Is it appropriate to contact faculty post submission?

[meow]

I have applied to PhD programs in engineering in the US. I have been early in applying to most programs (deadlines range from mid-December to mid-January). I have not contacted any professors prior to applying, as I have been told by many that it would largely be a waste of time. Instead I reached out to several grad students and read/skimmed through many recent published papers by faculty. In my statement of purpose, I talked about at least two professors that I thought had interesting topics for research and why they interested me. I tailored this part of the statement for each program.

My question is, is it appropriate to contact faculty post application submission? I would like to say hello, let them know that I have applied and that I would be interested in working with them if I am fortunate enough to get in, and ask them to keep me in mind if they are looking for students.

I thought it might be a good idea to just put myself out there. Who knows? Maybe they'll recognize my name once they start going through applications and give it another good look. But I also do not want to annoy them. I recognize that these professors are busy and that they get many of these emails every day. I want to be considerate of their time too. I also do not want to come across as being arrogant or silly or needy..

What is your expert advice? Would you recommend that I sit tight? With my application already out of my hands, I understand that there is nothing I can do now to improve my chances. But I am always hoping. Thanks in advance for your advice!

[Extra Espresso]

@loveor I've asked this question of a lot of people lately, and my best understanding is that it depends very heavily on the field you are going into whether or not it would help you to contact faculty at this point. The best advice I was given was to ask people applying in my field (BioE) what they are doing/did when they applied. You may want to check out some of the boards in your field and see what advice people have!