# Wavenumber-Frequency Analysis of Internal Aerodynamic Noise in Constriction-Expansion Pipe
## 2 Theoretical Background  
### Valve Flow and Its Quasi-1D Approximation  
Commercially available valves may be categorized into globe valves, butterfly valves, pressure relief valves, and so on. A schematic diagram of a typical pressure relief valve is shown in Figure 1.  
A relief valve is designed to open at a set pressure to protect a pressure vessel or other equipment. When the inlet pressure exceeds this set pressure, the relief valve opens and a portion of the gas is diverted through the valve system. As the fluid begins to move faster, its compressibility plays an increasingly important role, and it undergoes a significant change in density. If the pressure ratio across the constriction area exceeds a critical value, a choked flow with shock waves occurs [9], which causes additional aerodynamic noise. The flow passage through a valve can be considered as a simple constriction and expansion pipe in terms of its cross-sectional area variation, whereby the area is reduced to a minimum at the throat of the valve and is then increased. An orifice plate works by essentially the same principle [10], in that it reduces the local pressure by providing a mechanical constriction in the pipe, although a valve is much more sophisticated than an orifice. The high-speed flow through a pipe with an orifice can be simplified as a quasi-1D compressible flow if the cross-sectional area is smoothly varied: the velocity components normal to the axial direction of the pipe (in the \(x\) and \(y\) directions) are small when compared to the velocity in the axial (\(z\)) direction, such that the flow field can be assumed to be a function of \(z\) only. In other words, it can be assumed that the flow is uniform across the cross-sectional plane of the pipe, which is called a quasi-1D flow. In this quasi one-dimensional flow, internal flow can be predicted analytically by solving the relevant equations presented by [PERSON] [9]. As the pressure difference between the upstream and downstream regions increases, the mass flow rate also increases until the system reaches a critical condition, under which the mass flow no longer increases with an increase in the pressure difference. If the pressure difference increases further, the flow velocity in the throat remains constant at Mach 1. This is also called a choked condition [11]. When the outlet pressure is below the critical pressure, normal shock waves occur in the pipe, which are generated perpendicular to the flow direction. Generally, normal shock waves occur in a very thin region on the scale of \(10^{-7}\) m, and are characterized by sudden changes in the flow quantities [12]. In a steady inviscid adiabatic 1D normal shock wave problem, the flow quantities downstream of the shock wave can be calculated from the upstream flow properties, which are used as the inlet conditions. The changes in the pressure and velocity across the shock wave are expressed as follows:  
\[\frac{p_{2}}{p_{1}}=\frac{1}{\gamma+1}\left(2\gamma\,M_{1}^{2}-(\gamma-1)\right) \tag{1}\]  
\[M_{2}^{2}=\frac{(\gamma-1)\,M_{1}^{2}+2}{2\gamma\,M_{1}^{2}-(\gamma-1)} \tag{2}\]  
where \(p_{1}\) and \(p_{2}\) are the pressures before and after shock wave, respectively. Because the shock wave is an adiabatic and irreversible process, an energy loss occurs and the entropy increases. The entropy variation across the shock wave is expressed as follows:  
Figure 1: Schematic diagram of typical pressure relief valve. \[s_{2}-s_{1} =c_{p}\ln\frac{T_{2}}{T_{1}}-R\ln\frac{p_{2}}{p_{1}}\] \[=c_{p}\ln\left[\left\{1+\frac{2\gamma}{\gamma+1}(M_{1}^{2}\ -\ 1) \right\}\frac{2+(\gamma-1)M_{1}^{2}}{(\gamma+1)M_{1}^{2}}\right]\] \[-R\ \ln\left\{1+\frac{2\gamma}{\gamma+1}(M_{1}^{2}-1)\right\}\]  
The entropy variation can be expressed using the total pressure change as follows:  
\[s_{2}-s_{1}=-R\ln\frac{p_{0,2}}{p_{0,1}} \tag{4}\]  
In the case of a gas with \(T<1000\) K, the specific heat can be assumed to be constant. A perfect gas with a specific heat constant is referred to as a calorically perfect gas.