Laminar flow in a two-dimensional plane channel with local pressure-dependent crossflow

2007 ◽  
Vol 593 ◽  
pp. 463-473 ◽  
Author(s):  
P. HALDENWANG
Keyword(s):  
Laminar Flow ◽  
Stokes Equations ◽  
Biological Fluids ◽  
Plane Channel ◽  
Axial Flow ◽  
Two Dimensional ◽  
Permeate Flux ◽  
Local Pressure ◽  
Porous Walls ◽  
Pressure Independent

Long ducts (or pipes) composed of transpiring (e.g. porous) walls are at the root of numerous industrial devices for species separation, as tangential filtration or membrane desalination. Similar configurations can also be involved in fluid supply systems, as irrigation or biological fluids in capillaries. A transverse leakage (or permeate flux), the strength of which is assumed to depend linearly on local pressure (as in Starling's law for capillary), takes place through permeable walls. All other dependences, as osmotic pressure or partial fouling due to polarization of species concentration, are neglected. To analyse this open problem we consider the simplest situation: the steady laminar flow in a two-dimensional channel composed of two symmetrical porous walls.First, dimensional analysis helps us to determine the relevant parameters. We then revisit the Berman problem that considers a uniform crossflow (i.e. pressure-independent leakage). We expand the solution in a series of Rt, the transverse Reynolds number. We note this series has a rapid convergence in the considered range of Rt (i.e. Rt ≤ O(1)). A particular method of variable separation then allows us to derive from the Navier–Stokes equations two new ordinary differential equations (ODE), which correspond to first and second orders in the development in Rt, whereas the zero order recovers the Regirer linear theory. Finally, both new ODEs are used to study the occurrence of two undesirable events in the filtration process: axial flow exhaustion (AFE) and crossflow reversal (CFR). This study is compared with a numerical approach.

Axial flow in a two-dimensional microchannel induced by a travelling temperature wave imposed at the bottom wall

2018 ◽  
Vol 848 ◽  
pp. 1040-1072 ◽  
Author(s):  
Chenguang Zhang ◽  
Harris Wong ◽  
Krishnaswamy Nandakumar
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Axial Flow ◽  
Temperature Wave ◽  
Bottom Wall ◽  
Moving Frame ◽  
Global Maximum ◽  
System Of Equations ◽  
Two Dimensional ◽  
Perturbation Solutions

Fluid flow in microchannels has wide industrial and scientific applications. Hence, it is important to explore different driving mechanisms. In this paper, we study the net transport or fluid pumping in a two-dimensional channel induced by a travelling temperature wave applied at the bottom wall. The Navier–Stokes equations with the Boussinesq approximation and the convection–diffusion heat equation are made dimensionless by the height of the channel and a velocity scale obtained by a dominant balance between buoyancy and viscous resistance in the momentum equation. The system of equations is transformed to an axial coordinate that moves with the travelling temperature wave, and we seek steady solutions in this moving frame. Four dimensionless numbers emerge from the governing equations and boundary conditions: the Reynolds number $Re$, a Reynolds number $Rc$ based on the wave speed, the Prandtl number $Pr$ and the dimensionless wavenumber $K$. The system of equations is solved by a finite-volume method and by a perturbation method in the limit $Re\rightarrow 0$. Surprisingly, the leading and first-order perturbation solutions agree well with the computed axial flow for $Re\leqslant 10^{3}$. Thus, the analytic perturbation solutions are used to study systematically the effects of $Re$, $Rc$, $Pr$ and $K$ on the dimensionless induced axial flow $Q$. We find that $Q$ varies linearly with $Re$, and $Q/Re$ versus any of the three remaining dimensionless groups always exhibits a maximum. The global maximum of $Q/Re$ in the parameter space is subsequently determined for the first time. This induced axial flow is driven solely by the Reynolds stress.

Numerical simulation of the evolution of Tollmien–Schlichting waves over finite compliant panels

1997 ◽  
Vol 335 ◽  
pp. 361-392 ◽  
Author(s):  
CHRISTOPHER DAVIES ◽  
PETER W. CARPENTER
Keyword(s):  
Numerical Simulation ◽  
Laminar Flow ◽  
Stokes Equations ◽  
Plane Channel ◽  
Solution Procedure ◽  
Good Prospect ◽  
Flow Perturbation ◽  
Compliant Wall ◽  
Compliant Walls ◽  
Tollmien Schlichting

The evolution of two-dimensional Tollmien–Schlichting waves propagating along a wall shear layer as it passes over a compliant panel of finite length is investigated by means of numerical simulation. It is shown that the interaction of such waves with the edges of the panel can lead to complex patterns of behaviour. The behaviour of the Tollmien–Schlichting waves in this situation, particularly the effect on their growth rate, is pertinent to the practical application of compliant walls for the delay of laminar–turbulent transition. If compliant panels could be made sufficiently short whilst retaining the capability to stabilize Tollmien–Schlichting waves, there is a good prospect that multiple-panel compliant walls could be used to maintain laminar flow at indefinitely high Reynolds numbers.We consider a model problem whereby a section of a plane channel is replaced with a compliant panel. A growing Tollmien–Schlichting wave is then introduced into the plane, rigid-walled, channel flow upstream of the compliant panel. The results obtained are very encouraging from the viewpoint of laminar-flow control. They indicate that compliant panels as short as a single Tollmien–Schlichting wavelength can have a strong stabilizing effect. In some cases the passage of the Tollmien–Schlichting wave over the panel edges leads to the excitation of stable flow-induced surface waves. The presence of these additional waves does not appear to be associated with any adverse effect on the stability of the Tollmien–Schlichting waves. Except very near the panel edges the panel response and flow perturbation can be represented by a superposition of the Tollmien–Schlichting wave and two other eigenmodes of the coupled Orr–Sommerfeld/compliant-wall eigensystem.The numerical scheme employed for the simulations is derived from a novel vorticity–velocity formulation of the linearized Navier–Stokes equations and uses a mixed finite-difference/spectral spatial discretization. This approach facilitated the development of a highly efficient solution procedure. Problems with numerical stability were overcome by combining the inertias of the compliant wall and fluid when imposing the boundary conditions. This allowed the interactively coupled fluid and wall motions to be computed without any prior restriction on the form taken by the disturbances.

Simulation of a Cantilevered Thin-Elastic Plate with Large Deformation Subjected to Axial Flow

2014 ◽  
Vol 1065-1069 ◽  
pp. 2069-2075
Author(s):  
Wei Bin Hong ◽  
Chang Qing Guo ◽  
Ye Zhou Sheng
Keyword(s):  
Flow Velocity ◽  
Large Deformation ◽  
Elastic Plate ◽  
Stokes Equations ◽  
Axial Flow ◽  
Navier Stokes ◽  
Thin Elastic Plate ◽  
Two Dimensional ◽  
Vibration Responses ◽  
Displacement Based

The instability and dynamics behavior of a cantilevered thin-elastic plate with large deformation subjected to axial flow is studied numerically. The structural dynamics equation is discretized by isoparametric displacement-based finite, and the motion of a continuous fluid domain is governed by two-dimensional incompressible viscous Navier-Stokes equations, which discretized by finite volume method. The two-dimensional numerical model of two-way fluid-structure coupling is established combined with moving mesh technology, realizing the interaction of thin-elastic plate and axial fluid. Firstly, under given different flow velocity, the stability of limit-cycle oscillations has been studied through Hopf bifurcation, time trace, vibration responses. Secondly, the fluid domain features are analyzed qualitatively by separately comparing with vorticity under given different flow velocity, and cloud diagram of pressure and velocity are also analyzed at U=3.6m/s.

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.

scholarly journals Unsteady Reversed Stagnation-Point Flow over a Flat Plate

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Vai Kuong Sin ◽  
Chon Kit Chio
Keyword(s):  
Laminar Flow ◽  
Flow Field ◽  
Asymptotic Solution ◽  
Stagnation Point ◽  
Stokes Equations ◽  
Stagnation Point Flow ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Total Flow ◽  
Navier Stokes Equations

This paper investigates the nature of the development of two-dimensional laminar flow of an incompressible fluid at the reversed stagnation-point. Proudman and Johnson (1962) first studied the flow and obtained an asymptotic solution by neglecting the viscous terms. Robins and Howarth (1972) stated that this is not true in neglecting the viscous terms within the total flow field. Viscous terms in this analysis are now included, and a similarity solution of two-dimensional reversed stagnation-point flow is investigated by solving the full Navier-Stokes equations.

Prediction of Flow Behavior on Diffuser Angle in an Enclosure

2005 ◽  
Author(s):  
B. Tripathi ◽  
R. C. Arora ◽  
S. G. Moulic
Keyword(s):  
Laminar Flow ◽  
Energy Equation ◽  
Stokes Equations ◽  
Flow Behavior ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Specific Location ◽  
Temperature Distributions ◽  
Diffuser Angle

The present investigation deals with numerical prediction of airflow pattern in a room (enclosure) with a specific location of inlet and outlet with different values of Gr/Re2. Two-dimensional, steady, incompressible, laminar flow under Boussinesq’s approximation has been considered. The velocity and temperature distributions in a room have been found by solving Navier Stokes equations and energy equation numerically by SIMPLE and SIMPLEC algorithms.

Laminar Flow in a Uniformly Porous Channel with Large Injection

1965 ◽  
Vol 16 (4) ◽  
pp. 323-332 ◽  
Author(s):  
R. M. Terrill
Keyword(s):  
Laminar Flow ◽  
Axial Velocity ◽  
Complete Solution ◽  
Two Dimensional ◽  
Series Solutions ◽  
Viscous Layer ◽  
High Injection ◽  
Large Injection ◽  
Porous Walls ◽  
Derivatives Of

SummaryThe author has previously discussed the flow in a two-dimensional channel with porous walls through which fluid is uniformly injected or extracted. Yuan has given a solution for high injection in which certain derivatives of the axial velocity become infinite at the centre of the channel. Physically this means that there must be a viscous layer at the centre of the channel and that Yuan’s solution is ignoring the shear layer. In this paper the complete solution for large injection is obtained by the method of inner and outer expansions and therefore includes the viscous layer. The resulting series solutions are confirmed by numerical results.

Laminar Flow in a Uniformly Porous Channel

1964 ◽  
Vol 15 (3) ◽  
pp. 299-310 ◽  
Author(s):  
Thein Wah
Keyword(s):  
Differential Equation ◽  
Reynolds Number ◽  
Laminar Flow ◽  
Numerical Solutions ◽  
Large Reynolds Number ◽  
Two Dimensional ◽  
Series Solutions ◽  
Porous Walls ◽  
Linear Differential ◽  
Good Agreement

SummaryThe flow in a two-dimensional channel with porous walls through which fluid is uniformly injected or extracted is considered. Following Berman, a solution is obtained giving a fourth-order non-linear differential equation which depends on a suction Reynolds number R. Numerical solutions of this equation have been obtained. Series solutions of this equation for small and large Reynolds number are given and are shown to give good agreement with the numerical solutions.

Sign in / Sign up

Export Citation Format

Share Document