The ExasSMR project focuses on the exascale application of single and coupled Monte Carlo (MC) and computational fluid dynamics (CFD) physics. Work is based on the Shift MC depletion, OpenMC temperature-dependent MC, and Nek5000 CFD codes. The application development objective is to optimize these applications for exascale execution of full-core simulations and to modularize and integrate them into a common framework for coupled and individual execution. Given the sheer scale of nuclear systems, the main algorithmic driver on the CFD side is weak scaling. The focus for the first four years of the project is on demonstrating scaling up to a full reactor core for high-fidelity simulations of turbulence. Full-core fluid calculations aimed at better predicting the steady-state performance will be conducted with a hybrid approach in which large eddy simulation is used to simulate a portion of a core and unsteady Reynolds-averaged Navier-Stokes handles the rest. This zonal hybrid approach provides an additional scaling dimension besides the number of assemblies. The present manuscript focuses on performance assessment using assembly-level simulations with Nek5000. We discuss the development of two benchmark problems: a subchannel (single-rod) problem to assess internode performance and a larger full-assembly problem representative of a small modular reactor (SMR). We note that current SMR assemblies are considerably simpler than pressurized water reactor assemblies since they contain no mixing vanes. This feature allows for considerable reduction in the degrees of freedom required to simulate the full core. We discuss profiling and scaling results with Nek5000, describe current bottlenecks and potential limitations of the approach, and suggest optimizations for future investigation.