Today's giant data centers are power hungry. Data center energy saving not only helps control the operational cost, but also benefits the sustainable growth of cloud services. Due to the adoption of much more switches in modern data centers as well as the mature server-side power management techniques, energy saving for the data center network is becoming increasingly important. Most previous works on saving data center network energy focus on aggregating flows to as few switches as possible. However, in this paper we argue that this method may not work for network-limited flows, the throughputs of which are elastic based on the competing flows. To save the network energy consumed by this kind of elastic flows, we propose a flow scheduling approach called Willow, which takes both the number of switches involved and their active working durations into consideration. We formulate this problem by programming and design a greedy approximate algorithm to schedule flows in an online manner. Simulations based on MapReduce traces show that Willow can save up to 60 percent network energy compared with ECMP scheduling in typical settings, and outperforms other classical heuristic algorithms such as simulated annealing and particle swarm optimization. Testbed Experiments demonstrate that this kind of dynamic energy-efficient flow scheduling causes negligible impact on upper-layer applications.