Free surface in fluid dynamics In Computational Fluid Dynamics, if the fluid does not cross the free surface, the relation between fluid velocity at free surface and free surface velocity is given by$$
{\vec{V}}_{\text{surface}} {\cdot} \vec{n} = \vec{v}_{\text{fluid}} {\cdot} \vec{n}
\,,$$where:


*

*$\vec{V}_{\text{surface}}$ is free surface velocity;

*$\vec{v}_{\text{fluid}}$ is the fluid velocity; and

*$\vec{n}$ is the normal to the surface.
How does such a relation occur?
 A: I think it should be $$\mathbf{V}_{surface}=(\mathbf{v}_{fluid}\cdot\mathbf{n})\mathbf{n}$$ Above equation says that the shape of the free surface of a fluid can change only due to fluid motion normal to the free surface. Fluid motion "in the surface", i.e. tangential to the surface, does not cause a change in the shape of the free surface. 
This is so because a free surface is composed of fluid particles, and does not have an independent existence. Think of a flat free surface of water at rest; if the fluid particles lying in the surface move only in the plane of the free surface, then the shape of the surface does not change and consequently the free surface cannot be said to have moved.
Forming the dot-product of the previous expression with $\mathbf{n}$ gives the particular formula in your question. However the formula in your question gives only partial information about $\mathbf{V}_{surface}$, and must be supplemented by the second equation $\mathbf{V}_{surface}\cdot\mathbf{t}=0$, in which $\mathbf{t}$ is the tangent vector to the surface.
