Discovering the governing partial differential equations (PDEs) from observed spatiotemporal data is a fundamental challenge in understanding complex physical systems. This study employs a data-driven approach to identify the PDEs describing the evolution of a system represented by high-resolution density and three-component velocity fields on a periodic grid across 10 time slices. Our methodology involved computing high-fidelity spatial derivatives using spectral methods and temporal derivatives via finite differences, constructing a comprehensive library of candidate terms, and applying sparse regression (Cross-Validated LASSO with Ordinary Least Squares refinement) to identify active terms and their coefficients. Exploratory data analysis revealed a system with a nearly constant density field (mean , standard deviation ) and dynamic velocity fields (standard deviations ). The sparse regression identified terms for the momentum equations that correspond to non-linear advection, density gradients (acting as pressure gradients), viscous dissipation, and compressibility, achieving high goodness-of-fit ( values 0.57-0.71). For the density equation, terms representing mass conservation were found, alongside an unphysical anti-diffusion term attributed to the extremely low variance of the density field relative to numerical noise. Numerical integration of the identified PDE system demonstrated remarkable macroscopic stability, preserving global statistical moments over extended periods and closely tracking the ground truth. Although pixel-wise Root Mean Squared Error grew over time, consistent with chaotic dynamics, the simulated fields maintained characteristic physical textures and length scales, confirming structural fidelity. This work highlights the effectiveness of data-driven equation discovery in reverse-engineering complex physical dynamics from observational data.