  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
$D^2(T_{2,2m+1})$ are linearly independent in the smooth knot concordance group
$\mathcal{C}$ for each $m\geq 2$. Further, $D(T_{2,5})$ and $D^2(T_{2,5})$
generate a $\mathbb{Z}\oplus\mathbb{Z}$ summand in the subgroup of
$\mathcal{C}$ generated by topologically slice knots. We use the concordance
invariant $\delta$ of Manolescu and Owens, using Heegaard Floer correction
term. Interestingly, these results are not easily shown using other concordance
invariants such as the $\tau$-invariant of knot Floer theory and the
$s$-invariant of Khovanov homology. We also determine the infinity version of
the knot Floer complex of $D(T_{2,2m+1})$ for any $m\geq 1$ generalizing a
result for $T_{2,3}$ of Hedden, Kim and Livingston.
