Will the water go inside the moving water bottle? Let's say that there is a empty bottle in the water moving at a high speed like this:
 
My question is: Will the water go inside the the empty bottle when the bottle is moving at a high speed? If yes, from what this speed is affected? and if this bottle is 1/2 full will its water come out? Thanks in advance.
 A: The opening sees water at stagnation pressure. Water will flow in until the air in the bottle is squeezed to that pressure.
A: I'm appreciating your thinking and question. 
This question will be even more challenging question if the working fluid is air instead of water and fluid is moving. In this answer I'm considering bottle is moving and fluid is moving in opposite direction of bottle. (because stationary fluid is sub case of  moving fluid where $v_1$=0)
Let us assume the the bottle is empty and moving from zero velocity to a constant velocity, because of relative motion between bottle and fluid ( here I'm considering air because compressibility of air is more than water, fluid will enter the bottle at the mass flow rate
$$\dot{m} = c_d \rho A_{c} v_1$$ here 
$c_d$ is discharge coefficient depends on lid design of bottle. 
$\rho, A_{c}$ and $v_1$ are density of fluid, cross sectional area normal to flow angle and relative Velocity of external fluid and bottle respectively.
If there is a contraction then we should use continuity equation
$\rho_{1} A_1 v_1=\rho_{2} A_2 v_2$
here $_2$ is properties after diverging section and $_1$ is free steam properties.
This is valid till the flow touches the other end of bottle. After that this continuity equation is not valid because no mass flow rate outside of bottle. Then the velocity inside the bottle can be calculated from Bernoulli equation. please note that till I'm in incompressible region of flow and that velocity is calculated from,  
$v_2= \sqrt[]{\frac{2*(p_1-p_2)}{\rho_1}}+v_1$
One that fluid hit the end of the bottle, "hammer effect" (similar to "water hammer effect") will take place and one shock will move from end of bottle to lid of the bottle at about the speed of sound in incompressible region and more than that in compressible region. This shock will increase the pressure of the fluid inside the bottle and reduces the previous inflow velocity calculated from continuity. This shock also makes the velocity of fluid behind the shock, zero. Please refer the following link to get an idea how to calculate the flow properties in moving shock. This is a dynamic process, this shock will tell information to the upstream fluid that "there is a wall so we can't move". After the pressure inside the bottle reaches stagnation pressure every where (flow become quasi-steady state) the fluid layer at the lid of bottle act like a solid wall and it will not allow further new fluid elements to go into the bottle.
Will the fluid inside the bottle go out?


*

*We may think stagnation of pressure is more than the external
pressure so fluid may come out but the possibility is less because.
Next layer outside the fluid lid is stagnated because of solid nature
of fluid layer in lid ( already mentioned), so leakage is impossible
unless some complex process takes place.

*That reflected shock from the bottle end will from a normal shock in
front of the bottle and reduce the velocity of fluid entering the
bottle. Because of that reflected shock a factious layer of the
immobile fluid layer (w.r.t bottle) that covers the bottle where
pressure is less than or equal to the stagnation pressure of external
fluid so fluid inside the bottle can't go out. 

*If compressive shock is formed at upstream of bottle that will reduce
stagnation pressure at downstream and make the fluid inside the
bottle difficult to come outside.

*Fluid may go out because of inertial effect, if velocity of moving
bottle is reducing but this will happen only if inertial force is
grater than pressure force.

*It will go outside if  its stagnation pressure is more than external
stagnation pressure    this is not possible unless we supply some
energy to bottle to    increase pressure inside the bottle.
Method to verify this process 


*

*Transient CFD simulation this is relatively easy one

*Mathematical modelling this process in differential form and solving.   This is tough one

*Experimental process using flow visualization like  schlieren or
shadowgraph with PIV. It is very expensive.

