# Model for Wall Shear Stress from Obliquely Imping Planar Underexpanded Jets
## 2 Methodology
### Computational Method  
The scientific literature does not report the experimental data or DNS related to the wall jet developed from compressible impinging jets. To compute the flow properties needed for the estimation of shear stresses, the CFD simulations solve the steady-state Favre-Averaged Navier-Stokes equations:  
\[\frac{\partial(\overline{\rho}\widetilde{u}_{i})}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial\big{(}\overline{\rho}\widetilde{u}_{i}\widetilde{u}_{j}\big{)} }{\partial x_{i}}=-\frac{\partial\overline{p}}{\partial x_{i}}+\frac{\partial \overline{\tau_{ij}}}{\partial x_{j}}-\frac{\partial\big{(}\overline{\rho} \widetilde{u}_{i}^{\prime\prime}\widetilde{u}_{j}^{\prime\prime}\big{)}}{ \partial x_{j}} \tag{2}\]  
\[\frac{\partial}{\partial x_{j}}\bigg{(}\overline{\rho}\widetilde{u}_{j}\bigg{(} \widetilde{u}+\frac{1}{2}\widetilde{u}_{i}\widetilde{u}_{j}\bigg{)}+ \widetilde{u}_{j\beta}\widetilde{u}_{i}^{\prime\prime}\widetilde{u}_{j}^{ \prime\prime}\bigg{)}=\frac{\partial}{\partial x_{j}}\bigg{(}\widetilde{u}_{i }\big{(}\overline{\tau_{ij}}-\overline{\rho}\widetilde{u}_{i}^{\prime\prime} \widetilde{u}_{j}^{\prime\prime}\big{)}-\overline{\vartheta}-\overline{\rho u }_{j}^{\prime\prime}\widetilde{u}^{\prime\prime}+\overline{\tau_{ij}} \widetilde{u}_{i}^{\prime\prime}-\frac{1}{2}\overline{\rho}\widetilde{u}_{j} ^{\prime\prime}\widetilde{u}_{i}^{\prime\prime}\bigg{)} \tag{3}\]  
Numerical simulations were performed using ANSYS FLUENT 17.2 software (Ansys, Canonsburg, PA, USA). A second-order scheme was used to solve the pressure. A third-order monotonic upstream-centered scheme was used for density, momentum, and turbulence, which was necessary to avoid the effects of numerical viscosity (associated with the low-order schemes) on the jet dissipation, as well as pressure-strain relationship. Since the flow contains non-negligible changes in temperature, the Sutherland model, based on the kinetic theory of ideal gases and an idealized intermolecular force potential, was used for viscosity calculation. Turbulence closure models are challenged in modeling complex flow phenomena; however, [PERSON] et al. [37] demonstrated that \(k-\omega\) models result in good agreement with DNS when calculating the mean flow of planar impinging jets. The \(k-\omega\) shear stress transport (SST) model used in this work, uses \(k-\epsilon\) away from the wall in the free stream and free jet portions of the flow while using \(k-\omega\) near the wall to resolve the boundary layer. As demonstrated by [PERSON] et al. [38] and discussed by [PERSON] et al. [23], the \(k-\omega\) SST model [39] is a good choice for modeling underexpanded impinging jets while resolving the wall jet boundary layer. [PERSON] and [PERSON]. [40] also found \(k-\omega\) SST to be superior to other closure models when considering planar impinging jets. The computational grids contain \(\sim\)500,000 quadrilateral elements. The grid is constructed so that, along the impingement surface, the first node in the wall-normal direction is located within a \(y^{+}=1\) at the maximum shear stress location in every case. This is ensured by evaluating the maximum wall shear stress for the normal impingement case at the maximum NPR for each H/W ratio found in our previous work [24] and calculating the distance from the wall that yields \(y^{+}=1\) for this maximum wall shear. The \(\Delta x\) of the mesh is generated via biasing from \(\Delta x=5\Delta y_{wall}\) at the impingement point to \(\Delta x=50\Delta y_{wall}\) at the edge of the domain. This grid resolution ensures that the viscous sublayer is resolved for the entire domain. The maximum x-direction spacing gives an element aspect ratio of less than 50:1. Mesh independence was confirmed by a simulation with the highest wall shear stress case and doubled number of elements in each direction, such that the first node was placed at \(y^{+}=0.5\) still with a maximum aspect ratio of 50:1.