How to bring bar and \sim symbol at the same level

By all means replace \overset{\sim}{u} with \tilde{u}. Optionally, replace \overline{\rho} with \bar{\rho}.

Here's a screenshot of u with various math-mode adornments.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
$\bar{u} \tilde{u} \hat{u} \dot{u} \ddot{u} \acute{u} \grave{u} \check{u} \breve{u}$ 
$\bar{\bar{u}} \tilde{\tilde{u}} \hat{\hat{u}} \hat{\tilde{u}} \tilde{\bar{u}}$
\end{document}

I have used accents package to put correctly (using the negative space \kern-.2em) the \sim symbol over the u_i. See the command \accentset{\kern-.2em\sim}{u_i}.

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath}
\usepackage{accents}
\begin{document}
\begin{equation}
\frac{\partial \bar{\rho}}{\partial{t}}+\frac{\partial (\bar{\rho}\accentset{\kern-.2em\sim}{u_i})} 
{\partial{t}}
\end{equation}
\end{document}

Or you can use the "natural" code without the negative space:

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath}
\usepackage{accents}
\begin{document}
\begin{equation}
\frac{\partial \bar{\rho}}{\partial{t}}+\frac{\partial (\bar{\rho}\accentset{\sim}{u_i})} 
{\partial{t}}
\end{equation}
\end{document}

This addition is due to the comment of the very good user @egreg that you can see below: Following his advice putting

\accentset{\sim}{u}_i

you will have:

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath}
\usepackage{accents}
\begin{document}
\begin{equation}
\frac{\partial \bar{\rho}}{\partial{t}}+\frac{\partial (\bar{\rho}\accentset{\sim}{u}_i)} 
{\partial{t}}
\end{equation}
\end{document}

Just from comment by the user @barbara beeton into chat I use also the \widetilde command: here there is the output.

\documentclass[a4paper,12pt]{article}
\usepackage{mathtools}
\usepackage{amssymb}

\begin{document}
\begin{equation}
\frac{\partial \bar{\rho}}{\partial{t}}+\frac{\partial (\bar{\rho}{\widetilde u}_i)} 
{\partial{t}}
\end{equation}
\end{document}