The hammerhead shark possesses a unique head morphology that is thought to facilitate enhanced olfactory performance. The olfactory chambers, located at the distal ends of the cephalofoil, contain numerous lamellae that increase the surface area for olfaction. Functionally, for the shark to detect chemical stimuli, water-borne odors must reach the olfactory sensory epithelium that lines these lamellae. Thus, odorant transport from the aquatic environment to the sensory epithelium is the first critical step in olfaction. Here we investigate the hydrodynamics of olfaction in Sphyrna tudes based on an anatomically-accurate reconstruction of the head and olfactory chamber from high-resolution micro-CT and MRI scans of a cadaver specimen. Computational fluid dynamics simulations of water flow in the reconstructed model reveal the external and internal hydrodynamics of olfaction during swimming. Computed external flow patterns elucidate the occurrence of flow phenomena that result in high and low pressures at the incurrent and excurrent nostrils, respectively, which induces flow through the olfactory chamber. The major (prenarial) nasal groove along the cephalofoil is shown to facilitate sampling of a large spatial extent (i.e., an extended hydrodynamic "reach") by directing oncoming flow towards the incurrent nostril. Further, both the major and minor nasal grooves redirect some flow away from the incurrent nostril, thereby limiting the amount of fluid that enters the olfactory chamber. Internal hydrodynamic flow patterns are also revealed, where we show that flow rates within the sensory channels between olfactory lamellae are passively regulated by the apical gap, which functions as a partial bypass for flow in the olfactory chamber. Consequently, the hammerhead shark appears to utilize external (major and minor nasal grooves) and internal (apical gap) flow regulation mechanisms to limit water flow between the olfactory lamellae, thus protecting these delicate structures from otherwise high flow rates incurred by sampling a larger area.