A cost-effective FE method for 2D Navier–Stokes equations

Abstract

A cost-effective approach to the solution of 2D Navier-Stokes equations for incompressible fluid flow problems is presented. The aim is to reach a good compromise between numerical properties and computational efficiency. In order to achieve the set goal, the nonlinear convective terms are approximated by means of characteristics and spatial approximations of equal order are performed by polynomials of degree two. In this way, the computational kernels are reduced to elliptic ones for which solution very efficient techniques are available. The time-advancing is afforded by a fractional step method combined with a stabilization technique suitably simplified, so that the inf-sup condition is easily overcome. The algebraic systems generated by the new technique are solved by an iterative solver (Bi-CGSTAB), preconditioned by means of a suitable Schwarz additive scalable preconditioner. The properties of the new method have been confirmed from the comparison among the results obtained by it, and those obtained from other methods in the solution of some well known test problems. The obtained results, both in terms of accuracy and computational efficiency, make realistic the possibility to extend the method to 3D problems and to develop a multidomain approach.

A Hybrid Analytical Approximate Technique for Solving Two-dimensional Incompressible Flow in Lid-driven Square Cavity Problem

“…In table 1 we compare the results obtained from the suggested method with the methods KRDRM and KPIM of velocity 𝑢 along the vertical line and 𝑣 velocity along the horizontal line through the geometric center of the square cavity, the findings demonstrate that these values are identical with values stated in KRDRM and KPIM for various values of Renolds numbers at 𝑡 = 0.1 𝑎𝑛𝑑 𝑀𝑎 = 0.01 ,was q-HALPM solutions from two iterations .Table 2 represented the results 𝑢 𝑎𝑛𝑑 𝑣 that are obtained from q-HALPM with KRDRM, KPIM and finite volume method in Ref [9]. The 𝐿 ∞ − 𝑒𝑟𝑟𝑜𝑟𝑠 shows in Table 3 for the stream function 𝜓 and the vorticity 𝜔 by using the current method and comparted with these provided methods in [1] and [9] and rational fourth-order compact finite difference method in [33] , for three different values of Reynolds numbers 𝑅𝑒 = 10,100 𝑎𝑛𝑑1000 𝑎𝑡 𝑀𝑎 = 0.001, We see that the number of grid points has no bear on the estimated errors, which are small for all Reynolds number values. The analytical approximate solution produced by 𝑣(0.9375; 0.5) −0.15380859374 −0.1429446548 −0.1429451057…”

“…Now, before we start applying the q-HALPM on KRLNS equation taking the initial conditions of 𝑢, 𝑣, 𝑃 as [31][32][33];…”

Rotation and orientation of irregular particles in viscous fluids using the gradient smoothed method (GSM)

“…Since two decades ago, researchers have done lots of work on dynamics of particles' freefalling in fluids both experimentally and numerically (Ardekani, Dabiri, & Rangel, 2008;Feng & Michaelides, 2004;Goyal & Derksen, 2012;Hu, 1996;Huang, Hu, & Joseph, 1998;Pu, Shao, Huang, & Hussain, 2013). Among the numerical methods, finite element method (FEM) (Jotsa & Pennati, 2015;Liu & Quek, 2013), finite difference method (FDM), finite volume method (FVM), and even meshfree methods (Liu, 2009;Pu et al, 2013) are most widely used. Hu, Joseph, and Crochet (1992) used FEM to model the Navier-Stokes equations for liquid and the Newton's equations of motion for solids.…”