To measure the loss reduction and identify where the loss reduction has occurred due to LE optimization, loss breakdown is carried out to investigate the effect of LE modification on the change in loss at different regions . The strategy of loss breakdown is achieved by separating the domain into several fixed zones; the loss audit is calculated through the volume integration of entropy production rates in each zone. Then the loss coefficient ξi of each zone is defined in Eq. (2), which is normalized by the isentropic kinetic energy at a specified location in Fig. 7.
is a mean temperature calculated with
is the static temperature at the Plane 2;
are the total temperature at Plane 1 and Plane 2 respectively; m1
is the mass flow at Plane 1; νi
is the volume of each fixed zone; cp
is the specific heat capacity at constant pressure;
are the total and the static pressure at Plane 2 respectively; γ
is the heat capacity ratio.
The entropy production rates
consist of four aspects :
: entropy production rates by direct (time-averaged) mechanical dissipation;
: entropy production rates by turbulent mechanical dissipation with SST k
: entropy production rates by heat conduction due to direct (time-averaged) temperature gradients;
: entropy production rates by heat conduction due to turbulent temperature gradients with the SST turbulence model.
is the strain-rate tensor; k
is the turbulent kinetic energy; ω
is the specific dissipation rate;
is the direct (time-averaged) temperature; According to SST k-ω
turbulence model theory, β
is thermal diffusivity; αt
is turbulent thermal diffusivity.
The zones for volume integration of entropy production rates include six parts, i.e., upstream zone, endwall zone, pressure side zone, suction side zone, passage zone and downstream zone. And all are illustrated in Fig. 7(b) and Fig. 7(c). All the zones for
integration are located in the region between Plane 1 and Plane 2, as showed in Fig. 7(a), excluding the regions of repeating trailing wakes and inlet freestream from the whole computational domain. The hub and tip endwall zone include all the grid elements within 5% of the span near each endwall. The upstream zone starts from Plane 1 that 23.5% axial chord ahead of the leading edge, meanwhile the downstream zone before Plane 2 is 25% axial chord downstream of the TE. The pressure and suction side zones are located in the regions that both extend 2mm away from the profile surface. The remaining region is the passage zone.
Considering that the accuracy of direct volume integration of entropy production rates is sensitive to the grid resolution, further validation on the grid number of N2
= 5.71 million is carried out through comparing the total loss coefficients wtot_sw
(Eq. (6)), which is transformed from the relational expression of entropy increase, against the total pressure loss wtot
(Eq. (4)) and the total energy loss ζtot
(Eq. (5)). The entropy increases
can be transformed into total pressure loss with Eq. (6) in an adiabatic flow process, across a stationary component. Noting that the entropy increase
is used to compare with that of the volumetric integration of entropy production rates in a direct way, which is obtained with the total outlet pressure
in Eq. (8). Fig. 8 illustrates the comparisons at all the operating points. The ordinates of total pressure loss coefficients wtot_sw1
in Fig. 8(c) and Fig. 8(d) are obtained through Eq. (7) and Eq. (8), respectively. Obviously, the loss coefficients wtot_sw1
of “Ori” and “CST” LE match well with the other three audit methods. Apart from the condition with the incidence of 12°, the benefit of the wtot_sw1
reduction from LE optimization is a little larger than that of wtot_sw2
, which is mainly caused by the larger region occupied by secondary flow at larger incidence. Nevertheless, as an aerodynamic loss audit parameter, the integration of entropy production rates is suitable for the present study with the grid number of 5.7 million based on the quantitative relation between ζtot
Fig. 7 The different loss breakdown zones for entropy production rate integral calculation
Fig. 8 The comparison of loss in different definitions at the condition of Ma = 0.65
3.2 Influence on total loss reduction and loss proportion
Firstly, a global review of the impact of LE optimi- zation on the change in total losses is described in Fig. 9. It is found that the performance of the cascade is improved in a certain amount under all conditions, especially at the condition of nonzero incidences. The LE optimization with the CST method can significantly maintain the flow in a state of lower loss over a wider operating range. It should be noted that the relation between the total losses and the Reynolds number varies with Mach number and incidence. The total loss with “Ori” LE gets higher when the Reynolds number increases from 2.71×105 to 5.55×105 at zero incidence when the Mach number is less than 0.8. Unlike the results of “Ori” LE, the total losses are significantly reduced under the conditions with higher Reynolds number, because of the delay of the transition on the suction surface with “CST” LE. The effects of Reynolds number, Mach number, and incidence on profile loss are provided in Ref. . More numerical information about the boundary layer behavior of HD profile at different Reynolds number can be seen in Ref. . Fig. 9 also demonstrates that the benefit reduces as the Mach number increases from 0.45 to 0.80 when the other two independent variables remain unchanged.
The cases with Ma
=0.65 are chosen for the quantitative study, considering the similar tendency of the relation of total loss and incidence or total loss and Reynolds number at different Mach number. It should be noted that the benefits are weakened as the Mach number increases. Fig. 10 presents the loss reduction percentage of each zone relative to the corresponding total loss
of “Ori” LE, and the loss of each zone is calculated through Eq. (2). Obviously, the relative loss reductions of the pressure side, the suction side, and the endwall zones are the mostly affected three ones among the six loss components.
Regarding profile loss, most loss reduction comes from the pressure side at negative incidence condition, while comes from the suction side at positive incidence condition. At the incidence of ‒12°, the relative loss reduction of the pressure side and the endwall are approximately 3.5%–10% and 2.5%–5% respectively. At the incidence of ‒6°, the relative loss reduction of the pressure side is 2.5%, and the percentage of loss reduction seems not to be influenced by the Reynolds number. At the incidence of +6°, the loss reduction of suction side increases by 16%–26% as the increase of Reynolds number. While at the incidence of +12°, the loss reduction of the suction side decreases nearly by 10%–17% compared with that at +6° incidence.
Regarding the relative loss reduction of endwall zone, it decreases markedly by 2.5%–5.0% with the LE optimization at the incidence of ‒12°, while decreases by 1.0% at the incidence of +12°. To summarize, the profile zone and the endwall zone are the two main sources of the total loss reduction. To further identify the local position of the loss reduction, the flow mechanism in these two regions needs to be paid more attention to.
Secondly, the loss proportion of each zone (
) is calculated at different conditions. In the present study, the profile loss is more than 50% of the total loss and about 2 times of the endwall loss. Considering the similar tendency of the loss proportion against the Mach number, the cases at the Mach number of 0.65 are chosen for further investigation. A bar chart is created to visualize the contribution of each loss component to the total loss in Fig. 11. The incidence has a more significant influence on the loss proportion of each zone compared with Mach number and Reynolds number. As the increase of incidence, the loss proportion of the pressure side reduces, and the suction side loss on the contrary decreases. Meanwhile, the loss proportion of endwall reduces as the increase of the incidence, especially at the conditions of positive incidence.
In addition, the absolute difference of loss proportion of each zone (
) is shown in Fig. 12. Obviously, the difference between the “CST” and “Ori” cascade can be found variation with the incidence and Reynolds number. Overall, the changes in the loss proportion of endwall are all enlarged with the “CST” LE, which is unlike the behavior of the two sides loss. At the nonzero incidence, the loss proportion on the suction side and the pressure side changes inversely, i.e., the pressure side loss proportion is decreased with the suction side increased at the negative incidence, and vice versa. At the incidence of +6°, it is interesting to find that the increase of the difference of downstream loss proportion is about 6%, which is just a little less than that of the endwall. The reason of this phenomenon will be discussed in the later section, since the downstream loss is not as high as the endwall loss at other incidences.
Fig. 9 Total loss variations along with Reynolds number, Mach number, and incidence (based on entropy production rates integration)
Fig. 10 The loss reduction percentage of each zone at the condition of Ma = 0.65
In Fig. 12, each side difference of the loss proportion is affected remarkably at nonzero incidence, which is unlike the loss variation as discussed above in Fig. 10. In respect to the suction side, both the loss and loss proportion are decreased significantly at the positive incidence. However, at negative incidence, the variations are smaller than that of positive incidences. In respect to the endwall loss at negative incidence, a slight increment in the loss proportion difference is shown in Fig. 12, while the relative loss is reduced obviously. As to the cases at positive incidences, the difference in loss proportion can be found to be increased significantly with only a little reduction in the relative loss.
To investigate the loss distribution further in two- dimension, Fig. 13 provides the comparison of the spanwise loss distribution before and after the LE optimization at the incidences of ‒12°, 0°, and +12°. Clearly, the location of the loss reduction varies in the spanwise direction at different incidences, i.e., the reduction of total loss is mainly located close to the endwall and mid-span region at the incidence of ‒12°. while located at the regions of passage vortex and mid-span for the incidence of +12°.
Fig. 11 The proportion of each loss component to the total loss at Ma=0.65
Fig. 12 The absolute difference of the proportion of each zone at the condition of Ma = 0.65
Fig. 13 The comparison of the spanwise loss distribution before and after the LE optimization
As discussed above, the profile and endwall loss are both influenced significantly by the LE optimization over the whole working range, which in general are the two major losses in turbomachines. The reasons for the reduction of these two kinds of losses will be discussed below respectively.