Grid sensitivity and role of error in computing a lid-driven cavity problem

The investigation on grid sensitivity for the bifurcation problem of the canonical lid-driven cavity (LDC) flow results is reported here with very fine grids. This is motivated by different researchers presenting different first bifurcation critical Reynolds number (${\text{Re}}_{\text{cr}1}$), which appears to depend on the formulation, numerical method, and choice of grid. By using a very-high-accuracy parallel algorithm, and the same method with which sequential results were presented by Lestandi et al. [Comput. Fluids 166, 86 (2018)] [for (257 $\times$ 257) and (513 $\times$ 513) uniformly spaced grid], we present results using ($1025\times 1025$) and ($2049\times 2049$) grid points. Detailed results presented using these grids help us understand the computational physics of the numerical receptivity of the LDC flow, with and without explicit excitation. The mathematical physics of the investigated problem will become apparent when we identify the roles of numerical errors with the ambient omnipresent disturbances in real physical flows as interchangeable. In physical or in numerical setups, presence of disturbances cannot be ignored. In this context, the need for explicit excitation for the used compact scheme arises for a definitive threshold amplitude, below which the flow relaxes back to quiescent state after the excitation is removed in computations. We also implement the present parallel method to show the physical aspects of primary and secondary instabilities to be maintained for other numerical schemes, and we show the results to reflect the complex physics during multiple subcritical Hopf bifurcation. Also, we relate the various sources of errors during computations that is typical of such shear-driven flow. These results, with near spectral accuracy, constitute universal benchmark results for the solution of Navier-Stokes equation for LDC.

Figure 1

The schematic of the computational domain and location of sampling point (0.95,0.95) with vortical excitation applied at (0.02,0.98) for a ($257\times 257$) grid.

Figure 2

Comparison of computed wall vorticity distribution—${\omega }_b$ at the top wall for the LDC flow at $\text{Re}=9600$ at time $t=1750$ for ($257\times 257$), ($513\times 513$), and ($1025\times 1025$) grids.

Figure 3

Illustration of the vorticity dynamics at supercritical $\text{Re}=8750$ for the LDC problem solved using a ($257\times 257$) grid. The plotted vorticity time series is sampled at (0.95,0.95). Vorticity contours plotted at the indicated times shows a well defined triangular vortex during transient stages.

Figure 4

Illustration of the vorticity dynamics at super-critical $\text{Re}=9600$ for the LDC problem solved using a ($1025\times 1025$) grid. The plotted vorticity time series is sampled at $(x=0.95,y=0.95)$. Vorticity contours plotted at the indicated times show well defined pentagonal and triangular vortices during the transient stages.

Figure 5

Plot of vorticity time series and vorticity contours showing the dynamics of flow inside LDC at different times for $\text{Re}=9600$ for (a) ($257\times 257$), (b) ($513\times 513$), and (c) ($1025\times 1025$) grids.

Figure 6

Plot of vorticity time series and vorticity contours showing the dynamics of flow inside LDC at different times for $\text{Re}=9700$ for (a) ($257\times 257$), (b) ($513\times 513$), and (c) ($1025\times 1025$) grids.

Figure 7

Plot of vorticity time series obtained at location (0.95,0.95) and its corresponding FFT spectrum for (a) $\text{Re}=9600$ and (b) $\text{Re}=9700$.

Figure 8

Plot of vorticity time series at location (0.95,0.95) for the external vortical excitation imposed at (0.02,0.98) for subcritical Reynolds numbers. The results are for ($1025\times 1025$) grid and the amplitude of excitation is indicated by $A_{\text{ex}}$.

Figure 9

Vorticity time series for $\text{Re}=9400$ at the indicated excitation amplitudes and the band of threshold amplitude of vortical excitation for the ($1025\times 1025$) grid.

Figure 10

Threshold amplitudes of pulsating vortical excitation needed for the three grids used here, having ($257\times 257$), ($513\times 513$), and ($1025\times 1025$) points.

Figure 11

Fast Fourier transform of the vorticity time series for $\text{Re}=8500$ and amplitude of excitation $A_{\text{ex}}=1$ for different excitation frequencies: $f_o=0.173$, $f_o=0.41$, and $f_o=0.6133$, for $257\times 257$ grid.

Figure 12

Bifurcation diagram for the three grids with ($257\times 257$), ($513\times 513$), and ($1025\times 1025$) points.

Figure 13

Comparison of wall vorticity distribution at the top lid for $\text{Re}=10000$ at $t=1800$—($1025\times 1025$) and ($2049\times 2049$) grid points by using NCCD and Lele scheme.

Figure 14

Plot of vorticity time series and vorticity contours showing the dynamics of flow inside LDC at different times for $\text{Re}=10000$ for (a) ($1025\times 1025$—NCCD Scheme), (b) ($1025\times 1025$—Lele Scheme), and (c) ($2049\times 2049$—NCCD Scheme).

Figure 15

Fourier transform of vorticity time series for $\text{Re}=10000$ using two grids: ($1025\times 1025$) and ($2049\times 2049$) with NCCD and Lele's schemes.