Rapid surrogate models are attractive for iterative computational fluid dynamics (CFD) design loops, though defining their operating envelope remains crucial. This study proposes RenaNet, a Reynolds-aware convolutional gated recurrent unit (ConvGRU) surrogate, for predicting two-dimensional laminar and transitional flows past cylinder and square obstacles. Using two initial flow snapshots and a Reynolds-number map, the model predicts spatiotemporal flow states up to 2000 time steps into the future, with Lattice Boltzmann Method (LBM) simulations serving as ground truth. Trained on Reynolds numbers of $1\le \mathrm{Re}\le 500$ (cylinder) and $1\le \mathrm{Re}\le 250$ (square), RenaNet achieves a minimum validation mean squared error (MSE) of $1.47\times {10}^{-5}$. A Reynolds-number ablation shows that removing the conditioning channel increases the validation MSE to $1.17\times {10}^{-3}$, while a ConvLSTM baseline gives $9.94\times {10}^{-4}$ with 24% more parameters. RenaNet also uses a direct long-horizon prediction interface for distant target frames. Auxiliary physics diagnostics confirm that predictions trained via MSE maintain acceptable continuity residuals across fitting, interpolation, and extrapolation cases. The average inference time for a 1000-step prediction horizon is approximately 1.25 s, delivering a 500-fold speedup over the reference LBM solver. Interpolation errors range from ${10}^{-4}$ to ${10}^{-2}$ depending on Reynolds number and geometry, while extrapolation beyond the training regime increases errors to the order of ${10}^{-2}$. These results establish RenaNet as a robust, parameter-efficient surrogate for laminar and transitional flows, with a clearly characterized operational boundary that informs future extensions into turbulent regimes.