Steady Compressible Navier-Stokes Equations Research Articles

Asymptotic expansion for 1D steady nonuniform enthalpy compressible supersonic flow at high Reynolds number

The exact solution of one‐dimensional (1D) steady compressible Navier–Stokes (N‐S) equations at high Reynolds number has not been given yet under nonuniform total enthalpy when the Prandtl number ( ) is not equal to 0.75 since Becker's work in 1922. In this paper, we give an asymptotic expansion of the solution of the above equations at high Reynolds number by using the method of matched asymptotic expansions and prove convergence of the asymptotic solution under some assumptions. By analyzing the dimensionless form of one‐dimensional steady compressible Navier–Stokes equations, we find that if there are extreme points inside the boundary layer, the number of extreme points of velocity is at most one more than that of total enthalpy and the extreme points of each flow variables are different from each other. Based on the second‐order asymptotic expansion solution, we show that there are extreme points inside the thin boundary layer under some special conditions. Examples are given to verify theoretical analysis. The present asymptotic expansion solution is valuable for verifying the efficiency of high‐order numerical methods in flow simulation of high Reynolds number.

Exponential boundary-layer approximation space for solving the compressible laminar Navier-Stokes equations

In general, it needs to take about nearly 10 grid points inside a wall boundary layer for low accuracy order methods to get satisfactory wall-normal gradient related results, such as friction coefficient and wall heat flux. If there exist extreme points inside the boundary layer, this situation becomes even worse. In this work, with the help of the analytic solution of the one-dimensional steady compressible Navier-Stokes equations under some restrictions, we show that the flow variables actually vary in the form of an exponential function instead of a polynomial one inside the boundary layer. Then, we propose an exponential space for approximating the solutions inside the boundary layer and numerically implementing it in the frame of direct discontinuous Galerkin (DDG) method. We show that the DDG methods based on the exponential boundary-layer space give much better numerical results for both conservative variables and wall-normal gradients than those with the standard polynomial space. Generally only 1–2 grid points inside the boundary layer are demanded to resolve the wall boundary layer to obtain satisfactory wall-normal gradients under the exponential space. Preliminary extension to two-dimensional laminar boundary-layer flow shows a similar performance of the proposed exponential boundary-layer space, exhibiting its potential applications in high dimensions.

Effect of Streamlined Nose Length on the Aerodynamic Performance of a 800 km/h Evacuated Tube Train

The aerodynamic resistance of a train running in the open air limits the maximum speed that can be attained by the train. For this reason, evacuated tube trains (ETT) are being considered as valid alternatives to be implemented in the future. The atmosphere in the tube, the so-called blocking ratio and the length of the streamlined nose are the key factors affecting the aerodynamic performances of these trains. In the present work, we investigate evacuated tube trains with different lengths of the streamline nose on the basis of computational fluid dynamics (CFD). The three-dimensional steady compressible Navier-Stokes equations are solved. The running speed of the ETT is 800 km/h and the blocking ratio is 0.2. Results show that with the increase of the streamlined nose length, the aerodynamic drag and lift forces of the head car decrease gradually, and the drag and lift forces of the middle car change slightly. For the tail car, the drag force decreases, whereas the absolute value of the lift force increases. At a speed of 800 km/h, a slight shock wave appears at the rear of the tail car, which affects the aerodynamic forces acting on the train.

A lower bound for the density of the steady Navier–Stokes equations

In this paper, we study the approximation system to the steady compressible Navier–Stokes equations. It is proved that the density has a positive lower bound. And we use the lower bound to study the distribution of the density for a family of elliptic equations. A further investigation gives a typical upper bound of the density, and its limit functions only fit the solutions with the velocity divergence free.

Numerical investigation and optimization of airfoil flow control using passive air-jet

In this paper, a numerical investigation of airfoil flow control using passive air jet is presented. The study of generate a passive jet from pressure side to suction side was conducted in order to improve the airfoil characteristics. Such characteristics include boundary layer separation and stall inception. The numerical simulations conducted using CFDRC software. Different structured and unstructured finite volume technique is used to solve the steady compressible Navier—Stokes equations. The code results were validated with the experimental testing results for different turbulence model. Parametric study is performed for the location, slot width and angle of a synthetic jet on the suction side of a subsonic flow over a NACA 23012C airfoil. The maximum lift is achieved with the jet flow being normal to suction side surface but this come with penalty of the high drag. The maximum lift to drag ratio was accompanying with synthetic jet being located at 43% chord and 30° jet angle. Finally, optimum geometry of jet is obtained using simplex algorithm with initial shape obtained from the parametric study.

Weak solutions to the two-dimensional steady compressible Navier–Stokes equations

We study motions of the steady compressible viscous isothermal fluids in a bounded two dimensional domain governed by the Navier–Stokes equations. We obtain that there exists at least one weak solution to such a problem satisfying some integrability. The proof of this result is based on delicate estimates for the momentum equations and a bootstrapping argument.

Linear and non-linear high order accurate residual distribution schemes for the discretization of the steady compressible Navier–Stokes equations

A robust and high order accurate Residual Distribution (RD) scheme for the discretization of the steady Navier–Stokes equations is presented. The proposed method is very flexible: it is formulated for unstructured grids, regardless the shape of the elements and the number of spatial dimensions. A continuous approximation of the solution is adopted and standard Lagrangian shape functions are used to construct the discrete space, as in Finite Element methods. The traditional technique for designing RD schemes is adopted: evaluate, for any element, a total residual, split it into nodal residuals sent to the degrees of freedom of the element, solve the non-linear system that has been assembled and then iterate up to convergence. The main issue addressed by the paper is that the technique relies in depth on the continuity of the normal flux across the element boundaries: this is no longer true since the gradient of the state solution appears in the flux, hence continuity is lost when using standard finite element approximations. Naive solution methods lead to very poor accuracy. To cope with the fact that the normal component of the gradient of the numerical solution is discontinuous across the faces of the elements, a continuous approximation of the gradient of the numerical solution is recovered at each degree of freedom of the grid and then interpolated with the same shape functions used for the solution, preserving the optimal accuracy of the method. Linear and non-linear schemes are constructed, and their accuracy is tested with the method of the manufactured solutions. The numerical method is also used for the discretization of smooth and shocked laminar flows in two and three spatial dimensions.

Time-discretized steady compressible Navier–Stokes equations with inflow and outflow boundaries

The time-discretized steady compressible Navier–Stokes equations in n-dimensional bounded domains with the velocity specified only at the inflow boundary are considered. The existence and uniqueness of L p solutions are proved for p > n. For time-discretized steady flows, results of Kweon and Kellogg and of Kweon and Song are extended in a manner that allows for more general domains and for density-dependent viscosity coefficients. Moreover, we only require p > n which is a critical barrier in the previous works.

Analysis of semidiscretization of the compressible Navier–Stokes equations

The objective of this work is to investigate the time discretization of two-dimensional Navier–Stokes system with the slip boundary conditions. First, the existence of weak solutions for fixed time step △ t > 0 is presented and then the limit passage as △ t → 0 + is carried out. The proof is based on a new technique established for the steady Navier–Stokes equations by P.B. Mucha and M. Pokorný [P.B. Mucha, M. Pokorný, On a new approach to the issue of existence and regularity for the steady compressible Navier–Stokes equations, Nonlinearity 19 (8) (2006) 1747–1768] which enables to estimate the growth of L ∞ norm of the density when △ t goes to 0.

Existence of weak solutions to the three-dimensional steady compressible Navier–Stokes equations

We prove the existence of a spatially periodic weak solution to the steady compressible isentropic Navier–Stokes equations in R3 for any specific heat ratio γ>1. The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in some higher integrability of the density, and the method of weak convergence.

Preconditioning methods for discontinuous Galerkin solutions of the Navier–Stokes equations

A Newton–Krylov method is developed for the solution of the steady compressible Navier–Stokes equations using a discontinuous Galerkin (DG) discretization on unstructured meshes. Steady-state solutions are obtained using a Newton–Krylov approach where the linear system at each iteration is solved using a restarted GMRES algorithm. Several different preconditioners are examined to achieve fast convergence of the GMRES algorithm. An element Line-Jacobi preconditioner is presented which solves a block-tridiagonal system along lines of maximum coupling in the flow. An incomplete block-LU factorization (Block-ILU(0)) is also presented as a preconditioner, where the factorization is performed using a reordering of elements based upon the lines of maximum coupling. This reordering is shown to be superior to standard reordering techniques (Nested Dissection, One-way Dissection, Quotient Minimum Degree, Reverse Cuthill–Mckee) especially for viscous test cases. The Block-ILU(0) factorization is performed in-place and an algorithm is presented for the application of the linearization which reduces both the memory and CPU time over the traditional dual matrix storage format. Additionally, a linear p-multigrid preconditioner is also considered, where Block-Jacobi, Line-Jacobi and Block-ILU(0) are used as smoothers. The linear multigrid preconditioner is shown to significantly improve convergence in term of number of iterations and CPU time compared to a single-level Block-Jacobi or Line-Jacobi preconditioner. Similarly the linear multigrid preconditioner with Block-ILU smoothing is shown to reduce the number of linear iterations to achieve convergence over a single-level Block-ILU(0) preconditioner, though no appreciable improvement in CPU time is shown.

On a new approach to the issue of existence and regularity for the steady compressible Navier–Stokes equations

We prove the existence of weak solutions to the steady compressible Navier–Stokes equations in the isentropic regime. We consider the two-dimensional problem with the slip boundary conditions. The proof is based on a new idea of the construction of approximative solutions which guarantees that the sought density belongs to the L∞ (up to the boundary). Our approach improves the known methods, reducing the number of technical tricks.

A Remark on the Smoothness of Bounded Regions Filled with a Steady Compressible and Isentropic Fluid

For convenient adiabatic constants, existence of weak solutions to the steady compressible Navier-Stokes equations in isentropic regime in smooth bounded domains is well known. Here we present a way how to prove the same result when the bounded domains considered are Lipschitz.

On the Existence of Weak Solutions to the Steady Compressible Navier–Stokes Equations in Domains with Conical Outlets

We investigate the steady compressible Navier–Stokes system of equations in the isentropic regime in a domain with several conical outlets and with prescribed pressure drops. Existence of weak solutions is proved and estimate of these solutions with respect to the pressure drops is derived under the hypothesis γ > 3 where γ is the adiabatic constant.

Steady compressible Navier–Stokes equations in domains with non‐compact boundaries

AbstractWe consider the steady compressible Navier–Stokes equations of isentropic flow in three‐dimensional domains with several exits to infinity with prescribed pressure drops. On the one hand, when each exit is supposed to contain a cone inside, we shall construct bounded energy weak solution for adiabatic constant γ>3. On the other hand, when the exits do not open sufficiently rapidly, we shall prove a non‐existence result. Copyright © 2005 John Wiley & Sons, Ltd.

On the existence of weak solutions to the steady compressible Navier-Stokes equations when the density is not square integrable

On the existence of weak solutions to the steady compressible Navier-Stokes equations when the density is not square integrable

Physically Reasonable Solutions to Steady Compressible Navier-Stokes Equations in 2d Exterior Domains with Nonzero Velocity at Infinity

In a previous paper, the present authors studied the asymptotic behaviour of solutions to steady compressible Navier-Stokes equations in barotropic and isothermal regime with sufficiently small external data, in particular, in the whole plane. Here, the same problem is investigated in a two dimensional exterior domain with the prescribed velocity at infinity \( v_{\infty}\ne 0 \). Similar results as in Dutto and Novotny [DuNo] are found; in particular, it is proved that there exists a unique solution which possesses the similar pointwise decay and the wake structure as the fundamental Oseen tensor.

Steady compressible Navier-Stokes equations with large potential forces via a method of decomposition

We investigate the steady compressible Navier–Stokes equations near the equilibrium state v = 0, ρ = ρ0 (v the velocity, ρ the density) corresponding to a large potential force. We introduce a method of decomposition for such equations: the velocity field v is split into a non-homogeneous incompressible part u (div (ρ0u) = (0) and a compressible (irrotational) part ∇ϕ. In such a way, the original complicated mixed elliptic–hyperbolic system is split into several ‘standard’ equations: a Stokes-type system for u, a Poisson-type equation for ϕ and a transport equation for the perturbation of the density σ = ρ − ρ0. For ρ0 = const. (zero potential forces), the method coincides with the decomposition of Novotny and Padula [21]. To underline the advantages of the present approach, we give, as an example, a ‘simple’ proof of the existence of isothermal flows in bounded domains with no-slip boundary conditions. The approach is applicable, with some modifications, to more complicated geometries and to more complicated boundary conditions as we will show in forthcoming papers. © 1998 B.G. Teubner Stuttgart–John Wiley & Sons Ltd.

Coupled solution of the steady compressible Navier-Stokes equations and the k− ε turbulence equations with a multigrid method

A relaxation method for the steady turbulent compressible Navier-Stokes equations is developed. The flow equations are fully coupled to the turbulence equations. The principles of the method are illustrated for three different k− ε models. The relaxation method can be used in multigrid form. Multigrid results are given for one turbulence model on a flat plate test case.

On steady compressible Navier-Stokes equations in plane domains with corners

On steady compressible Navier-Stokes equations in plane domains with corners