An Adaptive Mesh Refinement–Rotated Lattice Boltzmann Flux Solver for Numerical Simulation of Two and Three-Dimensional Compressible Flows with Complex Shock Structures

Abstract

An adaptive mesh refinement–rotated lattice Boltzmann flux solver (AMR-RLBFS) is presented to simulate two and three-dimensional compressible flows with complex shock structures. In the method, the RLBFS, which has a strong shock-capturing capability and can effectively eliminate the shock instability phenomenon, is applied to solve the flow filed by reconstructing the fluxes at each cell interface adaptively with the mesoscopic lattice Boltzmann model. To locally and dynamically improve the resolution of intricate shock structures and optimize the required computational resources, a block-structured adaptive mesh refinement (AMR) technique is introduced. The validity and effectiveness of the proposed method are confirmed through a range of two and three-dimensional numerical cases, including the shock tube problem, the four-wave Riemann problem, explosion within a rectangular box, and the vorticity induced by a shock. The results obtained using the AMR-RLBFS exhibit excellent agreement with published data and demonstrate high accuracy in capturing complex shock structures. The computational efficiency of the AMR-RLBFS can be also improved significantly compared to the RLBFS on uniform grids. Furthermore, the numerical outcomes underscore the capability of the AMR-RLBFS to eliminate shock instability effects while efficiently capturing a broader spectrum of small-scale vertical structures. These findings highlight the ability of AMR-RLBFS to improve the computational efficiency and capture intricate shock structures effectively, making it a valuable tool for studying a wide range of compressible flows from aerodynamics to astrophysics.

1. Introduction

With the rapid development of computer technology, computational fluid dynamics (CFD) was introduced and soon became widely used in medical, energy, marine, and aerospace fields due to its low cost and high applicability. It deals with the governing equations (Navier–Stokes equations) using various numerical methods, and the finite volume method (FVM) [1] is one of the most widely used numerical methods. Through direct discretization of the conservation laws, the physical quantities in FVM are conserved. Mass, momentum, and energy are also preserved through the numerical treatment, which maintains a high degree of accuracy and can be used for problems with complex geometry. The key to the success of the FVM for simulating compressible flows is the development of an appropriate flux solver to evaluate the fluxes at the cell interface.

Many effective Riemann solvers have been proposed by approximating the macroscopic fluxes directly. For instance, the Roe scheme [2] is a good choice for simulating compressible inviscid flow due to its high accuracy and good resolution for shock structures. However, the traditional Roe approach [3] is impotent in the face of the problem of shock instability in hypersonic flows. Quirk [4] delineated three instances of shock instability: the carbuncle phenomenon, kinked Mach stem, and odd–even decoupling. Many experts posit that the primary catalyst for shock instability lies in the inadequate numerical dissipation within the shock region. Numerous strategies have been proposed to address this issue. These include amalgamating dissipative schemes with less dissipative yet reasonably accurate schemes [5], employing entropy fixation [6] to augment dissipation through restrictions on the minimum eigenvalue of the system, enhancing fundamental upwind dissipation [7], and introducing shear viscosity in the momentum flux [8]. Some have tackled the problem by curbing local dissipation [9]. Others attribute the shock instability to the momentum interpolation mechanism (MIM) in the Roe scheme and consequently advocate for an enhanced Roe scheme [10]. A prevailing viewpoint attributes shock instability to the multi-dimensional character of compressible flow, prompting the development of a multi-directional Riemann solver along each cell interface, known as the rotated Roe scheme [11,12]. This novel approach employs the Roe scheme in two directions by decomposing the cell interface’s normal vector into two components: one aligned with the velocity difference vector, and the other orthogonal to it. Notably, the rotated Roe scheme exhibits a robust capacity to capture strong shocks and effectively eliminate the phenomenon of shock instability.

In contrast to macroscopic Riemann flux solvers, recent years have seen the development of mesoscopic numerical methods based on lattice Boltzmann [13,14] and gas kinetic schemes. Utilizing the Maxwellian equilibrium distribution function as a foundation, various compressible lattice Boltzmann models [15,16,17,18,19,20] have been introduced, differing in lattice velocities and equilibrium distribution functions [21,22,23,24]. Building upon these models, the lattice Boltzmann flux solver (LBFS) [25] was formulated, accompanied by subsequent refinements [26,27,28,29,30,31,32]. This novel approach effectively establishes a local mesoscopic kinetic flux solver when addressing both the Euler equation and Navier–Stokes equations through the FVM. For the simulation of incompressible flows, the LBFS is known to successfully circumvent the limitations of the traditional lattice Boltzmann method (LBM), such as lattice uniformity, coupling of time intervals with grid spacing, and the difficulty of implementing boundary conditions. For the simulations of compressible flows, the inviscid fluxes in the LBFS are reconstructed from the local solution of the one-dimensional compressible lattice Boltzmann model [30]. The viscous fluxes in the LBFS are approximated by a smoothing function, and the numerical dissipation is controlled by a selection function.

Recently, a rotated lattice Boltzmann flux solver (RLBFS) with enhanced stability was introduced in [31,32] for scenarios involving high Mach numbers and strong shock waves in compressible flows. This innovative technique decomposes the outer normal direction of each cell interface into two components, determined by the velocity difference vector and the direction perpendicular to it, respectively. Inviscid fluxes in both of these directions are assessed using the LBFS, with the final fluxes at each interface derived from a weighted combination of these two fluxes.

For the simulation of compressible flows featuring intricate shock structures, the applications of fixed or uniform grids across the entire flow domain can often entail substantial computational resources, although they are advantageous in terms of simplifying implementation and reducing modeling complexity. This is especially true when investigating unsteady flows with small-scale vortex phenomena. In contrast, addressing the challenge of complex shock structures can be accomplished with greater efficiency by employing adaptive mesh refinement (AMR) methods. Adaptive mesh refinement enables the enhancement of or reduction in mesh resolution where it is needed, offering a dynamic and localized approach for reducing overall computational resource demands. Various AMR approaches [33,34,35,36,37,38,39,40] have been developed and utilized, each with their own strengths and suitability for particular applications. Among them, the block-based approach [36,37,38,39,40] has demonstrated strong performance in diverse applications. This approach involves dividing the computational domain into blocks or patches of cells, and then selectively refining or coarsening entire blocks based on specified criteria. Block-based AMR is often favored for its simplicity and efficient parallelism since each block can be processed independently. It is particularly effective in scenarios where localized refinement is sufficient to capture key flow features, such as intricate shock structures and vortices.

In sight of the advantages of the RLBFS and the AMR technology, the combination of these two methods may present a powerful synergy that significantly reduces the overall computational cost and maintains high resolution in critical areas simultaneously, which motivates the present work. Particularly, a block-structured AMR-RLBFS is presented and validated by simulating both two and three-dimensional compressible flows with intricate shock structures and vortices. The method is developed in the framework of AMROC [37,38,39,40], which dynamically refines the computational mesh in regions with complex flow features, allowing for the allocation of computational resources precisely where they are needed the most. When coupled with the recently proposed RLBFS [31,32], which accurately captures the complex wave interactions and shock structures in compressible flows, the simulation becomes more efficient and robust. This combination enables one to achieve a balance between computational resources and simulation accuracy, making it a valuable tool for studying a wide range of compressible flow phenomena, from aerodynamics to astrophysics, with improved efficiency. This rest of this paper is organized in the following way. Section 2 presents the details of the AMR-RLBFS, including the governing equations, LBFS and RLBFS principles, and the block-structured AMR technology. Section 3 validates the AMR-RLBFS and compares it with the original LBFS in the AMR framework. Applications of the AMR-RLBFS to simulate two and three-dimensional problems are also presented to further investigate its performance. Section 4 concludes and summarizes this paper.

2. Methodology

In this section, the development of the AMR-RLBFS is presented, covering the governing equations for compressible flows, the utilization of LBFS and RLBFS to compute fluxes at cell interfaces, and the application of block-structured AMR techniques to update computational grids.

2.1. Governing Equations

The compressible, inviscid flows are considered in the present study, whose governing equations of mass, momentum and energy can be written in the following conserved form:

$$\frac{\partial Q}{\partial t}+\frac{\partial F}{\partial x}+\frac{\partial G}{\partial y}+\frac{\partial H}{\partial z}=0,$$

(1)

$$Q=\left[\begin{array}{c}\rho \\ \rho u\\ \rho v\\ \rho w\\ \rho E\end{array}\right],F=\left[\begin{array}{c}\rho u\\ \rho u^2+p\\ \rho uv\\ \rho uw\\ \rho uH\end{array}\right],G=\left[\begin{array}{c}\rho v\\ \rho vu\\ \rho v^2+p\\ \rho vw\\ \rho vH\end{array}\right],H=\left[\begin{array}{c}\rho w\\ \rho wu\\ \rho wv\\ \rho w^2+p\\ \rho wH\end{array}\right].$$

(2)

where ρ is density; p is pressure; and u, v, and w represent the velocity components in the Cartesian coordinate system. Considering the ideal gas model, the equation of state is given as follows:

$$p=(\gamma -1)\left[\rho E-\frac{\rho }{2}\left(u^2+v^2+w^2\right)\right],$$

(3)

In Equation (3), E and H represent the total specific internal energy and total specific enthalpy, respectively, which are obtained by the following:

$$\begin{array}{l}E=\frac{1}{\gamma -1}\frac{p}{\rho }+\frac{1}{2}\left(u^2+v^2+w^2\right),\\ H=\frac{p}{\rho }+E,\end{array}$$

(4)

Here, γ is the specific heat ratio, taken as γ = 1.4 in air. By applying the cell-centered finite volume discretization of Equation (1) in a control volume Ω, the integral of the fluxes in Equation (1) is approximated by a sum of the fluxes that cross the cell interface:

$$\frac{d{W}_I}{dt}=-\frac{1}{{\mathsf{\Omega }}_I}{\displaystyle \sum _{i=1}^{N_f}\left(F_{ci}{S}_i\right)}.$$

(5)

where I is the index of the control volume. Ω_I and N_f represent the volume and the number of faces of the control volume I, respectively. S_i denotes the area of the i-th face of the control volume. In the present work, the fluxes in Equation (5) can be obtained using LBFS or RLBFS.

2.2. LBFS with D1Q4 Model

It has been shown that the non-free parameter lattice Boltzmann model [27,28] can effectively model inviscid compressible flows. In this method, the FVM is used to discretely solve the governing equations for updating the flow variables at the cell centers, and the D1Q4 model is used to evaluate the inviscid fluxes at each cell interface. A typical non-free parameter D1Q4 model is shown in Figure 1, which contains six unknowns (four equilibrium distribution functions, g₁, g₂, g₃, g₄, and two lattice velocities, d₁, d₂), and all are determined by Equation (6).

$$\begin{array}{l}{g}_1=\frac{\rho \left(-d_1{d}_2^2-d_2^{2}u+d_1{u}^2+d_1{c}^2+u^3+3u{c}^2\right)}{2d_1\left(d_1^2-d_2^2\right)},\\ g_2=\frac{\rho \left(-d_1{d}_2^2+d_2^{2}u+d_1{u}^2+d_1{c}^2-u^3-3u{c}^2\right)}{2d_1\left(d_1^2-d_2^2\right)},\\ g_3=\frac{\rho \left(d_1^2{d}_2+d_1^{2}u-d_2{u}^2-d_2{c}^2-u^3-3u{c}^2\right)}{2d_2\left(d_1^2-d_2^2\right)},\\ g_4=\frac{\rho \left(d_1^2{d}_2-d_1^{2}u-d_2{u}^2-d_2{c}^2+u^3+3u{c}^2\right)}{2d_2\left(d_1^2-d_2^2\right)},\\ d_1=\sqrt{u^2+3c^2-\sqrt{4u^2{c}^2+6c^4}},\\ d_2=\sqrt{u^2+3c^2+\sqrt{4u^2{c}^2+6c^4}}.\end{array}$$

(6)

where c stands for the specific velocity of particles defined as $c=\sqrt{Dp/\rho }$ and D denotes the space dimension for the D1Q4 model, D = 1. Once the equilibrium distribution functions and lattice velocities are in hand, the flow variables can be calculated from the following relationships [30]:

$$\begin{array}{l}\rho ={\displaystyle \sum _{i=1}^4{g}_i},\\ \rho u={\displaystyle \sum _{i=1}^4{g}_i{\xi }_i},\\ \rho uu+p={\displaystyle \sum _{i=1}^4{g}_i{\xi }_i{\xi }_i},\\ \rho E={\displaystyle \sum _{i=1}^4{g}_i\left(\frac{1}{2}{\xi }_i{\xi }_i+e_p\right)},\\ \left(\rho E+p\right)u={\displaystyle \sum _{i=1}^4{g}_i\left(\frac{1}{2}{\xi }_i{\xi }_i+e_p\right)}{\xi }_i.\end{array}$$

(7)

where ${\xi }_i$ is the particle velocity in the i-direction, ${\xi }_1=d_1,{\xi }_2=-d_1,{\xi }_3=d_2,{\xi }_4=-d_2$. $e_p$ is the potential energy of particles, $e_p=\left[1-\left(\gamma -1\right)D/2\right]e$. The above 1D model must be applied along the normal direction of the interface when considering multi-dimensional problems. For example, for the 2D case, as shown in Figure 2, we can use the normal velocity U_n to replace the velocity u in the D1Q4 model given in Equation (7) and take the tangential velocity U_τ into account.

The flux is reconstructed by the LBFS in the direction perpendicular to the cell interface after the introduction of the non-free parameter D1Q4 model [27,28]. Under the assumption that this cell interface is located at x = 0, the inviscid flux of the interface contributed by the normal velocity is calculated according to the following equation [30]:

$$F_{c,i+1/2}^{*}={\left[\begin{array}{ccc}\rho U_n& \rho U_n{U}_n+p& \left(\rho \left(\frac{1}{2}{U}_n{U}_n+e\right)+p\right)U_n\end{array}\right]}^T={\displaystyle \sum _{i=1}^4{\xi }_i}{\phi }_{\alpha }{f}_i\left(0,t\right),$$

(8)

$${\phi }_{\alpha }={\left(1,{\xi }_i,\frac{1}{2}{\xi }_i^2+e_p\right)}^T.$$

(9)

where φ_α represents the momentum, the notation * means numerical value, and $f_i\left(0,t\right)$ is the distribution function at the cell interface. Normally, $f_i$ is composed of an equilibrium part $f_{{}_i}^{eq}$ and a non-equilibrium part $f_{{}_i}^{neq}$.

$$f_i\left(0,t\right)=f_i^{eq}\left(0,t\right)+f_i^{neq}\left(0,t\right),$$

(10)

The non-equilibrium part $f_{{}_i}^{neq}$ can be approximated by the equilibrium distribution function at the cell interface and its surrounding point [30]. According to Chapman–Enskog analysis and the simplification using Taylor expansion, the distribution function of the cell interface is obtained:

$$f_i\left(0,t\right)=g_i\left(0,t\right)-{\tau }_0\left[g_i\left(0,t\right)-g_i\left(-{\xi }_i\delta t,t-\delta t\right)\right]+\mathrm{O}\left(\delta t\right).$$

(11)

where $g_i\left(0,t\right)$ is the equilibrium distribution function at the cell interface, and $g_i\left(-{\xi }_i\delta t,t-\delta t\right)$ is the equilibrium distribution function at the surrounding point of the cell interface. ${\tau }_0=\tau /\delta t$ is the dimensionless collision time and $\delta t$ is the streaming time step. The total inviscid fluxes at the cell interface can be obtained by substituting Equation (11) into Equation (8) and by considering the tangential inviscid flux.

$$\begin{array}{c}{F}_{c,i+1/2}=\left(1-{\tau }_0\right){\mathrm{F}}_{c,i+1/2}^{\left(I\right)}+{\tau }_0{\mathrm{F}}_{c,i+1/2}^{\left(II\right)},\\ F_{c,i+1/2}^{\left(I\right)}={\left[\begin{array}{c}{F}_{c,i+1/2}^{\left(I*\right)}\left(1\right)\\ F_{c,i+1/2}^{\left(I*\right)}\left(2\right)n_x-\rho U_n{U}_{\tau }{n}_y\\ F_{c,i+1/2}^{\left(I*\right)}\left(2\right)n_y+\rho U_n{U}_{\tau }{n}_x\\ F_{c,i+1/2}^{\left(I*\right)}\left(3\right)+\rho U_n{U}_{\tau }^2/2\end{array}\right]}_{i+1/2}^{*},F_{c,i+1/2}^{\left(II\right)}={\left[\begin{array}{c}{F}_{c,i+1/2}^{\left(II*\right)}\left(1\right)\\ F_{c,i+1/2}^{\left(II*\right)}\left(2\right)n_x-\rho U_n{U}_{\tau }{n}_y\\ F_{c,i+1/2}^{\left(II*\right)}\left(2\right)n_y+\rho U_n{U}_{\tau }{n}_x\\ F_{c,i+1/2}^{\left(II*\right)}\left(3\right)+\rho U_n{U}_{\tau }^2/2\end{array}\right]}_{i+1/2}^{*}.\end{array}$$

(12)

According to Equation (12), there are two parts $F_{c,i+1/2}^{\left(I\right)}$ and $F_{c,i+1/2}^{\left(II\right)}$ of the inviscid fluxes $F_{c,i+1/2}$: the first part $F_{c,i+1/2}^{\left(I\right)}$ is the flux at the cell interface while the second part $F_{c,i+1/2}^{\left(II\right)}$ is the flux on the surrounding points of the interface. Therefore, an accurate evaluation of $g_i\left(0,t\right)$ and $g_i\left(-{\xi }_i\delta t,t-\delta t\right)$ is the key to calculating total inviscid fluxes $F_{c,i+1/2}$. By taking ${\tau }_0=0$, Equation (11) can be reduced to the following:

$$f_i\left(0,t\right)=g_i\left(0,t\right),$$

(13)

In order to obtain the equilibrium distribution function $g_i\left(0,t\right)$, the conserved variables at the cell interface should be computed in advance. According to Equation (7), at the cell interface, the density, momentum in the normal direction, and energy attributed to normal velocity can be computed by the following equation:

$$W_{c,i+1/2}^{*}={\left[\begin{array}{ccc}\rho & \rho U_n& \rho \left(\frac{1}{2}{U}_n{U}_n+e\right)\end{array}\right]}^T={\displaystyle \sum _{i=1}^4{\phi }_{\alpha }{f}_i\left(0,t\right)}={\displaystyle \sum _{i=1}^4{\phi }_{\alpha }{g}_i\left(0,t\right)},$$

(14)

According to the compatibility condition, the non-equilibrium part of distribution function has no contribution in calculation of conserved variables. Thus, the conserved variables of the cell interface change to the following:

$$W_{c,i+1/2}^{*}={\displaystyle \sum _{i=1}^4{\phi }_{\alpha }{g}_i\left(0,t\right)}={\displaystyle \sum _{i=1}^4{\phi }_{\alpha }{g}_i\left(-{\xi }_i\delta t,t-\delta t\right)}.$$

(15)

From Equation (15), it can be seen that the conserved variables at the cell interface can be computed from $g_i\left(-{\xi }_i\delta t,t-\delta t\right)$, which is the function of the conserved variables at the surrounding point of the cell interface. Specifically, the equilibrium distribution function $g_i\left(-{\xi }_i\delta t,t-\delta t\right)$ can be written as follows:

$$g_i\left(-{\xi }_i\delta t,t-\delta t\right)=\{\begin{array}{l}{g}_i^L,\mathrm{if} {\xi }_i\ge 0,\\ g_i^R,\mathrm{if} {\xi }_i\le 0.\end{array}$$

(16)

where $g_i^L$ and $g_i^R$ are the equilibrium distribution functions at the left and right side of the cell interface, respectively. For the non-free parameter D1Q4 model [27,28], Equation (16) can be further reduced:

$$g_i\left(-{\xi }_i\delta t,t-\delta t\right)=\{\begin{array}{l}{g}_i^L,\mathrm{if} i=1,3,\\ g_i^R,\mathrm{if} i=2,4.\end{array}$$

(17)

The streaming and collision process of the D1Q4 model at the cell interface can be found in Figure 3. Substituting Equation (17) into Equation (15), we can obtain the following:

$$W_{c,i+1/2}^{*}={\displaystyle \sum _{i=1,3}^{}{\phi }_{\alpha }{g}_i^L}+{\displaystyle \sum _{i=2,4}^{}{\phi }_{\alpha }{g}_i^R},$$

(18)

To evaluate the tangential velocity $U_{\tau }^{\ast }$ at the cell interface, one of the feasible ways can be expressed as follows:

$$\begin{array}{l}{\rho }^{\ast }{U}_{\tau }^{\ast }={\displaystyle \sum _{i=1}^4{g}_i}\cdot U_{\tau }^{\ast }={\displaystyle \sum _{i=1,3}^{}{g}_i^L}\cdot U_{\tau }^L+{\displaystyle \sum _{i=2,4}^{}{g}_i^R}\cdot U_{\tau }^R,\\ {\rho }^{\ast }{\left(U_{\tau }^{\ast }\right)}^2={\displaystyle \sum _{i=1}^4{g}_i}\cdot {\left(U_{\tau }^{\ast }\right)}^2={\displaystyle \sum _{i=1,3}^{}{g}_i^L}\cdot {\left(U_{\tau }^L\right)}^2+{\displaystyle \sum _{i=2,4}^{}{g}_i^R}\cdot {\left(U_{\tau }^R\right)}^2,\end{array}$$

(19)

where $U_{\tau }^{\ast }$, $U_{\tau }^L$, and $U_{\tau }^R$ are the tangential velocity at the cell interface and at the left and right side of the cell interface, respectively. Using Equations (18) and (19), we plug all of the primitive variables into Equation (6) to obtain the equilibrium distribution function $g_i\left(0,t\right)$ for the cell interface. As such, the inviscid flux $F_{c,i+1/2}^{\left(I\right)}$ at the cell interface can be expressed as follows:

$$F_{c,i+1/2}^{\left(I\right)}={\left[\begin{array}{c}\rho U_n\\ \left(\rho U_n{U}_n+p\right)n_x-\rho U_n{U}_{\tau }{n}_y\\ \left(\rho U_n{U}_n+p\right)n_y+\rho U_n{U}_{\tau }{n}_x\\ \left(\rho E+p\right)U_n+\rho U_n{U}_{\tau }^2/2\end{array}\right]}_{i+1/2}^{*}.$$

(20)

In this case, the numerical dissipation of the inviscid fluxes calculated with LBFS is very small, and thus the results are very accurate for boundary layer flows. However, when simulating hypersonic flows with strong excitation waves, oscillations and even dispersions tend to occur. By taking ${\tau }_0=1$, Equation (11) can be reduced to the following:

$$f_i\left(0,t\right)=g_i\left(-{\xi }_i\delta t,t-\delta t\right),$$

(21)

In Equation (21), the distribution function $g_i\left(-{\xi }_i\delta t,t-\delta t\right)$ at the cell interface is determined by the equilibrium distribution function at the surrounding point of the cell interface. Since $g_i\left(-{\xi }_i\delta t,t-\delta t\right)$ has been determined by Equation (17), by substituting Equation (21) into Equation (8), we can obtain the mass flux, momentum flux in the normal direction and energy flux attributed to the normal velocity across the interface:

$$F_{c,i+1/2}^{*}={\left[\begin{array}{ccc}\rho U_n& \rho U_n{U}_n+p& \left(\rho \left(\frac{1}{2}{U}_n{U}_n+e\right)+p\right)U_n\end{array}\right]}^T={\displaystyle \sum _{i=1}^4{\xi }_i}{\phi }_{\alpha }{g}_i\left(-{\xi }_i\delta t,t-\delta t\right),$$

(22)

In addition, the contribution of the tangential velocity to the momentum flux in the tangential direction and energy flux can be approximated by the following:

$$\begin{array}{l}{\rho }^{\ast }{U}_n^{\ast }{U}_{\tau }^{\ast }={\displaystyle \sum _{i=1}^4{\xi }_i{g}_i}\cdot U_{\tau }^{\ast }={\displaystyle \sum _{i=1,3}^{}{\xi }_i{g}_i^L}\cdot U_{\tau }^L+{\displaystyle \sum _{i=2,4}^{}{\xi }_i{g}_i^R}\cdot U_{\tau }^R,\\ {\rho }^{\ast }{U}_n^{\ast }{\left(U_{\tau }^{\ast }\right)}^2={\displaystyle \sum _{i=1}^4{\xi }_i{g}_i}\cdot {\left(U_{\tau }^{\ast }\right)}^2={\displaystyle \sum _{i=1,3}^{}{\xi }_i{g}_i^L}\cdot {\left(U_{\tau }^L\right)}^2+{\displaystyle \sum _{i=2,4}^{}{\xi }_i{g}_i^R}\cdot {\left(U_{\tau }^R\right)}^2,\end{array}$$

(23)

With Equations (22) and (23), the inviscid flux $F_{c,i+1/2}^{\left(II\right)}$ at the cell interface can be expressed as follows:

$$F_{c,i+1/2}^{\left(II\right)}={\left[\begin{array}{c}{\displaystyle \sum _{i=1}^4{\xi }_i{g}_i}\\ {\displaystyle \sum _{i=1}^4{\xi }_i{\xi }_i{g}_i{n}_x-{\displaystyle \sum _{i=1}^4{\xi }_i{g}_i\cdot U_{\tau }{n}_y}}\\ {\displaystyle \sum _{i=1}^4{\xi }_i{\xi }_i{g}_i{n}_y+{\displaystyle \sum _{i=1}^4{\xi }_i{g}_i\cdot U_{\tau }{n}_x}}\\ {\displaystyle \sum _{i=1}^4{\xi }_i\left(\frac{1}{2}{\displaystyle \sum _{i=1}^4{\xi }_i{\xi }_i+e_p}\right)g_i+\frac{1}{2}{\displaystyle \sum _{i=1}^4{\xi }_i{g}_i}\cdot {\left(U_{\tau }\right)}^2}\end{array}\right]}_{i+1/2}^{*}.$$

(24)

Obviously, a large numerical dissipation is introduced in this case, and the LBFS can simulate the hypersonic inviscid flow with a very high Mach number very well, but the large numerical dissipation will not provide an accurate solution in the boundary layer.

In order to accurately capture strong shock waves and thin boundary layers, ${\tau }_0^{\ast }$ can be regarded as the weight of the numerical dissipation, and the value of ${\tau }_0^{\ast }$ should be a variable rather than a constant, so we use a switch function developed by [30] that can control ${\tau }_0^{\ast }$. This switch function can make the value of ${\tau }_0^{\ast }$ close to zero in the boundary layer and close to one around the strong shock wave. ${\tau }_0^{\ast }$ is shown in the following equations.

$$\begin{array}{l}{\tau }_0=\tanh \left(C\frac{\left|p_L-p_R\right|}{p_L+p_R}\right),\\ {\tau }_0^{\ast }=\max \left\{{\tau }_L,{\tau }_R\right\},\\ {\tau }_L={\max }_{i=1,N_{fL}}\left\{{\tau }_i\right\},\\ {\tau }_R={\max }_{i=1,N_{fR}}\left\{{\tau }_i\right\}.\end{array}$$

(25)

where $\tanh \left(x\right)$ is the hyperbolic tangent function and P_L and P_R are the pressure on the left side and the right side of the cell interface, respectively. C is an amplifying factor that usually ranges from 10 to 100 when simulated. In this paper, C = 100 is used. ${\tau }_L$ and ${\tau }_R$ are the maximum values of the switch function at the left and right control volumes. $N_{fL}$ and $N_{fR}$ are the numbers of the faces of the left and right control volumes, respectively. Then, Equation (12) is changed to the following:

$$\begin{array}{c}{F}_{c,i+1/2}=\left(1-{\tau }_0^{\ast }\right)F_{c,i+1/2}^{\left(I\right)}+{\tau }_0^{\ast }{F}_{c,i+1/2}^{\left(II\right)},\\ F_{c,i+1/2}^{\left(I\right)}={\left[\begin{array}{c}\rho U_n\\ \left(\rho U_n{U}_n+p\right)n_x-\rho U_n{U}_{\tau }{n}_y\\ \left(\rho U_n{U}_n+p\right)n_y+\rho U_n{U}_{\tau }{n}_x\\ \left(\rho E+p\right)U_n+\rho U_n{U}_{\tau }^2/2\end{array}\right]}_{i+1/2}^{*},\\ F_{c,i+1/2}^{\left(II\right)}={\left[\begin{array}{c}{\displaystyle \sum _{i=1}^4{\xi }_i{g}_i}\\ {\displaystyle \sum _{i=1}^4{\xi }_i{\xi }_i{g}_i{n}_x-{\displaystyle \sum _{i=1}^4{\xi }_i{g}_i\cdot U_{\tau }{n}_y}}\\ {\displaystyle \sum _{i=1}^4{\xi }_i{\xi }_i{g}_i{n}_y+{\displaystyle \sum _{i=1}^4{\xi }_i{g}_i\cdot U_{\tau }{n}_x}}\\ {\displaystyle \sum _{i=1}^4{\xi }_i\left(\frac{1}{2}{\displaystyle \sum _{i=1}^4{\xi }_i{\xi }_i+e_p}\right)g_i+\frac{1}{2}{\displaystyle \sum _{i=1}^4{\xi }_i{g}_i}\cdot {\left(U_{\tau }\right)}^2}\end{array}\right]}_{i+1/2}^{*}.\end{array}$$

(26)

Meanwhile, the two sides of the interface conserved variables are processed using a three-point MUSCL interpolation with a slope limiter before the flux calculation to improve the accuracy of the calculations.

$$\begin{array}{l}{W}_{i+1/2}^L=W_i+\frac{1}{2}\mathsf{\Psi }\left(r_{i-1/2}^{+}\right){\mathsf{\Delta }}_{i-1/2},\\ W_{i-1/2}^R=W_i-\frac{1}{2}\mathsf{\Psi }\left(r_{i+1/2}^{+}\right){\mathsf{\Delta }}_{i+1/2},\\ {\mathsf{\Delta }}_{i-1/2}=W_i-W_{i-1},{\mathsf{\Delta }}_{i+1/2}=W_{i+1}-W_i,\\ r_{i-1/2}^{+}=\frac{{\mathsf{\Delta }}_{i+1/2}}{{\mathsf{\Delta }}_{i-1/2}},r_{i+1/2}^{-}=\frac{{\mathsf{\Delta }}_{i-1/2}}{{\mathsf{\Delta }}_{i+1/2}},\\ \mathsf{\Psi }\left(r\right)=\frac{1}{2}\left[\left(1-\omega \right)\mathsf{\Phi }\left(r\right)+\left(1+\omega \right)r\mathsf{\Phi }\left(\frac{1}{r}\right)\right].\end{array}$$

(27)

where $W_{i-1/2}^R$ and $W_{i+1/2}^L$ represent the conserved variables on the left and right sides of cell i, respectively. $\omega$ is the interpolation parameter, and in this paper, $\omega =0$ represents linear interpolation. $\mathsf{\Phi }\left(r\right)$ is the slope limiter function, which satisfies $\mathsf{\Phi }\left(r\right)=r\mathsf{\Phi }\left(\frac{1}{r}\right)$. For the one-dimensional Riemann problem, we use the van Leer slope limiter function [41]:

$$\mathsf{\Phi }\left(r\right)=\frac{r+\left|r\right|}{1+\left|r\right|}.$$

(28)

For the two-dimensional and three-dimensional cases, we use the minmod slope limiter function:

$$\mathsf{\Phi }\left(r\right)=\max (0,\min (r,1)).$$

(29)

2.3. The Rotated LBFS

The LBFS is an effective numerical method for the simulation of compressible flows, as described in the previous section. In order to further improve the performance of the LBFS for solving the carbuncle, kinked Mach stem, and odd–even decoupling problems encountered by conventional Riemann solvers when simulating compressible flows with complex shock structures, the RLBFS proposed in [31,32] has been used in the present study. Instead of applying the LBFS directly in the direction $n$ perpendicular to the cell interface as shown in Figure 4, the RLBFS first decomposes the normal vector of the cell into two orthogonal vectors, $n_1$ and $n_2$, and then applies the LBFS in these two directions separately to calculate the fluxes. The relationship between these vectors at each cell boundary can be given as follows:

$$\begin{array}{l}n={\alpha }_1{n}_1+{\alpha }_2{n}_2,\\ n_1\cdot n_2=0.\end{array}$$

(30)

where $\left|n_1\right|=\left|n_2\right|=1$, ${\alpha }_1=n\cdot n_1$ and ${\alpha }_2=n\cdot n_2$. In order to make $n_1$ and $n_2$ have the same left and right states at the cell interface, ${\alpha }_1\ge 0$ and ${\alpha }_2\ge 0$ need to be guaranteed. In Figure 4, if the separation vector is in the green line state, it can be identified as an optional separation scheme, and if any of the components are in the red dashed line state, they must all be reversed to the blue line state before it can be identified as an optional separation scheme.

Therefore, the flux of the rotated LBFS can be calculated by the following equation:

$$F_{RLBFS}=F_{RLBFS}\left(n\right)={\alpha }_1{F}_{LBFS}\left(n_1\right)+{\alpha }_2{F}_{LBFS}\left(n_2\right).$$

(31)

Both $F_{LBFS}\left(n_1\right)$ and $F_{LBFS}\left(n_2\right)$ can be obtained using Equation (26). According to [11,12] and [31,32], the choice of $n_1$ has a great influence on numerical stability. The pressure–gradient direction, velocity–magnitude–gradient direction, and velocity–difference vector are some common choices, while the velocity–difference vector can effectively eliminate the shock instabilities. Therefore, in this paper, $n_1$ is also determined by the velocity–difference vector, which is obtained by the following:

$$n_1=\{\begin{array}{cc}n,& \mathrm{if}\sqrt{{\left(\mathsf{\Delta }u\right)}^2+{\left(\mathsf{\Delta }v\right)}^2}\le \epsilon ,\\ \frac{\mathsf{\Delta }u\mathrm{i}+\mathsf{\Delta }v\mathrm{j}}{\sqrt{{\left(\mathsf{\Delta }u\right)}^2+{\left(\mathsf{\Delta }v\right)}^2}},& \mathrm{otherwise}.\end{array}$$

(32)

where $\mathsf{\Delta }u=u_R-u_L$, $\mathsf{\Delta }v=v_R-v_L$, and $\epsilon$ is a small positive number. According to Equation (32), $n_1$ can be obtained. Moreover, since $n_1\cdot n_2=0,$ $n_2$ can be calculated in the following form:

$$n_2=k\times n_1,$$

(33)

where $k=i\times j$. The rotated flux function can be easily extended to three-dimensional cases. $n_1$ can also be obtained by Equation (32), then $n_2$ can be calculated in the following form:

$$n_2=\left(n_1\times n\right)\times n_1.$$

(34)

Other parameters are calculated in the same way as in two-dimensional cases. Regarding the values of $\epsilon$, in this paper, $\epsilon ={10}^{-12}{U}^{*}$ is used for the Sod shock tube problem and the four-wave two-dimensional Riemann problem to make a large enough difference between the RLBFS and the LBFS, and $\epsilon ={10}^{-3}{U}^{*}$ is used for the three-dimensional problem, where $U^{*}$ is the free stream velocity. By performing this, we obtain the rotated LBFS scheme, which is proposed in [31,32] to obtain a strong shock-capture capability, to eliminate the shock instability that occurs in the LBFS, and to obtain better stability than the LBFS scheme.

2.4. Block-Structured Adaptive Mesh Refinement

The LBFS and RLBFS presented in the previous two sub-sections are implemented on the block-structured adaptive mesh so that the AMR-RLBFS is developed, which is able to accurately capture complex flow phenomena, such as shock waves, contact discontinuities, shear layers, and vortices in compressible flows. A simplified version [42] of the block-structured adaptive mesh refinement technique, i.e., AMROC [37,38,39,40], is introduced to improve the resolution of complex shock structures and reduce the required computational resources. AMROC follows a block refinement strategy by clustering the flagged grids into rectangular boxes of an appropriate size and then generating sub-grids with the same refinement coefficients in the spatial and temporal directions, such that the refinement is recursive from the coarser grids, thus constructing the entire hierarchy of continuous embedded grid patches, see Figure 5.

Unlike other refinement techniques that allow only one parent grid, AMROC considers arbitrary parent/child relationships, and in Figure 5, this generality is represented by the grid G_2,2 covering both parents. At each hierarchy level, a separate refinement factor r_i can be used, and in this paper, we unify all refinement factors with r_i = 2. In AMROC, the grids are stored independently at each hierarchy level, avoiding additional communication overhead and improving parallel performance, and each hierarchy level evolves independently. In this case, the fine grids are free from the excessive constraints of the CFL condition, and the fine grids can evolve in smaller time steps than the coarse grids. Then, the fine grids’ information overwrites the coarse grids that generated it, and the average values of the fine grids replace the values of the coarse grids, which generally results in the loss of important conserved properties. To ensure conservation, a flux correction is required to replace the coarse grid fluxes on the affected side of neighboring grids with the accumulated fine grid fluxes.

Depending on the flow problem to be solved, the computational domain is initially covered with one or more root blocks of the same grid size and resolution. Here, we denote the grid refinement level by l, the root blocks have l = 0, and the maximum refinement level is denoted by l_max. For AMR methods, computational efficiency and accuracy are issues to focus on. The parallel AMR algorithm can improve the computational efficiency, and detailed information about the parallel algorithm of AMROC can be found in [39,40]. The computational efficiency can be measured by the speed-up ratio, which is defined as the ratio between the computation time on a uniform grid and on an AMR mesh of the same resolution at the highest refinement level. Obviously, the speed-up ratio is mainly determined by the size of the AMR mesh used for computation, which is closely related to the refinement strategy. The rigorously derived error estimation is only applicable to scalar equations for the conservation law of the FVM [43]. This paper adopts two common refinement indices, using only density and pressure. The scaled gradients refer to an adaptive refinement criterion along the discontinuity. Taking the two-dimensional case as an example, grid$(i,j)$ is refined if one of the following conditions is satisfied:

$$\left|\omega (W_{i+1,j})-\omega (W_{i,j})\right|>{\epsilon }_{\omega },\left|\omega (W_{i,j+1})-\omega (W_{i,j})\right|>{\epsilon }_{\omega },\left|\omega (W_{i+1,j+1})-\omega (W_{i,j})\right|>{\epsilon }_{\omega }.$$

(35)

where, in this paper, $\omega (W_{i,j})$ represents ρ or p derived from conserved variables $W_{i,j}(t)$ at any time. ${\epsilon }_{\omega }$ is the limiting index for scaling gradient for any moment in time, and for different problems, appropriate scaled gradients should be selected. Another adaptive refinement criterion is the heuristic error estimation of local truncation errors in the smooth solution domain by Richardson extrapolation [42,44,45]. The local truncation error of the o-order difference scheme satisfies the following condition:

$$q\left(x,t+\mathsf{\Delta }t\right)-H^{(\mathsf{\Delta }t)}\left(q\left(\cdot ,t\right)\right)=C\mathsf{\Delta }{t}^{o+1}+O\left(\mathsf{\Delta }{t}^{o+2}\right),$$

(36)

If q is sufficiently smooth, we have for the local error at $t+\mathsf{\Delta }t$ after two steps with $\mathsf{\Delta }t$:

$$q\left(x,t+\mathsf{\Delta }t\right)-H_2^{(\mathsf{\Delta }t)}\left(q\left(\cdot ,t-\mathsf{\Delta }t\right)\right)=2C\mathsf{\Delta }{t}^{o+1}+O\left(\mathsf{\Delta }{t}^{o+2}\right),$$

(37)

and for the local error at $t+\mathsf{\Delta }t$, after one step with $2\mathsf{\Delta }t$:

$$q\left(x,t+\mathsf{\Delta }t\right)-H^{(2\mathsf{\Delta }t)}\left(q\left(\cdot ,t-\mathsf{\Delta }t\right)\right)=2^{o+1}C\mathsf{\Delta }{t}^{o+1}+O\left(\mathsf{\Delta }{t}^{o+2}\right),$$

(38)

Subtracting Equation (37) from Equation (38), one obtains the following relation:

$$H_2^{(\mathsf{\Delta }t)}\left(q\left(\cdot ,t-△t\right)\right)-H^{(2\mathsf{\Delta }t)}\left(q\left(\cdot ,t-\mathsf{\Delta }t\right)\right)=(2^{o+1}-2)C\mathsf{\Delta }{t}^{o+1}+O\left(\mathsf{\Delta }{t}^{o+2}\right).$$

(39)

which can be employed to approximate the leading-order term $C\mathsf{\Delta }{t}^{o+1}$ of the local error at $t+\mathsf{\Delta }t$. The implementation of a criterion based on Equation (39) requires a discrete solution $Q^l$ defined on a grid two times coarser than the grid of level l, where level l represents the finer grid, see Figure 6. A coarse approximation ${\overline{Q}}^l\left(t_l+\mathsf{\Delta }{t}_l\right)=H_2^{\mathsf{\Delta }{t}_l}{Q}^l\left(t_l-\mathsf{\Delta }{t}_l\right)$ is obtained by taking one step of $2\mathsf{\Delta }{t}_l$ (green arrow in Figure 6). Then, a second coarse solution $Q^l\left(t_l+\mathsf{\Delta }{t}_l\right)=H^{2\mathsf{\Delta }{t}_l}{Q}^l\left(t_l-\mathsf{\Delta }{t}_l\right)$ is derived by advancing $Q^l\left(t\right)$ by $\mathsf{\Delta }{t}_l$ (blue arrows in Figure 6). The following is an approximation to the leading-order term of the local error of quantity $\omega$ of the grid$(i,j)$.

$${\tau }_{i,j}^{\omega }:=\frac{\left|\omega \left({\overline{Q}}_{i,j}^l\left(t+\mathsf{\Delta }{t}_l\right)\right)-\omega \left(Q_{i,j}^l\left(t+\mathsf{\Delta }{t}_l\right)\right)\right|}{2^{o+1}-2}>{\eta }_{\omega }.$$

(40)

where, in this paper, $\omega$ represents ρ or p derived from the coarse approximate solution $Q_{i,j}^l\left(t+\mathsf{\Delta }{t}_l\right)$ at any time, and ${\eta }_{\omega }$ is a limiting index of error. If this inequality is satisfied, all four grids below the coarse grid $(i,j)$ will be refined.

3. Numerical Validations and Applications

In this section, the proposed AMR-RLBFS is validated and applied to study the classical shock tube problem, the four-wave Riemann problem, and two unsteady three-dimensional compressible flows with complex shock structures. The performances of the LBFS and RLBFS on the adaptive mesh are also investigated.

3.1. Shock Tube Problem

The shock tube problem is a fundamental physical problem to examine the reliability and capability of different numerical schemes, whose exact solutions are available for comparison. The Sod shock tube [46], which consists of a left rarefaction wave, a contact discontinuity, and a right shock wave, is considered in the study. To visualize the variation of the adaptive mesh, we simulated the Sod shock tube problem with a computational domain of [0,1] × [0,0.2]. The outflow boundary condition is applied on the left and right sides, while the slip wall condition is used on top and bottom. A basic mesh resolution of size 200 × 40 (l = 0) with a maximum adaptive level of l_max = 2 is used. The initial conditions are given below.

$$\begin{array}{c}{\rho }_L=1.0,u_L=0.0,p_L=1.0, 0.0<x<0.5\hfill \\ {\rho }_R=0.125,u_R=0.0,p_R=0.1,0.5<x<1.0\hfill \end{array}$$

(41)

Figure 7 shows the numerical results of density, pressure, and velocity at t = 0.25, obtained by the AMR-RLBFS and the AMR-LBFS. Also included in the figure is the theoretical solution. It can be seen that good agreements among all mentioned solutions are achieved, which verifies the reliability of the proposed method. It is also noticed that, with the same computational grids and limiters, the AMR-RLBFS has smaller oscillations around x = 0.49 than the AMR-LBFS. Figure 8 shows the adaptive mesh of the Sod shock tube at t = 0 and t = 0.25, confirming the location of the dramatic changes in physical quantities in Figure 7. And Figure 9 also shows the adaptive mesh variation with time, in which the red triangles represent the total amount of adaptive mesh, and the black line is the fit of these triangles. In addition, the average number of adaptive grids used by the AMR-RLBFS is about 33,456, which is about 26.14% of the uniform grids. Meanwhile, the computation time of the adaptive grid is 71.37 s, while the computation time of the uniform grid is 140.69 s, with a speed-up ratio of 1.97. This indicates that both the required computational resources and consumptions are reduced by the present proposed method.

3.2. Two-Dimensional Four-Wave Riemannian Problem

Two-dimensional Riemann problems are a special class of initial value problems. The study of two-dimensional Riemann problems is important for verifying the capability and accuracy of numerical simulations because very complex phenomena, such as shock reflection, vortex generation, spiral structure, and vortex–shock interaction, exist. To simplify the problem, the current research focuses on a special class of two-dimensional Riemann problems, i.e., each quarter region of the two-dimensional plane is constant, so four initial discontinuities constitute four one-dimensional Riemann problems, and each one-dimensional Riemann problem generates a simple wave, i.e., rarefaction waves (R), contact discontinuities (J), or shock waves (S), and the four simple waves will form a two-dimensional structure near the center point, and this two-dimensional Riemann problem is called a four-wave two-dimensional Riemann problem. For the four-wave two-dimensional Riemann problem, there are many theoretical and numerical research works [47,48,49,50,51]. The construction of the initial values of the four-wave two-dimensional Riemann problem and the meaning of the notation according to the type of the one-dimensional Riemann problem on the boundary can be seen in detail from the appendix of [51].

In this section, to verify the performance and the accuracy of the AMR-RLBFS for solving Riemann problems, as well as the improvement in the AMR-RLBFS over AMR-LBFS algorithm, two four-wave two-dimensional Riemann problems are considered. The computational domain is [0,1] × [0,1] with a base mesh of size 200 × 200 (l = 0). A maximum refinement level of l_max = 3 and γ = 1.4 are used for the simulation.

3.2.1. Interaction of Rarefaction Waves $\left[R_{12}^{+}{R}_{23}^{-}{R}_{34}^{+}{R}_{41}^{-}\right]$

The initial values of this four-wave two-dimensional Riemann problem are given below, and the interaction of the rarefaction waves produces rarefaction waves at all four initial discontinuities.

$$\begin{array}{c}{\rho }_1=1{.0,\mathrm{u}}_1=0.0,v_1=0.0,p_1=1.0, 0.5<x<1.0,0.5<y<1.0,\hfill \\ {\rho }_2=0{.5197,\mathrm{u}}_2=-0.7259,v_2=0.0,p_2=0.4, 0.0<x<0.5,0.5<y<1.0,\hfill \\ {\rho }_3=1{.0,\mathrm{u}}_3=-0.7259,v_3=-0.7259,p_3=1.0, 0.0<x<0.5,0.0<y<0.5,\hfill \\ {\rho }_4=0{.5197,\mathrm{u}}_4=0.0,v_4=-0.7259,p_4=0.4, 0.5<x<1.0,0.0<y<\mathrm{0.5.}\hfill \end{array}$$

(42)

Figure 10 shows the comparison of density contours of the interaction of rarefaction waves at t = 0.2. As can be seen, the results are in good agreement with [49]. It may be highlighted that the AMR-RLBFS eliminates the oscillating effects caused by the interaction of rarefaction waves near the center and improves the accuracy. Figure 11a further shows the adaptive mesh at the same time, and Figure 11b presents the overall grid size varying with time. The average number of grids for the adaptive mesh is about 351,984, which is about 15.31% of the uniform grid with the same resolution. The computation time of the adaptive mesh by the present AMR-RLBFS is 0.42 h, while that of the uniform grid is 1.56 h, resulting in a speed-up ratio of 3.71. This indicates that the proposed method is able to not only improve the computational efficiency, but also effectively capture rarefaction waves without generating unphysical oscillations.

3.2.2. Interaction of Contact Discontinuities and Shock Waves $\left[S_{12}^{+}{J}_{23}^{-}{J}_{34}^{+}{S}_{41}^{-}\right]$

Contact discontinuities and shock waves are further considered, and the initial conditions of this flow problem are given as follows:

$$\begin{array}{c}{\rho }_1=0{.5313,\mathrm{u}}_1=0.0,v_1=0.0,p_1=0.4, 0.5<x<1.0,0.5<y<1.0,\hfill \\ {\rho }_2=1{.0,\mathrm{u}}_2=0.7276,v_2=0.0,p_2=1.0, 0.0<x<0.5,0.5<y<1.0,\hfill \\ {\rho }_3=0{.8,\mathrm{u}}_3=0.0,v_3=0.0,p_3=1.0, 0.0<x<0.5,0.0<y<0.5,\hfill \\ {\rho }_4=1{.0,\mathrm{u}}_4=0.0,v_4=0.7276,p_4=1.0, 0.5<x<1.0,0.0<y<\mathrm{0.5.}\hfill \end{array}$$

(43)

The comparison of the density contour lines for the interaction of rarefaction waves and contact discontinuities is shown in Figure 12. As shown in Figure 12, the results of the AMR-LBFS are basically consistent with [49], and the AMR-LBFS captures many small-scale vortex structures at the interface, and for the resolution of small-scale vortex structures at the interface, the AMR-RLBFS still shows a higher resolution, which improves the precision and accuracy of the numerical results. The adaptive mesh for the interaction of rarefaction waves and contact discontinuities at t = 0.25, and the adaptive mesh change with time are shown in Figure 13. Based on the parallel computation, the average number of grids for the adaptive mesh is about 276,133, which is about 10.79% of the uniform grid. Meanwhile, the computation time of the adaptive mesh is 0.35 h, while that of the uniform mesh is 1.55 h, with a speed-up ratio of 4.43. The reduction in the computational grid and the improvement in the computational efficiency of the AMR-RLBFS are more obvious in the two-dimensional Riemann problem compared to the shock tube problem.

3.3. Three-Dimensional Problems

After studying the various two-dimensional Riemann problems mentioned above, the potential of the AMR-RLBFS has been demonstrated. Therefore, this section further presents numerical simulations of some three-dimensional compressible problems using the AMR-RLBFS.

3.3.1. Explosion in a Box

The explosion in a box is described as a spherical shock propagating in a closed box, where the reflected shock waves have complex interactions. The computational domain, [0,1] × [0,1] × [0,1], is schematically shown in Figure 14, in which all six faces are reflecting walls with the following values: center of sphere: (0.4, 0.4, 0.4); radius of sphere: 0.3. The number of the base mesh is 30 × 30 × 30 (l = 0), with a maximum adaptive level of l_max = 3, γ = 1.4. The adaptive mesh is used to improve the accuracy and efficiency of the RLBFS algorithm, and the initial conditions are as follows:

$$\left(\rho ,u,v,w,p\right)=\{\begin{array}{l}\left(5,0,0,0,5\right) if \sqrt{{\left(x-0.4\right)}^2+{\left(y-0.4\right)}^2+{\left(z-0.4\right)}^2}\le 0.3,\\ \left(1,0,0,0,1\right) else.\end{array}$$

(44)

Figure 15 shows the density contours on the plane z = 0.4 for the explosion in a box at t = 0.5. The present results are obtained by the AMR-RLBFS using both two and three refinement levels. Also included in this figure are the results of [21,52]. It can be seen from Figure 15 that the density contours obtained by the AMR-RLBFS with two refinement levels have a higher resolution of complex features than the results in [21,52]. Furthermore, the results obtained by the AMR-RLBFS with three refinement levels show a more accurate identification of the fine flow structure and shock waves. Comparison of the isosurface of the density is further given in Figure 16, demonstrating good consistency with the results of [52]. These comparisons indicate that the AMR-RLBFS proposed in this paper is able to simulate the three-dimensional compressible flow with complex shock structures on adaptive meshes. Figure 17 illustrates the block-structured adaptive mesh at z = 0.4 from t = 0 to t = 0.5. It can be seen that the mesh near the shock wave is refined to improve the ability to capture the shock wave. Figure 18 shows the mesh size variation over time. The average number of mesh size is about 5,757,220, which is about 41.65% of the uniform grids with the same finest resolution. The computation time on this mesh by the AMR-RLBFS is 0.48 h, while the computation time on the uniform grid integration is 0.83 h, resulting in a speed-up ratio of 1.73.

3.3.2. Vorticity Generated by a Shock

The vorticity generated by a shock [53] is a challenging flow problem by setting different densities and pressures in two cylindrical regions in a rectangular domain as shown in Figure 19. This problem is further studied by the AMR-RLBFS. A cylindrical region along the z-axis generates a cylindrical shock wave due to the large pressure difference, while another cylindrical region of low density, parallel to the y-axis, is hit by the cylindrical shock wave and generates a large number of vortices. The radius of both cylinders is r = 0.2. In the present study, the computational domain is set as [0,1.5] × [0,1] × [0,0.5], and the base mesh size is taken as 45 × 30 × 15 (l = 0). The planes x = 0, y = 0, and z = 0 are symmetry surfaces and outflow conditions that are applied on the other three surfaces. A maximum adaptive level is l_max = 3. The initial conditions are as follows:

$$\left(\rho ,u,v,w,p\right)=\{\begin{array}{l}\left(1,0,0,0,10\right) if\sqrt{x^2+y^2}\le 0.2,\\ \left(0.1,0,0,0,1\right) if \sqrt{{\left(x-0.4\right)}^2+z^2}\le 0.2,\\ \left(1,0,0,0,1\right) else.\end{array}$$

(45)

Figure 20 shows the magnitude of the density gradient of the vorticity generated by a shock by the AMR-RLBFS at t = 0.1–0.5. It can be seen that the incident shock wave at t = 0.1 wraps around part of the low-density cylindrical region, which begins to collapse. At t = 0.3, the passing shock wave causes the low-density cylindrical region to wind into two vortex spirals, and the contact discontinuity generated by the high-pressure region is also swept up in this vortex motion. The interaction produces a triple-shock structure with distinct vortices visible at t = 0.4. Vortex spirals form on the envelope of the vortex spirals at t = 0.5, which is a purely three-dimensional feature and creates a cycle of vortex spirals in the y-direction that will stretch, move, and eventually break up to form a complex region. Figure 21 shows the detailed information of the density and pressure variations with time on the x-axis. The high-pressure cylindrical region on the left collapses to form a shock wave at nearly 0.1, which starts to impinge on the low-density cylindrical region, and then the complex shock structures formed around 0.15 reach the peak of density and pressure. And the complex shock structures begin to propagate to the right and gradually break up and escape, accompanied by dissipation of the intensity of the left shock wave. Due to the turbulence-like behavior, the symmetry assumption is incorrect, the details of the vortex dynamics depend on the limiter used, and the results for this region can only be viewed qualitatively.

Figure 22 further shows the contours of the magnitude of the density gradient and block-structured adaptive mesh on the plane y = 0 at different times. It can be seen that, as the vortex moves, the mesh around it is also refined dynamically in order to obtain a complex flow structure in the flow field. Figure 23 shows the mesh size over time. The average adaptive mesh is about 1,889,829 which is about 18.23% of the uniform grid with the same resolution. The computation time on the adaptive mesh is 0.72 h, while the computation time of the uniform grid is 2.73 h, leading to a speed-up ratio of 3.79. This demonstrates the improvement in the computational efficiency of the AMR-RLBFS.

4. Conclusions

In this paper, a block-structured AMR-RLBFS method has been proposed and validated by simulating two and three-dimensional unsteady compressible flows with complex shock structures. The AMR-RLBFS applies the flow field solver RLBFS locally on each block of the adaptive mesh and refines the computational grids dynamically. With the advantages the block-structured AMR technique, the present method is able to obtain solutions on feature-specified adaptive meshes with higher numerical accuracy and relatively less computational resources.

By simulating the shock tube problem and the four-wave two-dimensional Riemann problem, it is shown that the AMR-RLBFS successfully captures rarefaction waves, shock waves, and contact discontinuities and effectively resolves small-scale vortex structures on adaptive meshes. The effects caused by shock instability are removed as well. Comparisons also highlight the superior accuracy and stability of the AMR-RLBFS over the AMR-LBFS. The capability of the proposed method in solving three-dimensional problems has also been examined by simulating the explosion in a box and vorticity generated by a shock. It is shown again that the results obtained by the AMR-RLBFS are well structured and clear and in good agreement with the literature results, with sufficient precision and accuracy. Numerical results also show acceleration of the simulation with a max speed-up ratio up to 4.43 for all cases in this work. All these observations demonstrate of the potential of the AMR-RLBFS in simulating high Mach number compressible flows with strong shock waves, and therefore it can be used as a promising numerical method to continue its optimal development and then be useful in practical aerospace applications.