This article was downloaded by: [University of Saskatchewan Library] On: 06 October 2012, At: 08:32 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK International Journal of Computational Fluid Dynamics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gcfd20 Steady viscous incompressible flow driven by a pressure difference in a planar T-junction channel Nikolay Moshkin a & Damrongsak Yambangwi a a School of Mathematics, Suranaree University of Technology, Nakhon Ratchasima, 30000, Thailand Version of record first published: 31 Mar 2009. To cite this article: Nikolay Moshkin & Damrongsak Yambangwi (2009): Steady viscous incompressible flow driven by a pressure difference in a planar T-junction channel, International Journal of Computational Fluid Dynamics, 23:3, 259-270 To link to this article: http://dx.doi.org/10.1080/10618560902815204 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
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This article was downloaded by: [University of Saskatchewan Library]On: 06 October 2012, At: 08:32Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
International Journal of Computational Fluid DynamicsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/gcfd20
Steady viscous incompressible flow driven by apressure difference in a planar T-junction channelNikolay Moshkin a & Damrongsak Yambangwi aa School of Mathematics, Suranaree University of Technology, Nakhon Ratchasima, 30000,Thailand
Version of record first published: 31 Mar 2009.
To cite this article: Nikolay Moshkin & Damrongsak Yambangwi (2009): Steady viscous incompressible flow driven by apressure difference in a planar T-junction channel, International Journal of Computational Fluid Dynamics, 23:3, 259-270
To link to this article: http://dx.doi.org/10.1080/10618560902815204
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions
This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form toanyone is expressly forbidden.
The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses shouldbe independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims,proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly inconnection with or arising out of the use of this material.
Steady viscous incompressible flow driven by a pressure difference in a planar
T-junction channel
Nikolay Moshkin* and Damrongsak Yambangwi
School of Mathematics, Suranaree University of Technology, Nakhon Ratchasima, 30000, Thailand
(Received 2 November 2008; final version received 8 February 2009)
The purpose of this article is to present a mathematical formulation and to develop a computational method for theprediction of incompressible flow in a domain with pressure (/or total pressure) specified on boundaries where fluidenters or leaves the domain. The developed method is applied to an investigation of laminar steady flow in a two-dimensional T-junction. Calculations are performed for a wide range of pressure differences between T-junctionbranches. Flow patterns and pressure fields in the region of parameters where transition between different flowregimes occurs are carefully analysed.
Flow through branching channels is widely used inengineering construction, such as in piping andventilation systems, and is encountered in the humanbody, for example in blood flow in veins and arteries.The mechanics of such flow are complex and not wellunderstood, exhibiting non-trivial flow patterns whichinclude zones of recirculation and streamwise vortices.One important and very popular flow geometry is thatof flow in bifurcating 908 T-junctions.
In the large number of numerical and experimentalinvestigations on viscous incompressible flow inbifurcating T-junctions, both Newtonian and non-Newtonian fluids are considered. The flow is analysedin terms of topology pattern, particle path and wallshear stress. Let us mention just some of the morerecent articles which will provide an adequate review ofthis problem. A computational method for the predic-tion of incompressible flow in domains with specifiedpressure boundaries is developed in Kelkar andChoudhury (2000). The proposed method is appliedfor predicting incompressible forced flow in branchedducts. Fully developed velocity boundary conditions(i.e. a parabolic profile) are applied at the inlet, andconstant pressure boundary conditions are applied atthe exit of the two downstream channels of planarT-shape and Y-shape branch configurations. Numer-ical predictions of the laminar fluid flow and heattransfer characteristics in planar (two-dimensional)
impacting T-junctions have been reported by El-Shboury et al. (2003). Fully developed velocity andtemperature profiles were assumed at the inlet face ofthe impacting junction. The results include wall shearstress distributions, streamlines showing the number,location, and size of the recirculation zones, thepressure loss coefficient, wall heat flux distributions,isotherms and the overall rate of heat transfer.
In the study of Tsui and Lu (2006) a numericalmethod is employed to examine the flow in symmetrical,two-dimensional branches of Y-shape and T-shape. Themethodology is based on a pressure-correction proce-dure within the frame of unstructured grids. Specifiedpressures are imposed at the outlet of the two branches.The area ratio of the branches is allowed to vary in therange of 2–3. The effects of slightly different pressuresprescribed on the outlets are investigated.
An investigation of laminar steady and unsteadyflows in a two-dimensional T-junction was carried outby Miranda et al. (2008) for Newtonian and non-Newtonian fluid analogues to blood. Under steadyflow, calculations were performed for a wide range ofReynolds numbers and extraction flow rate ratios, andaccurate data for the recirculation sizes were obtainedand tabulated. At the inlet, the streamwise velocitycomponent and the shear stress component areprescribed, based on available analytical solutions.The velocity follows a parabolic shape for the steadyflows and the Womersley solution for the pulsatingflows. At the walls the no-slip condition is applied
International Journal of Computational Fluid Dynamics
Vol. 23, No. 3, March 2009, 259–270
ISSN 1061-8562 print/ISSN 1029-0257 online
� 2009 Taylor & Francis
DOI: 10.1080/10618560902815204
http://www.informaworld.com
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directly, as a Dirichlet condition, and the shear stress iscalculated from the local velocity distribution. At thetwo outlets of the T-junction the flow rates in each ofthe outlets are prescribed. Blood flow dynamics in theaneurysm dome and neck have been studied inValencia (2006). A steady, sinusoidal and physiologi-cally representative waveform of inflow for a meanReynolds number is utilised as boundary conditions atthe entrance. Symmetric and asymmetric outflowconditions in the branches were also studied.
There are three types of boundary conditionsrelevant to the present flow problems: inlets, outletsand solid walls. As a rule, at the inlet, the streamwisevelocity component is prescribed, based on availableanalytical solutions. At the solid walls the no-slipcondition is applied. At the outlets (maybe more thantwo) of the junction either the pressure is prescribed,and the flow split, or the flow rates in each of theoutlets are prescribed. In all these cases the direction ofthe flow at the domain boundary is assumed to beknown. To the best of authors knowledge there is lackof work on steady flow of fluids through T-junctions incases where only the pressure difference betweenbranches is known and the flow within the domainneeds to be determined. Such kinds of problems arewell-posed from the mathematical point of view andhave physical meanings.
Antontsev et al. (1990), Ragulin (1976), Ragulinand Smagulov (1980) have studied the solvability ofinitial boundary value problems where, on through-flow parts of the domain boundary (inlet and outletparts), the values of pressure or total pressure areprescribed. In Ragulin (1976) and Ragulin andSmagulov (1980) the problems for the homogeneousNavier-Stokes equation have been considered. Thewellposedness of the non-homogeneous Navier-Stokesequation has been investigated in Antontsev et al.(1990). Because these results are not well known, weshall shortly represent the well-posed statement ofinitial boundary value problems with specified pressureon through-flow boundaries. Let us call the problemwhere fluid can enter or leave the domain throughparts of the boundary as the ‘flowing through problem’for viscous incompressible fluid flow. The objective ofthis work is twofold:
(i) To develop and validate the numerical methodto solve flowing through problems.
(ii) To analyse the numerical solution in order toelucidate the flow topology patterns dependingon a pressure drop between branches of theplanar T-junction channel.
In the following section of this article, a briefoverview of various kinds of well-posed flowing
through problems for the incompressible Navier-Stokes equation is presented. This is followed by adescription of the finite volume numerical method withemphasis on implementation of boundary conditionson through-flow parts. The numerical method is thenvalidated by comparing our numerical results withknown experimental and computational data for thetwo-dimensional laminar flow through a 908 T-junc-tion channel. The computed solutions are analysed interms of flow topology patterns depending on pressuredrop between T-junction branches.
2. Mathematical models
Let us consider the flow of viscous liquid through abounded domain Q of R3. Let �1
k, k ¼ 1, . . . , Kdenote parts of the boundary �¼@Q where the fluidenters or leaves the domain. Let �0
l , l ¼ 1, . . . , L beimpermeable parts of the boundary, D ¼ Q 6 (0,T),S ¼ �6(0,T), Sa ¼ �a 6 (0,T), a ¼ 0, 1. The flowingthrough problem is to find a solution to the Navier-Stokes system
r@~u
@tþ ð~u � rÞ~u
� �¼ mD~u�rp; r �~u ¼ 0;
in the domain D ¼ Q 6 (0,T) with appropriate initialand boundary conditions, where ~u is the velocityvector, p is the pressure, r is the density, and m is theviscosity. The initial data are
~u jt¼0¼ ~u0ð~xÞ; r �~u0 ¼ 0; ~x 2 Q:
On the solid wall �0l , the no-slip condition holds
~u ¼ 0; ð~x; tÞ 2 S0l ; l ¼ 1; . . . ; L:
On through-flow parts �1k; k ¼ 1; . . . ; K three types
of boundary conditions can be set up to make theproblem well-posed. As shown in Antontsev et al.(1990) and Ragulin (1976) the conditions are thefollowing:
. On the through-flow parts �1j ; j ¼ j1; . . . ; jn the
tangent components of the velocity vector andthe total pressure are prescribed
~u �~tm ¼ Gmj ð~x; tÞ; m ¼ 1; 2;
pþ 1
2rj~u2j ¼ Hjð~x; tÞ; ð~x; tÞ 2 S1
j ; j ¼ j1; . . . ; jn:
Here, ~t1;~t2 are the linearly independent vectorstangent to �1
j , and Gmj ð~x; tÞ, andHjð~x; tÞ are given
on S1j ¼ �1
j � ð0;TÞ functions.
260 N. Moshkin and D. Yambangwi
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. On the through-flow parts �1l ; l ¼ l1; . . . ; ln the
tangent components of the velocity vector andpressure are known
~u �~tm ¼ Gml ð~x; tÞ; m ¼ 1; 2;
p ¼ Hlð~x; tÞ; ð~x; tÞ 2 S1l ; l ¼ l1; . . . ; ln:
Here, Gml ð~x; tÞ, and Hlð~x; tÞ are given on
S1l ¼ �1
l � ð0;TÞ functions.. On the through-flow parts �1
s ; s ¼ s1; . . . ; sn thevelocity vector (all three components) has to beprescribed
It should be mentioned that various combinationsof boundary conditions on S1
k; k ¼ 1; . . . ; K give thewell-posed problem.
3. Discretisation
Let us represent a numerical algorithm for the flowingthrough problem. Although some of the main aspectsare well known in the literature, for the sake ofcompleteness the issue is illustrated.
3.1. Time discretisation
The time discretisation used here is based upon avariation of the projection scheme. Using explicit Eulertime stepping, the algorithm is as follows: Set~u jt¼0¼ ~u0,then for n � 0 compute ~u�;~unþ1; pnþ1 by solving
First sub-step:
~u� �~unDt
þ ð~un � rÞ~un ¼ nD~un: ð1Þ
Second sub-step:
~unþ1 �~u�Dt
¼ �rpnþ1; ð2Þ
r �~unþ1 ¼ 0; ð~unþ1Þ�0 ¼ 0; ð3Þ
where Dt ¼ T/N is the time step, N is the integer,~un � ~uð~x; nDtÞ, and pnþ1 � pð~x; ðnþ 1ÞDtÞ. Withoutloss of generality, density is equal to one, r ¼ 1.
3.2. Space discretisation
For the sake of simplicity and without losing generality,the formulation of the numerical algorithm is illu-strated for a two-dimensional domain. Let ~u ¼ ðux; uyÞbe the velocity vector, where ux and uy are the Cartesiancomponents in x and y direction, respectively. The finitevolume discretisation is represented for non-orthogonal quadrilaterals grid. The collocated variablearrangement is utilised. Each discrete unknown isassociated with the centre of the control volume O.Integrating Equation (1) on each control volume O,followed by the application of the Gauss theorem yields
ZO
f� � fn
DtdOþ
IS
fnð~un �~nÞdS ¼ nIS
rfn �~ndS;
ð4Þ
where the variable f can be either ux or uy,~un is such that
r �~un ¼ 0, S is the boundary of the control volumeO (forexample, in the case shown in Figure 1a), S is the union ofthe control volume faces s, e, n, w) and ~n is the outwardunit normal vector to S. The midpoint rule approxima-tion of the surface and volume integrals leads to
ZO
f� � fn
DtdO � f� � fn
Dt
� �P
DO; ð5Þ
Figure 1. A typical 2D control volume and the notation used.
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IS
fnð~un �~nÞdS �X
c¼e;s;n;wfncð~u
n �~nÞc Sc; ð6Þ
IS
rfn �~ndS ¼IS
DnfndS �
Xc¼e;s;n;w
ðDnfnÞc Sc; ð7Þ
where DO is the volume of the control volume O, Sc isthe area of the ‘c’ control volume face, and (Dnf)c is thederivative of Cartesian velocity components in thenormal direction at the centre of the ‘c’ control volumeface. To estimate the right-hand side in Equations (6)and (7), we need to know the value of the Cartesianvelocity components and its normal derivative on theboundary of each control volume. The 2D interpola-tion of irregularly spaced data (see Donald 1968) isused to interpolate Cartesian velocity components onthe boundary of each control volume in Equation (6).Only the east side of the 2D control volume shown inFigure 1a will be considered. The same approachapplies to other faces; only the indices need be changed.For example, let fk be the value of the Cartesianvelocity components at node k where k ¼ N, P, S, SE,E, NE and L�2ðe;kÞ ¼ 1=½ðxe � xkÞ2 þ ðye � ykÞ2�. Using2D interpolation yields
fe ¼Xk
L�2ðe;kÞfk
!� Xk
L�2ðe;kÞ
!;
k ¼ N;P;S;SE;E;NE:
ð8Þ
The derivative of Cartesian velocity componentsin the normal direction at the centre of the controlvolume face in Equation (7) can be calculated byusing the central difference approximation (seeFigure 1a)
ðDnfÞe �fE0 � fP0
LðP0;E0Þ:
The auxiliary nodes P0 and E0 lie at the intersectionof the line passing through the point ‘e’ in the directionof normal vector ~n and the straight lines which connectnodes P and N or E and NE, respectively; L(P0,E0)
stands for the distance between P0 and E0. The valuesof fE0 and fP0 can be calculated by using the gradientat the control volume centre
where ~xP, ~xE, ~xP0 and ~xE0 are the radius vectors ofP, E, P0 and E0, respectively. The kth Cartesian
component of rfP is approximated using Gauss’stheorem
rfP �~ik ¼@f@xk
� �P
¼ 1
DO
Xc¼e;s;n;w
fcSkc ; S
kc ¼ Scð~n � ik
!Þ;
ð9Þ
where ~n is the outward unit normal vector to Sc, and~ikis the unit basis vector of Cartesian coordinate system(x1, x2) ¼ (x, y). Using Equations (5)–(9) to approx-imate Equation (4), we can determine intermediatevelocity field ~u� (which is not solenoidal) at each gridnode even on the boundary.
In the first sub-step (1) the continuity Equation (3)is not used so that the intermediate velocity field is, ingeneral, non-divergency free. Equation (2) applies bothin continuous and discrete senses. Taking the diver-gence of both sides of Equation (2) and integratingover a control volume O, after applying the Gausstheorem and setting the update velocity field, ~unþ1, tobe divergence free, we get the equation
0 ¼ 1
DO
IS
~unþ1 �~ndS
¼ 1
DO
IS
~u� �~ndS� Dt1
DO
IS
rpnþ1 �~ndS; ð10Þ
that has to be discretised while collocating thevariables in the control volume centres. At this stageof the projection procedure, only the discrete values ofu�x and u�y are already known and represent the sourceterm in Equation (10). A second-order discretisation ofthe surface integrals can be obtained by utilising themean value formula
1
DO
IS
~u� �~ndS � 1
DO
Xc¼e;s;w;n
ð~u� �~nÞcSc;
1
DO
IS
rpnþ1 �~ndS � 1
DO
Xc¼e;s;w;n
ðrpnþ1 �~nÞcSc ð11Þ
It follows that, by substituting Equation (11) intoEquation (10), we get the discrete pressure equation
1
DO
Xc¼e;s;w;n
ð~u� �~nÞcSc �DtDO
Xc¼e;s;w;n
ðDnpnþ1ÞcSc ¼ 0:
ð12Þ
The normal-to-face intermediate velocitiesð~u� �~nÞc; c ¼ e; s;w; n are not directly available. Theyare found using interpolation. The derivative ofpressure with respect to outward normal direction nthrough the cell face ‘c’, ðDnpÞnþ1c is approximated by
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an iterative technique (see Muzaferija 1994 ) to reach ahigher order of approximation and preserve thecompact stencil in the discrete Equation (12). Onlythe east face of the 2D control volume shown inFigure 1a will be considered. The same approachapplies to other faces. Using the second upper index ‘s’to denote the number of iteration, we write
where x is the local coordinate along the grid line con-necting nodes P and E (see Figure 1a). The terms insquare brackets are approximated with high order andare evaluated by using values from the previous itera-tion. Once the iterations converge, the low order appro-ximation term ðDxpÞnþ1;sþ1e drops out and the solutionobtained corresponds to the higher order of approx-imation. The derivatives of pressure are written as
ðDnpÞnþ1;se ¼ ðrp �~nÞnþ1;se ;
ðDxpÞnþ1;se ¼ ðrp �~xÞnþ1;se ;
where ~n is the outward unit normal vector to cell face‘‘e’’, and~x is the unit vector in the direction from pointP to point E. The term ðrpÞnþ1;se is approximatedsimilar to (8)
ðrpÞnþ1;se ¼Xl
L�2ðe;lÞrpnþ1;sl
!� Xl
L�2ðe;lÞ
!;
l ¼ N;P;S;SE;E;NE;
where rpnþ1;sl is the gradient of pressure at grid node l.The kth component of rpnþ1;sl is discretised using theGauss theorem (see Equation (9)). The first term on theright-hand side of Equation (13) is treated implicitly,and a simple approximation is used (that gives acompact stencil)
ðDxpÞnþ1;sþ1e � pnþ1;sþ1E � pnþ1;sþ1P
LðP;EÞ:
The final expression for the approximation of thederivative of pressure with respect to ~n through the cellface ‘e’ (13) can now be written as
ðDnpÞnþ1;sþ1e ¼ pnþ1;sþ1E � pnþ1;sþ1P
LðP;EÞþ rpnþ1;s � ð ~n�~x Þe:
ð14Þ
The terms labeled ‘n þ 1, s’ become zero when~x ¼ ~n. Repeating the computation similar to (13)–(14)
for the other faces of the control volume andsubstituting the result into Equation (12), we generatethe equation for finding the pressure at the nextiteration (n þ 1, s þ 1)
1
DO
Xc¼e;s;w;n
ð~u� �~nÞcSc �DtDO
Xc¼e;s;w;n
ðrpnþ1;sÞcð~n�~xÞc
¼ DtDO
pE � pPLðE;PÞ
� �nþ1;sþ1� pP � pW
LðP;WÞ
� �nþ1;sþ1(
þ pnþ1;sN � pnþ1;sþ1P
LðN;PÞ
!� pP � pS
LðP;SÞ
� �nþ1;sþ1):
ð15Þ
We use pnþ1;sN instead of pnþ1;sþ1N to make the matrixof the algebraic system tridiagonal.
3.3. Implementation of boundary conditions
The Finite Volume Method requires that the boundaryfluxes either be known or expressed through knownquantities and interior nodal values.
Impermeable wall: The following condition isprescribed on the impermeable wall ~u ¼ ~uwall. Thiscondition follows from the fact that a viscous fluidsticks to a solid wall. Since there is no flow through thewall, mass fluxes and convective fluxes of all quantitiesare zero. Diffusive fluxes in the momentum equationare approximated using known boundary values of theunknown and one-sided finite difference approxima-tion for the gradients.
Through-flow part: The implementation of threekinds of boundary conditions on the through-flowparts is addressed here. Only the case where the eastface of the control volume aligns with through-flowboundary �1 will be considered. Other cases aretreated similarly. A sketch of the grid near thethrough-flow boundary and notations are shown inFigure 1b.
. The velocity is set up
~u�1 ¼ ~up ð16Þ
. Since the velocity vector is given, the mass flowrate and the convective fluxes can be calculateddirectly. The diffusive fluxes are not known,but they are approximated using knownboundary values of the unknowns and one-sided finite differences for the gradient. It isimportant to note how boundary condition(16) is involved in the derivation of the discrete
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pressure equation. Because ð~unþ1Þe is given by(16) the approximation of Equation (10)becomes
1
DOð~unþ1 �~nÞe þ
Xc¼s;w;n
ð~u� �~nÞcSc
" #
� DtDO
Xc¼s;w;n
ðDnpnþ1ÞcSc ¼ 0:
. There is no need to approximate (Dnpnþ1)e at
face ‘‘e’’. However, if pressure at the boundary�1 is needed at some stage it is obtained byextrapolation within the domain.
. The tangential velocity and pressure areprescribed
ð~u �~tÞ�1 ¼ G; p�1 ¼ p; ð17Þ
When the tangential velocity and pressure arespecified on the through-flow part the mass andconvective fluxes are not known and must befound during the solution process. The solenoi-dal constraint, r �~u ¼ 0 has to be applied at theboundary where the pressure is specified. Be-cause the through-flow boundaries may not bealigned with the Cartesian coordinates, we shallrefer to a local coordinates system (n, t), which isa rotated Cartesian frame with n in the directionof the normal vector to the through-flowboundary and t in the direction of the tangentialvector to the through-flow boundary. Thevelocity vector ~u ¼ ðux; uyÞ can be expressed interms of velocity components in local orthogonalcoordinates ~u ¼ ðUn;UtÞ, where Un ¼ ~u �~n is thenormal velocity component to the through-flowboundary, and Ut ¼ ~u �~t is the tangential velo-city component to the through-flow boundary,which is known at �1 from boundary condition(17). The continuity equation in terms of localorthogonal coordinates (n, t) reads
@Unþ1n
@nþ@U
nþ1t
@t¼ 0; or
@Un
@n
� �nþ1
�1
¼�@G@t:
ð18Þ
To find the flux on the through-flow part, weneed to calculate the normal velocity (Un)e at theeast cell face ‘e’ (see Figure 1b). The normalderivative of Un at the east cell face is approxi-mated by the one-side difference
@Un
@n
� �nþ1
e0¼ ðUnÞe0 � ðUnÞP
LðP;e0Þ; ð19Þ
where e0 is the point of intersection of the linepassing through node P parallel to the normalvector to �1 and the line coinciding withboundary �1 (see Figure 1b). Following Equa-tions (18) and (19), the normal velocity compo-nent at point e0 is approximated as
ð~unþ1 �~nÞe0 ¼ ðUnþ1n Þe0 ¼ ðU
nþ1n ÞP � LðP;e0Þ
@G
@t
� �e0:
ð20Þ
The discrete pressure equation for controlvolume O near the through-flow boundary hasthe following form
1
DOð~unþ1 �~nÞe0Se þ
Xc¼s;w;n
ð~u� �~nÞcSc
" #
� DtDO
Xc¼s;w;n
ðDnpnþ1ÞcSc ¼ 0: ð21Þ
Here, the point ‘e0’ is used instead of ‘e’ toapproximate the flux through the east face. Inthis case, the order of approximation is reduced tofirst order. Moreover, in many cases the grid canbe arranged so that ‘e0’ coincides with the centre ofthe east face. Substituting Equation (20) intoEquation (21) and utilising (10) at node P yields
1
DO
�ð~u� �~nÞP � Dtðrpnþ1;sþ1 �~nÞP � LðP;e0Þ
@G
@t e0
�
� Se þ1
DO
� Xc¼s;w;n
ð~u� �~nÞcSc
�
� DtDO
Xc¼s;w;n
ðDnpnþ1;sþ1ÞcSc ¼ 0: ð22Þ
The derivative of pressure with respect to theoutward normal direction n at node P approxi-mated by the one-side difference is
ðDnpÞnþ1;sþ1P ¼ pnþ1e0 � pnþ1;sþ1P
LðP;e0Þ;
where L(P,e0) is the distance between nodes P ande0 on the boundary �1.
. The tangential velocity and total pressure areprescribed on through-flow part �1
ð~u �~tÞ�1 ¼ G; pþ 1
2j~uj2 ¼ H; ð23Þ
There is a situation in which mass flux, con-vective flux and pressure are not known. Let us
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use a local coordinates system (Z, t) as in theprevious case. Since we do not know the pressureterm on the through-flow boundary �1 (seeFigure 1b), it is necessary to approximate thepressure on the through-flow part by usingEquation (23). We need to calculate the pressureat point e0. The total pressure on the through-flow part can be expressed in terms of the localorthogonal coordinates (n, t) at point e0 as
We have the previous case where the pressureand tangent component of the velocity vector aregiven on the through-flow parts.
4. Validation tests
The T-junction flow geometry is schematically repre-sented in Figure 2. The origin of the coordinate system
is located in the lower horizontal boundary oppositethe left corner of the branch as demonstrated. The lefthand side branch, the upper branch, the right handside branch and the junction area are denoted by �1,�2, �3 and �4, respectively. All branches have the samewidth, w. The flow rate ratio is defined as b:Q3/Q1,where Q1 and Q3 are the inlet duct and branch ductflow rates per unit span, respectively. The mainrecirculation region is in the branch duct starting aty ¼ ys and ending at y ¼ yr, thus defining a normal-ised recirculation length of YR ¼ (yr 7ys)/w. Thisnomenclature is adapted for the secondary recircula-tion found in the main duct and aligned with the x-direction with the necessary adaptations leading toXR ¼ (xr 7xs)/w.
Four kinds of flowing through problems have beenconsidered in this study. They are the following
. T1. On through-flow part �11 a laminar, fully
developed, parabolic velocity profile is pre-scribed. On through-flow parts �1
2 and �13 the
tangent component of the velocity vector and thepressure are specified.
. T2. On through-flow part �11 a laminar, fully
developed, parabolic velocity profile is pre-scribed. On through-flow parts �1
2 and �13 the
tangent component of the velocity vector and thetotal pressure are specified.
. T3. On through-flow part �11 a laminar, fully
developed, parabolic velocity profile is pre-scribed. On through-flow part �1
2 the tangentcomponent of the velocity vector and the totalpressure are specified. On through-flow part �1
3
the tangent component of the velocity vector andthe pressure are specified.
. T4. On through-flow parts �1i ; i ¼ 1; 2; 3 the
tangent component of the velocity vector andthe pressure are specified.
The first set of calculations is compared with thoseof Hayes et al. (1989), Kelkar and Choudhury (2000),and FLUENT Inc. (1998). The Navier-Stokes equa-tion is dimensionalised with the width, w, as char-acteristic length, the inlet centreline velocity Uc as thecharacteristic velocity and rU2
c as the scale of pressure.A range of Reynolds numbers Re ¼ wUc/n, where n isthe kinematic viscosity, is studied with Re 2 [10,400].The computational domain is set to have lengths ofL1/w ¼ 2 and L2/w ¼ L3/w ¼ 3 according to theresults represented in FLUENT Inc. (1998). Thesquare meshes containing 20, 30 and 40 cells fromwall to wall are used. The studied cases start from amotionless state. A steady flow is achieved if thefollowing condition is held k~unþ1 �~unk e ¼ 10�8.The maximum norm of grid function is used. A flowing
Figure 2. Schematic geometry of T-junction bifurcationand coordinate system.
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through problem T1 with ux(y) ¼ 4y 74y2, and equalstatic pressure p2 ¼ p3 ¼ 0 at �1
2 and �13 is considered.
Figure 3 shows the effect of increasing the Reynoldsnumber on the flow split between the main and the sideexit branches. The value of b increases from 0.5 for asmall Reynolds number, Re 5 10, to about 0.9 atRe ¼ 400. In Table 1, the flow split predicted by thepresent method in the sequence of grids is comparedwith the computation results of FLUENT Inc. (1998)software.
The second set of calculations is compared with theexperiments of Liepsch et al. (1982) and the numericalcalculations of Miranda et al. (2008). The experimentalcase of Liepsch et al. (1982) pertains to a Reynoldsnumber of ReQ ¼ 248 (ReQ:r�u1w/m, �u1 ¼ Q1/w is thebulk velocity) and a fixed flow rate ratio b* ¼ Q2/Q1 ¼ 0.44. The relation between the Reynolds numberdefined by bulk velocity, ReQ and the Reynoldsnumber defined by centreline velocity, Rec, isRec ¼ rUcw
m ¼32ReQ. In our numerical experiments,
the computational domain was set to have lengthsL2/w ¼ L3/w ¼ 12, whereas for the inlet duct L1/w ¼ 2. The flow through problem T1 with the fullydeveloped parabolic velocity profile ux(y) ¼ 4(y 7y2),p2 ¼ 0, and various p3 is considered. To carry out thecomparison with the experiments of Liepsch et al.(1982) and numerical calculations of Miranda et al.(2008), we have to find the value of p3 whichcorresponds to the flow rate ratio b* ¼ 0.44. Therelation between b* and p3 is shown in Figure 4 forRec ¼ 372 (ReQ ¼ 248). From the results of the serial
computations represented by Figure 4 we found thatb* ¼ 0.44 corresponds to the value p3 ¼ 0.129. Fig-ure 5 shows a comparison of velocity profiles with theexperimental and numerical results represented inMiranda et al. (2008). The velocity profiles comparedwell with the numerical results of Miranda et al.(2008), which are shown by solid and dashed lines.Triangle signs (D) are used to represent the experi-mental data of Liepsch et al. (1982). Circle signs ()are chosen to represent our computational results.Everywhere within the main channel the predictions ofMiranda and our results are in excellent agreementwith each other and with the experimental data ofLiepsch et al. The measurements indicate the existenceof a separation bubble along the outer wall of the mainchannel (Zone XR ¼ (xr 7xs)/w) which is accuratelycaptured by Miranda and our calculations. However,in the branch channel the two-dimensional numericalprediction appears to consistently overpredict thevelocities within Zone YR (profiles in Figure 5d,e).They also predict that the flow becomes fully-developed in the downstream region earlier than themeasurements would indicate (profiles in Figure 5f).These discrepancies are reduced by accounting for theside solid boundaries in the three-dimensional simula-tion. Table 2 lists values of the lengths of twoseparation regions, XR and YR, for three refined gridswith h1 ¼ 1/20, h2 ¼ 1/30, and h3 ¼ 1/40. The bench-mark data of Miranda et al. (2008) are represented insecond column of Table 2.
There are no numerical and experimental results tovalidate the algorithm for flowing-through problemsT2, T3 and T4. We used the solution of problem T1 togenerate test solutions for problems T2, T3 and T4.The values of total pressure or pressure on through-flow parts was specified by the numerical solution offlowing through problem T1. The steady state solutionof problems T2, T3, and T4 compared well with thegenerating solution of problem T1.
The above results demonstrate the ability of thedeveloped numerical method to predict complexlaminar flow through a bounded domain with
Figure 3. Fractional flow rate b in main branch as afunction of Reynolds number Re.
Table 1. Flow rate ratio b, Re ¼ 10, 100, 200, 300, 400.
Figure 4. The relation between p3 and b* for ReQ ¼ 248.
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pressure (or total pressure) specified at through-flowparts.
5. Flow driven by pressure differences in 908 Planar T-junctions
The flowing through problem with pressure andtangent component of velocity vector given onthrough-flow parts �1
1, �12 and �1
3 of T-junction iswell-posed (see for example Ragulin (1976), Antontsevet al. (1990)). This problem has a physical meaning.The physical situation consists of a T-junction channelconnected to three reservoirs with different levels offluid. We have to find the flow direction and volumerate through each branch of the T-junction, as well asthe pattern of flow in the junction region. A schematicrepresentation of the T-branch channel along with
relevant nomenclature is shown in Figure 2. To takeadvantage of symmetry, the sizes of the T-junctionbranches are chosen the same: L1/w ¼ L2/w ¼ L3/w ¼ 3. The Navier-Stokes equation is reduced tonon-dimensional form with width w, as characteristiclength, the dynamic velocity n /w as the characteristicvelocity, and r(n /w)2 as the scale of pressure. Withoutloss of generality we assume that the pressure atthrough-flow boundary �1
2 is equal to zero. No-slipboundary conditions are given on solid walls of theT-junction. The following boundary conditions areprescribed on the through-flow parts
~u �~t ¼ uy ¼ 0; p ¼ p1; ðx; yÞ 2 �11;
~u �~t ¼ ux ¼ 0; p ¼ 0; ðx; yÞ 2 �12;
~u �~t ¼ uy ¼ 0; p ¼ p3; ðx; yÞ 2 �13;
where pi ¼ ðpdi w2Þ=ðrn2Þ, i ¼ 1, 3 are the dimen-sionless pressure (superscript ‘d’ denotes dimensionalquantity).
A schematic diagram of flow regimes on thep1 7 p3 plane is symbolised in Figure 6. Because ofthe symmetry (L1 ¼ L3), we have studied flow patternsonly below the line p1 ¼ p3. Each studied case startsfrom the motionless state. Let us define the split
Figure 5. Comparison between experimental data of Liepsch et al. (1982) and numerical data of Miranda et al. (2008)(Re ¼ 248 and b21 ¼ 0.44).
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number bij as the ratio of flow rate through branch i tothe flow rate through branch j
bij ¼ Qi=Qj; Qi ¼Z
�1i
~u �~n ds;
where ~n is outward normal to through-flow part �1i .
The Reynolds number Rei based on the flow rate Qi iscomputed as steady state was reached with given p1and p3. As shown in Figure 6, six kinds of flow patternsexist over the range of 75000 5 p1, p3 5 5000. Theyare denoted by the Roman numbers from I to VI.Regime I is the impacting T-junction. Laminar flowenters the T-junction through the region �2. Atjunction region �4, the inlet flow stream divides intotwo outlet streams through regions �1 and �3. RegimeII corresponds to the case where fluid enters theT-junction through region �1 and �2. At junctionregion �4 these inlet streams merge and leave theT-junction through region �3. Regime III denotes theflow pattern in which fluid enters the T-junction
through boundary �11, separates in junction region �4
and leaves the domain through through-flow parts �12
and �13. Regime IV represents the case in which fluid
enters through boundaries �11 and �1
3, merges in �4 andleaves the domain through �1
2. Regime V and VI aresymmetric with respect to the line GA to the RegimesII and III, respectively. Dashed lines in Figure 6indicate equal volume flow rate through two inlet oroutlet sections. Along solid lines OB, OD and OF, theflow changes pattern (or there is zero volume flow ratethrough some branch of the T-junction). For example,there exists flow pattern III in the region between linesOB and OD. Along line OC split number b21 ¼ 70.5.
The streamlines and pressure contours of theparticular case are shown in Figure 7a,b for p1 ¼ p3.In the p1 7p3 plane, cross sign (6) points to this case.In Figure 7a,b and thereinafter an arbitrary stepbetween different streamlines and pressure contourswas used in order to illustrate the main features of theflow field. Figure 7 shows the case of the impactingT-junction, Regime I, for b12 ¼ b32 ¼ 70.5. A recir-culation zone can be seen in the top part of outletregions �1 and �3. A large pressure gradient is observednear stagnation point in region �4. There is a pressuregradient opposite to the flow direction along the wallsin regions �1 and �3 where recirculation occurs.
Figure 8 shows the streamline and pressurecontours for the case pointed out by the triangle sign(D) in Figure 6. Regime III for b21 ¼ b31 ¼ 70.5 isrepresented in Figure 8. The main flow from inletregion �1 divides so that a half portion enters thebranch channel �2 and the remaining half continuesdownstream in the region �3. A second recirculationzone is observed along the lower horizontal wall of the�3 region. A large pressure gradient is observed nearthe junction corners, and a positive pressure gradientexists along the walls where recirculation occurs.
It is important to analyse the flow structure nearthe lines OF, OD and OB where the fluid flow switchesregime. For example, from the left hand side of curve
Figure 7. The streamline and pressure contours for the cases correspond to cross signs (6).
Figure 6. Diagram of flow regimes.
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OF, we observe Regime I, while from the right handside, we observe Regime II. Figure 9 shows streamlinepatterns near line OF. In the p1 7p3 plane (Figure 6)signs (¤) are used to indicate this. The recirculationzone in the region �1 blocks the passage of the fluidinto the branch �1 (see Figure 9b). Figure 9a showsthe flow of the Regime I for b12 ¼ 70.012 andb32 ¼ 70.988. Fluid enters into branch �1 throughthe narrow gap between the recirculation zone and the
wall y ¼ 0 of the T-junction. Figure 9c shows the caseof flow belonging to Regime II for b12 ¼ 0.005 andb32 ¼ 71.005. Fluid enters into region �4 through thegap near the left corner of the T-junction.
Figure 10 shows streamline patterns near line OD.The down triangular signs (r) denote these cases inFigure 6 in the p1 7p3 plane. Figure 10b correspondsto the case b21 ¼ 0 due to the zero flow rate throughboundary �1
2. The recirculation zone closes branch �2
Figure 8. The streamline and pressure contours for the cases corresponding to triangle sign (D).
Figure 9. The streamline and pressure contours for the cases corresponding to square signs (¤).
Figure 10. The streamline and pressure contours for the cases corresponding to down triangle signs (r).
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for fluid flow. Figure 10a represents the streamlinepattern for Regime II for b21 ¼ 0.003 and b31 ¼71.003. Flow stream from �2 merges with the mainstream through the small gap near the left corner of theT-junction. Flow of Regime III is shown in Figure 10cfor b21 ¼ 70.003 and b31 ¼ 70.997. Fluid enters intothe branch �2 through the small gap between therecirculation zone and the right corner of the sidebranch. The secondary recirculation zone along wally ¼ 0 was not observed in these cases.
6. Conclusion
Numerical algorithms to simulate viscous incompres-sible fluid flow driven by a pressure difference havebeen developed and validated. These algorithms wereapplied to simulate viscous incompressible flow in theplanar T-junction channel. The comparisons withknown experimental and computational data for thetwo-dimensional flow through 908 T-junction demon-strate the robustness and accuracy of the developedcode. The method allows us to determine the directionof flow through each branch and the structure of theflow pattern in the junction region in cases wherepressure (/or total pressure) is given on the through-flow parts of domain. A schematic diagram of flowpatterns depending on pressure differences betweenT-junction branches has been constructed. In particu-lar, the present paper carefully analyses flow patternsand pressure fields in the region of parameters wheretransition between different flow regimes occurs.
Acknowledgements
This work was supported by Royal Golden Jubilee Ph.D.program of Thailand (PHD/0006/2548).
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