Linear stability analysis of compressible vortex flows considering viscous effects

Abstract

This study investigates the stability of compressible swirling wake flows including the viscous effects using linear stability theory. A spatial stability analysis is performed to evaluate the influence of the axial velocity deficit and circulation as well as the Reynolds number and Mach number as the main parameters that affect the instability. The growth rates of the unstable modes at several azimuthal wavenumbers are compared. The maximum growth rates and their dependency with respect to each parameter are analyzed. It is confirmed that the instability monotonically increases as the axial velocity deficit increases. For small axial velocity deficit, characteristics that are different from the results reported using inviscid analysis are identified and analyzed. Additionally, a decrease in instability is observed as the viscous and compressibility effects become stronger. In terms of circulation, it is confirmed that there is a certain region of circulation that exhibits maximum instability. The stability analysis is expected to serve as a part of a useful methodology for preliminary design and parametric study for engineering problems such as vortex generators in high-speed flows, owing to both efficiency and accuracy.

Graphical abstract

Appendices

Appendix A: Matrix elements of linear disturbance equations

The elements of coefficients matrices of linear disturbances equation (using $l_{j} =j+\frac{\lambda }{\mu })$

$$\begin{aligned}&\Gamma _{11} =\frac{\gamma M^{2}}{{{\overline{T}}} }\,\,\,\,\,\,\,\,\,\Gamma _{15} =-\frac{\bar{{\rho }}}{\bar{{T}}}\,\,\,\,\,\,\,\,\,\Gamma _{22} ={\overline{\rho }} \,\,\,\,\,\,\,\,\,\Gamma _{33} ={\overline{\rho }} \,\,\,\,\,\,\,\,\,\Gamma _{44} ={\overline{\rho }} \,\,\,\,\,\,\,\,\,\Gamma _{51} =-\left( {\gamma -1} \right) M^{2}\,\,\,\,\,\,\,\,\,\Gamma _{55} ={\overline{\rho }} \\&A_{11} =\frac{\gamma M^{2}}{{{\overline{T}}} }U\,\,\,\,\,\,\,\,\,A_{12} ={\overline{\rho }} \,\,\,\,\,\,\,\,\,A_{15} =-\frac{\bar{{\rho }}}{\bar{{T}}}U \\&A_{21} =1\,\,\,\,\,\,\,\,\,A_{22} ={\overline{\rho }} U\,\,\,\,\,\,\,\,\,A_{23} =-\frac{1}{Re}\frac{\partial \mu }{\partial T}\frac{\partial \bar{{T}}}{\partial r}-\frac{l_{1} }{Re}\frac{1}{r}{\overline{\mu }} \,\\&A_{32} =-\frac{l_{0} }{Re}\frac{\partial \mu }{\partial T}\frac{\partial {{\overline{T}}} }{\partial r}\,\,\,\,\,\,\,\,\,A_{33} ={\overline{\rho }} U\,\,\,\,\,\,\,\,\,A_{35} =-\frac{1}{Re}\frac{\partial \mu }{\partial T}\frac{\partial U}{\partial r}\\&A_{44} ={\overline{\rho }} U\\&A_{51} =-\left( {\gamma -1} \right) M^{2}U\,\,\,\,\,\,\,\,\,A_{53} =-\frac{2{\overline{\mu }} }{Re}\left( {\gamma -1} \right) M^{2}\frac{\partial U}{\partial r}\,\,\,\,\,\,\,\,\,A_{55} ={\overline{\rho }} U\\&B_{13} ={\overline{\rho }}\\&B_{22} =-\frac{1}{Re}\frac{\partial \mu }{\partial T}\frac{\partial \bar{{T}}}{\partial r}-\frac{{\overline{\mu }} }{R_{0} }\frac{1}{r}\,\,\,\,\,\,\,\,\,B_{25} =-\frac{1}{Re}\frac{\partial \mu }{\partial T}\frac{\partial U}{\partial r}\\&B_{31} =1\,\,\,\,\,\,\,\,\,B_{33} =-\frac{l_{2} }{Re}\frac{\partial \mu }{\partial T}\frac{\partial {{\overline{T}}} }{\partial r}-\frac{{\overline{\mu }} l_{2} }{Re}\frac{1}{r}\\&B_{44} =-\frac{1}{Re}\frac{\partial \mu }{\partial T}\frac{\partial \bar{{T}}}{\partial r}-\frac{{\overline{\mu }} }{R_{0} }\frac{1}{r}\,\,\,\,\,\,\,\,\,B_{45} =-\frac{1}{Re}\frac{\partial \mu }{\partial T}\left( {\frac{\partial V_{\theta } }{\partial r}-\frac{V_{\theta } }{r}} \right) \\&B_{52} =-\frac{2{\overline{\mu }} }{Re}\left( {\gamma -1} \right) M^{2}\frac{\partial U}{\partial r}\,\,\,\,\,\,\,\,\,B_{54} =-\frac{2{\overline{\mu }} }{Re}\left( {\gamma -1} \right) M^{2}\left( {\frac{\partial V_{\theta } }{\partial r}-\frac{V_{\theta } }{r}} \right) \,\,\,\,\,\,\,\,\,B_{55} =-\frac{2}{\Pr Re}\frac{\partial k}{\partial T}\frac{\partial \bar{{T}}}{\partial r}-\frac{{{\overline{k}}} }{\Pr Re}\frac{1}{r}\\&C_{11} =\frac{\gamma M^{2}}{{{\overline{T}}} }V_{\theta } \frac{1}{r}\,\,\,\,\,\,\,\,\,C_{14} ={\overline{\rho }} \frac{1}{r}\,\,\,\,\,\,\,\,\,C_{15} =-\frac{\bar{{\rho }}}{\bar{{T}}}V_{\theta } \frac{1}{r}\\&C_{22} ={\overline{\rho }} V_{\theta } \frac{1}{r}\,\,\,\,\\&C_{33} ={\overline{\rho }} V_{\theta } \frac{1}{r}\,\,\,\,\,\,\,\,\,C_{34} =-\frac{l_{0} }{Re}\frac{\partial \mu }{\partial T}\frac{\partial \overline{T} }{\partial r}\frac{1}{r}+\frac{{\overline{\mu }} l_{3} }{Re}\frac{1}{r^{2}}\,\,\,\,\,\,\,\,\,C_{35} =-\frac{1}{Re}\frac{\partial \mu }{\partial T}\left( {\frac{\partial V_{\theta } }{\partial r}-\frac{V_{\theta } }{r}} \right) \frac{1}{r}\\&C_{41} =\frac{1}{r}\,\,\,\,\,\,\,\,\,C_{43} =-\frac{1}{Re}\frac{\partial \mu }{\partial T}\frac{\partial \bar{{T}}}{\partial r}\frac{1}{r}-\frac{{\overline{\mu }} l_{3} }{Re}\frac{1}{r^{2}}\,\,\,\,\,\,\,\,\,C_{44} ={\overline{\rho }} V_{\theta } \frac{1}{r}\,\,\,\,\\&C_{51} =-\left( {\gamma -1} \right) MV_{\theta } \frac{1}{r}\,\,\,\,\,\,\,\,\,C_{53} =-\frac{2{\overline{\mu }} }{Re}\left( {\gamma -1} \right) M^{2}\left( {\frac{\partial V_{\theta } }{\partial r}\frac{1}{r}-\frac{V_{\theta } }{r^{2}}} \right) \,\,\,\,\,\,\,\,\,C_{55} ={\overline{\rho }} V_{\theta } \frac{1}{r}\\&D_{13} =\frac{\partial {\overline{\rho }} }{\partial r}+\frac{{\overline{\rho }} }{r}\\&D_{23} ={\overline{\rho }} \frac{\partial U}{\partial r}\,\,\,\,\,\,\,\,\,D_{25} =-\frac{1}{Re}\frac{\partial ^{2}\mu }{\partial T^{2}}\frac{\partial \bar{{T}}}{\partial r}\frac{\partial U}{\partial r}-\frac{1}{Re}\frac{\partial \mu }{\partial T}\left( {\frac{\partial ^{2}U}{\partial r^{2}}+\frac{1}{r}\frac{\partial U}{\partial r}} \right) \\&D_{31} =-\frac{\gamma M^{2}}{{{\overline{T}}} }\frac{V_{\theta } ^{2}}{r}\,\,\,\,\,\,\,\,\,D_{33} =-\frac{l_{0} }{Re}\frac{\partial \mu }{\partial T}\frac{\partial {{\overline{T}}} }{\partial r}\frac{1}{r}+\frac{{\overline{\mu }} l_{2} }{Re}\frac{1}{r^{2}}\,\,\,\,\,\,\,\,\,D_{34} =-2{\overline{\rho }} \frac{V_{\theta } }{r}\,\,\,\,\,\,\,\,\,D_{35} =\frac{\bar{{\rho }}}{\bar{{T}}}\frac{V_{\theta }^{2}}{r}\\&D_{43} ={\overline{\rho }} \frac{\partial V_{\theta } }{\partial r}+\overline{\rho }\frac{V_{\theta } }{r}\,\,\,\,\,\,\,\,\,\,D_{44} =\frac{1}{Re}\frac{\partial \mu }{\partial T}\frac{\partial \bar{{T}}}{\partial r}\frac{1}{r}+\frac{{\overline{\mu }} }{R_{0} }\frac{1}{r^{2}}\\&D_{45} =-\frac{1}{Re}\frac{\partial ^{2}\mu }{\partial T^{2}}\frac{\partial \bar{{T}}}{\partial r}\left( {\frac{\partial V_{\theta } }{\partial r}-\frac{V_{\theta } }{r}} \right) -\frac{1}{Re}\frac{\partial \mu }{\partial T}\left( {\frac{\partial ^{2}V_{\theta } }{\partial r^{2}}+\frac{1}{r}\frac{\partial V_{\theta } }{\partial r}-\frac{V_{\theta } }{r^{2}}} \right) \\&D_{53} ={\overline{\rho }} \frac{\partial {{\overline{T}}} }{\partial r}-\left( {\gamma -1} \right) M^{2}\frac{\partial {{\overline{p}}} }{\partial r}\,\,\,\,\,\,\,\,\,D_{54} =-\frac{2{\overline{\mu }} }{Re}\left( {\gamma -1} \right) M^{2}\left( {\frac{V_{\theta } }{r^{2}}-\frac{\partial V_{\theta } }{\partial r}\frac{1}{r}} \right) \\&D_{55} =-\frac{1}{\Pr }\frac{1}{Re}\left\{ {\frac{\partial ^{2}k}{\partial T^{2}}\left( {\frac{\partial \bar{{T}}}{\partial r}} \right) ^{2}+\frac{\partial k}{\partial T}\left( {\frac{\partial ^{2}{{\overline{T}}} }{\partial r^{2}}+\frac{\partial {{\overline{T}}} }{\partial r}\frac{1}{r}} \right) } \right\} \\&\qquad \quad \quad -\frac{1}{Re}\left( {\gamma -1} \right) M^{2}\frac{\partial \mu }{\partial T}\left\{ {\left( {\frac{\partial U}{\partial r}} \right) ^{2}+\left( {\frac{\partial V_{\theta } }{\partial r}} \right) ^{2}+\left( {\frac{V_{\theta } }{r}} \right) ^{2}-2\frac{\partial V_{\theta } }{\partial r}\frac{V_{\theta } }{r}} \right\} \\&V_{xx,22} =\frac{{\overline{\mu }} l_{2} }{Re}\,\,\,\,\,\,\,\,\,V_{xx,33} =\frac{{\overline{\mu }} }{Re}\,\,\,\,\,\,\,\,\,V_{xx,44} =\frac{{\overline{\mu }} }{Re}\,\,\,\,\,\,\,\,\,V_{xx,55} =\frac{{{\overline{k}}} }{\Pr Re} \\&V_{rr,22} =\frac{{\overline{\mu }} }{Re}\,\,\,\,\,\,\,\,\,V_{rr,33} =\frac{{\overline{\mu }} l_{2} }{Re}\,\,\,\,\,\,\,\,\,V_{rr,44} =\frac{\overline{\mu }}{Re}\,\,\,\,\,\,\,\,\,V_{rr,55} =\frac{{{\overline{k}}} }{\Pr Re}\\&V_{\theta \theta ,22} =\frac{{\overline{\mu }} }{Re}\frac{1}{r^{2}}\,\,\,\,\,\,\,\,V_{\theta \theta ,33} =\frac{\overline{\mu }}{Re}\frac{1}{r^{2}}\,\,\,\,\,\,\,\,\,V_{\theta \theta ,44} =\frac{{\overline{\mu }} l_{2} }{Re}\frac{1}{r^{2}}\,\,\,\,\,\,\,\,\,V_{\theta \theta ,55} =\frac{{{\overline{k}}} }{\Pr Re}\frac{1}{r^{2}}\\&V_{xr,23} =\frac{{\overline{\mu }} l_{1} }{Re}\,\,\,\,\,\,\,\,\,V_{xr,32} =\frac{{\overline{\mu }} l_{1} }{Re}\\&V_{x\theta ,24} =\frac{{\overline{\mu }} l_{1} }{Re}\frac{1}{r}\,\,\,\,\,\,\,\,\,V_{x\theta ,42} =\frac{{\overline{\mu }} l_{1} }{Re}\frac{1}{r}\,\\&V_{r\theta ,34} =\frac{{\overline{\mu }} l_{1} }{Re}\frac{1}{r}\,\,\,\,\,\,\,\,\,V_{r\theta ,43} =\frac{{\overline{\mu }} l_{1} }{Re}\frac{1}{r}\,\, \end{aligned}$$

Appendix B: Linear stability equation with $\alpha$ as an eigenvalue

$$\begin{aligned} \alpha ^{2}\left[ {{{\overline{V}}}_{2} \frac{\partial ^{2}\widehat{\phi }}{\partial r^{2}}+{{\overline{B}}}_{2} \frac{\partial {\widehat{\phi }}}{\partial r}+{{\overline{D}}}_{2} {\widehat{\phi }}} \right] +\alpha \left[ {{{\overline{V}}}_{1} \frac{\partial ^{2}{\widehat{\phi }}}{\partial r^{2}}+{{\overline{B}}}_{1} \frac{\partial {\widehat{\phi }}}{\partial r}+{{\overline{D}}}_{1} \widehat{\phi }} \right] +\left[ {{{\overline{V}}}_{0} \frac{\partial ^{2}\widehat{\phi }}{\partial r^{2}}+{{\overline{B}}}_{0} \frac{\partial {\widehat{\phi }}}{\partial r}+{{\overline{D}}}_{0} {\widehat{\phi }}} \right] =0 \end{aligned}$$

where $\bar{{V}}_{0} =-V_{rr} \quad \bar{{B}}_{0} =B-i(mV_{r\theta })$ $\bar{{D}}_{0} =D+m^{2}V_{\theta \theta } +i(mC-\omega \Gamma )$

$$\begin{aligned}&\bar{{V}}_{1} =0 \quad \bar{{B}}_{1} =-V_{xr} \quad \bar{{D}}_{1} =iA+mV_{x\theta }\\&\bar{{V}}_{2} =0 \quad \bar{{B}}_{2} =0 \quad \bar{{D}}_{2} =V_{xx} \end{aligned}$$