2.1 Solver
The flow field through this axial fan stage extended upstream and downstream is solved by means of the code ANSYS-CFX [25] which uses the finite volume method. The governing equations are integrated over each control volume defined by joining the centres of edges and the element centres surrounding each node. Gauss divergence theorem is applied to convert the volume integrals involving divergence and gradient operators to the surface integrals. Volume integrals are discretized within each element sector and accumulated to the control volume to which the sector belongs, whereas the surface integrals are discretized at the central integration points of each surface segment. After discretizing the volume and surface integrals, the integral equations become:
\(V\left(\frac{\rho-\rho_{0}}{\Delta t}\right)+\sum_{\text{ip}}{{\dot{m}}_{\text{ip}}=0}\)(4)
\(V\left(\frac{\rho U_{i}-\rho^{0}U_{i}^{0}}{\Delta t}\right)+\sum_{\text{ip}}{{\dot{m}}_{\text{ip}}{(U_{i})}_{\text{ip}}=\sum_{\text{ip}}{{(P\Delta n_{i})}_{\text{ip}}+\sum_{\text{ip}}{\left(\mu_{\text{eff}}\left(\frac{\partial U_{i}}{\partial x_{j}}+\frac{\partial U_{j}}{\partial x_{i}}\right)n_{j}\right)_{\text{ip}}+\overset{\overline{}}{S_{U_{i}}}V}}}\)(5)
\(V\left(\frac{\rho\varphi-\rho^{0}\varphi^{0}}{\Delta t}\right)+\sum_{\text{ip}}{{\dot{m}}_{\text{ip}}{(\varphi)}_{\text{ip}}=\sum_{\text{ip}}{\left(\Gamma_{\text{eff}}\frac{\partial\varphi}{\partial x_{j}}n_{j}\right)_{\text{ip}}+}}\overset{\overline{}}{S_{\varphi}}V\)(6)
Where \({\dot{m}}_{\text{ip}}={(\rho U_{j}n_{j})}_{\text{ip}}\) ,\(t\) is the time step, \(n_{j}\ \)is the discrete outward surface vector, ip denotes an integration point and (° ) refers to an old time level. Where \(\mu_{\text{eff}}=\mu+\mu_{\tau}\) and\(\Gamma_{\text{eff}}=\Gamma+\Gamma_{\tau}\) are successively the effective viscosity and effective diffusivity. The solution field parameters are stored at the mesh nodes and by using finite-element linear shape function in terms of parametric coordinates\(\varphi=\sum_{i=1}^{N_{\text{node}}}{N_{i}\varphi_{i}}\) the approximation of all flow properties, gradients and diffusion terms at the integration points are obtained. To prevent the pressure field oscillations as a result of the non-staggered collocated grid arrangement, a coupled solver solves the flow equations as a single system. The high resolution scheme is used for the advection terms in the equations of momentum and the turbulent model casted in the form\(\varphi_{\text{ip}}=\varphi_{\text{up}}+\beta\nabla\varphi\overrightarrow{r}\)[25], where \(\varphi_{\text{up}}\) is the value at upwind node and\(\overrightarrow{r}\) is the vector from the upwind node to the point ip. For the high resolution β is computed to be less or equal to 1.
For the steady state simulations the solver applies a false time step as a means of under-relaxing the equations as they iterate towards the final solution. This can be adjusted as an internally calculated physical time scale based on the domain geometry, boundary conditions and flow conditions, or a local time scale factor in different regions and finally a fixed value over the entire flow domain equal to\(\frac{1}{\omega}\) (\(\omega\) rotational speed). This last option was selected in the steady flow computations based on the high resolution scheme for the advection terms while the convergence residual was set at a value of 10-6.
For unsteady flow computations the high resolution scheme [25] operates as a second order backward Euler scheme (shown as below is robust and implicit) wherever and whenever possible and reverts to the first order backward Euler scheme when is required to maintain a bounded solution.
\(\frac{\partial}{\partial t}\int_{V}{\rho\varphi dV=V}\frac{1}{t}\left(\frac{3}{2}\left(\text{ρφ}\right)-\left(\text{ρφ}\right)^{0}+\frac{1}{2}\left(\text{ρφ}\right)^{00}\right)\)(7)
During the computations, at each time step the convergence is controlled by the minimum and maximum number of iterations, but the maximum number of iterations per a time step may not always be reached if the residual target level is achieved first. The solver performs a number of 15 iterations for each time step to reach a residual inferior to a value of 10-5. The transient rotor/stator interface is used to account for the transient interactions between the IGV vanes and rotor blades-row.
The time step has to be small enough to get the necessary time resolution depending on the speed of rotation. However, to have a good resolution of unsteady RSI the time step is chosen to satisfy all the time periods characterizing the aerodynamic operation of this axial fan stage. For two blade-rows (vanes and rotor blades) of N1vanes and N2 blades, the different characteristic time scales are estimated as follows:
For the IGV and rotor blades\(T_{\min}=\frac{T_{\text{round}}\text{GCD}\left(19,11\right)}{19*11}=0.00478{t}_{\text{round}}\text{.\ }\)When the fan operates at the nominal point (N= 6000 rpm, m= 5.06 kg/s)\({t}_{\text{round}}=\frac{60}{N}=0.01\text{\ s}\), thus\(T_{\min}=47.84\ \mu\text{s\ }\) equivalent to 1.72 deg and represents the upper limit of the time to be used. However, to resolve the high frequencies, the computational time step should correspond to a rotation less or equal 1.5 deg as stated in some references such as [26], and thus the time step was set at \(\ 41.66\ \text{μs}\ \)and for one round the total time is equal to\(\ 10\ ms\).
The transient relative motion on each side of the general grid interface (GGI) connection is simulated and the interface position updated at each time step.