By incorporating the traditionally overlooked advective term in the wall-normal momentum equation, a new momentum integral equation is developed for two-dimensional incompressible turbulent boundary layers under arbitrary pressure gradients. The classical Kármán's integral arises as a special instance of the new momentum integral equation when the pressure gradient is weak. The new momentum integral equation's validity is substantiated by direct numerical simulation data. Unlike the classical Kármán's integral, which is limited to predicting wall shear stress within mild pressure gradients, the new momentum integral equation accurately computes wall shear stress across a broad range of pressure gradients, even in the presence of strong adverse pressure gradients that lead to flow separation. Moreover, a new pressure parameter $\beta _\kappa$ is introduced through examining terms in the new momentum integral equation. This parameter naturally quantifies the pressure gradient's influence on turbulent boundary layers and offers guidance for applying the classical Kármán's integral. Additionally, to facilitate experimental determination of wall shear stress under strong pressure gradients, an approximate integral equation is proposed that relies solely on easily measurable variables. Validation against direct numerical simulation data demonstrates that this simplified equation provides reasonably accurate estimates of wall shear stress in turbulent boundary layers experiencing strong pressure gradients.

中文翻译：

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适用于任意压力梯度下边界层流动的新动量积分方程

通过将传统上被忽视的平流项纳入壁法向动量方程中，针对任意压力梯度下的二维不可压缩湍流边界层建立了新的动量积分方程。当压力梯度较弱时，经典卡门积分是新动量积分方程的一个特例。直接数值模拟数据证实了新动量积分方程的有效性。与仅限于预测温和压力梯度内的壁面剪应力的经典卡门积分不同，新的动量积分方程可以在广泛的压力梯度范围内准确计算壁面剪应力，即使存在导致流动的强逆压力梯度分离。此外，还有一个新的压力参数 $\beta _\kappa$ 通过检查新动量积分方程中的项引入。该参数自然地量化了压力梯度对湍流边界层的影响，并为应用经典卡门积分提供了指导。此外，为了便于在强压力梯度下实验确定壁面剪应力，提出了一个仅依赖于易于测量的变量的近似积分方程。针对直接数值模拟数据的验证表明，这个简化的方程可以对经历强压力梯度的湍流边界层中的壁剪切应力进行相当准确的估计。