Navier–Stokes equations

About: Navier–Stokes equations is a research topic. Over the lifetime, 18180 publications have been published within this topic receiving 552555 citations. The topic is also known as: Navier-Stokes equations.

A Stable Penalty Method for the Compressible Navier--Stokes Equations: III. Multidimensional Domain Decomposition Schemes

Abstract: This paper, concluding the trilogy, develops schemes for the stable solution of wave-dominated unsteady problems in general three-dimensional domains. The schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established. The complex computational domains are constructed by using nonoverlapping quadrilaterals in the two-dimensional case and hexahedrals in the three-dimensional space. To illustrate the ideas underlying the multidomain method, a stable scheme for the solution of the three-dimensional linear advection-diffusion equation in general curvilinear coordinates is developed. The analysis suggests a novel, yet simple, stable treatment of geometric singularities like edges and vertices. The theoretical results are supported by a two-dimensional implementation of the scheme. The main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible Navier--Stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented. Well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume, with special attention given to a hexahedral domain. The treatment includes a stable procedure for dealing with boundary conditions at a solid wall. The efficacy of the scheme for the compressible Navier--Stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions. The results are found to agree with the solution of the compressible boundary layer equations.

Relativistic third-order dissipative fluid dynamics from kinetic theory

Abstract: We present the derivation of a novel third-order hydrodynamic evolution equation for the shear stress tensor from kinetic theory. The Boltzmann equation with a relaxation time approximation for the collision term is solved iteratively using a Chapman-Enskog-like expansion to obtain the nonequilibrium phase-space distribution function. Subsequently, the evolution equation for the shear stress tensor is derived from its kinetic definition up to third order in gradients. We quantify the significance of the new derivation within a one-dimensional scaling expansion and demonstrate that the results obtained using the third-order viscous equations derived here provides a very good approximation to the exact solution of the Boltzmann equation in a relaxation time approximation. We also show that the time evolution of pressure anisotropy obtained using our equations is in better agreement with transport results than that obtained with an existing third-order calculation based on the second law of thermodynamics.

Stable transport equations for rarefied gases at high orders in the Knudsen number

Abstract: An approach is presented to derive transport equations for rarefied gases from the Boltzmann equation within higher orders of the Knudsen number. The method focuses on the order of magnitude of the moments of the phase density, and the order of accuracy of the transport equations, both measured in powers of the Knudsen number. The method is developed up to the third order, and it is shown that it yields the Euler equations at zeroth order, the Navier–Stokes–Fourier equations at first order, Grad’s 13 moment equations (with omission of a nonlinear term) at second order, and a regularization of these at third order. The method is discussed in detail, and compared with the classical methods of kinetic theory, i.e., Chapman–Enskog expansion and Grad moment method. The advantages of this method above the classical approaches are discussed conclusively. An important feature of the method presented is that the equations of any order are stable, other than in the Chapman–Enskog method, where the second and third ...

Explicit multigrid algorithm for quasi-three-dimensional viscous flows in turbomachinery

Abstract: A rapid quasi-three-dimensional analysis has been developed for blade-to-blade flows in turbomachinery. The analysis solves the unsteady Euler or thin-layer Navier-Stokes equations in a body-fitted coordinate system. It accounts for the effects of rotation, radius change, and stream surface thickness. The BaldwinLomax eddy viscosity model is used for turbulent flows. The equations are solved using a two-stage RungeKutta scheme made efficient by the use of vectorization, a variable time step, and a flux-based multigrid scheme, all of which are described. Results of a stability analysis are presented for the two-stage scheme. Results for a flat-plate model problem show the applicability of the method to axial, radial, and rotating geometries. Results for a centrifugal impeller and a radial diffuser show that the quasi-three-dimensional viscous analysis can be a practical design tool.

Well-Posedness of the Euler and Navier-Stokes Equations in the Lebesgue Spaces $L^p_s(\mathbb R^2)$

Abstract: In this paper we show that the Euler equation for incompressible fluids in R2 is well posed in the (vector-valued) Lebesgue spaces Lsp = (1 - ?)-s/2 Lp(R2) with s > 1 + 2/p, 1 < p < 8 and that the same is true of the Navier-Stokes equation uniformly in the viscosity ?.