Modified Zweifel Coefficient and Lift Coefficient Definition Considering Compressible Effect

Abstract

The accurate prediction of blade loading is crucial to designing efficient turbomachinery, but traditional methods often neglect the impact of compressibility, leading to inaccuracies at high speeds. This study investigates the effect of compressibility on the blade loading parameters, particularly the Zweifel coefficient (Z_w) and lift coefficient (C_L), in turbine cascades. A novel intermediate method (IM), with averaged flow properties derived from both inlet and outlet conditions, is proposed to enhance the accuracy of Z_w and C_L calculations in compressible flow regimes. This method is based on the extended Kutta–Joukowski theorem for compressible flow and incorporates the Mach number directly into the modified definitions of Z_w and C_L. The analysis reveals that the averaged flow angle (α_m), calculated by using a velocity-weighted approach, serves as a crucial parameter for blade similarity studies. The proposed correction method is applied and validated based on CFD simulations of the VKI-RG turbine cascade. The IM and modified definitions provide a robust framework for accurately predicting blade loading at high speeds, enabling improved design and analysis of turbomachinery.

1. Introduction

The term “aerodynamic loading”, which is widely discussed in turbomachinery, can be further divided into stage loading and blade loading. Stage loading shows the capacity of energy transformation in a stage, which is often confirmed by the change in enthalpy and is normalized by the square of the rotational speed, namely, $\psi =\Delta h/U^2$, where U can be chosen as either the speed at the blade tip or the mid-span [1]. The concept of blade loading is a little bit more complex, encompassing the blade spanwise loading and blade-to-blade (B2B) loading, where the latter, B2B loading, is the main interest of this paper.

When referring to B2B loading in more detail, the dimensionless number, the Zweifel coefficient ($Z_w$) [2], is often introduced in turbines to measure the magnitude of the circumferential force or tangential force on a blade, and the force is normalized by an ideal force composed of the product of the ideal outlet dynamic pressure head and axial chord length [3]. The application of $Z_w$ is indeed closely linked to various design considerations in turbomachinery. For instance, it plays a vital role in determining the optimal solidity or the choice of the blade number, which influences the overall performance of the turbomachinery system [2,4,5,6]. Moreover, the correlation between losses and the choices of inlet and outlet flow angles is an essential facet of turbomachinery design. By considering different flow angle configurations, designers can evaluate the resulting losses within the system. The utilization of $Z_w$ aids in determining suitable flow angle combinations that minimize losses, enhancing the performance and efficiency of the turbomachinery design [7].

In academia, $Z_w$ is a standard for research when measuring the influence of loading on a blade [8,9]. Additionally, $C_L$ is widely employed in the literature [5,10,11,12] to describe B2B loading, as it directly originates from the definition of airfoil loading in external flow [13]. Figure 1 depicts the knowledge domains associated with $Z_w$ and $C_L$, illustrating the interconnected relationships between their concepts and definitions. This visual representation shows that the lift force originates from the circulation around the blade. Additionally, by employing the Kutta–Joukowski theorem, the lift forces acting on the blade can be precisely calculated. The lift coefficient, a dimensionless quantity, quantifies the lift force generated by the blade. Significantly, the Zweifel coefficient exclusively considers the circumferential force component, making it a special case of the lift coefficient.

Historically, numerous studies on B2B loading have relied on the incompressible hypothesis, which assumes equal axial velocities at the inlet and outlet. This simplification facilitated the computational process and definition [14,15]. However, with the growing demand for and development of highly loaded turbines aimed at reducing the number of stages, weight, and cost of modern gas turbine engines, the velocity and temperature differences between the inlet and outlet of a turbine stage have increased significantly. However, there are currently few studies that provide a specific distinction for $Z_w$ at high and low speeds. The incompressible assumption can introduce deviations in calculating the actual loading coefficient, leading to an overestimation of the turbine’s loading when using incompressible flow calculations for B2B loading. The specific reasons for this overestimation will be analyzed in subsequent sections.

In order to address the existing research gap about the influence of gas compressibility on B2B loading, many researchers have investigated and explored potential solutions. The current progress made for addressing this issue is summarized in

Table 1. Current definitions and variants of $Z_w$ and $C_L$ in the literature.

First Author and Year	Reference	Definition	Features
Zweifel 1945	[2]	$Z_w={\displaystyle \frac{2t}{c_z}}{sin}^2{\alpha }_2(cot{\alpha }_1+cot{\alpha }_2)$	Incompressible, original Form
Glassman 1972	[16]	$$\begin{gathered} {\displaystyle \frac{Z{w}_{comp}}{Z{w}_{inc}}}={\displaystyle \frac{\frac{\gamma }{\gamma +1}{(1-\frac{\gamma -1}{\gamma +1}{\lambda }_2^2)}^{\frac{1}{\gamma -1}}{\lambda }_2^2{D}_s}{1-{[1-\frac{\gamma -1}{\gamma +1}{\lambda }_2^2{D}_s]}^{\frac{\gamma }{\gamma -1}}}}\approx \\ 1-{\displaystyle \frac{{\lambda }_2^2}{\gamma +1}}+{\displaystyle \frac{D_s{\lambda }_2^2}{2(\gamma +1)}},D_s={\left({\displaystyle \frac{V_{max}}{V_2}}\right)}^2 \end{gathered}$$	Compressible, PM
Moustapha 1987	[17]	$Z_w={\displaystyle \frac{2t}{c_z}}{sin}^2{\alpha }_3[\left({\displaystyle \frac{V_{z2}}{V_{z3}}}\right)cot{\alpha }_2+cot{\alpha }_3]$	Rotor $Z_w$, compressible, axial velocity modification, IM
Song 2000	[18]	$Z_w={\displaystyle \frac{2(t/c)}{(H/c)}}{sin}^2{\alpha }_2(cot{\alpha }_1+cot{\alpha }_2)$	Incompressible, aspect ratio modification
Cardamone 2006	[24]	$C_L={\displaystyle \frac{2t}{c_z}}{\displaystyle \frac{{\rho }_2{V}_{2}sin{\beta }_2(V_{2}cos{\beta }_2+V_{1}cos{\beta }_1)}{(P_{t1}-P_2)(h_1/h_2+1)}}$	Compressible, IM
McQuilling 2007	[19]	$Z_w={\displaystyle \frac{2S}{c_z}}{sin}^2{\alpha }_2[\left({\displaystyle \frac{V_{z1}}{V_{z2}}}\right)cot{\alpha }_1+cot{\alpha }_2]$	Compressible, axial velocity modification, IM
Gier 2010	[20]	$Z_w={\displaystyle \frac{t}{c_z}}·{\displaystyle \frac{{\rho }_2{V}_{z2}^2(\frac{V_{z1}}{V_{z2}}·\frac{A_{z2}}{A_{z1}}cot{\alpha }_1+cot{\alpha }_2)}{P_{t1}-P_2}}$	Compressible, axial velocity modification, IM
Schobeiri 2012	[5]	$C_L={\displaystyle \frac{t_2}{c}}{\displaystyle \frac{sin{\alpha }_2^2}{sin{\alpha }_m}}(1+\mu )(cot{\alpha }_2+\nu \mu {\alpha }_1),\mu ={\displaystyle \frac{V_{z1}}{V_{z2}}},\nu ={\displaystyle \frac{t_1}{t_2}}$	Compressible, IM
Coull 2013a	[8]	$C_o=\oint \left({\displaystyle \frac{V}{V_s}}\right)dz$	Compressible, implicit form, circulation
Coull 2013b		$C_o={\displaystyle \frac{t}{S}}sin{\alpha }_2(cot{\alpha }_1+cot{\alpha }_2)$	Incompressible, circulation
Babajee 2013	[21]	$Z_w={\displaystyle \frac{2t}{c_z}}{\displaystyle \frac{|\rho V_u{V}_z{|}_2+{\left|\rho V_u{V}_z\right|}_1}{{\rho }_2{V}_2^2}}$	Compressible, direct form
Schmitz 2016	[22]	$Z_w={\displaystyle \frac{2F_u}{\rho V_2^2{c}_z}}$	Compressible, implicit form
Yang 2020	[23]	$Z_w={\displaystyle \frac{{\int }_0^1[{(1-\frac{\gamma -1}{\gamma +1}{\lambda }_{ps}^2)}^{\frac{\gamma }{\gamma -1}}-{(1-\frac{\gamma -1}{\gamma +1}{\lambda }_{ss}^2)}^{\frac{\gamma }{\gamma -1}}]dz}{1-{(1-\frac{\gamma -1}{\gamma +1}{\lambda }_{is2}^2)}^{\frac{\gamma }{\gamma -1}}}}$	Compressible, blade design stage, IM

, and the comments and differences among them will be further discussed in the later part of this paper. Glassman [16] conducted modifications on the incompressible $Z_w$ based on the critical velocity ratio relationship. Moustapha et al. [17] defined a rotor blade $Z_w$ and performed single-stage turbine performance experiments with $Z_w$ ranging from 0.77 to 1.18. Song et al. [18] provided a modified $Z_w$ with aspect ratio correction, which acts as one of the parameters in the tip leakage flow model. McQuilling [19] considered the differences in inlet and outlet velocities and provided a modified definition of $Z_w$, thereby establishing a quantified standard definition for high-lift turbine blade profiles. Gier et al. [20] observed and emphasized that $Z_w$ only partially considers the compressibility effects. When $Z_w$ is used to evaluate blade loading, caution must be exercised. Gier also proposed a modification to the compressibility effects on $Z_w$ and provided a revised definition formula incorporating inlet and outlet axial velocities and areas. Coull et al. [8] introduced a dimensionless coefficient called the circulation coefficient ($C_o$), which is defined as the ratio of the actual circulation to the ideal circulation on the blade surface, and the value of $C_o$ is always smaller than $Z_w$. Babajee et al. [21] employed a direct formulation of $Z_w$ and compared the cases of $Z_w$ = 0.95 and 1.46 to investigate the laminar separation-induced transition on the turbine airfoil. Schmitz et al. [22] employed an implicit form to calculate the blade circumferential force to calculate the compressible $Z_w$ and verified the performance of a low-pressure turbine stage with loading parameter $\psi =2.8$ and $Z_w=1.35$. Yang et al. [23] utilized the isentropic velocity coefficient to describe the acceleration and deceleration distribution along the suction and pressure surfaces of the blade profile to calculate $Z_w$. Cardamone [24] introduced a modified formula that takes into account variations in inlet/outlet height and gas compressibility to assess the aerodynamic performance of three different turbine cascades at high-subsonic condition. Schobeiri [5] introduced a definition of the generalized lift–solidity coefficient, considering the radius and the meridional velocity changes, to quantify the blade loading in both axial-flow and radial-flow turbomachinery.

In conclusion, the efforts shown above can be simply divided into two main categories following the traditional aerodynamic modification framework:

• The intermediate method (IM): This method involves modifying the blade forces directly, offering a means to manipulate the aerodynamic performance. It offers advantages such as providing more accurate blade force calculation but also has certain limitations that need to be noticed and addressed, such as the constant density assumption.
• The post-processing method (PM): This approach involves modifying the coefficient itself directly and operates within the Prandtl–Glauert modification framework. It provides a more comprehensive means of adjusting the overall characteristics of B2B loading but might have limitations in terms of accuracy and applicability under higher flow speed, especially in the transonic flow regime.

In this paper, an alternative modification will be presented under the IM approach. This modification revolves around extending the traditional incompressible Kutta–Joukowski equation into the compressible flow regime, focusing on enhancing the blade force calculation. The development of the compressible inviscid Kutta–Joukowski equation was initiated by Kasahara in 1955 [25], laying the foundation for this research study. By exploring the compressible flow domain and utilizing an extended Kutta–Joukowski equation, enhancing the understanding and manipulation of blade forces while addressing some limitations associated with traditional approaches is anticipated. The modification in this paper can offer potential enhancements for future turbomachinery design and performance.

2. Extending the Definitions of ${\mathit{Z}}_{\mathit{w}}$ and ${\mathit{C}}_{\mathit{L}}$ into the Compressible Regime

2.1. Brief Comparison and Discussion on Current Definitions

The definition of cascade geometry is provided in Figure 2. In the given context, the positive direction “k” is defined as the direction where the circulation points into the plane. The flow angle is universally defined as the angle between the flow direction and the tangential direction at the inlet or outlet. All the definitions presented in Table 1 have been adapted and converted into the cascade system of this paper, emphasizing the consistency and relevance of the terminology employed in this study. In the following context, the different definitions of $Z_w$ and $C_L$ will be represented in the format (First Author Year). Additionally, the original definition of $Z_w$ is revisited in Appendix A.1.

2.1.1. Incompressible

Formulas (Zweifel 1945), (Song 2000), and (Coull 2013b) are all for calculating $Z_w$ in incompressible flow. Definition (Song 2000) specifically introduced an aspect ratio correction to the original $Z_w$ formula, making it suitable for cases that focus on endwall flow. Formula (Coull 2013b) has a direct connection with (Zweifel 1945), and by dividing the two formulas, $C_o$ is always smaller than $Z_w$ in value. It is important to note that (Coull 2013b) is considered applicable to both compressible and incompressible domains in the original paper. This is achieved by using PM to simulate low-speed behavior and ensure equivalency in high-speed and low-speed pressure distribution (circulation). In essence, the representation in (Coull 2013b) takes on an incompressible form.

2.1.2. Implicit Form

Definitions (Schimtz 2016) and (Coull 2013a), from different perspectives, capture the core essence of calculating B2B loading. The former identifies the role of tangential force on the blade, while the latter highlights the fundamental contribution of circulation as the source of the blade forces. The calculations derived from the implicit formulation are inherently correct. As the implicit formulation does not provide explicit equations, it can be applied to both compressible and incompressible domains simultaneously. However, it is worth noting that the implicit formulation, while robust, requires further analysis and derivation of equations based on the inflow and outflow conditions to determine the exact loading.

2.1.3. Direct Form

The work (Babajee 2013) proposed a direct formulation that provides an explicit calculation for $Z_w$, incorporating inlet and outlet densities and velocities. This formulation offers significant advantages for performing CFD calculations, since all the necessary parameters can be directly extracted from the results of CFD analysis. However, when it comes to design and experimental applications, further elaborations are required. For instance, local density can be computed by using Mach number isentropic relationships, while the outlet tangential velocity can be determined through the utilization of velocity triangles or through the implementation of wake probes. It is worth mentioning that the implicit form and direct form remain accurate and applicable to compressible and incompressible fields simultaneously.

2.1.4. PM

Prandtl and Glauert [26] employed linearization assumptions to derive the basic transformation equations, establishing a connection between low-speed and high-speed regimes. However, this assumption has two significant limitations:

• Gas compressibility can accurately be linearized only for relatively low Mach numbers, typically below 0.6. The nonlinearity of gas behavior intensifies beyond this range.
• Near the sonic point, mathematical singularities arise. Despite the availability of several correction methods, such as the Von Karman–Tsian method, effectively managing the singularity and selecting appropriate similarity relationships in transonic regions pose challenges. Furthermore, there is a lack of a unified physical interpretation for the singularity in aerodynamics.

Consequently, methods encompassing compressibility corrections under PM, such as (Glassman 1972), are less commonly utilized. Therefore, the present study aims to avoid the Prandtl–Glauert transformation to prevent the import of its singularity and linearization limitation.

2.1.5. IM

At present, most of the compressibility corrections for $Z_w$ and $C_L$ primarily focus on the IM, which involves modifying the aerodynamic forces themselves. This is also the key focus of this paper, as the comparison and analysis of different correction approaches are addressed in this part. In Appendix A.1, the classical definition of the incompressible $Z_w$ is briefly reviewed. Its core application equation is Equation (A6), which assumes that the axial velocities and densities at the inlet and outlet are consistent. This equation can be interpreted as mass flow conservation, i.e., Equation (1). By substituting Equation (1) into the formulation of (Babajee 2013), the equation presented in Equation (2) can be derived. This equation forms the foundation of (Cardamone 2006) and (McQuilling 2007). In (Cardamone 2006), the influence of the height difference between the inlet and outlet is considered. Although it is defined as $C_L$ by the authors, the derivation suggests that it is essentially related to $Z_w$. Similarly, (Gier 2010) also takes a similar approach. In (McQuilling 2007), the computation of circulation adopts the suction surface length. However, this choice is not rigorous, and it is recommended to utilize the pitch length to satisfy the circulation theorem. Moreover, (Yang 2020) approaches the problem of compressibility corrections from another perspective by using the velocity coefficient distribution on the suction side and pressure side of the blade to determine $Z_w$. This method is applicable during specific blade design stages to control the acceleration and deceleration of fluid but may not be suitable for the preliminary design stage where the inlet and outlet flow angles remain the primary controlling parameters.

In general, the correction methods based on the IM indeed incorporate the density factor in the formula, thus accurately capturing the compressibility characteristics of gas. Compared with the original definition of $Z_w$, these correction methods account for the effect of density changes in the calculation results. By integrating the density term into the equation, these correction methods consider the compressibility of fluids and result in more accurate computations.

$${\rho }_1{V}_{z1}={\rho }_2{V}_{z2}$$

(1)

$$\begin{array}{cc}\hfill Z_w& ={\displaystyle \frac{2t}{c_z}}{\displaystyle \frac{{\rho }_2{V}_{z2}(V_{u1}+V_{u2})}{{\rho }_2{V}_2^2}}\hfill \\ \hfill & \hfill ={\displaystyle \frac{2t}{c_z}}{\displaystyle \frac{{\rho }_2{V}_{z2}^2(\begin{array}{|c|}\hline \frac{{\rho }_1{V}_{z1}}{{\rho }_2{V}_{z2}}\\ \hline\end{array}\frac{V_{z1}}{V_{z2}}cot{\alpha }_1+cot{\alpha }_2)}{{\rho }_2{V}_2^2}}\hfill \end{array}$$

(2)

2.2. The Influence of Compressibility on $Z_w$

Figure 3 illustrates the density ratio across a range of inlet and outlet Mach numbers, demonstrating that larger velocity differences correspond to greater density variations. Therefore, under such conditions, applying the incompressible flow assumption across the entire flow domain becomes unreasonable, necessitating the consideration of gas compressibility.

According to the previous discussion by (McQuilling 2007), the influence of compressibility was considered by using the axial velocity ratio between the inlet and the outlet. In the revised formula, the suction surface length S is replaced by the pitch length t, denoted by $Z{w}_{comp}$. The original incompressible calculation by (Zweifel 1945) is denoted by $Z{w}_{inc}$. The difference D between the two definitions is defined as shown in Equation (3). In the comparison, the blade size, pitch, and inlet and outlet flow angles are kept constant. The boxed part of the formula in Equation (3) is now defined separately as a function f, considering mass conservation (Equation (1)) and the isentropic relationship between density and the Mach number (Equation (4)). The function f can be viewed as a function of the inlet Mach number and the outlet Mach number. The analysis is as follows:

• D is always non-negative in the turbine regime, meaning that $Z{w}_{inc}\ge Z{w}_{comp}$. As mentioned in the Introduction section, calculating $Z_w$ based on the incompressible assumption while keeping other parameters constant results in a higher estimate compared with the compressible case.
• D is proportional to f, i.e., $D\propto f$. According to Figure 3 for turbine cases, the larger the Mach number difference between the outlet and inlet, the smaller the ${\displaystyle \frac{{\rho }_2}{{\rho }_1}}$, and the larger the f, leading to a greater D. This indicates that the larger the Mach number difference between the inlet and outlet in the turbine cascade, the greater the overestimation of $Z{w}_{inc}$.

  $$D=Z{w}_{inc}-Z{w}_{comp}={\displaystyle \frac{2t}{c_z}}{sin}^2{\alpha }_2\begin{array}{|c|}\hline (1-{\displaystyle \frac{V_{z1}}{V_{z2}}})\\ \hline\end{array}cot{\alpha }_1\ge 0$$

  (3)

  $${\displaystyle \frac{\rho }{{\rho }_{\infty }}}={(1+{\displaystyle \frac{\gamma -1}{2}}M{a}^2)}^{{\displaystyle \frac{1}{1-k}}}$$

  (4)

  $$f(M{a}_1,M{a}_2)=1-{\displaystyle \frac{V_{z1}}{V_{z2}}}=1-{\displaystyle \frac{{\rho }_2}{{\rho }_1}}=1-{\left({\displaystyle \frac{1+\frac{\gamma -1}{2}M{a}_1^2}{1+\frac{\gamma -1}{2}M{a}_2^2}}\right)}^{{\displaystyle \frac{1}{1-k}}}$$

  (5)

Specifically, in the process of blade profile modular design from high speed to low speed, the effect of compressibility on $Z_w$ has not been explicitly considered. Ensuring that the modular design conditions at both high and low speeds, particularly the blade load $Z_w$, are truly equivalent remains a challenge. Is the smaller $Z_w$ under high-speed conditions compared with low-speed conditions ensuring that the absolute values of $Z_w$ at high and low speeds are consistent? Or is it sufficient to ensure that the flow angle and the velocity triangles at the inlet and outlet are consistent, considering the two conditions to be similar, even though the absolute values of $Z_w$ at high and low speeds are not equal? This issue of modular design similarity remains controversial and requires further exploration, which is beyond the scope of this paper. The purpose of this paper is merely to supplement the definition of blade load and to consider the impact of gas compressibility. However, at least, this paper clearly presents the key controversy in the high- and low-speed modular design issue: the blade load calculation under high-speed conditions is inconsistent with that under low-speed conditions, even when the geometric dimensions and inlet and outlet airflow angles are similar. The greater the difference in inlet and outlet speeds, the smaller the $Z_w$ considering compressibility compared with the incompressible $Z_w$.

2.3. The Extended Kutta–Joukowski Theorem

While existing research has partially addressed compressibility corrections in $Z_w$ and $C_L$ definitions, further investigation is crucial due to the following limitations:

• The current commonly used method for calculating lift is based on the Kutta–Joukowski theorem, as shown in Equation (A2), which is derived from incompressible flow. The reference velocity and density are typically chosen as parameters related to the freestream, assuming they remain constant throughout the flow. However, in transonic and supersonic flows, velocity and density experience significant variations. As for the reference velocity and density for compressor and turbine cascades, the inlet and outlet velocity and density are often chosen separately. The calculation of lift using an inlet or outlet parameter alone is inaccurate, leading to errors in blade loading.
• Among other metrics related to compressibility, the Mach number is frequently used as a dimensionless quantity. While $Z_w$ and $C_L$ calculations typically use velocity instead, the generality of the Mach number is more apparent in the modular design of cold and hot states, as well as loss calculations. Currently, there is a lack of blade loading definitions that directly incorporate Mach number effects.

In this section, based on Kasahara’s work [25] about the compressible activities of inviscid Kutta–Joukowski theorem derivation, i.e., the extended Kutta–Joukowski theorem, compressibility considerations and Mach number-based equations for calculating $Z_w$ and $C_L$ will be derived.

Based on the calculation path of the non-dimensional numbers shown in Figure 1, the core of the IM is to accurately determine the aerodynamic forces on the blade. Kasahara’s correction also falls within this framework. In the traditional incompressible Kutta–Joukowski theorem, either the inlet or outlet conditions are typically used independently to calculate the aerodynamic forces. In contrast, the extended method replaces the calculation parameters with average parameters, thereby comprehensively considering the differences between the inlet and outlet states. This becomes particularly evident in transonic flows, where using either the inlet or outlet conditions alone may lead to an overestimation of the aerodynamic forces. By introducing average parameters, the extended Kutta–Joukowski theorem can more accurately predict the aerodynamic loading on the blade under transonic flow conditions.

The key of the extended Kutta–Joukowski theorem is still based on the conservation of mass. By considering an averaged state, Equation (6) is obtained.

$${\rho }_1{V}_{z1}t={\rho }_2{V}_{z2}t={\rho }_m{V}_{zm}t$$

(6)

Through algebraic manipulations, the mass conservation relation is obtained as shown in Equation (7). In engineering practice, during the process of calculating turbine velocity triangles, the negative sign is usually ignored due to the velocity characteristics being on opposite sides. The definition of inlet and outlet velocity vectors typically uses the commonly accepted unsigned vectors. Referring to Figure 2, the vector calculation relationship for the average velocity is given in Equation (8). By substituting Equation (8) into Equation (7), the calculation formula for the average density, Equation (9), can be derived. It is noteworthy that Equation (8) indicates that the average velocity is based on geometric averaging, while Equation (9) shows that the average density is a harmonic average. The boxed equation in Equation (8) represents the z-component of the average velocity, which is essential to calculating the averaged properties.

$$\left\{\begin{array}{c}\hfill {\rho }_1{\rho }_2{V}_{1}sin{\alpha }_{1}t={\rho }_2{\rho }_m{V}_{m}sin{\alpha }_{m}t\hfill \\ \hfill {\rho }_1{\rho }_2{V}_{2}sin{\alpha }_{2}t={\rho }_1{\rho }_m{V}_{m}sin{\alpha }_{m}t\hfill \end{array}\right.\Rightarrow {\rho }_1{\rho }_2(V_{1}sin{\alpha }_1+V_{2}sin{\alpha }_2)=({\rho }_1+{\rho }_2){\rho }_m{V}_{m}sin{\alpha }_m$$

(7)

$$\mathbf{Vm}={\displaystyle \frac{1}{2}}({\mathbf{V}}_{\mathbf{1}}+{\mathbf{V}}_{\mathbf{2}})\Rightarrow \left\{\begin{array}{c}\begin{array}{|c|}\hline V_{zm}={\displaystyle \frac{1}{2}}(V_{z1}+V_{z2})\\ \hline\end{array}\hfill \\ V_{um}={\displaystyle \frac{1}{2}}|V_{u1}-V_{u2}|\hfill \\ V_m={\displaystyle \frac{1}{2}}\sqrt{{(V_{z1}+V_{z2})}^2+{(V_{u1}-V_{u2})}^2}\hfill \end{array}\right.$$

(8)

$$({\rho }_1+{\rho }_2){\rho }_m=2{\rho }_1{\rho }_2\Rightarrow {\displaystyle \frac{1}{{\rho }_m}}={\displaystyle \frac{1}{2}}({\displaystyle \frac{1}{{\rho }_1}}+{\displaystyle \frac{1}{{\rho }_2}})$$

(9)

The lift force in Equation (A1) can be modified with the averaged properties, as shown in Equation (10).

$${\mathbf{F}}_{\mathbf{L}}={\rho }_m{\mathbf{V}}_{\mathbf{m}}\times \mathbf{\Gamma }$$

(10)

2.4. Modified Definitions of $Z_w$ and $C_L$

With the basic relationships between the average parameters and the inlet and outlet parameters established through the derivations and the modified $F_L$, the core modified definitions of $Z_w$ or $C_L$ in this paper can be obtained by incorporating their original definitions without simplifying assumptions, along with the isentropic Mach relationship.

For further analysis, the key dimensionless average parameters need to be derived. Based on the previously obtained average pressure and density, the average Mach number, denoted by $M{a}_m$, can be obtained. The derivation process involves the selection of upstream parameters: for experimental investigations, the stagnation parameters within the settling chamber are chosen, whereas for design and CFD simulations, the stagnation parameters at the inlet are chosen. By utilizing the isentropic relation between density and the Mach number in Equation (4) and referencing Equation (8), the expression for $M{a}_m$ is Equation (11). To simplify subsequent calculations, consider a function $\mathcal{M}\left(Ma\right)$, as defined in Equation (12). By introducing a reference temperature, Equation (12) enables the transformation of velocity terms into equivalent Mach number expressions.

$$\left\{\begin{array}{c}{\displaystyle \frac{{\rho }_m}{{\rho }_{\infty }}}=2{\displaystyle \frac{\frac{{\rho }_1}{{\rho }_{\infty }}\frac{{\rho }_2}{{\rho }_{\infty }}}{\frac{{\rho }_1}{{\rho }_{\infty }}+\frac{{\rho }_2}{{\rho }_{\infty }}}}\hfill \\ M{a}_m=\sqrt{{\displaystyle \frac{2}{\gamma -1}}[{\left({\displaystyle \frac{{\rho }_m}{{\rho }_{\infty }}}\right)}^{1-k}-1]}\hfill \end{array}\right.$$

(11)

$$\mathcal{M}\left(Ma\right)={\displaystyle \frac{V}{\sqrt{\gamma R{T}_{\infty }}}}={\displaystyle \frac{V}{\sqrt{\gamma RT}}}{\displaystyle \frac{\sqrt{\gamma RT}}{\sqrt{\gamma R{T}_{\infty }}}}={\displaystyle \frac{Ma}{\sqrt{1+\frac{\gamma -1}{2}M{a}^2}}}$$

(12)

The average flow angle, denoted by ${\alpha }_m$, is another crucial average parameter with a clear physical interpretation. For instance, $tan{\alpha }_m$ represents the ratio of axial velocity to tangential velocity, which can be approximated as the ratio of axial force to tangential force. By analyzing ${\alpha }_m$, the dominant factors influencing blade aerodynamic forces can be effectively identified: when ${\alpha }_m>{45}^{\circ }$, axial force dominates; when ${\alpha }_m\approx {45}^{\circ }$, axial and tangential forces are equally important; and when ${\alpha }_m<{45}^{\circ }$, tangential force prevails. To simplify calculations, previous studies often employed simple arithmetic averaging methods, such as $tan{\alpha }_m={\displaystyle \frac{1}{2}}(tan{\alpha }_1+tan{\alpha }_2)$ or $tan{\alpha }_m=tan{\displaystyle \frac{{\alpha }_1+{\alpha }_2}{2}}$. However, these simplifications originate from the incompressible flow assumption and neglect the influence of inlet and outlet velocity differences. In this paper, as referring to Equation (13), the authors argue that the calculation of ${\alpha }_m$ should consider the weighted effects of different inlet and outlet velocities.

$${\alpha }_m={tan}^{-1}{\displaystyle \frac{V_{zm}}{V_{um}}}={\displaystyle \frac{V_{1}sin{\alpha }_1+V_{2}sin{\alpha }_2}{|V_{1}cos{\alpha }_1-V_{2}cos{\alpha }_2|}}={\displaystyle \frac{\mathcal{M}\left(M{a}_1\right)sin{\alpha }_1+\mathcal{M}\left(M{a}_2\right)sin{\alpha }_2}{|\mathcal{M}\left(M{a}_1\right)cos{\alpha }_1-\mathcal{M}\left(M{a}_2\right)cos{\alpha }_2|}}$$

(13)

Finally, based on the preceding knowledge, the modified definitions of $Z_w$ and $C_L$ can be obtained, as shown in Equation (14) and Equation (15), respectively. With the help of the extended Kutta–Joukowski theorem, the relationship between $Z_w$ and $C_L$ can still follow Equation (18), i.e., the ratio of $Z_w$ and $C_L$ is only in $sin{\alpha }_m$, where ${\alpha }_m$ is a function of ${\alpha }_1$, ${\alpha }_2$, $M{a}_1$, and $M{a}_2$, making it a valuable parameter in preliminary blade modular design.

It is important to note that unlike conventional definitions that utilize either the inlet or outlet dynamic pressure head, the denominator terms in Equations (14) and (15) employ the dynamic pressure head of the average flow state. Consequently, the calculated values may differ from those obtained by using traditional definitions. However, the fundamental definition remains consistent: the ratio of the aerodynamic force on the blade (or a specific force component) to a reference ideal force. Utilizing the average-state dynamic pressure head offers the advantage of simplifying the calculation formulas by eliminating the explicit density term. This simplification does not imply neglecting gas compressibility effects; instead, these effects are implicitly incorporated both within the calculation of the average-state parameters and the Mach numbers.

$$Z_w(M{a}_1,M{a}_2,{\alpha }_1,{\alpha }_2)={\displaystyle \frac{2t}{c_z}}{\displaystyle \frac{{\rho }_m{V}_{zm}(V_{u1}+V_{u2})}{{\rho }_m{V}_m^2}}={\displaystyle \frac{2t}{c_z}}sin{\alpha }_m{\displaystyle \frac{\mathcal{M}\left(M{a}_1\right)cos{\alpha }_1+\mathcal{M}\left(M{a}_2\right)cos{\alpha }_2}{\mathcal{M}\left(M{a}_m\right)}}$$

(14)

$$C_L(M{a}_1,M{a}_2,{\alpha }_1,{\alpha }_2)={\displaystyle \frac{2t}{c_z}}{\displaystyle \frac{{\rho }_m{V}_m(V_{u1}+V_{u2})}{{\rho }_m{V}_m^2}}={\displaystyle \frac{2t}{c_z}}{\displaystyle \frac{\mathcal{M}\left(M{a}_1\right)cos{\alpha }_1+\mathcal{M}\left(M{a}_2\right)cos{\alpha }_2}{\mathcal{M}\left(M{a}_m\right)}}$$

(15)

Equations (16) and (17) present modified definitions for $Z_w$ and $C_L$, respectively, incorporating gas compressibility and utilizing the outlet dynamic pressure head for normalization. These definitions are consistent with existing conventions, enabling the direct utilization of the established empirical loading choice without the need for further transformations.

$$Z_w^{\prime }(M{a}_1,M{a}_2,{\alpha }_1,{\alpha }_2)={\displaystyle \frac{2t}{c_z}}{\displaystyle \frac{{\rho }_m{V}_{zm}(V_{u1}+V_{u2})}{{\rho }_2{V}_2^2}}={\displaystyle \frac{2t}{c_z}}{\displaystyle \frac{{\rho }_2{V}_{z2}(V_{u1}+V_{u2})}{{\rho }_2{V}_2^2}}={\displaystyle \frac{2t}{c_z}}sin{\alpha }_2{\displaystyle \frac{\mathcal{M}\left(M{a}_1\right)cos{\alpha }_1+\mathcal{M}\left(M{a}_2\right)cos{\alpha }_2}{\mathcal{M}\left(M{a}_2\right)}}$$

(16)

$$C_L^{\prime }(M{a}_1,M{a}_2,{\alpha }_1,{\alpha }_2)={\displaystyle \frac{2t}{c_z}}{\displaystyle \frac{{\rho }_m{V}_m(V_{u1}+V_{u2})}{{\rho }_2{V}_2^2}}={\displaystyle \frac{2t}{c_z}}{\displaystyle \frac{{\rho }_m{V}_{zm}/sin{\alpha }_m(V_{u1}+V_{u2})}{{\rho }_2{V}_2^2}}={\displaystyle \frac{2t}{c_z}}{\displaystyle \frac{sin{\alpha }_2}{sin{\alpha }_m}}{\displaystyle \frac{\mathcal{M}\left(M{a}_1\right)cos{\alpha }_1+\mathcal{M}\left(M{a}_2\right)cos{\alpha }_2}{\mathcal{M}\left(M{a}_2\right)}}$$

(17)

Equation (18) presents the ratio between $Z_w$ and $C_L$. Notably, this ratio remains consistent, equating to $sin{\alpha }_m$, regardless of whether the average-state definitions (Equations (14) and (15)) or the outlet-based definitions (Equations (16) and (17)) are employed. This consistency highlights the fundamental relationship captured by Equation (18): $Z_w$ represents a specialized case of $C_L$, considering only the circumferential force component. Given that ${\alpha }_m$ typically ranges from 0° to 90°, the value of $sin{\alpha }_m$ falls between 0 and 1, implying that $Z_w$ is always less than or equal to $C_L$. This aligns with the physical interpretation of these parameters. Furthermore, this definition offers a more concise and intuitive representation of the relationship between $Z_w$ and $C_L$ compared with the relationship between $C_o$ and $Z_w$ presented by Coull et al. [8].

$${\displaystyle \frac{Z_w}{C_L}}={\displaystyle \frac{Z_w^{\prime }}{C_L^{\prime }}}=sin{\alpha }_m$$

(18)

Additionally, it is also important to note the limitations of the method presented in this paper. In the derivation process, whether it is Equation (1) or Equation (6), the assumption of two-dimensional flow is made without considering the impact of changes in the inlet and outlet areas. In other words, the method presented in this paper is applicable to situations where $AVDR\approx 1$, such as two-dimensional blade design and planar cascade tests. For actual turbomachinery, there are complex effects, such as endwall influence and changes in the streamtube along the meridional plane, which require additional corrections and cannot be directly applied by using the method presented in this paper.

In conclusion, in this section, based on Kasahara’s work on the derivation of the extended Kutta–Joukowski theorem, the compressible $Z_w$ and $C_L$ definitions are proposed:

• The core of the IM is to accurately determine the aerodynamic forces on the blade. Kasahara’s correction work also falls within this framework. Unlike the traditional approach of using either the inlet or outlet state parameters independently, the IM employs average parameters for the calculations, thereby comprehensively considering the differences between the inlet and outlet states. This becomes particularly important in transonic flows, as using a single state parameter may lead to an overestimation of the aerodynamic loading.
• The improvement offered by the IM lies in the introduction of average parameters, combined with the effects of the Mach number, enabling more accurate predictions of blade aerodynamic loading under compressible flow conditions.
• The theoretical foundation of the IM remains the principle of mass conservation. By considering an averaged state, the mass conservation equation, as well as the calculation formulas for average velocity and average density, based on geometric and harmonic averaging, respectively, can be derived.
• With the basic relationships between the average parameters and the inlet/outlet parameters established, by incorporating the definitions of the lift coefficient without simplifying assumptions, along with the isentropic Mach relationship, the core modified formulas of the IM can be obtained.

3. Application and Validation of Modified Definitions

3.1. Research Objective and Numerical Methods

This paper focuses on the VKI-RG (also known as VKI-LS 59) turbine cascade, which was extensively tested in four European cascade wind tunnels [27]. Its key parameters are presented in

Table 2. Key parameters of the VKI-RG cascade.

Parameter	Value
Chord length c, mm	60
Solidity $\sigma$	1.408
Stagger angle, °	33.3
Inlet angle ${\alpha }_1$, °	30
Design state $M{a}_{is2}$	≈0.8

.

The proposed corrections for $Z_w$ and $C_L$ in Section 2.4 are applicable to various scenarios, including design, experimental, and simulation settings. Due to the richness and accessibility of data, this study utilizes numerical simulations for investigation, with Figure 4 illustrating the computational domain and mesh employed for the VKI-RG cascade. To increase the stability of the simulation, the blade height for the computational model is chosen as $12.5\%C$, which is 7.5 mm. A hybrid unstructured mesh consisting of prisms and hexahedra is employed. As shown in Figure 4, a hexahedral mesh is used in the boundary layer region for better resolution near the blade surface, while the mainstream region is filled with a prismatic mesh.

Grid independence is validated by comparing results from different mesh scales, ranging from 14 thousand to 2.04 million cells, as shown in

Table 3. Grid independency validation setup.

No.	Total Grid Number, Million	Single Layer Grid Number, Thousand	Layers
1	0.14	28	5
2	0.28	28	10
3	0.44	44	10
4	0.88	44	20
5	1.50	75	20
6	2.04	102	20

. The table illustrates the setup for grid independence validation, where various cases with increasing grid numbers are tested to ensure that the results are not significantly affected by further mesh refinement. As depicted in Figure 5, the loss and outlet flow angle (${\theta }_2$) are chosen as the validated parameters. The loss definition is shown in Equation (19). It can be observed that the loss and ${\theta }_2$ values stabilize as the grid number increases, indicating that further refinement beyond 1.5 million cells does not result in significant changes in the results.

Finally, the mesh contains over 75 thousand cells per layer with a total of 20 layers, resulting in a total computational mesh size of approximately 1.5 million cells to meet the grid independence requirements, providing a balance between computational efficiency and accuracy.

$$Loss={\displaystyle \frac{P_{t,inlet}-P_{t,outlet}}{P_{t,outlet}-P_{s,outlet}}}$$

(19)

The simulations are conducted by using Ansys CFX 18.2 for steady-state RANS simulations. Both convective and diffusive terms are discretized with the High Resolution scheme which is a blended second-order scheme that combines accuracy and stability. The solution method is implicit, and the convergence criterion is set to a maximum residual of $1\times {10}^{-5}$. The k-$\omega$ SST turbulence model is employed for its suitability in capturing the complex flow behavior within the turbine cascade.

The basic CFD boundary conditions are listed in

Table 4. Basic CFD configurations for validation.

Configuration	Value
Inlet total pressure, kPa	155
Inlet total temperature, K	290
Inlet velocity vector	[0.866, 0.50, 0]
Outlet static pressure, kPa	100.6
$M{a}_{is2}$	0.810
Max $Y^{+}$	<2.5
Turbulence model	k-$\omega$ SST
Turbulence intensity	1%

. The inlet boundary conditions are specified as total temperature, total pressure, and flow direction; the outlet boundary condition is set as static pressure. The blade surface is defined as a no-slip wall, and the max $Y^{+}$ is kept at less than 2.5 along the wall to ensure proper resolution of the boundary layer. The pitchwise direction is set as a translational periodic boundary, and both spanwise directions utilize symmetry boundary conditions.

3.2. Numerical Method Validation

The operating condition with an outlet isentropic Mach number of 0.810 is selected for validation purposes, with experimental data obtained from [27]. During the post-processing of the computational results, the inlet plane was positioned $41.6\%$ of the chord length upstream of the leading edge, while the outlet plane was located $35\%$ of the chord length downstream of the trailing edge. This choice aligns with the measurement locations used in the reference experimental study. Area averaging was employed for calculating average flow parameters to ensure consistency between the simulation and experimental result processing methods, thereby enhancing the comparability of the results.

The correction method presented in this paper relies on key parameters, including inlet and outlet flow angles, as well as inlet and outlet Mach numbers. To ensure comparability with experimental data, the outlet flow angle was chosen as the primary quantitative parameter for validating the numerical method. Figure 6 illustrates the isentropic Mach number distribution on the VKI-RG cascade, where “CFD” represents the CFX simulation results and “EXP” represents the experimental results. On the suction surface, within the dimensionless distance range of 0–0.6, the CFD results are slightly lower than the EXP data, with a more pronounced discrepancy observed in the 0–0.15 range. However, good agreement between CFD and EXP is achieved within the 0.6–1.0 range. The isentropic Mach number distribution on the pressure surface reveals a close match between CFD and EXP results. In the blade trailing edge region, the CFD results exhibit significant fluctuations and deviate from the EXP data. Overall, the simulated B2B loading distribution demonstrates good agreement with the experimental results.

Table 5. Outlet flow angle comparison between CFD and experiment.

Conditions	Outlet Flow Angle, °
EXP	23.03
CFD	23.24
Absolute Error	0.21

presents a comparison of outlet flow angles obtained from experimental measurements and CFD. The table shows that the CFD prediction for the outlet flow angle (23.24°) closely matches the experimentally measured value (23.03°), with an absolute error of only 0.21°.

In conclusion, the CFX simulation results for the VKI-RG flow field data exhibit a high degree of consistency with the experimental results, confirming the reliability of the numerical simulation methodology.

3.3. Computational Setup and Operating Conditions

This study investigates seven distinct isentropic Mach number conditions, ranging from 0.6 to 1.2, as detailed in

Table 6. Isentropic Mach numbers and corresponding back pressures for investigated configurations.

No.	${\mathbf{Ma}}_{\mathbf{is}2}$	Back Pressure, kPa
1	0.6	121.5
2	0.7	111.7
3	0.81	100.6
4	0.9	91.5
5	1.0	81.7
6	1.1	72.5
7	1.2	64.3

. Throughout the simulations, only the back pressure is varied, while all other settings remain consistent with those outlined in the previous section.

4. Results and Discussion

Figure 7 presents a comparative analysis of various $Z_w$ calculation methods and their relative errors, highlighting the impact of compressibility and input parameter choices.

Figure 7a illustrates how $Z_w$ varies with the isentropic outlet Mach number ($M{a}_{is2}$) for five distinct methods: (Zweifel 1945), (Babaejee 2013), (McQuilling 2007), and Equations (14) and (16). While all methods show a decreasing trend in $Z_w$ as $M{a}_{is2}$ increases, Equation (14) consistently yields the highest values. This difference stems from its unique normalization approach: Equation (14) utilizes the averaged dynamic pressure head, while the other methods employ the outlet dynamic pressure head, which is generally larger and results in lower normalized $Z_w$ values.

Figure 7b focuses on the relative error of three methods—(Babaejee 2013), (McQuilling 2007), and Equation (16)—compared with a baseline method, (Zweifel 1945). The relative error, calculated by using Equation (20), becomes more pronounced with the increase in $M{a}_{is2}$ for all three methods. This trend underscores the growing deviation from the incompressible (Zweifel 1945) method, as compressibility effects become more significant at higher Mach numbers. Notably, all three methods consistently produce lower $Z_w$ values than (Zweifel 1945), indicating that incorporating compressibility modifications leads to a reduction in the predicted Zweifel coefficient. The near-identical relative errors observed across the three methods further validate the accuracy and robustness of the proposed compressibility corrections.

The choice of the $Z_w$ calculation method also involves considering the trade-off between simplicity and accuracy, particularly at high speeds. The incompressible (Zweifel 1945) method, requiring only ${\alpha }_1$ and ${\alpha }_2$, is straightforward but less accurate at high Mach numbers. In contrast, (McQuilling 2007) necessitates four input variables (${\alpha }_1$, ${\alpha }_2$, and inlet/outlet axial velocities), while (Babaejee 2013) demands six (inlet/outlet densities, axial velocities, and tangential velocities).

Equations (14) and (16) strike a balance by utilizing four parameters (${\alpha }_1$, ${\alpha }_2$, $M{a}_1$, and $M{a}_2$), aligning with (McQuilling 2007) in terms of data acquisition complexity. Importantly, they achieve this without introducing difficult-to-obtain parameters, making them practical choices for accurate $Z_w$ calculations, especially in high-speed scenarios.

Furthermore, Equation (16) exhibits good compatibility with existing empirical guidelines for selecting $Z_w$ values due to its use of a conventional non-dimensionalization approach. In contrast, Equation (14)’s different non-dimensionalization reference results in a numerical range that deviates from established empirical data. However, Equation (14) is particularly well suited for airfoil similarity studies because it derives both blade forces and the non-dimensionalization baseline from a weighted average of inlet and outlet velocities, providing a more physically representative model of flow conditions and blade loading.

$$ϵ={\displaystyle \frac{Z_w-Z_{w,baseline}}{Z_{w,baseline}}}\times 100\%$$

(20)

Figure 8 presents a comparative analysis of the trends exhibited by $C_L$ and $Z_w$ for different $M{a}_{is2}$, calculated by using two distinct sets of equations. Figure 8a,b illustrate the contrasting trends of $C_L$ and $Z_w$ calculated by using average-state and outlet-based definitions, respectively.

Figure 8a shows both $C_L$ (Equation (15)) and $Z_w$ (Equation (14)) decreasing with the increase in the outlet isentropic Mach number ($M{a}_{is2}$), with $C_L$ being consistently higher than $Z_w$. This trend aligns with fundamental flow physics: as $M{a}_{is2}$ increases, local supersonic regions and shock waves may emerge on the blade surface, introducing shockwave drag. This shockwave drag is a pressure drag, effectively reducing $C_L$ and $Z_w$.

While Figure 8b shows $C_L$ (Equation (17)) increasing with $M{a}_{is2}$, the trend for $Z_w$ (Equation (16)) is less pronounced than in Figure 8a, making it less suitable for analyzing the influence of compressibility on blade loading.

Figure 9 illustrates the relationship between the averaged flow angle (${\alpha }_m$) and the isentropic Mach number at the outlet plane ($M{a}_{is2}$), comparing different calculation methods. The light-blue line with circular markers represents the variation in ${\alpha }_m$ with the increase in $M{a}_{is2}$ calculated by using Equation (13), while the orange dashed line with x markers shows the corresponding trend of $\tan \left({\alpha }_m\right)$, also derived from Equation (13), which represents the ratio of axial force ($F_z$) to tangential force ($F_u$). As the outlet Mach number increases, the averaged flow angle decreases, indicating a shift in flow direction towards a more tangential orientation. However, the rate of decrease in ${\alpha }_m$ slows down as $M{a}_{is2}$ increases.

Two additional methods for determining the averaged flow angle linearly are presented. The blue line with triangular markers represents a simple arithmetic average of the flow angles at inlet and outlet, denoted by (${\displaystyle \frac{{\alpha }_1+{\alpha }_2}{2}}$); it remains consistently below 45° across all $M{a}_{is2}$, suggesting axial force dominance throughout. Conversely, the purple line with arrow markers, representing the averaged flow angle calculated by using the arctangent function (${\tan }^{-1}{\displaystyle \frac{\tan {\alpha }_1+\tan {\alpha }_2}{2}}$), stays above 45° for all $M{a}_{is2}$, indicating tangential force dominance. These two methods, while similar in trend, exhibit an increasing averaged flow angle as the outlet Mach number increases, contrasting with the decreasing trend observed for the methods based on Equation (13).

These contrasting trends highlight the significant influence of the averaging method, particularly the inclusion or exclusion of velocity weighting, on the flow angle and the resulting interpretation of force dominance.

The velocity-weighted approach of Equation (13) appears to capture a more agreeable and potentially more accurate representation of the flow physics, revealing a transition in force dominance that is not apparent in the non-weighted methods. This finding underscores the importance of carefully considering the averaging method and its underlying assumptions when analyzing flow angles and inferring force characteristics, especially in scenarios with varying Mach numbers.

Based upon the preceding discussion, a strict set of similarity criteria for blade modular design are proposed, encompassing three primary aspects:

• Mach number-based velocity similarity: This criterion ensures that the flow velocities are scaled appropriately between the model and the real blade, maintaining similar compressibility effects.
• Blade surface pressure distribution similarity: This criterion emphasizes matching the pressure distribution on the blade surface, either through the isentropic Mach number or the pressure coefficient. This ensures similar aerodynamic loads acting on the blade.
• Averaged flow angle-based blade force similarity: This criterion ensures that the overall force vectors acting on the blade are scaled proportionally between the model and the real blade. This can be achieved by matching the averaged flow angle (${\alpha }_m$). Notably, as demonstrated in Equation (18), this criterion is interchangeable with simultaneously satisfying the similarity of both the Zweifel coefficient ($Z_w$) and the lift coefficient ($C_L$). This interchangeability arises because the ratio of $Z_w$ to $C_L$ is solely a function of ${\alpha }_m$.

5. Conclusions

This study investigated the impact of compressibility on the blade loading parameters, specifically the Zweifel coefficient ($Z_w$) and lift coefficient ($C_L$), in turbine cascades, and was validated based on CFD simulations of the VKI-RG turbine cascade. Traditional incompressible methods for calculating these parameters were found to be inadequate at high Mach numbers, necessitating the development of compressibility corrections. The key conclusions of this work are as follows:

• Incorporating compressibility corrections is crucial to accurately predicting $Z_w$ and $C_L$ at high Mach numbers, as demonstrated by the increasing deviation of traditional methods from compressible results.
• The IM, based on an extension of the Kutta–Joukowski theorem for compressible flow, provides a more physically representative model for blade loading calculations by utilizing averaged flow properties. It is worthy to note that the basic assumption is 2D flow. For complex real-world applications, such as inconsistent inlet and outlet areas, the calculation method proposed in this paper cannot be directly applied and requires additional corrections.
• The direct integration of the Mach number into the modified definitions of $Z_w$ and $C_L$ simplifies the equations and aligns them with common practices in compressible flow analysis.
• The averaged flow angle (${\alpha }_m$), calculated by using a velocity-weighted approach, emerges as a key parameter for blade similarity studies. It ensures the proportional scaling of overall force vectors and reflects the influence of compressibility on blade loading.
• The modified definitions offer a robust and practical method for accurately predicting blade loading in compressible flow scenarios, ultimately contributing to the improved modular design of turbines and cascade experiments.