Open Set Domain Adaptation (OSDA) aims to adapt a model trained on a source domain to a target domain that undergoes distribution shift and contains samples from novel classes outside the source domain. Source-free OSDA (SF-OSDA) techniques eliminate the need to access source domain samples, but current SF-OSDA methods utilize only the known classes in the target domain for adaptation, and require access to the entire target domain even during inference after adaptation, to make the distinction between known and unknown samples. In this paper, we introduce Unknown Sample Discovery (USD) as an SF-OSDA method that utilizes a temporally ensembled teacher model to conduct known-unknown target sample separation and adapts the student model to the target domain over all classes using co-training and temporal consistency between the teacher and the student. USD promotes Jensen-Shannon distance (JSD) as an effective measure for known-unknown sample separation. Our teacher-student framework significantly reduces error accumulation resulting from imperfect known-unknown sample separation, while curriculum guidance helps to reliably learn the distinction between target known and target unknown subspaces. USD appends the target model with an unknown class node, thus readily classifying a target sample into any of the known or unknown classes in subsequent post-adaptation inference stages. Empirical results show that USD is superior to existing SF-OSDA methods and is competitive with current OSDA models that utilize both source and target domains during adaptation.