3. CAPÍTULO 3
3.4. La violencia escolar como experiencia social
3.3.1 Unsteady pressure fluctuation
The pressure was monitored at the mid-point of the cavity floor for the two cases and the predicted history is shown in Figure 3.2(a). The pressure fluctuations were normalized
with respect to free stream speed of sound a∞ and density ρ∞. The cavity length was
employed as the relevant length scale to non-dimensionalise the time.
It can be seen from Figure 3.2(a) that the flow predicted on both the the fine and coarse computational grids has settled in a self-sustained oscillatory state. The variation of the pressure fluctuations with time for the coarse mesh is similar to that of the fine mesh. More insight can be obtained by performing a discrete Fourier transform (DFT) of the
signals to analyze their frequency contents. The∆St=0.0003 constant bandwidth power
spectra obtained are shown in Figure 3.2(b). The striking feature of the power spectra for both grid densities is that that they agree over a wide range of frequencies. This shows
that, firstly, the numerics are performing as required and, secondly, the same physical phenomenon is modelled with both computational meshes. The only major difference be- tween the power spectra of the coarse and of the fine mesh is that they appear to diverge at higher frequencies. This is due to the numerical dispersion and dissipation in the scheme that affects higher frequency waves, but it is to be expected that the finer mesh provides more spatial resolution in the higher end of the frequency-wavenumber spectrum. How-
ever, the underlying physics is within the Strouhal range of St =0.1 to St =1.0. It is
in this range of frequencies that the dominant shear layer oscillation modes appear. In fact, from Figure 3.2(b), the close agreement between the spectra extends up to the third
visible peak at St =0.99. The peaks at St =0.66 and St =0.99 are harmonics of the
fundamental mode. The predicted fundamental mode from equation (1.6) is St =0.34
and the computed Rossiter Mode 1 from both the fine and coarse meshes is 0.33. The amplitude of the first mode in the coarse mesh computation is 148.6 dB and the fine mesh calculation indicates an amplitude for the first mode of 149.4 dB. Therefore, the coarse mesh performs adequately with respect to the computed frequencies and amplitudes.
3.3.2 Time averaged mean flow
This section is concerned with comparing the mean flow results obtained from the coarse and fine grids.
Figure 3.3 shows the time-mean velocity profiles at the cavity upstream edge obtained from the fine and coarse grids. The predictions show that the oncoming boundary layer profiles are similar. Hence there is no dependency in the upstream mean flow on the level of mesh refinement at the separation point.
Figure 3.4 shows the pressure coefficient plotted along the cavity floor at y/D=−1.0.
The solid line shows the predictions from the fine mesh and the dashed line, the ones from the coarse mesh. It can be seen that the trends in the variation of the pressure coefficient are similar. The peak and trough are indicative of the presence of a large
time-mean re-circulating flow structure inside the cavity. The position of the minimum
pressure is computed at the same location for both the coarse and fine meshes at x/L=
0.69. Similarly, the position of the maximum pressure in both cases is the same atx/L=
0.83. However, theCp is lower in the fine mesh predictions at the maximum and the
minimum points. Specifically the difference in the predicted pressure minimum is 10.1%
and for the pressure maximum the difference is 14.8%. This shows that the aerodynamic
predictions are mesh dependent. Whereas the predictions show that the numerical model of the laminar cavity flow is not quantitatively mesh-converged, the fine mesh predictions do not show any deviation in trend with respect to the coarse mesh predictions. This indicates that the two models are reproducing the same flow structure and this adequate for the purpose of the current work, which is an investigation on the laminar cavity flow
instability mode shape atM∞=0.4,ReD=3000,L/D=2 andδ/D=0.2.
The local Mach number variation along y/D=−0.9 is shown in Figure 3.5. There is
a good agreement between the Mach number predictions from the coarse and fine mesh
computations, except atx/L=0.6 andx/L=0.83. Due to the lower due to the lowerCp
distribution predicted by the fine mesh model, the local flow speeds at these locations are slightly higher than in the calculation performed on the coarse mesh.
The streamlines for the coarse and fine mesh are shown in Figure 3.6(a) and Figure 3.6(b) respectively. The streamlines in the two cases show a good similarity in flow pattern. The separation and re-attachment points along the cavity floor can be observed to be at the same location. This is confirmed by the similar trends in pressure distribution along the cavity floor. The downstream clockwise and the upstream anti-clockwise re-circulation patterns are similar on both mesh refinement levels.
Figure 3.7(a) and 3.7(b) show the the time mean vorticity contours predicted on the coarse and fine mesh respectively. The vorticity is non-dimensionalised with respect to the free stream flow speed and the cavity length as the reference length scale. The shape and size of the large-scale structure inside the cavity are similar in both cases.
3.4 Summary
A grid sensitivity study is conducted on a two-dimensional laminar cavity CAA model us-
ing two mesh densities. The unsteady pressure predicted at x/L=0.5 on the cavity floor
show the same dominant oscillation mode of the cavity, which is the Rossiter mode 1. A DFT analysis of the predicted pressure history on the coarse and fine meshes show that the spectra are characterized by the same harmonic sequence and there is only a marginal difference in the amplitude of the dominant tones. The time averaged predictions showed that on both mesh densities, using the same solver and flow parameters, the same flow features are produced. The shear layer spans the mouth of the cavity with a dominant
re-circulating region centered at approximatelyx/L=0.65 from the leading edge. There-
fore, it has been shown that, for the selected test case, a grid independent flow pattern is obtained from the implemented flow solver. The results of the this study are used in the subsequent computations.