Entry and exit flows in curved pipes

Solutions are presented for both laminar developing flow in a curved pipe with a parabolic inlet velocity and laminar transitional flow downstream of a curved pipe into a straight outlet. Scalings and linearized analyses about appropriate base states are used to show that both cases obey the same governing equations and boundary conditions. In particular, the governing equations in the two cases are linearized about fully developed Poiseuille flow in cylindrical coordinates and about Dean’s velocity profile for curved pipe flow in toroidal coordinates respectively. Subsequently, we identify appropriate scalings of the axial coordinate and disturbance velocities that eliminate dependence on the Reynolds number $Re$ and dimensionless pipe curvature $\unicode[STIX]{x1D6FC}$ from the governing equations and boundary conditions in the limit of small $\unicode[STIX]{x1D6FC}$ and large $Re$ . Direct numerical simulations confirm the scaling arguments and theoretical solutions for a range of $Re$ and $\unicode[STIX]{x1D6FC}$ . Maximum values of the axial velocity, secondary velocity and pressure perturbations are determined along the curved pipe section. Results collapse when the scalings are applied, and the theoretical solutions are shown to be valid up to Dean numbers of $D=Re^{2}\unicode[STIX]{x1D6FC}=O(100)$ . The developing flows are shown numerically and analytically to contain spatial oscillations. The numerically determined decay of the velocity perturbations is also used to determine entrance/development lengths for both flows, which are shown to scale linearly with the Reynolds number, but with a prefactor ${\sim}60\,\%$ larger than the textbook case of developing flow in a straight pipe.

针对具有抛物线入口速度的弯管中的层流发展流动以及从弯管下游到直管出口的层流过渡流动，均给出了解决方案。通过对适当基态进行标度和线性化分析，表明这两种情况都遵循相同的控制方程和边界条件。特别是，这两种情况下的控制方程分别是在圆柱坐标系中围绕充分发展的泊肃叶流以及在环面坐标系中围绕弯管流的迪恩速度剖面进行线性化的。随后，我们确定了轴向坐标和扰动速度的适当标度，使得在小曲率$\unicode[STIX]{x1D6FC}$和大雷诺数$Re$的极限情况下，控制方程和边界条件中对雷诺数$Re$和无量纲管道曲率$\unicode[STIX]{x1D6FC}$的依赖性得以消除。直接数值模拟在一系列$Re$和$\unicode[STIX]{x1D6FC}$值下证实了标度论证和理论解。确定了沿弯管段的轴向速度、二次速度和压力扰动的最大值。当应用标度时，结果具有一致性，并且理论解在迪恩数$D = Re^{2}\unicode[STIX]{x1D6FC} = O(100)$范围内均有效。通过数值和解析方法表明发展流动包含空间振荡。通过数值确定的速度扰动衰减也用于确定两种流动的入口/发展长度，结果表明其与雷诺数呈线性比例关系，但前置因子比直管中发展流动的教科书案例大约$60\%$。