We present a tetrahedral finite cell method for the simulation of incompressible flow around geometrically complex objects. The method immerses such objects into non-boundary-fitted meshes of tetrahedral finite elements and weakly enforces Dirichlet boundary conditions on the objects’ surfaces. Adaptively-refined quadrature rules faithfully capture the flow domain geometry in the discrete problem without modifying the non-boundary-fitted finite element mesh. A variational multiscale formulation provides accuracy and robustness in both laminar and turbulent flow conditions. We assess the accuracy of the method by analyzing the flow around an immersed sphere for a wide range of Reynolds numbers. We show that quantities of interest such as the drag coefficient, Strouhal number and pressure distribution over the sphere are in very good agreement with reference values obtained from standard boundary-fitted approaches. We place particular emphasis on studying the importance of the geometry resolution in intersected elements. Aligning with the immersogeometric concept, our results show that the faithful representation of the geometry in intersected elements is critical for accurate flow analysis. We demonstrate the potential of our proposed method for high-fidelity industrial scale simulations by performing an aerodynamic analysis of an agricultural tractor.