[Paper Review] Time dependent simulation of the Driven Lid Cavity at High Reynolds Number

Nuno Vieira Cardoso, Pedro Bicudo|ArXiv.org|Sep 18, 2008

Computational Fluid Dynamics and Aerodynamics11 references3 citations

TL;DR

This study presents time-dependent simulations of the 2D driven lid cavity flow at Reynolds numbers up to 30,000 using a 1024×1024 grid and the SSPRK(5,4) scheme in the vorticity-stream function formulation. With double-precision CUDA-accelerated computation, the results converge to a steady-state solution, resolving prior controversy by showing that unsteady solutions in earlier studies were numerical artifacts of coarse grids, not physical behavior.

ABSTRACT

In this work, numerical solutions of the two dimensional time dependent incompressible flow, in a driven cavity at high Reynolds number Re, are presented. At high Re, there is a controversy. Some studies predicted that the flow is steady, others found time dependent non-steady flow, either periodic or aperiodic. In this study, the driven lid cavity is successfully solved using a very fine grid mesh, for Re up to $30 000$. We discretize the Vorticity-Stream formulation of the Navier-Stokes equation with the SSPRK(5,4) scheme in a $1024 imes1024$ grid. Using this very fine grid, the results obtained converge to a stationary solution. Detailed results for Re between $5 000$ and $30 000$ are presented. The driven lid cavity problem is solved with a NVIDIA GPU using the CUDA programming environment with double precision.

Motivation & Objective

• To resolve the long-standing controversy over whether the driven lid cavity flow becomes unsteady at high Reynolds numbers.
• To determine whether time-dependent simulations with a very fine grid converge to a steady solution, as opposed to oscillatory or periodic behavior.
• To investigate whether previously reported unsteady solutions in the literature are numerical artifacts due to insufficient spatial resolution.
• To validate the claim that a sufficiently fine grid is necessary to capture the true physical solution, avoiding spurious oscillations from coarse meshes.
• To demonstrate the effectiveness of high-resolution direct numerical simulation using GPU-accelerated double-precision computation.

Proposed method

• Numerical solution of the 2D incompressible Navier-Stokes equations using the vorticity-stream function formulation to enforce continuity automatically.
• Discretization of the time-dependent vorticity transport equation with the SSPRK(5,4) Runge-Kutta scheme for temporal accuracy and stability.
• Use of a 1024×1024 structured Cartesian grid to ensure high spatial resolution, especially in corner regions where small vortices form.
• Implementation on NVIDIA GPU using CUDA with double-precision arithmetic to minimize truncation errors and improve solution fidelity.
• Application of consistent boundary conditions: no-slip on all walls except the top lid, which moves with constant velocity to drive the flow.
• Time integration until the solution reaches a stationary state, monitored via residual norms and vorticity field evolution.

Experimental results

Research questions

• RQ1 Does the driven lid cavity flow at high Reynolds numbers (up to 30,000) exhibit time-dependent behavior, or does it converge to a steady state?
• RQ2 Are the unsteady solutions reported in previous direct numerical simulations (DNS) physically real or numerical artifacts due to coarse grid resolution?
• RQ3 Can a very fine grid mesh (1024×1024) resolve all relevant vortical structures, including small corner vortices, and stabilize the solution?
• RQ4 Does the use of high-precision double-precision arithmetic on a GPU improve the accuracy and convergence of the simulation compared to lower-precision methods?
• RQ5 Is the existence of a steady solution in the stationary Navier-Stokes equations sufficient to conclude physical reality, or is time-dependent simulation required to confirm it?

Key findings

• For Reynolds numbers from 5,000 to 30,000, the time-dependent simulation converges to a steady-state solution on a 1024×1024 grid, indicating the true physical solution is steady.
• At Re = 5,000, only the main vortex and some corner vortices (BL1, BR1, TL1, TR1) are clearly visible; BL3 and TL2 vortices are missing due to insufficient resolution.
• By Re = 15,000, all major vortices, including BL3 and TL2, are fully resolved and stable in the solution.
• The solution remains stationary for Re = 20,000, 25,000, and 30,000, confirming convergence to a steady state at high Re.
• The study attributes prior reports of unsteady or periodic flow to numerical oscillations caused by coarse grids, not physical instability.
• The results support Erturk’s claim that a very fine grid is essential to resolve small-scale vortices and obtain the correct steady solution.