Flipped classrooms support learner-centered approaches to improve conceptualization, comprehension, and problem solving skills by delivering content outside the classroom and actively engaging students inside the classroom. While literature in engineering and science education supports and encourages the use of inverted instruction, many core engineering courses continue to utilize the traditional lecture-based format. This report describes the design, development, implementation, and assessment of the flipped core course "Heat Transfer" in the undergraduate mechanical engineering curriculum. In this study, the course was restructured for flipped instruction, utilizing custom electronic media for out-of-class learning and student-centered activities for in-class engagement. Open-ended case studies were created to motivate learning and provide opportunities to apply learned knowledge to real world problems. Comparisons of student performance in flipped and traditional classrooms, as well as student observations and perspectives, are presented to demonstrate the effectiveness of flipped instruction. The report outlines an approach for transforming traditional lecture-based core mechanical engineering courses into flipped courses.