We study the development potential of ultrasonic thermometry in a liquid fluctuating sodium environment similar to that present in a Sodium-cooled Fast Reactor, and thus investigate if and how ultrasonic thermometry could be used to monitor the sodium flow at the outlet of the reactor core. In particular we study if small temperature variations in the sodium flow of e.g. about 1% of the sodium temperature, i.e., about 5°C, can have a reliably-measurable acoustic signature. Since to our knowledge no experimental setups are available for such a study, and considering the practical difficulties of experimentation in sodium, we resort to a numerical technique for full wave propagation called the spectral-element method, which is a highly accurate finite-element method owing to the high-degree basis functions it uses. We obtain clear time-of-flight variations in the case of a small temperature difference of one percent in the case of a static temperature gradient as well as in the presence of a random fluctuation of the temperature field in the turbulent flow. The numerical simulations underline the potential of ultrasonic thermometry in such a context.