The internal flow field and loss distributions are quite complicated in the axial-inflow turbine. It is necessary to develop physical understandings of the flow and losses. The synergy relationship between the velocity and velocity gradient, which can be revealed by mathematical expressions in the cylindrical coordinates, is innovatively applied to the internal flow field of axial-inflow turbine. According to the mathematical expression, the smaller the synergy angle is, the higher flow resistance and loss should be. To assess the validity of the synergy principle, the relationship between the synergy angle and flow characteristics is analyzed in detail. It is found that the regions where the high energy loss and entropy generation locate correspond to the relatively low synergy angle, which agrees well with the mathematical analysis. The relatively low synergy angle in the axial direction could represent high-loss regions near the blade surface and wake on the $\theta -z$ stream surface. The relatively low synergy angle in the radial and circumferential direction could represent high-loss regions near the endwall and wake on the $\theta -r$ stream surface, respectively. An apparent negative correlation between the loss coefficient and synergy angle could be perceived under off-designed conditions. It is recommended the synergy angle should be kept in relatively low magnitudes to reduce losses for the design or optimization routines of axial-inflow turbines.

在轴流式涡轮机中，内部流场和损失分布非常复杂。有必要对流量和损失进行物理理解。可以通过圆柱坐标中的数学表达式来揭示速度与速度梯度之间的协同关系，并将其创新地应用于轴流式涡轮机的内部流场。根据数学表达式，协同角越小，流动阻力和损失应该越高。为了评估协同作用原理的有效性，详细分析了协同作用角度与流动特性之间的关系。发现高能量损失和熵产生所在的区域对应于相对较低的协同角，这与数学分析非常吻合。$\theta -ž$流表面。沿径向和周向的相对较低的协同角可能代表着端壁附近的高损耗区域，并在水流中唤醒$\theta -\mathrm{[R}$流面。在偏离设计的条件下，可以感觉到损耗系数和协同角之间的明显负相关。建议将协同角保持在较低的水平，以减少轴流式涡轮机设计或优化程序的损失。