Coanda Effect for Incompressible Flows in Moving Domains

科安达效应是一种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的现象。最近，Coanda效应在心脏病学中被用来描述二尖瓣关闭不全时的贴壁射流：返流血流流过漏水的二尖瓣有时会拥抱左心房的壁，这使得使用经典的彩色多普勒成像技术评估二尖瓣关闭不全的严重程度变得困难。尽管有大量的心血管和生物医学文献报道了超声心动图中的Coanda效应，但缺乏与流体动力学研究的联系，这些研究可以帮助识别和了解相应血流条件的主要特征。该项目建立了这种联系，并探索了在一个新的环境中导致科安达效应的流体动力学特性：移动的几何形状和时间周期流动条件，其中将包括二尖瓣关闭不全患者遇到的情况。相应的流体动力学问题与雷诺数低于湍流时孔口的流动行为有关。Coanda效应对应于在一定的雷诺数和一定的孔口形状下，Navier-Stokes方程的解的对称性的破缺(分叉)。虽然在固定孔口和固定流体域的背景下，孔板流动已经得到了广泛的研究(数值和实验)，但还没有关于在时间周期压力载荷下移动孔板时导致Coanda效应的流动条件。这个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体-结构相互作用(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求解基本FSI问题的任意拉格朗日-欧拉(ALE)公式，并基于相应的能量估计。数学模型的实验验证和计算机模拟将在休斯顿的DeBakey心脏和血管中心与医疗合作者一起进行。尽管超过50%的美国人口有不同程度的心脏瓣膜功能障碍，但大多数病例不需要任何治疗。在瓣膜返流严重的情况下，治疗失败可能会导致心律失常、充血性心力衰竭和死亡。多普勒超声心动图是内科医生诊断和评估二尖瓣返流严重程度的常规方法。然而，使用超声心动图准确评估瓣膜返流是一个持续的挑战。特别是，流经漏水的二尖瓣的返流血流有时会拥抱左心房的壁(称为科安达效应)，这使得难以看到和测量返流体积。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调查导致科安达效应的血液流动状况和反流瓣的形状。这是一个以前从未被研究过的新的流体动力学问题，因为很难解决血液流动与运动的返流瓣膜之间的相互作用。通过使用最新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的心内血流状况，并在设计二尖瓣反流的超声心动图评估新方案方面处于领先地位。学生参与本研究的方方面面。他们正在接受计算机模拟、数学建模和数学模型的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.