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A new approach was recently proposed to construct equilibrium distribution functions of the lattice Boltzmann method for simulation of compressible flows. In this approach, the Maxwellian function is replaced by a simple function which satisfies all needed relations to recover compressible Euler equations. With Lagrangian interpolation polynomials, the simple function is discretized onto a fixed velocity pattern to construct . In this paper, the finite volume method is combined with the new lattice Boltzmann models to simulate 1D and 2D shock-wave propagation. The numerical results agree well with available data in the literatures.

We propose an implicit-explicit finite-difference lattice Boltzmann method for compressible flows in this work. The implicit-explicit Runge–Kutta scheme, which solves the relaxation term of the discrete velocity Boltzmann equation implicitly and other terms explicitly, is adopted for the time discretization. Owing to the characteristic of the collision invariants in the lattice Boltzmann method, the implicitness can be completely eliminated, and thus no iteration is needed in practice. In this fashion, problems (no matter stiff or not) can be integrated quickly with large Courant–Friedriche–Lewy numbers. As a result, with our implicit-explicit finite-difference scheme the computational convergence rate can be significantly improved compared with previous finite-difference and standard lattice Boltzmann methods. Numerical simulations of the Riemann problem, Taylor vortex flow, Couette flow, and oscillatory compressible flows with shock waves show that our implicit-explicit finite-difference lattice Boltzmann method is accurate and efficient. In addition, it is demonstrated that with the proposed scheme non-uniform meshes can also be implemented with ease.

A lattice Boltzmann model is developed for viscous compressible flows with flexible specific-heat ratio and Prandtl number. Unlike the Maxwellian distribution function or circle function used in the existing lattice Boltzmann models, a polynomial kernel function in the phase space is introduced to recover the Navier–Stokes–Fourier equations. A discrete equilibrium density distribution function and a discrete equilibrium total energy distribution function are obtained from the discretization of the polynomial kernel function with Lagrangian interpolation. The equilibrium distribution functions are then coupled via the equation of state. In this framework, a model for viscous compressible flows is proposed. Several numerical tests from subsonic to supersonic flows, including the Sod shock tube, the double Mach reflection and the thermal Couette flow, are simulated to validate the present model. In particular, the discrete Boltzmann equation with the Bhatnagar–Gross–Krook approximation is solved by the finite-difference method. Numerical results agree well with the exact or analytic solutions. The present model has potential application in the study of complex fluid systems such as thermal compressible flows.

In this paper, a new smoothness indicator is proposed to improve the finite-difference lattice Boltzmann method (FDLBM). The necessary and sufficient conditions for convergence are derived. A detailed analysis reveals that the convergence order is higher than that of the previous finite-difference scheme. The coupled double distribution function (DDF) model is used to describe discontinuity flows and verify the improvement. Numerical simulations of compressible flows with shock wave show that the improved finite-difference lattice Boltzmann scheme is accurate and has less dissipation. The numerical results are found to be in good agreement with the analytical results and better than those of the previous scheme.

A lattice Boltzmann model on Hermite basis for compressible viscous flows is presented in this paper. The model is developed in the framework of double-distribution-function approach, which has adjustable specific-heat ratio and Prandtl number. It contains a density distribution function for the flow field and a total energy distribution function for the temperature field. The equilibrium distribution function is determined by Hermite expansion, and the D3Q27 and D3Q39 three-dimensional (3D) discrete velocity models are used, in which the discrete velocity model can be replaced easily. Moreover, an artificial viscosity is introduced to enhance the model for capturing shock waves. The model is tested through several cases of compressible flows, including 3D supersonic viscous flows with boundary layer. The effect of artificial viscosity is estimated. Besides, D3Q27 and D3Q39 models are further compared in the present platform.

The recently proposed boundary condition-enforced immersed boundary-gas kinetic flux solver (IB-GKFS) is a new approach for simulation of compressible flows with curved and moving boundaries. In the previous application of IB-GKFS, only the Dirichlet boundary condition is considered, which cannot be applied directly to the Neumann boundary condition. In this paper, an auxiliary layer of Lagrangian points is introduced to tackle Neumann boundary condition. Two test cases, including flow around a circular cylinder and flow around a NACA0012 airfoil, are carried out for validation. The results obtained by the present scheme are compared with the reference data available in the literature. Good agreements are achieved, which indicate that the developed method can effectively simulate the compressible flow with Neumann boundary condition.

The convection and diffusion processes of free vortex in compressible flows are simulated by using high precision numerical method to solve for the Navier–Stokes equations. Accurate treatment of the boundary condition is extremely important for simulation of vortex flows. The developed numerical methods are well presented by combining six-order non-dissipation compact schemes with Navier–Stokes characteristic boundary condition having transverse and viscous terms, and can accurately simulate the movement of free vortex. The numerical reflecting waves at the boundaries are well controlled.

We present an immersogeometric analysis (IMGA) approach for the simulation of compressible flows around complex geometries. In this method, compressible flow simulations are performed directly on various boundary representations (B-reps) of mechanical designs, circumventing the labor-intensive and time-consuming cleanup of complex geometric models. A new formulation for the weak imposition of essential boundary conditions in the context of non-body-fitted meshes is proposed. The formulation employs the non-symmetric Nitsche method, which yields good performance especially when the penalty parameters are difficult to estimate. We test the proposed immersogeometric formulation on benchmark problems for a wide range of Reynolds and Mach numbers, showing its robustness and accuracy. Finally, the methodology is applied to the simulation of a UH-60 Black Hawk helicopter in forward flight, illustrating the ability of the proposed approach to support the design of real-world engineering systems through high-fidelity aerodynamic analysis.

In this paper, a new flux solver was developed based on a lattice Boltzmann model. Different from solving discrete velocity Boltzmann equation and lattice Boltzmann equation, Euler/Navier-Stokes (NS) equations were solved in this approach, and the flux at the interface was evaluated with a compressible lattice Boltzmann model. This method combined lattice Boltzmann method with finite volume method to solve Euler/NS equations. The proposed approach was validated by some simulations of one-dimensional and multi-dimensional problems.

A scheme located between the Lax-Wendroff and Roe schemes is presented for the multidimensional compressible Euler and Navier-Stokes equations. The scheme is second-order accurate at steady-state with a low internal dissipation. It is easy to code and involves no tuning parameter. Accuracy and efficiency are demonstrated for several flow problems including a turbulent transonic flow over a wing.