# Investigation on Unsteady Flow Characteristics of a SCO\({}_{2}\) Centrifugal Compressor
## 3 Computational Method
### Governing Equations  
The governing equations were derived from the conservation of mass, momentum, and energy. The general form of these equations can be written as [29]:  
\[\frac{\partial(\rho\phi)}{\partial t}+\mathrm{div}(\rho u\phi)=\mathrm{div}( \Gamma\mathrm{grad}\phi)+S \tag{1}\]  
The terms in the above equation are respectively transient, convective, diffusive, and source term. Where \(\phi\) is the universal variable, and \(\phi=1\) indicates continuity equation. When \(\phi\) is temperature, it represents energy equation, and when \(\phi\) is set as the velocity of an arbitrary direction, Equation (1) corresponds to being a momentum equation. In addition, \(\Gamma\) is generalized diffusion coefficient and \(S\) is generalized source term. Turbulence modeling is a process to make Reynolds averaged equations closed. Compared to the standard \(k\)-\(\epsilon\) turbulence model, RNG \(k\)-\(\epsilon\) is demonstrated to be more accurate by [PERSON] et al. [30], which appears high precision. Hence, the RNG \(k\)-\(\epsilon\) turbulence model was employed for the investigation in this paper. The transport equations of turbulent kinetic energy \(k\) and turbulent dissipation rate \(\epsilon\) respectively perform as follows [29]:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_{ i})=\frac{\partial}{\partial x_{j}}(\alpha_{k}\mu_{\mathrm{eff}}\frac{\partial k }{\partial x_{j}})+G_{k}+G_{b}-\rho\epsilon-Y_{M} \tag{2}\]  
\[\frac{\partial}{\partial t}(\rho\epsilon)+\frac{\partial}{\partial x_{i}}( \rho\epsilon u_{i})=\frac{\partial}{\partial x_{j}}(\alpha_{\epsilon}\mu_{ \mathrm{eff}}\frac{\partial\epsilon}{\partial x_{j}})+C_{1\epsilon}\frac{ \epsilon}{k}(G_{k}+C_{3\epsilon}G_{b})-C_{2\epsilon}\rho\frac{\epsilon^{2}}{k }-R_{\epsilon} \tag{3}\]  
where \(\alpha_{k}\), \(\alpha_{\epsilon}\), \(C_{1\epsilon}\), \(C_{2\epsilon}\), and \(C_{3\epsilon}\) are empirical constants. The abovementioned \(G_{k}\) is the production of turbulence kinetic energy caused by averaged velocity gradient, \(G_{b}\) is the production of turbulence kinetic energy caused by buoyancy, and the dilatation dissipation term \(Y_{M}\) can exert the compressibility effect. \(G_{k}\), \(G_{b}\), and \(Y_{M}\) respectively perform [31]:  
\[G_{k}=u_{t}(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{ \partial x_{i}}) \tag{4}\]  
\[G_{b}=\beta g_{i}\frac{\mu_{t}}{\mathrm{Pr}_{t}}\frac{\partial T}{\partial x_ {i}} \tag{5}\]  
\[Y_{M}=2\rho\epsilon\mathrm{M}_{t}^{2} \tag{6}\]  
where \(\beta\) is the coefficient of thermal expansion, \(\mathrm{Pr}_{t}\) is the turbulence Prandtl number, and \(\mathrm{M}_{t}\) is the turbulence Mach number. When analyzing the unsteady flow characteristics, the transient rotor stator method in ANSYS-CFX was applied to the interface between the impeller and the diffusers. Virtual and true time step were introduced to the time discretization in order to avoid the restriction of time step length in explicit scheme and the complication transform of implicit scheme [32]. \[\frac{U^{q+1}-U^{q}}{\Delta\tau}+\frac{3U^{q}-U^{n}+U^{n-1}}{2\Delta t}=-R(U^ {q}) \tag{7}\]  
where the time level is denoted by the superscript \(n\), \(q\) is the symbol of backward difference operator, and \(R\) is the residual. In order to describe the practical property of the working fluid SCO\({}_{2}\) in the designed centrifugal compressor, a modified Soave Redlich Kwong equation of state in CFX database MATERIALS-redkw was adopted [33]. It has been proven to be precise and effective by [PERSON] et al. [25]. In the database, \"CO2 RK\" represents the working fluid carbon dioxide. With the restriction of pressure and temperature, it is able to be limited in the supercritical region, thus validly simulating the property of SCO\({}_{2}\) in the designed compressor [34]. The explicit state equation is shown below in Equation (8).