After using the bathroom some folks let the water run (without opening and closing the faucet) and pass each hand under the faucet 3 times alternating between the right and left hands. Since there's no cup involved, is this futile?
With a cup you get koach gavra (human power) but with water running, does passing your hands under the water multiple times have any effect?
Rav David Yosef (Halacha Berurah 7:3) is lenient to not require koach gavra or a kli (however he says that one who is machmir to use a kli should be blessed).
He also mentions (ad. loc.) that it's sufficient to wash each hand once, but it's appropriate to be machmir and wash 3 times alternating. The language that he uses to describe this chumra seems (to me) slightly stronger than the language he uses to describe the chumra of koach gavra.
According to this opinion, this "charade" is not "futile", and seems to be a reasonable middle-ground if you can't be machmir for both. (Finding a kli in a public bathroom is not always easy, so I totally understand not being machmir for both.)
(Note: certain things require you to wash with koach gavra, and 3 times alternating, so what I've written doesn't necessarily apply to anything other than using the bathroom. The halacha I've quoted comes from the siman about the beracha of Asher Yatzar.)
Perhaps it is futile, but considering that there is no real source to wash one's hand three times after leaving the bathroom, there is probably nothing lost. Once is enough.
At best some achronim, Reb Yaakov Kaminetsky in the Emes Liyaakov on Shulchan Aruch comes to mind, say washing three times after the bathroom is a nice thing to do.
