Transcatheter aortic valve replacement (TAVR) is the implantation of an artificial aortic heart valve without an open-heart surgery. Computer modeling and simulation is an important tool in the process of transcatheter aortic valve (TAV) device design, regulatory approval, and indication in the care of specific patients, since there are still many open questions surrounding post-implantation complications. Improved computational models beyond those in the existing literature have the ability to provide more accurate performance predictions for individual patients. We present computational fluid-structure interaction models of TAVs using the immersed finite element-difference method. We perform dynamic simulations of crimping and deployment of the devices as well as their behavior across the cardiac cycle in a patient-specific aortic root anatomy reconstructed from CT image data. These IFED simulations incorporate biomechanics models fit to experimental tensile test data and automatically capture the contact within the devices and between the stents and native anatomies. We apply realistic driving and loading conditions based on clinical measurements of human ventricular and aortic pressures and flow rates, and our models provide informative clinical performance predictions, such as pre- and post-procedure transvalvular pressure differences, detailed flow patterns, leaflet dynamics, and valve orifice areas.