Lid-Driven Cavity Flow — A CFD Benchmark Verification Guide

In short, it's the most widely used benchmark problem for CFD code verification. When only the top wall of a square box (cavity) slides horizontally at a constant velocity, a recirculating flow is generated inside. By computing this flow field and comparing it with the reference solution obtained by Ghia, Ghia & Shin (1982) using a high-accuracy finite difference method, we check if the solver is implemented correctly.

Good question. There are three reasons. First, the geometry is extremely simple (square, walls only), so there is no ambiguity in mesh generation. Second, it yields a non-trivial flow field—including recirculating vortices and secondary vortices in the corners, involving the competition between convection and viscosity. Unlike parallel flows like pipe flow, the quality of the discretization scheme is clearly reflected in the results. Third, thanks to Ghia's paper publishing numerical tables of velocity profiles for 7 conditions from Re=100 to 10000, quantitative comparison is easy.

It's the 2D, incompressible, steady Navier-Stokes equations. We solve the continuity and momentum equations as a set:

The boundary conditions are as follows:

• Top wall (lid): $u = U$, $v = 0$ (sliding at constant velocity)
• Bottom, left, and right walls: $u = 0$, $v = 0$ (no-slip condition)

The Reynolds number is defined as $\text{Re} = UL/\nu$. $L$ is the cavity side length, $U$ is the lid velocity, and $\nu$ is the kinematic viscosity coefficient.

Ghia's original paper did exactly that. It uses the stream function $\psi$ and vorticity $\omega$ to eliminate the pressure term:

Ghia computed using a multigrid method on a $129 \times 129$ uniform grid. This method is suitable for 2D problems because it doesn't require a pressure equation, but it's difficult to extend to 3D. Current commercial CFD solvers mainly use the velocity-pressure form (primitive variable method).

The Reynolds number decisively governs the nature of the flow field. Let's look at it step by step:

• Re=100: The primary vortex is located slightly above the cavity center. Viscosity is dominant, and the vortex has a nearly symmetrical, round shape. Secondary vortices are almost invisible.
• Re=400: The center of the primary vortex approaches the geometric center. Small secondary vortices (corner vortices) begin to appear at the bottom-left and bottom-right.
• Re=1000: The primary vortex center moves near $(0.5313, 0.5625)$. The bottom-left (BL) and bottom-right (BR) secondary vortices become clearly identifiable.
• Re=5000: Secondary vortices grow further. A small tertiary vortex (Top-Left, TL) also appears at the top-left. The shape of the primary vortex becomes biased upward and more elliptical.
• Re=10000: All corner vortices grow significantly. The existence of a steady solution is borderline, and it can transition to unsteady with slight disturbances.

In 2D, Ghia's steady solution has been obtained up to Re=10000. However, convergence is very slow. According to research by Erturk et al. (2005), obtaining the steady solution at Re=10000 required hundreds of thousands of iterations on a $512 \times 512$ grid. It is said that a Hopf bifurcation occurs around Re=20000, transitioning to periodic flow. In 3D, unsteadiness can appear at much lower Re numbers (around Re=1000).

Ghia et al. presented in tabular form the horizontal velocity $u$ on the vertical centerline and the vertical velocity $v$ on the horizontal centerline at 17 points each. The $u$ velocity profile on the vertical centerline at Re=1000 is the most commonly used data:

Looking at this table, the minimum value of $u/U \approx -0.38$ occurs around $y \approx 0.17$. This is the position of strongest reverse flow on the lower side of the primary vortex. If you can plot your CFD results overlaid with this profile and achieve a relative error within 1% for all points, your solver implementation is trustworthy.

Of course. The stream function value at the primary vortex center for Re=1000 is $\psi_{\min} = -0.1189$. This is convenient because it's a single scalar value for verification, but comparing the entire velocity profile provides a more rigorous verification.

For obtaining a steady solution, SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) is the standard. It solves the pressure correction equation once per iteration and updates the velocity field. With appropriate under-relaxation factors, it converges stably. For Re around 1000, a starting point might be velocity under-relaxation of 0.7 and pressure under-relaxation of 0.3.

PISO (Pressure-Implicit with Splitting of Operators) is for transient analysis. It performs pressure correction twice per time step to ensure temporal accuracy. It's used when an unsteady solution is needed at Re=10000 or higher. If you're only seeking the steady solution at Re=1000 for cavity flow, SIMPLE is more computationally efficient.

In cavity flow, the choice of scheme has a pronounced effect. Let me show you a concrete example. Using the same mesh (64×64) at Re=1000:

First-order upwind has significant numerical diffusion (artificial viscosity), causing errors over 7% on a 32×32 grid. In practice, for Re=1000, you should use at least a second-order scheme + 64×64 or finer mesh. The QUICK scheme (third-order upwind interpolation) provides sufficient accuracy even on a 64×64 grid.

It's unsuitable for verification purposes. However, it has very good convergence properties, so it can be used for initial solution estimation or debugging when second-order schemes diverge. A staged approach—"first run with first-order upwind to confirm the general flow pattern, then use that result as an initial guess and switch to second-order"—is commonly used in practice.

It depends on the Reynolds number. For Re=100–400, uniform spacing is sufficient, but for Re=1000 and above, steep velocity gradients occur near the walls, so a non-uniform mesh that clusters cells geometrically towards the walls is effective. Typically, use an expansion ratio of 1.05–1.1, with the first cell thickness near the wall being about 1/200 of the total.