Investigation of the effects of main geometric parameters and flow characteristics on secondary flow losses in a turbine cascade 研究主要几何参数和流动特性对涡轮机级联中二次流损失的影响
To cite this article: M Mesbah et al 2021 J. Phys.: Conf. Ser. 2131032081 引用本文：M Mesbah et al 2021 J. Phys.： Conf. Ser. 2131032081
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247th ECS Meeting Montréal, Canada May 18-22, 2025
Palais des Congrès de Montréal 第 247 届 ECS 会议 加拿大蒙特利尔 2025 年 5 月 18 日至 22 日
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Abstracts due December 6th 摘要截止日期为 12 月 6 日
Investigation of the effects of main geometric parameters and flow characteristics on secondary flow losses in a turbine cascade 研究主要几何参数和流动特性对涡轮机级联中二次流损失的影响
M Mesbah ^(1^(**)){ }^{\mathbf{1}^{*}}, V G Gribin ^(1){ }^{1}, and K Souri ^(2){ }^{2} M Mesbah ^(1^(**)){ }^{\mathbf{1}^{*}} 、 V G Gribin ^(1){ }^{1} 和 K Souri ^(2){ }^{2}^(1){ }^{1} National Research University "Moscow Power Engineering Institute", Department of Steam and Gas Turbines, Krasnokazarmennaya Street, Building No. 14, Moscow, 111250, Russian Federation ^(1){ }^{1} 国立研究型大学“莫斯科电力工程研究所”，蒸汽和燃气轮机系，克拉斯诺卡扎尔门纳亚街，14 号楼，莫斯科，111250，俄罗斯联邦^(2){ }^{2} Peoples' Friendship University of Russia, Department of Mechanical and Instrumental Engineering, Miklukho-Maklaya Street, Building No. 6, Moscow, 105005, Russian Federation ^(2){ }^{2} 俄罗斯人民友谊大学机械与仪器工程系，Miklukho-Maklaya 街 6 号楼，莫斯科，105005，俄罗斯联邦E-mail: mo.mesbah64@gmail.com 电子邮件： mo.mesbah64@gmail.com
Abstract 抽象
This paper presents numerical simulation results of a three-dimensional (3D) transitional flow in a stator cascade of an axial turbine. The influences of the main geometric parameters and flow characteristics including, the blade aspect ratio, pitch-to-chord ratio, inlet flow angle, and exit Mach number, on secondary flows development and end-wall losses, were studied. The numerical results were validated by the results of experiments conducted in the laboratory of the steam and gas turbine faculty of the Moscow Power Engineering Institute. The maximum difference between computed and experimental results was 2.4%2.4 \%. The total energy losses decrease by 20%20 \% when the exit Mach number changes from 0.38 to 0.8 . Numerical results indicated that the blade aspect ratio had the most effect on secondary flow losses. The total energy losses increase by 46.6%46.6 \% when the aspect ratio decreases from 1 to 0.25 . The total loss of energy by 13.2 % decreases by increasing the inlet flow angle from 60 degrees to 90 degrees. Then by increasing the inlet flow angle from 90 to 110 degrees, the total loss rises by 3.6%3.6 \%. As the pitch-to-chord ratio increases from 0.7 to 0.75 , the total energy losses are reduced by 12.2%12.2 \%. Then by increasing the pitch-to-chord ratio from 0.75 to 0.8 , the total energy losses increase by 6%6 \%. As with experimental data, the numerical results showed that the optimal inlet flow angle and relative pitch for the cascade are 90 degrees and 0.75 , respectively. 本文介绍了轴向涡轮机定子级联中三维 （3D） 过渡流的数值模拟结果。研究了主要几何参数和流动特性（包括叶片纵横比、节弦比、入口流角和出口马赫数）对二次流发展和端壁损失的影响。在莫斯科电力工程学院蒸汽和燃气轮机学院实验室进行的实验结果验证了数值结果。计算结果和实验结果之间的最大差异为 2.4%2.4 \% 。 20%20 \% 当出口马赫数从 0.38 变为 0.8 时，总能量损失会减少。数值结果表明，叶片纵横比对二次流损失的影响最大。 46.6%46.6 \% 当纵横比从 1 减小到 0.25 时，总能量损失会增加。通过将入口流角从 60 度增加到 90 度，总能量损失减少了 13.2 %。然后，通过将入口流角从 90 度增加到 110 度，总损失增加 3.6%3.6 \% 。当音高与弦比从 0.7 增加到 0.75 时，总能量损失减少 12.2%12.2 \% 。然后通过将音高与弦比从 0.75 增加到 0.8 ，总能量损失增加 6%6 \% 。与实验数据一样，数值结果表明，级联的最佳入口流角和相对螺距分别为 90 度和 0.75 度。
1. Introduction 1. 引言
When the viscous fluid turns in the curvilinear blade-to-blade turbine passage, centrifugal forces act on all its particles. The response to the centrifugal forces in a continuous medium is the appearance of a transverse pressure gradient [1, 2]. This transverse pressure gradient is defined in Equation 1 as: 当粘性流体在曲线叶片到叶片涡轮通道中转动时，离心力作用在其所有颗粒上。在连续介质中对离心力的响应是横向压力梯度的出现 [1， 2]。该横向压力梯度在公式 1 中定义为：
where p_(s)p_{s} is the static pressure, rr is the radius of curvature, vv is the velocity and rho\rho is the density. According to the second Prandtl equation for the boundary layer, both the end-wall boundary layer and 其中 p_(s)p_{s} 是静压， rr 是曲率半径， vv 是速度， rho\rho 是密度。根据边界层的第二个 Prandtl 方程，端壁边界层和
the mainstream flow experience the same pressure gradient but the particles within the boundary layer have a lower velocity. According to Equation 1, since the velocity in the boundary layer is lower than mainstream velocity, the boundary layer streamlines must have a smaller radius of curvature than mainstream. Therefore, the boundary layer fluid is turned more than the mainstream flow, leading to a cross-flow near the end walls from the concave side of one blade to the convex side of the adjacent blade. Finally, this fluid hits the suction surface, and then it moves over the suction surface in the span-wise direction away from the end-wall. 主流流动经历相同的压力梯度，但边界层内的粒子具有较低的速度。根据公式 1，由于边界层中的速度低于主流速度，因此边界层流线的曲率半径必须小于主流。因此，边界层流体的转动幅度大于主流流动，导致端壁附近从一个叶片的凹侧向相邻叶片的凸侧发生交叉流。最后，这种流体撞击吸入表面，然后沿翼展方向在吸入表面上移动，远离端壁。
A compensating return flow must occur at a certain distance from the end-wall, giving rise to the recirculating flow from which a passage vortex is formed. Due to the interaction of two secondary flows from the upper and lower end-walls in the region of the convex surface of the blades, the flow is curtailed into two vortex cords rotating in opposite directions [1-6]. 补偿回流必须发生在距端壁一定距离处，从而产生再循环流，从而形成通道涡流。由于叶片凸面区域上下端壁的两股次级气流相互作用，该流被缩减为两条方向相反旋转的涡旋线[1-6]。
The actual vortex structure of the flow turns out to be even more complicated at the end-wall regions since, when the inlet boundary layer meets the leading edge, an inlet vortex is formed, which encompasses the blade in the form of a horseshoe [1]. Another element of secondary flows is the corner vortex. It is generated by the interference of the cross-flow and the near suction side flow in the baled passage [4]. 在端壁区域，流动的实际涡旋结构更加复杂，因为当入口边界层与前缘相遇时，会形成入口涡旋，该涡旋以马蹄形的形式包围叶片 [1]。次级流的另一个元素是角涡。它是由草捆通道中的错流和近吸入侧流的干扰产生的 [4]。
Different sources have quoted various reports on the share of the secondary losses in the total energy losses in turbine cascades. For example, reported by Sharma and Butler that end-wall losses can exceed 30-50%30-50 \% of the total aerodynamic losses in a single blade row [7]. Also reported by Denton that the secondary losses in turbines can be up to 1//31 / 3 of the total pressure losses [8]. 不同的来源引用了关于二次损失在涡轮机级联总能量损失中所占份额的各种报告。例如，Sharma 和 Butler 报告说，在单叶片排中，端壁损失可能超过 30-50%30-50 \% 总空气动力学损失 [7]。Denton 还报告说，涡轮机中的二次损失可以达到 1//31 / 3 总压力损失 [8]。
Because of the significant influence of secondary flows on turbomachines efficiency, they have been studied extensively over the past decades. Herzig et al. used flow-visualization techniques to determine the streamline patterns of the secondary flows. They discovered the three-dimensional deflection of the end-wall boundary layer that results in a vortex formation in each cascade passage [9]. 由于二次流对涡轮机效率的重大影响，它们在过去几十年中得到了广泛的研究。Herzig 等人使用流动可视化技术来确定次级流的流线模式。他们发现了端壁边界层的三维偏转，导致每个级联通道中形成涡旋 [9]。
The classical model showing the secondary flows was presented for the first time by Hawthorne. This model introduced the secondary flow vortex system [10]. But a complete model was offered by Langston et al. (see Figure 1 (a)). Their model was able to display better the vortex structures in the blade passage and was the first model that visually presents the secondary flow. This model shows how the inlet boundary layer separates at the blade leading edge, then one leg of the horseshoe vortex enters into the passage and forms the passage vortex, while the other leg enters into the adjacent passage and forms the counter vortex [11]. Some years later, Sieverding and Van den Bosche [12] and Sharma and Butler (see Figure 1 (b)) [13] proposed their models. The only difference between their models and the model of Langston et al. was that the counter vortex was located on the mid-span side of the passage vortex and not on the end-wall side, as reported by Langston et al… 显示次级流的经典模型由 Hawthorne 首次提出。该模型引入了二次流涡流系统 [10]。但 Langston 等人提供了一个完整的模型（见图 1 （a））。他们的模型能够更好地显示叶片通道中的涡流结构，并且是第一个直观地呈现二次流的模型。该模型显示了入口边界层如何在叶片前缘分离，然后马蹄形涡的一条腿进入通道并形成通道涡旋，而另一条腿进入相邻通道并形成反涡 [11]。几年后，Sieverding 和 Van den Bosche [12] 以及 Sharma 和 Butler （见图 1 （b）） [13] 提出了他们的模型。他们的模型与 Langston 等人的模型之间的唯一区别是，正如 Langston 等人所报告的那样，反涡位于通道涡流的跨中侧，而不是端壁侧......
In the last decades, by the improvement of computer capabilities, CFD techniques increasingly were employed to model 3D flows in turbomachines. Ananthakrishnan and Govardhan carried out steady and transient numerical simulations in the first stage of a high-loaded low aspect ratio transonic turbine to study the aerodynamic loss mechanisms [14]. Numerical simulation of a 3D flow through a turbine cascade was performed by Yershov et al. to investigate the effect of laminar-to-turbulent transition on secondary flow formation, kinetic energy loss, and flow parameter distributions [15]. Winkler et al. numerically investigated a 3D turbulent flow structure and end-wall heat transfer characteristics in a turbine vane passage [16]. A literature review of recent experimental and numerical researches on physical analysis of the aerodynamic loss models in turbine cascades, particularly secondary flow losses, is presented by da Trindade et al. [17] and Ligrani et al. [18]. 在过去的几十年里，随着计算机能力的提高，CFD 技术越来越多地被用于对涡轮机中的 3D 流动进行建模。Ananthakrishnan 和 Govardhan 在高负载低纵横比跨音速涡轮机的第一阶段进行了稳态和瞬态数值模拟，以研究空气动力学损失机制 [14]。Yershov 等人对通过涡轮机级联的三维流动进行了数值模拟，以研究层流到湍流转变对二次流形成、动能损失和流动参数分布的影响 [15]。Winkler 等人对涡轮叶片通道中的 3D 湍流结构和端壁传热特性进行了数值研究 [16]。da Trindade等[17]和Ligrani等[18]对涡轮机级联空气动力损失模型的物理分析，特别是二次流损失，进行了近期实验和数值研究的文献综述。