## 2 Hybrid TPJB Lubricated with sCO\({}_{2}\) and Its Computational Model
### Thermohydrodynamic Model for Turbulent Compressible Fluid Flows  
The Reynolds equation for turbulent and compressible fluid flows governs pressure distribution in the film land area. Note that sCO\({}_{2}\) is a highly compressible fluid, and its large density and low viscosity result in a large Reynolds number:  
\[\frac{\partial}{\partial x}\left(\frac{G_{x}\rho h^{3}}{\mu}\frac{\partial p}{ \partial x}\right)+\frac{\partial}{\partial z}\left(\frac{G_{z}\rho h^{3}}{ \mu}\frac{\partial p}{\partial z}\right)=\frac{U}{2}\frac{\partial(\rho h)}{ \partial x}+\frac{\partial(\rho h)}{\partial t} \tag{1}\]  
where \(\rho\), \(h\), \(\mu\), \(p\), \(U\), \(t\), \(x\), and \(z\) denote the lubricant density, film thickness, viscosity, pressure, journal surface speed, time, circumferential coordinate, and axial coordinate, respectively. \(G_{x}\) and \(G_{z}\) are the turbulent coefficients empirically determined using the Hirs bulk flow model [25], which considers the inertia effects inherent to the flow turbulence:  
\[G_{x}=\frac{1}{2+m_{0}}G_{z},\ G_{z}=\frac{2^{1+m_{0}}}{n_{0}}Re^{-(1+m_{0})} \tag{2}\]  
where \(m_{0}\) (=\(-0.25\)) and \(n_{0}\) (=\(0.066\)) are empirical constants applicable exclusively to smooth surfaces. \(Re\) is the local flow Reynolds number dependent on the local density \(\rho\), viscosity \(\mu\), and film thickness \(h\):  
\[Re=\frac{\rho Llh}{\mu} \tag{3}\]  
The flow becomes turbulent when \(Re\) is greater than or equal to the critical Reynolds number \(Re_{c}\), which is equal to 1000 [15; 26; 27]. For \(Re\) less than the critical value, \(G_{x}\) and \(G_{z}\) are equal to 1/12 for the laminar flow. The equation of film thickness is modified from that in Ref. [28] by including the recess depth \(D_{r}\) to account for the depth of the recess:  
\[h=C_{p}+e\cos(\theta_{c}-\theta)-\left(r_{p}-\zeta\right)\cos(\theta-\theta_{ p})-\delta\big{(}R_{p}+t_{p}\big{)}\sin(\theta-\theta_{p})+D_{r} \tag{4}\]  
Figure 1: Schematic of a four-pad tilting-pad journal bearing (TPJB) in a load-between-pad (LBP) configuration with recesses for high-pressure supercritical carbon dioxide (sCO\({}_{2}\)) supply. where \(C_{p}\) (=\(R_{p}-R\)), \(e\), \(\theta_{c}\), \(\theta\), \(r_{p}\) (=\(R_{p}-R_{b}\)), \(\zeta\), \(\theta_{p}\), \(\delta\), \(t_{p}\), and \(D_{r}\) are the pad radial clearance, journal eccentricity, angle between the line passing through the bearing and journal centers and the horizontal axis \(X\), circumferential coordinate starting from the axis X, preload, pivot deflection, pivot circumferential location, pad tilt angle, pad thickness, and depth of the recess, respectively. \(R_{p}\), \(R_{b}\), and \(R_{j}\) are the pad, bearing, and journal radii, respectively. The mathematical model for the pivot deflection \(\zeta\) is detailed in Refs. [28; 29; 30]. Note that \(D_{r}\) is null at the film land region. In the recess region, externally pressurized sCO\({}_{2}\) with pressure \(p_{s}\) is provided through the orifice restrictor, causing a pressure drop from \(p_{s}\) to \(p_{re}\), which then flows over the land area on each pad's surface. The gaseous mass flow rate through the orifice restrictor \(Q_{o}\) for the choked and unchoked conditions are derived from the Bernoulli's equation considering mechanical energy balance between two arbitrary points on a streamline. The relations for both choked and unchoked conditions can be given from [18; 31] as follows:  
\[ \begin{split}&\text{Un}-\text{choked}:\frac{p_{re}}{p_{s}}> \big{(}\frac{2}{\kappa+1}\big{)}^{\kappa/(\kappa-1)}\\ & Q_{o}=C_{d}A_{o}\sqrt{R_{g}T_{s}}\sqrt{\frac{2\kappa}{\kappa-1} \bigg{[}\left(\frac{p_{re}}{p_{s}}\right)^{2/\kappa}-\left(\frac{p_{re}}{p_{s }}\right)^{(\kappa+1)/\kappa}\bigg{]}}\\ &\text{Choked}:\frac{p_{re}}{p_{s}}\leq\big{(}\frac{2}{\kappa+1} \big{)}^{\kappa/(\kappa-1)}\\ & Q_{o}=C_{d}A_{o}\sqrt{R_{g}T_{s}}\sqrt{\frac{2\kappa}{\kappa+1} }\big{(}\frac{2}{\kappa+1}\big{)}^{1/(\kappa-1)}\end{split}