Direct Numerical Simulation of Flow and Heat Transfer in a Compressor Blade Passage Across a Range of Reynolds Numbers

Abstract

This study employs Direct Numerical Simulation (DNS) to investigate the flow and heat transfer characteristics in a compressor blade passage at five Reynolds numbers ($Re=1.091\times {10}^5$, $1.229\times {10}^5$, $1.367\times {10}^5$, $1.506\times {10}^5$, and $1.645\times {10}^5$). A recent method based on local inviscid velocity reconstruction is applied to define and calculate boundary layer parameters, whereas the Rortex vortex identification method is used to analyze turbulent vortical structures. Results indicate that $Re$ significantly affects separation bubble size, transition location, and reattachment behavior, thereby altering wall heat transfer characteristics. On the pressure surface, separation and early transition are observed at higher $Re$, with the Nusselt number ($Nu$) remaining high after transition. On the suction surfaces, separation occurs such that large-scale separation at low $Re$ reduces $Nu$, while reattachment combined with turbulent mixing at high $Re$ significantly increases $Nu$. Turbulent vortical structures enhance near-wall fluid mixing through induced ejection and sweep events, thereby promoting momentum and heat transport. As $Re$ increases, the vortical structures become denser with reduced scales and the peaks in heat flux move closer to the wall, thus improving convective heat transfer efficiency.

1. Introduction

In aerospace engines and gas turbines, the flow and heat transfer characteristics within blade passages are critical factors that influence aerodynamic performance and thermal efficiency [1]. The flow environment in these regions is extremely complex, especially under moderate to low Reynolds number ($Re$) conditions ($Re\sim {10}^5$). Under such conditions, phenomena such as boundary layer separation, laminar-to-turbulent transition, and the formation of laminar separation bubbles are particularly prevalent [2]. These flow phenomena are typically influenced by numerous factors.

In experimental research, Roberts [3] investigated the performance of NACA 65 airfoils at various $Re$ (as low as $Re=0.4\times {10}^5$) using low-speed cascade wind tunnel experiments, and examined the effects of laminar separation. Schreiber et al. [4] studied boundary layer transition on CDA airfoils using oil flow and liquid crystal coating techniques at $Re$ from $0.7\times {10}^5$ to $3\times {10}^6$ and turbulence intensities between $Tu=0.7\%$ and $4\%$. Their results indicated that transition occurs within separation bubbles at low turbulence levels and advances at high turbulence levels and high $Re$. Hilgenfeld and Pfitzner [5] examined the effects of wake passing on unsteady boundary layer development in V103 high-load compressor cascades using hot-film and hot-wire measurements, and observed forced transition along with trailing calm regions. Sun et al. [6] combined experiments and numerical simulations to investigate the unsteady characteristics of horseshoe vortex structures at the leading edge of high-lift low-pressure turbine endwalls and their effects on secondary flow, discovering that the instability frequency of these structures correlates with boundary layer thickness.

Numerical simulation approaches, such as Large Eddy Simulation (LES) and Detached Eddy Simulation (DDES), have been widely applied. Duan et al. [7] used LES to simulate the effects of various inflow turbulence intensities on low-pressure turbine boundary layer development within the $Re$ range of $1\times {10}^5$ to $4\times {10}^5$, indicating that the influence of $Re$ exceeds that of inflow turbulence. Xu et al. [8] compared unsteady flows in compressor cascades at $Re=0.8\times {10}^5$ and $Re=1.1\times {10}^5$ using LES and highlighted that the flow characteristics at low $Re$ are dominated by Kelvin–Helmholtz (K-H) instabilities. Wang et al. [9] compared transition processes in compressor blades with different load distributions (V103-B and V103-F) using LES, finding that front-loaded blades experience an earlier transition and smaller separation bubbles. Medic et al. [10] employed wall-resolved LES to simulate separation-induced transition and losses in NACA 65 series cascades at $Re=2.5\times {10}^5$, thereby validating the effectiveness of the LES method. Yin and Durbin [11] simulated transitional flow in V103 compressor cascades at $Re=1.385\times {10}^5$ using an adaptive DDES model. Their study demonstrated the model’s ability to capture disturbance development within laminar boundary layers and to predict transition fronts under various turbulence levels, including scenarios of separation-induced transition. Michelassi et al. [12] conducted a comparative study using LES and Direct Numerical Simulation (DNS) on low-pressure turbine cascades with incoming wakes (T106, $Re=0.518\times {10}^5$), confirming that LES can effectively reproduce DNS results, including the suppression effect of wakes on flow separation. Collectively, these studies indicate that flow separation, transition, and the formation of separation bubbles in cascade passages are influenced by a complex interplay of $Re$, inflow turbulence, pressure gradients, blade load distribution, and endwall effects.

In summary, although considerable research has focused on flow characteristics within blade passages—particularly separation and transition phenomena under low $Re$ conditions—most studies have concentrated on the suction surface, with limited attention given to flow on the pressure surface and its corresponding impact on heat transfer characteristics. Regarding heat transfer, Calzada et al. [13] investigated separated flow on turbine blade pressure surfaces using Reynolds-Averaged Navier–Stokes (RANS) simulations. They found that local extremes in heat transfer coefficients are closely related to separation reattachment points and emphasized the importance of wall-normal velocity components. Choi et al. [14] demonstrated through experiments ranging from $Re=0.157\times {10}^5$ to $1.05\times {10}^5$ that increasing inflow turbulence significantly enhances heat transfer in turbine blades by suppressing separation and promoting transition. Lu et al. [15] indicated through LES research that wall cooling (relative to adiabatic conditions) can advance the transition on compressor blades and reduce laminar separation bubbles (LSB), thereby decreasing aerodynamic losses. Kiss and Spakovszky [16] analyzed the effects of transient heat transfer in compressors on stability, arguing that heat exchange between components and the main flow significantly affects stage matching and stall margin. Bassi et al. [17] emphasized the challenges in accurately simulating transition for heat transfer prediction at low $Re$, particularly when using high-order RANS methods to study turbine blades with film cooling. These studies suggest that the mechanisms and predictions for convective heat transfer in separated flow regions—especially under low $Re$ conditions and varied boundary conditions—warrant further in-depth investigation.

High-precision numerical simulation methods are powerful tools for investigating such complex flow and heat transfer problems. DNS, due to its independence from turbulence model assumptions, offers the most detailed information about the flow field. However, the application of DNS is also constrained by its inherent limitations. The most significant is its extremely high computational cost, which typically scales with a high power of $Re$ [18], making it difficult to widely apply under high $Re$ and complex geometries commonly encountered in engineering and often necessitating the use of simplified geometric models, even in academic research [2]. Zaki et al. [19] conducted DNS on V103 compressor cascades at $Re=1.385\times {10}^5$, analyzing transition mechanisms on both pressure and suction surfaces under varying inflow turbulence intensities. Wheeler et al. [20] performed DNS on high-pressure turbine guide vanes at high $Re$ ($Re=5.7\times {10}^5$) under transonic conditions, studying the enhancement of heat transfer by turbulence-induced near-wall streaks, K-H instabilities, and loss generation in the trailing edge region. Wu and Durbin [21] used DNS to reveal how distorted wakes evolve into longitudinal vortex structures in low-pressure turbine passages. Ventosa-Molina et al. [22] employed DNS to explore the effects of relative endwall motion on secondary flow structures in compressor cascades. Jiang and Fu [23] investigated the secondary instability mechanism associated with separation-induced transition in low-pressure turbine cascades using DNS. Zhu and Li [24] employed high-order finite difference schemes for two-dimensional DNS of V103 cascades, analyzing vortex structures and shedding frequencies within the separation region. However, relatively few studies have used DNS to investigate flow and heat transfer in compressor blade passages.

Based on prior explorations [25,26], this study employs DNS to investigate the flow and heat transfer characteristics within a compressor blade passage under five different $Re$ conditions ($Re=1.091\times {10}^5$, $1.229\times {10}^5$, $1.367\times {10}^5$, $1.506\times {10}^5$, and $1.645\times {10}^5$). The research focuses on elucidating the development of the boundary layer, flow separation, and transition processes, as well as their impacts on wall heat transfer efficiency on the blade’s pressure and suction surfaces under varying $Re$. A recent method [27] for defining the flow boundary layer thickness—based on reconstructing the local inviscid velocity profile—is applied to the compressor blade passage to accurately calculate boundary layer parameters, thereby revealing the influence of $Re$ on transition and heat transfer efficiency. Additionally, the Rortex method [28] is used to identify turbulent vortex structures, which are then analyzed in relation to convective heat transfer to provide deeper physical insights for optimizing blade design and enhancing thermal efficiency.

2. Numerical Methodology and Validation

2.1. Governing Equations and Computational Domain

The DNS performed in this study simulates the conservation of mass and momentum by solving the time-dependent incompressible continuity and Navier–Stokes (N-S) equations, corresponding to Equation (1) and Equation (2), respectively. The temperature field is obtained by solving the energy equation, Equation (3). For computational convenience and generality of the results, these governing equations are transformed into nondimensional form as follows [29]:

$$\nabla ·\mathbf{u}=0,$$

(1)

$${\displaystyle \frac{\partial \mathbf{u}}{\partial t}}+\nabla ·(\mathbf{u}\otimes \mathbf{u})=-\nabla P+\nabla ·\left({\displaystyle \frac{1}{Re}}\left[\nabla \mathbf{u}+{(\nabla \mathbf{u})}^s\right]\right),$$

(2)

$${\displaystyle \frac{\partial \theta }{\partial t}}+\nabla ·\left(\theta \mathbf{u}\right)={\displaystyle \frac{1}{RePr}}\Delta \theta ,$$

(3)

where $\mathbf{u}$ represents the velocity vector, and P and $\theta$ denote the pressure and temperature, respectively. The superscript s indicates the transpose of a tensor. The dimensionless temperature $\theta$ is defined as $\theta =(T-T_w)/(T_i-T_w)$, where T is the local physical temperature, and $T_i$ and $T_w$ are the physical temperatures at the inlet and the wall, respectively. The computational domain employed in the simulation consists of two V103 compressor blades, as depicted in Figure 1. The axial chord length of the blade, $L_x$, is chosen as the reference length, while the incoming flow velocity at the inlet, U, is selected as the reference velocity. Based on these, the $Re$ is defined as $Re=U{L}_x/\nu$, where $\nu =\mu /\rho$ is the kinematic viscosity, and $\rho$ and $\mu$ are the fluid density and dynamic viscosity, respectively. The Prandtl number, $Pr=c\mu /k$ (where c is the specific heat capacity and k is the thermal conductivity), is set to 0.71 in the simulations. Note that for incompressible flow, the Eckert number has a negligible effect, and therefore, the viscous dissipation term is omitted in Equation (3).

As shown in Figure 1, the computational domain extends $0.5L_x$ upstream of the blade leading edge to ensure fully developed inflow and $1.5L_x$ downstream of the trailing edge to capture the wake characteristics. The incoming flow velocity, U, at the inlet is kept constant, with its angle of inclination relative to the axial direction set to 42°. The height of the domain in the y direction is $0.55L_x$, and periodic boundary conditions are applied to the top and bottom boundaries (excluding the blade surfaces). In the z direction, the domain length is set to $0.2L_x$, and periodic boundary conditions are applied in this direction as well. This choice was informed by Zhang and Samtaney [30], who demonstrated that a spanwise domain width of $0.2L_x$ provides reasonable spanwise decorrelation for most velocity components and locations, and was specifically based on previous DNS calculations for compressor cascade [19], where $0.2L_x$ was found to be adequate for capturing the essential spanwise flow physics. For the blade surfaces, a no-slip wall condition is imposed. The main conditions of this study are listed in

Table 1. Conditions for the present study.

Parameter	Symbol	Value	Unit/Description
Reference velocity	U	1	Dimensionless reference value
Axial chord length	$L_x$	1	Dimensionless reference value
Inlet flow angle	$\alpha$	42	Degree (°)
Prandtl number	$Pr$	0.71	Dimensionless value
Reynolds number	$Re$	1.091, 1.229, 1.367, 1.506, 1.645	Dimensionless value ($\times {10}^5$)
Wall temperature	${\theta }_w$	0	Dimensionless value
Inlet temperature	${\theta }_i$	1	Dimensionless value

.

This study aims to investigate the influence of the $Re$ on the boundary layer characteristics within the blade passage; therefore, five different $Re$ were simulated: $Re=1.091\times {10}^5$, $1.229\times {10}^5$, $1.367\times {10}^5$, $1.506\times {10}^5$, and $1.645\times {10}^5$. To eliminate the influence of grid density on the results, the same computational grid was used for all simulations. A structured grid covering the three-dimensional domain was generated by solving the Poisson equation, with the number of grid nodes being $N_x\times N_y\times N_z=1001\times 514\times 128$. As illustrated in Figure 2, the grid is refined near the wall, leading edge, and trailing edge regions to ensure good orthogonality and accurately capture the flow and heat transfer details near the boundaries. To evaluate grid resolution, Figure 3 presents the distributions of the dimensionless normal distance $\Delta y_n^{+}$ of the first grid layer on the blade surface, as well as the streamwise $\Delta x_t^{+}$ and spanwise $\Delta z^{+}$ distributions, under the highest Reynolds number condition $Re=1.645\times {10}^5$. Under these conditions, the wall unit values of a fixed physical grid typically reach their maximum, which represents the grid’s resolution capability under the most stringent flow conditions [31].

According to [30], the DNS grid resolution $\Delta y_{n,\max }^{+}<1$, $\Delta z_{\max }^{+}$ typically ranges between 5 and 10, and $\Delta x_{t,\max }^{+}$ typically ranges between 10 and 20. As shown in Figure 3, at $Re=1.645\times {10}^5$, the $\Delta y_n^{+}$ across the entire blade surface remains below 0.5, with its maximum value far less than 1, fully satisfying the DNS requirement for wall-normal resolution. For the spanwise resolution $\Delta z^{+}$, in most regions of the pressure surface (Figure 3a) and suction surface (Figure 3b), the maximum value of $\Delta z^{+}$ generally stays below 12. For the streamwise resolution $\Delta x_t^{+}$, the maximum value generally remains below 13. At the foremost ends of the suction and pressure surfaces, a peak in $\Delta z^{+}$ can be observed, which results from the combined effect of the high wall friction velocity $u_{\tau }$ in the extremely thin boundary layer of the strong acceleration region at the leading edge and the local physical grid size. However, the flow state in this region is primarily laminar, which is not the focus of our analysis of turbulent structures.

Table 2. Comparison of DNS mesh sizes and simulation settings *.

Study	Mesh Size	$\mathit{Re}$ ($\times {10}^5$)	FST	Domain (Axial; Pitch; Span)	$\Delta {\mathit{y}}_{\mathit{n}}^{+}$	$\Delta {\mathit{x}}_{\mathit{t}}^{+}$	$\Delta {\mathit{z}}^{+}$
Present study	$1001\times 514\times 128$	1.091, 1.229, 1.367, 1.506, 1.645	No	$(-0.5,2.5)L_x$; $0.55L_x$; $0.2L_x$	<0.5	<12.95	<12.53
Zaki et al. [19]	$1025\times 641\times 129$	1.385	Yes	$(-0.4,1.5)L_x$; $0.59L_x$; $0.2L_x$	<1	5–10	5–10
Zhu and Li [24]	$1021\times 201$	1.38	No	$(-1.0,5.0)L_x$; $0.55L_x$	/	/	/

* FST stands for free-stream turbulence. $\Delta y_n^{+}$, $\Delta x_t^{+}$, $\Delta z^{+}$ are wall-normal, streamwise, and spanwise grid resolutions in wall units, respectively. For “Present studies”, $\Delta x_t^{+}$ and $\Delta z^{+}$ values are maximums at the highest $Re$. “/” indicates not reported.

compares the current simulation setup with DNS studies on the V103 blade found in the literature.

As listed in Table 2, the intermediate $Re$ used in this study ($Re=1.506\times {10}^5$) is comparable to the values reported by Zhu and Li [24] and Zaki et al. [19]. The scale of the three-dimensional (3D) grid employed is also similar to that used in the fine grid study by Zaki et al. [19]. However, a key difference lies in the inlet conditions: Zaki et al.’s study used a turbulent inlet, whereas both the present study and that of Zhu and Li set the inlet condition to zero free-stream turbulence (zero-FST). Although real compressor environments typically contain FST, the zero-FST condition was deliberately chosen to isolate the pure effects of $Re$ on the boundary layer instabilities and natural transition pathways. Previous experimental studies [32] have shown that even in high-FST environments, the transition characteristics driven by inherent boundary layer instabilities remain detectable and play a fundamental role. Therefore, investigating these phenomena under zero-FST conditions provides an essential basis for understanding their underlying physical mechanisms.

Furthermore, Zhu and Li’s study was two-dimensional (2D), which limits its ability to capture the three-dimensional effects of turbulence. Since the DNS in this study is three-dimensional and no artificial free-stream turbulence was introduced at the inlet, the observed flow separation and transition phenomena result from the natural evolution of the flow, unaffected by external or artificial disturbances. This approach enables clear identification of $Re$ effects on the intrinsic flow physics, establishing a crucial reference for future studies that will incorporate FST effects.

2.2. Numerical Methods

The numerical method employed in this study is based on the finite volume method, which discretizes the conservative form of the governing equations (Equations (1)–(3)) on a staggered grid in curvilinear coordinates. For spatial discretization, and to ensure both accuracy and stability, the convective terms in the governing equations are reconstructed using the Minmod TVD scheme [33], which effectively suppresses numerical diffusion. Meanwhile, the diffusive terms are discretized using a second-order central difference scheme. For temporal advancement, the convective terms are treated explicitly using the second-order Adams–Bashforth scheme, while the diffusive terms are handled implicitly using the second-order Adams–Moulton scheme. This spatio-temporal discretization strategy achieves a balance between computational efficiency and numerical stability and is consistently applied to both the velocity and temperature fields.

To solve for the pressure field while ensuring the divergence-free condition of the velocity field, a fractional step method [34] is employed to decouple the velocity and pressure. This method executes the following four steps within each time step n:

First, an intermediate velocity field ${\tilde{u}}_i^{n+1}$ without the pressure gradient is computed:

$${\displaystyle \frac{{\tilde{u}}_i^{n+1}-u_i^n}{\Delta t}}={\displaystyle \frac{1}{2}}(3C_i^n-C_i^{n-1})+{\displaystyle \frac{1}{2}}({\tilde{D}}_i^{n+1}+D_i^n),$$

(4)

where $C_i=-\partial \left(u_i{u}_j\right)/\partial x_j$ and $D_i={\displaystyle \frac{1}{Re}}{\partial }^2{u}_i/\partial x_j^2$ represent the spatial discretization operators for the convective and diffusive terms, respectively.

Second, a Poisson equation for the pressure potential ${\varphi }^{n+1}$ is solved:

$${\displaystyle \frac{1}{\Delta t}}{\displaystyle \frac{\partial {\tilde{u}}_i^{n+1}}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_i}}\left({\displaystyle \frac{\partial {\varphi }^{n+1}}{\partial x_i}}\right)$$

(5)

Third, the intermediate velocity field is corrected using the computed pressure potential ${\varphi }^{n+1}$ to obtain the final divergence-free velocity field $u_i^{n+1}$:

$${\displaystyle \frac{u_i^{n+1}-{\tilde{u}}_i^{n+1}}{\Delta t}}=-{\displaystyle \frac{\partial {\varphi }^{n+1}}{\partial x_i}}$$

(6)

Finally, the physical pressure field $p^{n+1}$ is obtained from the pressure potential ${\varphi }^{n+1}$:

$$p^{n+1}={\varphi }^{n+1}-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\Delta t}{Re}}{\displaystyle \frac{\partial }{\partial x_i}}\left({\displaystyle \frac{\partial {\varphi }^{n+1}}{\partial x_i}}\right)$$

(7)

To ensure an efficient and convergent solution of the pressure Poisson equation, a hybrid strategy combining the Fast Fourier Transform (FFT) spectral method and the Flexible-Cycle Additive-Correction Multigrid (FCAC-MG) technique [35] is implemented. Throughout the simulation, the time step $\Delta t$ is strictly controlled to maintain a Courant–Friedrichs–Lewy (CFL) number below 0.2, ensuring the stability of the temporal integration. For boundary conditions, Dirichlet conditions are applied at the inlet with a fixed inflow velocity U and a specified inlet temperature $T_i$. At the computational domain outlet, Neumann conditions are applied for both velocity and temperature, with zero normal gradients ($\partial u_i/\partial x=0$, $\partial T/\partial x=0$). This setup has been proven to possess good non-reflecting properties, minimizing numerical disturbances from the boundaries into the internal flow field [19,21]. No-slip conditions ($u=v=w=0$) are imposed on the wall surfaces. For thermal simulations, isothermal conditions are specified at the walls. For the discretized pressure Poisson equation, homogeneous Neumann conditions ($\partial \varphi /\partial n=0$) are applied to all non-periodic boundaries [21]. The DNS consumed a total of about $4.6\times {10}^5$ core hours for each 3D case using 64-core CPUs operating at 2.7 GHz. This substantial computational expenditure underscores the practical constraints of the DNS, generally limiting its application to moderately complex geometries and $Re$ ranges where fundamental flow physics can be investigated with available resources, as is the case of the present study.

2.3. Validation of DNS Codes

The reliability of the DNS codes employed in this study has been thoroughly validated through numerical simulations of typical flow and heat transfer configurations, such as tandem cylinders, square annular ducts, and film-cooling setups [29,36,37].

To validate the fundamental predictive capabilities of the 3D DNS solver employed in this study for the current blade passage configuration, the quasi-two-dimensional (quasi-2D) validation case was first conducted. This involved utilizing the full 3D DNS code but with a significantly simplified grid setup in the z direction (specifically, using only 8 grid points with periodic boundary conditions applied), at a Reynolds number of $Re=1.38\times {10}^5$. The distributions of the blade surface static pressure coefficient $C_p$ (defined as $C_p=2P/\left(\rho U^2\right)$) and the wall friction coefficient $C_f$ (defined as $C_f=2{\tau }_w/\left(\rho U^2\right)$, where ${\tau }_w$ is the wall shear stress) are plotted in Figure 4 and compared with the reference data provided by Zhu and Li [24].

As shown in Figure 4a, the $C_p$ distribution on the blade surface obtained in the current simulation shows good agreement with the data of Zhu and Li [24]. On the pressure side, due to the influence of the concave curvature, $C_p$ initially increases and then decreases. This causes the boundary layer (BL) to first experience an adverse pressure gradient (APG), which extends over approximately 80% of the chord length. Near the trailing edge, as $C_p$ decreases, the boundary layer is subsequently subjected to a favorable pressure gradient (FPG).

In contrast, the $C_p$ distribution on the suction side is more complex. After a brief initial decrease, $C_p$ increases, indicating the presence of an adverse pressure gradient. Under the combined influence of this APG and the suction surface curvature, the boundary layer decelerates and eventually separates. The separation point is identified by the location where the wall friction coefficient $C_f$ changes from positive to negative. Figure 4b shows the $C_f$ distribution on the suction side, indicating that the separation point is located at $x/L_x=0.38$.

After boundary layer separation on the suction side, the $C_p$ distribution typically exhibits a plateau region, a phenomenon consistent with the observations of Pauley et al. [38]. Subsequently, at $x/L_x=0.58$, the pressure begins to decrease, forming an FPG region that occupies approximately 10% of the chord length. The formation of this FPG region is induced by a clockwise separation bubble [19], as evidenced by the corresponding region in Figure 4b, where $C_f$ changes from negative back to positive. Finally, the reattachment point of the boundary layer on the suction side is located at $x/L_x=0.88$, which is in excellent agreement with the data of Zhu and Li [24]. The separation region occupies approximately 49% of the axial chord length $L_x$, slightly higher than the 45% reported by Zhu and Li [24]. In their study, the separation and reattachment points are located at $x/L_x=0.40$ and $x/L_x=0.85$, respectively, while in the present study, the reattachment point is found at $x/L_x=0.88$, slightly downstream. These differences may arise from variations in numerical algorithms and discretization schemes. Nevertheless, the current numerical method successfully captures the key flow features, including the location and extent of boundary layer separation on the suction side as well as the time-averaged secondary vortex structures. Based on the validation results, it can be concluded that the current numerical simulation has good accuracy and credibility. The flow and heat transfer characteristics in the blade passage will be analyzed and discussed under different $Re$ conditions below.

3. Results and Discussion

3.1. Average Flow and Heat Transfer Characteristics

To investigate the average flow and heat transfer characteristics on the blade surface, this section analyzes the distributions of the $C_p$, $C_f$, and Nusselt number ($Nu$) under different $Re$. All the averaged results presented in this section, including the surface distributions and boundary layer profiles, were obtained by averaging the instantaneous 3D DNS data in both time and the spanwise (z) directions. The spanwise averaging is appropriate since we employed periodic boundary conditions in the z-direction, making these quantities statistically two-dimensional. For time averaging, the flow was first allowed to reach a statistically steady state at approximately $t\approx 8L_x/U$ (where $L_x/U$ is the flow-through time). Subsequently, time averaging was performed over an additional period of nine flow-through times to ensure adequate statistical convergence [26].

Figure 5 shows the distributions of $C_p$ and $C_f$ on the pressure surface under various $Re$ conditions. As shown in Figure 5a, the boundary layer initially experiences an adverse pressure gradient followed by a favorable pressure gradient. This general trend in the $C_p$ distribution remains consistent over the range of $Re$ values. It is worth noting that in the interval $0.65<x/L_x<0.75$, the three higher $Re$ cases ($Re=1.367\times {10}^5$, $Re=1.506\times {10}^5$, and $Re=1.645\times {10}^5$) exhibit a sharp increase in $C_p$, which is primarily due to the pressure recovery caused by flow reattachment.

Combining the friction coefficient $C_f$ curves shown in Figure 5b with the data listed in

Table 3. Separation and reattachment locations ($x/L_x$) at different $Re$ for the pressure and suction surfaces.

Re ($\times {10}^5$)	S (PS)	R (PS)	S (SS)	R (SS) *
1.091	/	/	0.36	/
1.229	/	/	0.36	/
1.367	0.45	0.73	0.38	0.95
1.506	0.41	0.66	0.39	0.87
1.645	0.40	0.65	0.39	0.85

* S and R represent separation and reattachment points; PS and SS represent pressure and suction surfaces, respectively.

, it is evident that the flow separation positions vary with $Re$. Specifically, the two lower $Re$ cases ($Re=1.091\times {10}^5$ and $Re=1.229\times {10}^5$) do not exhibit flow separation. For the three higher $Re$ cases, as $Re$ increases, the separation point on the pressure surface boundary layer moves upstream. According to Table 3, the separation points are located at approximately $x/L_x=0.45$, 0.41, and 0.40 for $Re=1.367\times {10}^5$, $1.506\times {10}^5$, and $1.645\times {10}^5$, respectively, whereas the corresponding reattachment points shift forward to approximately $x/L_x=0.73$, 0.66, and 0.65. This pressure recovery occurs because the reattachment of separated flow changes the streamline curvature near the wall. As a result, the near-wall fluid experiences reduced deceleration and may even regain some kinetic energy, leading to an increase in static pressure, and consequently, an increase in the pressure coefficient $C_p$. These reattachment positions roughly correspond to the regions where $C_p$ shows a sharp increase, as previously reported [39]. Under these higher $Re$ conditions, the lengths of the laminar separation bubbles on the pressure surface are similar, occupying roughly 25%, 26%, and 27% of the chord length $L_x$, respectively.

Figure 6 depicts the distributions of $C_p$ and $C_f$ on the suction surface for various $Re$ values. Compared to the pressure surface, the flow state on the suction surface is more complex. As shown in Figure 6a, the $C_p$ on the suction surface first decreases rapidly, after which the flow enters a pronounced adverse pressure gradient region. Under the influence of this strong adverse pressure gradient and curvature effects, the boundary layer separates. The separation positions under different $Re$ conditions show little variation; according to Table 3, the separation points are located between approximately $x/L_x=0.36$ and $x/L_x=0.39$. After boundary layer separation, the near-wall flow decelerates such that the wall pressure remains nearly constant, forming a “pressure plateau” region [19] similar to that observed in two-dimensional flow (see Figure 4). In Figure 6b, the $C_f$ values for the two lower $Re$ cases ($Re=1.091\times {10}^5$ and $Re=1.229\times {10}^5$) remain negative, indicating no flow reattachment. In contrast, reattachment occurs for all three higher $Re$ cases. Consistent with the pressure surface observations, as $Re$ increases, the reattachment position on the suction surface also moves upstream [9,31].

Figure 7 shows the distribution of the $Nu$ on the blade surfaces. As depicted in Figure 7a, on the pressure surface, the $Nu$ distribution exhibits similarities to the $C_f$ distribution. Although flow separation occurs on the pressure surface under the three higher Reynolds number conditions ($Re=1.367\times {10}^5$, $1.506\times {10}^5$, and $1.645\times {10}^5$), the separation intensity is relatively weak. This weak separation has a limited impact on disrupting the qualitative similarity between the $Nu$ and $C_f$ distributions, with both still maintaining a similar streamwise evolution trend, which aligns qualitatively with the classical Reynolds Analogy [40]. In contrast, the strong flow separation on the suction surface (Figure 7b) significantly affects the heat transfer characteristics. As shown in Figure 7b, $Nu$ initially decreases as the boundary layer develops. When the flow enters the separation region, $Nu$ drops markedly to its minimum value; however, once the separated shear layer undergoes transition, $Nu$ rises sharply [13]. For the three higher $Re$ cases ($Re=1.367\times {10}^5$, $Re=1.506\times {10}^5$, and $Re=1.645\times {10}^5$) in which reattachment occurs, the $Nu$ values remain relatively high in the downstream reattachment region. For the two lower $Re$ cases ($Re=1.091\times {10}^5$ and $Re=1.229\times {10}^5$), where reattachment does not occur, the $Nu$ values in the latter half of the suction surface stay low as the flow remains separated.

To more intuitively demonstrate the effects of $Re$ on the flow and temperature fields, Figure 8 and Figure 9 show the ensemble-averaged velocity magnitude ($\overline{U_m}$) contours and temperature ($\overline{\theta }$) contours, respectively. All velocity values in Figure 8 have been nondimensionalized using the inlet velocity U. The cases with the lowest $Re$ ($Re=1.091\times {10}^5$) and the highest $Re$ ($Re=1.645\times {10}^5$) are selected as representative examples.

Comparing Figure 8a,b, the significant influence of $Re$ on the size of separation bubbles on the suction surface is evident. At the lower $Re$, a large-scale separation region covers an extensive area on the suction surface, with low-speed fluid occupying a large portion of the near-wall region in the mid and rear parts of the airfoil. At the higher $Re$, the separation region is notably reduced, and the boundary layer on the pressure surface also appears thinner. These observations agree with the quantitative data in Table 3 and the separation and reattachment behavior revealed by the $C_f$ distributions in Figure 5b and Figure 6b.

The corresponding temperature fields in Figure 9 also reflect these differences in flow structure. At the lower $Re$ (Figure 9a), associated with the large separation bubble, a very thick thermal boundary layer forms on the suction surface, directly leading to the low $Nu$ values observed in Figure 7b. At the higher $Re$ (Figure 9b), with the reduction in the separation bubble and the thinning of the velocity boundary layer, the thermal boundary layer thickness decreases significantly. This reduction results in increased near-wall temperature gradients and, consequently, higher $Nu$ values (see Figure 7b).

3.2. Velocity and Thermal Boundary Layers Analysis

The flow separation and heat transfer phenomena on the blade surface are directly influenced by the state of the near-wall boundary layer. This section provides a detailed analysis of the velocity and thermal boundary layers near the blade. First, it is necessary to precisely determine the extent of the boundary layer. For a traditional zero-pressure-gradient boundary layer on a flat plate, the nominal thickness $\delta$ is defined as the normal distance from the wall where the fluid velocity u reaches 99% of the free-stream velocity U, i.e., $u(y=\delta )=0.99U$ [41]. However, in complex wall flows with pressure gradients (such as on blade surfaces), this definition is often no longer applicable.

Taking the case with $Re=1.367\times {10}^5$ as an example, Figure 10 shows the ensemble-averaged velocity field at this $Re$ and marks six characteristic locations (L1 to L6) for boundary layer analysis. Among these, L1 ($x/L_x=0.12$), L2 ($x/L_x=0.3$), and L3 ($x/L_x=0.8$) are on the pressure surface, corresponding to laminar, separated, and reattached regions, respectively. Similarly, L4 ($x/L_x=0.12$), L5 ($x/L_x=0.3$), and L6 ($x/L_x=0.8$) are on the suction surface, corresponding to the laminar, separated, and reattached regions. Figure 11 and Figure 12 then present the ensemble-averaged tangential velocity ($\overline{u_t}$) and ensemble-averaged temperature ($\overline{\theta }$) profiles at these six characteristic locations, where the velocity is nondimensionalized by the inlet velocity U, and the wall-normal distance $y_n$ is nondimensionalized by $L_x$.

As shown in Figure 11, near the edge of the boundary layer, the velocity profiles indicate that the $\overline{u_t}$ no longer asymptotically approaches a constant value. Unlike the flat-plate boundary layer where velocity approaches a uniform free-stream value, the edge velocity varies along the blade surface due to the pressure gradient and flow acceleration in the cascade passage. In contrast, as demonstrated in Figure 12, the $\overline{\theta }$ in the profiles does asymptotically approach a constant at the edge of the thermal boundary layer. Therefore, the thermal boundary layer thickness ${\delta }_T$ can still be defined by the traditional method, that is, as the normal distance from the wall where $\overline{\theta }$ reaches 99% of the inlet reference temperature $\overline{{\theta }_i}$.

Existing methods for defining the flow boundary layer based on average vorticity [42] or shear stress thresholds [43] typically involve complex iterations, numerical integration, or differential calculations. These techniques increase computational complexity and can introduce additional errors and instabilities. In this study, a new method is adopted to calculate the flow boundary layer thickness $\delta$, based on the reconstruction of local inviscid velocity profiles [27].

For incompressible steady flow, the total pressure $P_0$ is given by

$$P_0=P+{\displaystyle \frac{1}{2}}\rho {\overline{U_m}}^2$$

(8)

where P is the static pressure, $\rho$ is the fluid density, and the square of the velocity magnitude, ${\overline{U_m}}^2={\overline{u_t}}^2+{\overline{v_n}}^2$, is determined by the mean tangential velocity $\overline{u_t}$ and the mean wall-normal velocity $\overline{v_n}$ in the two-dimensional local coordinate system. Since $P_0$ reflects the total energy (or work capacity) of the fluid, in regions far from the wall, the diminishing viscous effects cause $P_0$ to approach a constant value. Therefore, the maximum value of $P_0$ in this region is used as the inviscid reference total pressure: $P_{0,\mathrm{ref}}=\max \left(P_0\right)$. By applying the Bernoulli equation in the wall-normal direction, the square of the corresponding inviscid flow velocity $\overline{u_{tI}}$ can be reconstructed:

$${\overline{u_{tI}}}^2\left(y\right)={\displaystyle \frac{2}{\rho }}(P_{0,\mathrm{ref}}-P\left(y\right))-{\overline{v_n}}^2\left(y\right)$$

(9)

Based on this, the flow boundary layer thickness $\delta$ is determined by the more general condition:

$${\overline{u_t}}^2{|}_{y=\delta }={\left(0.99\overline{u_{tI}}\right)}^2\approx 0.98{\overline{u_{tI}}}^2.$$

(10)

Figure 13 shows the curves of the actual velocity squared, ${\overline{u_t}}^2$, and the reconstructed inviscid velocity squared, ${\overline{u_{tI}}}^2$, as functions of the wall-normal distance $y_n$ at the six characteristic locations L1–L6. In the figure, both velocity-squared terms are normalized by $U^2$.

As shown in Figure 13, in the outer region of the boundary layer, the reconstructed inviscid velocity squared ${\overline{u_{tI}}}^2$ agrees well with the actual viscous velocity squared ${\overline{u_t}}^2$. Thus, the flow boundary layer thickness $\delta$ is clearly defined as the wall-normal distance at which ${\overline{u_t}}^2$ reaches 98% of ${\overline{u_{tI}}}^2$. The square markers in Figure 13 indicate the calculated boundary layer thickness $\delta$ at the six representative locations. The results show that on the pressure surface $\delta \left(L3\right)>\delta \left(L2\right)>\delta \left(L1\right)$, and on the suction surface $\delta \left(L6\right)>\delta \left(L5\right)>\delta \left(L4\right)$. This trend is consistent with the physical expectation that the flow boundary layer typically grows thicker in the streamwise direction. Therefore, this new method conveniently and consistently determines the flow boundary layer thickness in both attached and separated regions.

In velocity boundary layer theory, the displacement thickness ${\delta }^{*}$ and momentum thickness ${\delta }_m$ are two important integral measures that characterize the boundary layer from different perspectives [39,41]. The displacement thickness ${\delta }^{*}$ measures the mass flow deficit due to the boundary layer velocity profile, while the momentum thickness ${\delta }_m$ reflects the momentum loss. In this method, these quantities are defined as ${\delta }^{*}={\int }_0^{\delta }\left(1-\overline{u_t}/\overline{u_{\delta }}\right)d{y}_n$, ${\delta }_m={\int }_0^{\delta }\overline{u_t}/\overline{u_{\delta }}\left(1-\overline{u_t}/\overline{u_{\delta }}\right)d{y}_n$, where $\delta$ is the nominal boundary layer thickness obtained from the local inviscid velocity profile reconstruction, $\overline{u_{\delta }}$ is the mean tangential velocity at the boundary layer edge, and $y_n$ is the wall-normal distance. Combining these, a dimensionless shape factor H is defined: $H={\delta }^{*}/{\delta }_m$, which serves as an important indicator for assessing the boundary layer state and predicting transition. The velocity boundary layer parameters and thermal boundary layer thickness on the pressure surface for different $Re$ are shown in Figure 14, where the ${\delta }^{*}$, ${\delta }_m$, and ${\delta }_T$ are all nondimensionalized using the axial chord length $L_x$.

Figure 14a indicates that the displacement thickness ${\delta }^{*}$ at the two lower $Re$ is smaller than at the three higher $Re$, which is attributed to flow separation in the latter, causing greater mass loss. As the $Re$ increases, the peak value of ${\delta }^{*}$ decreases, suggesting a weakening of the separation. In Figure 14b, the momentum thickness ${\delta }_m$ increases slowly along the flow direction at the lower $Re$. However, for the three higher $Re$, ${\delta }_m$ exhibits a sharp increase near $x/L_x=0.6$, indicating a rapid rise in near-wall momentum loss typically associated with transition phenomena [44].

Figure 14c shows that the initial value of the shape factor H is close to 2.5, a typical value for an attached laminar boundary layer [41,45]. At the two lower $Re$, H remains nearly constant in the first half of the flow and begins to decrease after $x/L_x=0.65$, yet remains relatively high, indicating an essentially laminar state on the pressure surface. For the three higher $Re$, H initially changes slowly and then exhibits a pronounced peak. When the separated shear layer becomes unstable and triggers transition, intensified momentum exchange causes a rapid increase in ${\delta }_m$, which in turn leads to a sharp decrease in H. Consequently, the peak in H can be regarded as the approximate location of transition onset [46]. Based on these peaks, the transition locations for $Re=1.367\times {10}^5$, $1.506\times {10}^5$, and $1.645\times {10}^5$ are approximately at $x/L_x=0.63$, $0.59$, and $0.57$, respectively, indicating that the transition position moves upstream with increasing $Re$. In the region $x/L_x>0.85$, the H values for these higher $Re$ tend to stabilize at levels lower than typical turbulent boundary layers (around 1.4–1.7) [45], implying that the boundary layer has developed into a turbulent state in the rear part of the pressure surface. Correspondingly, Figure 14d shows that after transition at higher $Re$, the thermal boundary layer thickness ${\delta }_T$ is significantly greater than the laminar thermal boundary layer thickness observed at the two lower $Re$ due to enhanced turbulent mixing. For the suction surface, Figure 15 presents the boundary layer parameters and thermal boundary layer thickness at different $Re$.

Figure 15a shows that due to significant separation, the displacement thickness ${\delta }^{*}$ is markedly increased, reflecting substantial mass flow loss. However, as $Re$ increases, the degree of separation weakens and ${\delta }^{*}$ correspondingly decreases. Figure 15b shows that, similar to the pressure surface, even with separation, the momentum thickness ${\delta }_m$ increases slowly before transition because the initial momentum loss is relatively small; once transition occurs, momentum loss intensifies and ${\delta }_m$ increases rapidly. As indicated in Figure 15c, the initial value of H on the suction surface is also approximately 2.5, typical of a laminar boundary layer. Since flow separation occurs on the suction surface for all investigated $Re$, the instability of the separated shear layer ultimately results in turbulent transition, and the H curves for all cases exhibit distinct peaks. For the two lower $Re$ ($Re=1.091\times {10}^5$ and $Re=1.229\times {10}^5$), the transition peaks occur at approximately $x/L_x=0.74$. For the three higher $Re$ ($Re=1.367\times {10}^5$, $1.506\times {10}^5$, and $1.645\times {10}^5$), the transition locations are at $x/L_x=0.77$, 0.75, and 0.74, respectively, indicating a slight upstream shift with increasing $Re$. For $Re=1.506\times {10}^5$ and $1.645\times {10}^5$, after transition, the H value stabilizes around 2. In contrast, for $Re=1.367\times {10}^5$ and the two lower $Re$, the reattachment point is further downstream (or even incomplete), so H continues to decrease toward the trailing edge without stabilization, suggesting that the flow is still transitioning toward turbulence. This is attributed to the combined effects of the upstream large-scale separation history and the insufficient turbulent development length offered by the finite blade chord. Consequently, for these lower $Re$ cases, the boundary layer on the suction surface fails to achieve a fully developed turbulent state by the trailing edge of the blade. Regarding the thermal boundary layer (Figure 15d), ${\delta }_T$ increases slowly before separation and grows rapidly thereafter. Unlike on the pressure surface, on the suction surface, the thermal boundary layer thickness ${\delta }_T$ decreases monotonically over the entire streamwise range as the $Re$ increases.

3.3. Turbulent Vortex Structure and Heat Transfer Enhancement Mechanism

Based on the preceding analysis of the average heat transfer characteristics and the boundary layer state on the blade surface, it is observed that once the flow transitions to a turbulent state, its convective heat transfer efficiency is significantly enhanced compared to that in the laminar region. To elucidate the physical mechanism underlying this phenomenon, an in-depth investigation of the three-dimensional flow structures and the associated turbulent characteristics within the turbulent boundary layer is essential. Because the flow and heat transfer characteristics exhibit a monotonic trend with increasing $Re$ within the studied range, the most representative conditions were selected for presentation. For the pressure surface, conditions at $Re=1.367\times {10}^5$, where transition has occurred and turbulent structures were formed, and the highest Reynolds number $Re=1.645\times {10}^5$ are presented. For the suction surface, the lowest Reynolds number $Re=1.091\times {10}^5$, which exhibits large-scale separation, and the highest Reynolds number $Re=1.645\times {10}^5$, which shows reattached flow, are presented to fully demonstrate the range of influence of the $Re$ effects.

To effectively identify three-dimensional vortex structures, this study employs the Rortex method proposed by Liu et al. [28]. This method decomposes the vorticity vector into a Rortex vector $\mathbf{R}$, representing rigid rotation, and a non-rotating shear vector $\mathbf{S}$. The Rortex vector is expressed as

$$\mathbf{R}=R·\mathbf{r}=R_x\mathbf{i}+R_y\mathbf{j}+R_z\mathbf{k},$$

(11)

where $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are the unit vectors in the Cartesian coordinate system $(X,Y,Z)$, respectively. The magnitude of the Rortex vector, $\mathbf{R}$, which characterizes the strength of the rigid rotation, is defined as follows:

$$\left|\mathbf{R}\right|=\left\{\begin{array}{cc}2(\beta -\alpha ),\hfill & \mathrm{if}{\alpha }^2-{\beta }^2<0,\beta >0\hfill \\ 2(\beta +\alpha ),\hfill & \mathrm{if}{\alpha }^2-{\beta }^2<0,\beta <0\hfill \\ 0,\hfill & \mathrm{if}{\alpha }^2-{\beta }^2\ge 0\hfill \end{array}\right.$$

(12)

here, $\alpha$ and $\beta$ are computed as

$$\alpha ={\displaystyle \frac{1}{2}}\sqrt{{\left({\displaystyle \frac{\partial U}{\partial X}}-{\displaystyle \frac{\partial V}{\partial Y}}\right)}^2+{\left({\displaystyle \frac{\partial U}{\partial Y}}+{\displaystyle \frac{\partial V}{\partial X}}\right)}^2},\beta ={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial U}{\partial Y}}+{\displaystyle \frac{\partial V}{\partial X}}\right)$$

(13)

where $(U,V,W)$ denote the velocity components in the principal coordinate system $(X,Y,Z)$. Compared with traditional vortex identification methods, the isosurfaces of $\left|\mathbf{R}\right|$ provide a more accurate characterization of vortex structures by eliminating shear effects and retaining only the pure rigid rotation of the Rortex as a clear vortex identifier [47].

Figure 16 illustrates the instantaneous flow structure and thermal field characteristics on the pressure surface under two $Re$ conditions ($Re=1.367\times {10}^5$ and $Re=1.645\times {10}^5$). The three-dimensional vortex structures are visualized by the $\left|\mathbf{R}\right|=4$ isosurface and are colored by the temperature $\theta$. To reveal the direct correlation between these structures and the near-wall thermal field, the figure simultaneously displays the instantaneous $\theta$ distribution and velocity vector field on two characteristic cross-sections. The spanwise cross-section S1, located at $z=0.05L_x$, was chosen to avoid the influence of periodic boundaries and to capture typical spanwise flow characteristics. The wall-parallel cross-section S2, located at $y_n=0.003L_x$, corresponds to a $y^{+}\approx 10-15$ in the turbulent region, placing it within the buffer layer where near-wall streak structures are most prominent.

In Figure 16a for the $Re=1.367\times {10}^5$ case, early two-dimensional Kelvin–Helmholtz (K-H) vortices appear in the region $x/L_x=0.65$–0.7. Subsequently, these K-H vortices become unstable and evolve into three-dimensional structures [19]. As the flow reattaches, hairpin vortex structures, similar to those observed during natural transition, form near the wall. These structures consist of streamwise streaks in the near-wall region and arched vortex heads further away from the wall [48]. In this manner, K-H vortices and hairpin vortices are the primary vortex structures on the pressure surface. Section S1 displays the instantaneous morphology of the thermal boundary layer at the $z=0.05L_x$ position, along with the cross-sectional features observed when K-H or hairpin vortex heads pass through this plane (indicated by white dashed circles). The presence of these vortex structures significantly distorts the isotherms, resulting in localized downward displacement of hot fluid. Section S2 more clearly reveals the thermal effects of the ejection and sweep events induced by hairpin vortices in the near-wall region, that is, the upwash motion beneath the hairpin vortex legs ejects near-wall cold fluid, while sweeps on the outer sides of the legs transport mainstream hot fluid toward the wall. This is manifested as alternating blue (cold) and red (hot) streaks on the S2 plane.

A comparison of Figure 16a,b indicates that at $Re=1.645\times {10}^5$, the onset of K-H vortices on the pressure surface shifts upstream. The flow visualization also reveals that the hairpin vortex structures exhibit smaller scales and denser spatial distribution. Correspondingly, the temperature fields on planes S1 and S2 exhibit smaller-scale, more intricate mixing patterns, signifying intensified turbulent mixing at the higher $Re$. The persistent influence of these instantaneous vortex structures and their induced ejection and sweep events is clearly reflected in the turbulent statistics. Figure 17 presents the ensemble-averaged turbulent statistics near the pressure surface for the two $Re$, including the root mean square (RMS) of velocity fluctuations ($u_{t,rms}^{\prime }$ and $v_{n,rms}^{\prime }$), RMS of temperature fluctuations (${\theta }_{rms}^{\prime }$), Reynolds shear stress ($\overline{u_t^{\prime }{v}_n^{\prime }}$), and turbulent heat fluxes ($\overline{{\theta }^{\prime }{u}_t^{\prime }}$ and $\overline{{\theta }^{\prime }{v}_n^{\prime }}$). These statistical quantities are all presented in dimensionless form: the RMS of velocity fluctuations are normalized by U; Reynolds stress is normalized by $U^2$; and turbulent heat fluxes, which are the product of dimensionless velocity fluctuations and dimensionless temperature fluctuations, are inherently dimensionless.

In Figure 17a, for the $Re=1.367\times {10}^5$ case, the high-value regions of the RMS velocity and temperature fluctuations ($u_{t,rms}^{\prime }$, $v_{n,rms}^{\prime }$, and ${\theta }_{rms}^{\prime }$) are primarily concentrated in the region following turbulent transition, with peaks located near the wall. This corresponds to the active region of hairpin vortices shown in Figure 16a, reflecting the strong fluctuations induced by these vortex structures. In the region where $x/L_x>0.75$, the Reynolds shear stress $\overline{u_t^{\prime }{v}_n^{\prime }}$ exhibits negative values in the near-wall region, while it approaches zero farther away from the wall. This distribution indicates the transport of turbulent momentum toward the wall, which is a typical characteristic of an attached turbulent boundary layer closely associated with bursting and sweeping events [45]. Similarly, the normal turbulent heat flux $\overline{{\theta }^{\prime }{v}_n^{\prime }}$ also shows negative values in the near-wall region and exhibits a distribution pattern similar to that of $\overline{u_t^{\prime }{v}_n^{\prime }}$. In contrast, the streamwise turbulent heat flux $\overline{{\theta }^{\prime }{u}_t^{\prime }}$ is positive, indicating a tendency for streamwise velocity fluctuations and temperature fluctuations to be positively correlated, with its peak occurring closer to the wall than those of $\overline{u_t^{\prime }{v}_n^{\prime }}$ and $\overline{{\theta }^{\prime }{v}_n^{\prime }}$. When the $Re$ increases to $Re=1.645\times {10}^5$ (Figure 17b), the peak locations of all statistical quantities shift upstream, in line with the earlier transition location. Simultaneously, the peak intensities are significantly enhanced, indicating that both turbulence intensity and turbulent transport efficiency (for momentum and heat) increase with $Re$.

The instantaneous flow structures and associated thermal field characteristics on the suction surface under two $Re$ conditions, namely $Re=1.091\times {10}^5$ and $Re=1.645\times {10}^5$, are depicted in Figure 18. The three-dimensional vortex structures are visualized by the $\left|\mathbf{R}\right|=4$ isosurface and colored by the temperature $\theta$. The figure includes four sections for the present analysis: S3, S4, S5, and S6. Section S3 is a spanwise plane located at $z=0.1L_x$; this position was chosen over the $z=0.05L_x$ used for the pressure surface as it better captures the characteristic vortex structures on the suction side. Section S4 is a near-wall parallel plane at $y_n=0.003L_x$. Finally, considering the large-scale flow separation, two streamwise-normal sections, S5 (at $x/L_x=0.79$) and S6 (at $x/L_x=0.92$), are also included.

In Figure 18a, at the lowest $Re$ ($Re=1.091\times {10}^5$), large-scale separation produces prominent spanwise K-H roll-up structures originating from the instability of both the mainstream and the separated shear layer. Their spanwise cross-sectional features are displayed in section S3 (the leftmost white dashed circle indicates the K-H vortex core region). Their rotation drives the early transport of high-temperature fluid toward the wall and the lifting of low-temperature fluid, in agreement with the findings on the pressure surface. As the flow develops downstream, the K-H vortices break down into a series of complex three-dimensional vortex structures, primarily consisting of quasi-streamwise vortices and distorted arch-like vortices. The near-wall flow is dominated by quasi-streamwise vortices, which manifest as temperature streaks in the wall-parallel section S4 (i.e., band-like distributions formed by the stretching of high- and low-temperature fluids along the streamwise direction). In contrast to the pressure surface, the suction surface separation zone exhibits smaller-scale and fragmented vortex structures. These fragmented vortices significantly shorten the streamwise streaks. Additionally, their random orientations and three-dimensional interactions generate spanwise streaks. In the streamwise sections S5 and S6, the red high-temperature regions become more prominent, reflecting further enhancement of thermal mixing and thickening of the thermal boundary layer. These sections simultaneously display the vortex core structures and the associated complex temperature distributions (as indicated by the white dashed circles), where strong rotational motion significantly affects the $\theta$ field. Vortex rotation typically occurs in regions where the instantaneous $\theta$ isolines change abruptly, and these rotational zones are often accompanied by interfaces between high- and low-temperature fluids, indicating that the rotational effect locally increases the instantaneous temperature gradient $\nabla \theta$.

When the $Re$ increases to $Re=1.645\times {10}^5$ (Figure 18b), the flow on the suction surface reattaches, and the turbulent mixing layer becomes more confined to the near-wall region. Consequently, in sections S3, S5, and S6 the overall thickness of the mixing layer is reduced compared with the low $Re$ case. Despite this reduction, turbulent activity within the layer is significantly enhanced due to the increased $Re$. The vortex structures exhibit greater fragmentation and complexity, with the scale of K-H vortices decreasing and small-scale, distorted arch-like vortex structures becoming more prevalent. Moreover, the temperature field in the near-wall section S4 displays finer, more chaotic streak patterns. Notably, in sections S5 and S6, although the mixing layer is thinner, the proportion of red high-temperature regions increases, particularly near the vortex cores (as indicated by white dashed circles). This suggests that under high $Re$ and reattachment conditions, the vortex rotation and turbulent mixing in the near-wall region are significantly enhanced. These enhanced turbulent activities entrained and transported high-temperature mainstream fluid toward the wall more effectively. Compared to the low $Re$ case with large-scale separation, the thermal boundary layer at high $Re$ is inherently thinner. Within this thinner boundary layer, the intensified turbulent transport results in steeper local temperature gradients and more intense heat exchange near the wall.

The ensemble-averaged turbulent statistics on the suction surface for $Re=1.091\times {10}^5$ and $Re=1.645\times {10}^5$ are shown in Figure 19.

In Figure 19a, at the lowest $Re$, $Re=1.091\times {10}^5$, the large-scale open separation produces a significantly elevated shear layer, with the peak regions of the RMS velocity fluctuations ($u_{t,rms}^{\prime }$ and $v_{n,rms}^{\prime }$) located far from the wall (concentrated at $y_n=0.04$–0.08). This corresponds precisely to the core region of vortex breakdown and interaction observed in Figure 18a. The strong fluctuations away from the wall are a direct manifestation of the separated shear layer instability. However, the RMS temperature fluctuation (${\theta }_{rms}^{\prime }$) exhibits a distinctive dual-peak distribution: one peak within the elevated shear layer ($y_n=0.04$–0.08) associated with large-scale mixing and K-H instability, and another stronger peak in the downstream near-wall region ($x/L_x>0.7$). The formation mechanism of the latter differs from that of the former; as the separated shear layer develops downstream and undergoes transition, the resulting small-scale turbulent structures can penetrate into the near-wall region. Because of the extremely steep mean temperature gradient near the wall, even small wall-normal velocity fluctuations can generate intense temperature fluctuations, contrasting with velocity fluctuations which are suppressed by the no-slip condition. Notably, the distribution of the Reynolds shear stress $\overline{u_t^{\prime }{v}_n^{\prime }}$, although predominantly negative, exhibits a weak positive peak along the shear layer (roughly corresponding to the location of the K-H vortices in Figure 18a). In attached boundary layers, such as on the pressure surface, $\overline{u_t^{\prime }{v}_n^{\prime }}$ is typically negative, indicating momentum transport toward the wall. Here, the positive region indicates momentum transport in the opposite direction within the separated shear layer, driven by the entrainment and flapping motions of large-scale K-H vortices—that is, momentum transport from the near-vortex-core region toward the outer edge of the shear layer. Additionally, the normal turbulent heat flux $\overline{{\theta }^{\prime }{v}_n^{\prime }}$ exhibits a similar dual-peak pattern, with peaks observed in both the shear layer and the near-wall region, reflecting the combined effects of $v^{\prime }$ fluctuations and the temperature gradient distribution discussed above.

In contrast, at the higher Reynolds number $Re=1.645\times {10}^5$ (Figure 19b), the peak intensities of all turbulent statistical quantities are markedly enhanced, and their peak regions shift closer to the wall. The positive peak region of $\overline{u_t^{\prime }{v}_n^{\prime }}$ is smaller, and the negative peak is also located nearer to the wall, reflecting intense near-wall momentum exchange driven by smaller-scale vortex structures. This enhanced momentum exchange enables the flow to overcome the adverse pressure gradient and reattach. Furthermore, the peaks of the turbulent heat fluxes $\overline{{\theta }^{\prime }{u}_t^{\prime }}$ and $\overline{{\theta }^{\prime }{v}_n^{\prime }}$ are stronger and occur closer to the wall, indicating significantly improved heat transport in the near-wall region compared to the low $Re$ case, in agreement with [49].

In summary, the vortex structures on the blade surface (such as K-H vortices, quasi-streamwise vortices, hairpin vortices, and arch-like vortices) play a crucial role in promoting mixing between the mainstream and the near-wall fluid by inducing ejection and sweep events. The cumulative effect of these instantaneous flow behaviors is ultimately reflected in the turbulent statistics, which show enhanced velocity and temperature fluctuations, increased Reynolds stresses, and higher turbulent heat fluxes. As the $Re$ increases, these vortex structures become smaller in scale and more densely distributed, leading to enhanced peak values of turbulent statistics closer to the wall and thereby effectively improving the convective heat transfer efficiency on the blade surface. These detailed physical mechanisms revealed under zero-FST conditions establish an essential baseline for understanding the flow and heat transfer phenomena in blade passages. To extend these insights to practical compressor applications, it is important to acknowledge that FST is ubiquitous in real environments and can modify the observed flow characteristics. Specifically, FST promotes earlier transition through bypass mechanisms, injects energy into separated shear layers to accelerate reattachment and reduce separation bubble size, and causes vortical structures to become more fragmented and complex, thereby altering the friction and heat transfer distributions on blade surfaces [19]. Future studies will systematically investigate the coupled effects of $Re$ and FST to predict cascade flow and heat transfer under realistic operating conditions more accurately.

4. Conclusions

This study investigated the flow and heat transfer characteristics in a blade passage under different Reynolds numbers ($Re=1.091\times {10}^5$, $1.229\times {10}^5$, $1.367\times {10}^5$, $1.506\times {10}^5$, and $1.645\times {10}^5$) using Direct Numerical Simulation (DNS). The research revealed differences in boundary layer behavior between the pressure and suction surfaces and their effects on heat transfer. The main conclusions are as follows:

• In the absence of free-stream turbulence, the $Re$ significantly affects flow separation. On the pressure surface, no separation occurred at $Re=1.091\times {10}^5$ and $1.229\times {10}^5$, whereas at $Re=1.367\times {10}^5$, $1.506\times {10}^5$, and $1.645\times {10}^5$, the separation bubble length slightly increased with increasing $Re$. On the suction surface, separation occurred to varying degrees at all $Re$; however, the degree of separation decreased with increasing $Re$, with reattachment observed at $Re=1.367\times {10}^5$, $1.506\times {10}^5$, and $1.645\times {10}^5$.
• The velocity boundary layer thickness identification method based on Bernoulli’s principle accurately determined the boundary layer thickness under internal flow conditions where the free-stream velocity is not fixed. This method effectively identified boundary layer thickness in laminar, separated, and reattached regions by reconstructing inviscid velocity profiles, demonstrating high accuracy, particularly in complex flow environments.
• Significant differences exist in the boundary layer states between the pressure and suction surfaces. On the pressure surface, when the Reynolds number is low ($Re=1.091\times {10}^5$ and $1.229\times {10}^5$), the boundary layer remains in a laminar state. In contrast, under higher Reynolds number conditions ($Re=1.367\times {10}^5$, $1.506\times {10}^5$, and $1.645\times {10}^5$), the boundary layer undergoes transition following a weak separation. As the Reynolds number increases, the transition location shifts upstream, and the boundary layer subsequently develops into a turbulent state. Turning to the suction surface, transition was induced by flow separation at all $Re$ examined, with the transition location moving slightly upstream as the $Re$ increased; however, due to large-scale separation, the boundary layer did not reach a fully developed turbulent state.
• Turbulent vortex structures enhanced near-wall fluid mixing through induced ejection and sweep events, thereby improving momentum and heat transport efficiency. As the $Re$ increases, the boundary layer becomes thinner, the scale of turbulent vortex structures decreases, and their distribution becomes denser. These changes lead to a significant enhancement of turbulent fluctuations in the near-wall region, with increased Reynolds stresses promoting momentum transport. Meanwhile, the peak position of the turbulent heat flux shifts closer to the wall. This ultimately increases the wall Nusselt number and enhances convective heat transfer.