Dear Mangilal, 

I am not sure what exactly you understand of co-rotating vortices. Just to be sure: if you really have two identical vortices side by side, there would be a discontinuity where they touch each other. Else, the Taylor--Green vortex is a well-known and often-referred-to exact solution of the Navier stokes equations. 

In some sense, you have there co-rotating vortices (just with the restriction that there is a counter-rotating one in between the two) 

Hoping I could help... 

Cedrick

https://nrg.iag.uni-stuttgart.de/research/turbulence/taylor-green-vortex

 Dear Cedrick Ansorge,

                                           Thanks for your reply.  Co-rotating means all the vortexes have flow in same direction (either clockwise or anti-clockwise) and interface of the vortexes are discontinuous (anti parallel flow exist at interface). In the case of three, all will be in same directions. There will be not a counter-rotating vortex between of two co--rotating vortexes. So reconsider the problem and give your opinion. 

I do not know in terms of which fluids you think (zero-viscosity?), but generally-speaking, interfaces of anti-parallel flow are not maintained under Navier-Stokes. In presence of viscosity, there would be immense shear and even if you by some kind of forcing manage to force a system into such a state, it would be highly unstable. The instability would immediately create exactly those counter-rotating vortices which you want to avoid.

I, however, doubt that a configuration with discontinuities in the velocity field is physically realizable given any finite viscosity of the fluid. 

The medium is made of the negatively charged particles, electrons, and ions, which definitely has finite viscosity. This medium (negatively charged grain medium) is under the shear flow of  ions then possibility to excite co--rotating vortices. All the vortices are well separated and have anti-parallel flow at interface. These  vortices are stable and never turn to counter-rotating.