Specifying positions of floats exactly (e.g. top of right hand column of page 2)

Question

Short version: how can I disable LaTeX's automatic figure placement and instead specify their positions manually, relative to the page layout rather than their position in the text?

Long version: Once I've finished writing a document I often have to do a lot of fiddling to get the figures where I want them. (This happens especially when there's a page limit, e.g. for a conference paper.) Typically this involves a lot of trial-and-error moving of the figure environments in the document source, as well as editing the captions and rescaling the figures etc. in order to try and coax them into the right places. This process is inefficient and I'm wondering if there might be a better way.

Specifically, I'm wondering if there's a way to specify the figures' positions directly, without using LaTeX's automatic positioning. I still want them to appear as floating figures in the document, but I want to say exactly where they belong, e.g. at the top of the right hand column on page 2. Is this possible?

Here's an MWE. The goal is to position the two figures at the top of the right-hand column on the second page, regardless of where they appear in the source document.

\documentclass[twocolumn]{article}
\usepackage{lipsum}
\usepackage{graphicx}
\begin{document}
% this figure should appear at the top of the right-hand column on page 2, no matter what.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{example-image-a}
\caption{A figure.}
\end{figure}
% this figure should appear at the top of the right-hand column on page 2, underneath the
% first figure, no matter what.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{example-image-b}
\caption{Another figure.}
\end{figure}
\lipsum[1-11]
\end{document}