tensor and the pressure. FCFV accurately predicts engineering quantities of interest, such as drag and lift, on unstructured meshes and, by avoiding gradient reconstruction, the method is less sensitive to mesh quality than other FV methods, even in the presence of highly distorted and stretched cells. A monolithic and a staggered solution strategies for the RANS-SA system are derived and compared numerically. Numerical benchmarks, involving laminar and turbulent, steady and transient cases are used to assess the performance, accuracy and robustness of the proposed FCFV method.
Paper Structure (23 sections, 51 equations, 25 figures, 4 tables)

This paper contains 23 sections, 51 equations, 25 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Face-centred finite volume formulation
  4. Integral form of the local problems
  5. Integral form of the global problem
  6. Numerical fluxes and stabilisation
  7. Face-centred finite volume solution
  8. Spatial and temporal discretisation
  9. Newton-Raphson linearisation
  10. Numerical examples
  11. Couette flow
  12. Lid-driven cavity flow
  13. Assessment of the convective stabilisation
  14. Unsteady laminar flow past a circular cylinder
  15. Turbulent flow over a flat plate
  16. ...and 8 more sections

Figures (25)

  • Figure 1: Detail of a triangular mesh highlighting a cell $\Omega_e$ where the velocity $\bm{u}_e$, velocity gradient $\bm{L}_e$, mean pressure $p_e$, turbulent variable $\tilde{\nu}_e$ and gradient of the turbulent variable $\tilde{\bm{q}}_e$ are defined and internal edges where the hybrid velocity $\bm{\hat{u}}$ and hybrid turbulent variable $\hat{\nu}$ are defined.
  • Figure 2: Couette flow: (a,b) Triangular and (c,d) quadrilateral meshes used for the convergence study.
  • Figure 3: Couette flow: Mesh convergence of the error of the cell velocity, face velocity, gradient of the velocity and pressure for regular and distorted triangular and quadrilateral meshes using the HLL convection stabilisation.
  • Figure 4: Lid-driven cavity flow: (a) Regular and (b) distorted triangular meshes used for the convergence study.
  • Figure 5: Lid-driven cavity flow: Mesh convergence of the error of the cell velocity, face velocity, gradient of the velocity and pressure for (a) regular and (b) distorted triangular meshes using different convection stabilisations.
  • ...and 20 more figures

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4