# The Role of Buoyancy Induced Instability in Transpirational Cooling Applications
## 2 Mathematical Model
### Governing Equations
In many transpiration cooling applications, the fuel is often repurposed as the coolant. These fuels, e.g., hydrogen, methane, and other heavier alkanes, are delivered as liquids at high pressures at or above the thermodynamic critical point of the fluid. As the fluid is transpired through the porous material, heating occurs which often causes the fluid to become supercritical (if it is not supercritical to begin with) and causes a reduction in the fluid density. Density gradients are highly nonlinear near the thermodynamic critical point and in the supercritical regions. Accordingly, the stability behavior can be significantly different from systems with linear dependencies. In order to capture these nonlinear effects, density is modeled as a third order polynomial, which qualitatively captures the real-fluid behavior of the coolant as illustrated in Figure 2. This is often considered as the nonlinear Boussinesq approximation [22; 23]. Figure 2 shows non-dimensional reduced density, \(\rho_{r}=\frac{\rho}{\rho c}\), versus non-dimensional reduced temperature, \(T_{r}=\frac{T^{*}}{T_{c}}\), of methane and hydrogen generated using real-fluid equations of state implemented through REFPROP [24]. The subscript 'c' represents the property at the critical point. Figure 2 shows methane and hydrogen for sweep of reduced pressures from \(P_{r}=1\) to \(P_{r}=\)5 - a typical pressure range for coolant delivery [25]. A least squares fit using a third order polynomial is shown in Figure 2 as the solid black line. Figure 3 shows typical conditions cooling the throat of a rocket engine using methane with an at in inlet temperature of 105 K and a wall temperature of 755 K with the density (red-dashed line), the polynomial curve fit (black solid line), and the linear fit (blue small-dashed line) [25]. The values are also used for the fluid properties in Table 1. Equation (2) shows the third order polynomial:  
\[\rho=\rho_{0}(1-\beta\Delta T\big{(}T+c_{1}T^{2}+c_{2}T^{3}\big{)} \tag{2}\]  
where the non-linear portion can be separated as the function \(f(T)=c_{1}T^{2}+c_{2}T^{3}\). In Equation (2), \(T\) is the non-dimensional scaling of temperature where \(T=\frac{T^{*}-T_{0}}{\Delta T}\), \(\beta\) is an average coefficient of thermal expansion, and \(\Delta T=T_{w}-T_{0}\). It is worth noting that \(\beta\) is different for the linear case compared to the non-linear case. As a result, it varies from the linear case, where \(\beta=0.00145\) to the non-linear curve fit where \(\beta_{2}=0.0057\). To appropriately compare to linear and non-linear cases, a scale factor, \(\beta/\beta_{2}=3.97\), must be accounted for in the results and will be discussed in the later sections. Figure 3: Dimensionless density versus dimensionless temperature is shown (red-dashed line) for methane at typical conditions of \(T_{0}=105\) K and \(T_{\text{\tiny{tr}}}=755\) K [25]. A third order polynomial is used to fit the data (black solid line). The linear fit is shown as the blue small-dashed line. Figure 2: Reduced density (\(\frac{\rho}{\rho_{\text{\tiny{r}}}}\)) versus reduced temperature (\(\frac{\Gamma}{T_{\text{\tiny{r}}}}\)) for hydrogen (blue circle) and methane (red square) for a sweep of reduced pressures range from 1 to 5. The black solid line is a third order approximation of the thermodynamic data. The system of equations in Equation (1) are non-dimensionalized by introducing the following scaling variables:  
\[\mathbf{x}=\mathbf{x}^{*}/d\quad,\mathbf{u}=\mathbf{u}^{*}\frac{v^{2}}{Kg\beta\Delta T},\quad T= \frac{T^{*}-T_{0}}{\Delta T},\quad t=t^{*}\frac{Kg\beta\Delta T}{\
u L}. \tag{3}\]  
The system of equations in Equation (1) are rewritten as:  
\[ \begin{cases}\
abla\cdot\mathbf{u}=0\\ \
abla\times\mathbf{u}+\xi\
abla\times(\mathbf{u}|\mathbf{u}|)=\
abla\times T \mathbf{e_{z}}+\
abla\times f\mathbf{e_{z}}\\ \frac{\partial T}{\partial t}+\frac{1}{\sigma}\mathbf{u}\cdot\
abla T=\gamma \
abla^{2}T\\ z=0:\quad\mathbf{u}=<0,0,-\lambda>,\quad T=1\\ z=1:\quad\mathbf{u}=<0,0,-\lambda>,\quad T=0\end{cases}