A Three-Dimensional Design to Study the Shock Waves of Linear Cascade with Reduced Mass Flow Requirements

Featured Application

Design of linear cascades for high-speed bladed machinery.

Abstract

This paper presents the development of high-specific-speed mixed flow/centrifugal compressor vaned diffusers. Specifically, the design of a test rig that will make the visualization of shock waves on diffuser vanes manageable is addressed in the current study. In this particular case, linearization of an existing state-of-the-art compressor stage was used. For the computational modeling, a series of RANS analyses were conducted to examine the flow characteristics of the two cases explored: a complete transonic cascade and an idealized periodic passage. The distinct behavior exhibited by each vane passage within the entire cascade offers the opportunity to analyze the shockwave structures across a mass flow range of ±9% around the design point. Overall, the pressure coefficient distributions and flow field patterns appear to align with the single-passage conditions, although there are some minor lateral wall influences, particularly in the first passage close to the suction lateral wall. However, because of the nature of the flow, which is characterized by high velocity and density differences near the vanes, the equivalent mass flow per individual passage was difficult to estimate. This may also be attributed to the small endwall axial vortices; nonetheless, for the purposes of this paper, this was of little consequence.

1. Introduction

Wind tunnel testing of linear cascades provides crucial experimental data to understand the aerodynamic behavior of compressor blades and to further enhance the overall performance of compressor systems. Moreover, linear cascade experimental campaign is relatively cheap compared to full-scale annular geometries as it allows rapid prototyping and uses cheaper, less complex manufacturing techniques.

Nevertheless, it is necessary to obtain reliable results from experiments, meaning that the flow pattern in a linear cascade should correspond to the flow pattern inside a turbomachinery. Therefore, a Computational Fluid Dynamic (CFD) analysis of the test rig is required to establish a suitable configuration.

Blade periodicity, which is a crucial factor for reproducing real turbomachinery fluid flow conditions, has been extensively studied in various sources. Björkman et al. [1] focused on the design of a linear cascade test rig for turbine components. Their research revealed that blade periodicity was obtained at first at two thirds of the axial chord length, while for full periodicity, further analysis that involved different angles of the tailboards was necessary. Another study by Tian et al. [2] emphasized the sensitivity of tailboard angles as a significant factor influencing the periodicity of a cascade. Besides the tailboard angle, the outlet pressure also plays a crucial role in determining the position of shock waves.

A comprehensive investigation of fluid flow periodicity under both steady-state and unsteady conditions was conducted by Tian and Salas [3]. Their investigation specifically focused on a transonic cascade and revealed that unsteady pressure reflections on the tailboards play a significant role and must be taken into account during wind tunnel testing. Furthermore, the bottom tailboard was found to cause more pronounced unsteady reflections compared to the upper tailboard [4]. This finding highlights the importance of considering the specific characteristics and effects of each tailboard to accurately analyze fluid flow periodicity in transonic/supersonic cascades [5,6,7,8].

In [9], Shaukat investigated the impact of side cascade walls on periodicity and developed a parametric model that encompassed the entire geometry, including the inlet/outlet side walls and blade row. The study findings indicated that the outlet wall angles exerted the most significant influence on the periodicity of the test rig.

Gao’s study [10] presents a series of numerical analysis using URANS (Unsteady Reynolds-Averaged Navier–Stokes) and LES (Large Eddy Simulation) models. The simulations were performed based on a linear compressor cascade and validated against the experimental results obtained by Ma [11]. When comparing the two simulation approaches, it was found that the results obtained by LES were closer to the experimental data, while the RANS approach tended to overestimate the occurrence of corner separation. Despite this discrepancy, the RANS approach still holds value in the industrial sector, providing a reasonably accurate representation of the mean trend relationship with experimental data [12]. While RANS simulations may not attain the same level of precision as LES/DNS methods, they are standing as a practical and valuable tool for engineering applications. For this research purpose, RANS models offer valuable insights into the overall flow structure of linear cascades, yielding reliable predictions [13,14]. Additionally, when coupled with essential components such as grid resolution, numerical schemes, and turbulence models, the accuracy of a model can be significantly enhanced.

Ma’s paper [11] explores the influence of incidence angles ranging from −2° to 6° on pressure and losses. This study revealed that the maximum total pressure loss occurred at an incidence angle of 4°, which resulted in a large separation bubble in the corner region. Although the RANS approach overestimated the size of the separation bubble, it captured the overall pattern accurately. The misprediction could be attributed mainly to the unsteady characteristics of corner separation [15,16].

In the analysis of transonic/supersonic compressor cascades, it is crucial to accurately capture the interaction between shock waves and boundary layers. This interaction significantly affects the performance estimation of turbomachinery [17,18]. Therefore, establishing a correlation between grid resolution, turbulence model, and solver settings is necessary to achieve a precise representation of the fluid flow and ensure proper validation against experimental data [19]. A study by Pievesan et al. [20] demonstrated the substantial impact of the turbulence model on performance prediction. This influence was primarily attributed to the high sensitivity of shockwave patterns to the development of the boundary layer.

In the field of supersonic cascades, Liu et al. [21] developed a one-dimensional analytical model for predicting shock losses, aiming to facilitate the rapid evaluation and reduction in these losses in the optimization design of a supersonic compressor cascade. Their model was applied to two supersonic cascades, resulting in a significant reduction of 20% and 16% in the total pressure loss coefficient. Furthermore, Sun et al. [22] conducted research to determine the primary source of shock losses. By developing a shock loss mechanism and validating the results, it was identified that the main source of loss in the supersonic cascade is the terminal passage shock loss. Other related studies focusing on the development of shock loss prediction methods are found in [23,24,25], and most of them were developed based on experimental empirical correlations.

These studies contribute to the understanding of shock losses in supersonic cascades and provide insights into the development of analytical models and optimization approaches to mitigate these losses, ultimately enhancing the performance of supersonic compressor cascades.

To control the flow field inside a cascade wind tunnel and establish a procedure for this approach, Teteishi et al. [8] developed a numerical method capable of simulating complex configurations involving movable mechanisms. Using that method, steady and unsteady simulations of a transonic cascade wind tunnel were performed. Their study investigated the effects of wall suction, throttle opening angle, and tailboard angle in the case of steady analysis, as well as blade oscillation in the case of unsteady analysis. The findings revealed that the differences between the ideal configuration and the wind tunnel configuration become more prominent as the pressure amplitude increases downstream of the lower blades. Furthermore, this research study highlights the significance of various parameters on the flow characteristics and contributes to understanding the control and assessment of flow fields within cascade wind tunnels using numerical simulations. Similar studies focusing on advanced analytical techniques and computational tools to address complex problems are presented in [26,27,28].

The current work examines the effects on the flow structure of an asymmetric transonic linear cascade in contrast with single-passage mapping, with the study being performed for a mixed flow/centrifugal compressor vaned diffuser.

The primary focus of this paper is centered around the development of an axisymmetric test rig designed to facilitate the visualization of shock waves on diffuser vanes. Managing the interaction between shock waves and the boundary layer is crucial for minimizing drag and shock losses. The presence of such an interaction introduces aerodynamic instabilities that significantly impact the overall performance and operational capabilities of turbomachinery. Using an asymmetric cascade configuration to scrutinize flow patterns and the structure of shock waves has proven to be fruitful, leading to reduced computational and time loads. The distinctive behavior exhibited by each vane passage within the complete cascade provides an opportunity to examine the shockwave structures across a specific mass flow range around the designated operating point.

2. Methodology

The objective of this analysis is to develop a transonic linear cascade wind tunnel. The design arises from the need to test numerically and experimentally different intensities of shock wave systems within vaned diffusers, while minimizing computational and experimental resources. With a mass flow imposed on the linear cascade inlet, it was anticipated to achieve at least three operating points on the cascade channels. This is due to the distinct flow characteristics and mass flow passing through every channel.

Figure 1 shows the initial configuration of the mixed flow/centrifugal compressor, designed as part of a microgas turbine engine application.

To obtain the infinitely long linear cascade, the cylindrical intersection of the blade row needs to be topologically unwrapped into a plane. Figure 2a presents the middle plane of a vaned diffuser from which the geometry of the blades is extracted, while Figure 2b shows a part of the initial linear cascade that resulted from this procedure.

Due to the very small vane height (i.e., 4 mm), it was decided to scale up the model 2.5× in height and 2× in the other two dimensions. This is acceptable since the unwrapping is equivalent to a 2D blade-to-blade mapping of the original vane, which means that the height of the cascade does not influence the flow features to be investigated.

The baseline cascade model defined in Figure 2b was used for a preliminary numerical analysis of the flow structure. To capture different intensities of shock waves, multiple working points were simulated using ANSYS CFX. The flow angle was adjusted to prevent boundary-layer separation, and, eventually, it was defined as a boundary condition. No further changes were made to the geometry to correct the flow pattern in this initial stage.

Since the stator flow enters the cascade at a high angle (α_inlet = 68°), for practical purposes, the blade row itself was rotated by the said angle α_inlet in the actual test rig. Therefore, the final experimental configuration considered for the transonic cascade is the one presented in Figure 3.

The geometry consists of five groups (main blade and splitter) whose parameters and reference flow conditions are defined in

Table 1. Cascade parameters and flow conditions.

Symbol	Parameters	Main Blade/Splitter Blade
c	Blade chord length [mm]	71.16	45.22
s	Pitch width [mm]	52
l	Blade span [mm]	10
σ	Solidity	1.368	0.869
γ	Stagger angle [°]	51.55	49.23
	Inlet total pressure [Pa]	200,000
	Inlet temperature [K]	293

.

This geometry was defined to observe the nominal inlet flow angle. However, each blade assembly (main blade and splitter) works on a different mass flow. This approach allows the observation of shockwave structure and behavior over a ±9% mass flow interval around the nominal point. This leverages the natural tendency of the rig to induce different local mass flows per individual passage, thereby facilitating the study of the flow differences between the CFD simulations and the experiments.

Two sets of simulations were performed. The first simulation assumed full periodicity of the passage and utilized 2× instancing of the main blade + splitter (to eliminate the possibility of interface artefacts). The second simulation was performed for the full cascade, with the wall interference and vane inter-dependencies that arise from the experimental setup.

The numerical simulations were performed using the commercial software Ansys CFX (https://www.ansys.com/products/fluids/ansys-cfx, accessed on 23 August 2023), employing the steady-state RANS approach. The solver model implemented by CFX is a pressure-based implicit coupled solver. To accurately capture shock waves and adequately resolve advection and turbulence model equations, high-order schemes were employed. These schemes are the integral components of Total Variation Diminishing (TVD) algorithms and have demonstrated their accuracy in handling sharp gradients and intricate flow phenomena [13,14]. For the transonic/supersonic simulations involving steep gradients, such as shock waves or turbulent boundary layer, numerical schemes capable of providing accurate results, along with a good balance between stability and accuracy, are essential. Therefore, in addition to the numerical schemes, other factors such as turbulence modeling and specific software settings play a crucial role in determining the accuracy of the solution.

The mesh density for the idealized periodic passage consisted of approximately 1.5 million nodes. This configuration was established based on earlier research [29], which identified the optimal balance between resolution and numerical stability and convergence time for studies related to turbomachinery.

To ensure an accurate solution for the examined problem, a grid independence study was conducted for the entire cascade. This process involved six different grid resolutions, ranging from coarse to fine meshes. All generated grids are composed of unstructured tetrahedral cells with low skewness. The coarse meshes (7-, 11-, and 17-million-cell mesh) have refinements around the airfoil and in the wake, achieving a y+ distribution close to 1. This variation was achieved through modifications of the number of layers in the radial and axial directions, as well as the overall sizing of the geometry. For the next set (24, 30, and 36 million cells), an additional level of refinement was applied in the blades’ test area to enhance mesh density, thus ensuring an accurate representation of the shockwave structures. The transition from a denser grid to a far-field grid was executed gradually, avoiding sudden transitions between adjacent cells. Also, the number of layers from the inflation layer was increased to assure proper definition of the boundary layer.

For each grid resolution, the performance was assessed using CFD simulations, and the optimal grid was determined by comparing the results of these analyses.

Figure 4 illustrates the mesh convergence study, using total pressure drop as the comparative parameter to evaluate grid density. Beyond 30 million nodes, further mesh refinement has minimal impact on the solution, but it does increase the solver time. Consequently, this grid resolution was selected for subsequent analyses, and Figure 5 and Figure 6 present the quality of the grid and the corresponding y+ distribution.

Turbulence was modeled using the k-ω SST (Shear Stress Transport) formulation with streamline curvature acceleration corrections [30]. The SST turbulence model was chosen due to its extensive applicability in simulating turbomachinery flows, allowing accurate performance predictions across a wide range of operating conditions [31,32]. This two-equation turbulence model leverages blending functions to utilize the two underlying turbulence models that define it. It employs the k-epsilon model in the far field and transitions to the k-omega model near the walls. For the studied case, it was crucial to accurately capture both the flow structure near the blade walls and in the far field to determine the structure of shock waves.

The computation for the full transonic cascade was performed for an imposed mass flow (0.4 kg/s) and total temperature (293 K) at the inlet patch, while at the outlet, atmospheric static pressure was assumed. The endwalls, as well as the main and splitter vanes, were modeled as adiabatic walls with no slip and without surface roughness (machined surface). For this simulation, the mass flow rate for each channel was extracted and served as the input boundary condition (i.e., inlet mass flow rate) for the single-passage simulations (as presented in

Table 2. Flow conditions for the single-passage cascade.

Position	Inlet Mass Flow [kg/s]
Passage 1	0.089865
Passage 2	0.0898
Passage 3	0.08835
Passage 4	0.08675
Passage 5	0.08475

).

To assure the accuracy and stability of the solution, different convergence criteria were evaluated. For the continuity equation and velocity components, the residuals were set to 1 × 10⁻⁶. Additionally, the overall imbalance among pressure, energy, and velocity was less than 0.15%. Due to the asymmetry of the full cascade, it was anticipated that distinct operating points would be achieved in each passage. As a result, numerical simulations were conducted for each passage (idealized periodic passage) using the corresponding conditions from the linear cascade (as detailed in Table 2). This validation primarily aimed to confirm whether the same flow structure and pressure losses would be attained under those conditions.

Note that there is a higher mass flow per passage (idealized periodic) than the one estimated by the individual sections in the linear cascade (non-idealized periodic), as shown in Figure 7. It is important to acknowledge that this technique suffers from inaccuracies, particularly when dealing with cross sections featuring high velocity and density gradients. In essence, when computing the mass flow passing through a section, the orientation of the particular virtual boundary in that section influences the result. Even if the cutplane is normal to the vane geometry, there is still the influence of the flow pattern (vortices, density, and velocity distribution), which may lead to inaccuracies. For this particular case, the results were deemed to be sufficiently close to the original estimations, and therefore, the geometric normal cross section was used. This is mainly because the geometry of the cutplanes does not cut through the vortices, as seen in Figure 8. In other cases where differences become more pronounced, one might consider a cross section that is also determined by the flow features observed, particularly if the vortices do not convect or if they occupy a large area of the cutplane.

In the first two passages of the cascade, higher operating conditions, along with blade geometry, lead to the generation of shock waves. These, in turn, give rise to the formation of three-dimensional vortices as a consequence of boundary-layer separation. As depicted in Figure 8, the zone impacted by these vortices is situated in proximity to the trailing edge of the blades. Here, vorticity stretching induces flow separation on both sides of the blades. The blade curvature in this vicinity, combined with the high velocity of the three-dimensional fluid flow, creates the optimal conditions for the development of three-dimensional vortices. In the examined scenario, the vortices exhibit diminished magnitude, thus exerting a localized effect primarily in the vicinity of the trailing edge. Consequently, as long as the affected region remains restricted to the proximity of the trailing edge, it exerts a minimal impact on the intended objectives of the study.

3. Results and Discussions

Figure 9 illustrates the distribution of density gradients for the complete cascade geometry. Inherently, each passage passes a slightly different mass flow fraction; therefore, it becomes possible to test multiple working points simultaneously, enabling the assessment of how different shockwave patterns influence fluid flow. It is worth mentioning that since the flow angle is identical for all passages, the differences can be regarded as surrogates for slightly different speedlines.

Notably, in the first two passages, the shockwave intensity is more pronounced. This diminishes as the mass flow passing through the channel decreases, specifically toward passage 5. Passages 1 and 2 exhibit robust normal shock waves on the splitter and near the leading edge of the main blade. The shock patterns in these two channels indicate that the working conditions are approaching the physical choke point of the passage. However, this is not necessarily equivalent to the speedline choke, in which a negative incidence could also be observed. Therefore, it is more correct to think of these passages as part of the nominal point of a higher speedline.

At supersonic speeds, as is the case of the first two blade passages, due to the sudden change in the profile geometry a local disturbance of the flow is created, generating a shock wave. In this case, the shock wave is influenced also by the blade angle of attack and the flow velocity. The intensity of the shock waves formed near the leading edge is relatively mild when contrasted with those generated on the splitter. Consequently, the flow separation induced by this shock wave results in a thin separation layer.

The flow structure of the cascade is examined at midspan, and Figure 10 and Figure 11 present the variation in the pressure coefficient for the main blade and splitter, respectively. In Figure 10, the first three channels of the cascade display three zones with significantly reduced pressure coefficients, corresponding to the presence of shock waves attached to the airfoil. In these cases, the intensity of these attached shock waves is notably stronger compared to the lower part of the cascade (i.e., toward passage 5), where the shock waves are only formed in proximity to the leading edge of the blade.

Figure 11 depicts the variation of the pressure coefficient for the splitter blade. For this airfoil, shock waves are formed on both the suction and pressure side. Specifically, the first two passages exhibit strong shock intensity, whereas the magnitude of the shock waves reduces for the subsequent passages in the cascade.

The distribution of the Mach number on the near-blade flow field, as shown in Figure 12, indeed confirms the variation in the pressure coefficient mentioned earlier. Notably, areas of local acceleration are observed on the leading edge of the main blade and the mid-cord of the splitter. These regions of acceleration lead to different intensities of shock structures, ranging from strong structures (near the first passage) to progressively weaker ones as the passage index increases. Since each passage works at a different mass flow, it is worth noting that the difference between the higher and lower mass flow is approximately 9%. This variability in working conditions is the main contributor to the differences in shock intensities and pressure coefficients observed across the cascade geometry.

Figure 13 and Figure 14 provide a comparison between the idealized (single-passage flow) and non-idealized periodic passage (full cascade). Each blade row in the full cascade represents a distinct working condition of the vaned diffuser. In all cases, it is evident that the flow structure in the single passage exhibits the same overall pattern as the full cascade. However, there are also some differences between the cases, with a more pronounced discrepancy observed for the first two channels, particularly for the main blade, as depicted in Figure 12. This trend may be also due to lateral wall influence, but it is much more pronounced in passage 1 than in passage 5. Perhaps, since passage 5 is much less loaded, this influence is less prominent. It is also interesting how the regions with a high Mach number themselves are quite well captured, whereas the areas downstream of the shock waves appear to be more discrepant in passage 1.

With the exception of passage 1, all main blade surface Cp distributions are well replicated in the linear cascade setup. A concern that this would also happen in passage 5 was put to rest because of the low Mach number observed there and the positive pressure gradient of the lateral wall. This decreased the chance of local accelerations and left a “cleaner” suction side for the fifth main blade.

These discrepancies in the shockwave patterns suggest that the behavior of the blade in passage 2 in the linear cascade replicates the single-passage flow well but not entirely perfectly. The variations observed in the Mach number distribution, as seen in Figure 15, can be attributed to the flow interactions present in the full cascade as opposed to the idealized single-passage simulation. These distinctions highlight the importance of considering the cascade’s complete geometry when analyzing the flow characteristics and shockwave patterns accurately.

A visual inspection of the flow field in passage 2 reveals some differences, particularly on the suction side of the splitter—as shown in Figure 15b. The pressure coefficient distribution reveals that the single-passage simulation exhibits more pronounced shock waves than the full cascade (in the first two passages). It is important to mention that this behavior is not a result of mesh effects.

This may indicate that the assumption of being able to perfectly replicate the single-passage flow by using a linear cascade is not without limitations. However, the overall flow features are well represented in terms of location, number of high Mach spots, and, to good extent, the shock intensity and pressure coefficient.

4. Conclusions

The paper presents the development and numerical analysis of a high transonic linear cascade designed to study shockwave structures under different working conditions. A notable advantage of this cascade design lies in its ability to concurrently accommodate different operating conditions, leading to reduced experimental expenses. Additionally, since the flow angle is largely identical for each passage in the cascade, the experiment covers a CFD uncertainty in mass flow as high as 9%—the difference between the lowest and highest mass flows registered in the linear cascade assembly. This is quite conservative, given that modern RANS models are employed. Conversely, if the estimation of the nominal mass flow is regarded as perfect, the differences between the passages could be seen as nominal points on slightly different speedlines. Therefore, the first row would correspond to a higher speedline and row 5 would be associated with an inferior speedline.

The computational aspect of this study involved employing the Ansys CFX solver, which facilitated steady simulations for both the linear cascade and single-passage geometry.

The results primarily center around the asymmetrical nature of the flow field and can be summarized as follows:

(1)

In terms of the position, strength, and number of regions with a high Mach number, the cascade replicated the five different variations in the nominal operation point of the diffuser well.

(2)

The pressure coefficient distribution matches both the main and splitter vanes. The observed discrepancies are not passage-related in the sense that if a difference in Cp is observed on the splitter, this is not necessarily also observed in the Main vane. Therefore, this indicates that these are some global, cascade-related, 3D effects that warrant further study.

(3)

The differences between the flow of the upper and lower sections of the complete cascade are approximately 9%. However, the equivalent passage flows—simulated as idealized periodic passages—show differences of only +/- 5% in mass flow. This feeds into the hypothesis that some 3D effects influenced by the cascade walls and the interaction between the passages exists.

(4)

All things considered, when designing a similar test rig, users must expect and account for differences between individual flow passages (early in the design stage) and the final rig. Depending on the investigated parameters, such a fixed linear cascade can be useful in the assessment of flow field characteristics under various operating conditions using both numerical and experimental tests, even for higher Mach numbers.

Moving forward, this research will be extended to further investigations involving the same linear cascade. Such studies will eventually aim to analyze the effectiveness of passive control methods in mitigating the effects of shock waves in high-speed turbomachinery applications.