1. Introduction

2. WENO-ZQ Scheme

2.1. WENO-ZQ Scheme in One Dimension

This section introduces the higher-order WENO-ZQ scheme of the finite volume form of Navier-Stokes (N-S) equations. We first consider the one-dimensional N-S equation:

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

(1)

To simplify matters, the computational mesh is divided into several cells $I_i=\left[x_{i-1/2},x_{i+1/2}\right]$, each with a uniform size $h=x_{i+1/2}-x_{i-1/2}$ and associated cell centers are $x_i=\frac{1}{2}\left(x_{i+1/2}+x_{i-1/2}\right)$. Utilizing the intervals I_i as our control volumes, we can reformulate the semi-discretization of Eq. (1) as follows:

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

(2)

where $\overline{u}\left(x_i,t\right)=\frac{1}{h}{\displaystyle {\int }_{x_{i-1/2}}^{x_{i+1/2}}u}(x,t)dx$ is the cell average. We represent Eq. (2) with the following conservative formulation:

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

(3)

where ${\overline{u}}_i(t)$ is the numerical approximation to the cell average $\overline{u}\left(x_i,t\right)$, and the non-viscous numerical flux ${\widehat{f}}_{i+1/2}^{inv}$ is defined by ${\widehat{f}}_{i+1/2}^{inv}={\widehat{f}}^{inv}\left(u_{i+1/2}^{-},u_{i+1/2}^{+}\right)$ cedures narrated latter. A sixth order scheme [38] is adopted for the viscous numerical flux ${\widehat{f}}_{i\pm 1/2}^{vis}$.

If we can reconstruct $u_{i+1/2}^{\pm }=u\left(x_{i+1/2},t\right)+O\left(h^r\right)$, then Eq. (3) is the r-th order approximation to Eq. (2). Now we provide a detailed explanation of the reconstruction procedure of $u_{i+1/2}^{-}$to approximate $u\left(x_{i+1/2},t\right)$ up to third-order and fifth-order, respectively. Additionally, the reconstruction procedure for $u_{i+1/2}^{+}$is symmetrically mirrored with respect to $x_{i+1/2}$ of that for $u_{i+1/2}^{-}$.

Step 1. Select the larger stencil $T_1=\{I_{i-\mathcal{l}},\dots ,I_{i+\mathcal{l}}\},\mathcal{l}=1$ for the third-order case (or $\mathcal{l}=2$ for the fifth-order case). Then, a quartic polynomial is derived from the cell averages of $T_1$ is obtained by ensuring

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

(4)

Step 2. Then, select two smaller stencils $T_2=\left\{I_{i-1},I_i\right\}$ and $T_3=\left\{I_i,I_{i+1}\right\}$. Subsequently, two linear polynomials based on the cell averages of $T_2$ and $T_3$are obtained by requiring

(5)

and

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

(6)

Step 3. According to the construction idea of the central WENO scheme [20,21], we reformulate $p_1\left(x\right)$ as:

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

(7)

Note that Eq. (7) holds for any choice of the linear weights ${\gamma }_1$,${\gamma }_2$, ${\gamma }_3$with ${\gamma }_1\ne 0$. In order to ensure the numerical stability of spatial discretization, we require these linear weights to be positive numbers and satisfy ${\gamma }_1+{\gamma }_2+{\gamma }_3=1$.

Step 4. Compute the smoothness indicators ${\beta }_{\mathcal{l}},\mathcal{l}=1,2,3$ to assess the smoothness of the functions $p_{\mathcal{l}}(x),\mathcal{l}=1,2,3$ within the target cell $I_i$. The smaller ${\beta }_{\mathcal{l}}$ is, the smoother the polynomial $p_{\mathcal{l}}(x)$ is on the target grid $I_i$. The computation of these smoothness indicators follows the same methodology outlined in [18,28,29]:

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

(8)

where r=2 (or r=4 ) for r=1 and for $\mathcal{l}=2,3$. The associated explicit expressions are

$${\beta }_1=\frac{13}{12}{\left({\overline{u}}_{i-1}-2{\overline{u}}_i+{\overline{u}}_{i+1}\right)}^2+\frac{1}{4}{\left({\overline{u}}_{i-1}-{\overline{u}}_{i+1}\right)}^2$$

(9)

or

$$\begin{array}{ccc}\begin{array}{l}{\beta }_1\\ \\ \\ \end{array}& \begin{array}{l}=\\ \\ \\ \end{array}& \begin{array}{l}\frac{1}{144}{\left({\overline{u}}_{i-2}-8{\overline{u}}_{i-1}+8{\overline{u}}_{i+1}-{\overline{u}}_{i+2}\right)}^2\\ \frac{1}{15600}{\left(-11{\overline{u}}_{i-2}+174{\overline{u}}_{i-1}-326{\overline{u}}_i+174{\overline{u}}_{i+1}-11{\overline{u}}_{i+2}\right)}^2\\ \frac{781}{2880}{\left(-{\overline{u}}_{i-2}+2{\overline{u}}_{i-1}-2{\overline{u}}_{i+1}+{\overline{u}}_{i+2}\right)}^2\\ \frac{1421461}{1310400}{\left({\overline{u}}_{i-2}-4{\overline{u}}_{i-1}+6{\overline{u}}_i-4{\overline{u}}_{i+1}+{\overline{u}}_{i+2}\right)}^2\end{array}\end{array}$$

(10)

$${\beta }_2={\left({\overline{u}}_{i-1}-{\overline{u}}_i\right)}^2$$

(11)

and

$${\beta }_3={\left({\overline{u}}_i-{\overline{u}}_{i+1}\right)}^2$$

(12)

Step 5. Calculate the nonlinear weights using the linear weights and smoothness indicators. Specifically, we employ τ, defined as the absolute difference between β₁, β₂, and β₃. Notably, this definition differs from the formula outlined in [30,31,32] due to variations in the reconstruction stencils. The expression for τ was initially introduced in [22] for the finite difference approach and subsequently adopted in [29]:

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

(13)

Then the non-linear weights are expressed as

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

(14)

where, ε represents a small positive value introduced to prevent the denominator of Eq. (14) from becoming zero. Therefore, if $\epsilon \ll {\beta }_{\mathcal{l}}$, then within the smooth region ${\omega }_{\mathcal{l}}={\gamma }_{\mathcal{l}}+O\left(h^4\right)$, where the nonlinear weights are essentially consistent with the linear weights, the WENO scheme can achieve third-order or fifth-order numerical accuracy. In this study, the value of ε is set to ${10}^{-6}$.

Step 6. Substituting the linear weights in Eq. (7) with the nonlinear weights specified in Eq. (14), we derive the updated reconstruction formulation for the conservative values u(x, t) at the point $x_{i+1/2}$ within the target cell $I_i$, as follows:

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

(15)

Step 7. The semi-discretization scheme presented in Eq. (3) undergoes temporal discretization using a third-order TVD Runge-Kutta method [33]:

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

(16)

2.2. WENO-ZQ Scheme in Two Dimensions

Next we consider the two-dimensional N-S equation:

$$\left\{\begin{array}{c}{u}_t+f{(u)}_x+g{(u)}_y=0\\ u\left(x,y,0\right)=u_0\left(x,y\right)\end{array}\right.$$

(17)

To simplify matters, the grid meshes are uniformly partitioned into cells, with each cell having sizes represented by $x_{i+1/2}-x_{i-1/2}=\mathrm{\Delta }x,$$y_{k+1/2}-y_{k-1/2}=△y,$$(\mathrm{\Delta }x=\mathrm{\Delta }y=h),$ and cell centers denoted by $\left(x_i,y_k\right)=\left(\frac{1}{2}\left(x_{i-1/2}+x_{i+1/2}\right),\left(\frac{1}{2}\left(y_{k-1/2}+y_{k+1/2}\right)\right)\right.$. Two dimensional cells are labeled as $I_{i,k}=I_i\times I_k=\left[x_{i-1/2},x_{i+1/2}\right]\times \left[y_{k-1/2},y_{k+1/2}\right]$ and the semi-discrete finite volume scheme is defined as follows：

$$\begin{array}{ccc}\frac{d{\overline{u}}_{i,k}\left(t\right)}{dt}& =& L\left(u_{i,k}\right)\\ & =& -\frac{1}{h^2}({{\displaystyle \int }}_{y_{k-1/2}}^{y_{k+1/2}}f\left(u\left(x_{i+1/2},y,t\right)\right)dy-{{\displaystyle \int }}_{y_{k-1/2}}^{y_{k+1}}f\left(u\left(x_{i-1/2},y,t\right)\right)dy\\ & & +{{\displaystyle \int }}_{x_{i-1/2}}^{x_{i+1/2}}g\left(u\left(x,y_{k+1/2},t\right)\right)dx-{{\displaystyle \int }}_{x_{i-1/2}}^{x_{i+1/2}}g\left(u\left(x,y_{k-1/2},t\right)\right)dx).\end{array}$$

(18)

We approximate Eq. (18) through a conservative formulation as follows:

(19)

where the numerical fluxes ${\widehat{f}}_{i+1/2,k}^{inv}$ and ${\widehat{g}}_{i,k+1/2}^{inv}$ are defined as

$$\begin{array}{c}{\widehat{f}}_{i+1/2,k}^{inv}={\displaystyle {\displaystyle \sum }_{\mathcal{l}=1}^3}{\varpi }_{\mathcal{l}}{\widehat{f}}^{inv}\left(u_{i+1/2,k+{\sigma }_{\mathcal{l}}}^{-},u_{i+1/2,k+{\sigma }_{\mathcal{l}}}^{+}\right)\\ {\widehat{g}}_{i,k+1/2}^{inv}={\displaystyle {\displaystyle \sum }_{\mathcal{l}=1}^3}{\varpi }_{\mathcal{l}}{\widehat{g}}^{inv}\left(u_{i+{\sigma }_{\mathcal{l}},k+1/2}^{-},u_{i+{\sigma }_{\mathcal{l}},k+1/2}^{+}\right)\end{array}$$

(20)

to approximate $\frac{1}{h}{\displaystyle {\int }_{y_{k-1/2}}^{y_{k+1/2}}{f}^{inv}}\left(u\left(x_{i+1/2},y,t\right)\right)dy$ and $\frac{1}{h}{\displaystyle {\int }_{x_{i-1/2}}^{x_{i+1/2}}{g}^{inv}}\left(u\left(x,y_{k+1/2},t\right)\right)dx$, respectively. ${\varpi }_{\mathcal{l}}$, and ${\sigma }_{\mathcal{l}}$

represent the weights and nodes of a three-point Gaussian within the cell $\left[-1/2,1/2\right]$, respectively. Here $u_{i+1/2,k+{\sigma }_{\mathcal{l}}}^{\pm }$ and $u_{i+{\sigma }_{\mathcal{l}},k+1/2}^{\pm }$ denote the fifth-order approximations of $u\left(x_{i+1/2}^{\pm },y_k+{\sigma }_{\mathcal{l}}h\right)$ and $u\left(x_i+{\sigma }_{\mathcal{l}}h,y_{k+1/2}^{\pm }\right)$, respectively, which will be reconstructed by WENO procedures. The numerical fluxes ${\widehat{f}}^{inv}(a,b)$ and ${\widehat{g}}^{inv}(a,b)$ are determined as one-dimensional monotone fluxes.

Now, employing a dimension-by-dimension approach, we elucidate the reconstruction process for point values $u_{i\pm 1/2,k+{\sigma }_{\mathcal{l}}}^{\mp }$ from cell averages $\left\{{\overline{u}}_{i,k}\right\}$.

• In the x-direction, utilizing $\left\{{\overline{u}}_{i,k}\right\}$, we execute the WENO reconstruction procedure described earlier for the one-dimensional case to reconstruct

$$\overline{u}{\left(x_{i\mp 1/2}^{\pm }\right)}_j=\frac{1}{h}{{\displaystyle \int }}_{y_{j-1/2}}^{y_{j+1/2}}u\left(x_{i\mp 1/2}^{\pm },y\right)dy,j=k-2,\cdots ,k+2,$$

(20)

where $\overline{u}{\left(x_{i\mp 1/2}^{\pm }\right)}_j$ represents point values in the x-direction and cell averages in the y-direction. During the WENO reconstruction process, we compute the nonlinear weights only once for each $j$, as we employ the same linear weights and smoothness indicators at the reconstructed points $x_{i\mp 1/2}^{\pm }$.

• In the y-direction, utilizing $\overline{u}{\left(x_{i\mp 1/2}^{\pm }\right)}_j$, we apply the WENO reconstruction method outlined above for the one-dimensional scenario to reconstruct the point values $u_{i\pm 1/2,k+{\sigma }_{\mathcal{l}}}^{\mp }$. Once again, we calculate the nonlinear weights only once for all three Gauss points in the y-direction. In contrast, for classical WENO reconstruction, nonlinear weights must be computed at least three times. Additionally, negative linear weights occur at Gauss point σ₂ = 0 in classical WENO reconstruction, necessitating the application of techniques to address negative linear weights in WENO reconstruction [34].

Reconstructing the point values $u_{i+{\sigma }_{\mathcal{l}},k\pm 1/2}^{\mp }$ from cell averages $\left\{{\overline{u}}_{i,k}\right\}$is similar to that for $u_{i\pm 1/2,k+{\sigma }_{\mathcal{l}}}^{\mp }$. Notably, in this step, the intermediate results should be a point value in the $y$ direction, but the cell averages in the x direction.

After the abovementioned steps, the semi-discretization scheme Eq. (19) is temporally discretized using a third-order TVD Runge-Kutta method Eq. (16).

Extending the third-order and fifth-order finite volume WENO-ZQ schemes to the three-dimensional scenario follows a dimension-by-dimension approach, akin to the two-dimensional counterpart. Similar to the two-dimensional case, it becomes evident that the novel third-order and fifth-order finite volume WENO-ZQ schemes outlined in this study surpass the classical same-order finite volume WENO-JS schemes in [28]. This superiority stems from the ability to employ identical linear weights across various quadrature points along the boundaries of the target cell, thereby significantly reducing the reconstruction process's computational burden.

2.6. Unsteady Convergence Compared with WENO-JS Scheme

The three-dimensional viscous Taylor-Green vortex (TGV) [43] is a typical benchmark case in fluid mechanics, commonly used to assess the capabilities of viscous flow solvers in simulating vortex structure evolution, turbulent transition, and kinetic energy dissipation. Due to its simple initial and boundary conditions and known analytical solutions, it can be employed to evaluate the accuracy and convergence of numerical methods. The physical domain is a cube with a side length of 2πL, which contains the vorticity of the initial flow field with a smooth distribution. It is solved with the following initial conditions to present the convergence of the unsteady solution by various high-order schemes.

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

(24)

Detailed flow and initial conditions used here were specified in [43]. The Reynolds number of the flow is defined as $\mathrm{Re}=\frac{{\rho }_0{V}_{0}L}{\mu }$ and equals 1600. In this field, kinetic energy is transferred from the large scale down to the smaller scale until it reaches a sufficiently small length scale. Finally, it dissipates into internal energy under physical viscosity. The largest dissipation rate occurs at $t=8\frac{L}{V_0}=8t_c$. The purpose here is to evaluate the convergence property of the schemes. This paper investigates the evolution of the Taylor-Green vortex with time using four different mesh sizes, 32×32×32, 64×64×64, 128×128×128, and 256×256×256, under different WENO schemes. Coherent structures in the field are observed employing a Q-criterion, with iso-surfaces set at $Q{\left(L/V_0\right)}^2=0.005$, and velocity magnitude colors the iso-surfaces.

Figure 5, Figure 6, Figure 7 and Figure 8 illustrate the Taylor-Green vortex flows computed with different WENO schemes for comparative analysis. Computational results show that the new WENO-ZQ schemes can capture more turbulence structures than those of the same order WENO-JS schemes across identical meshes. Moreover, the computed flows demonstrate favorable symmetry in the vortex structures, suggesting that the schemes effectively preserve the flow field's symmetry.

The evolution of kinetic energy and its dissipation rate is contrasted with the reference solution acquired from the pseudo-spectral code [44]. The evolution of kinetic energy calculated on different meshes is shown in Fig. 9. The results of the WENO-ZQ scheme on 32×32×32 cells are similar to that of the WENO-JS scheme on 64×64×64 cells. It is shown that the convergence rate of the WENO-ZQ scheme is faster than that of the WENO-JS scheme. The numerical results show that these new high-order finite volume WENO schemes reduce the computational cost and capture more details of flow physics than classical WENO-JS schemes [18]. The kinetic energy dissipation rates calculated using different schemes on 256×256×256 cells are shown in Fig. 10. It can be observed from the figure that the WENO-ZQ scheme is closer to the reference solution.

Figure 10. Evolution of kinetic energy dissipation rates by various WENO schemes

Figure 10. Evolution of kinetic energy dissipation rates by various WENO schemes

3. Numerical Tests of Turbomachinery Flow

4. Conclusions

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