For $η\in S_3$, let $S_n^{\text{av}(η)}$ denote the set of permutations in $S_n$ that avoid the pattern $η$, and let $E_n^{\text{av}(η)}$ denote the expectation with respect to the uniform probability measure on $S_n^{\text{av}(η)}$. For $n\ge k\ge2$ and $τ\in S_k^{\text{av}(η)}$, let $N_n^{(k)}(σ)$ denote the number of occurrences of $k$ consecutive numbers appearing in $k$ consecutive positions in $σ\in S_n^{\text{av}(η)}$, and let $N_n^{(k;τ)}(σ)$ denote the number of such occurrences for which the order of the appearance of the $k$ numbers is the pattern $τ$. We obtain explicit formulas for $E_n^{\text{av}(η)}N_n^{(k;τ)}$ and $E_n^{\text{av}(η)}N_n^{(k)}$, for all $2\le k\le n$, all $η\in S_3$ and all $τ\in S_k^{\text{av}(η)}$. These exact formulas then yield asymptotic formulas as $n\to\infty$ with $k$ fixed, and as $n\to\infty$ with $k=k_n\to\infty$. We also obtain analogous results for $S_n^{\text{av}(η_1,\cdots,η_r)}$, the subset of $S_n$ consisting of permutations avoiding the patterns $\{τ_i\}_{i=1}^r$, where $τ_i\in S_{m_i}$, in the case that $\{τ_i\}_{i=1}^n$ are all simple permutations. A particular case of this is the set of separable permutations, which corresponds to $r=2$, $τ_1=2413,τ_2=3142$.