Particle Image Velocimetry (PIV) is a method for obtaining a fluid velocity field based on the translation of particles between images with a known time span between them. A potential limitation of PIV is that only two-dimensional velocity field along a single plane can be obtained from a two-dimensional image. This limitation is normally overcome by designing the experimental flow system so that the third velocity component is either zero or unimportant. In many applications, however, it is not possible to simplify the fully three-dimensional velocity field (e.g., in the left ventricle of the heart), and the two-dimensional limitations associated with PIV analysis are a significant problem that must be overcome. Would it be possible, however, to combine the two-dimensional PIV data together with a fully three-dimensional numerical approximation to the Navier-Stokes equations and obtain a sufficiently accurate three-dimensional velocity field in a domain such as the left ventricle of the heart? The use of least-squares finite element methods (LSFEMs) is proposed here to approximately solve the Navier-Stokes equations, and, significantly, to weakly constrain the solution along the PIV plane to match the experimental data. The PIV data would basically act as an internal boundary, and the numerical solution would weakly match the data with a variable weighting that determines the strength of the coupling between the data and numerical solution. LSFEMs are uniquely well suited for solving an over-constrained problem like this in a computationally efficient manner.
Echocardiologists have developed methods for introducing microbubbles into circulating blood that can be resolved using ultrasound. The location of the microbubbles combined with the high temporal resolution of ultrasound allows the local blood velocity to be determined using Particle Image Velocimetry, but the high temporal resolution requirement also limits the ultrasound scans (and, hence, the velocity field data) to two dimensions. One goal of using the FDA approved microbubbles is to use the blood velocity data to calculate physiologically important information such as pressure gradients and energy loss for the blood flow, but these calculations require a full three-dimensional velocity field. The goal of the proposed research is to develop mathematical techniques that combine computational fluid dynamics with experimental velocity data, such as that from microbubbles, to obtain a full, three-dimensional velocity field, thus providing greater insight into the dynamics of the flow.
