Just as a remark:

... so that people can really see the periodic temperature distribution.

So how about this:

img = ImageCrop@
   ContourPlot[ufun[x, z], {x, z} \[Element] mesh, MaxRecursion -> 4, 
     Contours -> 30, ColorFunction -> "BlueGreenYellow",
    Frame -> False, PlotRangePadding -> 0, ImageSize -> 600];
ParametricPlot3D[{Cos[\[Phi]], Sin[\[Phi]], z}, {\[Phi], 0, 2 Pi}, {z,
   0, 0.0001}, PlotStyle -> Texture[img], Mesh -> None, 
 BoxRatios -> {1, 1, 1.5}, Ticks -> {None, None, Automatic}]

giving:

Regards -- Henrik

Hi Markus,

You could just define a function that shifts the contour plot and then seam them together with Show. For example:

cp[i_, n_] := 
  ContourPlot[
   ufun[x - 2*\[CapitalLambda] i, z], {x, 2*\[CapitalLambda]*i, 
    2*\[CapitalLambda]*(i + 1)}, {z, 0, Lz}, MaxRecursion -> 4, 
   Contours -> 30, ColorFunction -> "BlueGreenYellow", 
   FrameStyle -> Directive[Black, Thickness[0.003]], 
   LabelStyle -> Directive[Black, 10], 
   FrameLabel -> {"x [m]", "z [m]"} , 
   PlotRange -> {{0, 2 * \[CapitalLambda] * (n)}, {0, Lz}}];
nplts = 5;
Show[Table[cp[i, nplts], {i, 0, nplts - 1}]]

Best regards and Happy New Year!

Tim