In this section, the gradient-based optimization of the volute is presented. The optimization loop that is used within this work is shown in Figure 6. Starting from a baseline design defined by initial parameters
${x}_{0}$, the volute geometry is created, meshed and CFD calculations are run. The gradients evaluation of the objective function with respect to the grid points position (
$\frac{d\omega}{dY}$) is performed by means of a discrete adjoint solver [
11]. In parallel, grid sensitivities (
$\frac{dY}{dx}$) with respect to the design variables are computed using a complex step method [
14]. The chain rule is then applied to obtain the gradient of the performance quantities of interest with respect to the design variables (
$\frac{d\omega}{dx}$). This information is provided to a SQP optimizer [
15] that updates the design variables. The process is repeated until an optimal solution is found. The accuracy of the gradients calculation method has been proved in previous studies ([
8,
11,
15]).
Figure 6.
Gradient-based design optimization loop implemented within the CADO software.
2.5. Optimization Results
The optimization history is presented in Figure 7. A reduction of 14% in loss coefficient is achieved within 13 major iterations, over which 17 CFD evaluations and 13 adjoint calculations are required. The data are normalized by the baseline performance, for which the absolute loss coefficient is equal to 0.347. During the optimization process, the L2 norm of the gradients is reduced by two orders of magnitude as shown in Figure 8. At the final iterate, the L2 norm is relatively close to zero, suggesting that the optimizer has found an optimal solution to the optimization problem.
Figure 7.
Optimization history.
Figure 7.
Optimization history.
Figure 8.
Gradients L2 norm history.
Figure 8.
Gradients L2 norm history.
Another important performance parameter for a volute is the pressure recovery coefficient, which characterizes the static pressure rise relative to the kinetic energy available at the inlet, such as defined in Equation (
2). Its value goes from 0.61 in the baseline case, to 0.665 in the optimized one, which represents an increase of 9%. This is due to the increased mass flow as a result of the decreased losses.
To understand how shape modifications lead to loss reduction, it is important to recall the major loss mechanisms in a volute. The largest contributions to volute losses are caused by friction losses and non-isentropic deceleration. Additional losses can be generated indirectly through interactions with the impeller by circumferential pressure distorsion generated by the volute [
17]. A volute converts the radial influx of mass into a tangential stream through the tangentially increasing cross sections. If the volute is too large, the cross-sectional area increases more than needed for the given influx, and an inefficient diffusion takes place in the azimuthal direction. A direct consequence is a circumferential distorsion at the volute inlet, which will introduce unsteadiness in the impeller. Conversely, if the volute is too small, the flow is accelerated along the circumference, which leads to a destruction of the pressure rise that was achieved in the impeller and thus, extra losses as well. Hence, the most important parameters are those that control the cross-sectional growth of the volute in the azimuthal direction. Zhenzhong Sun et al. [
18] showed that the minimum volute radius and radius ratio distribution play a significant role in the compressor performance. In the present case, the minimum volute radius is decreased from 23 to 17.3 centimeters, as shown in
Table 2, while the radius ratio distribution is changed. One can also observe that the inner wall radius stays at its upper bound value. The wall fillet radius is reduced to 1.0 mm which corresponds to its lower range limit, and the tongue radius is increased from 5 to 5.8 mm.
Table 2.
DESIGN PARAMETERS EVOLUTION.
Table 2.
DESIGN PARAMETERS EVOLUTION.
Index |
Parameter Name |
Baseline |
Optimized |
Units |
1 |
Minimum volute radius |
0.023 |
0.017 |
[m] |
2 |
Wall height |
0.008 |
0.0074 |
[m] |
3 |
Wall fillet radius |
0.0025 |
0.001 |
[m] |
4 |
Wall radius |
0.195 |
0.195 |
[m] |
5 |
Inlet fillet radius |
0.001 |
0.0012 |
[m] |
6 |
Radius ratio point 2 |
1.4 |
1.22844 |
[-] |
7 |
Radius ratio point 3 |
1.6 |
1.45406 |
[-] |
8 |
Radius ratio point 4 |
1.7 |
1.72688 |
[-] |
9 |
Radius ratio point 5 |
1.75 |
2.06383 |
[-] |
10 |
Radius ratio point 6 |
1.9 |
2.22392 |
[-] |
11 |
Radius ratio point 7 |
2.1 |
2.26456 |
[-] |
12 |
Radius ratio point 8 |
2.3 |
2.35792 |
[-] |
13 |
Radius ratio point 9 |
2.5 |
3.08543 |
[-] |
14 |
Tongue radius |
0.005 |
0.0058 |
[m] |
Figure 9 shows a cut at a constant Y plane in the main scroll (at an azimuthal position of 270 degrees), with total pressure map being displayed. The reduction in the cross-sectional area is apparent. The central low pressure zone corresponds to losses due to the shear forces at the center of the highly rotational flow, also called swirl losses. This contribution to the total losses can hardly be minimized due to the parametrization of the volute, as it doesn’t allow for modifications of the profile shape and inlet height. By extracting cross-sectional flow data at different azimuthal positions, the evolution of the mass flow averaged tangential velocity and area averaged static pressure along the volute circumference is computed, as presented in Figure 10. Following a simplified loss prediction model [
19], it is possible to estimate the total pressure losses due to the drop in tangential velocity from the volute inlet to the center of each section, as presented in Equation (
3):
with
$\sigma =1$ for decelerating flows and
$\sigma =0$ for accelerating flows. The quantity
$\rho $ is the area averaged density and
${V}_{t,in}$ is the mass flow averaged tangential velocity at the inlet. These flow variables are equal to
$2.39\phantom{\rule{0.166667em}{0ex}}kg/{m}^{3}$ and
$151.5\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}m/s$ for the baseline design. For the optimized design,
${V}_{t,in}$ is equal to
$165\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}m/s$. As an example, by taking an azimuthal position of 200 degrees, one can compute that the total pressure losses are
$\Delta {p}_{tot,200}=325.3\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}Pa$ in the baseline geometry, while they are at zero in the optimized case as there is no deceleration of the flow. This can be verified at different azimuthal positions and shows that the optimized area distribution allows to reduce the losses arising from inefficient diffusion along the volute. In contrast, the flow is decelerated in the baseline case which results in pressure rise along the circumference, as illustrated in Figure 10b, and additional diffusion losses.
Figure 9.
Cross section in Y plane.
Figure 9.
Cross section in Y plane.
Figure 10.
Cross-sectional averaged flow data along the circumference.
Figure 10.
Cross-sectional averaged flow data along the circumference.
Figure 11 shows the static pressure distribution at a constant X-plane corresponding to the middle height of the inlet (i.e. X = 3mm). One can observe that the circumferential flow distribution benefits from the optimized shape with less distorsion compared to the initial design. To demonstrate the reduced level of distorsion at the inlet, the tangential velocity is extracted at a constant radius $R=165\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}mm$, that is one centimeter downstream of the volute inlet. The results are displayed in Figure 12. Again one can see the reduced flow distorsion close to the inlet, thanks to the optimized volute cross sectional area distribution. It can be shown that the radial velocity follows the same trend, as the inlet flow angle is imposed.
Figure 11.
Cross section in X plane.
Figure 11.
Cross section in X plane.
Figure 12.
Tangential velocity distribution along circumference.
Figure 12.
Tangential velocity distribution along circumference.
Finally, Figure 13 shows a close up on the tongue profile in a constant X plane. From this figure, one can deduce that the geometry is modified such that the tongue position is adapted to the flow angle, which results in lower losses. Indeed, the stagnation point is nicely located on the tongue in the optimized design, while it is shifted downstream inside the outlet diffuser in the baseline case, which leads to a larger flow separation close to the tongue wall, as shown by the velocity vectors. Similar observations can be found in [
20].
Figure 13.
Zoom on tongue area - Baseline vs Optimized design.
Figure 13.
Zoom on tongue area - Baseline vs Optimized design.