Introduction

The value of the blade solidity, i.e., the ratio c/s between the chord and the pitch, strongly affects the cost and the performance of axial turbines. To reduce the weight and the cost of the machine, high blade spacing, and low solidity values are desired. However, efficient flow turning requires a sufficient number of blades. For low solidity values, the blade loading is high, causing poor fluid guidance and, therefore, flow separation over the rear suction side. Conversely, for high solidity values, the large cascade wetted area leads to considerable viscous dissipation. In summary, there is an optimum value of the axial solidity, defined as σX=cx/s, being cx the axial chord, that minimizes the passage losses. In axial turbines operating with air or steam, this value can be estimated using the empirical correlations proposed by Zweifel (1945) and Traupel (1966). The use of the Zweifel criterion is well documented also in recent literature (Giuffré and Pini, 2021). Although the Zweifel correlation is still largely used to estimate the solidity of air, gas, or steam turbines, its accuracy has not been thoroughly assessed in the case of turbines operating with more molecularly complex working fluids. Examples of such fluids are organic compounds, e.g., fluorocarbons, hydrocarbons, or siloxanes, used as working fluids in organic Rankine cycle (ORC) power systems (Colonna et al., 2015; Knowledge Center on Organic Rankine Cycle (KCORC), 2022). These fluids are characterized by large molecular complexity and relatively high critical temperature. Turbines of ORC systems operate with comparatively higher stage expansion ratios than those of more conventional turbines. Moreover, part of the expansion typically occurs in thermodynamic states close to that of the critical point, therefore the flow is affected by non-ideal effects. The magnitude of non-ideal effects in dense vapor flows can be assessed by evaluating the variation of the generalized isentropic pressure-volume exponent (Kouremenos and Antonopoulos, 1987), defined as

where γ is the ratio of the specific heats. If the fluid is in the dense vapor state, the γpv value changes over the expansion process and the internal flow field substantially depart from that characterizing turbo-machines operating with steam or air. For example, if the fluid is in the dense vapor state, the intensity of shock waves and expansion fans is quantitatively different from that characterizing ideal gas expansions (Romei et al., 2020). Moreover, previous research works (Giuffré and Pini, 2021; Tosto et al., 2021) showed that compressibility effects in turbines operating with the fluid at least partially in the dense vapor state can be enhanced or mitigated depending on whether the γpv value exceeds that of γ evaluated for ideal gas states (Baumgärtner et al., 2020; Romei et al., 2020).

Moreover, the non-ideality of the flow largely affects the loss breakdown, and, consequently, the blade design. All loss sources, i.e., those associated with viscous effects in boundary layers, with shocks, and with mixing at the blade trailing edge, depend on the thermodynamic state of the fluid and the level of flow compressibility (Giuffré and Pini, 2021). Design parameters, such as the optimal solidity, thus arguably differ from those that would be obtained by applying existing guidelines (Wilson, 1987; Giuffré and Pini, 2021). To the authors’ knowledge, no design guideline for the selection of the optimal solidity of turbomachinery operating with non-ideal compressible flows is documented in the literature. The only study on the accuracy of the Zweifel correlation in the case of gas turbines is that of Doughty et al. (1992), who performed an experimental campaign on transonic stator cascades. They found that the Zweifel loading coefficient estimated from their experimental data was significantly higher than the value commonly used to design high-pressure turbine nozzles.

Furthermore, both the Traupel and the Zweifel correlations take into account only the passage loss, while the effects of either wake mixing or shock losses are not considered. As a consequence, the value of the optimal solidity prescribed by these correlations for axial turbines might lead to a sub-optimal design even in the case of steam, air, or gas turbines.

The focus of this study is an investigation of the influence of compressibility, fluid molecular complexity, and thermodynamic non-ideality on the optimal solidity of axial turbine cascades. The research documented in this paper is a first step towards establishing guidelines for the selection of the optimal solidity for the preliminary design of turbomachines operating with non-ideal compressible flows. Based on the approach of Denton (Denton, 1993) and on Coull and Hodson method (Coull and Hodson, 2013), a first-principle reduced-order model (ROM) not requiring the use of empirical coefficients was developed to estimate passage losses as a function of axial solidity. The resulting charts provide the value of optimal solidity as a function of cascade flow angles. Numerical simulations of the flow through two exemplary turbine blades were performed to assess the influence of both flow compressibility and axial solidity on losses and to verify the accuracy of the ROM. Optimal solidity values calculated by taking into account either passage losses alone or the overall loss within the domain are compared with the ones resulting from both the ROM and Zweifel correlation. The influence of the working fluid and its thermodynamic state, flow compressibility, and solidity value on both the boundary layer state at the blade trailing edge and the base pressure is evaluated. Models to evaluate the mixing and the passage losses in the compressible flow regime as a function of the axial solidity are developed and discussed.

The paper is structured as follows. Section 2 describes the approach used to study the effect of solidity on the performance of a turbine cascade. The setup of the computational fluid dynamics (CFD) simulations performed on two representative turbine cascade geometries is also presented. In Section 3, two physics-based reduced-order models (ROMs) for the estimation of the optimal solidity in turbine cascades operating with non-ideal flows, based on the incompressible and compressible flow assumption, respectively, are introduced and described. Section 4 offers a comparison between the results obtained from the CFD and those obtained from the ROMs. An overview of the limitations of each model is provided, concluding that the models are unsuitable for design purposes. Based on the CFD results, the influence of the flow field in the proximity of the blade trailing edge and in the mixing region on the optimal solidity is also discussed. Finally, Section 5 lists the main conclusions drawn from this study and outlines possible next steps.

Effect of solidity on cascade performance

The effect of solidity on the fluid dynamic performance of turbine cascades can be investigated, either numerically or experimentally, with three different methods: (1) by changing the blade solidity and optimizing its shape for minimum loss at each operating condition; (2) by changing the solidity for a fixed blade shape, but keeping constant the throat-to-spacing ratio for given operating conditions, thus varying the stagger angle of the cascade; (3) by solely changing the solidity, while keeping the blade shape fixed, and varying the operating conditions. This study is numerical and based on the second method. This method enables one to arguably achieve aerodynamic similarity regardless of the solidity value and without re-designing the blade. Following the same approach proposed by Doughty et al. (1992), the blade is rotated about the trailing edge to keep the gauge angle constant, regardless of the solidity value, see Figure 1. The gauge angle χ is defined as

Figure 1.

Control volume for the estimation of the mixing losses downstream of the blade trailing edge.

where o denotes the throat length.

In turbine cascades, passage losses are not the only contribution to the overall loss. The solidity can arguably have an effect also on the mixing process occurring downstream of the blade trailing edge which, in turn, contributes to the overall irreversible entropy generation across the cascade. As pointed out by Denton (1993), mixing losses depend on the base pressure coefficient, as well as on the displacement and momentum thickness at the trailing edge of the boundary layers developing on the pressure and suction sides. For a compressible flow, mixing losses are also a function of the kinetic energy thickness of the boundary layer. With reference to the control volume shown in Figure 2, where the subscript e denotes the station at which the flow is completely mixed-out, the mass, axial momentum, tangential momentum, and energy conservation equations for the mixing process occurring downstream of a turbine cascade read

Figure 2.

Geometric rotation of the blades about the trailing edge used to obtain the value of optimal solidity for (see nomenclature of Table 1) the (a) iMM-Kis3 and (b) Aachen turbine blades. Solid lines denote the original geometries, dashed lines are the rotated geometries. The throat length (orange: original geometry, dashed blue: rotated geometry) is varied accordingly to the solidity value to keep the gauge angle constant.

where pt2 is the stagnation pressure at the exit of the vane. For an incompressible flow, an analytical equation for the loss within the domain can be obtained by simplifying Equations 3–5. The total pressure-based loss coefficient is (Denton, 1993)

From these equations, it emerges that the base pressure coefficient and the boundary layer integral quantities strongly influence the mixing process and associated losses, ultimately affecting the optimal solidity value.

Setup of the CFD simulations

To assess the influence of the axial solidity σx on the fluid-dynamic losses, we performed a set of steady-state Reynolds Averaged Navier-Stokes (RANS) simulations of the flow around two blades characterized by different geometrical features. In agreement with the established practice of evaluating the profile losses, to whom the solidity strongly depends, at the blade midspan, two-dimensional computations are deemed sufficient for the scope of this research. Figure 3 shows the computational domain of the two blade geometries. The first blade geometry, hereafter referred to as iMM-Kis3, is the mid-span section of the turbine stator also considered in the study documented in Giuffré and Pini (2021). This turbine stage has been designed to operate with hexamethyldisiloxane, commonly referred to as siloxane MM, at inlet total temperature and pressure ensuring a compressibility factor of Z∼1, i.e., in dilute gas conditions. In this analysis, only the fluid-dynamic performance parameters calculated at the mid-span section have been considered. The baseline design of the stator consists of 42 blades featuring an axial solidity value of σx,ref=0.73. This value has been calculated using the Zweifel criterion in the preliminary design phase, see Giuffré and Pini (2021). The second blade geometry is the mid-span section of the rotor of the so-called Aachen turbine (Stephan et al., 2000), which is representative of highly-loaded cascades. This is the case of, e.g., axial turbines of organic Rankine cycle power plants. The rotor of the Aachen turbine was designed to operate with air at a rotational speed of 3,500 rpm and a mass flow rate of m˙=6.8kg/s. The baseline design consists of 41 blades featuring an axial solidity value of σx,ref=1.3. Table 1 lists the main geometrical specifications of both blades. Compared to the iMM-Kis3 blade, the Aachen blade is front-loaded, i.e., characterized by a larger thickness in the proximity of the leading edge, whereas the iMM-Kis3 is aft-loaded. Moreover, in the Aachen blade, the extension of the straight part of the rear suction side is longer, the trailing edge is thicker, and the leading edge is more rounded than in the iMM-Kis3 blade.

Figure 3.

Computational grid for (a) the iMM-Kis3, an ORC vane expanding MM in the ideal gas state, and (b) the Aachen turbine blades. The axial solidity is the value at design conditions, i.e., σx,ref. Only one of every four grid lines is shown.

Table 1.

Key features of the iMM-Kis3 and of the Aachen turbine blades.

	c (m)	cx (m)	γ (°)	s (m)	α1	αm (°)	t (mm)	Mout,d
iMM-Kis3	0.0205	0.0126	50.72	0.0172	0	68.84	0.054	1.3
Aachen	0.0634	0.0543	31.04	0.0418	28.8	71.41	1.405	0.3

[i]  γ, αm, t, and Mout,d denote the blade stagger angle, the blade metal angle at the trailing edge, the trailing edge thickness, and the design outlet Mach number, respectively. For the Aachen blade, Mout,d refers to the relative Mach number at the rotor outlet.

To achieve constant χ for all axial solidity values, both blades are rotated around the trailing edge. Figure 1 illustrates the geometrical rotation of both blades. The inlet and the outlet of both domains have been placed ∼1.5cx upstream of the leading edge and ∼3.5cx downstream of the trailing edge, respectively to avoid upstream effects and to let the flow mix for several chord lengths downstream of the blade. Three sets of simulations per blade are considered, each characterized by a different working fluid: for the simulations denoted with iN₂, the working fluid is nitrogen, a compound formed by simple molecules, while for those denoted with iMM and niMM, the working fluid is siloxane MM, an organic compound used as the working fluid in several ORC turbogenerators. The inlet total temperature and pressure of the iN₂ and iMM test cases are prescribed to ensure that the compressibility factor at the inlet is equal to one (ideal gas), while the inlet thermodynamic state of the niMM test case is that of a dense vapor, i.e., of a fluid operating in the proximity of the vapor-liquid saturation line. Table 2 lists the boundary conditions for each set of simulations.

Table 2.

Setup of the numerical test cases. γ∞ denotes the ratio of specific heats for v→∞.

iMM-Kis3	Mmol (kg/kmol)	γ∞	pt1 (bar)	Tt1 (K)
iN2	28	1.4	15	473.15
iMM	162.38	1.025	9.66	534.26
niMM	162.38	1.025	25.1	542.04
Aachen	Mmol (kg/kmol)	γ∞	pt1 (bar)	Tt1 (K)
iN2	28	1.4	15	473.15
iMM	162.38	1.025	9.66	534.26
niMM	162.38	1.025	25.1	542.04
iMM-Kis3	Re2,is⋅106	βts,1	βts,2	βts,3
iN2	1.75−2.50	1.2	2	2.6
iMM	4.47−6.29	1.2	1.7	2.2
niMM	12.1−17.63	1.075	1.5	2.2
Aachen	Re2,is⋅106	βts,1	βts,2	βts,3
iN2	5.38−7.63	1.2	2	2.6
iMM	15.89−18.85	1.15	1.7	2.2
niMM	37.26−50.91	1.075	1.5	2.2

[i] The Reynolds number is computed using the blade chord c and the isentropic outlet conditions as reference. The last three columns list the values of three total-to-static expansion ratios investigated in each case.

To investigate the influence of compressibility on the optimal solidity value, three different values of the total-to-static expansion ratio βts are considered for each case. The first βts value leads to subsonic downstream Mach numbers (Mout∼0.5, Mout being the Mach number at the outlet of the domain), the second one leads to transonic flow (Mout∼1), while the third one is set to achieve supersonic flow (Mout∼1.2). For each value of βts, 16 different values of the axial solidity, ranging from σx=0.5 to σx=1.6 and from σx=0.6 to σx=1.6 for the iMM-Kis3 and the Aachen blades, respectively, are considered.

The domain is meshed with quadrilateral elements using a commercial program for turbomachinery blade meshing (ANSYS, 2019). A grid is generated for each value of the axial solidity σx, resulting in 8 different meshes for each of the two blades. Cell clustering is introduced near the blade walls to guarantee y+<1. Figure 4 shows the results of the mesh sensitivity analysis for the baseline iMM-Kis3 blade (σx,ref=0.73). The graph shows that the deviation in the value of the overall entropy loss coefficient ζs, defined as

Figure 4.

Variation of the overall dissipation coefficient ζs (Equation 9) as a function of the number of grid cells for the iMM-Kis3 test case.

(9)

ζs=T2(s2−s1)ht,2−h2.

between the 320 k and the 640 k cells meshes is 0.1%. For this study, the 320 k cells mesh ensures the best compromise between accuracy of the results and computational cost. All the mesh parameters, e.g., the growth rate of the cells in the proximity of the walls or the average cell size, are fixed and equal to those used for the baseline mesh also for the remaining 7 computational domains of the iMM-Kis3 blade, which differ only in the value of the solidity and the related geometry rotation. The same mesh parameters have also been fixed for the Aachen blade mesh: at σx,ref=1.3, the resulting mesh size consists of 440k cells.

A commercial CFD software (ANSYS, 2019) has been used to simulate the flow through the turbine cascades. The SST k−ω model (Menter, 1994) has been employed to compute the turbulence stresses. Stagnation pressure, stagnation temperature, and turbulent intensity are prescribed at the inlet, according to the values reported in Table 2. A turbulent intensity value of 5% and a turbulent viscosity ratio μT/μ equal to 10 were specified for all simulations. The turbulent Prandtl number was set to Prt=0.9, in accordance with the indications listed in Otero et al. (2018). The advective fluxes were discretized with a second-order accurate scheme. First-order upwind schemes were instead used to discretize turbulent fluxes. Fluid properties are calculated with a look-up table method and tabulated values were computed with a well-known program (Lemmon et al., 2018).

Reduced-order models for the estimation of the optimal solidity

Based on the approach proposed by Denton (1993), two physics-based reduced-order models for the estimation of the optimal solidity in axial turbines have been developed. Both models are based on a simplified blade loading distribution and provide an estimation of the chord-to-pitch ratio value at which the passage loss is minimized. The first model (iSol) is applicable to incompressible flows, whereas the second (cSol), is its extension to compressible flows. Results obtained from these models are then compared against those obtained from the Zwifel criterion and from the CFD simulations.

Physics-based model for incompressible flows (iSol)

The incompressible physics-based model has been developed by assuming an exemplary turbine blade geometry (Figure 5a) whose loading is described by the simplified velocity distribution shown in Figure 5b. The flow angles at the inlet and the outlet of the vane are α1 and α2, respectively, while δ is the deviation from the outlet metal angle αm. The blade features a parabolic camber line with a slope at both the leading and the trailing edge equal to the tangent of the flow angles. These assumptions result in the geometric relation

Figure 5.

(a) Reference blade geometry for the reduced-order model. Blade thickness is neglected in this study. Adapted from Coull and Hodson (2013). (b) Simplified velocity distribution along the blade axial direction.

(12)

ycx=−xcx(Axcx+B),

with A=(tanα2−tanα1)/2 and B=tanα1. The blade thickness is assumed negligible, as well as the flow deviation downstream of the blade trailing edge due to the incompressible flow assumption. Even if, in an actual turbine, the thickness distribution determines the blade loading, this assumption is deemed satisfactory for the given purposes. The velocity distribution, and, consequently, the blade loading, depend on the flow velocities V1 and V2 at the inlet and the outlet of the cascade. For an incompressible flow, V1 and V2 can be calculated knowing the axial velocity Vx, which is constant along the vane, and the flow angles α1 and α2. For a given set of flow angles and a given Vx, the blade loading solely depends on the values of the parameters k and dV, which are independent of V1 and V2.

The circulation around the blade suction and pressure sides is equal to that around a control volume encompassing the inlet, outlet and the periodic mid-line of the vane, i.e.,

(13)

∫01(Vss−Vps)1+(2Axcx+B)2d(xcx)=(V2sinα2−V1sinα1)σx,

where Vss and Vps are the values of the absolute velocity along the suction and pressure sides, respectively, and are defined by the distribution shown in Figure 5b. The equation for the balance of the tangential momentum reads

(14)

∫01(Vss2−Vps2)d(xcx)=2Vxσx(V2sinα2−V1sinα1)=2Vx2σx(tanα2−tanα1).

The pairs of k and dV values satisfying Equations 13 and 14 can be calculated by eliminating σx from the system of equations. For given values of the inlet and outlet flow angles, the optimal solidity value can finally be calculated by minimizing the normalized dissipation due to passage losses, defined as

(15)

ζP=T2S˙P1/2m˙Vx2=2CdσxVx3∫01(Vss3−Vps3)1+(2Axcx+B)2d(xcx),

where Cd is the average boundary layer dissipation coefficient over the blade. Following Denton (1993), Cd was assumed constant and equal to 0.002. Equation 15 takes thus into account only the contribution due to dissipation in the boundary layers developing over the blade, the contribution due to mixing and shock waves being neglected. Unlike the Zweifel correlation, the iSol model does not require any empirical closure coefficient, but only the values of the inlet and outlet flow angles, arguably ensuring improved accuracy.

Physics-based model for compressible flows (cSol)

To provide better estimations in the transonic and supersonic regimes, the incompressible model iSol has been modified to treat also compressible flow cases. In contrast to the incompressible case, the axial velocity is not constant but changes with the fluid density according to the mass conservation equation. Moreover, the thermodynamic and kinematic fields are now coupled, and the energy conservation equation is necessary to close the problem. Besides the flow angles α1 and α2, the model thus also requires as input the values of the inlet stagnation temperature Tt1 and pressure pt1, which fix the thermodynamic state of the fluid, and the total-to-static expansion ratio βts.

The cSol model is developed starting from the same simplified blade geometry described in Section 2 and shown in Figure 5. The blade loading is calculated by integrating the simplified pressure distribution over the pressure and suction sides depicted in Figure 6a, which varies according to the values of the two parameters c and dp. The total pressure pt1 and the static pressures p1 and p2 define the values of the pressure distribution at the leading and trailing edge: the outlet pressure p2 is retrieved from the expansion ratio βts=pt1/p2. The inlet density ρ1 value is set to 0.97ρt1, while the entropy at the outlet is set to s2=1.0001s1. These values have been chosen by averaging those obtained from the results of the CFD simulations performed on the iMM-Kis3 blade and described in Section 2. By fixing the inlet density and the outlet entropy, it is possible to estimate the trends of pressure, velocity and enthalpy along the blade. A sensitivity analysis on ρ1 and s2 has been conducted, resulting in a negligible influence on the values of the optimal solidity. The compressible form of the tangential momentum balance can be expressed as

Figure 6.

(a) Simplified pressure distribution along the blade axial direction. (b) Locus of c and Δp values satisfying Equations 13, 16 and 17 (c) Dissipation due to losses in the passage vs σx computed for the c and Δp values shown in graph (b). For graphs (b) and (c), α1=0∘, α2=68.84∘, βts=1.4. The thermodynamic inlet conditions match those of the iMM case.

(16)

∫(lss+lps)pn⋅dlcax=m˙sV2sinα2−V1sinα1σx,

where p is the pressure over the blade. The energy conservation equation for streamlines following the blade pressure and suction side is given by

(17)

ht1=hps+Vps22=hss+Vss22.

The combination of Equation 17 with the fluid thermodynamic model allows the velocity distribution over the blade to be retrieved from the pressure distribution. In this study, the thermodynamic properties of the fluid are calculated using a well-known program (Lemmon et al., 2018). The blade circulation equation (Equation 13), which also holds in the compressible flow case, closes the system of equations. For a given set of flow angles and boundary conditions, there are infinite combinations of c and dp values satisfying Equations 13, 16 and 17. The locus of such c and dp combinations is shown in Figure 6b. The dissipation due to passage losses, which reads

(18)

ζP=T2S˙P(1/2)m˙V22,

where

(19)

S˙P=∫lpsCdρpsVps3Tpsdl+∫lssCdρssVss3Tssdl,

is calculated for each combination of c and dp satisfying Equations 13, 16 and 17. The optimal solidity value corresponding to the minimum loss (Figure 6c) is then evaluated.

Comparison of model results

Figure 7a–c show the loss breakdown for the iMM-Kis3 blade obtained from the results of the CFD simulations, see Section 2. Each loss source is plotted as a function of the axial solidity for the iN₂, iMM and niMM cases, respectively, and for each value of the outlet Mach number corresponding to the βts reported in Table 2. The loss coefficient ζs=T2Δs/u22/2, where the reference temperature and the flow velocity are computed at the outlet boundary, is used to quantify the dissipation produced by each loss source. For all cases, the red and the black curves provide the overall entropy increase between the inlet and the outlet of the computational domain, calculated using both mass-flow (MF) and mixed-out (MO) averaging techniques, respectively. The contribution due to the passage losses (P) is calculated as the difference in mass-flow averaged entropy between the inlet section of the computational domain and a section placed in proximity of the trailing edge at x=0.98cx. The loss due to mixing (MIX) downstream of the blade trailing edge, instead, is obtained by subtracting the passage loss from the overall mass-flow averaged one.

Figure 7.

Normalized entropy generation vs blade axial solidity for the iMM-Kis3 blade. (a) iN₂, (b) iMM, and (c) niMM. Left graphs - subsonic, centre graphs - transonic, right graphs - supersonic outlet Mach number. MF: overall mass-flow averaged loss; MO: overall mixed-out averaged loss; P: passage loss; MIX: mixing loss downstream of the blade trailing edge.

Results show that the optimal solidity value increases with the flow compressibility, i.e., with the cascade Mach number, regardless of the working fluid and the thermodynamic conditions. At subsonic operating conditions (Mout∼0.5, left graphs on Figure 7a–c), passage losses prevail over the mixing ones and determine the value of the solidity at which the overall losses are minimized. For all the investigated cases, both the passage and the overall MF and MO averaged losses exhibit a minimum at σx,opt∼0.65, and no influence of the fluid and its thermodynamic state is observed. At transonic and supersonic Mout values (centre and right graphs in Figure 7a–c, respectively), mixing losses increase and prevail over the passage ones. Therefore, the optimal solidity value now strongly depends on the trend of the mixing losses. In the supersonic regime, in particular, a monotonically decreasing trend of the mixing losses with σx is observed for all the investigated cases. Conversely, little deviations are observed in the value of σx minimizing the passage losses (0.65−0.75 for all the investigated cases). At fixed solidity and operating conditions, larger mixing losses are observed if the expansion occurs in a complex organic compound (iMM and niMM cases) rather than in a fluid made of simple molecules (iN₂ case), and the overall loss is minimized at larger σx,opt values. The increase in mixing losses is even larger if the organic fluid is operated in the dense vapor state (niMM cases). In the transonic regime (central graphs in Figure 7a–c), the σx,opt value at which the overall MF and MO losses are minimized is ∼1.15 for the iN₂ and the iMM cases, and 1.25 for the niMM case. In the supersonic regime (graphs on the right in Figure 7a–c), instead, the σx,opt value is found within the range 1.2−1.3 for the iN₂ case, 1.3−1.5 for the iMM case, and 1.5−1.6 for the niMM case. However, for the supersonic niMM case (Figure 7b, on the right), the MF and MO averaged losses are arguably insensitive to variations of axial solidity in the proximity of σx,opt: a lower blade number could thus be used for this case, without sensibly affecting the performance of the cascade.

Figure 8 shows the loss breakdown for the Aachen blade. Despite the blade being characterized by a larger flow turning and, consequently, a higher loading, the observed trends match those obtained for the iMM-Kis3 blade: the σx,opt value minimizing the overall MF and MO losses increases with Mout, and the share of the mixing losses on the overall loss increases with Mout, exhibiting a decreasing monotonic trend with σx at supersonic regimes (right graphs in Figure 8a–c). Depending on the flow regime, the passage losses are minimized at different values of σx: if Mout<1 (left graphs in Figure 8a–c), the minimum is observed at σx∼0.6, while in the transonic and supersonic cases, the same minimum occurs at σx∼1 (centre and right graphs in Figure 8a–c). Note that an optimal solidity value for the niMM case in the highly transonic regime (rightmost graph in Figure 8c) has not been found: mixing losses are arguably minimized at larger σx values than those investigated in this study.

Figure 8.

Normalized entropy generation vs blade axial solidity for the Aachen blade. (a) iN₂, (b) iMM, and (c) niMM. Left graphs - subsonic, centre graphs - transonic, right graphs - supersonic outlet Mach number. MF: overall mass-flow averaged loss; MO: overall mixed-out averaged loss; P: passage loss; MIX: mixing loss downstream of the blade trailing edge.

The physical reason that allows explaining the trends of the losses as a function of the solidity value can be obtained by inspecting the entropy contours of the iMM-Kis3 blade displayed in Figure 9a–c. The contour plots refer to the iMM case at βts=1.7 or, equivalently, Mout∼1. At σx∼0.65 (Figure 9a), the passage loss is minimized because of the low overall wetted area. However, the poor flow guidance provided by the blades causes flow diffusion on the rear part of the suction side, which entails a considerable boundary layer growth in the vicinity of the trailing edge. As a result, losses are significantly high due to the mixing of a higher portion of low momentum flow with the core flow. At σx∼1.05 (Figure 9b), the wetted area increases, leading to an increase of the passage loss. However, mixing losses are lower than in the σx∼0.65 case due to the reduction of the flow diffusion in the rear suction side, thanks to the improved flow guidance. At σx∼1.4 (Figure 9c), the magnitude of the passage losses equals that of the mixing one, leading to an overall increase of the loss coefficient. The same physical explanation, albeit with a different share of the losses, also applies to supersonic cases. The effect of the solidity on the wake at the blade trailing edge is also visible in the Mach number contours, see Figure 10. The Mach number contour plots also highlight the presence of a shock wave in the proximity of the trailing edge. Similar considerations hold for the Aachen blade (not shown).

Figure 9.

Entropy field in the proximity of the blade trailing edge for the case iMM, βts=1.7. (a) σx≃0.65, (b) σx≃1.05, (c) σx≃1.4.

Figure 10.

Mach number field in the proximity of the blade trailing edge for the case iMM, βts=1.7. (a) σx≃0.65, (b) σx≃1.05, (c) σx≃1.4.

The previous discussions mostly focused on the results obtained from the CFD. In the following, the results obtained with the reduced-order models described in Section 3 are presented. Figure 11a–f depict the trends of the optimal solidity value as a function of the Mout for the iMM-Kis3 and the Aachen blades. Values of σx,opt calculated considering the overall MF and MO loss, and the passage losses only are reported in the charts. The optimal solidity values calculated with the Zweifel criterion assuming ZZw=0.85, ZZw=0.95 and ZZw=1.1, and that estimated by the iSol model are also depicted. For both models, α2=αm, αm being the outlet blade metal angle. For transonic and supersonic flows, both the iSol model and the Zweifel correlations do not provide the same optimal solidity value minimizing the overall loss that has been computed using CFD simulations. This is due to the inherent limitations of both models, which account for the passage loss only and rely on the incompressible flow assumption. For the Aachen blade, the transonic and supersonic cases also exhibit a higher value of the optimal solidity obtained by minimizing only the passage losses. In this blade, passage losses are affected by the presence of a reflected shock on the rear side of the suction side, which largely influences the value of the optimal solidity.

Figure 11.

Optimal solidity vs outlet Mach number for (a–e) the iMM-Kis3 and (b–f) the Aachen blades. The values predicted by the reduced-order model (ROM) and the Zweifel correlation for three different values of the ZZW are also displayed. The region of validity of the Zweifel correlation for ZZW=0.8−1.1 is also highlighted in blue. σx,opt values have been estimated considering (a) and (b) passage losses, (c) and (d) overall mass flow averaged losses and (e) and (f) overall mixed-out averaged losses. (a) iMM-Kis3 (b) Aachen (c) iMM-Kis3 (d) Aachen (e) iMM-Kis3 (f) Aachen.

Figures 12a and 13a show the trends of Cpbt/s as a function of the axial solidity obtained from the CFD simulations for all the cases listed in Table 2, and for the iMM-Kis3 and the Aachen blades, respectively. Figures 12b–d and 13b–d, instead, show the trends of θ/s, (δ∗+t)2/s2, and θ∗/s as a function of the axial solidity for the iMM-Kis3 and the Aachen blades, respectively. The trends of the mixing losses observed in Figures 7 and 8 are aligned with those of the three boundary layer parameters. No correlation is instead observed with the trends of the base pressure coefficient. It can then be argued that mixing losses are mainly a function of the state of the boundary layer at the trailing edge, which, in turn, is affected by the fluid molecular complexity and the thermodynamic state. Among the four parameters, the momentum and kinetic energy thickness trends exhibit a strong correlation with those observed for both the mixing and the overall loss: for both parameters, the minimum of the curves shifts towards larger solidity values at higher cascade Mach numbers, and the corresponding σx values are in line with the σx,opt ones obtained from the CFD results (Figures 7 and 8). Moreover, similarly to what has been observed for the loss breakdown, the fluid molecular complexity and the thermodynamic state move the minimum of both θ/t and θ∗/t towards larger σx values in the transonic and supersonic regimes. Regarding the displacement thickness, the subsonic and transonic cases follow the same trend as that of the overall loss and those of the other boundary layer integral quantities; conversely, the supersonic cases exhibit a minimum at cax/s∼0.8 for the iMM-Kis3 blade, while no minimum is observed in the Aachen blade case. However, the contribution of the displacement thickness is smaller in magnitude compared to that of the momentum and kinetic energy thickness.

Figure 12.

Normalized trends of (a) base pressure coefficient, (b) momentum thickness, (c) displacement thickness, and (d) kinetic energy thickness in the proximity of the trailing edge vs axial solidity for the iMM-Kis3 blade. The boundary layer integral parameters are estimated by summing the contributions of both the pressure and the suction side.

Figure 13.

Normalized trends of (a) base pressure coefficient, (b) momentum thickness, (c) displacement thickness, and (d) kinetic energy thickness in the proximity of the trailing edge vs axial solidity for the Aachen blade. The boundary layer integral parameters are estimated by summing the contributions of both the pressure and the suction side.

Figure 14a shows the contour plot of the computed optimal solidity values as a function of the flow angles calculated with the iSol model. These variations of optimal solidity are compared with those computed with the Zweifel correlation using two different values of the Zweifel coefficient (Figure 14b and c). It can be observed that the optimal σx trends are very similar, regardless of the model with which they are computed. At fixed α1 and α2, the optimal solidity value obtained with iSol is within the bounds defined by the values obtained with the Zweifel correlation for ZZw=0.8 and ZZw=1.1.

Figure 14.

Optimal solidity σx vs flow angles resulting from (a) the iSol model, and the Zweifel correlation assuming (b) ZZw=0.8 and (c) ZZw=1.1.

Figure 15 shows the results obtained with the cSol model for both the iMM-Kis3 and the Aachen blades for the test cases listed in Table 2. The optimal solidity values obtained with the model are compared against those obtained from CFD considering only the passage loss calculated up to the trailing edge. For the iN₂ case and the iMM one (iMM-Kis3 blade only), the trend with the Mach number is somewhat in agreement with that obtained with the CFD: both curves exhibit a maximum at transonic conditions, i.e., Mout∼1. However, the absolute values are incorrectly estimated. Predictions and values are instead inaccurate for all the remaining cases: the model fails to match the trends obtained from CFD. In particular, for both the investigated blades, the model does not provide reliable results for the niMM case beyond Mout>0.6, and, for the Aachen blade, iN₂ and iMM cases, the model fails if Mout>1. The outcome of this analysis points out that the physics of the problem is not correctly modelled: the possible causes of error can be attributed to (i) excessively simplified blade loading distribution; (ii) inaccurate Cd value, especially in the presence of strong stream-wise pressure gradients; (iii) inability to capture the effect of boundary layer-shock interaction in the compressible flow regime, which results in additional passage loss.

Figure 15.

Optimal solidity value estimations vs outlet Mach number for (a) the iMM-Kis3 and (b) the Aachen blades, respectively. Results obtained with the compressible reduced-order model and the numerical simulations are depicted in solid and dashed lines, respectively.

Finally, Figure 16a–c shows the mixing loss as a function of the solidity for the iN₂, iMM, and niMM cases, respectively. Calculations have been performed for the iMM-Kis3 blade only. The mixing loss has been estimated using the control volume-based approach described in Section 2 using as input the base pressure coefficient and the boundary layer integral parameters obtained from CFD and shown in Figure 12. For all three cases, the control volume-based model provides a trend of mixing losses with the solidity which is aligned with that obtained from the RANS calculations up to the transonic regime: the minimum of the mixing losses shifts towards higher σx as the outlet Mach number increases. This further corroborates the finding that the mixing losses, and, thus, the optimal solidity value, are strongly affected by the state of the boundary layer at the trailing edge. However, the results for the supersonic regime are not consistent with those obtained from CFD simulations: in particular, the minimum of the mixing losses is located at lower solidity values than those obtained for the transonic case. The model does not account for the loss induced by the fish-tail shock waves forming downstream of the blade trailing edge. In conclusion, the proposed control volume approach for the prediction of the optimal solidity value is incapable of modelling physical phenomena with a sufficient degree of detail, and the optimal solidity value can only be predicted by numerical simulations or experiments.

Figure 16.

Mixing losses computed using the control volume-based model described in Section 2 vs axial solidity for the iMM-Kis3 blade. Each graph refers to the (a) air, (b) iMM, and (c) niMM cases, respectively.

Conclusions and Outlook

The optimal solidity value is used to determine the number of blades of an axial turbine stage. The work here presented aimed at investigating the influence of the molecular complexity of the working fluid, its thermodynamic state, and flow compressibility on the optimal solidity, as well as developing reduced-order models which can be used to estimate its value during the preliminary design. The optimal solidity values provided by the Zweifel criterion have been compared against those obtained with two-dimensional CFD simulations of the flow around two representative turbine stator blade geometries. A loss breakdown analysis based on data obtained from the CFD simulations has also been carried out. Reduced-order models for the estimation of the optimal solidity value based on the minimization of the passage losses in the incompressible (iSol) and compressible (cSol) regimes have been developed and applied to selected test cases.

Based on the results of the study, the following conclusions can be summarized.

Nomenclature

Abbreviations