Blind sorting an array with a singular sort function?
Your approach by mimicking Bubble Sort is pretty good. 

Here is a hint. Since we can sort a block, we can raise half a block of the largest elements so far like a bubble each time. For example, when $u=5$, if we have done BlockSort on [0,5], [3,8] and [6,11] successively, we will have placed the largest 6/2=3 elements among the first 12 elements at index 9,10 and 11, the end of subarray so far.