On the Use of the Fully Compressible Navier-Stokes Equations for the Steady-State Solution of Natural Convection Problems in Closed Cavities

2006 ◽  
Vol 129 (3) ◽  
pp. 387-390 ◽  
Author(s):  
Sandip Mazumder
Keyword(s):  
Steady State ◽  
Stokes Equations ◽  
Pressure Rise ◽  
Steady State Solution ◽  
Boussinesq Approximation ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Simple Strategy ◽  
Reference Pressure

The steady-state compressible form of the Navier-Stokes equations, along with no-slip boundary conditions on walls, represents a boundary value problem. In closed heated cavities, these equations are incapable of preserving the initial mass of the cavity and predicting the pressure rise. A simple strategy to adjust the reference pressure in the system is presented and demonstrated. The strategy is similar to solving the transient form of the governing equations, but completely eliminates truncation errors associated with temporal discretization of the transient terms. Results exhibit good agreement with previous reports. Additional results are shown to highlight differences between the fully compressible formulation and the Boussinesq approximation.

scholarly journals Steady-State Navier–Stokes Equations in Thin Tube Structure with the Bernoulli Pressure Inflow Boundary Conditions: Asymptotic Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2433
Author(s):  
Rita Juodagalvytė ◽  
Grigory Panasenko ◽  
Konstantinas Pileckas
Keyword(s):  
Weak Solution ◽  
Steady State ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Thin Tube ◽  
Outflow Boundary ◽  
Tube Structure

Steady-state Navier–Stokes equations in a thin tube structure with the Bernoulli pressure inflow–outflow boundary conditions and no-slip boundary conditions at the lateral boundary are considered. Applying the Leray–Schauder fixed point theorem, we prove the existence and uniqueness of a weak solution. An asymptotic approximation of a weak solution is constructed and justified by an error estimate.

Parallel iterative finite-element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kangrui Zhou ◽  
Yueqiang Shang
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Parallel Algorithms ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Slip Boundary Conditions ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Parallel Iterative ◽  
Nonlinear Slip Boundary Conditions

AbstractBased on full domain partition, three parallel iterative finite-element algorithms are proposed and analyzed for the Navier–Stokes equations with nonlinear slip boundary conditions. Since the nonlinear slip boundary conditions include the subdifferential property, the variational formulation of these equations is variational inequalities of the second kind. In these parallel algorithms, each subproblem is defined on a global composite mesh that is fine with size h on its subdomain and coarse with size H (H ≫ h) far away from the subdomain, and then we can solve it in parallel with other subproblems by using an existing sequential solver without extensive recoding. All of the subproblems are nonlinear and are independently solved by three kinds of iterative methods. Compared with the corresponding serial iterative finite-element algorithms, the parallel algorithms proposed in this paper can yield an approximate solution with a comparable accuracy and a substantial decrease in computational time. Contributions of this paper are as follows: (1) new parallel algorithms based on full domain partition are proposed for the Navier–Stokes equations with nonlinear slip boundary conditions; (2) nonlinear iterative methods are studied in the parallel algorithms; (3) new theoretical results about the stability, convergence and error estimates of the developed algorithms are obtained; (4) some numerical results are given to illustrate the promise of the developed algorithms.

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