Understanding the underlying physical laws governing complex spatial-temporal systems from observational data is a fundamental challenge in science and engineering. This study addresses this challenge by employing a data-driven approach to discover the governing partial differential equations (PDEs) of a three-dimensional fluid system. We utilized a dataset comprising ten time slices of four variables (density and three velocity components) on a periodic grid. Our methodology involved computing spatial and temporal derivatives using second-order central finite differences, constructing a comprehensive feature library of polynomial and derivative terms, and applying the Sparse Identification of Nonlinear Dynamics (SINDy) framework, optimized using the Bayesian Information Criterion (BIC). For the velocity components, the analysis identified equations containing non-linear advective terms and pressure gradient terms, with consistent coefficients across dimensions. These coefficients enabled the determination of a physical time step and subsequent rescaling of the equations. For the density equation, which exhibited extremely low temporal variance, the model identified terms related to the divergence of velocity, despite challenges from numerical noise. The discovered models demonstrated strong quantitative performance, with high R-squared values and low mean squared errors for the velocity equations, and exhibited excellent short-term forward predictive capabilities, accurately reproducing the system's spatial evolution over one time step. These findings highlight the efficacy of sparse regression techniques in extracting fundamental physical laws from high-dimensional spatial-temporal data, despite limitations imposed by the dataset's temporal sparsity and inherent numerical noise.