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Tensors in Rectangular Rtesian Coordinate Systems
Published in C. Young Eutiquio, Vector and Tensor Analysis, 2017
This definition casts vectors in the setting of cartesian tensors. In fact, Definition 1 is precisely the definition of a cartesian tensor of order 1 (or rank 1). In the context of this definition, a scalar becomes a quantity that remains invariant (does not change in numerical value) under any coordinate transformation. In tensor language, scalars are tensors of order (or rank) zero.
Enhanced Conjugate Gradient Method for Heat Flux Estimation in Laminar and T-Junction Turbulent Mixing Flow
Published in Heat Transfer Engineering, 2022
Sanil Shah, Ajit Kumar Parwani
The tube is open to the atmosphere and heat is lost to the atmosphere by the natural convection. The Ansys Fluent CFD software is used for solving the RANS-based governing equations of the steady turbulent flow for determining the mean turbulent quantities and the peripheral heat flux at the fluid-solid interface. The continuity, momentum and energy equations in the Cartesian tensor form are given by: where i and j denote Einstein tensor convention indices, p is pressure, μ is the fluid dynamic viscosity and Pr is the Prandtl number. The primed quantities are fluctuating components and the overhead bar () denotes time average quantity. The terms (Eq. (27)) and (Eq. (28)) are known as Reynolds stress and turbulence heat flux respectively. The SST k- two-equation model is chosen to compute turbulence terms, where k stands for turbulence kinetic energy and stands for the specific dissipation rate. Hexahedral cells are chosen for meshing as shown in Figure 6. The inflation boundary condition is given at walls to resolve flow-field and temperature gradient near the wall. The first layer of inflation is set in such a way that the non-dimensional distance from the wall (y+) remains less than 1.
A circular cylinder in the main-channel/floodplain interface of a compound channel: effect of the shear flow on drag and lift
Published in Journal of Hydraulic Research, 2020
Miltiadis Gymnopoulos, Ana M. Ricardo, Elsa Alves, Rui M.L. Ferreira
Consider a multiply connected fluid control volume encompassing the cylinder, the approaching flow and the wake. The Reynolds-averaged integral equations of conservation of linear momentum (herein RAIM equations) for a stationary turbulent flow of an incompressible fluid, applied to this fluid control volume are: where Vc is the control volume, Sc is the total control surface, S0 is the part of Sc that coincides with the cylinder wall, represents the open control-surface, through which there may be mass and momentum fluxes, ni is the outward pointing normal unit-vector, Tik is the time-averaged viscous-stress tensor, is the Reynolds stress tensor, Ui is the time-averaged velocity, gi is the acceleration of gravity, P is the time-averaged pressure, and ρ is the density of the fluid. For the sake of conciseness, Eq. (1) is written is Cartesian tensor notation. The free index identifies the spatial direction while the dummy index is used in the summations inherent to the dot products in the 1st, 4th, 5th and 7th terms.
Numerical study of heat transfer and fluid flow of supercritical water in twisted spiral tubes
Published in Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 2022
Mahyar Najafian, Ali Esmaeili, Amirfarhang Nikkhoo, Hui Jin, M. R. Soufivand
In this study, the Navier-Stokes equations in 3D, incompressible and steady-state were discretized using finite volume approach. The diffusion flux was estimated using the central difference method. The equations were also solved using an implicit method and the accuracy of numerical solution was also second-order. The problem has been solved using mass, momentum and energy conservation equations. These equations in Cartesian tensor notation are as follows: