Plotting perpendicular lines to a contour plot (electric field from equipotential curves)

(1) Use Interpolation to get an interpolating function intF.

intF = Interpolation[Flatten[MapIndexed[{#2, #} &, Transpose@data, {2}], 1], 
      InterpolationOrder -> 7];

(2) Use intF to get a ContourPlot and a StreamPlot (using the approach from this answer by Michael E2 and combine the two plots with Show:

cp = ContourPlot[intF[x, y], {x, 1, 15}, {y, 1, 8}, 
   Contours -> Range[0, 7], AspectRatio -> Automatic, ImageSize -> Large];

sp = StreamPlot[Evaluate[-D[intF[x, y], {{x, y}}]], {x, 1, 15}, {y, 1, 8}, 
      StreamScale -> None, 
      StreamStyle -> {"Arrow", Directive[Thick, Red]}, 
      StreamPoints -> Fine, AspectRatio -> Automatic];

Show[cp, sp]

Due to @kgir 's response, I made it to get very decent results. I'd like to share it with you, in any case there's someone trying to do the same.

intF = Interpolation[
   Flatten[MapIndexed[{#2, #} &, Transpose@data1, {2}], 1], 
   InterpolationOrder -> {1, 1}];
    cp = ContourPlot[intF[x, y], {x, 1, 15}, {y, 0.9, 8}, 
   Contours -> Range[0.1, 12], AspectRatio -> Automatic, 
   ImageSize -> Medium, PlotLegends -> {Automatic}];

sp = StreamPlot[
   Evaluate[-D[intF[x, y], {{x, y}}]], {x, 1, 15}, {y, 1.5, 8} , 
   StreamScale -> Coarse, 
   StreamStyle -> {"Arrow", Directive[Thin, Blue]}, 
   StreamPoints -> Fine, AspectRatio -> Automatic, 
   VectorScale -> Automatic];
vp := VectorPlot[
  Evaluate[-D[intF[x, y], {{x, y}}]], {x, 1, 15}, {y, 1.5, 5.9}, 
  VectorScale -> Automatic, AspectRatio -> Automatic, 
  VectorStyle -> {LightGreen}, VectorPoints -> 15]
vp2 := VectorPlot[
  Evaluate[-D[intF[x, y], {{x, y}}]], {x, 1, 15}, {y, 6.1, 8}, 
  VectorScale -> Automatic, AspectRatio -> Automatic, 
  VectorStyle -> {LightGreen}, VectorPoints -> 15]
sp2 = MapAt[{Opacity[0.25], #} &, sp, 1];
Show[cp, vp, vp2, sp2]}

Show[%169, ImageSize -> Large, FrameLabel -> {"Posición en eje x (cm)",
HoldForm["Posición en eje y (cm)"], "", "valores de Voltaje (V)"}, Ticks ->
Automatic,FrameStyle -> Directive[Black, Thickness[Medium]],
LabelStyle -> {FontFamily->"Arial", 16, GrayLevel[0]}]

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