Venovenous extracorporeal membrane oxygenation is used for respiratory support in the most severe cases of acute respiratory distress syndrome. Blood is drained from the large veins, oxygenated in an artificial lung, and returned to the right atrium (RA). In this study, we have used large eddy simulations to simulate a single-stage “lighthouse” drainage cannula in a patient-averaged model of the large veins and RA, including the return cannula. We compared the results with previous experimental and numerical studies of these cannulas in idealized tube geometries. According to the simulations, wall proximity at the drainage holes and the presence of the return cannula greatly increased drainage through the tip (33% at 5 L/min). We then simulated a multi-stage device in the same patient-averaged model, showing similar recirculation performance across the range of extracorporeal membrane oxygenation (ECMO) flow rates compared to the lighthouse cannula. Mean and maximum time-averaged wall shear stress were slightly higher for the lighthouse design. At high ECMO flow rates, the multi-stage device developed a negative caval pressure, which may be a cause of drainage obstruction in a clinical environment. Finally, through calculation of the energy spectra and vorticity field, we observed ring-like vortices inside the cannula originating from the side holes, most prominent in the proximal position. Our work highlights the important differences between a patient-derived and simplified venous model, with the latter tending to underestimate tip drainage. We also draw attention to the different dynamics of single-stage and multistage drainage cannulas, which may guide clinical use.