Multi-dimensional Compressible Flows Research Articles

Global strong solutions for the multi-dimensional compressible viscoelastic flows with general pressure law

In this paper, we mainly focus on the compressible viscoelastic flows of Oldroyd type with the general pressure law, with one of the non-Newtonian fluids exhibiting the elastic behavior. For the viscoelastic flows of Oldroyd type with the general pressure law, P′(ρ̄)+α>0, with α > 0 being the elasticity coefficient of the fluid, we prove the global existence and uniqueness of the strong solution in the critical Besov spaces when the initial data u⃗0 and the low frequency part of ρ0, τ0 are small enough compared to the viscosity coefficients. In particular, when the viscosity is large, the part of the initial data can be large. The proof we display here does not need any compatible conditions. In addition, we also obtain the optimal decay rates of the solution in the Besov spaces.

Global existence in critical spaces for non Newtonian compressible viscoelastic flows

We are interested in the multi-dimensional compressible viscoelastic flows of Oldroyd type, which is one of non-Newtonian fluids exhibiting the elastic behavior. In order to capture the damping effect of the additional deformation tensor, to the best of our knowledge, the “div-curl” structural condition plays a key role in previous efforts. Our aim of this paper is to remove the structural condition and prove a global existence of strong solutions to compressible viscoelastic flows in critical spaces. In absence of compatible conditions, the new effective flux is introduced, which enables us to capture the dissipation arising from combination of density and deformation tensor. The partial dissipation in non-Newtonian compressible fluids, is weaker than that of classical Navier-Stokes equations.

The Lp-Lq decay estimate for the multidimensional compressible flow with free surface in the exterior domain

The aim of this paper is to develop the general Lp theory for the barotropic compressible Navier-Stokes equations with the free boundary condition in the exterior domain in RN (N≥3). By the spectral analysis, we obtain the classical Lp-Lq decay estimate for the linearized model problem (with variable coefficients) in view of the partial Lagrangian transformation.

A simplified approach for simulations of multidimensional compressible multicomponent flows: The grid-aligned ghost fluid method

In the present work the authors present a simplified formulation for the extension of a ghost fluid method in multidimensional space. In the proposed method, the Riemann problems at the interface are formulated along the grid rather than in a normal to the interface direction. The information that is required to construct these Riemann problems is acquired “on-the-fly” from the adjacent to the interface cells. With respect to existing multidimensional ghost fluid formulations, the method is computationally less expensive, as the procedures of determining ghost fluid regions, extending, interpolating and extrapolating variables and computing geometrical quantities are avoided. More importantly, it is markedly simple with respect to its implementation. By introducing the proposed formulation in a well-established front tracking framework we perform an extensive validation of the method and demonstrate that despite its simplicity it yields highly accurate results while remaining free of oscillations.

A Robust Rotated-Hybrid Riemann Scheme for Multidimensional Inviscid Compressible Flows

A robust algorithm is investigated for multidimensional inviscid compressible flows, based on rotated Riemann solver framework. Indeed, the proposed method combines the TV-HLL and Roe schemes. The upwind direction is imposed by the velocity-difference vector. Then, the TV-HLL solver is applied in the direction perpendicular to shocks in order to suppress carbuncle. In addition, the Roe solver is applied across shear layers to minimize the amount of dissipation. To assess the capabilities of the present method, numerous test problems (2D and 3D) are simulated.

Well-posedness in critical spaces for a multi-dimensional compressible viscous liquid-gas two-phase flow model

This paper is dedicated to the study of the Cauchy problem for a compressible viscous liquid-gas two-phase flow model in \begin{document}$\mathbb{R}^N\,(N≥2)$ \end{document} . We concentrate on the critical Besov spaces based on the \begin{document}$L^p$ \end{document} setting. We improve the range of Lebesgue exponent \begin{document}$p$ \end{document} , for which the system is locally well-posed, compared to [ 22 ]. Applying Lagrangian coordinates is the key to our statements, as it enables us to obtain the result by means of Banach fixed point theorem.

Interior local null controllability for multi-dimensional compressible flow near a constant state

We consider compressible flow with periodic boundary conditions. In a neighborhood of a state of constant density and nonzero velocity, we prove exact null controllability of the system with a control localized in space and acting only on the momentum equation.

The optimal convergence rates for the multi-dimensional compressible viscoelastic flows

In this paper, we are concerned with the multi-dimensional (N≥3) compressible viscoelastic flows in the whole space. We prove the optimal convergence rates of strong solutions to the system for the initial data close to a stable equilibrium state in critical Besov spaces. Our main ideas are based on the low-high frequency decomposition and the smooth effect of dissipative operator.

An extension of the TV-HLL scheme for multi-dimensional compressible flows

This paper investigates a very simple method to numerically approximate the solution of the multi-dimensional Riemann problem for gas dynamics, using the literal extension of the Toro Vazquez-Harten Lax Leer (TV-HLL) scheme as its basis. Indeed, the present scheme is obtained by following the Toro–Vazquez splitting, and using the HLL algorithm with modified wave speeds for the pressure system. An essential feature of the TV-HLL scheme is its simplicity and its accuracy in computing multi-dimensional flows. The proposed scheme is carefully designed to simplify its eventual numerical implementation. It has been applied to numerical tests and its performances are demonstrated for some two-dimensional and three-dimensional test problems.

An accurate multidimensional limiter on quadtree grids for shallow water flow simulation

ABSTRACTIn hydraulic research, numerical modelling of complex flows is essential for managing water risks. High-resolution finite volume schemes have become popular for shallow water flow modelling due to their mass and momentum balance characteristics and their ability to capture shocks. These methods use slope limiters to suppress numerical oscillations near discontinuities. However, one-dimensional limiters do not assure numerical accuracy in multidimensional applications, occasionally leading to excessive or insufficient numerical dissipation. For this reason, a multidimensional limiting process (MLP) was developed for oscillation control in multidimensional compressible flows. In this paper, we implement MLP on adaptive quadtree grids for shallow water flow simulations and compare MLP performance with simulations using conventional limiters. Four simulation cases show that MLP outperforms conventional limiters, and yield more accurate and stable solutions on adaptive quadtree grids. The capability of MLP for oscillation control is more noticeable on quadtree than on uniform grids.

Numerical Solution of Multidimensional Compressible Flow by High Order Nodal Discontinuous Galerkin Method

In this paper we present a robust, high order method for numerical solution of multidimensional compressible inviscid flow equations. Our scheme is based on Nodal Discontinuous Galerkin Finite Element Method (NDG-FEM). This method utilizes the favorable features of Finite Volume Method (FVM) and Finite Element Method (FEM). In this method, space discretization is carried out by finite element discontinuous approximations. The resulting semi discrete differential equations were solved using explicit Runge-Kutta (ERK) method. In order to compute fluxes at element interfaces, we have used Roe Approximate scheme. In this article, we demonstrate the use of exponential filter to remove Gibbs oscillations near the shock waves. Numerical predictions for two dimensional compressible fluid flows are presented here. The solution was obtained with overall order of accuracy of 3. The numerical results obtained are compared with experimental and finite volume method results.

On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain

We study the global existence of weak solutions to a multi-dimensional simplified Ericksen–Leslie system for compressible flows of nematic liquid crystals with large initial energy in a bounded domain Ω⊂RN, where N=2 or 3. By exploiting a maximum principle, Nirenbergʼs interpolation inequality and a smallness condition imposed on the N-th component of initial direction field d0 to overcome the difficulties induced by the supercritical nonlinearity |∇d|2d in the equations of angular momentum, and then adapting a modified three-dimensional approximation scheme and the weak convergence arguments for the compressible Navier–Stokes equations, we establish the global existence of weak solutions to the initial-boundary problem with large initial energy and without any smallness condition on the initial density and velocity.

Well-Posedness for a Multidimensional Viscous Liquid-Gas Two-Phase Flow Model

The Cauchy problem of a multidimensional ($d\geqslant 2$) compressible viscous liquid-gas two-phase flow model is studied in this paper. We investigate the global existence and uniqueness of the strong solution for the initial data close to a stable equilibrium and the local-in-time existence and uniqueness of the solution with general initial data in the framework of Besov spaces. A continuation criterion is also obtained for the local solution.

Global existence for the multi-dimensional compressible viscoelastic flows

The global solutions in critical spaces to the multi-dimensional compressible viscoelastic flows are considered. The global existence of the Cauchy problem with initial data close to an equilibrium state is established in Besov spaces. Using uniform estimates for a hyperbolic–parabolic linear system with convection terms, we prove the global existence in the Besov space which is invariant with respect to the scaling of the associated equations. Several important estimates are achieved, including a smoothing effect on the velocity, and the L 1 -decay of the density and deformation gradient.

Time analyticity and backward uniqueness of weak solutions of the Navier–Stokes equations of multidimensional compressible flow

We prove that solutions of the Navier–Stokes equations of three-dimensional compressible flow, restricted to fluid-particle trajectories, can be extended as analytic functions of complex time. As consequences we derive backward uniqueness of solutions as well as sharp rates of smoothing for higher-order Lagrangean time derivatives. The solutions under consideration are in a reasonably broad regularity class corresponding to small-energy initial data with a small degree of regularity, the latter being required for conversion to the Lagrangean coordinate system in which the analysis is carried out.

Multi-dimensional limiting process for three-dimensional flow physics analyses

The present paper deals with an efficient and accurate limiting strategy for multi-dimensional compressible flows. The multi-dimensional limiting process (MLP) which was successfully proposed in two-dimensional case [K.H. Kim, C. Kim, Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows. Part II: Multi-dimensional limiting process, J. Comput. Phys. 208 (2) (2005) 570–615] is modified and refined for three-dimensional application. For computational efficiency and easy implementation, the formulation of MLP is newly derived and extended to three-dimensional case without assuming local gradient. Through various test cases and comparisons, it is observed that the newly developed MLP is quite effective in controlling numerical oscillation in multi-dimensional flows including both continuous and discontinuous regions. In addition, compared to conventional TVD approach, MLP combined with improved flux functions does provide remarkable increase in accuracy, convergence and robustness in steady and unsteady three-dimensional compressible flows.

Wavenumber-extended high-order oscillation control finite volume schemes for multi-dimensional aeroacoustic computations

A new numerical method toward accurate and efficient aeroacoustic computations of multi-dimensional compressible flows has been developed. The core idea of the developed scheme is to unite the advantages of the wavenumber-extended optimized scheme and M-AUSMPW+/MLP schemes by predicting a physical distribution of flow variables more accurately in multi-space dimensions. The wavenumber-extended optimization procedure for the finite volume approach based on the conservative requirement is newly proposed for accuracy enhancement, which is required to capture the acoustic portion of the solution in the smooth region. Furthermore, the new distinguishing mechanism which is based on the Gibbs phenomenon in discontinuity, between continuous and discontinuous regions is introduced to eliminate the excessive numerical dissipation in the continuous region by the restricted application of MLP according to the decision of the distinguishing function. To investigate the effectiveness of the developed method, a sequence of benchmark simulations such as spherical wave propagation, nonlinear wave propagation, shock tube problem and vortex preservation test problem are executed. Also, throughout more realistic shock–vortex interaction and muzzle blast flow problems, the utility of the new method for aeroacoustic applications is verified by comparing with the previous numerical or experimental results.

Application of AUSM schemes to multi-dimensional compressible two-phase flow problems

This paper dicusses the extension of AUSM-type schemes to the solution of multidimensional compressible two-phase flow problems. We consider their first and second order approximations and several multidimensional problem have been analysed in order to illustrate the behaviour and performance of the schemes. Both two-fluid mixtures of water and air and phase change processes between water and steam have been considered. Thermodynamic variables have been calculated taking into account either analytical equations of state (water and air mixtures) or tabulated equations of state (water and steam mixtures). Non-conservative terms in the two-fluid system of equations have been discretized centrally. In addition, gravity forces, interfacial friction, interfacial heat and mass transfer, etc., have been taken into account using a point-wise evaluation.

Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows: Part II: Multi-dimensional limiting process

Through the analysis of conventional TVD limiters, a new multi-dimensional limiting function is derived for an oscillation control in multi-dimensional flows. And, multi-dimensional limiting process (MLP) is developed with the multi-dimensional limiting function. The major advantage of MLP is to prevent oscillations across a multi-dimensional discontinuity, and it is readily compatible with more than third order spatial interpolation. Moreover, compared with other higher order interpolation schemes such as ENO type schemes, MLP shows a good convergence characteristic in a steady problem and it is very simple to be implemented. In the present paper, third and fifth order interpolation schemes with MLP, named MLP3 and MLP5, are developed and tested for several real applications. Through extensive numerous test cases including an oblique stationary contact discontinuity, an expansion fan, a vortex flow, a shock wave/vortex interaction and a viscous shock tube problem, it is verified that MLP combined with M-AUSMPW+ numerical flux substantially improves accuracy, efficiency and robustness both in continuous and discontinuous flows. By extending the current approach to three-dimensional flows, MLP is expected to reduce computational cost and enhance accuracy even further.

Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows: Part I: Spatial discretization

The present papers deal with numerical methods toward the accurate and efficient computations of multi-dimensional steady/unsteady compressible flows. In Part I, a new spatial discretization technique is introduced to reduce excessive numerical dissipation in a non-flow-aligned grid system. Through the analysis of TVD limiters, a criterion is proposed to predict cell-interface states accurately both in smooth region and in discontinuous region. According to the criterion, a new way of re-evaluating the cell-interface convective flux in AUSM-type methods is developed. The resultant flux reduces numerical dissipation remarkably in multi-dimensional flows. Also, the monotonicity of AUSM-type methods is achieved by modifying the pressure splitting function directly based on the governing equations and the detection of sonic transition point with respect to a cell-interface. It is noted that the newly formulated AUSM-type flux for Multi-dimensional flows, named M-AUSMPW+, possesses many improved properties in term of accuracy, computational efficiency, monotonicity and grid independency. Through numerous test cases from contact and shock discontinuities, vortex flow, shock wave/boundary-layer interaction to viscous shock tube problems, M-AUSMPW+ proves to be efficient and about twice more accurate than conventional upwind schemes. The three-dimensional implementation of M-AUSMPW+ is expected to provide accuracy and efficiency improvement furthermore.