While whatever is feeding the brown algae is in the tank you wont be able to deal with it other than manual cleaning. As said it tends to clear up over time as either the nutrients are taken up by something else or they simply run out.
Silicates are one of the main things brown algae feed on, and one of the sources of silicates will be new substrate. Over time the silicates will leech out of the substrate and get used up. This is why brown algae often disappears on its own given time.
Over time, especially with your lighting period and no plants, green algae will develop and out compete the brown algae for nutrients. While lighting isnt as important for brown algae growth as it is for green algae, you could go the other way, and try and starve the brown algae of light. You might not like looking at a darkened tank though, so maybe isnt a good route for you.
Upping the water change frequency can remove nutrients from the water, depends on your tap water. You could be adding nutrients. Some tap water is high in nitrates and silicates for instance.
Try adding some live plants to take up nutrients. Java fern and anubias are really easy to keep. Just attach it to a rock, piece of driftwood or plant weight and drop it into the tank. They are very low maintenance, dont need strong lighting, dont need substrate at all, but may benefit from weekly liquid fertiliser.