Enhanced Runge-Kutta Discontinuous Galerkin Method for Ultrasound Propagation in Transit-Time Flow Meters

Abstract

We illustrate a time and memory efficient application of Runge-Kutta discontinuous Galerkin (RKDG) methods for the simulation of the ultrasounds advection in moving fluids. In particular, this study addresses to the analysis of transit-time ultrasonic meters which rely on the propagation of acoustic waves to measure fluids flow rate. Accurate and efficient simulations of the physics related to the transport of ultrasounds are therefore crucial for studying and enhancing these devices. Starting from the description of the linearized Euler equations (LEE) model and presenting the general theory of explicit-time DG methods for hyperbolic systems, we then motivate the use of a spectral basis and introduce a novel high-accuracy method for the imposition of absorbing and resistive walls which analyses the incident wave direction across the boundary surface. The proposed implementation is both accurate and efficient, making it suitable for industrial applications of acoustic wave propagation.

Figures (6)

• Figure 1: Convergence and efficiency study of the RKDG methods for different polynomial orders. a) All the methods have a correct convergence behaviour after grid refinements even if fourth and fifth orders are not fully achieved. b) Low-order methods are less accurate for the same calculation time and the $P\ge 3$ choice is recommended.
• Figure 2: Snapshots of the test case proposed in Section \ref{['totally absorbing wall tests']} where the same color ranges are used for the pressure $p$ (Pa) in the four pictures. As expected, the wave is not dissipated nor reflected on the walls.
• Figure 3: $L^2$ norm of the perturbation pressure. The red line corresponds to the algorithm in Equation (\ref{['filtering']}). The dotted blue line is obtained instead when the absorbing condition in Equation (\ref{['absorbing wall']}) is used with $\theta=0$, i.e., assuming that the incident waves are orthogonal to the wall.
• Figure 4: Perturbation pressure snapshots from the clamp-on test proposed in Section \ref{['A 2D clamp-on flow meter']}. The pressure values range in $\pm1.5\cdot10^3$ Pa.
• Figure 5: Pressure measurements at the two transducers for the forward and backward waves. As expected, the two waves overlap at the sender but they are not completely aligned at the receiver because of the water convection.

Theorems & Definitions (9)

• Definition 2.1: Discontinuous mean and jump Hesthaven
• Definition 2.2: Lax-Friedrichs numerical flux Hesthaven
• Definition 3.1: Acoustic impedance of a surface
• Proposition 3.1: Acoustic impedance on the interface between two homogeneous mediums
• Definition Appendix A.1: Jacobi polynomials jacobi
• Definition Appendix A.2: 2D Dubiner basis antonietti-houston
• Definition Appendix A.3: 3D Dubiner basis sherwin
• Proposition Appendix A.1: Orthonormality of the Dubiner basis
• proof