2.1 Case set-up

A schematic for the set-up of the HPT simulations is shown in figure 1. A linear cascade set-up is used to replicate the experiments performed at the Von-Karman Institute (VKI). In the present simulations, only one passage is calculated, and the computational domain is enclosed by the red lines on the axial and pitchwise ($x{-}y$) plane cut. Representative of a modern transonic HPT nozzle, the simulations are performed at a Reynolds number of $Re=U_{e}^{\ast }C_{ax}^{\ast }/\unicode[STIX]{x1D708}^{\ast }=5.7\times 10^{5}$ and an exit Mach number $Ma_{e}=0.92$, which are in close agreement with the MUR224 case in the VKI experiments (Arts, Lambertderouvroit & Rutherford Reference Arts, Lambertderouvroit and Rutherford1990). Here, the superscript $\ast$ denotes dimensional quantities. Accordingly, $C_{ax}^{\ast }$ is the axial chord length, and $\unicode[STIX]{x1D708}^{\ast }$ and $U_{e}^{\ast }$ are the kinematic viscosity and the velocity at the exit plane, respectively.

Figure 1. Schematic for HPT case set-up. The computational grid is showing every fifteenth line in $x$ and $y$ directions.

The non-dimensionalized three-dimensional compressible Navier–Stokes equations are solved using the in-house solver HiPSTAR:

(2.1)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}u_{j})}{\unicode[STIX]{x2202}x_{j}}=0,\\[12.0pt] \displaystyle \frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}u_{i})}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}u_{i}u_{j}+p\unicode[STIX]{x1D6FF}_{ij})}{\unicode[STIX]{x2202}x_{j}}=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}_{ij}}{\unicode[STIX]{x2202}x_{j}},\\[12.0pt] \displaystyle \frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}e_{0})}{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}[u_{j}(\unicode[STIX]{x1D70C}e_{0}+p)]}{\unicode[STIX]{x2202}x_{j}}=\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70F}_{ij}u_{i})}{\unicode[STIX]{x2202}x_{j}}-\frac{\unicode[STIX]{x2202}q_{j}}{\unicode[STIX]{x2202}x_{j}}.\end{array}\right\}\end{eqnarray}$$

Here, $\unicode[STIX]{x1D70C}$, $u_{i}$, $p$ and $T$ are the non-dimensionalized flow density, velocity components, pressure and temperature, respectively. The non-dimensionalization results in dimensionless parameters as $Re_{\infty }=(\unicode[STIX]{x1D70C}_{\infty }U_{\infty }L_{\infty })/\unicode[STIX]{x1D707}_{\infty }$ and $Ma_{\infty }=U_{\infty }/c_{\infty }$. The reference length scale $L_{\infty }$ is selected as $C_{ax}^{\ast }$, and the reference velocity $U_{\infty }$ and density $\unicode[STIX]{x1D70C}_{\infty }$ are selected as the mean velocity and density at the inlet. Therefore, as non-dimensional quantities we have $C_{ax}=1$, $U_{in}=1$ and $\unicode[STIX]{x1D70C}_{in}=1$, and the non-dimensionalized flow-over time is $t_{F}=C_{ax}/U_{in}=1$. Moreover, $\unicode[STIX]{x1D707}_{\infty }$ and $c_{\infty }$ are viscosity and acoustic velocity for the reference state which are only dependent on the reference temperature $T_{\infty }$. The total energy $e_{0}$ is given by

(2.2)$$\begin{eqnarray}e_{0}=\frac{1}{2}\unicode[STIX]{x1D70C}u_{i}u_{i}+\frac{T}{\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D6FE}-1)Ma_{\infty }^{2}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}=1.4$ is the specific heat ratio. Moreover, the stress tensor is written as

(2.3)$$\begin{eqnarray}\unicode[STIX]{x1D70F}_{ij}=\unicode[STIX]{x1D70E}_{ij}+\unicode[STIX]{x1D70F}_{ij}^{sgs},\end{eqnarray}$$

which includes the viscous stress $\unicode[STIX]{x1D70E}_{ij}=(\unicode[STIX]{x1D707}/Re_{\infty })((\unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}x_{j})+(\unicode[STIX]{x2202}u_{j}/\unicode[STIX]{x2202}x_{i})-\frac{2}{3}(\unicode[STIX]{x2202}u_{k}/\unicode[STIX]{x2202}x_{k})\unicode[STIX]{x1D6FF}_{ij})$ and the subgrid-scale stress $\unicode[STIX]{x1D70F}_{ij}^{sgs}$. The molecular viscosity $\unicode[STIX]{x1D707}$ is computed using Sutherland’s law (White Reference White1991), setting the ratio of the Sutherland constant over free-stream temperature to 0.36867. Similarly, the heat flux $q_{j}$ is written as

(2.4)$$\begin{eqnarray}q_{j}=-\frac{\unicode[STIX]{x1D707}}{(\unicode[STIX]{x1D6FE}-1)PrRe_{\infty }Ma_{\infty }^{2}}\frac{\unicode[STIX]{x2202}T}{\unicode[STIX]{x2202}x_{j}}+q_{j}^{sgs},\end{eqnarray}$$

with $Pr=0.72$ representing the Prandtl number. The subgrid-scale terms in the governing equations, i.e. subgrid-scale stress $\unicode[STIX]{x1D70F}_{ij}^{sgs}$ and heat flux $q_{j}^{sgs}$, are closed by the WALE model of Nicoud & Ducros (Reference Nicoud and Ducros1999). A fourth-order wavenumber-optimized compact finite-difference scheme (Kim & Sandberg Reference Kim and Sandberg2012) was applied in the $x$ and $y$ directions, while a spectral method using fast Fourier transforms was applied in the uniform spanwise direction $z$. Moreover, the ultra-low storage frequency optimized explicit Runge–Kutta method (Kennedy, Carpenter & Lewis Reference Kennedy, Carpenter and Lewis2000) was used for time integration. Therefore, the numerical schemes used in the present simulations ensure fourth- or higher-order accuracy of both spatial and temporal discretizations.

In order to efficiently adapt the complex geometry without affecting the numerical accuracy, the computational grid of the present configuration is decomposed into nine blocks, and the characteristic interface conditions (Kim & Lee Reference Kim and Lee2003) were employed at the block interfaces as presented by the blue dashed lines in figure 1. This multi-block set-up consists of an O-type grid around the blade and a background H-type grid, which allows one to accurately resolve the blade boundary layers and to enforce the pitchwise periodicity at the same time. Moreover, a characteristic boundary condition with a sponge layer is applied at the outlet to prevent spurious reflection (Sandberg & Sandham Reference Sandberg and Sandham2006), and the blade surface is set as non-slip isothermal wall. At the inlet, a Riemann boundary condition is applied, and fluctuations, as described in § 2.2, are added to simulate the turbulence coming from upstream combustors in real engines. It is noted that the numerical solver and the multi-block set-up for turbine cascades have been extensively validated in previous studies, and more details can be found in Sandberg et al. (Reference Sandberg, Michelassi, Pichler, Chen and Johnstone2015).

Table 1. Parameters for HPT cases.

Figure 2. Near-wall grid size along the blade surface measured in wall units for case A in (a) the tangential direction $\unicode[STIX]{x0394}s^{+}$, (b) the wall-normal direction $\unicode[STIX]{x0394}n^{+}$ and (c) the spanwise direction $\unicode[STIX]{x0394}z^{+}$.

Figure 3. Incoming turbulence at a plane cut upstream of the blade leading edge at $x=-0.5$. The contours of the instantaneous streamwise velocity component from (a) case A and (b) case B. (c) The spanwise correlation of the pitchwise fluctuating velocity $v^{\prime }$.

In order to investigate the effects of the incoming turbulence on the HPT performance, four large-eddy simulations with different turbulence intensities and length scales have been performed in the present study, and the parameters of each are summarized in table 1. It is noted that the integral turbulence length scale $L_{s}$ introduced in cases B and D is as large as $20\,\%C_{ax}$, which requires a much larger domain in the spanwise direction compared to previous studies (Bhaskaran & Lele Reference Bhaskaran and Lele2010; Wheeler et al. Reference Wheeler, Sandberg, Sandham, Pichler and Michelassi2016). Therefore, the number of grid points significantly increases, and the final meshes applied in the present simulations have up to $8.6\times 10^{8}$ grid points. Furthermore, the grid resolution around the blade boundary layer for case A is shown in figure 2. The grid spacings on the blade surface in the tangential, wall-normal and spanwise directions are normalized with the local viscous length scale $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}$ and denoted as $\unicode[STIX]{x0394}s^{+}$, $\unicode[STIX]{x0394}n^{+}$ and $\unicode[STIX]{x0394}z^{+}$, respectively. Here $u_{\unicode[STIX]{x1D70F}}=\sqrt{\unicode[STIX]{x1D70F}_{w}/\unicode[STIX]{x1D70C}}$ is the wall-friction velocity with the wall shear stress $\unicode[STIX]{x1D70F}_{w}$ and density $\unicode[STIX]{x1D70C}$. Although the grid sizes vary along the blade surface, they remain at a relatively low level, and the grid spacings averaged around the blade surface are also given in table 1. It is also noted that the spanwise extents of the present cases, with the smaller ones being $L_{z}=0.3$ for cases A and C, are around six times the maximum boundary layer thickness around the blade. By comparing case A to the direct numerical simulation results of the same configuration with a much narrower spanwise extent ($L_{z}=0.1$) (Wheeler et al. Reference Wheeler, Sandberg, Sandham, Pichler and Michelassi2016), the wall heat flux and other boundary layer statistics are in close agreement, showing that the current grid resolution is sufficient to accurately resolve the boundary layer and most of the energy-containing scales for the FST (Pichler et al. Reference Pichler, Sandberg, Laskowski and Michelassi2017).

2.2 Inlet turbulence

The turbulent fluctuations at the inlet are introduced by a digital filter method (Klein, Sadiki & Janicka Reference Klein, Sadiki and Janicka2003), in which the generated velocity fields can efficiently reproduce first- and second-order one-point statistics as well as a locally given autocorrelation function. Therefore, the turbulence intensities and integral length scales can be prescribed for the different cases given in table 1. In order to investigate the generated inlet turbulence formed in cases with different length scales, the contours of instantaneous streamwise velocity at a $y{-}z$ plane cut at $x=-0.5$ are presented in figure 3. It is shown that the turbulent structures formed in case B, with a prescribed $L_{s}=20\,\%C_{ax}$, indeed have much larger scales compared to the structures in case A, with a prescribed $L_{s}=5\,\%C_{ax}$. Furthermore, the spanwise correlation of the pitchwise fluctuating velocity $R_{v^{\prime }v^{\prime }}(r)$ is also computed at this plane cut for cases A and B and shown in figure 3(c). The integral turbulence length scales, therefore, can be estimated as $2\int _{0}^{L_{z}/2}R_{v^{\prime }v^{\prime }}(r)\,\text{d}r$, and the results for cases A and B are $4.8\,\%C_{ax}$ and $20.1\,\%C_{ax}$, respectively. Thus, it is shown that the integral length scales of the incoming turbulence are in close agreement with the prescribed values. In addition, it can also be inferred that the spanwise extents for the presented simulations are sufficient for the large-scale inlet turbulence as

(2.5)$$\begin{eqnarray}\lim _{r\rightarrow L_{z}/2}R_{vv}(r)\rightarrow 0.\end{eqnarray}$$

The length scale of the prescribed inlet turbulence not only affects the large-scale structures formed at the inlet, but also influences the decay rate of the incoming turbulence. This can be shown by the evolution of the pitchwise averaged turbulence intensity

$$\begin{eqnarray}T_{u}=\sqrt{\frac{u^{\prime 2}+v^{\prime 2}+w^{\prime 2}}{3}}\end{eqnarray}$$

in figure 4. The turbulence intensities at the inlet are set as $6.5\,\%U_{in}$ and $20\,\%U_{in}$, respectively. It can be observed that the cases with larger integral length scale ($L_{s}=20\,\%C_{ax}$) have a lower dissipation rate compared to the ones with smaller length scale ($L_{s}=5\,\%C_{ax}$). Before entering into the vane, the intensities at $x=-0.2$ for the different cases reach levels of 16.8 %, 9.5 %, 6.1 % and 5.2 % of the inlet velocity $U_{in}$, respectively.

Furthermore, the turbulence spectra at a monitor point upstream of the blade leading edge ($x=-0.2$, $y=0.0$) have been extracted, and the power spectral density of the three fluctuating velocity components for case A are presented in figure 4(b). The curves for all of the three components show a range of - 5/3 behaviour, indicating an inertial range for the incoming turbulence. Moreover, it is noted that the streamwise component is stronger at low frequencies compared to the other components, which implies that the turbulence turns anisotropic downstream of the inlet due to the streamwise flow acceleration, driven by the pressure ratio between the outlet and inlet.

Figure 4. Characterization of the inlet turbulence: (a) evolution of pitchwise averaged turbulence intensity $T_{u}$ along the streamwise direction; (b) power spectral density (PSD) of different velocity components normalized by $T_{u}^{2}$ in case A.

2.3 Validation with experiments

The results from the present simulations, particularly the accuracy of the prediction of the blade boundary layers, have been quantitatively validated against the available data from the VKI experiments (Arts et al. Reference Arts, Lambertderouvroit and Rutherford1990). The isentropic Mach number $Ma_{is}$, which is related to the pressure distribution around the blade boundary layer, is plotted against the surface length $s$ in figure 5(a). On the pressure side of the blade ($s<0$), $Ma_{is}$ increases monotonically from the leading edge at $s=0$, which indicates a FPG boundary layer across the pressure side of the blade, as shown in figure 5(a). On the other hand, $Ma_{is}$ on the suction-side surface ($s>0$) first increases and then decreases, and the peak of $Ma_{is}$ divides the suction-side boundary layer into the FPG and APG regions. While the $Ma_{is}$ distribution for the same $Re$ was not reported in the experiment, an experimental case with the same pressure ratio but at a higher $Re$ is also selected for comparison in figure 5(a). It is shown that the numerical results are in close agreement with the experimental data, and the cases with different inlet turbulence do not show significant deviation for the mean pressure distribution around the blade.

Figure 5. Validation of the results: (a) isentropic Mach number $Ma_{is}$ across the blade surface, for negative $s$ on the pressure side and for positive $s$ on the suction side; (b) blade surface heat flux $q_{n}$, symbols representing experimental datasets with $T_{u}$ levels at 6 % and 4 %.

Furthermore, the blade surface heat flux

$$\begin{eqnarray}q_{n}=-\frac{\unicode[STIX]{x1D707}}{(\unicode[STIX]{x1D6FE}-1)PrRe_{\infty }Ma_{\infty }^{2}}\frac{\unicode[STIX]{x2202}\widetilde{T}}{\unicode[STIX]{x2202}n}\end{eqnarray}$$

is also presented in figure 5(b). Here, $\widetilde{T}$ is the Favre-averaged temperature. The experimental datasets shown here are from two cases at the same $Re$ and $Ma$, while the turbulence intensities are at the levels of $4\,\%U_{in}$ and $6\,\%U_{in}$, respectively. As the turbulence length scales or dissipation rate were not provided in the experiments, it is not possible to reproduce the exact turbulence states in the numerical simulations (Bhaskaran & Lele Reference Bhaskaran and Lele2010). However, despite the potential differences of the boundary conditions, the heat flux distributions from cases A, B and C agree well with the experimental data, including the prediction of the transition onset. Moreover, case D with high-intensity and large-scale inlet turbulence, which is more representative of real engine environments, shows significant deviation from the other cases, and it is therefore of great interest in the present study.

4.1 Leading-edge structures

As shown by the blade geometry in figure 1, the HPT in the present simulations has a blunt leading edge, which is known to have direct effects on the receptivity process of the boundary layer (see Kendall Reference Kendall, Reda, Reed and Kobayashi1991). In addition, it has been reported that streamwise vortices can form around the leading edge, which is due to the tilting and stretching of the incoming wall-normal vortices (Goldstein & Wundrow Reference Goldstein and Wundrow1998; Wundrow & Goldstein Reference Wundrow and Goldstein2001). The streamwise vortices thus can induce wake-like disturbances similar to backward jets (Jacobs & Durbin Reference Jacobs and Durbin2001) or low-speed streaks (Brandt et al. Reference Brandt, Schlatter and Henningson2004), and the instability of these streamwise streaks finally results in the breakdown to turbulence. Other than the instability of streaks, Nagarajan et al. (Reference Nagarajan, Lele and Ferziger2007) introduced a different path to transition considering the effects of the leading edge, for which turbulent spots are formed via precursors which can be traced back to leading-edge structures. This breakdown mechanism due to leading-edge structures was observed to be dominant when the FST intensity or the leading-edge bluntness is large. In the present study, the extraordinarily blunt leading edge of the HPT, along with incoming turbulence of high amplitudes and large scales coming from the inlet, is considered. It is clearly shown in figure 8 that the boundary layer transition taking place in all of the cases is influenced by the vortical structures from the leading-edge region. Therefore, the leading-edge structures will be studied to help understand the downstream transition.

The basic boundary layer statistics, i.e. the profiles of the mean tangential velocity and the velocity fluctuations at different locations, are presented in figure 9. In particular, cases A and D, with the lowest and highest levels of inlet turbulence in the present study, are selected for comparison. As shown in figure 9(a), the mean velocity profiles for both cases are in close agreement with each other, despite the remarkably strong fluctuations to which case D is subjected, as presented in figure 9(b). Furthermore, the mean tangential velocity profiles in case A are plotted in wall units in figure 9(c), in comparison with the profiles from a two-dimensional case without any FST, and the close agreement implies that the boundary layers in the leading-edge region ($x<0.3$) stay laminar. This is due to the stabilizing effects of the strong FPG and high convex curvature (see Muck, Hoffmann & Bradshaw Reference Muck, Hoffmann and Bradshaw1985; Katz, Seifert & Wygnanski Reference Katz, Seifert and Wygnanski1990; Mukund et al. Reference Mukund, Viswanath, Narasimha, Prabhu and Crouch2006), which are prominent in the leading-edge region as identified in figure 7(b,c).

It is also demonstrated in figure 9 that compared to other components, the amplitude of the spanwise fluctuation is much larger near the wall in both cases. The strong spanwise disturbances at the leading edge are also shown by instantaneous contours of the spanwise velocity component in figure 10(a), and a schematic for this plane cut is given in figure 10(b). It appears that the large amplitude of tangential vorticity, presented by the iso-lines, is basically located at those regions showing strong gradients of the spanwise velocity fluctuation $w^{\prime }$. This relation between the large amplitude of spanwise disturbances and tangential vorticity can be expressed as $\unicode[STIX]{x1D714}_{t}^{\prime }\sim \unicode[STIX]{x2202}w^{\prime }/\unicode[STIX]{x2202}n$, with $n$ representing the wall-normal distance to the blade surface. Similar to the observation in Goldstein & Wundrow (Reference Goldstein and Wundrow1998), the tangential vortices at the leading edge will wrap around the blade surface, and the resulting streamwise vortices are expected to play a key role in downstream transition.

Figure 10. (a) Contour of spanwise velocity fluctuation at the leading-edge wall-normal plane cut in case A. Iso-lines of tangential vorticity $\unicode[STIX]{x1D714}_{t}^{\prime }$ (normal to the plane cut) are also shown, with solid and dashed lines indicating $\unicode[STIX]{x1D714}_{t}^{\prime }=100$ and $\unicode[STIX]{x1D714}_{t}^{\prime }=-100$, respectively. (b) Schematic for the wall-normal plane cut at the leading edge.

The effects of the leading-edge structures on the boundary layer are further investigated by scrutinizing flow variables in the wall-normal–spanwise planes, as shown in figure 11. In the wall-normal plane cut near the suction peak at $x=0.3$, the iso-lines of $Q$ represent the streamwise vortical structures wrapped around the leading edge, which are also shown in figure 8, while the contours show the tangential velocity fluctuation $u_{t}^{\prime }$ and the temperature fluctuation $T^{\prime }$. For case A, it is noted that the near-wall region as marked by the black dashed box in figure 11(a) is packed with strips of positive and negative velocity fluctuations. In particular, the low-speed fluid seems to be elevated and forms ‘mushroom-like’ structures, which are also shown by the contour of temperature fluctuation $T^{\prime }$ from figure 11(c). The lift-up effects of the vortical structures are clearly presented in figure 11(d,e), in which two ‘mushroom-like’ structures in figure 11(c) are extracted to provide a zoom-in view, with the vectors showing the velocity fluctuations in the spanwise $w^{\prime }$ and wall-normal $u_{n}^{\prime }$ directions. It appears that the vortical structures induce strong upward velocity near the wall, driving the fluid with relatively low temperature away from the blade and thus enhancing the heat flux. Therefore, the streamwise vortices are proven to be directly linked to the low-speed streaks, which has also been presented in several previous studies on flat-plate boundary layers with zero pressure gradients (Andersson et al. Reference Andersson, Brandt, Bottaro and Henningson2001).

Figure 11. Instantaneous snapshots for contours of spanwise and wall-normal plane cuts at $x=0.3$: (a) tangential velocity fluctuation $u_{t}^{\prime }$ for case A; (b) $u_{t}^{\prime }$ for case D; (c) temperature fluctuation $T^{\prime }$ for case A; (d,e) zoom-in views of ‘mushroom-like’ structures of $T^{\prime }$ with fluctuating velocity vectors. The iso-lines of $Q=4000$ are also presented in the contours. Note that the wall-normal coordinate $n$ is enlarged to clearly show the structures.

For comparison, the contour of $u_{t}^{\prime }$ in figure 11(b) shows the velocity fluctuations in case D, which are apparently much more violent compared to those in case A. Moreover, the iso-lines of $Q$ are nearly randomly distributed over the plane cut, corresponding to the ‘chaotic’ vortical structures as shown in figure 8(d). This indicates that the high-level FST in case D drives stronger fluctuations and thus ‘chaotic’ vortical structures in the boundary layer, which are expected to cause much earlier transition as presented in figure 7(d). The different states of the vortical structures obviously cause different paths of transition, and the details will be further discussed in § 4.2.

Figure 12. Instantaneous snapshots for streaks and the breakdown to turbulence at $t=t_{0}$ in case A. Contours of (a) streamwise component $u_{t}^{\prime }$, (b) wall-normal $u_{n}^{\prime }$ and (c) spanwise $w^{\prime }$ are presented at the surface which is $n=\frac{1}{2}\unicode[STIX]{x1D6FF}(x=0.5)$ from the wall. The blue dashed line indicates the boundary between FPG and APG at $x=0.8$, and an incipient turbulent spot is highlighted by the red dotted circle.

4.2 Breakdown mechanisms

Affected by the structures formed at the leading-edge region and the continuing exposure to the FST, the suction-side boundary layers in the present simulations show differences in their transition behaviours, e.g. different transition onsets indicated by the wall-friction coefficient in figure 7(d) and various structures shown in figure 8. In this section, we will investigate the various transition mechanisms, including the instability of streaks in the cases with relatively low-level incoming turbulence, and the breakdown caused by vortical structures in the case with higher-level turbulence.

4.2.1 Varicose breakdown

With the relatively low-level turbulence in cases A and B, the boundary layers show elongated streaks across the FPG region as presented in figure 8(a,b), and the transition onset is near the blade trailing edge under the APG. In figure 12, another view of the suction-side boundary layer for case A is given, displaying the contours of $u_{t}^{\prime }$, $u_{n}^{\prime }$ and $w^{\prime }$ on a surface at a distance of $\unicode[STIX]{x1D6FF}(x=0.5)$ from the wall. The low- and high-speed streaks in the streamwise direction can be shown by the $u_{t}^{\prime }$ contour. It is also apparent that for the streamwise streaks in the suction-side boundary layer, the tangential velocity fluctuation $u_{t}^{\prime }$ is much stronger than the other components. These streaks, which are usually considered as key features in bypass transition (Durbin & Wu Reference Durbin and Wu2007), are closely related to the lift-up effects of the streamwise vortical structures formed at the leading edge (Andersson et al. Reference Andersson, Brandt, Bottaro and Henningson2001).

The instability of the streaks in the APG region, as highlighted by the red dashed circle in figure 12, is more clearly shown by the wall-normal and spanwise velocity fluctuations, and it is further studied in a zoom-in view in figure 13. Corresponding to the snapshot in figure 12 at $t=t_{0}$, the contour of $u_{n}^{\prime }$ in figure 13(a) shows a symmetric distribution about the streamwise direction. In addition, the iso-lines of $w^{\prime }$ are also presented, showing an antisymmetric pattern, with solid lines and dashed lines indicating $w^{\prime }=0.2$ and $w^{\prime }=-0.2$, respectively. It is noted that although the symmetry presented in figure 13(a) is not perfect due to the strong upstream turbulence, this type of instability corresponds well with varicose or symmetric modes (Andersson et al. Reference Andersson, Brandt, Bottaro and Henningson2001; Asai, Minagawa & Nishioka Reference Asai, Minagawa and Nishioka2002).

Figure 13. Zoom-in view of the instability of the streak in case A indicated by the red circle in figure 12. Instantaneous snapshots are presented at (a) $t=t_{0}$, (b) $t=t_{0}+0.0115$ and (c) $t=t_{0}+0.023$, with $t_{0}$ representing the snapshot shown in figure 12. Contours of the wall-normal velocity fluctuation $u_{n}^{\prime }$ are presented on the surface $\unicode[STIX]{x1D6FF}/2$ away from the wall, and the iso-lines of spanwise velocity fluctuation $w^{\prime }$ are also shown, with solid lines and dashed lines indicating $w^{\prime }=0.2$ and $w^{\prime }=-0.2$, respectively.

Figure 14. The inception of the varicose mode instability of the streak indicated by the red circle in figure 12. The contours of the tangential velocity perturbations $u_{t}^{\prime }$ on a cross-spanwise plane cut across the streak are shown by the instantaneous snapshots at (a) $t=t_{0}-0.023$, (b) $t=t_{0}-0.0115$, (c) $t=t_{0}$ and (d) $t=t_{0}+0.0115$, with $t_{0}$ representing the snapshot shown in figure 12. The iso-surfaces of $u_{t}^{\prime }=1.0$ (red), $u_{t}^{\prime }=-1.0$ (blue) and $Q=4000$ (green) are also shown at (e) $t=t_{0}-0.023$, (f) $t=t_{0}-0.0115$, (g) $t=t_{0}$ and (h) $t=t_{0}+0.0115$.

The inception of the varicose instability, which was found to be driven by the wall-normal shear of the streamwise flow by Skote et al. (Reference Skote, Haritonidis and Henningson2002), can be represented by the contours of $u_{t}^{\prime }$ on a cross-spanwise plane cut across the streak. In figure 14(a–d), the snapshots are from a sequence of time instants at $t_{0}-2\unicode[STIX]{x0394}t$, $t_{0}-\unicode[STIX]{x0394}t$, $t_{0}$ and $t_{0}+\unicode[STIX]{x0394}t$, respectively, with $t_{0}$ corresponding to the snapshot shown in figure 12 and $\unicode[STIX]{x0394}t=0.0115$. It is clearly shown that the varicose instability here happens due to the interactions between the low- and high-speed streaks (Brandt et al. Reference Brandt, Schlatter and Henningson2004; Lundell & Alfredsson Reference Lundell and Alfredsson2004). Specifically, the fast-moving high-speed streak reaches the downstream low-speed region, and the instability is triggered at the interface in between. This instability grows rapidly at the tilting interface between the streaks, particularly with the presence of the APG which enhances the inflection for the streamwise velocity profiles. The development of the varicose mode instability can be further shown by the iso-surfaces of $u_{t}^{\prime }$ in figure 14(e–h), highlighting the formation of the vortical structures represented by the $Q$ iso-surface. Convecting downstream, the streaks quickly break down as also suggested by the contours at two subsequent snapshots at $t_{0}+\unicode[STIX]{x0394}t$ and $t_{0}+2\unicode[STIX]{x0394}t$ in figure 13(b,c).

4.2.2 Sinuous breakdown

For case C with higher-amplitude incoming turbulence, early-stage turbulent spots are observed in the FPG region, while the skin friction indicates that the mean flow stays laminar. In figure 15, a sequence of the contours of wall-normal velocity fluctuation $u_{n}^{\prime }$ is shown to track the formation of a turbulent spot, with the evolution of the spanwise-averaged wall-friction coefficient $C_{f}$ for the corresponding snapshots also plotted on the suction-side surface. The contour surface is inside the boundary layer, at a distance of $\unicode[STIX]{x1D6FF}/2$ away from the wall, where $\unicode[STIX]{x1D6FF}$ is the boundary layer thickness at $x=0.5$. In these snapshots, the spot precursor first stretches in the streamwise direction from figures 15(a) to 15(b). As the amplitude of the disturbances is still small, the variation of the $C_{f}$ value near the spot is nearly negligible. Travelling downstream, the disturbances rapidly amplify and the precursor starts to break down, with the $C_{f}$ value around the spot significantly increasing as shown in figure 15(c). Specifically, the breakdown of the spot precursor initiates at the tail and expands in the spanwise direction, resulting in a triangular shape, with the head towards the downstream direction. The spot continues to grow in both size and amplitude moving further downstream, which can be shown by the evolution of the $C_{f}$ wave-packet indicated by the blue arrow. Nevertheless, it is noted that the skin friction recovers to the undisturbed state again once the spot has passed by, suggesting that the disturbances for the formation and evolution of the turbulent spot are localized and there are calming effects trailing the spot. The temporal evolution of the displacement thickness of the boundary layer at $x=0.8$, which is indicated by the blue dashed line in figure 15, is presented in figure 16, showing that the displacement thickness relaxes to a value even smaller than that in the undisturbed state after the spot-caused peak. As suggested by Katz et al. (Reference Katz, Seifert and Wygnanski1990), the low displacement thickness trailing the turbulent spot indicates the existence of a calmed region which is presumably caused by the distortion of the local mean flow by some nonlinear process. Finally, the enlarged spot merges with the turbulent boundary layer near the blade trailing edge, while a new spot precursor forms in the FPG region again as shown in figure 15(e).

Figure 15. Formation of the turbulent spot in case C. Contours of wall-normal velocity fluctuation $u_{n}^{\prime }$ at the instantaneous snapshots (a) $t=t_{0}$, (b) $t=t_{0}+0.069$, (c) $t=t_{0}+0.138$, (d) $t=t_{0}+0.207$ and (e) $t=t_{0}+0.276$. The blue dashed lines indicate $x=0.3$ and $x=0.8$, respectively, and the spot precursor is highlighted by the red dashed circle. Meanwhile, the wall-friction coefficient $C_{f}$ on the suction-side surface is averaged in the spanwise direction and plotted against $x$ for the corresponding snapshots (red solid lines), with the mean $C_{f}$ represented by the black dashed lines for reference.

Figure 16. Temporal evolution of the displacement thickness of the boundary layer profile at $x=0.8$ in case C. The instantaneous profile is spanwise-averaged, starting from $t_{0}$ corresponding to figure 15(a).

In order to further investigate the incipient spot, a zoom-in view of the spot precursor is shown in figure 17, tracking the structures highlighted by the red dashed circle in the snapshots of figure 15(a–c). Here, the vortical structures are identified by the iso-surface of $Q$ in figure 17(a–c), and the iso-surfaces of $u_{t}^{\prime }$ are also shown to track the development of the streamwise velocity fluctuation streaks. It is shown in figure 17(a,d) that there exists a quasi-streamwise vortex filament located in the strong spanwise shear region between the low- and high-speed streaks. The streamwise-elongated vortex filament, which seems to be a remnant of the leading-edge vortical structures, is tilting with the wall-normal vorticity component $\unicode[STIX]{x1D714}_{n}^{\prime }\sim \unicode[STIX]{x2202}u_{t}^{\prime }/\unicode[STIX]{x2202}n$. A secondary instability of the streak is then observed, caused by the strong spanwise shear in figure 17(b,e), with the tail of the filament oscillating in the spanwise direction. Furthermore, the streak breaks down and forms the triangular spot seen in figure 17(c,f).

Figure 17. Zoom-in view of the instantaneous structures highlighted by the red dashed circle in figure 15 to show the breakdown in case C. Contours of the streamwise velocity fluctuation $u_{t}^{\prime }$ are shown on the surface at $\unicode[STIX]{x1D6FF}/2$ from the wall, and iso-surface of $Q=4000$ is also shown at (a) $t=t_{0}$, (b) $t=t_{0}+0.069$ and (c) $t=t_{0}+0.138$. Iso-surfaces of $u_{t}^{\prime }=1.0$ (red) and $u_{t}^{\prime }=-1.0$ (blue) at (d) $t=t_{0}$, (e) $t=t_{0}+0.069$ and (f) $t=t_{0}+0.138$.

It is noted that the spanwise oscillations observed in figure 17(b) are only on one side of the streak, which is in close agreement with the one-sided sinuous mode. As discussed by Brandt et al. (Reference Brandt, Schlatter and Henningson2004), the sinuous instability of low-speed streaks, though many are symmetric, can be dominated by single-direction vorticity and oscillate only in one direction. This one-sided instability is believed to be related to the spanwise asymmetry of the streaks, which is indeed present for case C with strong FST. Furthermore, contours of the streamwise velocity fluctuation $u_{t}^{\prime }$ in four cross-stream plane cuts at different streamwise positions, which correspond to the sinuous disturbances in figure 17(b), are shown in figure 18. Along with the vectors showing the fluctuating velocities in the wall-normal and spanwise directions, the meandering of the low-speed streak also gives further credibility to the hypothesis of the presence of a one-sided instability.

Figure 18. The inception of the varicose mode instability in case C, corresponding to the snapshot shown in figure 17(b) at the instantaneous snapshot $t=t_{0}+0.069$. The contours of the tangential velocity perturbations $u_{t}^{\prime }$ are shown on four cross-streamwise plane cuts between the two blue dashed lines in figure 17(b) at (a) $x=0.70$, (b) $x=0.705$, (c) $x=0.71$ and (d) $x=0.715$, and the vectors present the fluctuating velocities on the plane.

Figure 19. Formation of the turbulent spot in case D. Contours of wall-normal velocity fluctuation $u_{n}^{\prime }$ at the instantaneous snapshots (a) $t=t_{0}$, (b) $t=t_{0}+0.046$, (c) $t=t_{0}+0.092$ and (d) $t=t_{0}+0.138$. The blue dashed lines indicate $x=0.5$ and $x=0.8$, respectively, and the spot precursor is highlighted by the red dashed circle.

4.2.3 Breakdown induced by vortex interaction

For case D, the velocity fluctuations in the boundary layer are stronger than in other cases and the vortical structures are more ‘chaotic’ from the blade leading edge, as shown in figure 11. As a result, the transition onset shown by skin friction already occurs in the FPG region ($x<0.8$) for case D, which is much earlier than in the other cases. In order to investigate the transition mechanisms, the breakdown process is presented by the contours of the wall-normal velocity fluctuations $u_{n}^{\prime }$ in figure 19. As a result of the large-amplitude inlet turbulence, the contours of $u_{n}^{\prime }$ for case D show much stronger fluctuations, compared to case C shown in figure 15, that has the same turbulence amplitude at the inlet but a smaller length scale and thus exhibits more rapid decay of the incoming turbulence. In particular, once the flow enters into the region with relatively weak FPG and convex curvature, which is downstream of the first blue dashed line representing $x=0.5$, the fluctuations originating from the leading edge induce multiple structure breakdowns in the spanwise extent. One spot precursor, as highlighted by the red dashed circle in figure 19, is considered as an example and tracked through the formation process. The precursor is first observed just downstream of $x=0.5$, and it rapidly grows in size in both the streamwise and spanwise directions, travelling downstream. Thereafter, while the precursor breaks down to form a region packed with smaller-scale fluctuations, the shape of this region is obviously different from the spot shown in figure 15.

The formation of the spot is further investigated in the zoom-in view presented in figure 20, with the vortical structures identified by the iso-surface of the $Q$ criterion and the contours of $u_{t}^{\prime }$ also shown. The instability of streaks, as presented for cases A, B and C in §§ 4.2.2 and 4.2.1, is not clearly observed here. Rather, the spot precursor results from the vortical breakdown. The pair of vortex filaments, as highlighted by the blue dashed circle in figure 20(a), forms a $\unicode[STIX]{x1D6EC}$-shape with the tip pointing in the negative streamwise direction, which is the opposite from the $\unicode[STIX]{x1D6EC}$-like structures usually observed in natural transition (Zhao, Yang & Chen Reference Zhao, Yang and Chen2016a,Reference Zhao, Yang and Chenb; Zhao et al. Reference Zhao, Xiong, Yang and Chen2018). The resulting spot, as presented in figure 20(b), is therefore different from the ordinary turbulent spots with the head towards the downstream direction, as shown in figure 17(c).

Figure 20. Zoom-in view of the instantaneous structures highlighted by the red dashed circle in figure 19 to show the breakdown in case D, at (a) $t=t_{0}$ and (b) $t=t_{0}+0.046$. Contours of the streamwise velocity fluctuation $u_{t}^{\prime }$ are shown on the surface at $\unicode[STIX]{x1D6FF}/2$ from the wall, and iso-surfaces of $Q=4000$ are also shown.

Figure 21. The incipient of the spot in case D. The contours of the instantaneous tangential velocity perturbations $u_{t}^{\prime }$ on a cross-spanwise plane cut at $z=0.29$ as indicated by the blue dashed arrow in figure 20: (a) $t=t_{0}$, (b) $t=t_{0}+0.023$, (c) $t=t_{0}+0.046$ and (d) $t=t_{0}+0.069$, with $t_{0}$ representing the snapshot shown in figure 20(a). The window is moving at a speed of $c=1.74$ to track the spot precursor highlighted by the blue dashed circle in figure 20(a).

The breakdown of the precursor highlighted in figure 20(a) can be further shown by the contours of $u_{t}^{\prime }$ in the cross-spanwise plane cut at the tip of the precursor. In figure 21(a), the tip of the $\unicode[STIX]{x1D6EC}$-shaped vortex pair is located at the interface between the high- and low-speed region, as indicated by the black solid arrow. The vortex pair induces a downward movement of relatively high-speed fluid and elevates the upstream low-speed fluid from the wall, and the instability of the interface rapidly develops and then breaks down under the strong background turbulence, as shown in the subsequent snapshots in figure 21(b–d).

We remark here that the $\unicode[STIX]{x1D6EC}$-shaped vortex in figure 20(a) is just one example of the key structures. Due to the ‘chaotic’ nature of the vortical structures under the strong background turbulence, structures of different forms have also been identified in the formation of spot precursors in case D. The consistent mechanism for this type of breakdown, however, is that the precursors always originate from energetic vortical structures that can be traced back to the blade leading edge. The transition breakdown resulting from leading-edge structures seems to be in close agreement with the leading-edge effects discussed in Nagarajan et al. (Reference Nagarajan, Lele and Ferziger2007), in which the vortical effect is dominant when the bluntness is large and the FST is strong.

As a summary, the transition onsets in cases A and B are in the APG region close to the blade trailing edge. High- and low-speed streaks are observed, which are induced by the lift-up effects of the streamwise vortices formed at the leading edge. The structures then transform while convected downstream, due to a varicose mode instability that occurs as a result of the wall-normal inflectional instability under the APG, and which rapidly amplifies and causes the final breakdown to turbulence. For cases with stronger FST, however, the transition onsets occur much earlier. For case C, spot precursors form in the weak FPG region, and the formation process is through the secondary instability of streaks, which is observed to be the sinuous mode. Furthermore, with the even stronger FST entering the HPT cascade in case D, the vortical structures formed at the leading edge are in a chaotic manner and have a direct effect on the transition breakdown.