Evaporation is an endothermic process, meaning it takes energy to do so. When air is blown onto water, that water wants to evaporate, especially if that air is relatively dry and has the space for the water molecules. So when the water does evaporate it pulls energy from its surroundings to make the state change. There’s that, and water is a much better conductor of heat than air, so if that water in your skin is cold (perhaps because some of it is evaporating and sapping it of its heat energy) it will feel extra cold because water is just good at conducting heat from your body.
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