# Wavenumber-Frequency Analysis of Internal Aerodynamic Noise in Constriction-Expansion Pipe
## 1 Introduction  
High-temperature and high-pressure gas is produced by various processes in chemical, petrochemical, and offshore oil production plants. Excessive noise and vibration can be generated when such gases are discharged, usually as they pass through a valve in the piping system used to protect the system and equipment from being damaged by such high-pressure gases. In the 1980s, pipe failures due to acoustic-induced vibration (AIV) were reported. Thus, the assessment and control of noise and vibration in piping systems have become essential parts of the design stage. [PERSON] et al. [1] first investigated pipe failure using nine pipe failure cases and 27 normal pipe cases. They suggested a criteria curve for the sound power and pipe diameter. This curve has been used as the basic screening method for the AIV of pipes. [PERSON] [2] proposed simple piping design criteria by including the pipe wall thickness as an additional parameter. These were adopted in the Norsok Standard [3]. Recently, [PERSON] et al. [4] suggested new guidelines for the pipe AIV problem, in which the maximum vibration level is proportional to a nondimensional parameter that is given by the pipe diameter divided by thepipe wall thickness. The Energy Institute [5] published guidelines for the pipe AIV problem based on the likelihood of failure (LOF), which depends on parameters, such as the branch connection, main pipe diameter ratio, and material type. However, all of these studies assumed that the main noise sources in a piping system are the pressure relief valves and estimated their acoustic power levels using an empirical formula that was based on API521 [6] and IEC 60534-8-3 [7]. In addition, they used an empirical criteria curve, the main parameters of which were the sound power and pipe dimensions obtained from the available pipe failure cases. Although these methods are very efficient at quickly screening a large number of pipes, it is difficult to apply efficient mitigation measures, because these tools rely on a limited number of input parameters to quantify the LOF, without any understanding of the relevant physical mechanisms. Recently, [PERSON] et al. [8] proposed a finite element methodology based on fluid-structure coupling, which makes it possible to predict the dynamic stress in a complex piping system. However, even though this method could be used to quantitatively and effectively assess mitigation actions, the acoustic energy that was generated by a pressure-reducing device was estimated using a simple empirical formula, which inevitably limited its predictive accuracy with significant uncertainty. In this study, to overcome this limitation, a systematic first-principles-based method was developed by combining high-order computational fluid dynamics (CFD) techniques with a wavenumber-frequency analysis. This method makes it possible to separate the acoustic pressure waves from the total pressure fluctuations in a pipe flow induced by valves, and to compute the acoustic power that is generated by the valves. First, the valve flow is predicted using high-resolution CFD techniques, in which the valve is modeled as a simple constriction-expansion pipe for simplicity. The validity of the calculated steady pressure is confirmed by comparing it to the theoretical solution obtained from quasi-one-dimensional (1D) inviscid compressible flow equations. Then, an unsteady simulation is performed to predict the fluctuating total pressure inside a pipe. Finally, the acoustic pressure modes in the pipe are extracted by applying the wavenumber-frequency transform to the total pressure field. The results show that the acoustic pressure fluctuations can be separated from the incompressible ones; this can provide accurate information about the acoustic power spectrum, which is used as an important parameter to determine the LOF of a piping system due to AIV. The remainder of this paper is organized as follows.