The objective of this study is to understand the formation of vortices and other flow characteristics associated with the three-dimensional motions of an incompressible Navier–Stokes fluid in tubes containing a sinusoidal extension. The study has some bearing on two conjectures that da Vinci made concerning the flow of blood through the aortic root. We investigate how the flow attributes change with increasing sinus radius and with the nature of the slip at the tube wall characterized by the parameter $\theta$, $0\le \theta \le 1$, with $\theta$ values being 0 for free slip and 1 for the “no-slip” (adherence) condition. Two time-dependent solvers – one fully three-dimensional and the other based on the assumption that admissible flows are axially symmetric – are used to solve the equations governing the flow in a geometry associated with the inflow velocities and the other conditions closely related to flows of blood in a blood vessel containing the aortic root. Both these solvers passed a benchmark test based on steady flow in a cylinder with different slip conditions. Computing the flows systematically using both solvers, focusing first on problems with constant-in-time inflow, we have found that for the circular cylinder with constant cross-section, serving as the reference domain, the flow is steady and unidirectional for all boundary conditions that are considered, with maximal vorticity and dissipation for no-slip. For tubes with the sinus radius up to 16 mm the flow remains steady and axially symmetric, but the vorticity and the bulk dissipation in the sinus are maximal for $\theta$ between 0.6 and 0.7. For a tube with the sinus radius of 20 mm the flow remains steady and axially symmetric only for $\theta$ greater or equal than 0.9 including the case of no-slip. For $\theta$ below this value, not only does the solution become unsteady (oscillatory or even chaotic), it does not converge to the steady state and is thus different from the corresponding axially symmetric flows, and also the total dissipation and the vorticity depend on $\theta$ in a non-monotone manner. In particular, for a tube with sinus radius of 20 mm, the bulk dissipation and the vorticity are the highest for $\theta =0$, i.e. for (free) slip of the fluid at the wall. For tubes with the sinus radius of 20 mm and for $\theta$ below 0.9, we computed two different solutions (one steady, axially symmetric, the other unsteady, fully three-dimensional) to three-dimensional evolutionary incompressible Navier–Stokes equations for the same set of the initial and boundary data. This last result supports the idea that, under the conditions considered, the axially symmetric solution looses its stability. Finally, we computed the problems with a pulsatile inflow. For the tube with the largest sinus radius of 20 mm, we observed that the full problem is axially symmetric for the slip parameter $\theta \ge 0.9$, while for more significant slip, i.e. for $\theta <0.9$, we again obtained two different solutions under the same initial and boundary conditions.

本研究的目的是了解涡流的形成和其他与不可压缩 Navier-Stokes 流体在包含正弦延伸的管中的三维运动相关的流动特性。这项研究与达芬奇关于血液流经主动脉根部的两个猜想有一定的关系。我们研究了流动属性如何随着窦半径的增加以及管壁滑动的性质而变化，其特征在于参数$\theta$,$0\le \theta \le 1$， 和$\theta$值为 0 表示自由滑动，1 表示“无滑动”（粘附）条件。两个与时间相关的求解器——一个是完全三维的，另一个基于允许流动是轴对称的假设——用于求解与流入速度相关的几何中的流动方程以及与流动密切相关的其他条件包含主动脉根部的血管中的血液。这两个求解器都通过了基于具有不同滑动条件的气缸中的稳定流量的基准测试。使用两个求解器系统地计算流动，首先关注时间常数流入问题，我们发现对于具有恒定横截面的圆柱体，作为参考域，流动在所有边界条件下都是稳定且单向的被认为是，具有最大的涡度和耗散以防滑。对于窦半径高达 16 mm 的管，流动保持稳定且轴对称，但窦中的涡量和体积耗散最大$\theta$在 0.6 和 0.7 之间。对于窦半径为 20 mm 的管，流动仅在以下情况下保持稳定且轴对称$\theta$大于或等于 0.9，包括防滑情况。为了$\theta$低于这个值，不仅解变得不稳定（振荡甚至混沌），它不会收敛到稳态，因此不同于相应的轴对称流动，而且总耗散和涡量取决于$\theta$以非单调的方式。特别是，对于窦半径为 20  mm 的管，体积耗散和涡度最高$\theta =0$，即流体在壁上的（自由）滑动。对于窦半径为 20  mm 的管和$\theta$低于 0.9 时，我们为同一组初始数据和边界数据计算了三维演化不可压缩 Navier-Stokes 方程的两种不同解（一个稳定的、轴对称的、另一个不稳定的、完全三维的）。最后一个结果支持这样一种观点，即在所考虑的条件下，轴对称解失去了稳定性。最后，我们计算了脉动流入的问题。对于最大窦半径为 20  mm 的管，我们观察到整个问题对于滑移参数是轴对称的$\theta \ge 0.9$，而对于更显着的滑移，即对于$\theta <0.9$，我们再次在相同的初始和边界条件下获得了两个不同的解。