Original image always shows, even when changed

I’ve noticed recently with WordPress, that if I upload an image it shows without a problem, but if I then delete that image and upload a new one with the same name, it will show the original image instead of the new one.

This is a problem when trying to see how something looks on a new theme, such as creating the logo, as I have to give different names to each image. It is particularly frustrating, as I have a section I want to add to my site, where the image changes daily, and I just upload a different image with the same name, so it changes everywhere, but because of this problem, it doesn’t change, it just shows the original image.

I don’t have caching set on the site, as I originally thought that may be the problem.

Is there any way to get around this?

Thanks.