On the LBB condition in the numerical analysis of the Stokes equations

2005 ◽  
Vol 54 (3-4) ◽  
pp. 314-323 ◽  
Author(s):  
Manfred Dobrowolski
Keyword(s):  
Numerical Analysis ◽  
Stokes Equations ◽  
Lbb Condition

The numerical analysis of bifurcation problems with application to fluid mechanics

Acta Numerica ◽  
2000 ◽  
Vol 9 ◽  
pp. 39-131 ◽  
Author(s):  
K. A. Cliffe ◽  
A. Spence ◽  
S. J. Tavener
Keyword(s):  
Numerical Analysis ◽  
Fluid Mechanics ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Finite Difference Methods ◽  
Singularity Theory ◽  
Mixed Finite Element Method ◽  
Convergence Theory ◽  
Navier Stokes ◽  
Bifurcation Problems

In this review we discuss bifurcation theory in a Banach space setting using the singularity theory developed by Golubitsky and Schaeffer to classify bifurcation points. The numerical analysis of bifurcation problems is discussed and the convergence theory for several important bifurcations is described for both projection and finite difference methods. These results are used to provide a convergence theory for the mixed finite element method applied to the steady incompressible Navier–Stokes equations. Numerical methods for the calculation of several common bifurcations are described and the performance of these methods is illustrated by application to several problems in fluid mechanics. A detailed description of the Taylor–Couette problem is given, and extensive numerical and experimental results are provided for comparison and discussion.

On the Inf-Sup Constant of a Triangular Spectral Method for the Stokes Equations

2016 ◽  
Vol 16 (3) ◽  
pp. 507-522 ◽  
Author(s):  
Yanhui Su ◽  
Lizhen Chen ◽  
Xianjuan Li ◽  
Chuanju Xu
Keyword(s):  
Lower Bound ◽  
Error Estimate ◽  
Spectral Method ◽  
Stokes Equations ◽  
Necessary Condition ◽  
Well Posedness ◽  
Saddle Point Problems ◽  
Numerical Examples ◽  
Lbb Condition ◽  
Optimal Lower Bound

AbstractThe Ladyženskaja–Babuška–Brezzi (LBB) condition is a necessary condition for the well-posedness of discrete saddle point problems stemming from discretizing the Stokes equations. In this paper, we prove the LBB condition and provide the (optimal) lower bound for this condition for the triangular spectral method proposed by L. Chen, J. Shen, and C. Xu in [3]. Then this lower bound is used to derive an error estimate for the pressure. Some numerical examples are provided to confirm the theoretical estimates.

Variation of Hydraulic Performance by Impeller Trimming of a Mixed-Flow Pump

2015 ◽  
Author(s):  
Hyeonmo Yang ◽  
Sung Kim ◽  
Kyoung-Yong Lee ◽  
Young-Seok Choi ◽  
Jin-Hyuk Kim
Keyword(s):  
Numerical Analysis ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Total Efficiency ◽  
Mixed Flow ◽  
Navier Stokes Equations ◽  
Ansys Cfx ◽  
Total Head ◽  
Flow Pump

One of the best examples of wasted energy is the selection of oversized pumps versus the rated conditions. Oversized pumps are forced to operate at reduced flows, far from their highest efficiency point. An unnecessarily large impeller will produce more flow than required, wasting energy. In the industrial field, trimming the impeller diameter is used more than changing the rotation speed to reduce the head of a pump. In this paper, the impeller trimming method of a mixed-flow pump is defined, and the variation in pump performance by reduction of the impeller diameter was predicted based on computational fluid dynamics. The impeller was trimmed to the same meridional ratio of the hub and shroud, and was compared in five cases. Numerical analysis was performed, including the inlet and outlet pipes in configurations of the mixed-flow pump to be tested. The commercial CFD code, ANSYS CFX-14.5, was used for the numerical analysis, and a three-dimensional Reynolds-averaged Navier-Stokes equations with a shear stress transport turbulence model were used to analyze incompressible turbulence flow. The performance parameters for evaluating the trimmed pump impellers were defined as the total efficiency and total head at the designed flow rate. The numerical and experimental results for the trimmed pump impellers were compared and discussed in this work.

scholarly journals Effect of Impeller Parameters on the Flow inside the Centrifugal Blower using CFD

2020 ◽  
Vol 8 (6) ◽  
pp. 3977-3980
Keyword(s):  
Numerical Analysis ◽  
Stokes Equations ◽  
Control Volume ◽  
Three Dimensional ◽  
Flow Characteristics ◽  
Navier Stokes ◽  
Computational Domain ◽  
Centrifugal Blower ◽  
Impeller Blade ◽  
Navier Stokes Equations

A numerical analysis is carried out to understand the flow characteristics for different impeller configurations of a single stage centrifugal blower. The volute design is based on constant velocity method. Four different impeller configurations are selected for the analysis. Impeller blade geometry is created with point by point method. Numerical simulation is carried out by CFD software GAMBIT 2.4.6 and FLUENT 6.3.26. GAMBIT work includes geometry definition and grid generation of computational domain. This process includes selection of grid types, grid refinements and defining correct boundary conditions. Processing work is carried out in FLUENT. The viscous Navier-Stokes equations are solved with control volume approach and the k-ε turbulence model. In this three dimensional numerical analysis is carried out with steady flow approach. The rotor and stator interaction is solved by mixing plane approach. Results of simulation are presented in terms of flow parameters, at impeller outlet and various angular positions inside the volute. Also, the contours of flow properties are presented at the outlet plane of fluid domain. Results suggest that for the same configurations of centrifugal blower, as we change geometrical parameter of impeller the flow inside the blower get affected.

Numerical Analysis of Convection Heat Transfer From an Array of Circular Perforated Fins due to Variable Perforation Size

2012 ◽  
Author(s):  
A. H. Dastbelaraki ◽  
M. Yaghoubi
Keyword(s):  
Heat Transfer ◽  
Numerical Analysis ◽  
Stokes Equations ◽  
Experimental Studies ◽  
Convection Heat Transfer ◽  
Heat Removal ◽  
Industrial Applications ◽  
Working Fluid ◽  
Manufacturing Cost ◽  
Convection Heat

Among rapid advances of electrical systems electronic circuit boards have become more compact and heat production rate from their components increased considerably. Such developments raised more attention and tendency in tackling their cooling problems. A wide variety of cooling systems are developed such as various fins for decreasing the circuit board temperature. Extended surfaces or fins are good heat transfer equipments that are used for various industrial applications. The wide industrial applications of fins to enhance heat transfer rate highlighted the need for further research of optimization of fins to increase their thermal performances. Among various fin types, rectangular fins are commonly used due to their simplicity of manufacturing. Fin configurations affect the cooling rate significantly, hence a comprehensive parametric study on the fin geometries may be improve their performance. Rectangular fins show a good performance of increasing heat removal rate, while reducing the manufacturing cost. Moreover the inspiration of putting holes along the flow through the fins may be very helpful in increasing the heat removal and reduction of the needed material. The present study investigates a numerical analysis of three dimensional, turbulent convection heat transfer from an array of rectangular perforated fins with increasing the perforation size from bottom to top. The perforations considered are like circular channels along the length of fins and the number of perforations is 3. For investigation, incompressible air as working fluid is modeled using Navier–Stokes equations. RNG based k-ε turbulent model is used to predict turbulent flow parameters. Temperature field inside the fins is obtained by solving Fourier’s conduction equation. The conjugate differential equations for both solid and gas phase are solved simultaneously by finite volume procedure using SIMPLE algorithm. Flow and heat transfer characteristics are presented for Reynolds numbers from 2 × 104 to 4 × 104 based on the fin length and Prandtl number of Pr = 0.71. Numerical model is first validated with previous experimental studies and good agreements were observed. Based on the valid simulation model, numerical solution is made to find flow field and temperature distribution for various perforation size. Results show that for a specific type of perforated fins the fin effectiveness is higher than other types and drag coefficient decreases with increasing the perforation size.

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