I am testing the boundary conditions in the Euler solver(compressible inviscid flow). Based on the contexts in books, I set as

U(1)=U(2) U(nx)=U(nx-1)

where 1 denotes the last left end and nx is the last right end. Here wave is propagated to the right.

In many books, they explained above way to impose the NRBC but I experienced small reflection from the right wall.

In a book, it is said that this BC setup does indeed produce some reflection.

Does anyone experience about the NRBC before?

Thanks in advance!

Reflections will be produced and if you run for long time runs the reflections might contaminate the solution.If you need to run for long time runs,you might want to use artificial boundary conditions rather than NRBC.There is a lot of literature available on that topic.Some of the different kinds of ABC are buffer zone, perfectly matched layer , Energy transfer and annhilation.Out of the three,the first and third are very simple to implement.The second is a comprehensive ABC but requires a bit of work in the implementation.Look for papers with similar names or look in the aeroacoustic benchmark workshops.They contain a lot of details on the methods.

Thanks for your kind answer. But it's still not clear for me. I am solving the set of conservation laws(mass, momentum and energy) so that the boundary condition need to be defined in terms of density, velocity and energy.

But these you mentioned related with a kind of acoustic approach. Is it possible to impose like this way?

u(1)= something <-- u(nx)= something <--

I wonder if they need to couple with the equation-type. Or they need to be included in the system as an additional equation.

The Euler system is hyperbolic and supports three kinds of waves ( acoustic,entropy and vorticity).You can get a feel of this by writing the euler equation in matrix (a simple 1D analysis )form and finding the eigen values and vectors.So at the boundaries you have the 3 kinds of waves which have to be properly resolved.Also information is propagated in the form of waves and the reflections in the boundary condition occur when the waves are not properly resolved.To overcome this difficulty,people came up with radiation and outflow boundary conditions based on 1D analysis.But for long time runs,it was found that the 1D analysis does not also nullify the reflections as the waves exiting the domain might not necessarily obey the 1D equations.This was the reason for development of the artificial reflecting boundary conditions.

For your problem since you use Euler it becomes necessary to introduce some kind of ABC if you intend to run for a long time.For small time runs,radiation and outflow boundary condition or characteristic based boundary condition should be fine.

Regarding the possibility,yes all the people impose boundary condition for compressible systems in this way.The way you impose it depends on the method you chose.For some of them you might need to solve an auxillary equation while for some of them you might just need to add an artificial term to your equation similar to the artificial viscosity.