  In the present paper, we study the capability of interacting dark energy
model with pure momentum transfer in the dark sector to reconcile tensions
between low redshift observations and cosmic microwave background (CMB)
results. This class of interacting model with pure momentum exchange introduces
modifications to the standard model in the level of perturbation. We
investigate the model by comparing to observational data, including integrated
Sachs-Wolfe-galaxy cross-correlation, galaxy power spectrum, $f \sigma_8$, and
CMB data. It is shown that this model can alleviate the observed tension
between local and global measurements of $\sigma_8$. According to our results,
the best fit value of $\sigma_8$ for interacting model is $0.700$, which is
lower than the one for $\Lambda$CDM model and also is consistent with low
redshift observations. Furthermore, we perform a forecast analysis to find the
constraints on parameters of the interacting model from future experiments.
