Grow vigorously is a term that describes the process of something increasing in size, strength, or vigor. The two crossword solutions for this term are 'THRIVE' (6 letters) and 'FLOURISH' (8 letters). THRIVE is defined as to grow or develop well or vigorously, while FLOURISH is defined as to grow or develop in a healthy or vigorous way, especially as the result of a particularly favorable environment. Both words imply a strong, healthy growth, which is what is meant by the term 'grow vigorously'.