Boundary conditions for Navier-Stokes equations

In the most elementary treatment of plane-wave reflection at the open end of a duct system, it is often assumed that the ends are pressure nodes. This implies that pressure is assumed as a constant at the open end termination and that steady flow boundary condition is supposed as instantaneously established. While this simplifying assumption seems reasonable, it does not consider any radiation of acoustic energy from the duct into the surrounding free space; hence, an error in the estimation of the effects of the flow on the acoustical response of an open-end duct occurs. The actual flow conditions, in fact, lag behind those computed in the conventional manner; although the lag times are small,they may become significant. Pressure at the duct exit has been calculated as a function of the incident waves reaching it. If radiation at the open end is accounted, a complicated three-dimensional wave pattern near the duct end is established, which tends to readjust the exit pressure to its steady-flow level. This adjustment process is continually modified by further incident waves, so that the effective instantaneous boundary conditions which determine the reflected waves depend on the flow history. Results based on the application of the new developed procedure for reflected waves have been compared with experimental measurements on a shock tube.


Experimental setup (shock tube) used for the measurements.


Computational domain used for the simulations. Variables at the inner nodes were solved by the two-step Lax-Wendroff finite difference method with the TVD flux-limiting technique. The method of characteristics has been used to solve governing equations at the boundaries.


Comparison between simulations and experiments for shock the tube configuration. Simulations were performed by using either a steady flow model at the open end termination of the computational domain (plot a) and the PRBC model (plot b).

Results show that the new theoretical model (Partially Reflecting Boundary Condition, PRBC) is able to satisfactorily predict instant pressure waves along the duct. The model has been tested and validated also to simulate silencers mounted on exhaust systems of internal combustion engine.

Related publications and documents:
1 - F. Piscaglia, A. Onorati. "A novel theoretical approach to model partial reflection of waves of finite amplitude at the open end of a duct". Submitted to Journal of Sound and Vibration (Elsevier).
2 - F. Piscaglia, A. Montorfano, A. Onorati, G. Ferrari, "Modeling of Pressure Wave Reflection from Open-Ends in I.C.E. Duct Systems",, SAE paper n. 2010-01-1051, SAE 2010 Int. Congress & Exp. (Detroit, Michigan), April 11-14, 2010.