Preliminary study on Reynolds stress model based on <i>ν</i><sub><i>t</i></sub>-scale equation

Abstract

Reynolds stress model has always been the frontier and challenging problem in turbulence model theory research, where improving numerical robustness is the key to its wide application in engineering. Referring to the classical <inline-formula><tex-math id="M8">\begin{document}$k$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220417_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220417_M8.png"/></alternatives></inline-formula>-<inline-formula><tex-math id="M9">\begin{document}$kL$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220417_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220417_M9.png"/></alternatives></inline-formula> turbulence model, a new <inline-formula><tex-math id="M10">\begin{document}${\nu_t}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220417_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220417_M10.png"/></alternatives></inline-formula>-scale equation is constructed and used to couple the SSG/LRR model to form a so-called SSG/LRR-<inline-formula><tex-math id="M11">\begin{document}${\nu_t}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220417_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220417_M11.png"/></alternatives></inline-formula> Reynolds stress model. Four benchmark cases, including zero pressure gradient turbulent plate boundary layer, airfoil wake flow, supersonic square duck flow and separated flow over NACA0012 airfoil at 45 degree angle of attack, are carried out to test the new turbulence model. At the same time, high-order numerical schemes are used to discretize the turbulence equations in order to assess its numerical robustness. The results are compared with those of SA eddy viscosity model and SSG/LRR-<inline-formula><tex-math id="M12">\begin{document}$\omega$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220417_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220417_M12.png"/></alternatives></inline-formula> Reynolds stress model. It is shown that the <inline-formula><tex-math id="M13">\begin{document}${\nu_t} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220417_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220417_M13.png"/></alternatives></inline-formula>-scale equation is strictly equal to zero at the viscous wall boundary. Compared with the traditional <inline-formula><tex-math id="M14">\begin{document}$\omega $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220417_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220417_M14.png"/></alternatives></inline-formula>-scale, it has better numerical robustness. Along with this, the new model can be matched with the high-order numerical schemes and obtain a better efficiency in the mesh convergence. Moreover, the new model has the inherent advantage of Reynolds stress model in simulating the corner flow and has the potential in scale adaptive simulation of unsteady separated flow.