Improvement of the parabolized stability equation to predict the linear evolution of disturbances in three-dimensional boundary layers based on ray tracing theory

Three-dimensional (3D) boundary layers are common flows in aircraft but present many problems to instability analysis and transition prediction. The difficulties of 3D boundary layers are reviewed and a method is proposed to predict the linear evolution of infinitesimal perturbations in 3D boundary layers, named as RTPSE, in which the line-marching parabolized stability equation (PSE) is improved by applying the ray tracing (RT) theory. Two major improvements are achieved. One is that the marching line is predefined along the direction of group velocity, which is related to both the characteristic line of local dispersion relation and the direction of energy propagation. Another is that the variation of the real part of the spanwise wave number is predicted by RT theory, while its imaginary part is determined based on the conservation relation of generalized growth rate. The implementation of RTPSE for 3D boundary layers is given in detail and involves linear stability theory, the PSE, and RT theory. Both the tracing ray and spanwise wave number are calculated in the real number space, only leading to a second-order error. Direct numerical simulation is performed to verify and validate the prediction by RTPSE in a 3D supersonic boundary layer on a blunt cone with a half angle of $7^{\circ }$ and an angle of attack of $9^{\circ }$. Results show that RTPSE can accurately predict the variation of spanwise wave number and linear evolution of disturbances for the whole wave packet, for stationary crossflow waves and for traveling crossflow waves, while the traditional PSE cannot. The application condition of RT theory is investigated numerically, and the caustic does not occur for unstable disturbances, implying that RTPSE is fully applicable to predict the linear evolution of disturbances in 3D boundary layers.

Figure 1

Schematic of a blunt cone with an AoA.

Figure 2

Validation of DNS: (a) pressure at the wall for 2D flow compared with Zhong and Ma [38] and (b) density along the wall-normal direction for 3D flow compared with Balakumar and Owens [27].

Figure 3

Effect of grid resolution on (a) the basic flow and (b) the growth rate of the most unstable stationary crossflow wave at $x=187\mathrm{mm}$, $\varphi ={90}^{\circ }$, where $\varphi$ denotes the azimuthal angle. The relative error is defined as $\delta =({\sigma }_e-\sigma )/{\sigma }_e\times 100$, where ${\sigma }_e$ is the growth rate of the extra fine grid.

Figure 4

Sketch map of the (a) computation domain and (b) computation mesh in the $x\text{−}\varphi$ plane for disturbance simulation.

Figure 5

Wave form of the stationary crossflow wave in the axial direction. (a) Contours of $u^{\prime }$. (b) Value of $u^{\prime }$ (dashed line) and its amplitude (solid line) at $\varphi ={120}^{\circ }$.

Figure 6

Instability analysis and $N$-factor distribution: (a) growth rate distribution in the $n\text{−}\omega$ plane at $x=195\mathrm{mm}$, $\varphi ={105}^{\circ }$; (b) growth rate distribution for the second Mack mode instability with high frequency $\omega =2.9$; (c) $N$-factor distribution for the traveling crossflow instability with low frequency $\omega =0.1$; and (d) $N$-factor distribution for the stationary crossflow instability with zero frequency $\omega =0$. The line in (b)–(d) is the integral line or marching line the direction of which is in the direction of the group velocity.

Figure 7

DNS result of the stationary crossflow wave: (a) contour of the peak value in the wall-normal direction of $u^{\prime }$ and (b) contour of the amplitude of $u^{\prime }$. The symbols represent the location of the hump of the amplitude in the spanwise direction.

Figure 8

Distribution of $u^{\prime }$ at different locations obtained by DNS: (a) $x=93\mathrm{mm}$, (b) $x=140\mathrm{mm}$, (c) $x=190\mathrm{mm}$, and (d) $x=240\mathrm{mm}$. ${\varphi }_p$ is the location of the hump of the wave packet.

Figure 9

Comparison of the disturbance evolution paths between DNS, RT, and the outer streamline for stationary crossflow waves. Circle symbols: position of the wave crest in the spanwise direction. Triangle symbols: the outer streamline. Dashed lines: tracing rays starting from different points at the inlet.

Figure 10

Comparison of the azimuthal wave number $n$ among RTPSE, PSE, and DNS for stationary crossflow waves: (a) ray 1, (b) ray 2, (c) ray 3, and (d) ray 4.

Figure 11

Comparison of the amplitude ratio $N=ln(A/A_{x=110})$ among RTPSE, PSE, and DNS for stationary crossflow waves: (a) ray 1, (b) ray 2, (c) ray 3, and (d) ray 4.

Figure 12

DNS result of traveling crossflow waves: (a) transient contour of the peak value in the wall-normal direction of $u^{\prime }$ and (b) contour of the amplitude of $u^{\prime }$. The symbols represent the location of the hump of amplitude in the spanwise direction.

Figure 13

Comparison of the disturbance evolution path between DNS and RT for traveling crossflow waves. Circle symbols: position of the wave crest in the spanwise direction. Dashed lines: path along group velocity from different starts.

Figure 14

Comparison of the azimuthal wave number $n$ among RTPSE, PSE, and DNS for traveling crossflow waves: (a) ray 1, (b) ray 2, (c) ray 3, and (d) ray 4.

Figure 15

Comparison of the amplitude ratio $N=ln(A/A_{x=110})$ among RTPSE, PSE, and DNS for traveling crossflow waves: (a) ray 1, (b) ray 2, (c) ray 3, and (d) ray 4.

Figure 16

Analysis of the caustics condition by contours of $lg\left|\mathrm{\Delta }\right|$ at $x=200\mathrm{mm}$ for different $\varphi$: (a) $\varphi =0^{\circ }$, (b) $\varphi ={30}^{\circ }$, (c) $\varphi ={60}^{\circ }$, (d) $\varphi ={90}^{\circ }$, (e) $\varphi ={120}^{\circ }$, and (f) $\varphi ={150}^{\circ }$.