How can I draw this logarithmic plot with tikz

Question

I want to draw an similar curve for my bachelor thesis. How can I apply that tikz gives me an similar output as below? The arrows and the seperations (viscous sublayer : buffer layer : logaritmic layer) in the graphic aren't necessary.

Answer 1

If you are drawing plots that have a simple function, you can

declare function={kappa:0.8; C=10.8;
                  uplus(\yp)=ln(\yp)/kappa + C;
                  yplus(\up)=\up;}

Then you plot them with normal

\addplot[domain=a:b] {uplus(x)};

That covers both u+ and y+ dashed plots. For the red line, if this plot has real data, then use \addplot table[x=xcolname, y=ycolname]{data.dat}; or something similar (there are several examples here and in the manual about handling data files with PGFplots). If there's a function for it, than the previous mentioned method will also work.

If it's just a representative plot (not real data nor function) you can join both plots with a Bezier curve (..controls (coord)..). To do this you draw part of the first plot, then the Bezier curve and the next part of the other plot:

\draw plot[domain=a:b]{f1(x)} ..controls (<ctr pt>).. (c,{f2(c)}) plot[domain=c:d]{f2(x)}

The ace in the hole here (or the cat's leap for Brazilians) is the placement of the control point (<ctr pt>). A wise choice in this case is the intersection of both plots, but that gives a super smooth transition, so we can shift it a little to better suit our needs. The important part of our code now looks like:

\addplot[domain=a:b1, dashed, name path=uplus] {uplus(x)};
\addplot[domain=a:b2, dashed, name path=yplus] {yplus(x)};
\path[name intersections{of=yplus and uplus}];
\draw[red] plot[domain=a:b]{uplus(x)} ..controls ([shift={(1.5,0)}]intersection-1)..
           (c,{yplus(c)}) plot[domain=c:b2]{yplus(x)};

Unfortunately, for reasons I don't understand the above snippet produces a plot that doesn't perfectly match the \addplot one:

But we can do tha same Thing with \addplots of course, but the red line will be in fact three separate paths, in the previous case it was only one. Finally, to draw the upper labels, is just some clever usage of the coordinate system available (the black dot is just for us to see the <ctr pt>):

MWE

\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.14}
\usetikzlibrary{intersections}
\begin{document}
\begin{tikzpicture}
\begin{semilogxaxis}[title=Turbulen boundary layer near the wall, title style={yshift=0.5cm},
             xlabel=$\ln(y^+)$,
             ylabel=$\mathrm{u}^+$, ymin=0, ymax=20,
             axis x line=bottom, axis y line=left, axis line style={-stealth},
             declare function={kappa=0.8; C=10.8; uplus(\yp)=ln(\yp)/kappa+C; yplus(\up)=\up;},
             width=16cm, height=10cm, clip=false]
    \addplot[domain=0.01:300, dashed, name path=uplus] {uplus(x)};
    \addplot[domain=0.01:18, smooth, dashed, name path=yplus] {yplus(x)};
    \path[name intersections={of=uplus and yplus}];
    \fill ([shift={(1.5,0)}]intersection-1) circle(1pt); % Dot on the plot
    \addplot[domain=40:300,  red, thick] {uplus(x)} coordinate[pos=0] (i);
    \addplot[domain=0.01:8, red, thick] {yplus(x)} coordinate[pos=1] (j);
    \draw[red, thick, line cap=round] (i) ..controls ([shift={(1.5,0)}]intersection-1).. (j);
% For some reason the below commented code produces a plot different than that of the \addplot for the first plot
%     \draw[red, thick] plot[domain=0.01:8, smooth] (axis cs:\x,{yplus(\x)}) ..controls ([shift={(1.5,0)}]intersection-1).. (axis cs:40,{uplus(40)}) plot[domain=40:300] (axis cs:\x,{uplus(\x)});
    \draw[very thin, gray] (axis description cs:0,1)  coordinate (a) -| coordinate (b) (axis cs:8,0);
    \draw[stealth-stealth] (a) -- node[midway, above]{Viscous} (b);
    \draw[very thin, gray]  (b) -| coordinate (c) (axis cs:40,0);
    \draw[stealth-stealth] (b) -- node[midway, above]{Buffer} (c);
    \draw[stealth-] (c) -- node[midway, above]{Turbulent} (axis description cs:1,1);
\end{semilogxaxis}
\end{tikzpicture}
\end{document}