Coanda Effect for Incompressible Flows in Moving Domains

康达效应是一种现象，在科学文献中被描述为流体射流被吸引到附近表面的趋势。最近，Coanda效应在心脏病学中被用来描述二尖瓣反流中的壁夹持射流：通过泄漏的二尖瓣的反流有时会夹持左心房壁，这使得传统的彩色多普勒成像技术难以评估二尖瓣反流的严重程度。尽管大量的心血管和生物医学文献报道了超声心动图中的康达效应，但缺乏与流体动力学研究的联系，无法帮助识别和理解相应血流条件的主要特征。该项目建立了这种联系，并探索了在新环境中导致康达效应的流体动力学特性：移动几何形状和时间周期流动条件，包括二尖瓣反流患者所遇到的情况。相应的流体动力学问题与雷诺数低于湍流时通过孔板的流动行为有关。Coanda效应对应于特定雷诺数和特定孔口形状下的Navier-Stokes方程解的对称性破坏（分岔）。虽然在固定孔口和固定流体域的背景下，通过孔口的流动已经进行了广泛的研究（数值和实验），但没有结果揭示了在时间周期压力载荷下运动孔口中导致康达效应的流动条件。该项目通过结合复杂的计算方法和与粘性、不可压缩流体和弹性结构之间的流固相互作用（FSI）相关的分析技术来解决这个问题。该方法基于一个整体的半隐式算法来解决潜在FSI问题的任意拉格朗日-欧拉（ALE）公式，以及相应的能量估计。数学模型和计算机模拟的实验验证将与休斯顿DeBakey心脏和血管中心的医学合作者一起进行。虽然超过50%的美国人有一定程度的心脏瓣膜功能障碍，但大多数病例不需要任何药物治疗。在瓣膜返流严重的情况下，治疗失败可能导致心律失常、充血性心力衰竭和死亡。多普勒超声心动图通常被医生用来诊断和评估二尖瓣反流的严重程度。然而，使用超声心动图准确评估瓣膜反流是一个持续的挑战。特别是，通过二尖瓣泄漏的反流血液有时会包围左心房壁（称为Coanda效应），这使得很难看到和测量反流体积。通过使用复杂的数学、科学计算和实验验证，由数学家和超声心动图专家组成的跨学科团队正在研究导致康达效应的血流状况和反流瓣膜的形状。由于血流与运动反流瓣膜之间的相互作用难以解决，这是一个新的流体动力学问题，以前没有研究过。通过使用最近开发的能够解决这一问题的最先进的计算算法，并通过开发新的数学技术来捕捉与Coanda效应相关的血流分叉，该项目的结果将揭示与这一现象相关的复杂心内血流状况，并在设计超声心动图评估二尖瓣反流的新方案方面发挥主导作用。学生参与这项研究的各个方面。他们正在接受计算机模拟、数学建模和数学模型实验验证方面的培训。

Coanda effect is a phenomenon that has been described in scientific literature as a tendency of a fluid jet to be attracted to a nearby surface. Recently, Coanda effect has been used in cardiology to describe the wall-hugging jets in mitral regurgitation: regurgitant blood flow through a leaky mitral valve sometimes hugs the wall of the left atrium which makes it difficult to assess the severity of mitral regurgitation using classical color Doppler imaging techniques. Despite the large cardiovascular and biomedical literature reporting on the Coanda effect in echocardiography, a connection with the fluid dynamics studies that could help identify and understand the main features of the corresponding flow conditions is lacking. This project makes this connection and explores the fluid dynamics properties leading to the Coanda effect in a novel environment: moving geometries and time periodic flow conditions, which will include those encountered in patients with mitral regurgitation. The corresponding fluid dynamics problem is associated with the behavior of flow through an orifice at Reynolds numbers below turbulence. Coanda effect corresponds to the breaking of symmetry (a bifurcation) in the solution of the Navier-Stokes equations at certain Reynolds numbers and for certain orifice shapes. While flows through orifices have been extensively studied (numerically and experimentally) in the context of fixed orifices and fixed fluid domains, there have been no results that shed light on the flow conditions leading to Coanda effect in moving orifices under time-periodic pressure loads. This project addresses this problem by combining sophisticated computational methodology and analytical techniques associated with fluid-structure interaction (FSI) between a viscous, incompressible fluid, and an elastic structure. The methodology is based on a monolithic, semi-implicit algorithm to solve an Arbitrary Lagrangian-Eulerian (ALE) formulation of the underlying FSI problem, and on the corresponding energy estimates. Experimental validation of the mathematical models and computer simulations will be performed with the medical collaborators at the DeBakey Heart and Vascular Center in Houston.Although over 50% of the US population has some degree of heart valve dysfunction, most cases do not require any medical treatment. In the cases when valve regurgitation is severe, failure of treatment can lead to arrhythmias, congestive heart failure and death. Doppler echocardiography is routinely used by physicians to diagnose and assess the severity of mitral valve regurgitation. The accurate assessment of valve regurgitation using echocardiography is, however, an ongoing challenge. In particular, regurgitant blood flow through a leaky mitral valve sometimes hugs the wall of the left atrium (known as the Coanda effect), which makes it difficult to see and measure the regurgitant volume. By using sophisticated mathematics, scientific computing, and experimental validation, the interdisciplinary team consisting of mathematicians and echocardiographic specialists, is investigating the blood flow conditions, and the shape of the regurgitant valves, that lead to the Coanda effect. This is a novel fluid-dynamics problem that has not been studied before due to the difficulties in resolving the interaction between blood flow and the moving regurgitant valve. By using a recently developed state-of-the-art computational algorithm that is capable of resolving this problem, and by developing novel mathematical techniques that will capture the bifurcation in the flow associated with Coanda effect, the results of this project will shed light on the complex intracardiac flow conditions associated with this phenomenon, and lead the way in designing novel protocols in echocardiographic assessment of mitral regurgitation. Students participate in all the aspects of this research. They are being trained in computer simulations, mathematical modeling, and in experimental validation of the mathematical models.