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:
- The necessary time to accomplish one rotor round is\({t}_{\text{round}}=\frac{60}{N}\)
- The apparent blade passing period of the row \((3-i)\)in the row\((i)\) is \({t}_{i}=\frac{{t}_{\text{round}}}{N_{3-i}}\)\(i=1,2\) , where \(N_{3-i}\) is the blade number of row\((3-i)\)
- The necessary time to cover geometrical coincidence (the space-time
period of two blade rows (stator and rotor)) is given by\({t}_{\min}=\frac{{t}_{\text{round}}\text{GCD}\left({N_{i},N}_{3-i}\right)}{{N_{i}N}_{3-i}}\)where \(\text{GCD}\left({N_{i},N}_{3-i}\right)\) is the greatest
common divisor of the two blade number \(N_{i}\) and \(N_{3-i}\)..
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.