Let D(K) be the positively-clasped untwisted Whitehead double of a knot K, and T(p, q) be the (p, q) torus knot. We show that D(T(2, 2m+1)) and D(D(T(2, 2m+1))) differ in the smooth concordance group of knots for each m>1. In fact, they generate a Z2 summand in the subgroup generated by topologically slice knots. We use the concordance invariant δ by Manolescu and Owens. More generally, we show that the same result holds for a knot K with |δ(D(K))|>8 and give a formula to compute δ(D(K)) for some classes of knots. Interestingly, these results are not easily be shown using other concordance invariants such as the knot signature and τ-invariants of the knot Floer theory. We also determine the knot Floer complex of D(T(2, 2m+1)) for any m>0 generalizing a result for T(2, 3) of Hedden, Kim and Livingston.