I know that blood flow increases/decreases in response to temperature change, which is why (lighter-skinned) people go red when they are hot.
I know that the eye contains lots of blood vessels.
Why do our eyes not turn red when we are hot?
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Skin turns red when light skinned people flush because it contains few absorbers, so photons diffuse through it long distances. During this time green and blue light is absorbed as photons encounter blood vessels. Red photons are not and so the skin appears redder.
The whites of the eyes are strongly scattering and have relatively few blood vessels, so changes in blood flow has a small effect on color. The (wavelength-independent) scattering dominates over absorption.
The pupils open up on the approximately 22 mm gap between the cornea and the retina. You don't see changes in color here because the entire retina strongly absorbs all colors of light. The few red photons scattered by blood are very unlikely to be exactly aimed at the pupil. Instead most scatter back into the retina and are absorbed. This is compounded by the scattering anisotropy of blood, which causes most incoming red light to be scattered in the forward direction - directly into the strongly absorbing retina.