# Model for Wall Shear Stress from Obliquely Imping Planar Underexpanded Jets
## 3 Results  
The CFD data for the maximum wall shear stress was tabulated for each case and examined as a function of each jet parameter. As found in our previous work [24], supersonic  
Figure 3: Validation of \(\mathbf{k}-\omega\) SST for impinging jet modeling as shown originally in [PERSON] et al. [24]. (**a**) Comparison of the velocity profile at 8 jet widths downstream of impingement location against DNS from [PERSON] et al. [37]. Comparison with wall jet DNS simulation from [PERSON] et al. [41]. (**b**) Decay of half maximum velocity location. (**c**) Maximum velocity. (**d**) Wall shear stress. flow is still present in the boundary layer for \(H\)/\(W\) values of less than 30. As such, we did not include these cases in the analysis, and the relations developed in this work should only be applied for \(H\)/\(W\) values above 30. Figure 4 plots the normalized maximum wall shear stress against the nozzle pressure ratio for various conditions. The angle is coded by colors, and the height-to-width ratio is represented by marker type. When each independent parameter is fixed, the maximum wall shear stress increases with approximately the square root of NPR for all cases. Figure 4 plots the normalized maximum wall shear stress against the height to width ratio. As expected, the maximum wall shear stress decreases with increasing height to width ratio. When all other parameters are fixed, the wall shear stress is proportional to the inverse of the height to width ratio. Figure 4 illustrates that maximum wall shear stress decreases with the impingement angle. This relationship was found to resemble a function of the form: \(1-\sin\theta\). After analyzing the maximum wall shear stress as a function of jet parameters, we propose an equation for normalized maximum wall shear stress of the form  
\[\tau_{max}^{*}=\tau_{max,\ \theta=0}^{*}\left(1-a\left(\frac{H}{W}\right)^{ b}NPR^{c}\sin^{d}\theta\right). \tag{4}\]  
Figure 4: _Cont._  
To obtain the global fit based on the form of Equation (1), we first determine the normalized maximum wall shear stress for a normal impingement angle \(\theta=0^{\circ}\) as a function of height to width ratio, nozzle pressure ratio, and jet Reynolds number. Based on our previous work [23], we expect the Reynolds number term to be relevant as it accounts for the effective turbulent viscosity increase for jets with higher Reynolds numbers. This leads to increased energy dissipation in the free jet region and, thus, reduces the maximum wall shear stress after the impingement. Therefore, to describe the maximum wall shear stress at normal impingement, we propose the following equation:  
\[\tau^{*}_{max,\ \theta=0}=\alpha\left(\frac{H}{W}\right)^{\beta}NPR^{\gamma}Re^{\lambda} \tag{5}\]  
Least-squares fitting gives the following values for the coefficient and exponents:  
\[\alpha=9338.08\]  
\[\beta=-1.146\]  
\[\gamma=0.589\]  
\[\lambda=-0.301\]  
The prediction of maximum wall shear stress from a normal impingement of under-expanded planar jets using Equation (5) agrees with the CFD data with a maximum deviation of 5.7% and a mean deviation of 2.1%. Figure 5 plots the predicted maximum wall shear stress against the calculated value from CFD. After finding the normal impingement correlations, we calculated the coefficient and exponents for Equation (1). Least-squares fitting yields the following coefficients:  
\[a=0.208\]  
\[b=-0.276\]  
\[\tau_{max}^{*}=\tau_{max,\ \theta=0}^{*}\left(1-a\left(\frac{H}{W}\right)^{ b}NPR^{c}\sin^{d}\theta\right). \tag{4}\]0  
\[\tau_{max}^{*}=\tau_{max,\ \theta=0}^{*}\left(1-a\left(\frac{H}{W}\right)^{ b}NPR^{c}\sin^{d}\theta\right). \tag{4}\]1  
\[\tau_{max}^{*}=\tau_{max,\ \theta=0}^{*}\left(1-a\left(\frac{H}{W}\right)^{ b}NPR^{c}\sin^{d}\theta\right). \tag{4}\]2 Normalized maximum wall shear stress plotted against Nozzle parameters with impingement angle represented by shade and height to width ratio designated by the symbol (**a**) nozzle pressure ratio, (**b**) height-to-width ratio, (**c**) impingement angle.