Solution Of Three-dimensional Navier-Stokes Equations Research Articles

Eventual smoothness and stabilization in a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization

The quasilinear Keller-Segel-Navier-Stokes system modeling coral fertilization(⁎){nt+u⋅∇n=Δn−∇⋅(nS(x,n,c)∇c)−nm,x∈Ω,t>0,ct+u⋅∇c=Δc−c+m,x∈Ω,t>0,mt+u⋅∇m=Δm−nm,x∈Ω,t>0,ut+κ(u⋅∇)u+∇P=Δu+(n+m)∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 is considered in a bounded domain Ω⊂R3 with smooth boundary, where ϕ∈W2,∞(Ω), S denotes a given tensor-valued function fulfilling |S(x,n,c)|≤S0(c)(1+n)−α with α≥0, and S0 is a nonnegative nondecreasing function. It is shown that if α>0, then for any κ∈R, the corresponding initial-boundary value problem (⁎) possesses at least one global weak solution, which can be obtained as the pointwise limit of a sequence of solutions to appropriately regularized problems. Additionally, we also show that, after some time, these solutions become classical solutions, that is, there exists a positive constant T such that (n,c,m,u) is smooth in Ω×(T,∞). Finally, we can also prove that when t→∞, any such solution (n,c,m,u) would stabilize to the spatially uniform equilibrium (nˆ,mˆ,mˆ,0) in L∞(Ω), where nˆ=1|Ω|{∫Ωn0−∫Ωm0}+ and mˆ=1|Ω|{∫Ωm0−∫Ωn0}+. The results can provide an important reference for solving the existence of classical solutions of three-dimensional Navier-Stokes equations.

ЧИСЛЕННАЯ МОДЕЛЬ ОГНЕНЕПРОНИЦАЕМОСТИ ЭЛЕМЕНТОВ ГАЗОТУРБИННЫХ ДВИГАТЕЛЕЙ ИЗ АЛЮМИНИЕВЫХ СПЛАВОВ

For the early stages of design, a numerical model of fire resistance and fire permeability of GTE elements made of alu-minum alloys is proposed. The numerical model is based on the solution of three-dimensional Navier-Stokes equations, closed by a two-parameter isotropic SST – turbulence model. We used an implicit pressure solver and a scheme of second-order accuracy in High Resolution space, which allows lowering the approximation order to the first in the areas of solution discontinuities and high gradients to increase the monotonicity and stability of the solution. Additionally, to achieve the solution stability, the High Speed Numerics solver option was used, activating the solver settings Gradient Factor Relaxation = 0.1 and Blend Factor Relaxation = 0.1. The mathematical model for the thermal-hydraulic calculation was set up in the formulation of the coupled heat transfer, taking into account radiation based on the discrete transfer model of the Participating Media. The influ-ence of the degree of blackness on the results of the numerical model is investigated and the most rational radiation model is chosen. The numerical model was validated according to the results of engineering tests on the fire permeability of the plate not cooled by the air flow using a kerosene burner NexGen of the bench Ts17-G3 of FAU P.I. Baranov TsIAM. The obtained results of calculations on fire permeability are comparable with the experimental data both quantitatively and qualitatively. The developed un-steady numerical model of fire permeability of flat plates of aluminum alloys under conditions of free convection can be used in the de-sign of the fan housing by selecting the most rational structural and layout solutions, such as the thickness of the housing, the presence of fire protection coating, cooling intensity, etc.

Non-perturbative solution of three-dimensional Navier-Stokes equations for the flow near an infinite rotating disk

In this paper, we present Homotopy perturbation method (HPM) and Pade technique, for finding non-perturbative solution of three-dimensional viscous flow near an infinite rotating disk. We compared our solution with the numerical solution (fourth-order Runge–Kutta). The results show that the HPM–Pade technique is an appropriate method in solving the systems of nonlinear equations. The mathematical technique employed in this paper is significant in studying some other problems of engineering. Copyright © 2009 John Wiley & Sons, Ltd.

Analytic approximate solution of three‐dimensional Navier‐Stokes equations of flow between two stretchable disks

AbstractThe similarity transform for the steady three‐dimensional Navier‐Stokes equations of flow between two stretchable disks gives a system of nonlinear ordinary differential equations which is analytically solved by applying a newly developed method, namely, the homotopy analysis method. The analytic solutions of the system of nonlinear ordinary differential equations are constructed in the series form. The convergence of the obtained series solutions is analyzed. The validity of our solutions is verified by the numerical results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

Numerical solution of three-dimensional Navier-Stokes equations by a velocity-vorticity method

A finite difference method for solving the incompressible viscous flow in velocity–vorticity form is presented. A staggered mesh is employed to ensure continuity. To enforce the zero divergence of the vorticity, the computed vorticity is replaced at each time step with . An explicit three-level backward scheme is used to update the vorticity transport equation for the vorticity at the next time level. To solve the Poisson equations for velocity, a restarted version of the Generalized Minimal Residual method (GMRES) implemented with incomplete lower–upper (ILU) decompositions preconditioner has been adopted. Two- and three-dimensional driven cavity flows with impulsively started and oscillating lids are used to test the method. Detailed results and comparisons with the numerical literature data show that the proposed method is accurate and efficient. Copyright © 2001 John Wiley & Sons, Ltd.

Numerical Solution of Three-Dimensional Navier-Stokes Equations Showing Trailing Tip Vortices

Numerical solution of the incompressible Navier-Stokes equations in integro-differential form is applied to timedependent flow about a rectangular slab at an angle of attack. With this formulation the solution is obtained in the entire unbounded flowfield, but with actual computation required only in regions of significant vorticity. This allows considerable reduction in computer storage, since only points in regions of significant vorticity need be stored at any particular time. The computational field thus expands in time. (This method is not to be confused with vortex using discrete vortices and images.) The finite numerical calculation field in the integro-differential formulation is, in effect, infinite, and the necessity of locating infinity at a finite distance is avoided. Although it is not necessary in this numerical method to calculate the velocity at points outside the region of nonzero vorticity, the velocity at these points and, in fact, to infinity, is determined by the solution via an integral over the vorticity distribution. The method requires two orders of magnitude less computer storage than do methods based on the differential formulation. Results have been obtained for the development of trailing tip vortices and the force coefficients.