## 2 Methods  
### Baseline RANS Model  
In our study, we utilize the Spalart-Allmaras (SA) turbulence model without the trip terms \(f_{t1}\) and \(f_{t2}\). The original SA model allows for the imposition of transition locations via the trip term \(f_{t1}\) as defined in [28]. However, this approach is seldom used in the literature because the transition location is generally unknown a priori. Additionally, activating \(f_{t1}\) necessitates the use of a second term, \(f_{t2}\), to delay natural transition and make \(f_{t1}\) effective. By omitting these trip terms, the SA model performs well at high Reynolds numbers but struggles to predict transition and separation at lower Reynolds numbers. This often resultsin an overestimation of turbulent eddy viscosity for such flows, making the model a suitable baseline for testing the field inversion approach and evaluating potential improvements. To address situations where the turbulence eddy viscosity may become negative, particularly in coarse grids and transient states, we employ the negative Spalart-Allmaras model as proposed by [PERSON] et al. in [27]. The negative SA model is designed to handle undershoots in the turbulence solution that might occur at the edges of boundary layers and wakes. This is achieved through a continuation of the SA model into the realm of negative \(\tilde{v}\) solutions. This formulation ensures non-negative solutions and addresses issues related to under-resolved grids and non-physical transient states in discrete settings. The governing equations used in our simulations are the mass-averaged RANS equations, which are expressed as follows:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho\mathbf{u})=0 \tag{1}\]  
\[\frac{\partial\rho\mathbf{u}}{\partial t}+\
abla\cdot(\rho\mathbf{u}\mathbf{u })=-\
abla p+\
abla\cdot\boldsymbol{\tau} \tag{2}\]  
\[\frac{\partial E}{\partial t}+\
abla\cdot(\mathbf{u}(E+p))=\
abla\cdot( \boldsymbol{\tau}\cdot\mathbf{u}-\mathbf{q}) \tag{3}\]  
\[\frac{\partial\rho\tilde{v}}{\partial t}+\
abla\cdot(\rho\mathbf{u}\tilde{v})= \rho(P-D)+\frac{1}{\sigma}\
abla\cdot(\rho(v+\tilde{v})\
abla\tilde{v})+\frac {cb_{2}}{\sigma}\rho(\
abla\tilde{v})^{2}-\frac{1}{\sigma}(v+\tilde{v})\
abla \rho\cdot\
abla\tilde{v} \tag{4}\]  
Since we are using the negative S-A model, in cases where \(\tilde{v}\) becomes negative, the following equation is solved instead of (4):  
\[\frac{\partial\rho\tilde{v}}{\partial t}+\
abla\cdot(\rho\mathbf{u}\tilde{v}) =\rho(P_{n}-D_{n})+\frac{1}{\sigma}\
abla\cdot(\rho(v+\tilde{v}f_{n})\
abla \tilde{v})+\frac{cb_{2}}{\sigma}\rho(\
abla\tilde{v})^{2}-\frac{1}{\sigma}(v +\tilde{v}f_{n})\
abla\rho\cdot\
abla\tilde{v} \tag{5}\]  
where \(\rho\), \(\mathbf{u}\), \(p\), \(E\), \(\
u\), \(\tilde{v}\), \(\mathbf{x}\), and \(t\) denote density, velocity, pressure, total energy per unit volume, molecular viscosity, modified eddy viscosity, spatial position, and time, respectively. A fluid with a constant specific heat ratio \(\gamma\) and constant viscosity is considered. The equation for the energy is given by:  
\[E=\frac{p}{\gamma-1}+\frac{1}{2}\rho|\mathbf{u}|^{2} \tag{6}\]  
The viscous stress tensor \(\boldsymbol{\tau}\) includes contributions from both the molecular and eddy viscosity. Its components are given by:  
\[\tau_{ij}=2\rho(v+\tilde{v}f_{v1})\left(\frac{\partial u_{i}}{\partial x_{j} }+\frac{\partial u_{j}}{\partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u _{k}}{\partial x_{k}}\right) \tag{7}\]  
where \(v\) denotes molecular viscosity, \(\tilde{v}\) denotes modified eddy viscosity, and \(f_{v1}\) is a function of the viscosity ratio.