# Characterisation of the Pump-Suction Flow Field of Antarctic Krill and Key Influencing Factors
## 2 Materials and Methods
### Turbulence Modelling  
The \(k\)-\(\omega\) SST model is mainly used to deal with compressible free shear flow problems and can show the flow of a near-wall free stream well [43]. Due to its low dependence on the far-field conditions and the high accuracy of the near-wall simulation, and the corresponding low mesh number requirement, the model is considered suitable for the simulation of flow separated by a blunt structure and has been widely used in recent years.  
\[\left\{\begin{array}{c}\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_ {i})}{\partial x_{i}}=\frac{\partial}{\partial x_{i}}(\Gamma_{k}\frac{\partial k }{\partial x_{i}})+G_{k}-Y_{k}+S_{k}+G_{b}\\ \frac{\partial(\rho\omega)}{\partial t}+\frac{\partial(\rho\omega u_{i})}{ \partial x_{i}}=\frac{\partial}{\partial x_{i}}(\Gamma_{k}\frac{\partial k }{\partial x_{i}})+G_{\omega}-Y_{\omega}+S_{\omega}+G_{\omega\psi}\\ \Gamma_{k}=\frac{\omega}{\mu}+\frac{\mu_{i}}{\sigma_{\psi}}\\ \Gamma_{\omega}=\mu+\frac{\mu_{i}}{\sigma_{\psi}}\\ \mu_{i}=\alpha^{*}{}_{\infty}(\frac{\theta_{i}/3+\beta\omega/\mu_{i}\omega R_ {k}}{1+\rho k/\mu_{i}\omega R_{k}})\frac{\rho k}{\omega}\end{array}\right. \tag{3}\]  
where \(G_{k}\) and \(G_{\omega}\) are the turbulent kinetic energy generated by the mean velocity and the generation of a specific dissipation rate, respectively; \(\Gamma_{k}\) and \(\Gamma_{\omega}\) are the effective diffusivities of \(k\) and \(\omega\), respectively, \(Y_{k}\) and \(Y_{\omega}\) are the dissipations due to the turbulence of \(k\) and \(\omega\); \(S_{k}\) and \(S_{\omega}\) are the source terms; \(G_{b}\) and \(G_{\omega\psi}\) are the diffusivity of \(k\) and \(\omega\), which is the effect of buoyancy on turbulence. \(\
u_{k}\) and \(\sigma_{\omega}\) are the turbulence Prandtl numbers for \(k\) and \(\omega\); \(\mu_{i}\) is the turbulence viscosity; \(\alpha^{*}\) is a coefficient whose value must be reduced when performing turbulence analysis at low Reynolds numbers. The purpose of reducing the value of the coefficient is twofold; firstly, it enables the calculation of turbulent viscosity, and secondly, it ensures that a low Reynolds number model with \(\alpha^{*}=\alpha^{*}{}_{\infty}=1\) is realised at high Reynolds numbers.  
The \(k\)-\(\omega\) SST model combines the advantages of the \(k\)-\(\omega\) model in the near-wall region with those of the \(k\)-\(\varepsilon\) model in free flow, rendering the model more adaptable to the entire flow field. This hybrid characteristic enables the \(k\)-\(\omega\) SST model to provide more accurate simulation results when dealing with devices such as krill suction pumps, which have complex geometries and flow characteristics. Furthermore, the SST model incorporates the transport of turbulent shear stresses in the turbulent viscosity calculation, thereby enhancing the prediction accuracy of the separated flow under complex flow conditions, such as the presence of a reverse pressure gradient or flow separation. In the context of the suction krill pumping system examined in this study, given the presence of the centrifugal force, it is deemed more appropriate to employ the \(k\)-\(\omega\) SST turbulence model for the management of flow within the pump. Consequently, the \(k\)-\(\omega\) SST turbulence model is employed in this calculation to ensure the study's accuracy.