We propose a novel formulation of a moving boundary method to account for the motion of the intimal flap (IF) in a TBAD post-thoracic endovascular aortic repair using patient-specific compliant computational fluid dynamics simulations. The simulations were informed by non-invasive 4DMRI sequences. Predicted flow waveforms, aortic wall and IF displacements were validated against in vivo 4DMRI and cine MRI data. The patient-specific simulation showed that at peak systole, the dynamic interplay between high IF displacement and high transmural pressures promoted true lumen compression and false lumen expansion, while luminal patterns were reversed at the deceleration phase. High vorticity and swirling flow patterns were observed throughout the cardiac cycle at the primary entry tear, the descending aorta and proximal to the visceral aortic branches, correlating with high relative residence time, which could indicate an increased localised risk of aortic growth proximal to the IF. A rigid IF simulation revealed significant discrepancies in haemodynamic metrics, highlighting the potential mispredictions when using a rigid wall assumption to assess disease progression. Simulations assuming a more compliant IF highlighted increased risks of visceral branches malperfusion and localised aortic wall degeneration. The study underscores the necessity of patient-specific-compliant IF simulations for accurate TBAD hemodynamic assessments. These insights can improve disease understanding and inform future treatment strategies.