The simplest case to consider is when the flow is steady. Then a streamline, a pathline and a streak line all coincide.  Hence if imaginary line in a steady flow is  taken to be a streamline, then by definition the fluid velocity is always tangent to the streamline. Thus the velocity component normal to the streamline is zero, i.e., the kinematic condition holds: u.n=0 wher n is the unit normal to the streamline.

If the flow is unsteady and you select your imaginary line to be a pathline, then at any point on the pathline at a particular instant in time the velocity is tangent to the pathline. Obviously at another instant in time the pathline line location will be different.

The situation get more complicated if there is a stagnation point on a streamline, the signature for flow separation.  For more details on this case check out the following presentation/publication: Goenaga and Higgins: "Flow Separation at a Free Surface: Scenarios that promote liquid jetting". 

You can keep that middle surface as a interface if there are two different fluid on both the side of surface but from your problem statement i figured it out that you are talking about the same fluid flow across the far field boundary so in this case you can assume that surface as a interior and after simulation you can easily get all the details of flow particles like motion of particles, path lines etc  at particular frame of time on that surface.

If I understand your question correctly, the surface z = z1(x) is fixed in the x-z space and is NOT a material surface; in other words, any fluid parcels can pass through the surface.  Correct?  In that case, there aren't any conditions: the motion of the fluid has nothing to do with the shape or position of the surface, unlike the true free-surface, whose shape and position is determined by the fluid motion, and unlike the bottom, which the fluid parcels cannot pass through.   To repeat, the shape and position of your surface can be arbitrary regardless of the fluid motion; there cannot be any relations between the fluid motion and the shape or position of your surface.  (Well, I've repeated equivalent statements three times . . . )

Thanks all. @Ryo,  yes z=z1(x) is fixed, pre-determined and any fluid parcel can pass through. And your answer is clear thanks.

To describe the processes in a non uniform fluid (the system), it mentally (imaginary) is divided into locally-equilibrium domains. To use the locally-equilibrium macro parameters, such domain is chosen small size, within which acceptable accuracy there is a equilibrium. This description applies to weakly non uniform  systems that are justified in terms of local thermodynamic equilibrium. The analysis shows that the usual conditions of local thermodynamic equilibrium are performed with sufficient accuracy.

Thus, the answer to Your question is. In the area of the domain must be a local equilibrium.

Sensory Yours,  Lev.

Чтобы описать процессы в неоднородной среде (системе), она мысленно (воображаемо) разбивается на локально-равновесные домены. Чтобы использовать локально-равновесные макропараметры, такой домен выбирается малого размера, в пределах которого с приемлемой точностью существует равновесие. Такое описание применяется для слабо неоднородных систем, что оправдано в условиях локального термодинамического равновесия. Анализ показывает, что обычно условия локального термодинамического равновесия выполняются с достаточной точностью.

Таким образом, ответ на Ваш вопрос таков. В области домена должно наблюдаться локальное равновесие.  Левъ

If you lay our hand on an old Mechanics of Fluids text by Milne ,there is a section that deals with tjhis problem