# Model for Wall Shear Stress from Obliquely Imping Planar Underexpanded Jets
## 4 Conclusions  
We report empirical formulation for predicting the maximum wall shear stress resulting from obliquely impinging under-expanded planar jets as a function of four jet parameters: the standoff height (\(\mathbf{H}\)), the jet width (\(\mathbf{W}\)), the jet nozzle pressure ratio (\(\mathbf{ NPR}\)), and the jet impingement angle (\(\mathbf{\theta}\)). The empirical relationship for the normal wall shear stress was developed in the form \(\tau^{*}_{max,~{}\theta=0}=\alpha\left(\frac{H}{W}\right)^{\beta}\mathbf{ NPR}^{\gamma}Re^{\lambda}\) with the coefficients (\(\mathbf{\alpha}=9338.08\), \(\mathbf{\beta}=-\mathbf{1.146}\), \(\mathbf{\gamma}=\mathbf{0.589}\), \(\lambda=-\mathbf{0.301}\)). The jet angle correction was  
Figure 5: Predicted maximum normalized wall shear stress plotted against maximum normalized wall shear stress from CFD for normal impingement. Shade represents the nozzle pressure ratio, whereas the symbol represents the height-to-width ratio.  
Figure 6: Predicted maximum normalized wall shear stress plotted against maximum normalized wall shear stress from CFD for all impingement angles. Shade represents the nozzle pressure ratio, the symbol represents the impingement angle, and the marker size represents the height-to-width ratio.  
implemented in the form \(\tau^{*}_{max}=\tau^{*}_{max}\), \(\theta=0\left(\mathbf{1}-\boldsymbol{a}\left(\frac{H}{W}\right)^{\mathbf{b}} \mathbf{N}\boldsymbol{PR}^{c}\mathbf{sin}^{d}\boldsymbol{\theta}\right)\) with the coefficients (\(\boldsymbol{a}=\mathbf{0.208}\), \(\boldsymbol{b}=-\mathbf{0.276}\), \(\boldsymbol{c}=\mathbf{1.213}\), \(\boldsymbol{d}=\mathbf{1.512}\)) determined based on the least square fit of the shear stresses calculated using scale resolved CFD simulations. The global expression can predict wall shear stress within a mean error of 3.3%. The formulation can be used to estimate particle resuspension rates and heat transfer rates when correlated with wall shear stresses.  
There are limitations to the relationships developed in this work. These expressions will only hold for subsonic boundary layer flow and, as such, should not be extended to use for \(H/W\) of less than 30 or \(N\boldsymbol{PR}\) above 3.4 without the expectation of error induced due to compressibility. Additionally, the expressions should not be extended to impingement angles above \(60^{\circ}\) as the characteristics of the system change from an impinging jet to a wall jet, and the expressions will no longer apply.  
Conceptualization, P.F. and I.V.N.; methodology, P.F.; software, P.F. and A.V.; validation P.F.; formal analysis, P.F.; investigation, P.F. and A.V.; resources, I.V.N.; data curation, P.F.; writing--original draft preparation, P.F. and A.V.; writing--review and editing, I.V.N.; visualization, P.F.; supervision, I.V.N.; project administration, I.V.N.; funding acquisition, I.V.N. All authors have read and agreed to the published version of the manuscript.  
This research was funded by DHS Science and Technology Directorate, and UK Home Office; contract no. HSHQDC-15-C-B0033.  
Not applicable.  
The data used in this study will be made available upon reasonable request to the corresponding author.  
This work was facilitated by using advanced computational, storage, and the networking infrastructure provided by the Hyak supercomputer system at the University of Washington.  
The authors declare no conflict of interests.