This paper is devoted to the structural stability of a transonic shock passing through a flat nozzle for two-dimensional steady compressible flows with an external force. We first establish the existence and uniqueness of one-dimensional transonic shock solutions to the steady Euler system with an external force by prescribing suitable pressure at the exit of the nozzle when the upstream flow is a uniform supersonic flow. It is shown that the external force helps to stabilize the transonic shock in flat nozzles and the shock position is uniquely determined. Then we are concerned with the structural stability of these transonic shock solutions when the exit pressure is suitably perturbed. One of the new ingredients in our analysis is to use the deformation-curl decomposition to the steady Euler system developed by Weng and Xin [Sci. Sinica Math., 49 (2019), pp. 307–320] to deal with the transonic shock problem.

We introduce an approach and a software tool for solving coupled energy networks composed of gas and electric power networks. Those networks are coupled to stochastic fluctuations to address possibly fluctuating demand due to fluctuating demands and supplies. Through computational results, the presented approach is tested on networks of realistic size.

In this paper, we consider the Cauchy problem for an inviscid compressible Oldroyd-B model in three dimensions. The global well posedness of strong solutions and the associated time-decay estimates in Sobolev spaces are established near an equilibrium state. The vanishing of viscosity is the main challenge compared with [47] where the viscosity coefficients are included and the decay rates for the highest-order derivatives of the solutions seem not optimal. One of the main objectives of this paper is to develop some new dissipative estimates such that the smallness of the initial data and decay rates are independent of the viscosity. Moreover, we prove that the decay rates for the highest-order derivatives of the solutions are optimal, which is of independent interest. Our proof relies on Fourier theory and delicate energy method.

We study the small data global well-posedness and time-decay rates of solutions to the Cauchy problem for three-dimensional compressible Navier–Stokes–Allen–Cahn equations via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained, the $\dot {H}^{-s}$($0\leq s<\frac {3}{2}$) negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.

This paper concerns the energy conservation for the weak solutions of the compressible Navier–Stokes equations. Assume that the density is positively bounded, we work on the regularity assumption on the gradient of the velocity, and establish a Lp–Ls type condition for the energy equality to hold in the distributional sense in time. We mention that no regularity assumption on the density derivative is needed any more.

In lubrication problems, which concern thin film flow, cavitation has been considered as a fundamental element to correctly describe the characteristics of lubricated mechanisms. Here, the well-posedness of a cavitation model that can explain the interaction between viscous effects and micro-bubbles of gas is studied. This cavitation model consists of a coupled problem between the compressible Reynolds partial differential equation (PDE) (that describes the flow) and the Rayleigh–Plesset ordinary differential equation (that describes micro-bubbles evolution). A simplified form without bubbles convection is studied here. This coupled model seems never to be studied before from its mathematical aspects. Local times existence results are proved and stability theorems are obtained based on the continuity of the spectrum for bounded linear operators. Numerical results are presented to illustrate these theoretical results.

We consider the compressible Navier–Stokes system on time-dependent domains with prescribed motion of the boundary. For both the no-slip boundary conditions as well as slip boundary conditions we prove local-in-time existence of strong solutions. These results are obtained using a transformation of the problem to a fixed domain and an existence theorem for Navier–Stokes like systems with lower order terms and perturbed boundary conditions. We also show the weak–strong uniqueness principle for slip boundary conditions which remained so far open question.

With the use of temporal derivative of flux function, a two-stage temporal discretization has been recently proposed in the design of fourth-order schemes based on the generalized Riemann problem (GRP) [21] and gas-kinetic scheme (GKS) [28]. In this paper, the fourth-order gas-kinetic scheme will be extended to solve the compressible multicomponent flow equations, where the two-stage temporal discretization and fifth-order WENO reconstruction will be used in the construction of the scheme. Based on the simplified two-species BGK model [41], the coupled Euler equations for individual species will be solved. Each component has its individual gas distribution function and the equilibrium states for each component are coupled by the physical requirements of total momentum and energy conservation in particle collisions. The second-order flux function is used to achieve the fourth-order temporal accuracy, and the robustness is as good as the second-order schemes. At the same time, the source terms, such as the gravitational force and the chemical reaction, will be explicitly included in the two-stage temporal discretization through their temporal derivatives. Many numerical tests from the shock-bubble interaction to ZND detonative waves are presented to validate the current approach.

This paper develops Runge-Kutta PK -based central discontinuous Galerkin (CDG) methods with WENO limiter for the one- and two-dimensional special relativistic hydrodynamical (RHD) equations, K = 1,2,3. Different from the non-central DG methods, the Runge-Kutta CDG methods have to find two approximate solutions defined on mutually dual meshes. For each mesh, the CDG approximate solutions on its dual mesh are used to calculate the flux values in the cell and on the cell boundary so that the approximate solutions on mutually dual meshes are coupled with each other, and the use of numerical flux will be avoided. The WENO limiter is adaptively implemented via two steps: the “troubled” cells are first identified by using a modified TVB minmod function, and then the WENO technique is used to locally reconstruct new polynomials of degree (2K+1) replacing the CDG solutions inside the “troubled” cells by the cell average values of the CDG solutions in the neighboring cells as well as the original cell averages of the “troubled” cells. Because the WENO limiter is only employed for finite “troubled” cells, the computational cost can be as little as possible. The accuracy of the CDG without the numerical dissipation is analyzed and calculation of the flux integrals over the cells is also addressed. Several test problems in one and two dimensions are solved by using our Runge-Kutta CDG methods with WENO limiter. The computations demonstrate that our methods are stable, accurate, and robust in solving complex RHD problems.

In this work, we investigate the numerical approximation of the compressible Navier-Stokes equations under the framework of discontinuous Galerkin methods. For discretization of the viscous and heat fluxes, we extend and apply the symmetric direct discontinuous Galerkin (SDDG) method which is originally introduced for scalar diffusion problems. The original compressible Navier-Stokes equations are rewritten into an equivalent form via homogeneity tensors. Then, the numerical diffusive fluxes are constructed from the weak formulation of primal equations directly without converting the second-order equations to a first-order system. Additional numerical flux functions involving the jump of second order derivative of test functions are added to the original direct discontinuous Galerkin (DDG) discretization. A number of numerical tests are carried out to assess the practical performance of the SDDG method for the two dimensional compressible Navier-Stokes equations. These numerical results obtained demonstrate that the SDDG method can achieve the optimal order of accuracy. Especially, compared with the well-established symmetric interior penalty (SIP) method [18], the SDDG method can maintain the expected optimal order of convergence with a smaller penalty coefficient.

This paper is concerned with the construction of global, non-vacuum, strong, large amplitude solutions to initial–boundary-value problems for the one-dimensional compressible Navier–Stokes equations with degenerate transport coefficients. Our analysis derives the positive lower and upper bounds on the specific volume and the absolute temperature.

We propose here a class of numerical schemes for the approximation of weak solutions to nonlinear hyperbolic systems in nonconservative form—the notion of solution being understood in the sense of Dal Maso, LeFloch, and Murat (DLM). The proposed numerical method falls within LeFloch-Mishra's framework of schemes with well-controlled dissipation (WCD), recently introduced for dealing with small-scale dependent shocks. We design WCD schemes which are consistent with a given nonconservative system at arbitrarily high-order and then analyze their linear stability. We then investigate several nonconservative hyperbolic models arising in complex fluid dynamics, and we numerically demonstrate the convergence of our schemes toward physically meaningful weak solutions.

Multi-field coupling simulation method based on the physical principles is used to simulate the discharge characteristics of nanosecond pulsed plasma synthetic jet actuator. Considering the effect of the energy transferring for air, the flow characteristics of nanosecond pulsed plasma synthetic jet actuator are simulated. The elastic heating sources and ion joule heating sources are the two main sources of energy. Through the collisions, the energy of ions is transferred to the neutral gas quickly. The flow characteristics of a series of blast waves and the synthetic jet which erupt from the plasma synthetic jet (PSJ) actuator are simulated. The blast wave not only promotes outward, but also accelerates the gas mixing the inhaled gas from the outside cavity with the residual gas inside the cavity. The performances of PSJ actuator fluctuate in the first three incentive cycles and become stable after that.

Nanosecond (ns) pulsed dielectric barrier discharge (DBD) actuator in a laminar flat plate boundary layer is investigated numerically in an attempt to gain some new insights into the understanding of ns DBD actuation mechanism. Special emphasis is put on the examination, separation and comparison of behaviors of discharge induced micro shock wave and residual heat as well as on the investigation of response of external flow to the two effects. The shock wave is found to introduce highly transient, localized perturbation to the flow and be able to significantly alter the flow pattern shortly after its initiation. The main flow tends to quickly recover to close to its undisturbed state due to the transient nature of perturbation. However, with the shock decay and final disappearance, another perturbation source in the vicinity of discharge region, which contains contribution from both residual heat and shock, becomes increasingly pronounced and eventually develops into a perturbation wave train in the boundary layer. The perturbation is relatively weak and may not be a Tollmien-Schlichting (TS) wave and not trigger the laminar-turbulent transition of boundary layer. Instead, it is more likely to manipulate the flow stability to achieve the strong control authority of this kind of actuation in the case of flow separation control. In addition, a parametric study over the different electrical and hydrodynamic parameters is also conducted.

We propose an all regime Lagrange-Projection like numerical scheme for the gas dynamics equations. By all regime, we mean that the numerical scheme is able to compute accurate approximate solutions with an under-resolved discretization with respect to the Mach number M, i.e. such that the ratio between the Mach number M and the mesh size or the time step is small with respect to 1. The key idea is to decouple acoustic and transport phenomenon and then alter the numerical flux in the acoustic approximation to obtain a uniform truncation error in term of M. This modified scheme is conservative and endowed with good stability properties with respect to the positivity of the density and the internal energy. A discrete entropy inequality under a condition on the modification is obtained thanks to a reinterpretation of the modified scheme in the Harten Lax and van Leer formalism. A natural extension to multi-dimensional problems discretized over unstructured mesh is proposed. Then a simple and efficient semi implicit scheme is also proposed. The resulting scheme is stable under a CFL condition driven by the (slow) material waves and not by the (fast) acoustic waves and so verifies the all regime property. Numerical evidences are proposed and show the ability of the scheme to deal with tests where the flow regime may vary from low to high Mach values.

In this paper we focus on the existence of a weak solution to a system describing a self-propelled motion of a single deformable body in a viscous compressible fluid that occupies a bounded domain in the three-dimensional Euclidean space. The governing system considered for the fluid is the isentropic compressible Navier–Stokes equation. We prove the existence of a weak solution up to a collision.

This paper presents a new and better suited formulation to implement the limiting projection to high-order schemes that make use of high-order local reconstructions for hyperbolic conservation laws. The scheme, so-called MCV-WENO4 (multi-moment Constrained finite Volume with WENO limiter of 4th order) method, is an extension of the MCV method of Ii & Xiao (2009) by adding the 1st order derivative (gradient or slope) at the cell center as an additional constraint for the cell-wise local reconstruction. The gradient is computed from a limiting projection using the WENO (weighted essentially non-oscillatory) reconstruction that is built from the nodal values at 5 solution points within 3 neighboring cells. Different from other existing methods where only the cell-average value is used in the WENO reconstruction, the present method takes account of the solution structure within each mesh cell, and thus minimizes the stencil for reconstruction. The resulting scheme has 4th-order accuracy and is of significant advantage in algorithmic simplicity and computational efficiency. Numerical results of one and two dimensional benchmark tests for scalar and Euler conservation laws are shown to verify the accuracy and oscillation-less property of the scheme.

In this paper, a compact third-order gas-kinetic scheme is proposed for the compressible Euler and Navier-Stokes equations. The main reason for the feasibility to develop such a high-order scheme with compact stencil, which involves only neighboring cells, is due to the use of a high-order gas evolution model. Besides the evaluation of the time-dependent flux function across a cell interface, the high-order gas evolution model also provides an accurate time-dependent solution of the flow variables at a cell interface. Therefore, the current scheme not only updates the cell averaged conservative flow variables inside each control volume, but also tracks the flow variables at the cell interface at the next time level. As a result, with both cell averaged and cell interface values, the high-order reconstruction in the current scheme can be done compactly. Different from using a weak formulation for high-order accuracy in the Discontinuous Galerkin method, the current scheme is based on the strong solution, where the flow evolution starting from a piecewise discontinuous high-order initial data is precisely followed. The cell interface time-dependent flow variables can be used for the initial data reconstruction at the beginning of next time step. Even with compact stencil, the current scheme has third-order accuracy in the smooth flow regions, and has favorable shock capturing property in the discontinuous regions. It can be faithfully used from the incompressible limit to the hypersonic flow computations, and many test cases are used to validate the current scheme. In comparison with many other high-order schemes, the current method avoids the use of Gaussian points for the flux evaluation along the cell interface and the multi-stage Runge-Kutta time stepping technique. Due to its multidimensional property of including both derivatives of flow variables in the normal and tangential directions of a cell interface, the viscous flow solution, especially those with vortex structure, can be accurately captured. With the same stencil of a second order scheme, numerical tests demonstrate that the current scheme is as robust as well-developed second-order shock capturing schemes, but provides more accurate numerical solutions than the second order counterparts.

We consider a two-step Sal’nikov reaction scheme occurring within a compressible viscous gas. The first step of the reaction may be either endothermic or exothermic, while the second step is strictly exothermic. Energy may also be lost from the system due to Newtonian cooling. An asymptotic solution for temperature perturbations of small amplitude is presented using the methods of strained coordinates and multiple scales, and a travelling wave solution with a sech-squared profile is derived. The method of lines is then used to approximate the full system with a set of ordinary differential equations, which are integrated numerically to track accurately the evolution of the reaction front. This numerical method is used to verify the asymptotic solution and investigate behaviours under different conditions. Using this method, temperature waves progressing as pulsatile fronts are detected at appropriate parameter values.

We present an approximate relation for the effective slip length for flows over mixed-slip surfaces with patterning at the nanoscale, whose minimum slip length is greater than the pattern length scale.