# The Role of Buoyancy Induced Instability in Transpirational Cooling Applications
## 5 Results
### Case 2: \(g\
eq 0\) and \(\lambda=0\)  
\[\begin{cases}[D^{2}-\alpha^{2}]W=-\alpha^{2}(1+g)\Theta\\ [D^{2}-\alpha^{2}]\Theta=-\Gamma_{2}W\end{cases} \tag{14}\]  
Next, the effects of non-linear density are added and the resulting system is shown in Equation (14). As mentioned previously, when a non-linear base density distribution is used, \(\beta\) is different than in Case 1. Thus, a scaling factor is needed to equate the \(\Gamma\) from case 1 to the \(\Gamma\) in Case 2. The \(\Gamma\) for Case 2 will be referred to as \(\Gamma_{2}\) and is related to Case one by \(\Gamma=\Gamma_{2}\frac{\beta}{\beta_{2}}\).  
The above system is solved using the three numerical methods that were validated for Case 1 above. As shown in Figure 4, these methods again are all in agreement in their prediction of the marginal stability curve. Due to the additional terms in Equation (14), the power series method converges slower. The Chebyshev method converges the fastest out of the three methods and thus is used for comparison. To demonstrate the agreement between the finite difference and Chebyshev, the mesh is increase by a factor of 10. Table 3 demonstrates the degree of convergence between the three methods for Case 2.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Power Series** & **Chebyshev** & **Finite Difference** \\ \hline Maximum Error [\%] & \(1.92\times 10^{-13}\) & \(6.9\times 10^{-7}\) & \(0.74\) \\ RMS Error [\%] & \(8.6\times 10^{-14}\) & \(3.3\times 10^{-7}\) & \(0.44\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Relative error of the three different methods compared to the analytical solution.  
Figure 4: The stability curves are illustrated for Case 1 and 2 for the three different solution techniques. The right axis plots \(\Gamma_{2}\) where \(\Gamma_{2}=\frac{\beta_{2}}{\beta}\Gamma\).  
Figure 4 shows the marginal stability curve shift to a higher \(\Gamma\) compared to Case 1, indicating a stabilizing effect. Figure 5 helps provide insight to why the non-linear density profile results in a stabilizing effect. Accounting for non-linear changes in density yields a rapid decrease in the initial density near \(z=1\). Thus, there is a lower amount of heavy fluid over-top the lighter fluid at the bottom (\(z=0\)). The end results is a higher Grashof number occurring at a larger wavenumber. The \(\Gamma_{c}\) and \(\alpha_{c}\) for Case 2 are 47.51 and 3.57, respectively.  
Next, the normalized velocities for each case are compared in Figure 6. The non-linear density base profile causes a shift in the sinusoidal shape toward the top of the domain. The overturning convective velocity currents appears to be biased to where the steepest density gradient occur (see Figure 5). Thus, most of the convective mixing inside the porous media will be shifted towards the top of the domain. In addition, the greater wavenumbers in Case 2 promote tighter convective cell formation. These tighter packed cells will ultimately enhance mixing near the coolant side of the porous media once instability ensues.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Power Series** & **Finite Difference** \\ \hline Maximum Error [\%] & 0.93 & \(4.6\times 10^{-3}\) \\ RMS Error [\%] & 0.29 & \(3.22\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Relative error of the power series and finite difference method compared to the Chebyshev method.  
Figure 5: Density versus domain depth is plotted for the non-linear density (red-dashed line) and linear density (blue small-dashed line) profiles.