How to add arrow tips (to achieve flow direction) to a process diagram with TikZ datavisualization?

I have tried and shamefully failed to get your example to cooperate in the desired fashion. Would a solution with pgfplots be of use to you?

Result:

\documentclass[margin=2mm,tikz]{standalone}
\usepackage{pgfplots}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usetikzlibrary{decorations.markings}
\usepackage[per-mode = fraction]{siunitx}

\begin{document}
\begin{tikzpicture}

    \begin{axis}[
            cycle list name=linestyles,
            legend style={
                cells={anchor=east},
                legend pos=outer north east,
            },
            xlabel = {$\frac{v}{\si{\cubic\m\per\kg}}$},
            ylabel = {$\frac{p}{\si{\bar}}$},
            xlabel style = {at={(axis description cs:1.1,0.2)}},
            ylabel style = {at={(axis description cs:0.1,1.1)},rotate=-90},
        ]

        \addplot+[
            domain=2:0.5,
            decoration={
                markings,
                mark=at position 0.5 with {\arrow [scale=1.5]{stealth}}},
            postaction={decorate},
        ] {1/(x^1.4)};
        \addplot+[
            domain=0.5:1,
            decoration={
                markings,
                mark=at position 0.5 with {\arrow [scale=1.5]{stealth}}},
            %postaction={nomorepostaction,decorate},
            postaction={decorate},
        ] {1/(0.5^1.4)};
        \addplot+[
            domain=1:2,
            decoration={
                markings,
                mark=at position 0.5 with {\arrow [scale=1.5]{stealth}}},
            postaction={decorate},
        ] {1/((x-0.5)^1.4)};

        \draw (axis cs:2,1/2^1.4)     circle (1pt) node[right,font=\footnotesize] {1};
        \draw (axis cs:0.5,1/0.5^1.4) circle (1pt) node[above,font=\footnotesize] {2};
        \draw (axis cs:1.0,1/0.5^1.4) circle (1pt) node[above,font=\footnotesize] {3};
        \draw (axis cs:2,1/1.5^1.4)   circle (1pt) node[right,font=\footnotesize] {4};

        \legend{isentrope Verdichtung,isobare Abkühlung,isentrope Entspannung}
    \end{axis}

\end{tikzpicture}
\end{document}