Effects of an external constant pressure gradient on a steady incompressible laminar flow through a semi-porous annular pipe

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Guy Leopold Mbogba ◽  
Elisabeth Ngo Nyobe ◽  
Maurice Lamara ◽  
Yves Christian Mbono Samba ◽  
Elkana Pemha
Keyword(s):  
Laminar Flow ◽  
Pressure Gradient ◽  
Steady Flow ◽  
Constant Pressure ◽  
Stokes Equations ◽  
Outer Wall ◽  
Optimization Method ◽  
External Pressure ◽  
Injection Velocity ◽  
Annular Pipe

Abstract In this paper, we examine a steady laminar flow for an incompressible fluid located in a semi porous annular pipe and subjected to a favorable constant pressure gradient applied between the two borders of the pipe. The inner wall is impermeable and the fluid is sucked or injected at the outer wall at constant and uniform velocity, orthogonally to the wall. The problem under study depends on three parameters: the pipe gap ratio, the dimensionless external pressure gradient, and the Reynolds number defined from the sum of the suction or injection velocity and the maximum Hagen–Poiseuille velocity. The conservation of mass induces the zero-divergence velocity field which allows replacing the steady-flow Navier–Stokes equations with a single equation satisfied by the stream function and called the vorticity equation. Assuming the similarity-solution hypothesis, the problem under consideration is reduced to a fourth-order nonlinear ordinary differential equation with two boundary conditions at each wall. The numerical shooting technique including the Runge–Kutta algorithm and the Newton–Raphson optimization method is applied to obtain the solution for the steady flow. For various values of the dimensionless external pressure gradient, the profiles of the velocity components are found and investigations on the wall shear stress for both walls are performed. The results obtained are discussed and physical understandings for the problem studied are derived.

Time-Dependent Laminar Flow in Curved Channels

1970 ◽  
Vol 37 (3) ◽  
pp. 838-843 ◽  
Author(s):  
R. J. Nunge
Keyword(s):  
Laminar Flow ◽  
Pressure Gradient ◽  
Constant Pressure ◽  
Time Dependent ◽  
Pressure Gradients ◽  
Straight Channel ◽  
Curvature Ratio ◽  
Curved Channels ◽  
Developing Flow ◽  
Time Required

The velocity distribution for time-dependent laminar flow in curved channels is derived. The analysis applies to flows with pressure gradients which are arbitrary functions of time. Numerical results are obtained for developing flow due to a constant pressure gradient. Developing flow in a straight channel is also discussed and it is found that the curvature ratio has only a small effect on the time required to reach the fully developed state.

The Torsion Effect on Fully Developed Laminar Flow in Helical Square Ducts

1993 ◽  
Vol 115 (2) ◽  
pp. 292-301 ◽  
Author(s):  
Wen-Hwa Chen ◽  
Ray Jan
Keyword(s):  
Laminar Flow ◽  
Secondary Flow ◽  
Friction Factor ◽  
Stokes Equations ◽  
Axial Flow ◽  
Outer Wall ◽  
Square Duct ◽  
Torsion Effect ◽  
Inclined Angle ◽  
Curvature And Torsion

The continuity equation and Navier-Stokes equations derived from a non-orthogonal helical coordinate system are solved by the Galerkin finite-element method in an attempt to study the torsion effect on the fully developed laminar flow in the helical square duct. Since high-order terms of curvature and torsion are considered, the approach is also applicable to the problems with finite curvature and torsion. The interaction effects of curvature, torsion, and the inclined angle of the cross section on the secondary flow, axial velocity, and friction factor in the helical square duct are presented. The results show that the torsion has more pronounced effect on the secondary flow rather than the axial flow. In addition, unlike the flow in the toroidal square duct, Dean’s instability of the secondary flow, which occurs near the outer wall in the helical square duct, can be avoided due to the effects of torsion and/or inclined angle. In such cases, a decrease of the friction factor is observed. However, as the pressure gradient decreases to a small value, the friction factor for the toroidal square duct is also applicable to the helical square duct.

LES of Compressible Turbulent Annular Pipe Flow With a Rotating Wall

2003 ◽  
Author(s):  
Joon Sang Lee ◽  
Xiaofeng Xu ◽  
R. H. Pletcher
Keyword(s):  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Stokes Equations ◽  
Outer Wall ◽  
Navier Stokes ◽  
Subgrid Scale ◽  
Turbulent Structures ◽  
Navier Stokes Equations ◽  
Rotating Wall ◽  
Annular Pipe

Flow in an annular pipe with and without a wall rotating about its axis was investigated at moderate Reynolds numbers. The compressible filtered Navier-Stokes equations were solved using a second order accurate finite volume method. Low Mach number preconditioning was used to enable the compressible code to work efficiently at low Mach numbers. A dynamic subgrid-scale stress model accounted for the subgrid-scale turbulence. When the outer wall rotated, a significant reduction of turbulent kinetic energy was realized near the rotating wall and the intensity of bursting effects appeared to decrease. This modification of the turbulent structures was related to the vortical structure changes near the rotating wall. It has been observed that the wall vortices were pushed in the direction of rotation and their intensity increased near the non-rotating wall. The consequent effect was to enhance the turbulent kinetic energy and increased the intensity of the heat transfer rate there.

Similarity Analysis for Nonequilibrium Turbulent Boundary Layers*

2004 ◽  
Vol 126 (5) ◽  
pp. 827-834 ◽  
Author(s):  
Luciano Castillo ◽  
Xia Wang
Keyword(s):  
Pressure Gradient ◽  
Boundary Layers ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Turbulent Boundary Layers ◽  
External Pressure ◽  
Navier Stokes ◽  
Similarity Analysis ◽  
Classical Paper ◽  
Nonequilibrium Flows

In his now classical paper on pressure gradient turbulent boundary layers, Clauser concluded that equilibrium flows were very special flows difficult to achieve experimentally and that few flows were actually in equilibrium [1]. However, using similarity analysis of the Navier–Stokes equations, Castillo and George [2] defined an equilibrium flow as one where the pressure parameter, Λ=[δ/ρU∞2dδ/dx]dP∞/dx, was a constant. They further showed that most flows were in equilibrium and the exceptions were nonequilibrium flows where Λ≠constant. Using the equations of motion and similarity analysis, it will be shown that even nonequilibrium flows, as those over airfoils or with sudden changes on the external pressure gradient, are in equilibrium state, but only locally. Moreover, in the case of airfoils where the external pressure gradient changes from favorable to zero then to adverse, three distinctive regions are identified. Each region is given by a constant value of Λθ, and each region remains in equilibrium with Λθ=constant, respectively.

Steady flow in a curved tube of triangular cross section

1976 ◽  
Vol 352 (1669) ◽  
pp. 189-211 ◽  
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Steady Flow ◽  
Cross Section ◽  
Numerical Solutions ◽  
Constant Pressure ◽  
Layer Structure ◽  
Asymptotic Model ◽  
Triangular Cross Section ◽  
Curved Tube

The steady flow of a viscous fluid moving under a constant pressure gradient in a curved tube with a uniform triangular cross section is investigated. Numerical solutions of the equations of motion have been found for the range 100-12 000 of the Dean number D = Ga 3 √(2a/ L )/ μv , where G is the constant pressure gradient, a is a dimension of the triangle, L the radius of the circle in which the tube is coiled, μ the viscosity and v the coefficient of kinematic viscosity of the fluid. The results for low D have been checked by an independent numerical method in which the stream function is expanded in a series of powers of D following the method of Dean (1928). All the results have been checked for accuracy by varying the grid size used in the numerical computations. The trend of the results as D increases is examined for evidence of the development of a boundary-layer structure as D → ∞. Some indication is found of the formation of a boundary layer of thickness proportional to D –1/3 near the side walls of the tube with an associated inviscid core region in the centre of the tube. In particular, comparison is made with details of an asymptotic model as D → ∞ proposed by Smith (1976). A measure of agreement with the general characteristics of this model is obtained, although there are some discrepancies in the precise details. It is possible that the range of D considered in the present work is not great enough to form any definite conclusions regarding the precise nature of the flow as D → ∞. A feature of the present results which develops for D > 3000 is that the maximum axial velocity in the tube ceases to occur on the axis of symmetry of the cross section. This feature appears to be generally consistent with numerical results obtained by Cheng & Akiyama (1970) and Hocking (unpublished) for a tube of rectangular cross section. The sequence of corner vortices of the type identified by Moffatt (1964) is found to occur in the numerical solutions. A detailed study of the vortices has already been published (Collins & Dennis 1976).

Laminar Flow Over Wavy Walls

1991 ◽  
Vol 113 (4) ◽  
pp. 574-578 ◽  
Author(s):  
V. C. Patel ◽  
J. Tyndall Chon ◽  
J. Y. Yoon
Keyword(s):  
Boundary Layer ◽  
Laminar Flow ◽  
Steady Flow ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Flow Regimes ◽  
Navier Stokes ◽  
Wavy Wall ◽  
Fully Developed Flow ◽  
Navier Stokes Equations

The boundary layer over a wavy wall and fully-developed flow in a duct with a wavy wall are considered. Numerical solutions of the Navier-Stokes equations have been obtained to provide insights into the various steady flow regimes that are possible, and to illustrate the nuances of predicting flows containing multiple separation and reattachment points.

Three-Dimensional Steady Flow Through A Bifurcation

1990 ◽  
Vol 112 (2) ◽  
pp. 189-197 ◽  
Author(s):  
Chain-Nan Yung ◽  
Kenneth J. De Witt ◽  
Theo G. Keith
Keyword(s):  
Reynolds Number ◽  
Shear Stress ◽  
Steady Flow ◽  
Stokes Equations ◽  
Control Volume ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Simple Algorithm ◽  
Outer Wall ◽  
Navier Stokes

Steady flow of an incompressible, Newtonian fluid through a symmetric bifurcated rigid channel was numerically analyzed by solving the three-dimensional Navier-Stokes equations. The upstream Reynolds number ranged from 100 to 1500. The bifurcation was symmetrical with a branch angle of 60 deg and the area ratio of the daughter to the mother vessel was 2.0. The numerical procedure utilized a coordinate transformation and a control volume approach to discretize the equations to finite difference form and incorporated the SIMPLE algorithm in performing the calculation. The predicted velocity pattern was in qualitative agreement with experimental measurements available in the literature. The results also showed the effect of secondary flow which can not be predicted using previous two-dimensional simulations. A region of reversed flow was observed near the outer wall of the branch except for the case of the lowest Reynolds number. Particle trajectory was examined and it was found that no fluid particles remained within the recirculation zone. The shear stress was calculated on both the inner and the outer wall of the branch. The largest wall shear stress, located in the vicinity of the apex of the branch, was of the same order of magnitude as the level that can cause damage to the vessel wall as reported in a recent study.

Direct simulation of transition in an oscillatory boundary layer

1998 ◽  
Vol 371 ◽  
pp. 207-232 ◽  
Author(s):  
G. VITTORI ◽  
R. VERZICCO
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Turbulent Regime ◽  
Two Dimensional ◽  
Temporal Development ◽  
Direct Simulation ◽  
Navier Stokes Equations ◽  
Oscillatory Boundary Layer

Numerical simulations of Navier–Stokes equations are performed to study the flow originated by an oscillating pressure gradient close to a wall characterized by small imperfections. The scenario of transition from the laminar to the turbulent regime is investigated and the results are interpreted in the light of existing analytical theories. The ‘disturbed-laminar’ and the ‘intermittently turbulent’ regimes detected experimentally are reproduced by the present simulations. Moreover it is found that imperfections of the wall are of fundamental importance in causing the growth of two-dimensional disturbances which in turn trigger turbulence in the Stokes boundary layer. Finally, in the intermittently turbulent regime, a description is given of the temporal development of turbulence characteristics.

Turbulent diffusional flames in the presence of an external pressure gradient

1987 ◽  
Vol 23 (3) ◽  
pp. 266-274
Author(s):  
V. K. Baev ◽  
M. A. Gorokhovskii ◽  
I. G. Shpil'berg
Keyword(s):  
Pressure Gradient ◽  
External Pressure
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