Compressible Navier Stokes System Research Articles

Convergence and error estimates of a penalization finite volume method for the compressible Navier–Stokes system

Abstract In numerical simulations a smooth domain occupied by a fluid has to be approximated by a computational domain that typically does not coincide with a physical domain. Consequently, in order to study convergence and error estimates of a numerical method domain-related discretization errors, the so-called variational crimes, need to be taken into account. In this paper we apply the penalty approach to control domain-related discretization errors. We embed the physical domain into a large enough cubed domain and study the convergence of a finite volume method for the corresponding domain-penalized problem. We show that numerical solutions of the penalized problem converge to a generalized, the so-called dissipative weak, solution of the original problem. If a strong solution exists, the dissipative weak solution emanating from the same initial data coincides with the strong solution. In this case, we apply a novel tool of the relative energy and derive the error estimates between the numerical solution and the strong solution. Extensive numerical experiments that confirm theoretical results are presented.

Tollmien–Schlichting waves in the subsonic regime

AbstractThe Tollmien–Schlichting (T‐S) waves play a key role in the early stages of boundary layer transition. In a breakthrough work, Grenier, Guo, and Nguyen gave the first rigorous construction of the T‐S waves of temporal mode for the incompressible fluid. Yang and Zhang recently made an important contribution by constructing the compressible T‐S waves of temporal mode for certain boundary layer profiles with Mach number . In this paper, we construct the T‐S waves of both temporal mode and spatial mode to the linearized compressible Navier–Stokes system around the boundary layer flow in the whole subsonic regime , including the Blasius profile. Our approach is based on a novel iteration scheme between the quasi‐incompressible and quasi‐compressible systems, with a key ingredient being the solution of an Orr–Sommerfeld‐type equation using a new Airy–Airy–Rayleigh iteration instead of Rayleigh–Airy iteration introduced by Grenier, Guo, and Nguyen. We believe that the method developed in this work can be applied in solving other related problems for subsonic flows.

Inviscid limit for the compressible Navier-Stokes equations with density dependent viscosity

We consider the compressible Navier-Stokes system describing the motion of a barotropic fluid with density dependent viscosity confined in a three-dimensional bounded domain Ω. We show the convergence of the weak solution to the compressible Navier-Stokes system to the strong solution to the compressible Euler system when the viscosity and the damping coefficients tend to zero.

Local well-posedness of the 1d compressible Navier–Stokes system with rough data

Local well-posedness of the 1d compressible Navier–Stokes system with rough data

Some controllability results for linearized compressible Navier-Stokes system with Maxwell's law

In this article, we study the control aspects of the one-dimensional compressible Navier-Stokes equations with Maxwell's law linearized around a constant steady state with zero velocity. We consider the linearized system with Dirichlet boundary conditions and with interior controls. We prove that the system is not null controllable at any time using localized controls in density and stress equations and even everywhere control in the velocity equation. However, we show that the system is null controllable at any time if the control acting in density or stress equation is everywhere in the domain. This is the best possible null controllability result obtained for this system. Next, we show that the system is approximately controllable at large time using localized controls. Thus, our results give a complete understanding of the system in this direction.

Time decay of solutions for compressible isentropic non-Newtonian fluids

In this paper, we consider the Cauchy problem of a compressible Navier–Stokes system of Eills-type non-Newtonian fluids. We investigate the time decay properties of classical solutions for the compressible non-Newtonian fluid equations. More specifically, we construct a new linearized system in terms of a combination of the solutions, and then we investigate the long-time behavior of the Cauchy problem for the three-dimensional isentropic compressible Eills-type non-Newtonian fluids with an initial perturbation.

A semi-implicit finite volume scheme for dissipative measure-valued solutions to the barotropic Euler system

A semi-implicit in time, entropy stable finite volume scheme for the compressible barotropic Euler system is designed and analyzed and its weak convergence to a dissipative measure-valued (DMV) solution [Feireisl et al., Calc. Var. Part. Differ. Equ. 55 (2016) 141] of the Euler system is shown. The entropy stability is achieved by introducing a shifted velocity in the convective fluxes of the mass and momentum balances, provided some CFL-like condition is satisfied to ensure stability. A consistency analysis is performed in the spirit of the Lax’s equivalence theorem under some physically reasonable boundedness assumptions. The concept of Ƙ-convergence [Feireisl et al., IMA J. Numer. Anal. 40 (2020) 2227–2255] is used in order to obtain some strong convergence results, which are then illustrated via rigorous numerical case studies. The convergence of the scheme to a DMV solution, a weak solution and a strong solution of the Euler system using the weak–strong uniqueness principle and relative entropy are presented.

Optimal decay rate of the incompressible Navier–Stokes–Maxwell system with Ohm’s law

This work is to establish optimal time decay rate of global classical solutions to the two-fluid incompressible Navier–Stokes–Maxwell system with Ohm’s law and the incompressible Navier–Stokes–Maxwell system with Ohm’s law in R3. The results show that the optimal time decay rate for the velocity of these two systems achieves the same as that of the incompressible Navier–Stokes equation. In other words, Ohm’s law can offset the negative effects brought by the Maxwell equation on the decay rate of solutions. This is interesting compared to the compressible Navier–Stokes system, whose time decay rate decreases when it is coupled with the self-consistent Maxwell system.

Hard congestion limit of the dissipative Aw-Rascle system with a polynomial offset function

We study the Aw-Rascle system in a one-dimensional domain with periodic boundary conditions, where the offset function is replaced by the gradient of the function ρnγ, where γ→∞. The resulting system resembles the 1D pressureless compressible Navier-Stokes system with a vanishing viscosity coefficient in the momentum equation and can be used to model traffic and suspension flows. We first prove the existence of a unique global-in-time classical solution for fixed n. Unlike the previous result for this system, we obtain global existence without needing to add any approximation terms to the system. This is by virtue of a n-uniform lower bound on the density which is attained by carrying out a maximum-principle argument on a suitable potential, Wn=ρn−1∂xwn. Then we prove the convergence to a weak solution of a hybrid free-congested system as n→∞, which is known as the hard-congestion model.

Convergence of a stochastic collocation finite volume method for the compressible Navier–Stokes system

Convergence of a stochastic collocation finite volume method for the compressible Navier–Stokes system

Interface behavior and decay rates of compressible Navier-Stokes system with density-dependent viscosity and a vacuum

Interface behavior and decay rates of compressible Navier-Stokes system with density-dependent viscosity and a vacuum

On initial boundary value problems for the compressible Navier–Stokes system with temperature dependent heat conductivity

On initial boundary value problems for the compressible Navier–Stokes system with temperature dependent heat conductivity

Global well-posedness of the 1d compressible Navier–Stokes system with rough data

In this paper, we study the global well-posedness problem for the 1d compressible Navier–Stokes systems (cNSE) in gas dynamics with rough initial data. First, Liu and Yu (2022) [30] established the global well-posedness theory for the 1d isentropic cNSE with initial velocity data in BV space. Then, it was extended to the 1d full cNSE with initial velocity and temperature data in BV space by Wang et al. (2022) [31]. We improve the global well-posedness result of Liu and Yu with initial velocity data in W2γ,1 space; and of Wang–Yu–Zhang with initial velocity data in L2∩W2γ,1 space and initial data of temperature in W˙−23,65∩W˙2γ−1,1 for any γ>0arbitrarily small. Our essential ideas are based on establishing various “end-point” smoothing estimates for the 1d parabolic equation.

The initial value problem for the equations of motion of fractional compressible viscous fluids

In this paper we consider the initial value problem to the fractional compressible isentropic generalized Navier-Stokes equations for viscous fluids with one Levy diffusion process in which the viscosity term appeared in the fluid equations is described by the nonlocal fractional Laplace operator. We give one detailed spectrum analysis on a linearized operator and the decay law in time of the solution semigroup for the linearized fractional compressible isentropic generalized Navier-Stokes equations around a constant state by the Fourier analysis technique, which is shown that the order of the fractional derivatives plays a key role in the analysis so that the spectrum structure involved here is more complex than that of the classical compressible Navier-Stokes system. Based on this and the elaborate energy method, the global-in-time existence and one optimal decay rate in time of the smooth solution are obtained under the assumption that the initial data are given in a small neighborhood of a constant state.

Efficient GPU parallelization of adaptive mesh refinement technique for high-order compressible solver with immersed boundary

A new, highly parallelized, adaptive mesh refinement (AMR) library, equipped with an accurate immersed boundary (IB) method for solving the compressible Navier–Stokes system is presented. The library, named ADAM, is designed to efficiently exploit modern exascale GPU-accelerated supercomputers and it is implemented with a highly modular structure in order to make easy to leverage it for a wide range of CFD applications. The structured Cartesian grids at the basis of (octree) AMR approach allows to implement very high order accurate models retaining a low computational cost and high level of parallelization. The accurate IB method coupled with efficient AMR technique enables the simulation flows with complex (possibly moving and deforming) geometries. The library is applied to the simulation of a strong shock diffraction over a solid sphere and a detailed discussion concerning the physical results and the parallel performance obtained is presented.

Global smooth solution of compressible Navier–Stokes system with temperature-dependent transport coefficients and pressure condition

Global smooth solution of compressible Navier–Stokes system with temperature-dependent transport coefficients and pressure condition

Low Mach number limit of the global solution to the compressible Navier–Stokes system for large data in the critical Besov space

In this paper, we consider the compressible Navier–Stokes system around the constant equilibrium states and prove the existence of a unique global solution for arbitrarily large initial data in the scaling critical Besov space provided that the Mach number is sufficiently small, and the incompressible part of the initial velocity generates the global solution of the incompressible Navier–Stokes equation. Moreover, we consider the low Mach number limit and show that the compressible solution converges to the solution of the incompressible Navier–Stokes equation in some space time norms.

Incompressible limit for the free surface Navier-Stokes system

We establish uniform regularity estimates with respect to the Mach number for the three-dimensional free surface compressible Navier-Stokes system in the case of slightly well-prepared initial data in the sense that the acoustic components like the divergence of the velocity field are of size $$\sqrt{\varepsilon }$$ , $$\varepsilon $$ being the Mach number. These estimates allow us to justify the convergence towards the free surface incompressible Navier-Stokes system in the low Mach number limit. One of the main difficulties is the control of the regularity of the surface in presence of boundary layers with fast oscillations.

URANOS: A GPU accelerated Navier-Stokes solver for compressible wall-bounded flows

URANOS, a massively parallel GPU-accelerated compressible flow solver for high-fidelity modeling of compressible wall flows, is presented. URANOS is based on modern high-fidelity and high-resolution discretization strategies for time-accurate compressible flow predictions. The solver provides six different convective scheme implementations, a cutting-edge method for viscous terms treatment, and three different frameworks for turbulence modeling (DNS, LES, and innovative WMLES), while utilizing a high-order finite-difference approach, ranging from second to sixth order spatial accuracy. This paper details all of these numerical models and implementation issues. Computationally, URANOS combines multiple three-dimensional MPI parallelization strategies with the open standard, OpenACC, for machine wide, on node, and on GPU parallelism, tailoring the software to match the state-of-the-art HPC computing facilities. Special attention is given to the GPU porting and data management, detailing the solver scaling properties and performance for multi-node/-GPU systems on three distinct architectures. In these experiments the different MPI strategies are compared for both GPU-aware MPI/GPU (Host-based MPI) and results are provided proving, consistently, that the GPU-aware MPI outperforms the GPU (Host-based MPI) approach. Furthermore, the GPU version is compared to the CPU only demonstrating over three times speed-up for a node-to-node comparison, even at large scales, with an efficiency of about 80% while using 1024 GPUs. Finally, the manuscript presents several validation benchmarks, from simple academic comparisons to turbulent channel and boundary layer configurations. URANOS is compared with the best DNS data available in the literature across various Mach numbers, ranging from almost incompressible conditions to hypersonic regimes, illustrating the solver's capability in treating a wide range of complex wall-flows problems using DNS, wall-resolved, and wall-modeled LES approaches. Thus, using OpenACC as a paradigm, URANOS provides a straightforward and efficient platform that can efficiently exploit the most modern GPU-accelerated computing architectures in a totally open-source and non-vendor-specific way. The solver represents a framework where the CFD community can explore the limits of modeling, test new models/methods all across a broad range of Mach and Reynolds numbers flows. The solver is released under a BSD license. Program summaryProgram title: Unsteady Robust All-around Navier-StOkes Solver (URANOS)CPC Library link to program files:https://doi.org/10.17632/pw5hshn9k6.1Developer's repository link:https://gitlab.com/fralusa/uranos_gpuLicensing provisions: BSD License 2.0Programming language: Fortran 90, OpenACC, MPINature of problem: Solving the compressible Navier-Stokes system of equations in a three-dimensional Cartesian framework from low to high Mach and Reynolds numbers conditions.Solution method: Convective terms are treated with several high-order high-resolution shock-capturing schemes (WENO/TENO) in hybrid and non-hybrid versions. The system dynamics is advanced in time with a three-stage, third-order total variation diminishing Runge-Kutta method. Three-dimensional MPI parallelization is implemented in several optimized arrangements and coupling OpenACC directives in a genuinely multi-GPU logic.

Global weak solutions to the compressible Cucker–Smale–Navier–Stokes system in a bounded domain

A coupled kinetic-fluid model is investigated, which describes the dynamic behavior of an ensemble of Cucker–Smale flocking particles interacting with a viscous fluid in a three-dimensional bounded domain. This system consists of a kinetic Cucker–Smale equation and a compressible Navier–Stokes system with nonhomogeneous boundary conditions. The global existence of weak solutions to this system with adiabatic coefficient γ>3/2 is established.