4.1 Pressure and friction on the wall surface
A first comparison of the aerodynamic performance of T106A cascade is quantified by the analysis of the time- and spanwise-averaged pressure coefficient and skin friction coefficients. The static pressure coefficient Cp is defined as: Cp = (p − p2)/(pt1 − p2), where p, pt1, and p2 stand for the static pressure on the blade surface, the total inlet pressure and the outlet static pressure, respectively. Figure 3 shows the results subject to uniform and disturbed inflow for the case I and case II with the comparison of the results from Cassinelli et al. [23]. As can be seen from the simulation results of the time-averaged Cp distribution in Fig. 3(a), both uniform inflow and disturbed inflow obtain a similar pressure distribution along the pressure surface. It implies that the wake-type disturbance by such a small cylinder has negligible impact on the pressure side of the blade. As flow is developed along the suction surface towards the trailing edge, the pressure for the case I is gradually decreased in the presence of the favorable pressure gradient, and the pressure is changed into suction near the position where s/S0 ≈ 0.2. The pressure coefficient of the suction surface is rising toward the trailing edge after reaching a trough value at s/S0 ≈ 0.456. The pressure plateau that appears near the trailing edge indicates that separation occurs in the boundary layer and a separation bubble is formed. The results obtained from the present work are in good agreement with the numerical results reported by Cassinelli et al. [23], with only minor differences in the aft portion of pressure surface. It can be seen that the difference of the static pressure coefficient under different inflow conditions is mainly reflected around the leading edge and trailing edge of the suction surface, but has little influence on the middle part of the suction surface. For the rear side of the suction surface, the pressure on the blade surface recovers rapidly and the pressure platform disappears when the wake-type disturbance exists upstream. On the other hand, the upstream wake-disturbance slightly reduces the pressure on the front portion particularly close to the leading edge.
The skin-friction coefficient of blade is defined as \({C}_f={\uptau}_w/\left(0.5\rho {U}_{\infty}^2\right)\), and its distribution along the suction and pressure surfaces is shown in Fig. 3(b). For the case I, good agreement is observed in the distribution trend of Cf as compared with Cassinelli et al. [23]. For the case II, the skin friction coefficient is distorted along the whole blade on both suction and pressure surface. A small secondary recirculation bubble found at s/S0 ≈ 0.95 in the uniform inflow case, which is well matched again with Cassinelli et al. [23], is disappeared in the disturbed inflow. The incoming flow with disturbances energizes the boundary layer of the blade, which leads to an anticipated reattachment of the separated boundary layer before the trailing edge. This can be proved in the observation that the sign of Cf is changed from negative to positive near the trailing edge as compared to the uniform inflow case I. The disturbed inflow also modifies the Cf distribution along the front portion of the blade with respect to the uniform case I. In particular, the magnitude of Cf is decreased in the front portion where favorable pressure gradient dominates the boundary layer. The reason is probably that the impingement and stretching of streamwise vortices introduces significant reducing of pressure on the leading edge of the blade as evidenced in Fig. 3(a).
4.2 Boundary layer characteristics
The time- and spanwise-averaged boundary layer profiles from case I and case II are compared in Fig. 4. Ten points were selected along the suction surface of the cascade between s/S0 = 0.1 ~ 0.98 and the tangential velocity profile at each point was plotted with the perpendicular coordinate n. The profiles match well before s/S0 = 0.60, but it starts to deviate from each other at s/S0 = 0.70 and becomes more visible at larger s/S0. The boundary layer is detected to be more energetic when it is close to the trailing edge for the case of disturbed inflow as compared with that for the case of uniform inflow. An injection of extra momentum into the boundary layer therefore results in a shortening of separation bubble and reattachment again before the trailing edge, which is in-line with the friction coefficient behavior as shown in Fig. 3(b).
The parameters such as shape factor, H, and momentum thickness, θ, highly reflect the evolution characteristics of boundary layer of the turbine blade and are linked to profile loss estimation [24]. Figure 5 shows their distributions along the suction surface of the blade, with the inclusion of the results in Cassinelli et al. [23]. When s/S0 < 0.4, the shape factor shows a plateau shape around a value of 2.5, verifying the nature of laminar state in the front portion of the boundary layer. After this point, its value increases gradually along the suction surface and reaches a peak at around s/S0 = 0.94, and then a dramatic jump is evidenced for further increase in s/S0, which indicates that flow transition is triggered in the aft portion of the boundary layer. However, the value of shape factor is still greater than 6 at the trailing edge, illustrating that the transition to turbulence is not completed in the boundary layer at this Re. Some numerical oscillations are observed at lower polynomial order in the results in Cassinelli et al. [23], however the current predictions at the same polynomial order match well with the results at higher polynomial order in Cassinelli et al. [23]. The reason is mainly due to the high resolution of micro-element used in our simulations. The momentum thickness distribution shows an increasing trend along the suction surface, which is again in good agreement with the results from Cassinelli et al. [23].
The presence of a small cylinder upstream introduces striking modification in the evolution of these statistics of boundary layer. Since the cylinder’s wake increases the turbulence intensity of incoming flow, the momentum thickness increases rapidly after s/S0 = 0.54, so that the shape factor does not surge due to the rapid increase of the boundary layer thickness after separation, and the boundary layer velocity profile is fuller, which has a suppressive effect on the boundary layer separation. At the same time, we can see that the shape factor at the trailing edge of the blade suction surface has dropped to 2 under the disturbed inflow condition, indicating that the boundary layer has basically completed the transition when the blade is subject to wake inflow turbulence.
4.3 Wake profiles
The normalized distance from the trailing edge is defined as \(\hat{x}=\left(x-{x}_{TE}\right)/{C}_{ax}\), which is depicted in Fig. 1. The profiles of wake pressure loss coefficient, which is defined as Wu = (pt1 − pt2)/(pt1 − p2) (where pt1 is the spanwise averaged total pressure at the inlet, pt2 is the spanwise averaged total pressure at the outlet, and p2 is the spanwise averaged static pressure at the outlet), are extracted by interpolating the time-averaged pressure fields from the unstructured mesh to traverses of equispaced points in the pitchwise direction; three extraction planes with streamwise locations \(\hat{x}=\) 0.1, 0.25, 0.4 are analyzed in Fig. 6. It can be seen that the pressure loss near the wake of the cascade is very obvious under the condition of both uniform and disturbed inflows. The center of the wake moves to the side of the suction surface. This is due to the existence of separation bubbles at the rear of the suction surface. At \(\hat{x}\) = 0.1, the peak deficit reaches the maximum value. As \(\hat{x}\) increases, the loss coefficient decreases and the width of corresponding wake increases. A similar trend is detected for the case with disturbed inflow at the measured wake positions. More importantly, it is also noticed that the pressure loss is visibly reduced for the disturbed inflow case as compared with that for the uniform inflow case.
In Fig. 7(a) and (b), we compare the time- and spanwise-averaged pressure contour of the cascade under uniform inflow and disturbed inflow conditions. The streamline is also given in this image. From the comparison, we can notice that the presence of the small cylinder alternates the pressure field around the blade surface. In particular, the wake momentum deficit behind the small cylinder slightly reduces the pressure magnitude along the pressure surface and the front portion of the suction surface of the blade. Rising up of the pressure around the trailing edge is also visible in this figure, which is consistent with the pressure coefficient shown in Fig. 3(a). The streamline with the marked separation point in the figure shows the formation of the separation bubble on the aft portion of the blade, which looks significantly shorter in closed shape in the disturbed inflow case as compared with that in the uniform inflow case, which is again in-line with the variation of the skin friction coefficient shown in Fig. 3(b).
Figures 8 and 9 show the time- and spanwise-averaged Reynolds stress contours in the cascade channel for the uniform inflow and disturbed inflow cases, respectively. For uniform flow case, Reynolds normal stresses occur in the shear layer behind the suction surface of the cascade and reach maximum values in the wake region of the blade. Except for the shear layer behind the suction surface and the wake region of the blade, the Reynolds normal stresses in other places are close to zero. While in disturbed case, due to the high turbulence intensity and strong velocity pulsation of the cylinder’s wake, there are also large Reynolds normal stress values in the strip region behind the cylinder. The influence of this phenomenon is that the position of Reynolds stress in the boundary layer behind the suction surface begins to increase in advance, the area of high Reynolds stress is closer to the cascade surface, the wake is filled with fully developed turbulence, and the area of high pulsation increases. For Reynolds shear stress, since the magnitudes of u’w’ and v’w’ are small, we only analyze for u’v’. It can be seen that, compared to the uniform flow case, the u’v’ values of disturbed inflow case are smaller in the wake region of the blade.
The visualization of the instantaneous flow field is shown in Fig. 10, where the coherent structure is identified using the Q-criterion and rendered in streamwise velocity. The effect of disturbed inflow can be qualitatively illustrated in this figure. It can be appreciated that the uniform inflow case sheds coherent large-scale vortical structures from the trailing edge, while the disturbed inflow modifies the transition mechanism for which the TS wave is no longer available and therefore it prohibits the formation of Karman vortex shedding and increases the content of small coherent structures in the wake of the blade. Subsequently, the separation bubble is closed before the trailing edge in the disturbed inflow case otherwise open in the uniform case, which is very similar to that observed in Cassinelli et al. [23].