3.1. Convergence Analyses
In this section the convergence characteristics of the implemented method, both in terms of iterations and discretization resolution, are investigated with the objective of verifying the stability of the method and establishing the grid resolution requirements.
The iterative convergence behavior of the inverse design method is illustrated in
Figure 6 for a representative Francis turbine case, using a mesh resolution that guarantees negligible discretization error.
Figure 6a presents the
-norm of the variations
and
between successive iterations as a function of the iteration number. The convergence is mostly monotonic, except for a small bump close to the tenth iteration after which the full periodic velocity is considered by having
. About 25 iterations are required to achieve a converged solution; a faster convergence is possible for this particular case by setting the relaxation factor
, although the more conservative adaptive approach is preferred in order to ensure convergence for all cases.
Figure 6b evidences the monotonic convergence of the camber surface angle
f at five points; the results are normalized by the respective angle value at the last iteration,
. As expected, the points closer to the leading edge (
) change significantly less than the points further down, in relation to their initialization value
, given that the camber surface angle at the leading edge is fixed by the stacking condition selected.
The convergence of the method with respect to the mesh resolution has been studied in detail. The grid resolution
R is defined such that the number of nodes in the spanwise direction is
. For reference, the mesh illustrated in
Figure 3 has
, implying 65 nodes across the span.
A representative Francis turbine case is used for the grid convergence analysis. The pressure difference across the blade as well as the blade angle are selected to assess the convergence, given that these variables depend on the mean and periodic flow velocity components, as well as on the camber surface angle f. A total of 25 points uniformly distributed over the blade and four different resolutions are considered.
Figure 7a presents the relative error for
with respect to the corresponding values computed on the finest grid,
. The relative error for each of the 25 points analyzed is illustrated by thin lines, whereas the bold lines represent the average relative error, with error bars computed from the standard deviation. A clear convergence of the solution is evidenced, with the relative error monotonically decreasing with the mesh resolution. For the coarsest mesh the average relative error for the pressure difference is about
, whereas for the blade angle it is less than
.
Richardson extrapolation is used to compute the average discretization error. Any given solution metric can be described by the general model
, where
is the discretization size,
is the convergence rate,
is a constant and
is the converged metric [
18]; as such, the discretization error is equal to
. This error model has been independently fitted to the 25 points analyzed, each of which includes the resulting
and
at four resolutions.
Figure 7b presents the averaged results of the Richardson extrapolation analysis. The average convergence rate computed is
, whereas the average discretization error for the coarsest resolution is
. Note that the error presented in
Figure 7b is relative to the finest resolution, which is not necessarily converged; on the contrary, the error presented in
Figure 7b is with respect to the converged solution with
, implying that it truly is the magnitude of the discretization error.
Even though all the operators in the present implementation are discretized with second-order accurate approximations, the effective convergence rate is
, below the formal 2.0. This behavior is probably due to the fact that the number of harmonics
N that can be resolved is proportional to the mesh resolution: the source terms of the governing equation for the potential function of the periodic flow, Equation (
11), are imaginary exponential functions of frequency proportional to
, with
, and therefore the highest order harmonic that can be resolved is subject to the Nyquist–Shannon sampling criterion. If higher order harmonics are used, the source terms suffer from aliasing and inject spurious information into the periodic velocity solution. In short, since the number of harmonics considered increases with the mesh resolution, the convergence rate is slightly below 2.0.
The time to solution, averaged over five runs, for each of the four resolution levels is presented in
Table 1. The computations are performed on a single i7-4810MQ CPU core running at 3.8 GHz on Ubuntu Linux 16.04 LTS. It is evidenced that the time to solution increases very significantly with
R, given that the number of equations being solved increases with the resolution, as explained above. Although these data are representative of the code performance, the time to solution does depend on other factors such as the turbine geometry: low specific speed runners with long blades tend to have more grid nodes for a given resolution level, since
R only depends on the number of nodes along the span, and are therefore relatively more expensive to compute.
Based on these results, it can be concluded that a resolution can be used to efficiently explore the design space at a computational cost of about 10 s per simulation, whereas a resolution can be used to compute the definite mesh-independent results in just over one minute. A considerable speed-up is expected if the code is ported to C++ or another high-performance programming language.
3.2. Implementation Validation
The inverse design method implementation is validated on a Francis turbine case characterized by a specific speed
and a hydraulic power of
MW. The blade loading and blade thickness distributions specified are similar to the ones presented in
Figure 2, whereas the mesh resolution is selected as
to ensure converged results; the corresponding meridional channel discretization is illustrated in
Figure 3. The number of blades is set to 16, and a quadratic camber surface angle distribution is specified at the leading edge as the stacking condition, with a
of 8.2° at the shroud with respect to the hub. This stacking condition corresponds to an average leading edge slant of
, where
is the leading edge radius at the shroud and
is the leading edge height. A render of the resulting runner is shown in
Figure 8, whilst
Figure 9 illustrates the distributions of the blade camber surface wrap angle
f and the blade angle
.
The runner geometry output by the method is evaluated using the commercial CFD software ANSYS CFX 19.2, where the Navier–Stokes equations coupled with the turbulence model are solved in steady-state. Only one blade-to-blade channel is simulated. A mesh with 8.54 million elements and an average dimensionless wall distance over the blade of is employed. The velocity orientation and magnitude is prescribed at the inlet, and an average static pressure of 0 Pa is imposed at the outlet.
This simulation configuration is commonplace in the field of hydraulic turbomachines, both in academia and in industry, and has been shown to provide numerical results that are in good agreement with experiments [
19,
20,
21,
22,
23], at least at the best efficiency operating conditions. Using this well-established CFD methodology is adequate to validate the correctness and physical soundness of the solution obtained with the implemented inverse design method, although it is acknowledged that both numerical solutions will have some degree of discrepancy with respect to physical reality.
3.2.1. Qualitative Validation
First the velocity fields are used to qualitatively validate the output of the inverse design method implementation. On the following figures, even though the color scale is not identical in
matlab and CFX, the same variable range is enforced and both color bars are presented. To further aid in the assessment of the results, the meridional projections of the blade leading and trailing edges, as well as of the hub and shroud, are superimposed on the CFX flow fields. There is a significant agreement for the mean flow velocity magnitude
, as evidenced in
Figure 10. Both the main flow features and the velocity magnitude are well approximated by the present method, compared to the fully-3D Navier–Stokes solution, in spite of the underlying inviscid and axisymmetric mean flow assumptions. The same can be said about the radial and axial mean velocity components, presented in
Figure 11 and
Figure 12, respectively.
The greatest discrepancy is evidenced in the flow near the shroud, where the mean axial velocity is somewhat overpredicted by the present method. This behavior is most likely a consequence of the inviscid flow assumption. Another divergence occurs for the mean radial velocity prediction near the intersection between the leading edge and the shroud, where the magnitude of the flow acceleration in this high-curvature region is slightly underpredicted by the present method. Overall, a good agreement between both approaches is evidenced.
The comparison of the mean tangential velocity
is presented in
Figure 13. This velocity component illustrates the manner in which the inflow angular momentum is extracted by the runner. As such, the notable agreement achieved validates the proposed methodology by demonstrating that the designed runner accomplishes an essentially complete extraction of the flow angluar momentum. Moreover, not only is the momentum extraction thorough, it is also performed according to the prescribed blade loading distribution, as revealed by the equivalent gradient of
on both solutions.
3.2.2. Quantitative Validation
For a quantitative validation of the inverse design method implementation, a comparison of the pressure distribution on the blade along five representative span locations is performed.
The pressure difference
across the blade is presented in
Figure 14. As revealed by Equation (
13), the pressure difference is proportional to
, i.e., proportional to the blade loading, and to the meridional projection of the relative flow velocity, which is greatest towards the shroud. Both of these factors are encompassed in the pressure difference distributions obtained.
There is an overall agreement between the present method results and those calculated with the Navier–Stokes solver, although there are also some noticeable discrepancies that are probably due to a combination of factors, including some error in the computed flow velocity, the assumptions inherent to Equation (
13), and the simplifications of the present model. Perhaps the assumption of infinitely thin blades has the greatest impact: it is likely responsible for the underprediction of the pressure difference in the first
of the chord, i.e., for
; the method then tends to compensate for this by an overprediction of the pressure difference for
, especially near the shroud. Based on the five profiles presented in
Figure 14, the average relative error of the pressure difference computed with the present method amounts to
, whereas the relative error in the computed torque is equal to
. Since the torque is calculated by integration of the pressure difference, a smaller error is expected because, to a certain extent, regions where the pressure difference is underpredicted will cancel out the overcontribution of regions where it is overpredicted. Moreover, at least part of the
torque overprediction by the current method is explained by the neglect of all energy losses in the flow.
A comparison of the pressure profiles over the blade is presented in
Figure 15. Whereas in the Navier–Stokes solver the pressure is actually computed on each of the blades surfaces, in the current implementation Equation (
14) is used to estimate the average meridional pressure
, and then the computed pressure difference across the blade is symmetrically superimposed:
.
Although there is some agreement in the pressure profiles over the blade between the inverse design method implementation and the Navier–Stokes solution, there are at least two significant discrepancies. The first one has to do with the leading and trailing edges: On the one hand, as already discussed, the assumption of infinitely thin blades hinders the rapid build-up of the pressure at the leading edge, and renders the formation of a stagnation point impossible. On the other hand, there is a pressure jump at the trailing edge near the shroud that is not supposed to be there according to the simplified model. Two possible explanations for that discrepancy are the accumulation of viscous effects or the influence of secondary structures, although no conclusive explanation has been found.
The second significant discrepancy has to do with the repartition of the pressure difference between the suction and pressure sides. It is evident that the simplified approach used to calculate the pressure on each surface, namely a symmetric repartition of the pressure difference, is only approximative. The Navier–Stokes solution suggests that, relative to the mean meridional pressure, the pressure drop on the suction side is greater than the increment on the pressure side.
The energetic efficiency of the runner, computed using the Navier–Stokes solution, is equal to . This value does not consider the energy losses in the guide vanes, which were not modeled, and by definition it does not include the disk friction or leakage discharge losses either. Acknowledging this, the calculated efficiency is considerably high, implying that the implemented inverse design method is actually capable of providing a good hydraulic design that follows the blade loading distribution prescribed as input.