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Numerical Methods for the “Parabolized” Navier–Stokes Equations
Published in Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar, Computational Fluid Mechanics and Heat Transfer, 2020
Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar
The complete Navier–Stokes equations are an obvious set of equations that can be used to solve the flow fields in Figure 8.1 as well as all other viscous flow fields for which the boundary-layer equations are not applicable. In some cases, they are the only equations that apply. Unfortunately, the Navier–Stokes equations are very difficult to solve in their complete form. In general, a very large amount of computer time and storage is necessary to obtain a solution with these equations. This is particularly true for the compressible Navier–Stokes equations, which are a mixed set of elliptic–parabolic equations for a steady flow and a mixed set of hyperbolic–parabolic equations for an unsteady flow. The time-dependent solution procedure is normally used when a steady-flow field is computed. That is, the unsteady Navier–Stokes equations are integrated in time until a steady-state solution is achieved. Thus, for a three-dimensional (3-D) flow field, a four-dimensional (4-D) (three space, one time) problem must be solved when the compressible Navier–Stokes equations are employed. Methods for solving the complete Navier–Stokes equations are discussed in Chapter 9.
Convection
Published in Greg F. Naterer, Advanced Heat Transfer, 2018
The Euler equations represent a reduced form of the Navier–Stokes equations under the idealized conditions of inviscid (frictionless) flow. An inviscid fluid refers to an idealized fluid with a zero viscosity. Viscous effects are significant near a solid boundary but often can be neglected at a sufficient distance away from the surface. The Euler equations are obtained by neglecting the viscous and body force terms in the Navier–Stokes equations, ρ∂ui∂t+ρuj∂ui∂xj=−∂p∂xi
Vector Calculus
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
The Navier-Stokes equations pose challenging problems purely in mathematical sense, as shown by a U.S.$1 million prize offered by the Clay Mathematics Institute for proving or disproving existence of smooth solution in three dimensions. Books by Doering and Gibbon [1], Galdi [2], and Temam [6] provide detailed account; interested readers should consult them to gain deeper knowledge about the Navier-Stokes equations.
Analysis of skid resistance and braking distance of aircraft tire landing on grooved runway pavement
Published in International Journal of Pavement Engineering, 2022
In the model, the Coupled Eulerian-Lagrangian method was used to capture the performance of water and to furthermore calculate the friction coefficient and braking distance. The Coupled Eulerian-Lagrangian (CEL) method has been utilised to capture the response of water and air when water and air flow underneath or alongside the tire, which has been proved to be reliable in modelling tire-water-pavement interaction in previous research (Cho et al. 2006, Ding and Wang 2018). In the CEL method, the fluid flow was described using the Eulerian method and the fluid domain was meshed with Eulerian elements. The Euler equations to describe the behaviour of fluid flow are derived from the widely used Navier-Stokes equations, assuming the fluids as Newtonian fluid (viscosity is constant) and the effects of viscosity and temperature change of water are negligible. To fully describe the fluid behaviour, the Mie-Gruneisen form of Equation of State (EOS) was introduced to provide a connection between pressure and fluid density, so that the detailed fluid properties could be defined through the EOS. In the CEL method, the material is simulated to flow through the computed Eulerian mesh, and volume of fraction (VOF) is introduced to track the flow boundary between air and water (ABAQUS 2014).
Optimization of structure parameters in a coal pyrolysis filtration system based on CFD and quadratic regression orthogonal combination and a genetic algorithm
Published in Engineering Applications of Computational Fluid Mechanics, 2021
Jinjin Liu, Tong Zhao, Kai Liu, Bo Sun, Chuanxin Bai
After the fluid domain was meshed with GAMBIT, the velocity uniformity of the filter tubes in the filtration system was calculated with CFD in the ANSYS environment. The fluid flow in Fluent accords with the conservation of mass and energy. ANSYS is industry-leading fluid simulation software known for its advanced physics modeling capabilities and industry leading accuracy. It is not only widely used to solve the Navier–Stokes equations, but also problems in fluid flow (John et al., 1985). The finite volume method was used to make a spatial discretization in all fluid domains of the filtration system. The computational domain was divided into a series of non-repetitive control bodies, with a control body around each grid point. The differential equations to be solved could be integrated for each control volume. Thus, discrete equations were obtained and the corresponding parameters recorded in Table 3.
Numerical analysis on the film cooling of leading edge with laid back holes to determine the optimal angle for the holes
Published in Australian Journal of Mechanical Engineering, 2020
Akbar Mohammadi-Ahmar, Ali Solati, Jalal Ghasemi
Analytical solution of the Navier–Stokes equations which can describe any Newtonian fluid behaviour is not possible for a wide range of engineering problems; thus, researchers have applied numerical methods and repeated procedures to obtain the solution. Solving the Reynolds averaged Navier–Stokes equations (RANS) is a method that is commonly applied to solve general cases and particularly engineering applications of turbulent flow. RANS equations are obtained by total or time averaging the Navier–Stokes equations in order to achieve a set of transport equations and solving the averaged momentum equation; moreover, the effects of all scales of flow and heat transfer field are modelled in this method. In the RANS equations, the required computational time is considerably reduced due to the nature of the time averaging, but the instantaneous behaviour of the flow turbulence scales is not considered.