  We examine linear perturbation theory to evaluate the contribution of
viscosity coefficient in the growing of dark matter perturbations in the
context of the bulk viscous dark energy model inspired by thermodynamical
dissipative phenomena proposed by [B. Mostaghel et al., 2017, The European
Physical Journal C, 77, 541]. As the cosmological implementations, we
investigate the {\it Integrated Sachs-Wolfe} (ISW) auto-power spectrum, the
ISW-galaxy cross-power spectrum and derive limits on $f\sigma_8$. The
dimensionless bulk viscosity coefficient ($\gamma$) in the absence of
interaction between dark sectors, modifies the Hubble parameter and the growth
function, while the Poisson equation remains unchanged. Increasing $\gamma$
reduces the dark matter growing mode at the early epoch while a considerable
enhancement will be achieved at the late time. This behavior imposes
non-monotonic variation in the time evolution of gravitational potential
generating a unique signature on the CMB photons. The bulk viscous dark energy
model leads to almost a decreasing in ISW source function at the late time.
Implementation of the Redshift Space Distortion (RSD) observations based on
"Gold-2017" catalogue, shows $\Omega^0_{\rm m} =0.303^{+0.044}_{-0.038}$,
$\gamma=0.033^{+0.098}_{-0.033}$ and $\sigma_8= 0.769^{+0.080}_{-0.089}$ at
$1\sigma$ level of confidence. Finally, tension in the $\sigma_8$ is alleviated
in our viscous dark energy model.
