3.2.1. Sensitivity to constraint model

Temporally and spatially averaged physical quantities can be used to evaluate the performance of the proposed TrCLES method in prediction of T106C cascade flow. Shown in figures 12(a) and 12(b) are the distributions of time-averaged isentropic Mach number ($Ma_{is}$) around the cascade and skin friction coefficient ($C_f$) along the suction side of T106C, respectively. As is clearly seen from figure 12(a), the profile of isentropic Mach number predicted by the TrCLES method is almost identical to the experimental data. The isentropic Mach number calculated by the IDDES method shows strong deviation from the experimental data, especially near the plateau region. In fact, the IDDES method significantly overpredicts the width of the pressure plateau, which corresponds to a larger and delayed separation bubble ($C_f<0$) as can be observed in figure 12(b). For the CLES method, however, no plateau is captured in the pressure recovery area at the suction side of the blade. This indicates that the separation bubble cannot be successfully captured by the CLES method due to the constraint of the full-turbulence SST model.

Figure 12.

Distributions of time-averaged (a) isentropic Mach number around T106C and (b) skin friction coefficient along the suction surface of T106C at $Re_{2,is} = 2.5\times 10^5$ obtained using different methods: TrCLES (solid), CLES (dashed) and IDDES (dash-dotted). The experimental data from Michálek et al. (Reference Michálek, Monaldi and Arts2012) (red circles) are included for comparison.

The flow over the suction side of the T106C blade starts from a laminar state, and then separates at about $0.75c_x$. After that, transition occurs when the disturbances are amplified in the shear layer, and the flow finally turns into turbulent state. As mentioned previously, the turbulence eddy viscosity provided by the RANS model plays a key role for the correct prediction of the boundary layer flow. In laminar flow region ($x/c_x<0.75$), the turbulence viscosity should be sufficiently small, and otherwise the wrong Reynolds stresses may have an irreducible impact on the mean flow. Displayed in figure 13 are time-averaged streamlines superposed on contours of normalized turbulence viscosity $\mu _t/\mu$ obtained using different methods. The turbulence eddy viscosity with the same order of magnitude as the laminar (molecular) viscosity can be seen in the entire boundary layer on the suction side of the blade when the full-turbulence SST model is employed (see figure 13a,b). The larger eddy viscosity ratio in the laminar boundary layer even restrains the flow separation in the CLES method as seen from figure 12(b). In contrast, the smaller eddy viscosity ratio achieved by the IDDES method cannot suppress flow separation, leading to a much larger separation bubble. The turbulence viscosity is close to zero at the laminar boundary for the TrCLES method, and rapidly increases when separation occurs as can be seen in figure 13(c). Such larger turbulence viscosity effectively restrains the formation of a large separation bubble and enhances the flow reattachment.

Figure 13.

Time-averaged streamlines overlaid on contours of normalized turbulence viscosity $\mu _t/\mu$ calculated using (a) IDDES method, (b) CLES method and (c) TrCLES method.

The temporal evolutions of pressure fluctuations ($p_i^\prime =p_i-p_{i,mean}$) at different probe locations on the T106C suction surface are plotted in figure 14. Note that all the curves are modified by adding the corresponding normalized curvilinear distances from the leading edge in order to discriminate one from another. The IDDES method obtains the largest pressure fluctuations in amplitude at probes P$_7$ and P$_8$ due to the larger separation bubble. Such much larger flow structures encourage the formation of pressure disturbance waves, which travel upstream and cause large pressure fluctuations in the laminar boundary layer (see the curves at P$_1$–P$_5$ in figure 14a). For the CLES method, the amplitude of pressure disturbances increases at about the third probe (around $s/s_0=0.5$) as can be seen in figure 14(b), and then the flow rapidly develops into turbulent state. Compared with laminar flow, the turbulent boundary layer can resist much larger inverse pressure gradient and maintains an attached-flow regime. This is why no separation is observed in figure 13(b). As for the transition-predictive TrCLES method, the pressure fluctuations remain negligibly small in laminar attached region ($s/s_0 \le 0.65$), and then gradually increase after laminar separation ($s/s_0=0.8$).

Figure 14.

Local normalized pressure fluctuations with respect to dimensionless time at various ‘probe’ locations of T106C calculated using (a) IDDES, (b) CLES and (c) TrCLES methods. Different curves are distinguished from each other by adding the corresponding normalized curvilinear distance from the leading edge to the normalized pressure fluctuations. Time $t$ is non-dimensionalized using the chord $c$ and exit isentropic velocity $u_{2,is}$.

Shown in figure 15 are instantaneous isosurfaces of the second invariant of velocity gradient tensor ($Q$) on the T106C suction surface obtained using IDDES (figure 15a), CLES (figure 15b) and TrCLES (figure 15c). For IDDES, the large flow structures appear in a very small region near the trailing edge. For CLES, unphysical near-wall flow structures are formed within a large region of the blade surface after the local pressure reaches its minimum value, which confirms again that the traditional SST model cannot provide the correct Reynolds stress in laminar boundary layer region. By contrast, much smaller flow structures are observed in TrCLES after the quasi-two-dimensional region. The well-resolved flow structures in TrCLES benefit from the facts that LES is implemented in the entire computational domain, and the laminar to turbulent transition is appropriately monitored by the SST–$\gamma$ model.

Figure 15.

Instantaneous isosurfaces of the second invariant of dimensionless velocity gradient tensor ($Q$) over the suction surface near the trailing edge of T106C obtained using (a) IDDES, (b) CLES and (c) TrCLES methods.

3.2.2. Effect of Reynolds number

For modern aircraft, a LPT usually works at Reynolds numbers ranging from $0.5\times 10^5$ (cruise state at high altitude) to $5\times 10^5$ (take-off and landing states at sea level) (Hodson & Howell Reference Hodson and Howell2005). Most RANS transition models can capture the transition process accurately at high Reynolds numbers, yet they fail to predict well the transition phenomenon at low Reynolds numbers, especially for $Re < 1.0\times 10^5$ (Wang et al. Reference Wang, Cui and Xiao2021). Therefore, it is necessary to evaluate the performance of the proposed TrCLES method in a wider range of Reynolds numbers.

Shown in figure 16 are distributions of time-averaged isentropic Mach number around the T106C blade surface at three Reynolds numbers: $Re_{2,is}=0.8\times 10^5$, $Re_{2,is}=1.2\times 10^5$ and $Re_{2,is}=1.6\times 10^5$. Both the experimental data from Michálek et al. (Reference Michálek, Monaldi and Arts2012) (red circles) and numerical data from Pacciani et al. (Reference Pacciani, Marconcini, Fadai-Ghotbi, Lardeau and Leschziner2011) (dash-double dotted) based on their own RANS transition-predictive model are included in figure 16 for comparison. The results obtained in the present RANS simulations using SST–$\gamma$ are also presented. Laminar separation occurs on the suction side of the blade, which is characterized by the appearance of a plateau (during the deceleration stage) on the isentropic Mach number profiles for all cases. It can be seen that the TrCLES method can predict the isentropic Mach number distributions better than RANS simulations by Pacciani et al. (Reference Pacciani, Marconcini, Fadai-Ghotbi, Lardeau and Leschziner2011), especially on the pressure side for all Reynolds numbers. Compared with RANS simulation based on the SST–$\gamma$ model, the TrCLES method provides a better prediction in the pressure recovery region after the pressure plateau, which can be clearly seen in figure 16(a) for the case of low Reynolds number.

Figure 16.

Distributions of time-averaged isentropic Mach number around the T106C cascade at different Reynolds numbers: (a) $Re_{2,is} = 0.8\times 10^5$, (b) $Re_{2,is} = 1.2\times 10^5$ and (c) $Re_{2,is} = 1.6\times 10^5$. The experimental data (red circles) are from Michálek et al. (Reference Michálek, Monaldi and Arts2012). The numerical data from Pacciani et al. (Reference Pacciani, Marconcini, Fadai-Ghotbi, Lardeau and Leschziner2011) (dash-double dotted), Hillewaert et al. (Reference Hillewaert, Carton de Wiart, Verheylewegen and Arts2014) (dash-dotted) and Garai et al. (Reference Garai, Diosady, Murman and Madavan2016) (dashed) and those given by CLES (dotted) are also included for comparison.

At $Re_{2,is} = 0.8\times 10^5$, the numerical data from Hillewaert et al. (Reference Hillewaert, Carton de Wiart, Verheylewegen and Arts2014) (dash-dotted) at an inlet flow angle of 34.7$^\circ$ and Garai et al. (Reference Garai, Diosady, Murman and Madavan2016) (dashed) at an inlet flow angle of 32.7$^\circ$ are also displayed in figure 16(a). Both of these numerical results are obtained based on high-order discontinuous Galerkin method without the aid of extra SGS models. Hillewaert et al. (Reference Hillewaert, Carton de Wiart, Verheylewegen and Arts2014) suggest that increasing the inlet flow angle from 32.7$^\circ$ to 34.7$^\circ$ can help improve the prediction accuracy for isentropic Mach number distributions in the acceleration region on the suction side. The underestimation of isentropic Mach number within the region $0 \le x/c_x \le 0.4$ can be noticed from the result of Garai et al. (Reference Garai, Diosady, Murman and Madavan2016) (dashed), which is believed to be exactly due to the setting of inlet flow angle. However, changing the inlet flow angle within a small range cannot reduce the peak of isentropic Mach number (at $x/c_x \approx 0.6$), which is overpredicted in all simulations by Hillewaert et al. (Reference Hillewaert, Carton de Wiart, Verheylewegen and Arts2014). Garai et al. (Reference Garai, Diosady, Murman and Madavan2016) argue that this mismatch may be due to the improper set-up of inlet and outlet boundary conditions. However, in the present simulations, the peak of isentropic Mach number is bounded the employed RANS models. Due to the constraint of the transition-predictive RANS model, the TrCLES method can predict the surface loadings accurately.

As is seen in figure 16, the pure RANS computation with the $\gamma$ transition model tends to give results almost as good as those of the proposed TrCLES method, especially as far as the statistical quantities are concerned. In fact, it is expected that the proposed TrCLES method can at least recover the results given by the pure RANS $\gamma$ transition model because the laminar separation and transition processes are important for the development of turbulent flow and the resultant aerodynamic forces. Moreover, it is anticipated that the TrCLES method can simulate unsteady resolved fluctuations (see figure 15c) other than the LSB and the induced transition event, which is worth the increased computational cost as a whole, especially when the detailed flow structures are desired. The skin friction coefficient for the T106C cascade is further calculated at several Reynolds numbers using these two methods, which are compared with each other in figure 17. As can be seen, the length of the separation bubble is reduced with an increase of Reynolds number and the turbulent flow after transition becomes reattached when approaching the trailing edge of the cascade. It is clear that the friction coefficient calculated by TrCLES method almost collapses onto that given by the SST–$\gamma$ model before the separation point, but the former predicts a smaller separation bubble in comparison with the SST–$\gamma$ model, which is attributed to the unsteady effect of fluctuating structures captured by the TrCLES method.

Figure 17.

Distributions of time-averaged skin friction coefficient $C_f$ along the suction surface of T106C cascade obtained from SST-$\gamma$ model and TrCLES method at different Reynolds numbers: $Re_{2,is} = 0.8\times 10^5$ (solid line and circles), $Re_{2,is} = 1.2\times 10^5$ (dashed line and squares), $Re_{2,is} = 1.6\times 10^5$ (dash-dotted line and down triangles) and $Re_{2,is} = 2.5\times 10^5$ (long-dashed line and diamonds).

The kinetic energy loss and outlet flow angle are important quantities for the aerodynamic performance of a LPT. The kinetic energy loss coefficient is defined as

(3.3)\begin{equation} \xi = 1 - \left[ {1 - {{\left( \frac{\overline{p}_{2} }{\overline{p}_{02} } \right)}^{{{\varGamma - 1}}/{\varGamma}}}} \right]/\left[ {1 - {{\left( {\frac{{\overline{p}_{2} }}{\overline{p}_{01} }} \right)}^{{{\varGamma - 1}}/{\varGamma }}}} \right]. \end{equation}

The mass-averaged kinetic energy loss coefficient and exit flow angle at $x/c_x=1.45$ downstream of the leading edge of T106C are compared with the experimental data given by Michálek et al. (Reference Michálek, Monaldi and Arts2012) in figures 18(a) and 18(b), respectively. The numerical data from Pacciani et al. (Reference Pacciani, Marconcini, Fadai-Ghotbi, Lardeau and Leschziner2011), Babajee & Arts (Reference Babajee and Arts2012) and the present RANS simulations based on the SST–$\gamma$ model are also included for comparison. As can be seen, the best comparisons are those of Pacciani et al. (Reference Pacciani, Marconcini, Fadai-Ghotbi, Lardeau and Leschziner2011), who employ a transition model based on a laminar kinetic energy concept. Both the kinetic energy loss coefficient and exit flow angle are seriously underestimated at low Reynolds numbers in the RANS simulation of Babajee & Arts (Reference Babajee and Arts2012), who employ the correlation-based two-equation transition model proposed by Langtry & Menter (Reference Langtry and Menter2009). In comparison with the SST–$\gamma$ RANS transition model, a better agreement is achieved between the numerical results predicted by the TrCLES method and the experimental data for exit mass-averaged kinetic energy loss coefficient. As for the exit flow angle, the values provided by all the numerical methods show large deviations from the experimental results, especially for high-Reynolds-number cases (about $1^{\circ }$ difference).

Figure 18.

(a) Mass-averaged kinetic energy loss coefficient and (b) exit flow angle with respect to Reynolds number at $x/c_x=1.45$ downstream of the leading edge of the T106C cascade. The experimental data from Michálek et al. (Reference Michálek, Monaldi and Arts2012) (solid line with open circles) and the numerical data from Pacciani et al. (Reference Pacciani, Marconcini, Fadai-Ghotbi, Lardeau and Leschziner2011) (dash-double dotted line with open diamonds), Babajee & Arts (Reference Babajee and Arts2012) (dash-dotted line with plus signs) and the present CLES (dashed line with solid down triangles) are included for comparison.

Depicted in figure 19 are time-averaged streamlines and contours of instantaneous spanwise vorticity in the separation region on the suction side of T106C for Reynolds number ranging from $0.8\times 10^5$ to $1.4\times 10^5$. As illustrated by the streamlines, the pattern of separation bubble is transformed from an ‘open bubble’ mode (without reattachment on the blade suction side) ($Re_{2,is} =0.8\times 10^5$) to a ‘long bubble’ mode ($Re_{2,is} = 1.4\times 10^5$). The smaller the LSB, the narrower the pressure plateau, as seen in figure 16. Upstream of the separation bubble, the shear layer with negative spanwise vorticity attaches to the suction surface in the laminar boundary layer, and then departs from the blade to form a free shear layer due to inverse pressure gradient, which breaks into small vortices with the growth of disturbances. When $Re_{2,is} =0.8\times 10^5$, the shear layer exhibits a large departure from the wall in the laminar separation zone, which greatly intensifies flow mixing with a tremendous augmentation in energy loss (see figure 18a) and a high under-turning in exit flow angle (see figure 18b). Although laminar separation still occurs at $Re_{2,is} = 1.4\times 10^5$, the shear layer is lifted up only a small distance from the wall. After transition, the flow reattaches to the surface again prior to the trailing edge, which leads to a modest rise of the energy loss and a limited impact on the exit flow turning.

Figure 19.

(a–d) Time-averaged streamlines and (e–h) contours of instantaneous spanwise vorticity in three $x$–$y$ plane slices over the suction side of T106C obtained using the TrCLES method at various Reynolds numbers: (a,e) $Re_{2,is} = 0.8\times 10^5$, (b,f) $Re_{2,is} = 1.0\times 10^5$, (c,g) $Re_{2,is} = 1.2\times 10^5$ and (d,h) $Re_{2,is} = 1.4\times 10^5$.

Some characteristic points such as the separation point, reattachment point and maximum bubble elevation point are important indicators for examining the correctness of predicted location and size of a separation bubble and gaining insight into the physical aspects of the bubble structures. As suggested by Hatman & Wang (Reference Hatman and Wang1999), a local (or distance) Reynolds number can be used to express the positions of these characteristic points, which is defined as $Re_{s}={s V_{\infty,s}}/{\nu _s}$ with $s$ being the curvilinear distance from the leading edge of the airfoil, $V_{\infty,s}$ the mean inlet velocity and $\nu _s=\mu _s/\rho _s$ the local kinematic viscosity. For example, one can define the local Reynolds number at separation point $Re_{s,S}$, reattachment point $Re_{s,R}$ and maximum bubble elevation point $Re_{s,M}$.

Hatman & Wang (Reference Hatman and Wang1999) suggest that the experimental data for low free-stream turbulence flat plate and airfoil expressed in terms of local Reynolds number should be correlated better than momentum or displacement thickness Reynolds number. For incompressible flows with low levels of turbulence intensity ($Tu=0.3\,\%\text {--}0.6\,\%$), Hatman & Wang (Reference Hatman and Wang1999) develop a correlation-based prediction model for separated-flow transition, which takes the following forms:

(3.4a)\begin{gather} Re_{s,M} = 1.0816Re_{s,S} + 26\,805, \end{gather}

(3.4b)\begin{gather}Re_{s,R} = 1.0608Re_{s,S} +34\,890. \end{gather}

Taking into account the compressibility effect of the flow, Michálek et al. (Reference Michálek, Monaldi and Arts2012) propose a modification of the above correlation model based on the experimental data of T106C cascade in the forms

(3.5a)\begin{gather} Re_{s,M} = 1.0167 Re_{s,S} + 32\,620, \end{gather}

(3.5b)\begin{gather}Re_{s,R} = 1.0356 Re_{s,S} +32\,435. \end{gather}

Shown in figures 20(a) and 20(b) are the Reynolds numbers for maximum bubble displacement ($Re_{s,M}$) and reattachment location ($Re_{s,R}$) as a function of the local Reynolds number at separation points ($Re_{s,S}$), respectively, obtained using the present TrCLES and reported by Pacciani et al. (Reference Pacciani, Marconcini, Fadai-Ghotbi, Lardeau and Leschziner2011) based on the transition-predictive RANS model. The prediction models proposed by Hatman & Wang (Reference Hatman and Wang1999) and Michálek et al. (Reference Michálek, Monaldi and Arts2012) are also included for comparison. As can be seen, the variations of local Reynolds numbers at the maximum bubble thickness point and reattachment point with respect to the separation point Reynolds number all exhibit a linear behaviour, similar to that of the two prediction models.

Figure 20.

Local Reynolds numbers at (a) maximum displacement location and (b) reattachment point as a function of the local Reynolds number at separation point obtained using the present TrCLES (open squares). The correlations provided by Hatman & Wang (Reference Hatman and Wang1999) (solid) and Michálek et al. (Reference Michálek, Monaldi and Arts2012) (dashed) and the numerical result given by Pacciani et al. (Reference Pacciani, Marconcini, Fadai-Ghotbi, Lardeau and Leschziner2011) (plus signs) are included for comparison.

In view of the formulations for the two prediction models, the correlation between maximum bubble displacement and separation points given by Michálek et al. (Reference Michálek, Monaldi and Arts2012) is a little smaller in slope and larger in intercept than that proposed by Hatman & Wang (Reference Hatman and Wang1999). Both the slope and intercept of the correlation function between reattachment and separation points are smaller in the compressible case than its incompressible counterpart. It should be mentioned again that all the experimental data used by Hatman & Wang (Reference Hatman and Wang1999) are from incompressible flows for flat plate and airfoil. However, the exit Mach number in the present simulations for T106C cascade is 0.65 and much higher Mach number can be achieved on the blade suction side. Under such condition, the impact of compressibility on the LSB cannot be ignored. Compared with data from Pacciani et al. (Reference Pacciani, Marconcini, Fadai-Ghotbi, Lardeau and Leschziner2011), the present TrCLES results are almost identical to the correlations proposed by Michálek et al. (Reference Michálek, Monaldi and Arts2012), especially at low local Reynolds numbers. These results indicate that the proposed TrCLES method can successfully capture the separation bubble behaviours and the Reynolds number effect.

3.2.3. Effect of Mach number

Mach number, which reflects the relative importance of kinetic energy to internal energy, can also be used to measure the compressibility of gas flows. Different exit Mach numbers inevitably lead to various wave and flow structures in an UHL LPT, and as a result, have significant influence on flow loss of the turbine. Three exit Mach numbers (0.6, 0.65 and 0.7) are first examined to validate the ability of the transition-predictive TrCLES method to recognize the levels of compressibility of turbine flows. Displayed in figures 21(a) and 21(b) are distributions of time-averaged isentropic Mach numbers for different exit Mach numbers obtained in the present RANS simulation based on the SST–$\gamma$ model and TrCLE, respectively, in comparison with the corresponding experimental data provided by Michálek et al. (Reference Michálek, Monaldi and Arts2012). The exit Reynolds number is $Re_{2,is} = 1.6\times 10^5$. With an increase of exit isentropic Mach number, the isentropic Mach number curve tends to be lifted upwards on the suction side and remains almost unchanged on the pressure side, which means that the loading of the blade is increased. The isentropic Mach number distributions obtained using transition-predictive RANS simulation and TrCLES are almost identical to the experimental data. At $Ma_{2,is} = 0.7$, the SST–$\gamma$ model cannot capture the isentropic Mach number accurately at the turning point from the pressure plateau to recovery region, which is similar to the performance at low Reynolds numbers. The TrCLES method, however, effectively improves the accuracy of prediction at this turning point though the mean modelled SGS quantities are constrained by the same SST–$\gamma$ model.

Figure 21.

Distributions of time-averaged isentropic Mach number at different exit Mach numbers and exit Reynolds number $Re_{2,is} = 1.6\times 10^5$. (a) The RANS method with the SST–$\gamma$ model and (b) TrCLES. The experimental data from Michálek et al. (Reference Michálek, Monaldi and Arts2012) (symbols) are included for comparison.

Furthermore, a wider range of the isentropic exit Mach numbers from 0.4 to 0.9 are considered at the same Reynolds number $Re_{2,is} = 1.6\times 10^5$ to study the compressibility effect on the flow patterns and aerodynamic performances of T106C. Shown in figures 22(a) and 22(b) are distributions of time-averaged isentropic Mach number around the blade and skin friction coefficient on the suction side at different exit Mach numbers, respectively. With an increase of exit Mach number, the isentropic Mach number undergoes more prominent uplift on the suction side than on the pressure side. Meanwhile, the location for the peak value of $Ma_{is}$ tends to move downstream. Thus, much stronger adverse pressure gradient acts on the rear part of the T106C suction side, causing much larger flow separation at high exit Mach number. As one can clearly see in figure 22(b), with an increase of exit Mach number, the separation point first moves downstream for $Ma_{2,is} \le 0.8$, and then travels upstream again for $Ma_{2,is} >0.8$. The reattachment point, however, moves monotonically towards the trailing edge. This can be seen more clearly in figure 24(a–d), where the time-averaged separation bubble passes from ‘short’ mode to ‘long’ mode when pushed downstream on the blade suction side with an increase of exit Mach number.

Figure 22.

(a) Distributions of time-averaged isentropic Mach number around the T106C cascade and (b) skin friction coefficient on the suction surface at different exit Mach numbers and $Re_{2,is} = 1.6\times 10^5$.

Displayed in figures 23(a) and 23(b) are mass-weighted kinetic energy loss coefficient and relative exit flow angle at $x/c_x = 0.45$ downstream of the trailing edge of T106C cascade, respectively. The exit kinetic energy loss coefficient first drops slightly with increasing $Ma_{2,is}$ until reaching its local minimum value near $Ma_{2,is} = 0.6$, and then the loss coefficient experiences a sharp growth, which is consistent with the observation by Abraham et al. (Reference Abraham, Panchal, Xue, Ekkad, Ng, Brown and Malandra2010). By contrast, the exit flow angle decreases much faster with the exit Mach number when $Ma_{2,is} <0.7$, and then reaches its minimum value at $Ma_{2,is} \approx 0.8$. Therefore, the exit flow angle is much more easily affected by the exit Mach number when $Ma_{2,is} \le 0.7$. In fact, the flow near the trailing edge on the suction side undergoes more rapid acceleration than that on the pressure side with an increase of exit Mach number, which leads to a decrease of the exit flow angle. Nevertheless, the kinetic energy loss is mainly caused by boundary layer mixing. The flow loss near the exit remains almost unchanged due to the small separation bubbles and thin boundary layers at low exit Mach numbers.

Figure 23.

(a) Mass-weighted kinetic energy loss coefficient $\xi$ and (b) relative exit flow angle $\theta -\theta _0$ with respect to the isentropic exit Mach number at $x/c_x = 0.45$ downstream of the trailing edge of T106C with $Re_{2,is} = 1.6\times 10^5$.

Figure 24.

(a–d) Time-averaged streamlines overlaid on contours of resolved TKE and (e–h) contours of instantaneous spanwise vorticity in three $x$–$y$ plane slices over the suction side of T106C obtained using the TrCLES method at $Re_{2,is} = 1.6\times 10^5$ and various isentropic exit Mach numbers: (a,e) $Ma_{2,is} = 0.4$, (b,f) $Ma_{2,is} =0.6$, (c,g) $Ma_{2,is} = 0.8$ and (d,h) $Ma_{2,is} = 0.9$. The resolved TKE $k_{res}$ is normalized by $u^2_{2,is}$.

Likewise, we show in figure 24 the time-averaged streamlines superposed on resolved TKE (figure 24a–d) and contours of instantaneous spanwise vorticity (figure 24e–h) in three $x$–$y$ plane slices near the separation bubble regions of the T106C flows at various exit Mach numbers. The time-averaged separation bubble tends to move downstream and experiences a transition from ‘short’ mode to ‘long’ mode as the exit Mach number increases. Meanwhile, the shear layer after separation breaks into much more complex flow structures which manifest that the transition point successively moves downstream with increasing exit Mach number. The resolved kinetic energy emerges due to the unsteadiness of the shear layer after separation, and increases to its maximum value near the reattachment point. The peak value of resolved kinetic energy tends to be intensified with an increase in size of the separation bubble. The region of non-zero resolved kinetic energy, in which turbulent mixing takes place, is much thicker than the corresponding separation bubble.

The power spectral densities (PSDs) of pressure fluctuations at different probe points on the blade suction surface are depicted in figure 25 for four exit Mach numbers. The PSD at the monitoring point in the laminar flow region (corresponding to the regions of low resolved TKE in figure 24) is much smaller than that at the monitoring point in turbulent boundary layer (corresponding to the regions of high resolved TKE in figure 24). At point P$_5$, the transition process is almost completed when $Ma_{2,is} = 0.4$, while the onset of transition is just observed when $Ma_{2,is} = 0.6$. When the exit Mach number increases to 0.8, the boundary layer still remains laminar at P$_5$, and transition takes place just ahead of the monitoring point P$_6$. This is because the separation point and transition point move downstream when $Ma_{2,is} \le 0.8$. When $Ma_{2,is} = 0.9$, the distribution of PSD at P$_6$ is almost identical to that at P$_8$, which means that the transition is almost completed at P$_6$. The transition starts earlier at $Ma_{2,is} = 0.9$ than at $Ma_{2,is} = 0.8$ because the separation point turns to move upstream when the exit Mach number increases from 0.8 to 0.9 as shown in figure 24.

Figure 25.

The PSD of the pressure fluctuations at different monitoring points for $Re_{2,is} = 1.6\times 10^5$ and various isentropic exit Mach numbers: (a) $Ma_{2,is} = 0.4$, (b) $Ma_{2,is} =0.6$, (c) $Ma_{2,is} = 0.8$ and (d) $Ma_{2,is} = 0.9$.

The resolved Reynolds stress $R_{uv}^{res}=-\langle \bar {\rho }\rangle (\{\tilde {u}_{i}\tilde {u}_{j}\}-\{\tilde {u}_{i}\}\{\tilde {u}_{j}\})$ in terms of distance to the nearest wall $d_w$ at several monitoring points is shown in figure 26 for different exit Mach numbers. At low background turbulence level, the boundary layer transition on the blade suction side usually starts from the free shear layer after separation. The free shear layer is affected by small disturbances and breaks up to develop into turbulent boundary layer. In light of the distribution of resolved Reynolds stress, the separated shear layer is located at regions of certain distances away from the wall as marked by the grey area in figures 26(c) and 26(d). When the separation bubble is large enough in a ‘long’ mode ($Ma_{2,is}\ge 0.8$), the disturbances (or velocity fluctuations) grow gradually in the shear layer after separation. When the exit Mach number is quite low (the separation bubble is small in a ‘short’ mode), only weak development of disturbances is observed in the shear layer as illustrated in figure 26(a,b).

Figure 26.

Resolved Reynolds stress $R_{uv}^{res}$ with respect to distance to the nearest wall $d_w$ at six successive monitoring points for $Re_{2,is} = 1.6\times 10^5$ and various isentropic exit Mach numbers: (a) $Ma_{2,is} = 0.4$, (b) $Ma_{2,is} =0.6$, (c) $Ma_{2,is} = 0.8$ and (d) $Ma_{2,is} = 0.9$. Each curve is plotted by adding a constant in order to distinguish one from another.