Real-time performance is critical for many time-sensitive applications of wireless sensor networks. We present a constrained flooding protocol, called CFlood that enhances the deadline satisfaction ratio per unit energy consumption of time-sensitive packets in sensor networks. CFlood improves real-time performance by flooding, but effectively constrains energy consumption by controlling the scale of flooding, i.e., flooding only when necessary. If unicasting meets the distributed sub-deadline of a hop, CFlood aborts further flooding even after flooding has occurred in the current hop. Our simulation-based experimental studies show that CFlood achieves higher deadline satisfaction ratio per unit energy consumption than previous multipath forwarding protocols, especially in sparsely deployed or unreliable sensor network environments.