## 1 Introduction  
In general, a complex pipe system is used to transport high-temperature and high-pressure gas in industries such as chemical, petrochemical, and offshore oil production plants. When the piping system is over-pressurized, the pressure relief valve is installed to prevent pipe failure due to high-pressure. When the fluid passes through the pressure relief valve, high acoustic energy is generated and radiated in the downstream direction due to the sudden pressure drop through the valve. Acoustic pressure waves induce the vibration of the pipe wall. This phenomenon is called \"Acoustic Induced Vibration\" (AIV). AIV in a piping system is known to be involved in the structural and acoustic fluctuations in a range of frequencies between 500 and 2000 Hz [1]. [PERSON] and [PERSON] [2] collected the failure cases until 1980 to separate the failures due to AIV and suggested the simple procedure of a pipe design to prevent AIV. [PERSON] [3; 4] collected more pipe failure cases and modified the criteria of [PERSON] and [PERSON]. NORSOK standard [5] adopted the method proposed by [PERSON] and [PERSON] to assess the acoustic power level in a design stage of pipe. [PERSON] et al. [6] summarized the history of empirical research about AIV in a piping system and suggested the modified empirical method newly adopting the thickness of pipe wall as an additional parameter to the diameter of the pipe. Because most of preceding studies on the AIV problems in piping systems depend on empirical approaches, the fundamental mechanism of AIV does not come out into the open, and thus it is difficult to devise effective measures for AIV even if the AIV problem is expected. To find the precise mechanism of AIV and to devise its effective measure, the first-principle-based method is essential. [PERSON] and [PERSON] [7] used a fluid-structure one-way coupling method using ACTRAN software based on the finite element method and suggested a methodology for AIV risk assessment. However, the method is still based on an empirical formula of [PERSON] and Mueller for the prediction of acoustic power. [PERSON] et al. [8] estimated the internal aerodynamic noise due to valve flow in a simple constriction-expansion pipe, by combining the large eddy simulation technique with a wavenumber-frequency analysis. However, the results were not confirmed by using any measured data. The main goal of this study is to propose the first-principle-based methodology to determine the acoustic power due to valve flow in a pipe. The method consists of three sequential steps: accurate internal flow simulation using Large Eddy Simulation (LES) techniques, the separation of compressible and incompressible pressure waves using the wavenumber-frequency analysis and the assessment of acoustic power using the duct acoustic theory. The critical nozzle is selected as a target system since it is commonly used as a flowmeter and thus there are a lot of relevant data for comparison. First, the steady RANS solver is used to predict the flow rate of the two-dimensional axisymmetric critical nozzles, and its validity is confirmed by comparing the predicted results with the measured ones. There is good agreement between the two results. Then, a high accuracy LES technique is performed on the three-dimensional critical nozzle, and the steady-state Reynold-Averaged Navier-Stokes (RANS) result is used as the initial condition to accelerate the convergence of the unsteady simulation. The validity of the unsteady LES results is also confirmed by comparing them with measured surface pressure data. The wavenumber-frequency transform is taken on the LES results, and the compressible surface pressure components matching the acoustical duct modes are identified in the wavenumber-frequency pressure diagram. The inverse wavenumber-frequency transform taken on the compressible pressure components leads to the acoustic power spectrum. The structure of the paper is organized as follows.