Questions about boundary TYPE_E

I would like to simulate the response of an object in an air fluid. For this, I have set the left side as the inlet, the right side as the outlet, and periodic boundary conditions for the top, bottom, front, and back. The left side has a velocity directed towards the right, while the right side is configured to allow free outflow.

However, I have observed that during the simulation, a backward shock wave appears at the left wall, as though the outflow condition is not functioning as expected. I would greatly appreciate any insights or suggestions as to why this might be occurring.

Here are the detailed settings I have used:

parallel_for(lbm.get_N(), [&](ulong n) {
    uint x=0u, y=0u, z=0u;
    lbm.coordinates(n, x, y, z);

    // Default initial values: rho=1, u=0, flags=0 (inside fluid)
    // Below, modifications will be made as needed
    // (1) Inlet (x=0): Set velocity boundary
    if(x == 0u) {
        lbm.flags[n] = TYPE_E;      // Fluid boundary, use Equilibrium boundary
        lbm.u.x[n]   = U_in;        // Set velocity in the x direction
        lbm.u.y[n]   = 0.0f;
        lbm.u.z[n]   = 0.0f;
    }

    // (2) Outlet (x=Nx-1): Can specify a near free outflow
    //     Method A: Approximate velocity as (U_in, 0, 0), or smaller
    //     Method B: Set density rho != 1
    //     The following demonstrates a simple "velocity" specification
    if(x == Nx_1) {
        lbm.flags[n] = TYPE_E;      // Equilibrium boundary
        // lbm.u.x[n]   = U_in;        // Can also be set to a smaller value, such as 0.8f*U_in
        // lbm.u.y[n]   = 0.0f;
        // lbm.u.z[n]   = 0.0f;
        lbm.rho[n]   = 1.0f;  // Or slightly less than 1.0f to simulate a slight pressure gradient
    }

    // (3) Top, bottom, front, and back boundaries: Rigid walls
    // if(y == 0u || y == Ny_1 || z == 0u || z == Nz_1) {
    //     lbm.flags[n] = TYPE_S;      // No-slip boundary
    //     // set the transparent boundary
    // }
});

Start with the TYPE_E boundary setup and inlet/outlet configuration shown in the issue, then review how those boundaries are handled in the project. The issue names no source file or test; completion would require identifying the cause of the backward shock and documenting or validating a corrected boundary setup.