Fourth-order compact formulation of Navier-Stokes equations and driven cavity flow at high Reynolds numbers

Abstract

SUMMARYA new fourth order compact formulation for the steady 2-D incompressible Navier-Stokes equations is presented. The formulation is in the same form of the Navier-Stokes equations such that any numerical method that solve the Navier-Stokes equations can easily be applied to this fourth order compact formulation. In particular in this work the formulation is solved with an efficient numerical method that requires the solution of tridiagonal systems using a fine grid mesh of 601×601. Using this formulation, the steady 2-D incompressible flow in a driven cavity is solved up to Reynolds number of 20,000 with fourth order spatial accuracy. Detailed solutions are presented.

Finite volume simulation of 2-D steady square lid driven cavity flow at high reynolds numbers

“…On the other hand, 2-D incompressible steady driven cavity flow can be computable at high Re (Erturk, 2009) with its peculiar computational challenges. Therefore, steady lid-driven cavity flow, especially at high Re (Re ≥ 10000), has been exploited by many researchers in order to test and improve robustness and stability of their computational methods (Erturk, 2009;Erturk et al, 2005;Ramšak and Škerget, 2004;Erturk and Gökçel, 2006;Sahin and Owens, 2003;Barragy and Carey, 1997;Schreiber and Keller, 1983;Ghia et al, 1982).…”

“…More specifically, in the case of flow in the lid-driven cavity, adopting a finer grid structure near the lid than in the bulk region enables one to resolve high gradients (Erturk and Gökçel, 2006) and to obtain an oscillation-free solution at high Re numbers (Erturk et al, 2005). Use of a non-uniform grid structure entails modification of the numerical schemes.…”

Discussions on driven cavity flow

“…al. [14], Erturk & Gokcol [15], Barragy & Carey [6], Schreiber & Keller [39], Benjamin & Denny [8], Liao & Zhu [30], Ghia et. al.…”

Numerical solutions of 2‐D steady incompressible flow in a driven skewed cavity

“…Even though the geometry is simple and easy to apply in programming point of view, the cavity flow has all essential flow physics with counter rotating recirculating regions at the corners of the cavity. Among numerous papers found in the literature, Erturk et al [6], Botella and Peyret [4], Schreiber and Keller [21], Li et al [12], Wright and Gaskel [30], Erturk and Gokcol [7], Benjamin and Denny [2] and Nishida and Satofuka [16] are examples of numerical studies on the driven cavity flow.…”

“…Using this numerical formulation, they have solved the very well known benchmark problem, the steady flow in a square driven cavity, up to Reynolds number of 21000 using a 601¯601 fine grid mesh. Their formulation proved to be stable and effective at very high Reynolds numbers ( [6], [7], [8]). …”