Full Compressible Navier-Stokes Equations Research Articles

Global convergence rates from relaxed Euler equations to Navier–Stokes equations with Oldroyd-type constitutive laws

In a previous work (Peng and Zhao 2022 J. Math. Fluid Mech. 24 29), it is proved that the 1D full compressible Navier–Stokes equations for a Newtonian fluid can be approximated globally-in-time by a relaxed Euler-type system with Oldroyd’s derivatives and a revised Cattaneo’s constitutive law. These two relaxations turn the whole system into a first-order quasilinear hyperbolic one with partial dissipation. In this paper, we establish the global convergence rates between the smooth solutions to the relaxed Euler-type system and the Navier–Stokes equations over periodic domains. For this purpose, we use stream function techniques together with energy estimates for error systems. These techniques may be applicable to more complicated systems.

Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier–Stokes equations

Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier–Stokes equations

Vanishing viscosity limit to the planar rarefaction wave with vacuum for 3-D full compressible Navier–Stokes equations with temperature-dependent transport coefficients

Vanishing viscosity limit to the planar rarefaction wave with vacuum for 3-D full compressible Navier–Stokes equations with temperature-dependent transport coefficients

A centered limited finite volume approximation of the momentum convection operator for low‐order nonconforming face‐centered discretizations

AbstractWe propose in this article a discretization of the momentum convection operator for fluid flow simulations on quadrangular or generalized hexahedral meshes. The space discretization is performed by the low‐order nonconforming Rannacher–Turek finite element: the scalar unknowns are associated with the cells of the mesh while the velocities unknowns are associated with the edges or faces. The momentum convection operator is of finite volume type, and its expression is derived, as in MUSCL schemes, by a two‐step technique: computation of a tentative flux, here, with a centered approximation of the velocity, and limitation of this flux using monotonicity arguments. The limitation procedure is of algebraic type, in the sense that its does not invoke any slope reconstruction, and is independent from the geometry of the cells. The derived discrete convection operator applies both to constant or variable density flows and may thus be implemented in a scheme for incompressible or compressible flows. To achieve this goal, we derive a discrete analogue of the computation (with the velocity, one of its component, the density, and assuming that the mass balance holds) and discuss two applications of this result: first, we obtain stability results for a semi‐implicit in time scheme for incompressible and barotropic compressible flows; second, we build a consistent, semi‐implicit in time scheme that is based on the discretization of the internal energy balance rather than the total energy. The performance of the proposed discrete convection operator is assessed by numerical tests on the incompressible Navier–Stokes equations, the barotropic and the full compressible Navier–Stokes equations and the compressible Euler equations.

A new blow-up criterion for the 2D full compressible Navier–Stokes equations without heat conduction in a bounded domain

This paper is to derive a new blow-up criterion for the 2D full compressible Navier–Stokes equations without heat conduction in terms of the density ρ and the pressure P. More precisely, it indicates that in a bounded domain the strong solution exists globally if the norm ||ρ||L∞(0,t;L∞)+||P||Lp0(0,t;L∞)<∞ for some constant p0 satisfying 1<p0≤2. The boundary condition is imposed as a Navier-slip boundary one and the initial vacuum is permitted. Our result extends previous one which is stated as ||ρ||L∞(0,t;L∞)+||P||L∞(0,t;L∞)<∞.

Calderón–Zygmund theory in Lorentz mixed‐norm spaces and its application to compressible fluids

AbstractIn this paper, it is shown that the general Calderón–Zygmund singular integral operators including Riesz transforms are bounded on anisotropic Lorentz spaces with and . As an application, we establish some new blowup criteria involving temperature in its scaling‐invariant anisotropic Lorentz space for the 3D full compressible Navier–Stokes equations allowing vacuum and without additional conditions on viscosity coefficients except physical restrictions, which improve the previous corresponding results.

Full compressible Navier-Stokes equations with the Robin boundary condition on temperature

We consider full compressible Navier-Stokes equations with the Robin boundary condition on temperature. Note that the viscosity is constant and the heat conductivity is proportional to a positive power of the temperature. It is shown that a unique global strong solution existed if the initial data belongs to H 1 . Subsequently, we find that the strong solution is nonlinearly exponentially stable as time tends to infinity. This result could be viewed as the first one on the global well-posedness of the strong solution to full Navier-Stokes equations in a bounded domain with the degenerate heat conductivity and the Robin boundary condition on temperature. The proofs are mainly based on the energy method and a special inequality.

Discrete unified gas kinetic scheme for continuum compressible flows.

In this paper, a discrete unified gas kinetic scheme (DUGKS) is proposed for continuum compressible gas flows based on the total energy kinetic model [Guo et al., Phys. Rev. E 75, 036704 (2007)1539-375510.1103/PhysRevE.75.036704]. The proposed DUGKS can be viewed as a special finite-volume lattice Boltzmann method for the compressible Navier-Stokes equationsin the double distribution function formulation, in which the mass and momentum transport are described by the kinetic equationfor a density distribution function (g), and the energy transport is described by the other one for an energy distribution function (h). To recover the full compressible Navier-Stokes equationsexactly, the corresponding equilibrium distribution functions g^{eq} and h^{eq} are expanded as Hermite polynomials up to third and second orders, respectively. The velocity spaces for the kinetic equationsare discretized according to the seventh and fifth Gauss-Hermite quadratures. Consequently, the computational efficiency of the present DUGKS can be much improved in comparison with previous versions using more discrete velocities required by the ninth Gauss-Hermite quadrature.

On Regular Solutions for Three-Dimensional Full Compressible Navier–Stokes Equations with Degenerate Viscosities and Far Field Vacuum

In this paper, the Cauchy problem for the three-dimensional (3-D) full compressible Navier–Stokes equations (CNS) with zero thermal conductivity is considered. First, when shear and bulk viscosity coefficients both depend on the absolute temperature $$\theta $$ in a power law ( $$\theta ^\nu $$ with $$\nu >0$$ ) of Chapman–Enskog, based on some elaborate analysis of this system’s intrinsic singular structures, we identify one class of initial data admitting a local-in-time regular solution with far field vacuum in terms of the mass density $$\rho $$ , velocity u and entropy S. Furthermore, it is shown that within its life span of such a regular solution, the velocity stays in an inhomogeneous Sobolev space, i.e., $$u\in H^3({\mathbb {R}}^3)$$ , S has uniformly finite lower and upper bounds in the whole space, and the laws of conservation of total mass, momentum and total energy are all satisfied. Note that, due to the appearance of the vacuum, the momentum equations are degenerate both in the time evolution and viscous stress tensor, and the physical entropy for polytropic gases behaves singularly, which make the study on corresponding well-posedness challenging. For proving the existence, we first introduce an enlarged reformulated structure by considering some new variables, which can transfer the degeneracies of the full CNS to the possible singularities of some special source terms related with S, and then carry out some singularly weighted energy estimates carefully designed for this reformulated system.

Global solutions to an initial boundary problem for the compressible 3D MHD equations with Navier-slip and perfectly conducting boundary conditions in exterior domains

An initial boundary value problem for compressible magnetohydrodynamics (MHD) is considered on an exterior domain (with the vanishing first Betti number) in in this paper. The global existence of smooth solutions near a given constant state for compressible MHD with the boundary conditions of Navier-slip for the velocity filed and perfect conduction for the magnetic field is established. Moreover the explicit decay rate is given. In particular, the results obtained in this paper also imply the global existence of classical solutions for the full compressible Navier–Stokes equations with Navier-slip boundary conditions on exterior domains in three dimensions, which was not available in literature prior to the work in this paper, to the best of knowledge of the authors’.

Optimal large-time behavior of solutions to the full compressible Navier–Stokes equations with Large initial data

Optimal large-time behavior of solutions to the full compressible Navier–Stokes equations with Large initial data

Energy conservation law for weak solutions of the full compressible Navier-Stokes equations

We consider a sufficient condition for the energy conservation law of a weak solution for the full compressible Navier-Stokes equations on the torus. We prove that a weak solution constructed by Feireisl with certain integrability conditions conserves the energy. Our assumption relaxes the regularity condition for the density compared with existing results.

Lagrange Multiplier-Based Optimal Control Technique for Streak Attenuation in High-Speed Boundary Layers

High-amplitude freestream turbulence and surface roughness elements can excite a laminar boundary-layer flow sufficiently to cause streamwise-oriented vortices to develop. These vortices resemble elongated streaks having alternate spanwise variations of the streamwise velocity. Downstream, the vortices “wobble” through an inviscid secondary instability mechanism and, ultimately, transition to turbulence. We formulate an optimal control algorithm to suppress the growth rate of the streamwise vortex system. Considering a high-Reynolds-number asymptotic framework, we reduce the full compressible Navier–Stokes equations to the nonlinear compressible boundary-region equations. We then implement the method of Lagrange multipliers via an appropriate transformation of the original constrained optimization problem into an unconstrained form to obtain the disturbance equations in the form of the adjoint compressible boundary-region equations (ACBREs) and corresponding optimality conditions. Numerical solutions of the ACBRE approach for high-supersonic and hypersonic flows reveal a significant reduction in the kinetic energy and wall shear stress for all considered configurations. We present contour plots to demonstrate the qualitative effect of increased control iterations. Our results indicate that the primary vortex instabilities gradually flatten in the spanwise direction thanks to the ACBRE algorithm.

Global Well-Posedness for the Full Compressible Navier-Stokes Equations

We are concerned with the Cauchy problem regarding the full compressible Navier-Stokes equations in ℝd (d = 2, 3). By exploiting the intrinsic structure of the equations and using harmonic analysis tools (especially the Littlewood-Paley theory), we prove the global solutions to this system with small initial data restricted in the Sobolev spaces. Moreover, the initial temperature may vanish at infinity.

Low Mach number limit of Navier-Stokes equations with large temperature variations in bounded domains

The low Mach number limit of full compressible Navier-Stokes equations with large temperature variations is verified rigorously in a three-dimensional bounded domain. Weighted uniform estimates of the solutions are derived in a time interval which is independent of the Mach number, in particular, for the high-order derivatives, when the initial data are well-prepared only in the sense of L2-norm. It can be viewed as the first result on the low Mach number limit of full Navier-Stokes equations with large temperature variations in bounded domains. The methods in the paper also apply to other fluid models with large temperature variations.

Periodic Perturbations of a Composite Wave of Two Viscous Shocks for 1-d Full Compressible Navier--Stokes Equations

This paper is concerned with the asymptotic stability of a composite wave of two viscous shocks under spatially periodic perturbations for the 1-D full compressible Navier-Stokes equations. It is proved that as time increases, the solution approaches the background composite wave with a shift for each shock, where the shifts can be uniquely determined if both the periodic perturbations and strengths of two shocks are small. The key of the proof is to construct a suitable ansatz such that the anti-derivative method works.

On Continuation Criteria for the Full Compressible Navier-Stokes Equations in Lorentz Spaces

In this paper, we derive several new sufficient conditions of the non-breakdown of strong solutions for both the 3D heat-conducting compressible Navier-Stokes system and nonhomogeneous incompressible Navier-Stokes equations. First, it is shown that there exists a positive constant ε such that the solution (ρ, u, θ) to the full compressible Navier-Stokes equations can be extended beyond t = T provided that one of the following two conditions holds: To the best of our knowledge, this is the first continuation theorem allowing the time direction to be in Lorentz spaces for the compressible fluid. Second, we establish some blow-up criteria in anisotropic Lebesgue spaces for the finite blow-up time T*: Third, without the condition on ρ in (0.1) and (0.3), the results also hold for the 3D non-homogeneous incompressible Navier-Stokes equations. The appearance of a vacuum in these systems could be allowed.

Global Strong Solution to the Two-Dimensional Full Compressible Navier–Stokes Equations with Large Viscosity

Global Strong Solution to the Two-Dimensional Full Compressible Navier–Stokes Equations with Large Viscosity

Low Mach Number Limit of Full Compressible Navier–Stokes Equations with Revised Maxwell Law

In this paper, we study the low Mach number limit of the full compressible Navier–Stokes equations with revised Maxwell law in \(\mathbb {R}^3\). By applying the uniform estimates of the error system, we prove that the solutions of the full compressible Navier–Stokes equations with time relaxation converge to that of the incompressible Navier–Stokes equations as the Mach number tends to zero. Moreover, the convergence rates are also obtained.

On globally large smooth solutions of full compressible Navier–Stokes equations with moving boundary and temperature-dependent heat-conductivity

The global existence and uniqueness of smooth solution to the vacuum free boundary problem of full compressible Navier–Stokes equations with large initial data and radial symmetry is established in this paper, when the fluid connects to vacuum continuously. The main difficulty lies in the fact that, the system is strongly nonlinear with unknown boundary variables, and degenerate near the free boundary separating the fluid and vacuum, thus general theory does not apply to this case. To overcome this trouble, we establish the point-wise estimates on the upper and lower bounds of the deformation variable ηx, the refined estimate for the temperature, and also the uniform-in-time weighted energy estimates for the solutions with high regularity, by careful analysis. Moreover, the expanding rate of the free interface is also proved. The main ingredient of this paper is that the high regularity of the solution is always up to the free boundary. Previous results are only for weak solutions, or the case that the fluids connect to the vacuum with a jump. The assumption on the heat conductivity coefficient κ≈1+θq,q≥2 is also improved to be q>0.