The correct answer is option 2: Vijayanagara. The Tungabhadra river provided sustenance to the Vijayanagara Empire.

Option 1: Chola - The Chola Empire, which existed from the 9th to the 13th century, was primarily located in Tamil Nadu and did not rely on the Tungabhadra river for sustenance.

Option 3: Vakataka - The Vakataka Empire, which existed from the 3rd to the 6th century, was centered in the Deccan region of India but did not have any direct association with the Tungabhadra river.

Option 4: Pandya - The Pandya Empire, which existed from the 6th to the 17th century, was located in the southern part of Tamil Nadu. While their kingdom may have extended close to the Tungabhadra river, it was not a significant source of sustenance for them.

Therefore, only option 2, the Vijayanagara Empire, is correct. This empire, which flourished from the 14th to the 17th century, was centered around the Tungabhadra river and heavily relied on its waters for agriculture, trade, and overall sustenance.