2.1. Direct numerical simulations

The Navier–Stokes equations are the governing equations for the DNS in this study, which are in the form of

(2.1)\begin{equation} \frac{\partial \boldsymbol{Q}}{\partial t}+\frac{\partial \boldsymbol{F}_{c,j}}{\partial x_j}=\frac{1}{Re} \frac{\partial \boldsymbol{F}_{v,j}}{\partial x_j} \quad (\,j=1,2,3). \end{equation}

The conservative variables $\boldsymbol {Q}$, convective flux vectors $\boldsymbol {F}_{c,j}$ and viscous flux vectors $\boldsymbol {F}_{v,j}$ in the DNS are described in more detail in Zhou et al. (Reference Zhou, Liu, Lu, Kang and Yan2022a), and Re is the Reynolds number. These simulations are carried out using the OpenCFD code developed by Li et al. (Reference Li, Fu, Ma and Liang2010b), which has been used to study high-speed boundary layer transitions (Li, Fu & Ma Reference Li, Fu and Ma2010a) and turbulent flows (Li et al. Reference Li, Tong, Yu and Li2019). The convective terms are discretized spatially using an optimized sixth-order monotonicity-preserving scheme (Li, Leng & He Reference Li, Leng and He2013) combined with the Steger–Warming splitting method. The viscous terms are discretized using a sixth-order central difference scheme. A third-order Runge–Kutta method (Jiang & Shu Reference Jiang and Shu1996) is used for time advancement. The non-reflection and supersonic outlet boundary conditions are used at the top and outlet of the computational domain, respectively. A no-slip isothermal wall is used at the bottom. Periodic boundary conditions are utilized on both sides of the domain. At the inlet, the profiles are solved using the compressible Blasius equations, validated in our previous study (Zhou et al. Reference Zhou, Liu, Lu, Wang and Yan2022b).

2.2. Stability analysis

As mentioned in § 2.1, the flow variables are described by the Navier–Stokes equations. To derive the governing equations of the fluctuation quantities, the flow variables $\phi =[\rho,u,v,w,T]$ can be divided into mean quantities $\bar {\phi }=[\bar {\rho },\bar {u},\bar {v},\bar {w},\bar {T}]$ and fluctuation quantities $\phi ^{'}=[\rho ^{'},u^{'},v^{'},w^{'},T^{'}]$, where $\rho$, $u$, $v$, $w$ and T are the density, streamwise velocity, wall-normal velocity, spanwise velocity and temperature, respectively. The mean quantities $\bar {\phi }$ are determined by the basic Navier–Stokes equations, while the fluctuations are calculated through stability analysis. The governing equations for the fluctuation quantities can be expressed as

(2.2) \begin{align} & \varGamma\frac{\partial \phi^{'}}{\partial t}+\boldsymbol{\mathsf{A}}\frac{\partial \phi^{'}}{\partial x}+\boldsymbol{\mathsf{B}} \frac{\partial \phi^{'}}{\partial y}+\boldsymbol{\mathsf{C}}\frac{\partial \phi^{'}}{\partial z}+\boldsymbol{\mathsf{D}}\phi^{'}+\boldsymbol{\mathsf{V}}_{xx} \frac{\partial^2 \phi^{'}}{\partial x^2}+\boldsymbol{\mathsf{V}}_{yy}\frac{\partial^2 \phi^{'}}{\partial y^2}+\boldsymbol{\mathsf{V}}_{zz} \frac{\partial^2 \phi^{'}}{\partial z^2} \nonumber\\ &\quad +\boldsymbol{\mathsf{V}}_{xy}\frac{\partial^2 \phi^{'}}{\partial x\partial y}+\boldsymbol{\mathsf{V}}_{xz} \frac{\partial^2 \phi^{'}}{\partial x\partial z}+\boldsymbol{\mathsf{V}}_{yz} \frac{\partial^2 \phi^{'}}{\partial y\partial z}=F {.} \end{align}

The matrices $\varGamma$, $\boldsymbol{\mathsf{A}}$, $\boldsymbol{\mathsf{B}}$, $\boldsymbol{\mathsf{C}}$, $\boldsymbol{\mathsf{D}}$, $\boldsymbol{\mathsf{V}}_{xx}$, $\boldsymbol{\mathsf{V}}_{yy}$, $\boldsymbol{\mathsf{V}}_{zz}$, $\boldsymbol{\mathsf{V}}_{xy}$, $\boldsymbol{\mathsf{V}}_{xz}$ and $\boldsymbol{\mathsf{V}}_{yz}$ are functions of the mean quantities $\bar {\phi }$. The nonlinear term $F$ contains both the mean-flow and fluctuation variables. The expressions for the matrices and nonlinear term $F$ can be found in Ren & Fu (Reference Ren and Fu2014).

2.2.1. Linear analysis theory

In LST, the mean quantities are solved for using the compressible Blasius equations (White Reference White2006). This approach has been shown to give results in good agreement with those obtained through DNS as reported in Zhou et al. (Reference Zhou, Liu, Lu, Wang and Yan2022b). The nonlinear term is not considered in the LST. Furthermore, it is assumed that the mean flow is quasi-parallel at a given location. The fluctuation quantities are assumed to be of the form

(2.3)\begin{equation} \phi^{'}=\hat{\phi}(y)\exp({{\rm i}(\alpha x+\beta z-\omega t)}), \end{equation}

where $\alpha$, $\beta$ and $\omega$ are the streamwise wavenumber, spanwise wavenumber and angular frequency, respectively. Substituting (2.3) into (2.2) and ignoring the lower-order terms and the wall-normal velocity of the mean flow, we can obtain the governing equations for the LST. The details of this derivation can be found in Chang (Reference Chang2004). The LST is solved as an eigenvalue problem, where the streamwise wavenumber $\alpha$ is a function of the spanwise wavenumber $\beta$ and the angular frequency $\omega$. In the spatial LST, both the spanwise wavenumber and the angular frequency are real, while the streamwise wavenumber $\alpha =\alpha _r+{\rm i}\alpha _i$ is complex. Given a fixed spanwise wavenumber and an angular frequency, we can determine the value of the streamwise wavenumber and the corresponding eigenfunction $\hat {\phi }$. The imaginary part $\alpha _i$ represents the amplification rate of the disturbance and a negative value of $\alpha _i$ indicates instability. The fourth-order central difference method is used for the wall-normal-direction discretization of the LST governing equations.

2.2.2. Nonlinear parabolized stability equations

In the NPSE, the form of disturbance is

(2.4)\begin{equation} \phi^{'}(x,y,z,t)=\sum_{{-}M}^{M}\sum_{{-}N}^{N}\hat{\phi}_{mn}(x,y) \exp\left({{\rm i}\int^x_{x_0}\alpha_{mn}(x)\,{\rm d}\kern 0.06em x}\right) \exp({{\rm i}(n\beta z-m\omega t)}). \end{equation}

Substituting (2.4) into (2.2) and disregarding the second derivative of the mode shape function $\hat {\phi }_{mn}$ in the streamwise direction, we obtain the nonlinear parabolized stability equations for mode $(m,\!n)$ as

(2.5)\begin{equation} \hat{D}_{mn}\hat{\phi}_{mn}+\hat{A}_{mn}\frac{\partial\hat{\phi}_{mn}}{\partial x}+ \hat{B}_{mn}\frac{\partial\hat{\phi}_{mn}}{\partial y}+\hat{C}_{mn} \frac{\partial^2\hat{\phi}_{mn}}{\partial y^2}=\hat{F}_{mn} \exp\left({-{\rm i}\int_{x_0}^{x}\alpha_{mn}\,\mathrm{d}x}\right). \end{equation}

The details of the coefficient matrices and nonlinear terms $F_{mn}$ in this equation can be found in Chang (Reference Chang2004). The calculation of the phase velocity for each mode is related to the real part of the streamwise wavenumber, which can be calculated as

(2.6)\begin{equation} c_{mn}=\frac{\omega_m}{{\rm Re}(\alpha_{mn})}. \end{equation}

The boundary conditions for each mode at the wall and far field are

(2.7)\begin{equation} y=0\ \mathrm{and}\ \infty: \hat{u}_{mn}=\hat{w}_{mn}=\hat{v}_{mn}=\hat{T}_{mn}=0. \end{equation}

For the mean-flow distortion $(0,\!0)$, the far-field boundary condition for the wall-normal velocity is (Chang & Malik Reference Chang and Malik1994)

(2.8)\begin{equation} y=\infty:\frac{\partial \hat{v}_{00}}{\partial y}=0, \end{equation}

which accounts for the evolution of the boundary layer. To solve for the shape function $\phi _{mn}$ and wavenumber $\alpha _{mn}$, the wavenumber is updated as

(2.9)\begin{equation} \alpha^{new}=\alpha^{old}-{\rm i}\frac{1}{E}\int_{0}^{\infty}\bar{\rho} \left(u^*\frac{\partial \hat{u}}{\partial y}+v^*\frac{\partial \hat{v}}{\partial y}+w^* \frac{\partial \hat{w}}{\partial y}\right)\mathrm{d}y, \end{equation}

where

(2.10)\begin{equation} E=\int_{0}^{\infty}\bar{\rho}(\vert{\hat{u}}\vert^2+\vert{\hat{v}}\vert^2+ \vert{\hat{w}}\vert^2)\,\mathrm{d}y \end{equation}

and superscript $*$ denotes the complex conjugate. The discretization of the NPSE in the wall-normal direction utilizes a fourth-order central difference method to accurately capture the flow dynamics. Furthermore, to maintain the stability of the calculations and prevent the residual ellipticity that can arise in numerical simulations, the streamwise direction is discretized using a first-order backward scheme with the Vigneron technique (Chang & Malik Reference Chang and Malik1994). This approach has been proven to be effective in producing reliable and accurate results in previous studies. The base flow for the NPSE simulations is obtained by the DNS, in which the inlet disturbance is not included. The validation of DNS and NPSE solvers is conducted in the Appendix.

2.3. Computational set-up

The current study focuses on a two-dimensional high-speed boundary layer and considers the fluid to be a perfect gas. The free-stream conditions are based on the high-altitude flight tests performed by Harvey (Reference Harvey1978) and Schneider (Reference Schneider1999), which have been proven to induce transition; see table 1. The viscosity of the fluid is calculated using Sutherland's law. The inlet of both NPSE and DNS is set at $x_{in}=0.45\,\mathrm {m}$, where the boundary layer thickness is $\delta _{in}\approx 3.20\times 10^{-3}\,\mathrm {m}$; see figure 1(a). For the simulation grids, the streamwise and spanwise distributions for DNS are equidistant, while the wall-normal direction for LST and NPSE uses 175 equidistant points, which is consistent with the DNS. The grid stretching in the wall-normal direction can be found in Zhou et al. (Reference Zhou, Lu, Liu and Yan2021). The NPSE uses at least 12 points within a wavelength of the fundamental wave in the streamwise direction. The details of the domain size and grid resolution for the DNS will be presented in § 6.

Table 1. Free-stream flow condition.

Figure 1. (a) Neutral curves obtained through linear stability analysis. (b) Contours of the linear amplitude, $\ln A_{lin}/A_{in}$, at $x=1.025\,\mathrm {m}$, where $A_{lin}$ is calculated using NPSE with only one oblique wave at the inlet. The first and second mode regions are labelled with Roman numerals I and II, respectively. The vertical dashed line indicates the inlet of DNS and NPSE.

4.1. Parametric study

For the parametric simulations, mode (18,2) is selected as the second mode because of its high linear amplification rate, which is very close to that of the planar second mode (18,0). In addition, it begins amplifying closer to the inlet than mode (14,0), which is amplified to higher amplitude at $x=1.05\,\mathrm {m}$; see figure 1(a,b). The second mode (18,2) and an oblique wave ($m,\!n$) are initiated at the inlet of the domain with amplitudes of the streamwise velocity fluctuation set to $1.5\times 10^{-3}u_{\infty }$ and $10^{-5}u_{\infty }$, respectively. The frequency extends from $m=1$ to $m=17$ and the spanwise wavenumber ranges from $n=1$ to $n=16$. In the NPSE simulations, only six modes are considered: the second mode (18,2), its harmonic (36, 4), the oblique wave ($m,\!n$), the difference mode ($18-m$, $2-n$), the sum mode ($18+m$, $2+n$) and the mean-flow distortion (MFD) (0,0). The main interactions can be listed as

(4.1)$$\begin{gather} (18,\!2)-(m,\!n)\rightarrow(18-m,\!2-n); \end{gather}$$

(4.2)$$\begin{gather}(18,\!2)+(m,\!n)\rightarrow(18+m,\!2+n); \end{gather}$$

(4.3)$$\begin{gather}(18+m,\!2+n)+(18-m,\!2-n)\rightarrow(36,\!4); \end{gather}$$

(4.4)$$\begin{gather}(18,\!2)+(18,\!2)\rightarrow(36,\!4). \end{gather}$$

It is worth noting that the interactions between the MFD and modes are also taken into account in the NPSE. After careful validation, it has been determined that this simple model is sufficient; see the Appendix. There are 272 cases covering 17 frequencies and 16 spanwise wavenumbers, which are same as those set in § 3. The amplitude of the streamwise velocity fluctuations of the other oblique wave is $10^{-5}u_{\infty }$. The wave angle is calculated as

(4.5)\begin{equation} \theta=\arctan\frac{\alpha_r}{\beta}, \end{equation}

where $\alpha _r$ and $\beta$ represent the real parts of the streamwise and spanwise wavenumbers, respectively. There are two scenarios for the interactions between the oblique second mode and other oblique waves, as shown in figure 2. The first scenario is where the evolution direction of the other oblique wave is in the same quadrant as the second mode, and the second scenario is where the evolution directions of the two oblique waves are in different quadrants, as shown in figure 2. When the evolution direction of the oblique wave is in the fourth quadrant, the value of the spanwise wavenumber is negative. As the frequency and spanwise wavenumber of the second mode are fixed, the wave angle $\theta _2$ of the second mode can be obtained through linear stability analysis and is set to a positive value in this study.

Figure 2. Diagram showing the direction of evolution of oblique waves. Here, $\theta _2$ and $\theta _1$ represent the wave angles of the oblique second mode and the other oblique wave, respectively. The first and fourth quadrants are designated as I and IV, respectively.

The results of the amplitude of the oblique wave with negative and positive spanwise wavenumbers in the $n\unicode{x2013}f$ contours at $x=1.025\,\mathrm {m}$ are displayed in figures 3 and 4, respectively. When the spanwise wavenumber of the oblique wave is negative, almost all of the oblique waves, except those with a large spanwise wavenumber, are amplified compared with the initial amplitude; see figure 3(a). However, as seen in figure 3(b), all of the oblique waves are amplified compared with the linear results, and oblique modes with a large spanwise wavenumber are amplified the most. This is due to the fact that the linear amplification rate of oblique waves with larger spanwise wavenumbers is damping, and the damping rate increases as the spanwise wavenumber increases, which will be further explained in the next section. Additionally, among the amplified oblique modes, the one with a frequency around $45\,\mathrm {kHz}$ and a negative spanwise wavenumber around $-7$ is the most amplified, as shown in region IV of figure 3(a). This is in contrast to the damping or low amplification rate seen in region IV of the linear results in figure 1. However, if the absolute spanwise wavenumber is less than 2.0, the oblique wave with a frequency around $45\,\mathrm {kHz}$ experiences less amplification, as indicated by region III in figure 3(a). Although the first and second modes exhibit high amplitudes in regions I and II, respectively, figure 3(b) shows that they are amplified weakly compared with the linear results.

Figure 3. The contours of the amplitude of the oblique waves with negative spanwise wavenumber at $x=1.025\,\mathrm {m}$: (a) contours of the nonlinear amplitude divided by the initial amplitude; (b) contours of the ratio of the nonlinear amplitude to the linear amplitude. The first and second mode regions are labelled with Roman numerals I and II, respectively, while Roman numerals III and IV indicate regions that are less and more amplified, respectively, compared with the initial amplitude.

Figure 4. The contours of the amplitude of the oblique waves with positive spanwise wavenumber at $x=1.025\,\mathrm {m}$: (a) contours of the nonlinear amplitude divided by the initial amplitude; (b) contours of the ratio of the nonlinear amplitude to the linear amplitude. The first and second mode regions are labelled with Roman numerals I and II, respectively, while Roman numerals III and IV indicate regions that are less and more amplified, respectively, compared with the initial amplitude.

The amplitude of oblique waves with a positive spanwise wavenumber at $x=1.025\,\mathrm {m}$ is presented in figure 4. In comparison with the oblique waves with a negative spanwise wavenumber shown in figure 3, there are some differences. Specifically, the absolute value of the spanwise wavenumber of the most amplified mode in region IV is greater than that in figure 3(a). The most amplified mode in figure 4(a) is (8,9), whereas in figure 3(a) it is (9,$-$7). However, in view of the wave angle difference with the oblique second mode, the most amplified modes are similar. The absolute initial wave angle difference between modes (18,2) and (8,9) is $51.2^{\circ }$ as calculated using the parabolized stability equations, whereas the difference between modes (18,2) and (9,$-$7) is $49.5^{\circ }$. Thus, the wave angle difference between the second mode and the most amplified mode is around $50^{\circ }$. Overall, the distribution of the oblique wave's amplitude in figure 4 is similar to that in figure 3. Therefore, the results of the interaction between the second mode and an oblique wave with only a negative value are presented in the following.

To further understand the interaction of the oblique second mode with the first oblique waves, the evolution of modes with low frequencies $3f_0$, $8f_0$ and $9f_0$ is depicted in figure 5. These frequencies belong to the first mode, as shown in figure 1. As illustrated in figure 5, almost all the modes are amplified when the amplitude of the second mode (18,2) reaches its highest level at $x\approx 0.64\,\mathrm {m}$, even the damping modes. The amplitude of modes with frequency $3f_0$ is slightly increased. Modes with frequencies $8f_0$ and $9f_0$ grow rapidly, particularly those with a spanwise wavenumber around $n=-7$, which aligns with the results in figure 3. The reasons behind the promotion effects on the single first mode will be discussed in § 4.2. The interactions between the oblique second modes are shown in figure 6. For frequencies $14f_0$ and $16f_0$, the linear results indicate that the growth rate of these modes decreases with increasing spanwise wavenumber. The ratio of nonlinear amplitude to linear amplitude of the oblique waves with a larger spanwise wavenumber is more amplified than that with a smaller wavenumber. However, the amplitude of oblique waves with a larger spanwise wavenumber cannot surpass that of waves with a smaller wavenumber, as these higher wavenumbers decrease to much lower amplitudes before starting to grow rapidly. Similarly, the cause of rapid growth of the second mode will be discussed in § 4.3.

Figure 5. Amplitude evolution of oblique waves with constant frequency but different spanwise wavenumbers: (a) $f=3f_0$; (b) $f=8f_0$; (c) $f=9f_0$. The red dashed lines indicate the linear evolution of the oblique waves.

Figure 6. Amplitude evolution of oblique waves with constant frequency but different spanwise wavenumbers: (a) $f=14f_0$; (b) $f=16f_0$. The red dashed lines indicate the linear evolution of the oblique waves.

4.2. Interactions between an oblique second mode and an oblique first mode wave

It has been observed that almost all the oblique waves are amplified using NPSE. However, to determine the cause of the rapid growth of these waves, two possible paths are explored (Chen et al. Reference Chen, Zhu and Lee2017). The first path is the transfer of energy from the second mode to the oblique waves. The second path is the high amplitude of the second mode acting as a catalyst and allowing the oblique wave to gain energy from the mean flow. To further understand this phenomenon, mode (8,$-$10) has been selected as a representative single first mode to study in detail. The streamwise velocity amplitude of mode (8,$-$10) has been set to $10^{-4}u_{\infty }$. Additionally, six modes, along with their higher-harmonic modes, have been considered, resulting in a total of fifteen modes being analysed.

The evolution of the interaction between a first and a second mode is illustrated in figure 7(a). As seen in figure 7(a), the high amplitude of the mode (18,2) results in the rapid growth of the oblique wave (8,$-$10). Additionally, the slope of the amplification rate of the difference mode (10,12) is similar to that of the oblique wave (8,$-$10) at a downstream position of $x\approx 0.8\,\mathrm {m}$, which indicates that these modes are in resonance. The phase velocity of these modes is depicted in figure 7(b). It can be observed that the phase speeds of the modes (8,$-$10) and (10,12) are nearly the same at $x\approx 0.8\,\mathrm {m}$, suggesting that they are phase locked. The role of the difference mode (10,12) will be further discussed in § 4.2.2.

Figure 7. Streamwise evolution of (a) the maximum amplitude of the streamwise velocity fluctuation and (b) the phase velocity of various modes in the streamwise direction. The higher-harmonic modes are denoted by the grey lines.

4.2.1. Kinetic energy budget

The kinetic energy equation, as described by Orszag & Patera (Reference Orszag and Patera1983) and Hajj, Miksad & Powers (Reference Hajj, Miksad and Powers1993), is used to investigate the cause of the rapid growth of mode (8,$-$10). Based on the work of Chen et al. (Reference Chen, Zhu and Lee2017), the streamwise momentum equations from (2.5) for each mode can be expressed as follows:

(4.6)\begin{equation} \bar{\rho}\frac{\mathrm{D}u}{\mathrm{D}t}={-}L_u\phi+N_u, \end{equation}

where $L_u$ and $N_u$ are the linear operator and nonlinear terms, respectively. Equation (4.6) multiplies the complex conjugate of the streamwise velocity, while the conjugate of (4.6) multiplies the streamwise velocity. Similar operations are performed on the wall-normal and spanwise momentum equations. The six equations are then added to obtain the kinetic energy equation for each mode:

(4.7)\begin{align} \bar{\rho}\frac{\mathrm{D}}{\mathrm{D}t}(\vert{u}\vert^2+\vert{v}\vert^2+\vert{w}\vert^2)={-}2\alpha_i \bar{\rho}\bar{U}(\vert{\hat{u}}\vert^2+\vert{\hat{v}}\vert^2+\vert{\hat{w}}\vert^2)\exp \left({-}2\int\alpha_i\,\mathrm{d}x\right)=L+N, \end{align}

where

(4.8)\begin{equation} L={-}u^*L_u\phi-v^*L_v\phi-w^*L_w\phi+c.c.\end{equation}

and

(4.9)\begin{equation} N=u^*N_u+v^*N_v+w^*N_w+c.c.\end{equation}

The linear term $L$ represents the energy transfer between the mode itself and the mean flow. However, the nonlinear term $N$ not only captures the energy transfer between different modes but also includes the energy transfer between the mode itself and the mean flow. The linear transfer can be divided into four parts:

(4.10)\begin{equation} L=P+\varXi+\varPi+\upsilon{.} \end{equation}

Here, $P$, $\varXi$, $\varPi$ and $\upsilon$ represent the production, energy transfer related to non-parallel terms, pressure dilatation, and diffusion and viscous dissipation, respectively. They are calculated as follows:

(4.11)\begin{equation} P={-}\bar{\rho}u^*v\frac{\partial\bar{U}}{\partial y}-\bar{\rho}v^*u\frac{\partial\bar{V}}{\partial x}+c.c., \end{equation}

(4.12)\begin{align} \varXi &={-}\bar{\rho}u^*u\frac{\partial\bar{U}}{\partial x}+ \left(u^*\frac{\partial u}{\partial y}+v^*\frac{\partial v}{\partial y}+w^* \frac{\partial w}{\partial y}\right)-\bar{\rho}v^*v\frac{\partial\bar{V}}{\partial y} \nonumber\\ &\quad -\rho u^*\left(\bar{U}\frac{\partial \bar{U}}{\partial x}+\bar{V} \frac{\partial \bar{U}}{\partial y}\right)+{-}\rho v^*\left(\bar{U} \frac{\partial \bar{V}}{\partial x}+\bar{V}\frac{\partial \bar{V}}{\partial y}\right)+c.c., \end{align}

(4.13)\begin{equation} \varPi={-}u^*\frac{\partial p}{\partial x}-v^*\frac{\partial p}{\partial y}-w^* \frac{\partial p}{\partial z}+c.c. \end{equation}

The viscous dissipation and diffusion term is not shown because of its lengthy expression. By integrating (4.7) in the wall-normal direction, the imaginary part of $\alpha$ can be expressed as

(4.14)\begin{equation} -\alpha_i=L_{\varGamma}+N_{\varGamma}=\int_{0}^{\infty}L\,\mathrm{d}y/\varGamma+ \int_{0}^{\infty}N\,\mathrm{d}y/\varGamma, \end{equation}

where

(4.15)\begin{equation} \varGamma=\int_{0}^{\infty}\bar{\rho}\bar{U}(\vert{\hat{u}}\vert^2+\vert{\hat{v}}\vert^2+ \vert{\hat{w}}\vert^2)\exp\left({-}2\int\alpha_i\,\mathrm{d}x\right)\mathrm{d}y. \end{equation}

Here, $L_{\varGamma }$ and $N_{\varGamma }$ indicate the growth rates associated with the linear and nonlinear terms, respectively.

The linear and nonlinear energy transfers of mode (8,$-$10) are depicted in figure 8(a,b), respectively. As observed, both the linear and nonlinear energy transfers are concentrated around the critical layer. The linear energy transfer is negative from the beginning to the end of the domain and is slightly above the critical layer. Downstream of $x\approx 0.7\,\mathrm {m}$, the positive linear energy transfer concentrates on the critical layer and is slightly lower than the boundary layer edge. The nonlinear energy transfer is slightly higher than the critical layer and has a negative value downstream of $x\approx 0.7\,\mathrm {m}$.

Figure 8. Contours of (a) linear and (b) nonlinear energy transfer of mode (8,$-$10). The value is normalized by the maximum value at each streamwise station. The boundary layer and critical layer are denoted by the solid and dashed lines, respectively.

To understand the contribution of the linear and nonlinear energy to the evolution of mode (8,$-$10), the evolution of the integrated linear and nonlinear energy transfer of modes (18,2) and (8,$-$10) is shown in figure 9(a). It is observed that the contribution of nonlinear energy transfer is small for both modes (18,2) and (8,$-$10), with the change in the amplification rate being primarily due to the linear energy transfer. The linear energy transfer of mode (18,2) is significant at the inlet of the domain, leading to the fast growth of mode (18,2). However, downstream of $x\approx 0.65\,\mathrm {m}$, the linear energy transfer becomes negative, resulting in a decrease in the amplitude of mode (18,2). Similarly, mode (8,$-$10) experiences a decrease in its amplitude upstream of $x\approx 0.65\,\mathrm {m}$ due to a negative linear energy transfer. However, downstream of $x\approx 0.65\,\mathrm {m}$, the significant linear energy transfer results in the rapid growth of mode (8,$-$10). It can be concluded that mode (18,2) acts as a catalyst, and the rapid growth of mode (8,$-$10) is directly caused by the energy transfer from the mean flow, which is similar to the conclusions of the study on a flared cone with hypersonic free stream (Chen et al. Reference Chen, Zhu and Lee2017). The linear energy budget of mode (8,$-$10) shown in figure 9(b) reveals that the production term $P_{\varGamma }$ plays a major role in the rapid growth of mode (8,$-$10), with a maximum value of approximately 0.006 at $x\approx 0.7\,\mathrm {m}$, which is several times larger than the value upstream at $x\approx 0.6\,\mathrm {m}$. These results indicate that mode (8,$-$10) gains energy directly from the mean flow, not from mode (18,2).

Figure 9. (a) Evolution of the integrated linear and nonlinear energy transfer of modes (18,2) and (8,$-$10); (b) linear energy budget of mode (8,$-$10).

4.2.2. Role of the mean-flow distortion and difference mode

The evolution of modes (18,2) and (8,$-$10) in a resonant system is shown in figure 10(a). The importance of the MFD (0,0) and the difference mode (10,12) is highlighted in this section. When the (0,0) mode is forced to zero, the amplitude of mode (18,2) decreases faster downstream of $x\approx 0.6\,\mathrm {m}$ compared with the baseline. Meanwhile, the amplitude of mode (8,$-$10) reaches its maximum value at $x\approx 0.9\,\mathrm {m}$ before decreasing. Figure 10(b) shows that the maximum value of the linear energy transfer of mode (8,$-$10) is lower than the baseline and the nonlinear term is less affected. This suggests that the MFD (0,0) plays a role in maintaining the amplitude of mode (18,2), which in turn promotes the growth of the oblique mode (8,10) through linear energy transfer. When the difference mode (10,12) is forced to zero, the amplitude of mode (8,$-$10) decreases downstream and the rapid growth disappears. The linear energy transfer is negative and the nonlinear energy transfer is close to zero, indicating that the rapid growth is mainly due to the interactions between the second mode (18,2), oblique first mode (8,$-$10) and difference mode (10,12). Both the second mode (18,2) and the difference mode (10,12) act as catalysts, promoting energy transfer from the mean flow to the oblique wave (8,$-$10). The role of the sum mode is not discussed here owing to its neglected impact in this promotion effect; see figure 18.

Figure 10. (a) Evolution of modes (18,2) and (8,$-$10) in the streamwise direction; (b) evolution of energy transfer of mode (8,$-$10). Black and red lines indicate that the modes (0,0) and (10,12) are forced to zero in the simulation, respectively.

4.3. Interactions between oblique second modes

In this section, we discuss the interaction between oblique second modes, with mode (14,$-$14) selected as the representative mode. Fifteen modes are considered in the interaction between modes (18,2) and (14,$-$14). The evolution of the amplitude and phase velocity of various modes is shown in figure 11. Like the results in figure 6(a), mode (14,$-$14) experiences two rapid growths. However, the promotion effect on the evolution of mode (14,$-$14) is weaker than the strong promotion effect on the first mode (8,$-$10). Additionally, there is no phase locking observed between mode (14,$-$14) and the difference mode (4,16), as seen in figure 11(b).

Figure 11. Streamwise evolution of (a) the maximum amplitude of the streamwise velocity fluctuation and (b) the phase velocity of various modes in the streamwise direction. The higher-harmonic modes are denoted by the grey lines.

To understand the cause of the rapid growth of mode (14,$-$14), the contours of linear and nonlinear energy transfer are shown in figure 12. The maximum linear and nonlinear energy transfer is around the critical layer, with linear energy transfer being negative upstream of $x\approx 0.64\,\mathrm {m}$ and positive downstream of $x\approx 0.64\,\mathrm {m}$. The nonlinear energy transfer is negative from the beginning to the end, with positive values observed at $x\approx 0.7\,\mathrm {m}$ and $x\approx 1.0\,\mathrm {m}$. The integrated linear and nonlinear energy transfers of modes (18,2) and (14,$-$14) are shown in figure 13. Unlike for the first mode, where linear energy transfer dominates the energy transfer during the rapid growth, both linear and nonlinear energy transfers are high at $x\approx 0.65$ where the second mode (14,$-$14) begins to grow rapidly. The nonlinear energy transfer gradually decreases to a negative value as it moves in the streamwise direction and becomes positive downstream of $x\approx 0.95$ where the amplitude of mode (14,$-$14) increases again. The linear energy transfer is positive downstream of $x\approx 0.65$ and experiences two rapid growths during the rapid growth of mode (14,$-$14). Hence, the rapid growth of mode (14,$-$14) is a result of both linear and nonlinear energy transfers. The linear energy transfer comes from the mean flow and mode itself, while the nonlinear transfer can come not only from the mean flow but also from other modes. To address this question, we forced only mode (18,2) at the inlet of NPSE. The energy transfer of mode (18,2) between only the (18,2) initiated and both the modes (18,2) and (14,$-$14) included coincides with the evolution in the streamwise direction, as shown in figure 14(c). Therefore, the rapid growth of mode (14,$-$14) is not from mode (18,2), but from the linear energy transfer and mean flow through nonlinear energy transfer.

Figure 12. Contours of (a) linear and (b) nonlinear energy transfer of mode (14,$-$14). The values are normalized by the maximum value at each streamwise station. The boundary layer and critical layer are denoted by the solid and dashed lines, respectively.

Figure 13. Evolution of the integrated linear and nonlinear energy transfer of modes (18,2) and (14,$-$14).

Figure 14. Evolution of (a) the amplitudes of modes (18,2) and (14,$-$14), (b) the integrated energy transfer of mode (14,$-$14), and (c) the integrated energy transfer of mode (18,2).

In the following, the effect of the mean-flow distortion mode (0,0) and difference mode (4,16) will be examined. Modes (0,0) and (4,16) are both forced to zero. The evolution of the maximum amplitudes of modes (18,2) and (4,16) is displayed in figure 14. When mode (0,0) is forced to zero, it can be seen that the mode (18,2) decreases to a lower amplitude compared with the baseline, and the amplitude of mode (14,$-$14) is also reduced in comparison with the baseline; see figure 14(a). Meanwhile, the maximum value of the linear energy transfer is lower than the baseline and is decreasing faster. The maximum value of the nonlinear energy transfer is relatively unaffected; see figure 14(b). This suggests that the role of mode (0,0) in interactions between the second modes and between the second mode and the first mode is the same, promoting the increase of the second mode (14,$-$14) by maintaining the amplitude of mode (18,2). When mode (4,16) is removed, the amplitude of mode (14,$-$14) steadily decreases until the end of the domain, and the overshoot of both linear and nonlinear energy transfers disappears; see figure 14(a,b). This indicates that the interactions between modes (18,2), (14,$-$14) and (4,16) are crucial in the rapid growth of mode (14,$-$14). Both modes (18,2) and (4,16) act as catalysts, facilitating energy transfer from the mean flow to the oblique mode (14,$-$14).

It is worth noting that, compared with the amplifying effect of the second mode oblique wave on the first mode, the amplifying effect of the second mode oblique wave on the oblique second mode is weaker, and it is difficult for the second mode oblique wave to enter the resonance state with the difference mode it produces.

6.1. Nonlinear interactions

Table 2 presents five cases for study. The reasons for using these five examples will be explained in the following paragraphs.

The first case, $S1$, is the second mode oblique breakdown, which is initiated by a pair of second mode waves with an initial amplitude of streamwise velocity perturbation of $0.0032u_\infty$. Case $S1_w$ is the second mode oblique breakdown with a weaker initial amplitude of 0.0016 than that of case $S1$. The second mode oblique breakdown has been previously studied in Franko & Lele (Reference Franko and Lele2013) and Zhou et al. (Reference Zhou, Liu, Lu, Wang and Yan2022b). The nonlinear stage of the second mode oblique breakdown results in the generation of streamwise streaks through the nonlinear interaction of oblique instability waves. These streaks can lead to the self-sustained breakdown of the boundary layer. For instance, the breakdown of streamwise streaks results in the breakdown of the boundary layer in figure 20(b). The high amplitude of modes (0,4) and (0,8) in case $S1$ is responsible for the high amplitude of streamwise streaks, as can be seen in figure 20(a). However, the self-sustained breakdown of the streaks can only occur when their amplitude reaches a certain threshold value. In case $S1_w$, the amplitude of the steady modes (0,4) and (0,8) is lower than in case $S1$, and the breakdown of streamwise streaks is not observed as in figure 20(c). This is because the weaker amplitude of streaks cannot lead to a self-sustaining state of the boundary layer breakdown. However, it is challenging to determine the exact amplitude of streaks that leads to the self-sustaining breakdown of the boundary layer.

Figure 20. (a) Evolution of modes (18,2), (0,4) and (0,8) for cases $S1$ and $S1_w$ in the streamwise direction. Instantaneous iso-surface of streamwise velocity at (b) $u=0.6$ for case $S1$ and (c) $u=0.6$ for case $S1_w$, coloured by the streamwise velocity fluctuations.

6.1.1. Path 1

It is interesting to see if the introduction of a damping first oblique mode with low amplitude in case $S1_w$ can trigger transition and result in a fully developed turbulent boundary layer, as a pair of low-amplitude oblique second modes or a pair of low-amplitude oblique first modes with damping amplification alone cannot cause the transition. This hypothesis is tested in case $S2$, which is called path 1 in the present study. In this path, both the second mode (18,2) and the first mode (8,10) are introduced at the inlet of the domain, with initial amplitudes of streamwise velocity disturbance of 0.0016 and 0.00075, respectively. The goal is to determine if this new transition path leads to a self-sustaining state of the breakdown of the boundary layer and if it results in a fully developed turbulent boundary layer. The results of case $S2$ will provide insights into the effect of combining the oblique second mode and a damping first oblique mode on the transition mechanism.

The breakdown of the boundary layer can be seen from figure 21(b–e). To determine the cause of the boundary layer breakdown, the evolution of the modes in the streamwise direction for case $S2$ is shown in figure 21(a). It is observed that the amplitude of mode (8,10) decreases upstream of $x\approx 0.6\,\mathrm {m}$ and then rapidly grows downstream of this streamwise station. This observation is consistent with the results of NPSE, which found that the oblique second mode leads to the rapid growth of the first oblique modes. Moreover, the interaction between modes (8,10) and (18,2) results in the generation of difference modes (10,8) and (10,12) and sum modes (26,8) and (26,12). The amplitude of difference modes is approximately one order of magnitude higher than that of sum modes.

Figure 21. (a) The evolution of various modes for case $S2$ in the streamwise direction. The instantaneous iso-surfaces of the streamwise velocity for (b) $u=0.9$, (c) $u=0.6$ and (d) $u=0.3$. (e) Instantaneous iso-surface of $Q=0.0002$ for case $S2$, coloured by the streamwise velocity fluctuations.

The evolution of oblique waves (18,2) and (8,10) can be visualized using the Q-criterion as shown in figure 21(e). The oblique wave induced by mode (18,2) dominates the flow field upstream of $x\approx 0.75$. As the amplitude of mode (18,2) decreases, this oblique wave disappears. Subsequently, a second type of oblique wave in the form of $\varLambda$ shapes is observed due to the high amplitude of mode (8,10). Although the amplitude of mode (8,10) grows to a high level, it is still lower than the amplitudes of modes (0,4) and (0,8), which are generated by the nonlinear interactions of mode (18,2). Due to the high amplitude of mode (8,10), the interaction between the oblique wave and streaks causes a weak distortion of the streaks from $x\approx 0.7$ to $x\approx 0.8\,\mathrm {m}$; see figure 21(c,d). However, this distortion is not enough to cause the breakdown of the boundary layer. The evolution of modes (0,4) and (0,8) between cases $S1_w$ and $S2$ is almost the same upstream of $x\approx 1.0\,\mathrm {m}$ where the boundary layer is about to break down in case $S2$, as shown in figures 22 and 21. This indicates that the rapid growth of mode (8,10) has limited effect on the steady modes (0,4) and (0,8) generated during the nonlinear interactions of the second mode oblique breakdown. The interactions between oblique waves and streaks generated by the oblique second mode are relatively weak; however, the interactions between modes (8,10) and (0,20) are quite strong. These nonlinear interactions between modes (8,10) cause the high amplitude of mode (0,20) in figure 21(a) from approximately $x=0.8$ to $x=1.0\,\mathrm {m}$, which is even higher than for modes (0,4) and (0,8). As a result, the high amplitude of mode (0,20) leads to high-amplitude streaks with smaller scale in the spanwise direction, as seen in figure 21(c,d). The strong interaction between the oblique wave and these streaks results in significant distortion of the streaks from approximately $x=0.8$ to $x=1.0\,\mathrm {m}$, which eventually leads to the breakup of streaks at $x=1.0\,\mathrm {m}$, as shown in figure 21(c,d). The breakdown of the $\varLambda$-shaped structure occurs slightly upstream of $x=0.95\,\mathrm {m}$ compared with the streaks, as seen in figure 21(c,e).

Figure 22. Evolution of modes (18,2), (0,4), (0,8) and (0,20) in the streamwise direction for cases $S2$ and $S1_w$.

6.1.2. Path 2

As mentioned above, the oblique second mode has a promotion effect on the oblique first mode. In addition, the unstable region of the second mode with lower frequency is more downstream than that of the second mode with higher frequency. Based on these two results, the addition of the oblique second mode (16,2) into case $S2$ results in the formation of case $S3$. The aim of this set-up is twofold: first, to investigate if the transition to turbulence can occur with a second mode of lower initial amplitude than in case $S2$. Second, to verify that the oblique mode (8,10) can be promoted by the two oblique second modes (18,2) and (16,2) at two different streamwise locations, creating a domino-like effect. To achieve these objectives, the modes (18,2), (16,2) and (8,10) are added at the inlet of case $S3$ with initial amplitudes of streamwise velocity perturbation of $0.0008u_\infty$, $0.0008u_\infty$ and $0.00075u_\infty$, respectively. Thus, case $S3$ represents path 2.

The evolution of modes, including the instantaneous iso-surface of the streamwise velocity and the iso-surface of the Q-criterion, are depicted in figure 23. As shown in figure 23(d), the boundary layer breaks down into smaller scales. Furthermore, compared with case $S2$, the breakup station of the boundary layer in case $S3$ occurs upstream. The reason behind the early boundary layer breakdown can be found in the evolution of various modes; see figure 23(a). The amplitude of mode (8,10) experiences a twofold increase, with the first beginning at $x\approx 0.6\,\mathrm {m}$, due to the promotion effect of mode (18,2), and the second beginning at $x\approx 0.8\,\mathrm {m}$, due to mode (16,2). As a result of the first promotion, the amplitude of mode (8,10) increases to a high level, leading to stronger interactions between modes (16,2) and (8,10) than between modes (18,2) and (8,10). For example, the amplitude of the difference modes (8,8) and (8,12) generated by the interactions between modes (16,2) and (8,10) is higher than that of the difference modes (10,8) and (10,12) generated by modes (18,2) and (8,10), which is almost the same as that of the steady modes (0,4) at $x\approx 0.87\,\mathrm {m}$. The two-stage promotion process on mode (8,10) results in a high amplitude of mode (0,20), leading to the boundary layer transition due to the high amplitude of oblique modes, difference modes and steady modes. In the flow field structure, the breakdown of streaks and $\varLambda$-shaped vortices leads to the breakdown of the boundary layer, which is in agreement with the modal results; see figure 23(c,e).

Figure 23. (a) The evolution of various modes for case $S3$ in the streamwise direction. The instantaneous iso-surfaces of the streamwise velocity for (b) $u=0.9$, (c) $u=0.6$ and (d) $u=0.3$. (e) Instantaneous iso-surface of $Q=0.0002$ for case $S3$, coloured by the streamwise velocity fluctuations.

In order to elucidate the specific roles played by modes (16,2) and (18,2) in the two-stage promotion processes, case $S2\text{-}1$ is employed as an illustrative example. The inlet disturbances consist of modes (16,2) and (8,10) with respective amplitudes of 0.0016 and 0.00075. The results pertaining to mode (8,10) for cases $S2$, $S2\text{-}1$ and $S3$ are depicted in figure 24. It is observed that both modes (18,2) and (16,2) can induce significant amplification of mode (8,10). In the case of $S2\text{-}1$, the rapid growth of mode (8,10) commences at $x\approx 0.75\,{\rm m}$, while in case $S2$ it initiates at $x\approx 0.6\,{\rm m}$. Notably, these two positions align with the locations where the two-stage promotion of mode (8,10) begins in case S3. Consequently, the two-stage promotion of mode (8,10) in case $S3$ can be attributed to modes (18,2) and (16,2).

Figure 24. Evolution of modes (18,2) and (8,10) for case $S2$, modes (16,2) and (8,10) for case $S2\text{-}1$, and mode (8,10) for case $S3$ in the streamwise direction.

In conclusion, the realization of path 1 by case $S2$ demonstrates that a pair of low-amplitude second oblique waves alone cannot lead to the second mode oblique breakdown. However, the addition of a pair of low-amplitude damping first oblique modes does lead to the breakup of the boundary layer. This suggests that the transition cannot be achieved by either of these two types of disturbances alone, but only through their combination. Case $S3$ is an example of path 2 and demonstrates that a combination of three low-amplitude modes can trigger a domino-like process, resulting in the rapid breakdown of the boundary layer with a lower initial amplitude of the second mode than in case $S2$.

6.2. Turbulent region

In the end, our goal is to determine if either of the two transition paths can lead to a fully developed turbulent boundary layer. At a fully developed turbulent boundary layer, the skin-friction coefficient and heat transfer on the wall can be described by theoretical correlations as documented in White (Reference White2006). The adiabatic wall temperature $T_{aw}$ is calculated using an approximation presented in White (Reference White2006),

(6.2)\begin{equation} T_{aw}=T_{\infty}\left(1+r\frac{\gamma-1}{2}Ma^2_\infty\right), \end{equation}

where $r=Pr^{1/3}$. The turbulent correlation of the skin-friction coefficient, as presented in White (Reference White2006) and Bradshaw (Reference Bradshaw1977), has been validated for the case of a non-adiabatic wall and is used in this study:

(6.3)\begin{equation} C_f=\frac{0.455}{S}\left[\ln \left(\frac{0.06}{S}(Re_x-Re_{x_{in}}) \frac{\mu_\infty}{\mu_w}\sqrt{\frac{T_\infty}{T_w}}\right)\right]^{{-}2}, \end{equation}

where

(6.4)\begin{equation} S=\frac{\sqrt{\left(T_{aw}/T_\infty-1\right)}}{\arcsin(A)+\arcsin(B)}. \end{equation}

The subscripts $\infty$ and $w$ denote the conditions at the free stream and wall, respectively. The parameters A and B are defined as

(6.5a,b)$$\begin{gather} A=\frac{2a^2-b}{\sqrt{b^2+4a^2}},\quad B=\frac{b}{\sqrt{b^2+4a^2}} \end{gather}$$

where

(6.6a,b)$$\begin{gather}a=\sqrt{r\frac{\gamma-1.0}{2}Ma^2_{\infty}\frac{T_{\infty}}{T_w}}, \quad b=\left(\frac{T_{aw}}{T_w}-1\right). \end{gather}$$

Based on the Reynolds analogy (White Reference White2006), the Stanton number is

(6.7)\begin{equation} St=\frac{C_f}{2Pr^{2/3}}. \end{equation}

The evolution of the skin-friction coefficient $C_f$ and Stanton number $St$ is plotted in figure 25. Owing to the high initial amplitude of the second mode, the transition onset in case $S1$ is earlier than in other cases. And the skin friction shows good agreement between the computational and theoretical results at the end of the domain. The skin friction of case $S1_w$ stays at the laminar value. Case $S3$ exhibits the most developed turbulent boundary layer owing to its rapid and early transition, with skin-friction values consistently aligning with theoretical values from $x\approx 1.05\,\mathrm {m}$. In contrast, case $S2$ exhibits a very short turbulent region due to its late transition and skin-friction values that align with theoretical values over a shorter range. The Stanton number shows an overshoot in heat transfer for cases $S2$ and $S3$ in the transition region, with case $S2$ failing to recover its value to the turbulent value due to the limited length of the computational domain. The DNS values of the Stanton number for both cases $S1$ and $S3$ are lower than the theoretical values. An additional overshoot in heat transfer is also observed in the laminar region for cases $S2$ and $S1$, with case $S1$ exhibiting a greater overshoot due to the higher initial amplitude of the second mode. The additional heat transfer overshoot is caused by the dilatation dissipation generated by the saturated oblique second mode, as reported in works such as Zhu et al. (Reference Zhu, Chen, Wu, Chen, Lee and Gad-el Hak2018) and Zhou et al. (Reference Zhou, Liu, Lu, Wang and Yan2022b). It should be noted that the additional heat transfer overshoot in the laminar region vanishes in case $S3$ due to the low initial amplitudes of the second modes (18,2) and (16,2), which cannot lead to strong dilatation dissipation.

Figure 25. Evolution of (a) the skin-friction coefficient $C_f$ and (b) the Stanton number $St$ for cases $S1$, $S1_w$, $S2$ and $S3$.

Finally, the mean-flow scaling transformation is utilized to examine the development of the mean streamwise profile, moving from the laminar to the turbulent boundary layer. Two transformation methods, namely the Van Driest transformation (Van Driest Reference Van Driest1951) and the total-stress-based transformation proposed by Griffin et al. (Reference Griffin, Fu and Moin2021), are employed in this study. In comparison with the Van Driest transformation, the mean-flow transformation introduced by Griffin et al. (Reference Griffin, Fu and Moin2021) demonstrates broader applicability, particularly for walls experiencing heat transfer. For the Van Driest transformation, the streamwise velocity and length are normalized based on the friction velocity and the length scale at the wall, respectively. They are defined as

(6.8)\begin{equation} U^+_{VD}=\frac{U_{VD}}{u_\tau}\end{equation}

and

(6.9)\begin{equation} y^+ = \frac{\rho_w u_\tau y}{\mu_w}, \end{equation}

respectively, where the friction velocity is calculated as

(6.10)\begin{equation} u_\tau=\sqrt{\frac{\tau_w}{\rho_w}}. \end{equation}

The Van Driest transform is in form of

(6.11)\begin{equation} U_{VD}=\int_{0}^{u}\sqrt{\frac{\rho}{\rho_w}}\,\mathrm{d}u. \end{equation}

For the total-stress-based transformation of Griffin et al. (Reference Griffin, Fu and Moin2021), a semilocal wall-normal coordinate is used, which is calculated as

(6.12)\begin{equation} Y^+ = \frac{y\sqrt{\tau_w \rho}}{\mu}. \end{equation}

The form of the transformation is

(6.13)\begin{equation} U^+ = \int_{0}^{Y^+}S^+_\tau \,\mathrm{d}Y^+. \end{equation}

The details of the generalized non-dimensional mean shear $S^+_\tau$ can be found in Griffin et al. (Reference Griffin, Fu and Moin2021). The scaled mean streamwise velocity for cases $S2$ and $S3$ is shown in figure 26. In the turbulent region, the scaled mean profile should match the theoretical formulas

(6.14a,b)\begin{equation} U^+_{VD}=y^+ \quad \mathrm{and} \quad U^+_{VD}=2.5\ln y^+ + 5.5{,} \end{equation}

at the viscous sublayer and the log layer (Coles Reference Coles1956), respectively.

Figure 26. Evolution of the scaled mean streamwise velocity at different streamwise stations for (c,d) case $S2$ and (a,b) case $S3$. The blue and red lines are results scaled by the Van Driest transformation (Van Driest Reference Van Driest1951) and total-stress-based transformation of Griffin, Fu & Moin (Reference Griffin, Fu and Moin2021), respectively.

The simulation results demonstrate that the mean streamwise velocity profile, when scaled using the approach proposed by Griffin et al. (Reference Griffin, Fu and Moin2021), progressively approaches the theoretical value as it evolves in the streamwise direction for cases $S2$ and $S3$, particularly within the log layer at $x=1.225\,\mathrm {m}$; see figure 26(b,d). In addition, the theoretical results closely align with the results obtained from the Van Driest transformation at $x=1.225\,\mathrm {m}$ for cases $S2$ and $S3$; see figure 26(a,c). However, a slight disparity is observed in the slope rate of the scaled velocity between the theoretical values and those obtained using the Van Driest transformation. Notably, a similar difference was reported in a previous study by Guo et al. (Reference Guo, Shi, Gao, Jiang, Lee and Wen2022). Overall, the results of the skin-friction distribution and the scaled mean streamwise velocity provide evidence that these two new transition paths can lead to a fully developed boundary layer.