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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
Hodograph Transformation and Limit Lines
Published in George Emanuel, Analytical Fluid Dynamics, 2017
Approximate analytical or numerical methods for solving the Euler equations are commonly encountered. For subsonic flow, for example, there is the RayleighJanzen method where an expansion in powers of M2 is used. For supersonic flow, there is the numerical method of characteristics. The situation for transonic flow, however, is more difficult. In this circumstance, part of the flow is subsonic and part is supersonic. From a mathematical viewpoint, the equations are sometimes elliptic and sometimes hyperbolic, this is referred to as a mixed system. The transonic equations, without the use of the hodograph transformation, are still nonlinear even when a small perturbation assumption is utilized. The hodograph equations, however, are linear for subsonic, transonic, and supersonic flows.
Fluid dynamics and wave-structure interactions
Published in Dezhi Ning, Boyin Ding, Modelling and Optimization of Wave Energy Converters, 2022
Malin Göteman, Robert Mayon, Yingyi Liu, Siming Zheng, Rongquan Wang
where Fext is the external net force acting on the fluid, usually only the gravity force Fext=∇(−ρgz). For inviscid flow, the Navier-Stokes equations reduce to the Euler equations.
Stability of a split-core configuration induced by saddle-splay elasticity in a submicron nematic liquid crystal spherical droplet
Published in Liquid Crystals, 2022
Hui Zhang, Hongen Liu, Zhidong Zhang, Guili Zheng
Because it is difficult to obtain analytical solutions of the Euler equation, we solve the numerical solution via finite-difference iteration [26]. The hemispherical cross section over the diameter is sufficient to describe the arrangement of LCs in a droplet. In the simulation, the hemispherical cross section is discretised into a network of lattice points. There are 65 lattice points along er and 129 lattice points along ez, so the relationship between the lattice point spacing h and droplet radius is h=R/64. The parameters of 5CB and other associated parameters are set to the following values in the calculation [27]: A0 = 0.195 × 106 J/m3, B = 7.155 × 106 J/m3, C = 8.82 × 106 J/m3, D* = 0.35 m2·N−1·s−1, L1 = 1.0125 × 10−13 J/m, characteristic length ξ = 2.64 nm, and scaled temperature à = 2/3, corresponding to . The homeotropic anchoring strength is w = 10−4 J/m2, and the reduced anchoring strength is .
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).
A localised dynamic closure model for Euler turbulence
Published in International Journal of Computational Fluid Dynamics, 2018
In present work, the three-dimensional Euler system of equations has been considered as underlying governing laws for flow simulation. Euler equations are a set of nonlinear hyperbolic conservation laws without the effects of body forces, heat flux or viscous stresses. Explicitly, they can be expressed in their conservative dimensionless form as with ρ, P, u, v and w are the density, pressure and velocity components of the flow field in the x, y and z Cartesian coordinates respectively, the quantities included in , , and are Here, e is the total energy and H is the total enthalpy. With the assumption of ideal gas law, the total enthalpy and total energy can be written as where and the ratio of specific heats γ is set as . The convective flux Jacobian matrices for the conservation laws above can be shown as (Laney 1998):