The Fourier-transformed spectra of the lift force along the circumferential direction of the cascade are given in Fig. 3. As shown in Fig. 3(a), f1, u=5.859 is the dominant frequency of the forces on the cascade under uniform flow condition, where 1 represents the main frequency and u denotes the case under uniform flow condition, it gives a dimensionless period of T1,u=0.171; As shown in Fig. 3(b), f1, c=5.493 is the dominant frequency of the lift force on the cascade under cylinder’s wake flow condition, where 1 represents the main frequency and c denotes the cylinder’s wake flow case, it gives a dimensionless period of T1,c=0.182. The periodic variation of the cascade force is caused by the alternate shedding of the trailing edge vortices, so f1, u=5.859 and f1, c=5.493 are the main frequencies of the shedding vortices at the trailing edge under the uniform flow and the cylinder’s wake flow condition, respectively.
The skin pressure coefficient Cp along the T106A blade surface is defined as \({C}_{p}=p-{p}_{2}/{p}_{t1-}{p}_{2}\), where \(p\) represents the skin static pressure value on the blade surface; \({p}_{t1}\) is the total inlet pressure; and \({p}_{2}\) is the outlet static pressure. As can be seen from the simulation results of the time-averaged Cp distribution in Fig. 3 (a), an abrupt change can be found on pressure when the surface of the blade is changed from a suction surface to a pressure surface in the vicinity of the leading edge. Meanwhile, pressure distribution characteristics similar to the T106A blade surface were also observed in experimental studies [7]. As flow is developed along the suction surface towards the trailing edge, the pressure value is gradually decreased in the presence of the favorable pressure gradient, and the pressure is changed into suction near the position where s/S0 =0.2366. The pressure coefficient of the suction surface is rising toward the trailing edge after reaching the valley value at s/S0=0.4564, indicating that the flow on the suction surface first undergoes the favorable pressure gradient and then the reverse pressure gradient. The pressure plateau that appears near the trailing edge indicates that separation occurs in the boundary layer here and a separation bubble is formed, which is produced by the combined effect of the inverse pressure gradient and viscous forces. As shown in Fig. 4(a), the results obtained from the present work are in good agreement with the experimental results reported by Cassinelli et al [7], with only minor differences in the pressure surface, which may be induced by the difference in the incoming turbulence or other settings for the experiment and the current simulation. Fig. 4 (b) shows the distribution of the time-averaged static pressure coefficient when the cylinder’s wake flows down the cascade surface. It can be seen that the difference of the static pressure coefficient under different flow conditions is mainly reflected in the leading edge and trailing edge of the suction surface, and has little influence on the middle part of the suction surface and the whole pressure surface side. At the suction leading edge, the cylinder’s wake makes the static pressure coefficient decrease, which may be caused by the change of the direction of the air flow through the cylinder. For the rear of the suction surface, the pressure on the cascade surface recovered rapidly and the pressure platform disappeared when the cylinder wake existed. It can be inferred that the cylinder wake suppressed the flow separation of the boundary layer to a large extent.
The skin-friction coefficient of blade is defined as \({C}_{f}={{\tau }}_{w}/\left(0.5\rho {U}_{\infty }^{2}\right)\), and its distribution along the suction and pressure surfaces is shown in Fig. 5. In the suction surface, the skin-friction coefficient reaches a local peak near the leading edge, which is caused by the impact of the incoming flow with the leading edge, then the friction coefficient decreases rapidly to a very small value with the increase of s and then increases slowly, and the joint analysis with the distribution of the static pressure coefficient in this range shows that in the interval 0.0359<s/S0 <0.1385, the flow has experienced the effect of the adverse pressure gradient and the favorable pressure gradient, but the friction coefficient is always greater than zero, indicating that no separation bubble is generated in this interval. At the range of 0.1385<s/S0 <0.3813, the skin-friction coefficient value is first varied slightly and then decreased gradually. Near s/S0 =0.6721 the skin-friction coefficient takes on an important change, the value changes from positive to negative, which indicates that flow separation occurs in the boundary layer at this location, which corresponds to the starting point of the pressure plateau in Fig. 3(a). Further when s/S0 =0.9397, the skin-friction coefficient takes a further decrease in value, which has something to do with the transition here and the Secondary separation bubble flow structure. The skin-friction coefficient remains less than zero at the trailing edge, indicating that the separated boundary layer does not re-attach. Also, the friction coefficient presents radically different distribution characteristics at the pressure surface, which is increased with the rising s/S0 in a large range from the leading edge to the trailing edge under the impact of the accelerated flow of the blade bending. What's more, the results obtained in this paper are also compared to the numerical calculation results of Cassinelli et al. [7] in the figure, proving to be in good agreement.
Figure 6 shows a comparison of the time-averaged skin-friction coefficients between the uniform flow and cylinder’s flow conditions. It can be seen that as the flow progresses from the leading edge to the trailing edge, the trends of the skin-friction coefficients at the pressure and suction surfaces under the two incoming flow states are basically the same, but the values at the leading and trailing edges are very different, this is mainly due to the integration effect of cylinder’s wake on air flow: It makes the incoming flow go to the leading edge at a more favorable angle of attack and the turbulence intensity of the incoming flow increased, resulting in a significant reduction of the shear stress on the leading edge surface. Looking at the skin-friction coefficient at the trailing edge of the suction surface, it can be seen that the skin-friction coefficient changes from positive to negative at s/S0 =0.7567 under the cylinder’s wake flow condition, indicating that the boundary layer starts to separate, but at s/S0 =0.9662, the skin-friction coefficient becomes positive again, indicating that the separated boundary layer reattaches to the suction surface here, forming a closed separation bubble. The comparison of the time-averaged surface skin-friction coefficient between the uniform flow and cylinder's wake flow conditions shows that the cylinder’s wake has a good suppression effect on the blade boundary layer separation.
In order to observe the development of the boundary layer at the suction surface under both incoming flow conditions, we have to explore the tangential velocity profile along the way. As shown in Fig. 7(a), 10 positions were selected along the suction surface of the cascade between s/S0 = 0.1-0.98, and a vertical line with a length of 0.1 times chord length was drawn at each position. The tangential velocity profile at each position was shown in Fig. 7(b), with the vertical coordinate n indicating the distance from the surface of the cascade. It can be seen that the tangential velocity profiles for both incoming flow conditions before s/S0 =0.50 are typical laminar profiles, and the boundary layer thickness grows slowly before s/S0 =0.60, which is due to the fact that the airflow in this region is in an accelerated state and the fluid is subjected to a strong stretching effect, which suppresses the generation of velocity pulsations to a large extent. The velocity profile at s/S0 =0.70 shows that the tangential velocity of the fluid near the wall under the uniform flow condition is less than zero, indicating that the boundary layer separation occurs here, indicating that the boundary layer separation point under the uniform flow condition is located between sS0 /=0.60 and 0.70, which is consistent with the results obtained from the skin-friction coefficient distribution at the suction surface; while the separation point under the cylinder’s wake flow condition is located between s/S0=0.70~0.80, indicating that the cylindrical wake is able to retard the arrival of the separation point. The velocity profile after separation shows that the thickness of the boundary layer increases rapidly after separation in both incoming flow conditions, but the tangential velocity of the fluid near the wall in the cylinder’s wake flow condition is already greater than 0 at s/S0=0.98, indicating that the boundary layer has attached again before that. Compared with the uniform flow case, the separation point of the boundary layer in the cylinder’s wake flow condition is more backward and can be attached again, indicating that the separation boundary layer in the cylinder’s wake flow condition is shorter, and the comparison of the tangential velocity profile shows that the separation boundary layer in the cylinder’s wake flow condition is thinner, which is because the tangential velocity profile in the cylinder’s wake flow condition is more full and the fluid in the boundary layer has higher energy amount to resist the inverse pressure gradient, thus enabling a better fit on the blade surface downstream.
The boundary layer momentum thickness and shape factor can highly reflect the evolution characteristics of the cascade boundary layer along the flow direction. The shape factor can reflect the shape of the velocity profile inside the boundary layer, from which the thickness of the separation bubble can be judged. As the shape factor increases, the thickness of the separation bubble becomes larger. Fig. 8 shows the boundary layer momentum thickness distribution and shape factor simulation results of the suction surface of the cascade under uniform flow condition, when s/S0 is less than 0.4, the shape factor is around 2.5, and the boundary layer is in laminar flow state, after s/S0=0.67, the shape factor increases rapidly along the flow direction, while the momentum thickness is only increasing slowly, indicating that the displacement thickness increases sharply after that point, combined with the change in the distribution of the time-averaged skin-friction coefficient at the suction surface and the analysis of the time-averaged tangential velocity profile show that the boundary layer separates after that point. The shape factor increases sharply to a peak point at s/S0 =0.94, indicating that there is a transition process in the boundary layer, and the shape factor is still greater than 6 at the trailing edge, indicating that the transition is not completed in the boundary layer. Also given in this figure are the simulation results of Cassinelli et al [7]. Although the orthogonal polynomial used in this paper is of relatively low order, it is still found that the simulation results of this paper agree well with the results of Cassinelli et al. The method adopted in this paper can well predict the variation trend of these two characteristic quantities along the surface, especially at the position where the value jumps near the trailing edge, which is in good agreement with Cassinelli's results. It shows that the spectral/hp element method can accurately predict the dynamic evolution of the T106A boundary layer.
Figure 9 shows the comparison of the time-averaged momentum thickness and shape factor of the blade suction surface under the uniform flow condition and the cylinder's wake flow condition. Since the cylinder’s wake increases the turbulence intensity of the incoming flow, the momentum thickness of the blade suction surface boundary layer under the cylinder’s wake 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 cylinder’s wake flow condition, indicating that the boundary layer has basically completed the transition under the cylinder’s wake flow condition.
In this paper, the wake flow position from the trailing edge is defined as , the specific position is shown in Fig. 1, and three different wake positions =0.1, 0.25, 0.4 under two incoming flow conditions are analyzed respectively. The change characteristics of the wake pressure loss coefficient (definition: Wu = Pt1 - Pt2/Pt1 - P2) are shown in Fig. 9. It can be seen from Fig. 9(a) that the pressure loss near the wake of the cascade is very obvious under the condition of uniform flow. The center of the wake, that is, the maximum loss coefficient value, moves to the side of the suction surface. This is due to the existence of separation bubbles at the rear of the suction surface, so that the degree of deflection of the air flow out of the cascade channel is reduced. At the three different positions analyzed, when =0.1, the peak deficit takes the maximum value, which is about 0.3342. As the value of increases, the maximum loss coefficient decreases and the width of the high-loss zone increases; the analysis of Fig. 9(b) shows that under the condition of cylinder’s wake flow, the pressure loss near the wake of the cascade is very large and the pressure loss coefficient distribution between the two positions =0.1 and 0.25 is very small, the maximum loss coefficient value is about 0.3836. Then with the increase of , the pressure loss coefficient decreases and the width of the high loss zone increases. At the same wake flow position, the pressure loss under the condition of the incoming flow of the cylinder’s wake is greater.
Figure 11(a), (b) shows the time-averaged streamline and velocity contour near the leading edge and trailing edge of the cascade under uniform flow condition, and (c), (d) show the time-averaged streamline and velocity contour near the leading edge and trailing edge of the cascade under cylinder’s wake flow condition. From the Figure 11(a), (c), it can be seen that there is a tendency for the fluid to flow from the pressure surface to the suction surface at the leading edge of the blade, which is driven by the air pressure difference between the pressure surface and the suction surface. From Figure 11 (b), it can be seen that open separation of the boundary layer occurs at the trailing edge of the blade under uniform flow condition; from Figure 11(d), it can be seen that there is a closed bubble at the trailing edge of the blade under cylinder's wake flow conditions, and the thickness of the boundary layer at the trailing edge is significantly smaller than that at the trailing edge of the blade under uniform incoming flow conditions.
Figure 12 shows the time-averaged Reynolds stress contour in the cascade channel under uniform flow condition, the Reynolds stress can reflect the turbulence intensity and energy dissipation magnitude of the fluid to a certain extent. It can be seen from the figure that both the Reynolds positive stress and the tangential stress show similar characteristics: the Reynolds stress is the largest in the wake region, which indicates that the fluid in the wake region has large velocity fluctuations and frequent energy exchange between large and small scale vortices, which will produce large energy dissipation.
Figure 13 shows the time-averaged Reynolds stress contour in the cascade channel under the incoming cylindrical wake flow. It can be seen that both positive stress and tangential stress distribution have the maximum values in the cylindrical wake region, indicating that energy exchange between large and small vortices is more frequent in the cylinder’s wake region.
The visualization of the transient flow field after the flow is fully developed under uniform flow condition is shown in Fig. 14(a), where the vortex structure is identified using the Q-criterion and rendered in stream velocity value u. As can be seen in the Fig. 14(a), the flow first separates near the trailing edge at the suction surface, after which the 3D effect increases leading to a transition from laminar to turbulent flow in the trailing region. The structural form of the Karman vortex street is evident in the near trailing region, and this trailing flow structure can also be distinguished in vorticity contour at the middle slice of the cascade, see Fig. 14(b). This boundary layer flow characteristic is in general agreement with the main features of the pressure distribution at the suction surface analyzed earlier.
The results of the transient flow field visualization after the fully developed flow under the cylinder’s wake flow condition are shown in Fig. 15; where the vortex structure identification is done using the Q-criterion and rendered in velocity value. As can be seen from the figure, most of the cylinder’s wake flow flows to the suction surface, and there is an obvious Karman vortex street structure in the cylinder’s wake region. The blade boundary layer is separated upstream of the trailing edge and then re-attached, and the flow at the trailing edge has obvious turbulence characteristics, indicating that transition has been basically completed at the trailing edge.