I think this problem could be modeled as the boundary flow past an isothermal plane surface. For laminar flow, the average Nu=0.68 Re^(1/2) Pr^(1/3). For turbulent, the average Nu=0.5 f * Re * Pr^(1/3) where the skin friction f = 0.079 / Re^0.25 or 0.0014 + 0.125/Re^0.32.
More correlation of Nu can be found in heat transfer books.
I don't have on my hand. But, I believe the heat transfer textbook should have. E.g., Fundamentals of Heat and Mass Transfer by Frank P. Incropera, David P. DeWitt.
Thanks much Shiuh. I have that text book. However, these are theoretical values. We recently ran experiment in the field and in the lab to try to derive a convective heat transfer coefficient to ue it later in equations for a model. But we came up with incredibly high values which are unlikely to be good. I wanted to see if others got the same problem in experiments.
Daniel, perhaps you have a mixed convection. Sec 9.9 discusses how to combine the free convection and force convection as the 1st approximation. You can also use CFD as a tool to find the convective heat transfer coefficient.
Hi Shiuh, again thanks much for your help in this matter. I will look into this. My feeling is that there isn't free convection as it is surface run-off water flowing over and into permafrost ground ice. There's no standing water that would allow for free convection. I received a paper from another researcher (experimental values) that shows, without explanation though, that ice surface inclination does change the ablation rate of ice. I haven't had time to read it n details but I hope it will help us to explain on very high melting rates. I will ask my PhD student who works on that topic to dig a bit more about that. I apreciate your help.
There is a phase change for ice melting as well as the force convection in the flowing water. If the ice surface is rough, and/or has some holes, it will be more complex. There is an review paper you might be interested in:
"Melting and Freezing" L.S. Yao ,J. Prusa "Advances in Heat Transfer, Volume 19, 1989, Pages 1–95
Hello there Shiuh, I will read through 'Melting and Freezing' soon - I think we may obtain some answers there regarding our specific context.
further, we have forced convection in our case as water is flowing over the ice. Well I am digging in thank you for referring these papers, I'll update eventually when I'll have something new