# ContourPlot — How do I color by contour curvature?

I'm plotting the stream lines of fluid flow past a cylinder, and I would like the colors to increase with contour curvature (i.e. increase as the velocity of the flow increases. Here's a MWE that seems to color it based on the the y-axis value:

ψ[r_, θ_] := U (r - a^2/r) Sin[θ]
r = Sqrt[x^2 + y^2];
θ = ArcSin[y/r];

stream = ContourPlot[
ψ[r, θ] /. {U -> 10, a -> 1},
{x, -5,5}, {y, -5, 5},
Contours -> 10 Table[i, {i, -10, 10, 0.025}]
];

cyl = Graphics[Disk[{0, 0}, 1]];

Show[stream, cyl] f = {ψ[r, θ]} /. {U -> 10, a -> 1};
gradf = D[f, {{x, y}, 1}];
Hessf = D[f, {{x, y}, 2}];
secondfundamentalform = -PseudoInverse[gradf].Hessf // ComplexExpand // Simplify;
tangent = RotationMatrix[Pi/2].normal;
curvaturevector = (secondfundamentalform.tangent).tangent // Simplify;
signedcurvature = curvaturevector.normal // Simplify;

stream = ContourPlot[
ψ[r, θ] /. {U -> 10, a -> 1}, {x, -5, 5}, {y, -5, 5},
Contours -> 10 Table[i, {i, -10, 10, 0.2}],
];
curvatureplot = DensityPlot[signedcurvature, {x, -5, 5}, {y, -5, 5},
ColorFunction -> "RedGreenSplit",
ClippingStyle -> Automatic,
PlotPoints -> 100,
PlotRange -> {-1, 1}
];
Show[
curvatureplot,
stream,
cyl
] 