Application of a Novel High-Order WENO Scheme in LES Simulations

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This study aims to extend the high-order WENO-ZQ scheme to large eddy simulation methodologies. A high-order WENO-ZQ scheme is integrated into a three-dimensional structured grid LES CFD solver for the high-fidelity simulation of complex flows, such as boundary layer separation, transition, and shockwave–turbulent boundary layer interactions. Validations under viscous flow conditions demonstrate that the WENO-ZQ scheme maintains high-resolution accuracy while reducing the number of grid points required, significantly lowering the computational cost of LES.

Abstract

To achieve high-fidelity large eddy simulation (LES) predictions of complex flows while keeping computational costs manageable, this study integrates a high-order WENO-ZQ scheme into the LES framework. The WENO-ZQ scheme has been extensively studied for its accuracy, robustness, and computational cost in inviscid flow applications. This study extended the WENO-ZQ scheme to viscous flows by integrating it into a three-dimensional structured grid LES CFD solver. High-fidelity simulations of turbulent boundary layer flow and supersonic compression ramp flows were conducted, with the scheme being applied for the first time to study laminar boundary layer transition and separation flows in the high-load, low-pressure turbine PakB cascade. Classic numerical case validations for viscous conditions demonstrate that the WENO-ZQ scheme, compared to the same-order WENO-JS scheme, exhibits lower dispersion and dissipation errors, faster convergence, and better high-frequency wave resolution. It maintains high-resolution accuracy with fewer grid points. In application cases, the WENO-ZQ scheme accurately captures the three-dimensional flow characteristics of shockwave–boundary layer interactions in supersonic compression ramps and shows high accuracy and resolution in predicting separation and separation-induced transition in low-pressure turbines.

1. Introduction

The accurate computation of turbulent separation [1,2], transition [3,4], and shockwave-turbulent boundary layer interactions (SWTBI) [5,6] is pivotal for the design and development of aero-engines [7,8]. The limitations of traditional Reynolds-averaged Navier–Stokes (RANS) approaches in accurately capturing these complex flow phenomena have been well-documented [9,10]. Advanced high-fidelity numerical strategies, particularly large eddy simulation (LES), are instrumental in providing a detailed understanding of these flows and their transient characteristics [11,12]. Therefore, to achieve high-fidelity LES predictions of such complex flows while maintaining manageable computational costs, it is essential to employ high-order numerical schemes that offer high resolution, low dissipation, and are suitable for compressible flows.

High-order numerical schemes have been the subject of intense investigation over recent decades. Harten [13] pioneered the notion of total variation diminishing (TVD) by analyzing the causes of non-physical oscillations near shockwaves and devised a TVD finite-difference scheme that maintains second-order accuracy in smooth regions. To rectify the degradation of the TVD scheme at smooth extrema, Harten and colleagues [14,15] subsequently developed the high-precision, essentially non-oscillatory (ENO) scheme. The ENO scheme maintains non-oscillatory properties in discontinuous regions by selecting the “smoothest” template among all candidates for function reconstruction, thereby avoiding accuracy degradation in smooth regions. However, the ENO scheme relies on information from only one template, resulting in some loss of flow field information while increasing computational cost and storage requirements. Moreover, the template selection process in the ENO scheme involves substantial logical judgments, which impairs the parallelization of the algorithm [16] and decreases the computational efficiency of multidimensional ENO schemes. To overcome these limitations, Liu et al. [17] proposed the weighted essentially non-oscillatory (WENO) scheme, which performs reconstruction using a weighted average of all templates. Jiang and Shu [16] further established a framework for developing finite-difference WENO schemes with arbitrary accuracy orders and introduced the fifth-order WENO (WENO-JS) scheme. They provided the essential methodology for creating smoothness indicators and nonlinear weights. A significant innovation of the WENO-JS scheme is its method of combining multiple local reconstructions using different-sized spatial stencils to achieve the final spatial reconstruction. However, the classical WENO schemes still present significant challenges [18], including high computational costs and process complexity, as well as the dependence of optimal (linear) weights on mesh topology, which can lead to negative weights and compromise robustness. These issues are particularly pronounced in three-dimensional contexts.

To enhance the effectiveness of WENO schemes, Levy and colleagues investigated the issue of non-existent linear weights when performing third-order WENO reconstruction at the target cell center, as discussed in their paper [19]. They observed that WENO schemes using linear weights exhibit slightly varying accuracy depending on the grid resolution. In contrast, schemes utilizing nonlinear weights show reduced errors and more stable accuracy under sparse grid conditions. They proposed a technique called the central WENO (CWENO) scheme [19,20] to deal with this difficulty. Based on the classical WENO scheme, Zhu and Qiu [21,22] introduced the core concept of constructing non-equidistant spatial templates and convex combinations of different-order polynomials. They designed a simple and practical high-precision finite-difference and finite-volume WENO scheme with unequal stencil sizes, referred to as the WENO-ZQ scheme. Recently, the WENO-ZQ scheme has seen advancements and applications across various engineering fields. Sheng et al. [23] and Zhong and Sheng [24] constructed a series of high-order WENO-ZQ schemes through the introduction of phantom points, enabling their deployment in both structured and unstructured grid systems.

In inviscid flow applications, the solution accuracy, robustness, and computational cost of the WENO-ZQ scheme have been extensively analyzed both qualitatively and quantitatively [21,22,24,25]. This study aims to extend this efficient high-order WENO-ZQ scheme to viscous flow and comprehensively evaluate its performance by integrating it into a structured-grid LES CFD solver. Viscous flow is a complex system, and the WENO scheme is only a small part; thus, the computation time of different schemes on the same grid cells will not significantly differ relative to the entire solver. However, since the WENO-ZQ scheme fundamentally eliminates the real-time calculation of nonlinear weights strongly correlated with grid shape, thereby addressing the issue of computational crashes caused by negative nonlinear weights, it can be qualitatively assumed that this scheme reduces computational costs. When applying the WENO-ZQ scheme in viscous flow, two particularly meaningful questions arise regarding the flow structure that can be captured under the same computational grid resolution and the number of grids needed to capture the same physical phenomenon. In light of this, this study uses the WENO-ZQ and classical WENO-JS schemes to conduct relevant calculations on these two points.

Numerous scholars have conducted LES research utilizing high-order WENO schemes, yielding a substantial body of significant findings. For instance, Ladeinde et al. [26] compared the performance of the ENO and WENO-JS schemes in LES for simulating the energy spectrum decay of two-dimensional isotropic compressible turbulence. Their results indicated that the standard ENO scheme exhibited numerical turbulence at high wavenumbers, resulting in insufficient energy dissipation, whereas the WENO scheme demonstrated moderate dissipation characteristics. Toh and Ragab [27] conducted LES studies on the collision of dual-jet under three-dimensional supersonic conditions using a fifth-order WENO scheme and a dynamic subgrid-scale turbulence model, revealing that the fifth-order WENO scheme is highly suitable for simulating turbulent flows with bow shocks and can effectively capture unstable shocks within the flow field. Hahn and Drikakis [28] compared three high-precision schemes (the three-order MUSCL, the fifth-order MUSCL, and the ninth-order WENO-JS) in the application of LES for a channel with hill-like curvature. Their findings showed that the WENO scheme achieved the highest resolution and was capable of capturing the finest flow details, though it also incurred the highest computational cost. Li et al. [29] applied a quasi-fourth-order hybrid central WENO scheme within a finite volume framework to perform LES on transonic turbulent flows over a protuberance, demonstrating the scheme’s ability to effectively distinguish between shocks and turbulent fluctuations, thereby exhibiting excellent shock-capturing and turbulence-resolving capabilities. However, the switch function within the hybrid scheme, the smoothness of transitions, and the tunable parameters within the function could impact the computational outcomes [30]. Hoffmann et al. [31] successfully predicted the transition process in flat plate LES using a fifth-order WENO scheme, though the accuracy of wall pressure predictions for a supersonic compression ramp was somewhat limited. It is evident that the current application of high-order WENO schemes in LES is primarily focused on flows with simple geometric configurations. Therefore, this study integrates the high-order, low-dissipation WENO-ZQ scheme into the LES framework to elucidate and evaluate the potential of the WENO-ZQ scheme in LES applications, thereby providing a clearer understanding of its practical applicability and effectiveness in engineering contexts.

The structure of this paper is outlined as follows: Section 2 constructs the high-order finite-volume WENO-ZQ scheme and integrates it into a three-dimensional LES CFD solver. The computational performance of the WENO-ZQ scheme for viscous flow fields is then systematically evaluated using multiple classical numerical experiments and compared with the classical WENO-JS scheme. Section 3 applies the fifth-order WENO-ZQ scheme to the prediction of flat-plate turbulent boundary layers, supersonic compression corner flows, and typical high-load, low-pressure turbine cascade transition and separation flows with high-fidelity. The advantages of the WENO-ZQ scheme in LES are further validated by comparing the results with those from the open literature. Finally, Section 4 concludes with some final remarks.

2. WENO-ZQ Scheme and Numerical Tests

2.1. Description of WENO-ZQ Scheme

This section introduces the high-order WENO-ZQ scheme using the one-dimensional Navier–Stokes (N-S) equations as an example:

$$\{\begin{array}{c}{u}_t+f_x^{inv}\left(u\right)+f_x^{vis}\left(u\right)=0\\ u\left(x,0\right)=u_0\left(x\right)\end{array}$$

(1)

The computational domain is uniformly discretized using grid cells $I_i=\left[x_{i-1/2},x_{i+1/2}\right]$, with a size set to $h=x_{i+1/2}-x_{i-1/2}$. Upon selecting cell $I_i$ as the current control volume, discretization of the spatial terms yields the following expression:

$$\begin{array}{cc}{\displaystyle \frac{d\overline{u}\left(x_i,t\right)}{dt}}& \begin{array}{cc}+& \begin{array}{cc}{\displaystyle \frac{1}{h}}\left(f^{inv}\left(u\left(x_{i+1/2},t\right)\right)-f^{inv}\left(u\left(x_{i-1/2},t\right)\right)\right)& \end{array}\end{array}\\ & \begin{array}{cc}+& \begin{array}{cc}{\displaystyle \frac{1}{h}}\left(f^{vis}\left(u\left(x_{i+1/2},t\right)\right)-f^{vis}\left(u\left(x_{i-1/2},t\right)\right)\right)& =0\end{array}\end{array}\end{array}$$

(2)

where the cell average of $u(x,t)$ is $\overline{u}\left(x_i,t\right)={\displaystyle \frac{1}{h}}{\displaystyle {\int }_{x_{i-1/2}}^{x_{i+1/2}}u}(x,t)dx$. The approximation of Equation (2) is rendered in a conservative form as follows:

$${\displaystyle \frac{d{\overline{u}}_i(t)}{dt}}=L\left(u_i\right)=-{\displaystyle \frac{1}{h}}\left({\widehat{f}}_{i+1/2}^{inv}-{\widehat{f}}_{i-1/2}^{inv}\right)-{\displaystyle \frac{1}{h}}\left({\widehat{f}}_{i+1/2}^{vis}-{\widehat{f}}_{i-1/2}^{vis}\right)$$

(3)

where ${\overline{u}}_i(t)$ represents the numerical approximation of $\overline{u}\left(x_i,t\right)$. The viscous numerical flux ${\widehat{f}}_{i\pm 1/2}^{vis}$ is computed using a sixth-order scheme [32]. The inviscid numerical flux ${\widehat{f}}_{i+1/2}^{inv}$ can be obtained using the flow parameters $u_{i+1/2}^{\pm }$ on either side of $x_{i+1/2}$, i.e., ${\widehat{f}}_{i+1/2}^{inv}={\widehat{f}}^{inv}\left(u_{i+1/2}^{-},u_{i+1/2}^{+}\right)$. The value of $u_{i+1/2}^{\pm }$ is reconstructed by the WENO scheme.

If $u_{i+1/2}^{\pm }=u\left(x_{i+1/2},t\right)+O\left(h^r\right)$ can be reconstructed, then Equation (3) is the r-th order approximation to Equation (2). Taking $u_{i+1/2}^{-}$ as an example, the reconstruction procedure of $u_{i+1/2}^{-}$ using third-order and fifth-order WENO schemes (denoted as WENO-ZQ3 and WENO-ZQ5, respectively) is described in detail. $u_{i+1/2}^{-}$ and $u_{i+1/2}^{+}$ are reconstructed using WENO schemes with left- and right-biased stencils, respectively.

Step 1. Select a larger stencil $T_1=\{I_{i-\ell },\dots ,I_{i+\ell }\}$, with $\ell =1$ for the WENO-ZQ3 (or $\ell =2$ for the WENO-ZQ5). Then, a quartic polynomial $p_1(x)$ can be constructed using the average values of the grid cell parameters within the stencil $T_1$:

$${\displaystyle \frac{1}{h}}{{\displaystyle \int }}_{I_{i+j}}{p}_1(x)dx={\overline{u}}_{i+j},j=-\ell ,\dots ,\ell ;\ell =1\begin{array}{cc}& \left(\begin{array}{cc}or& \ell =2\end{array}\right)\end{array}$$

(4)

Step 2. Selection of the two smaller stencils, $T_2=\left\{I_{i-1},I_i\right\}$ and $T_3=\left\{I_i,I_{i+1}\right\}$ is performed. Similarly, linear polynomials $p_2(x)$ and $p_3(x)$ are constructed using the cell averages within the stencils $T_2$ and $T_3$, respectively, by imposing the following conditions:

$${\displaystyle \frac{1}{h}}{{\displaystyle \int }}_{I_{i+j}}{p}_2(x)dx={\overline{u}}_{i+j},j=-1,0$$

(5)

$${\displaystyle \frac{1}{h}}{{\displaystyle \int }}_{I_{i+j}}{p}_3(x)dx={\overline{u}}_{i+j},j=0,1$$

(6)

Step 3. Unlike the classical WENO-JS scheme, which requires computing optimal linear weights, the WENO-ZQ scheme adopts a construction approach similar to the CWENO scheme [19,20]. It allows for the selection of arbitrary nonzero linear weights ${\gamma }_1$ and arbitrary linear weights ${\gamma }_2$ and ${\gamma }_3$. Thus, $p_1\left(x\right)$ can be rewritten as:

$$p_1(x)={\gamma }_1\left({\displaystyle \frac{1}{{\gamma }_1}}{p}_1(x)-{\displaystyle \frac{{\gamma }_2}{{\gamma }_1}}{p}_2(x)-{\displaystyle \frac{{\gamma }_3}{{\gamma }_1}}{p}_3(x)\right)+{\gamma }_2{p}_2(x)+{\gamma }_3{p}_3(x)$$

(7)

To ensure the numerical stability of spatial discretization, it is necessary for these linear weights to be positive numbers to satisfy the condition ${\gamma }_1+{\gamma }_2+{\gamma }_3=1$, and this expression achieves third- or fifth-order accuracy.

Step 4. Discontinuities may arise in the polynomial $p_{\ell }(x)$ within the cell. Therefore, the smoothness of reconstruction polynomials within the cell $I_i$ is denoted by the smoothness coefficients ${\beta }_{\ell }$:

$${\beta }_{\ell }={\displaystyle \sum }_{\kappa =1}^r{{\displaystyle \int }}_{I_i}{h}^{2\kappa -1}{\left({\displaystyle \frac{d^{\kappa }{p}_{\ell }(x)}{d{x}^{\kappa }}}\right)}^{2}dx,\ell =1,2,3$$

(8)

where $r=2$ (or $r=4$) for $\ell =1$ and $r=1$ for $\ell =2,3$.

Step 5. The nonlinear weights ${\omega }_{\ell }$ are calculated using a combination of linear weights ${\gamma }_{\ell }$ and smoothing coefficients ${\beta }_{\ell }$. The adaptive formula of τ can be defined in various ways [21,25]; in this study, it is defined based on the absolute differences between ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$:

$$\tau ={\left({\displaystyle \frac{\left|{\beta }_1-{\beta }_2\right|+\left|{\beta }_1-{\beta }_3\right|}{2}}\right)}^2$$

(9)

Notably, this definition differs from the formula outlined in [33,34,35] due to variations in the reconstruction stencils. Equation (10) defines the nonlinear weights, with a small positive value $\epsilon ={10}^{-6}$ to avert division by zero in the denominators [22,24,25]:

$${\omega }_{\ell }={\displaystyle \frac{{\overline{\omega }}_{\ell }}{{{\displaystyle \sum }}_{\ell \ell =1}^3{\overline{\omega }}_{\ell \ell }}},\begin{array}{cc}& {\overline{\omega }}_{\ell }={\gamma }_{\ell }\left(1+{\displaystyle \frac{\tau }{\epsilon +{\beta }_{\ell }}}\right)\begin{array}{cc},& \begin{array}{cc}& \ell =1,2,3\end{array}\end{array}\end{array}$$

(10)

Step 6. Substitute the nonlinear weights ${\omega }_{\ell }$ in Equation (10) for the linear weights ${\gamma }_{\ell }$ in Equation (7), yielding the final reconstruction formula for the conserved variable u(x, t) at the point $x_{i+1/2}$ within the target cell $I_i$:

$$u_{i+1/2}^{-}={\omega }_1\left({\displaystyle \frac{1}{{\gamma }_1}}{p}_1\left(x_{i+1/2}\right)-{\displaystyle \frac{{\gamma }_2}{{\gamma }_1}}{p}_2\left(x_{i+1/2}\right)-{\displaystyle \frac{{\gamma }_3}{{\gamma }_1}}{p}_3\left(x_{i+1/2}\right)\right)+{\omega }_2{p}_2\left(x_{i+1/2}\right)+{\omega }_3{p}_3\left(x_{i+1/2}\right)$$

(11)

Step 7. The semi-discretization scheme presented in Equation (3) undergoes temporal discretization using a third-order TVD Runge–Kutta (R-K) method [36]:

$$\{\begin{array}{ccc}{u}_i^{\left(1\right)}& =& u_i^n+\Delta tL\left(u_i^n\right)\\ u_i^{\left(2\right)}& =& {\displaystyle \frac{3}{4}}{u}_i^n+{\displaystyle \frac{1}{4}}{u}_i^{\left(1\right)}+{\displaystyle \frac{1}{4}}\Delta tL\left(u_i^{\left(1\right)}\right)\\ u_i^{n+1}& =& {\displaystyle \frac{1}{3}}{u}_i^n+{\displaystyle \frac{2}{3}}{u}_i^{\left(2\right)}+{\displaystyle \frac{2}{3}}\Delta tL\left(u_i^{\left(2\right)}\right)\end{array}$$

(12)

The WENO-ZQ schemes are readily adaptable to two-dimensional or three-dimensional scenarios through a dimension-by-dimension strategy. Unlike the classical WENO-JS schemes [37], the finite-volume WENO-ZQ schemes can employ identical linear weights at various quadrature points along the boundaries of the target cell. This approach eliminates the need for real-time calculation of linear weights and avoids concerns regarding negative linear weights. Consequently, this enhancement bolsters the computational process’s stability and diminishes the computational overheads associated with the reconstruction procedure [21,22,23,24]. Since the linear weights no longer depend on the mesh topology, this scheme is convenient for applications with a non-uniform and adaptive mesh [24,38]. Furthermore, it can be observed that constructing the third-order or fifth-order accuracy WENO-ZQ model requires the replacement of polynomials and smoothness coefficients of the large spatial stencil.

2.2. Governing Equations and Numerical Method

The high-order WENO-ZQ scheme, previously outlined, is incorporated into the in-house computational fluid dynamics (CFD) solver, NUAA-Turbo. NUAA-Turbo is a three-dimensional, viscous, unsteady, compressible, finite-volume RANS/LES-CFD framework solver that utilizes a multi-block structured grid topology. Designed for high-precision numerical simulations of complex flows in turbomachinery, this solver has been widely applied and endorsed by several aerospace engine design organizations in China. The formulation process for the RANS and LES governing equations within the NUAA-Turbo is detailed in reference [38], while references [39,40] illustrate the application of this solver within the RANS framework. To handle both low- and high-speed flows in aerodynamic analysis, a global preconditioning technique [41] is adopted. The governing equations in a curvilinear coordinate system are expressed in a conservative flux format, as delineated in Equation (13).

$$M{\mathrm{\Gamma }}_q^{-1}{\displaystyle \frac{\partial q}{\partial \tau }}+{\displaystyle \frac{\partial \left(F^{inv}+F^{vis}\right)}{\partial \xi }}+{\displaystyle \frac{\partial \left(G^{inv}+G^{vis}\right)}{\partial \eta }}+{\displaystyle \frac{\partial \left(H^{inv}+H^{vis}\right)}{\partial \xi }}=0$$

(13)

where $M\equiv \partial Q/\partial q$ represents the matrix responsible for the conversion from conservative variables $Q=(\rho ,\rho u,\rho v,\rho w,\rho e_t)$ to the primitive variable $q=(\rho ,u,v,w,p)$. The matrix ${\mathrm{\Gamma }}_q^{-1}$ is a constant diagonal matrix, and its elements are determined exclusively by the reference Mach number. Following the application of the Favre filtering technique to Equation (13), the final LES control equations are obtained, incorporating subgrid-scale (SGS) model closure relations [38]. The wall-adapting local eddy-viscosity (WALE) subgrid-scale model [42] is utilized for the LES numerical tests in this study. It can accurately recover the proper y³ near-wall scaling of eddy viscosity without the need for a dynamic procedure. The subgrid eddy viscosity is formulated as follows:

$${\mu }_{sgs}={\mathrm{Re}}_{ref}\rho {\left(C_w\Delta \right)}^2{\displaystyle \frac{{\left(S_{ij}^d{S}_{ij}^d\right)}^{3/2}}{{\left(S_{ij}{S}_{ij}\right)}^{5/2}+{\left(S_{ij}^d{S}_{ij}^d\right)}^{5/4}}}$$

(14)

where $S_{ij}$ represents the subgrid stress tensor and the grid scale is denoted by $\Delta$. The dissipation coefficient C_w is a model constant, typically calibrated using decaying homogenous isotropic turbulence (DHIT) [43], as detailed in Section 2.5. Given that the governing equations in this study are non-dimensionalized, the reference Reynolds number ${\mathrm{Re}}_{ref}$ is included in Equation (14).

This study employs a structured mesh for the discretization of the computational domain, and the governing equations are discretized using the FVM. For computing the numerical inviscid flux, the advection upstream splitting method (AUSM⁺-up) flux splitting scheme [44] is employed, as expressed below:

$${\widehat{f}}_{i+1/2}^{inv}=F_{i+1/2}^{\left(c\right)}+P_{i+1/2}={\dot{m}}_{i+1/2}{\overrightarrow{\psi }}_{i+1/2}+P_{i+1/2}$$

(15)

Unless otherwise specified, within the LES framework, the AUSM⁺−up scheme [44] is employed to solve the convective fluxes, while interface reconstruction is achieved with a fifth-order WENO-ZQ scheme and a sixth-order central difference scheme for viscous terms [32]. Time discretization is handled with a third-order TVD R-K method [36].

2.3. Spectral Properties of the WENO-ZQ Scheme

In high-precision numerical simulation, the ideal numerical scheme should minimize numerical dissipation and numerical dispersion to capture intricate wave structures and flow field details precisely. Spectral characteristic analysis is, therefore, a valuable tool for evaluating and comparing the advantages and limitations of various numerical schemes, especially for simulating high-frequency oscillations and nonlinear phenomena [38]. Pirozzoli [45] introduced a numerical approach for analyzing the spectral characteristics of the WENO scheme by means of the approximate dispersion relation (ADR). The present investigation utilizes the ADR technique to assess the dissipation and dispersion properties of diverse WENO schemes.

This section begins with an analysis of the spectral characteristics of the WENO-ZQ5 scheme, using various linear weight combinations to demonstrate that the choice of weights does not compromise the scheme’s optimal order of accuracy. Therefore, three sets of linear weight combinations are considered: (1) ${\gamma }_1=0.98$, ${\gamma }_2=0.01$, and ${\gamma }_3=0.01$; (2) ${\gamma }_1=1.0/3.0$, ${\gamma }_2=1.0/3.0$, and ${\gamma }_3=1.0/3.0$; (3) ${\gamma }_1=0.01$, ${\gamma }_1=0.495$, and ${\gamma }_3=0.495$. These combinations satisfy the linear weight selection criteria proposed in Section 2.1, namely ${\gamma }_1>0$, ${\gamma }_2>0$, ${\gamma }_3>0$, and ${\gamma }_1+{\gamma }_2+{\gamma }_3=1$. Figure 1 illustrates the spectral characteristics of the WENO-ZQ5 scheme with various linear weights. Here, ϕ denotes the wavenumber, and Φ_Re and Φ_Img represent the dispersion and dissipation properties, respectively. Despite variations in the choice of linear weights, curves 1, 2, and 3 display similar trends, with the dispersion and dissipation properties of the latter two sets of weights being nearly identical. It has been verified that when the linear weight ${\gamma }_1$ of the five-point large stencil is further reduced, the spectral characteristics remain very close to those of curve 3. This indicates that choosing any set of positive linear weights summing to one does not affect the optimal fifth-order accuracy of the WENO-ZQ5 scheme. Therefore, for simplicity, the WENO-ZQ scheme will utilize the first set of linear weights (${\gamma }_1=0.98$, ${\gamma }_2=0.01$, and ${\gamma }_3=0.01$) unless otherwise specified.

Figure 2 and Figure 3 depict the nonlinear analysis of spectral characteristics for different WENO schemes. Figure 2 illustrates that the WENO-ZQ3 scheme exhibits superior spectral properties to the third-order WENO-JS (WENO-JS3) scheme. Additionally, the spectral properties of the third-order MUSCL scheme with a minmod limiter are also included in this figure. The WENO-ZQ3 scheme demonstrates reduced dissipation and dispersion relative to the MUSCL scheme. However, the dispersion of the MUSCL scheme is higher than that of the spectral method, indicating that the calculated wave amplitude exceeds the theoretical value, which may result in an overestimation of prediction efficiency in internal flow. The WENO-ZQ5 scheme exhibits improved dispersion characteristics over the fifth-order WENO-JS (WENO-JS5) scheme across all wave number ranges. In the range of ϕ < 0.95, the WENO-ZQ5 scheme exhibits lower dissipation compared to the WENO-JS5 scheme, indicating a superior ability to resolve high-frequency waves. This implies that the WENO-ZQ5 scheme is more effective in capturing detailed flow structures and preserving waves or vortices within the flow field. As a result, the WENO-ZQ5 scheme can capture the same level of detail with fewer computational grid cells.

2.4. The One-Dimensional Burgers Equation

The Burgers equation, a fundamental equation in fluid mechanics, is commonly used to evaluate the computational accuracy of numerical schemes for solving flow problems. In this section, the WENO-JS and WENO-ZQ schemes are applied to solve the Burgers equation, allowing for a comparative analysis of their computational accuracy. Previous studies [22] have validated the computational precision of the WENO-ZQ and WENO-JS schemes using the inviscid Burgers equation, demonstrating that both achieve the designed high-order accuracy, with the WENO-ZQ scheme exhibiting a smaller numerical error. Consequently, these two schemes are applied in this section to solve the viscous Burgers equation. The one-dimensional viscous nonlinear Burgers equation is expressed as follows:

$$\{\begin{array}{c}{u}_t+u{u}_x=\upsilon u_{xx}\\ u(x,0)={\displaystyle \frac{1}{\upsilon }}\sin x\\ u(0,t)=u(2\pi ,t), t>0\end{array}$$

(16)

For this case, the simulation is conducted to a terminal time of t = 0.5 with a viscosity coefficient of $\upsilon =0.5$. The time step is set as $\Delta t=O\left(\Delta x\right)$, and time discretization is performed using a third-order explicit TVD-RK scheme. The numerical solution obtained on a fine grid with N = 2560 is defined as the “exact solution.” Numerical errors are evaluated using the L^∞ and L¹ norms to assess the accuracy of the schemes.

Table 1. Computational errors and convergence order in the one-dimensional viscous Burgers equation.

Schemes	N	L∞ Error	L∞ Order	L1 Error	L1 Order
WENO-JS	20	4.09 × 10−3		1.53 × 10−4
	40	1.29 × 10−4	4.9838	2.09 × 10−5	2.8746
	80	7.05 × 10−6	4.1977	2.80 × 10−6	2.8994
	160	3.33 × 10−7	4.4042	4.11 × 10−7	2.7698
WENO-ZQ	20	2.70 × 10−3		7.69 × 10−5
	40	1.34 × 10−4	4.3309	1.14 × 10−5	2.7530
	80	4.04 × 10−6	5.0576	1.65 × 10−6	2.7885
	160	1.17 × 10−7	5.1094	2.13 × 10−7	2.9535

presents the numerical errors and convergence orders for both schemes under different grid resolutions. The results indicate that the WENO-ZQ scheme achieves fifth-order accuracy. Moreover, WENO-ZQ exhibits smaller numerical errors compared to WENO-JS on the same mesh.

2.5. Calibration the Coefficient C_w for the WALE Model

Although the WENO-ZQ scheme has good spectral characteristics, as a three-dimensional viscous CFD solver, it is necessary to consider the dissipation characteristics of mutually coupled systems, such as the flux method, interface value reconstruction, the viscous flux calculation method, and the turbulence model. The grid-generated turbulence experiment conducted by Comte-Bellot and Corrsin [46] (referred to as the CBC experiment) provides a fundamental turbulence idealization known as DHIT. DHIT is a simple yet important test case used to calibrate numerous CFD solvers. Nicoud and Ducros [42] highlighted that the value of the dissipation coefficient C_w affects the magnitude of subgrid turbulent kinetic energy dissipation, necessitating its calibration to ensure the model accurately simulates the turbulence dissipation process. A straightforward method to determine C_w involves assuming that the WALE model provides the same subgrid turbulent kinetic energy dissipation as the classical Smagorinsky model, given by

$$C_w^2=C_s^2{\displaystyle \frac{\langle \sqrt{2}{\left(S_{ij}{S}_{ij}\right)}^{3/2}\rangle }{\langle S_{ij}{S}_{ij}\overline{O{P}_1}/\overline{O{P}_2}\rangle }}$$

(17)

where the constant C_s is typically set to 0.18. Therefore, C_w can be numerically assessed using homogeneous isotropic turbulence fields [42,43]. This section calibrates the dissipation coefficient C_w in the LES-WALE model of the NUAA-Turbo solver using energy spectra from the CBC experiment. The AUSM⁺-up flux splitting scheme, the fifth-order WENO scheme for reconstructing interface values, and a sixth-order central difference method for calculating viscous fluxes are used.

Since the measurements from the CBC experiment are dimensionless, a unit cube is employed as the computational domain. The field is initialized at the physical times 42M/U_o in the experiment. M = 0.0508 m and U_o = 10 m/s are the size of the turbulence-generating mesh and free stream velocity, respectively. The detailed generation method of the initial flow field is discussed by Rozema et al. [47]. The grid is distributed with 64 × 64 × 64 cells, and the simulations are conducted under periodic boundary conditions. The dimensionless time step is set to $\Delta t=5\times {10}^{-4}$. For the NUAA-Turbo solver, the results match well with experimental results when C_w = 0.45. Therefore, the dissipation coefficient C_w for the WALE model in this study is set to 0.45. Figure 4 compares the energy spectrum results obtained from different WENO schemes based on the WALE model with C_w = 0.45 at two subsequent physical times (98 and 171 M/U_o) with the CBC experiment data [46]. The WENO-JS5 scheme exhibits insufficient dissipation at high wavenumbers, as highlighted by the red circle in Figure 4. In contrast, the WENO-ZQ5 scheme shows good consistency with the experimental results within the same wavenumber range.

2.6. Unsteady Convergence Compared with WENO-JS Scheme

The convergence properties of high-order schemes are investigated using the three-dimensional viscous vortex (TGV) case, quantitatively addressing the two issues highlighted in the introduction. TGV [48] is a standard benchmark in fluid mechanics, commonly used to assess the capabilities of viscous flow solvers in simulating vortex structure evolution, turbulent transition, and kinetic energy dissipation. The simplicity of its initial and boundary conditions, combined with the availability of analytical solutions, make it an ideal benchmark for assessing the accuracy and convergence of numerical methods. The domain of interest is a cube with a side length of 2πL, which contains the vorticity of the initial flow field with a smooth distribution. The detailed flow and initial conditions used in this study are described in reference [48]. Simulations are conducted using the initial conditions outlined in Equation (18) to demonstrate the convergence of various high-order schemes in achieving an unsteady solution.

$$\{\begin{array}{l}\rho ={\rho }_0\\ u=V_0\sin \left({\displaystyle \frac{x}{L}}\right)\cos \left({\displaystyle \frac{y}{L}}\right)\cos \left({\displaystyle \frac{z}{L}}\right)\\ v=-V_0\cos \left({\displaystyle \frac{x}{L}}\right)\sin \left({\displaystyle \frac{y}{L}}\right)\cos \left({\displaystyle \frac{z}{L}}\right)\\ w=0\\ p=p_0+{\displaystyle \frac{1}{16}}{\rho }_0{V}_0^2\left(\cos \left({\displaystyle \frac{2z}{L}}\right)+2\right)\left(\cos \left({\displaystyle \frac{2x}{L}}\right)+\cos \left({\displaystyle \frac{2y}{L}}\right)\right)\end{array}$$

(18)

where $\rho$ denotes the density, while $u$, $v$, and $w$ represent the velocities in the three spatial directions. The pressure is indicated by $p$, and $L$ refers to the length of the computational domain. Subscript ‘0’ indicates reference values. The Reynolds number for the flow is defined as $\mathrm{Re}={\rho }_0{V}_{0}L/\mu =1600$, where $\mu$ is the viscosity coefficient. The Mach number is $Ma=V_0/\sqrt{\gamma p_0/{\rho }_0}=0.1$ and $\gamma =1.4$. In this field, kinetic energy transfers from the larger scales to the smaller scales until it reaches a sufficiently small scale, where it ultimately dissipates into internal energy due to physical viscosity. Periodic boundary conditions are employed. The evolution of the TGV over time is analyzed using four different mesh sizes: 32 × 32 × 32, 64 × 64 × 64, 128 × 128 × 128, and 256 × 256 × 256, to evaluate grid convergence and the accuracy of numerical methods for different mesh densities. For the 32³ case, the dimensionless time step is $\Delta t=0.001L/V_0=0.001t_c$, ensuring that the Courant number remains below 1. A total of 20,000 time gsteps were simulated, with the calculation extending to $t=20t_c$. The time step was halved with each doubling of the grid resolution. The approximate computational time required for each case utilizing various fifth-order schemes is provided in

Table 2. Computational time for different grid resolutions.

Grid Size	Dimensionless Time Step	Cores	WENO-JS5 Wallclock Time (h)	WENO-ZQ5 Wallclock Time (h)
323	0.001tc	2	0.9	0.8
643	0.0005tc	6	5	4.5
1283	0.00025tc	32	15	13
2563	0.000125tc	128	80	70

.

Figure 5, Figure 6, Figure 7 and Figure 8 illustrate the TGV structures computed using the WENO-JS and WENO-ZQ schemes at t_final = 20t_c. Coherent structures in the field are observed employing a Q-criterion, with iso-surfaces delineated at $Q{\left(L/V_0\right)}^2=0.005$ and color-coded based on velocity magnitude. The results reveal that the computed flow fields exhibit good symmetry in the vortex structures, indicating that both WENO schemes effectively preserve the flow field’s symmetry. The turbulence vortex structures obtained with the WENO-ZQ scheme show superior resolution compared to those obtained with the WENO-JS scheme on the same mesh. For example, Figure 7c and Figure 8b demonstrate that the WENO-ZQ5 scheme with a 64³ mesh captures turbulence vortex structures similar to those obtained with the WENO-JS scheme using a 128³ mesh. This indicates that the WENO-ZQ scheme can resolve more turbulence structures than the WENO-JS scheme with the same order and on the same mesh, achieving similar results with fewer grid cells.

Figure 9 displays the time evolution of average kinetic energy (AKE) for different WENO schemes compared to reference solutions obtained using a pseudo-spectral code [49]. Overall, higher-order WENO reconstruction schemes yield results closer to the reference solutions for the same grid resolution. Among schemes with the same order and grid resolution, the WENO-ZQ scheme consistently provides results that are closer to the reference values than the classical WENO-JS scheme. Notably, the AKE results for the WENO-ZQ scheme with a 64³ mesh are comparable to those of the WENO-JS scheme with a 128³ mesh, indicating superior grid convergence of the WENO-ZQ scheme. Since computational cost scales linearly with the number of grid cells, the WENO-ZQ scheme can significantly reduce computational costs while maintaining accuracy.

Figure 10 illustrates the kinetic energy dissipation rate (KEDR) results for various schemes using a 256³ mesh. The peak dissipation rate of the reference solution [49] occurs at $t=9t_c$, and both fifth-order WENO schemes successfully capture this peak. The differences between the WENO schemes are quantitatively assessed using the root mean square error (RMSE) relative to the reference solution. In Figure 10a, the RMSE for the WENO-JS3 scheme is 1.365 × 10⁻³, whereas the WENO-ZQ3 scheme achieves an RMSE of 6.25 × 10⁻⁴, indicating better accuracy. Similarly, in Figure 10b, the RMSE for the WENO-JS5 scheme is 5.353 × 10⁻⁴, while the WENO-ZQ5 scheme has an RMSE of 1.91 × 10⁻⁴, demonstrating closer agreement with the reference values. The WENO-ZQ scheme consistently yields predictions that closely approximate the reference values, with the WENO-ZQ3 scheme performing comparably to the WENO-JS5 scheme. These findings demonstrate that the high-order finite-volume WENO-ZQ scheme offers reduced computational costs while capturing more detailed fluid physics compared to the WENO-JS scheme [18].

3. Application Cases

The shockwave–turbulent boundary layer interaction is a prominent topic in LES research. This study investigates this issue using the WENO-ZQ scheme, with a comparison of the results with experimental data [50] and direct numerical simulation (DNS) [51]. The focus is on the formation of supersonic turbulent boundary layers, which is central to the analysis, including a comparative evaluation of the computational grid resolution required in this context. Finally, the WENO-ZQ5 scheme is applied to investigate the separation transition phenomenon in a Pak B turbine cascade. The test cases were executed on an H3C UniServer R4700 G3 high-performance rack server, manufactured by H3C Technologies Co., Ltd., headquartered in Hangzhou, China. It is equipped with 768 cores, featuring EPYC 7742 CPUs and 64 GB of 3200 MHz memory.

3.1. Turbulent Boundary Layer Simulation

To simulate the evolution of turbulent boundary layer (TBL) flows using LES and DNS, it is essential to provide dynamic inlet boundary conditions that accurately reflect the turbulence characteristics at the inlet. The obtained TBL with specified boundary layer thickness (δ) and friction velocity (u_τ) is pivotal for investigating SWTBLI. This study employs the recycling and rescaling method (RRM) for compressible flows, as refined by Sagaut et al. [52], to generate turbulent inflow conditions for the TBL. The simulation is conducted using a fifth-order WENO scheme, and the results are compared with those published by Dawson et al. [53].

Figure 11 depicts the configuration of the computational domain and mesh for the flat-plate TBL. The dimensions are non-dimensionalized with respect to the boundary layer thickness δ in the streamwise ($x$), spanwise ($y$), and wall-normal ($z$) directions. The recycling plane is positioned 7.31δ downstream from the inflow plane. Wagner et al. [54] summarized typical grid sizes for the DNS/LES of wall-turbulent boundary layers using a high-precision computational scheme, indicating that the first cell height near the wall in the wall-normal direction should satisfy $\Delta z^{+}\le 1$. This study adheres to these standards, with the grid dimensions at the first layer near the wall at the recycling plane being $\Delta x^{+}=8.2$, $\Delta y^{+}=5.3$, and $\Delta z^{+}=0.46$, with calculations using a local wall friction velocity u_τ = 32.66 m/s.

Table 3. Grid details and comparison.

	Calculation Domain Size Lx × Ly × Lz/δ3	Grid Resolution Δx+ × Δy+ × Δz+	Cells Nx × Ny × Nz
DNS [55]	8.3 × 2.0 × 8.2	7.5 × 4.3 × 0.2	410 × 160 × 112
Present LES	8.62 × 2.15 × 5.23	8.2 × 5.3 × 0.46	311 × 120 × 112

summarizes the grid details and compares them with the grid of the DNS [55] under similar inflow conditions, confirming that the grid meets the requirements for LES.

The boundary condition setting can be found in [51,53,56]. For the inlet, a Mach number of 2.9 and a static temperature of 108.1 K are specified. Supersonic outlet boundary conditions are imposed on the upper and right boundaries of the computational domain. The bottom wall adheres to a no-slip, isothermal condition, with the wall temperature constant at 307 K. Turbulent boundary conditions at the inlet are dynamically provided using the recycling and rescaling method (RRM).

Table 4. Comparison of key parameters of the TBL on the inflow plane.

	T∞/K	Tw/K	Mint	θ/mm	Reθ	δ/mm	δ∗/mm	Cf
Bookey et al. [50]	108.1	307	2.9	0.43	2400	6.7	2.36	0.00225
Wu et al. [51]	108.1	307	2.9	0.38	2300	6.4	1.8	0.00217
Tong et al. [56]	108.1	307	2.9	0.41	2300	6.5	2.06	0.00256
Dawson & Lele [53]	108.1	307	2.9	0.50	2400	6.7	2.85	0.00203
Present LES	108.1	307	2.9	0.47	2400	6.5	2.37	0.00209

summarizes the key parameters of the TBL at the inlet for both the present study and those reported in the literature.

To compare the results with those of different schemes, data for 200 dimensionless time flow fields are analyzed. Figure 12 shows the mean velocity profiles provided by Van Driest transformation [57] at the recycling plane, where $U_{vd}^{+}$ is dimensionless by u_τ. This figure illustrates that in the viscous sublayer, the velocity profiles predicted by the WENO-JS5 and WENO-ZQ5 schemes are consistent with the classical wall law. In the logarithmic layer, the average velocity profile distribution predicted by the WENO-ZQ5 scheme closely aligns with the classical wall law. The distribution of Reynolds stress $R_{ij}=\sqrt{{\displaystyle \frac{\rho }{\rho w}}}{\displaystyle \frac{\sqrt{\overline{\left(u_i^{\prime }{u}_j^{\prime }\right)}}}{u_{\tau }}}$ normal to the wall of the recycling plane is compared with Dawson’s LES results [53]. As shown in Figure 13, the distributions obtained from both schemes are essentially similar, with peaks predicted to be slightly higher than the reference values. A comparison of the RMSE for the predictions from both schemes reveals that the RMSE of the WENO-ZQ5 scheme is lower, indicating a superior alignment with the reference values in terms of the Reynolds stress distribution.

The visualization of coherent structures in Figure 14 is achieved through the Q-criterion ($Q{\left(\delta /U_{inlet}\right)}^2=0.1$), with iso-surfaces colored by the local Mach number. The TBL in this study has a momentum thickness Reynolds number ${\mathrm{Re}}_{\theta }=2400$, which classifies it as a moderate-Reynolds-number flow. Both WENO schemes capture the formation of elongated, asymmetric hairpin vortices, consistent with conclusions from the existing literature [58]. The WENO-ZQ5 scheme can capture a richer flow structure, as compared to the WENO-JS5 scheme. In summary, the comparison of velocity profiles, Reynolds stress, and coherent structures demonstrates that both the WENO-ZQ5 and WENO-JS5 schemes effectively generate realistic turbulent boundary layers.

3.2. Supersonic Compression Ramp

The SWTBLI represents a critical aerodynamic and thermodynamic process in the supersonic inlet flow. It can lead to significant alterations in heat transfer in the external flow and produce unsteady pressure loads, potentially shortening the lifespan of structures. Internally, it may trigger large-scale, unsteady separated flows, increasing total pressure loss and causing flow distortion that can lead to engine surge. The compression ramp model, despite its simplicity, captures essential SWTBLI phenomena such as boundary layer separation, reattachment, and turbulence enhancement due to inverse pressure gradients [50]. The highly three-dimensional flow field features intricate multi-scale interactions, making it an ideal test case for validating CFD methods. The WENO-ZQ5 scheme is employed for the LES of this case.

Figure 15 illustrates the configuration of the computational domain and mesh, where the inlet boundary layer thickness is specified as δ = 6.7 mm. The computational domain consists of a flat-plane section (auxiliary domain) and a 24° compression ramp section (principal domain) [50]. The coordinate system is origin-based at the ramp, with upstream and downstream lengths of 7.73δ, a spanwise width of 2.15δ, and a wall-normal height of 5.23δ. The TBL is generated in the auxiliary domain using the RRM, and the resulting turbulence data are utilized as the inlet boundary conditions for the principal domain. Figure 15b presents a simplified schematic of the grid for the ramp domain. The mesh is uniformly distributed in the spanwise direction and refined in the streamwise direction near the ramp. In the wall-normal direction, the grid spacing near the wall is designed to ensure that the height of the first cell satisfies $\Delta z^{+}\le 1$ [54]. The mesh dimensions at the inflow plane of the principal domain are identical to those at the recycling plane, with the flow field data from the auxiliary domain’s recycling plane directly employed as the inlet boundary condition for the principal domain.

To facilitate comparison with the DNS results,

Table 5. Grid details and comparison for the principal domain.

	Principal Domain Size Lx × Ly × Lz/δ3	Grid Resolution Δx+ × Δy+ × Δz+	Cells Nx × Ny × Nz
DNS [51]	(9 + 7) × 2.0 × 8.2	(3.4~7.2) × 4.1 × 0.2	1024 × 160 × 128
LES-G1	(7.73 + 7.73) × 2.15 × 5.23	(6.1~11.0) × 7.2 × 0.46	392 × 89 × 112
LES-G2		(4.5~8.2) × 5.3 × 0.46	529 × 120 × 112
LES-G3		(3.4~6.1) × 4.0 × 0.46	708 × 161 × 112

presents the grid information for the LES of the compression ramp flow, as well as for the DNS conducted by Wu et al. [51] under similar inflow conditions. In this study, three grid resolutions, namely LES-G1, LES-G2, and LES-G3, were employed to assess grid independence. The grid points in the wall-normal direction remain consistent across all three grids, while a uniform distribution is maintained in the spanwise direction. The refinement primarily affects the grid distribution near the corner region. Notably, the finest mesh resolution in the current simulations remains substantially lower than that of the DNS grid.

Once the SWBLI flow field reaches a fully developed state, sampling begins. Given the characteristic frequency of the turbulent boundary layer, $U_{\infty }/\delta$, the time interval $0.001U_{\infty }/\delta$ is selected to ensure adequate time resolution. A total of 400,000 time steps are computed, spanning a total time of $400U_{\infty }/\delta$. For instance, in the case of LES-G3, approximately 13 days were required to complete 400 time periods using 177 cores. To verify the accuracy of the computed results, the data from these $400U_{\infty }/\delta$ are averaged and compared with the DNS results [51] and experimental data [50]. Figure 16 illustrates the variation of the mean wall pressure ($p_w/p_{\infty }$) and the mean skin friction coefficient (C_f) along the streamwise direction within the principal domain. For the wall pressure distribution, it is evident that all three grid resolutions can accurately predict the pre-separation wall pressure, the pressure plateau within the separation region (a characteristic of separated flow), and the final inviscid pressure solution downstream. Notably, the results of LES-G2 and LES-G3 exhibit a high degree of consistency with the DNS results in terms of the streamwise distribution of wall pressure. By determining the location where the mean friction coefficient C_f equals zero, LES-G1 predicts a separation point at x_sep = −2.75δ and a reattachment point at x_rea = 0.84δ. In contrast, LES-G2 and LES-G3 predict an average separation point within the corner flow at x_sep = −2.92δ and a reattachment point near x_rea = 1.05δ. The separation region predicted by LES-G1 is slightly smaller than the DNS results, while those predicted by LES-G2 and LES-G3 are more closely aligned with the DNS data. Overall, the pressure and friction coefficient distributions computed by LES-G2 and LES-G3 exhibit better agreement with the experimental values and DNS results. Consequently, the LES-G2 grid is adopted for subsequent studies.

Figure 17 provides a visualization of the instantaneous flow structure within the principal domain. This figure delineates the quasi-ordered structure of the turbulent boundary layer through the application of the Q-criterion, with coloring based on streamwise velocity. The semi-transparent gray surface represents the dimensionless pressure iso-surface ($Q=1.5{\left(U_{\infty }/\delta \right)}^2$), showing the three-dimensional topology of the separation shockwave. This figure reveals pronounced three-dimensional features in the interaction region. Large vortex structures interacting with the shockwave’s root experience fragmentation at the ramp due to increased turbulent fluctuations. Concurrently, deformation of the shockwave occurs at its root, with surface wrinkling becoming evident in the spanwise direction. The shockwave maintains typical two-dimensional structural characteristics when located away from the interaction region.

3.3. Pak B Low-Pressure Turbine Cascade

The transition of the boundary layer from laminar to turbulent flow represents a pivotal and intricate aspect within the field of fluid mechanics. Understanding the physical mechanisms behind this transition is vital for studying turbulence formation and has significant practical value in engineering for predicting and controlling transitions. Due to the working environment of low-pressure turbines, the transition significantly influences the boundary layer state on the suction surface. Accurately predicting the separation of laminar flow and the boundary layer transition is crucial for designing high-load, low-loss, low-pressure turbines. The transition is driven by nonlinear instability, necessitating highly accurate methods to capture small-scale vortices.

The load factor Zweifiel coefficient $Z_w=1.8$ of the Pak B blade profile is selected to test the WENO-ZQ5 scheme, which is a typical high-load turbine blade profile [59]. It comes from the middle section of a Pratt and Whitney crewless air vehicle’s low-pressure blade, with the key parameters listed in

Table 6. The main parameters of Pak B.

Parameters	Values
Axial chord length (mm)	159.5
Blading pitch (mm)	141.2
Inflow angle (°)	55
Outflow angle (°)	30

. Blade profile data are sourced from reference [60], and the experimental data are from reference [61]. The free stream turbulence intensity (FSTI) is 0.08%, and the Reynolds number, based on the freestream velocity ($U_{\infty }$) and the axial chord length (C_x), is 100,000, according to the experiments.

The computational domain and grid distribution for the Pak B cascade are illustrated in Figure 18. The inlet to the leading edge of the blade is approximately one chord length, and the outlet to the trailing edge is about two chord lengths. An O-type mesh topology is employed around the blade, with mesh refinement applied near the blade surface, where the height of the first layer of cells in the wall-normal direction is set to Δy⁺ = 0.3. To ensure mesh independence, the computational grid was progressively refined along both the axial (x-direction) and spanwise directions (z-direction), to evaluate the impact of mesh resolution on the aerodynamic performance simulation results. The flow-through period is defined as $T=C_x/U_{\infty }$. A time step of 0.0001T is used to ensure that the Courant number remains below 1 throughout the simulation [62,63]. The LES covered 320 T, with the flow field achieving statistical stability after 100 T. Data collection began after advancing an additional 120 T. For the LES with 4.07 million grid points, utilizing 160 cores, the actual simulation process required approximately one month.

One of the challenges in numerical simulations of LPT turbines is the precise prediction of laminar separation bubbles. Figure 19a presents a comparison between the LES results and experimental data [61] for the time-averaged surface static pressure coefficient $C_p=\left(p_{\infty }-p_s\right)/\left(0.5\rho U_{\infty }\right)$ on the blade with varying grid resolutions. Here, $p_{\infty }$ is the freestream pressure, and p_s denotes the time-averaged local wall pressure. The LES outcomes across all three grid resolutions exhibit a high degree of concordance with the experimental data, particularly in predicting the location and size of the separation bubbles on the pressure surface (PS). The results obtained from the two finer grids are nearly identical, suggesting that, when balancing computational resources and accuracy, the grid with 4.07 million nodes is the optimal choice for subsequent flow field analyses. The figure further illustrates that the suction side (SS) of the blade experiences a favorable pressure gradient from 0.06C_x to 0.6C_x, followed by a strong adverse pressure gradient. A pressure plateau is observed at the 0.75C_x–0.85C_x region of the SS, where separation occurs before reattachment. Minor differences are noted in the separation region on the SS compared to experimental data.

Figure 19b further compares the time-averaged spanwise-averaged velocity distributions at various positions on the suction surface (50%, 60%, 70%, 75%, 80%, 85%, and 90% of C_x). Prior to the 60%C_x, position, the boundary layer thickness increases slowly due to the accelerating airflow in this region, which exerts significant stretching forces on the fluid, thereby suppressing velocity fluctuations. However, beyond 60%C_x, especially past the pressure peak on the SS, the boundary layer begins to experience adverse pressure gradients. At 75%C_x, the tangential velocity in the near-wall region is nearly zero, indicating an impending inflection point in the velocity profile and subsequent boundary layer separation. At the 80%C_x, the boundary layer thickness increases rapidly after separation. A slight deviation between the numerical simulation results and experimental data is observed at the reattachment point (85%C_x), which is attributed to increased flow field complexity. At 90%C_x, the turbulent boundary layer post-reattachment shows a good match with the experimental velocity distribution. Additionally, the time-averaged streamlines and pressure contours in Figure 19c clearly depict the locations of separation bubbles on both the PS and SS.

Figure 20 presents instantaneous coherent structures on the Pak B suction surface, identified using the Q-criterion and colored by the Mach number. It is evident that, under adverse pressure gradients, the boundary layer decelerates and separates, forming a full-span Kelvin–Helmholtz (K-H) structure. This structure gradually rolls up during downstream convection, evolving into a typical hairpin vortex structure and eventually developing into a turbulent boundary layer. The turbulent boundary layer sheds at the trailing edge and mixes with the pressure side boundary layer, forming a wake. Overall, the methods used in this study demonstrate high accuracy and resolution in predicting separation and transition induced by separation in low-pressure turbines, which is highly valuable for blade design.

4. Conclusions

This study extended the novel high-order finite volume WENO scheme to viscous flows. It was embedded into the three-dimensional unsteady viscous LES-CFD solver NUAA-Turbo. The new WENO scheme uses the same linear weights at each quadrature point on the boundary of the target cell in its spatial reconstruction process, which avoids the issue of negative linear weights. A series of classical benchmark cases were tested and validated using the LES-CFD solver. The results were compared with those obtained using the classical WENO-JS scheme and experimental data from the literature. The following conclusions are drawn:

(1)

Spectral characteristic analysis and DHIT case calibration reveal that the WENO-ZQ scheme exhibits lower dissipation and dispersion errors, providing better resolution for high-frequency waves.

(2)

Numerical simulations of the three-dimensional viscous TGV reveal that the WENO-ZQ scheme captures more turbulence structures on the grid compared to the same-order WENO-JS scheme. The flow field predictions using the third-order WENO-ZQ scheme are analogous to those of the WENO-JS5 scheme, with the results on 64³ cells for the WENO-ZQ scheme being comparable to those on 128³ cells for the WENO-JS scheme. This indicates that the WENO-ZQ scheme can deliver high-precision simulations with fewer grid points, significantly reducing computational resource requirements.

(3)

The WENO-ZQ scheme accurately predicts complex physical phenomena such as transition, separation-induced transition, and SWTBLI. Its time-averaged results are in strong agreement with those reported in the existing literature. At the same grid resolution, the WENO-ZQ scheme captures more detailed flow field structures, proving its potential for application in LES studies of complex compressible flows.