By using ‘turns up’ there is greater emphasis and immediacy on the hope of him arriving. If the arrival was further into the future, you would use ‘will turn up’ as in: I hope we’ll have better weather when we fly to Canada next year.
I am wondering if “show up” would be more natural for: Peter hasn’t arrived yet? – I hope he wiil show up soon. “Turn up” sounds to me a little to arrive too accidently to be hoped.
I know you like the subtleties of the English language and I would refer you to the character created by Charles Dickens in his novel David Copperfield, who goes under the name of Wilkins Micawber. Mr Micawber is an eternal optimist and believes that there is always the possibility of happiness just round the corner:
Present Simple tense - Use the Simple Present to express the idea that an action is repeated or usual. The action can be a habit, a hobby, a daily event, a scheduled event or something that often happens. It can also be something a person often forgets or usually does not do.
Future Simple tense - The simple future tense is often called will, because we make the simple future tense with the modal auxiliary will.
I see you advertise your services at the end of your message. I’m a bit worried about some of your use of English as shown in this part of your publicity: