Is there a normal force on an object submerged in water? I am not sure about whether or not there is a normal force acting on an object submerged in water. It might be because I have not good understanding about what a normal force actually is.
I know a force due to earth's gravity and a buoyancy force acting on the object. 
Could you let me know whether or not I am wrong? Any comments are welcome.
 A: All the submerged parts of the object are subject to a force from the surrounding fluid. This force is usual stated in terms of pressure (which is force per unit areas) and always acts normal to the local surface.
Buoyancy is the net of all the pressure-force acting on the body. SO for many purposes you can simply treat that complex mess as a single cohesive force acting upward.
Pressure (times area) is the liquid equivalent of the normal force (in so far as it is a contact force that acts normal to the surface) but it does not prevent interpenetration the way the normal force from a solid does.
A: Maybe this approach can also be helpful:
Forget all words or names, just keep the most important one, i.e., Newton's laws. A body will certainly move with some acceleration, if a net force acts on it. When a body is on the table, we see it doesn't move. So there must be another force equal and opposed to its weight (as you see, we don't care what is its origin). On the other hand, when a body is floating in water, again we see it doesn't move, so there must be another force equal and opposed to its weight.
Now just choose a name for each one. The former has been called "normal force" and the latter "buoyancy".
