Yes, every flue on a chimney should have a cap. A chimney cap is the mesh-sided cover that sits over the top of the flue opening, and its job is to keep rain, snow, birds, squirrels, and debris out of the flue while still allowing smoke and combustion gases to vent freely. A flue without a cap is an open pipe straight into the house, and it will take on water and animals eventually.
The cap is separate from the crown. The crown is the sloped slab covering the masonry around the flue; the cap sits above that, directly over the flue opening itself, usually attached with a clamp or set into mortar. The mesh sides keep sparks contained and animals out while leaving enough open area for smoke to draw properly, and most caps on wood-burning fireplaces also serve as a spark arrestor.
A chimney with more than one flue, common on a house with a fireplace and a separate furnace or water heater vent, needs a cap on each flue opening. Skipping a cap on even one flue leaves that one exposed to rain running straight down into the firebox or appliance below, and in a climate with regular freeze-thaw cycles, water sitting and freezing inside an uncapped flue does real damage to the liner over the course of a winter.
We check cap condition on chimney inspections and replace caps that have rusted through, lost their mesh, or were never installed correctly in the first place. It's one of the cheapest parts of a chimney to keep in good shape relative to the damage a missing or failed cap can cause to everything underneath it, from the liner down to the firebox.
More on this from Chelo's Brick & Stone.
Chelo's Brick & Stone serves Hillsville, Galax, Wytheville, Fancy Gap, and Independence, VA. Call for a straight answer and a free on-site estimate.