functions. Solutions of this system are obtained by the least-squares nite element method. A feature of this approach isthat the linearised system gives rise to a symmetric and positive-denite linear algebra problem at each Newton iteration.

The NavierStokes system of equations for an incompressible uid in steady ow and on which no body

of equations is non-linear, and the non-linear term comes to dominate for high values of the Reynolds number.Much eort has gone into obtaining nite-element solutions of these equations, see for example [1113,18,23].

* Corresponding author.E-mail address: p.bolton@manchester.ac.uk (P. Bolton).

Journal of Computational Physics 213 (2006) 174183

www.elsevier.com/locate/jcp0021-9991/$ - see front matter 2005 Elsevier Inc. All rights reserved.forces act is

1Re

r2~u~u r~urp 0 in X; 1r ~u 0 in X. 2

The enclosed ow boundary conditions for (1) and (2) are

~u ~g on C; 3ZXp dX 0. 4

ThequantityRe is theReynolds number, whichwedene as being the inverse of the viscosity parameter m. This setCare over handling the incompressibility term is needed to ensure good results are obtained. 2005 Elsevier Inc. All rights reserved.

Keywords: First-order; Least-squares; Finite elements; Conservation of mass; NavierStokes equations; Stress and stream functions

1. A rst-order reformulation of the NavierStokes equations

1.1. The NavierStokes equationsAvailable online 23 September 2005

Abstract

The NavierStokes equations for ow in a plane are reformulated as a rst-order system in terms of stress and streamdoi:10School of Mathematics, University of Manchester, P.O. Box 88, Sackville Street, Manchester M60 1QD, United Kingdom

Received 3 February 2005; received in revised form 28 July 2005; accepted 9 August 2005P. Bolton *, R.W. ThatcherA least-squares nite element method for theNavierStokes equations.1016/j.jcp.2005.08.015

1.2. T

Th

wherelook

neceshaveformu[14] ahead

1.3. A

Fo

Then

We in

P. Bolton, R.W. Thatcher / Journal of Computational Physics 213 (2006) 174183 175Let d denote the deformation tensor

d 12

2ou1ox

ou1oy

ou2ox

ou1 ou2 2 ou2

0BB@

1CCA.r R u1

ou2ox

u2 ou2oy@ A ~u r~u.R u21 u1u2

u1u2 u22

.

This matrix has a divergence with two components

r R 2u1

ou1ox

u2 ou1oy u1ou2oy

2u2ou2oy

u1 ou2ox u2ou1ox

0BB@

1CCA.

For incompressible uids, which satisfy Eq. (2), this can be simplied to

u1ou1ox

u2 ou1oy0BB

1CCRe ox2 oy2 ox ox oy

1Re

o2u2ox2

o2u2oy2

op

oy u1 ou2ox u2

ou2oy

0.

troduce R, the Reynolds stress tensor [19], dened as 1 o2u1 o

2u1

op u1 ou1 u2 ou1 0;r~u ox oxou1oy

ou2oy

BB@ CCA.(1) can be written explicitly asou1 ou20 1rst-order reformulation of the NavierStokes equations in terms of stress and stream functions

r a uid of velocity ~u u1; u2 in a Cartesian coordinate frame with axes x and ysary to recast the second-order system as a nonlinear rst-order one. A number of such formulationsbeen considered in the literature. One example of such a formulation is the velocityvorticitypressurelation, see [4]. Solutions of a backward facing step problem using this formulation can be found innd the driven cavity problem is solved in [1417]. Another example is the semi-linear velocityvorticityformulation, see [2,3,16].dimensional subspace of [H1(X)]n. Hence in obtaining solutions of the NavierStokes equations (1) and (2) it isXmi1

kLiU fik20;

Li is a rst-order dierential operator and U 2 [H1(X)]n satises appropriate boundary conditions. Wefor continuous, piecewise dierentiable nite element approximations Uh to the minimum U in a nite-found by minimising the least-squares functionalhe rst-order least-squares nite element method

e rst-order least-squares solution of a system of m equations in n unknowns holding over a region X isoy ox oy

We c

to ob

equat

secon

ariseA

176 P. Bolton, R.W. Thatcher / Journal of Computational Physics 213 (2006) 174183Jiang in [15]. The one we shall use is Newtons linearisation method, see for example [2,3,15,17]. This is anTiteratthem.number of linearisation techniques have been employed in the nite element literature, as highlighted byBefore moving on to develop a least-squares functional for the set of equations (6)(9), we shall rst line-i c 1 2 1

d point with a dierent x coordinate or U2 at a second point with a dierent y coordinate, see [21].where~g g1; g2. Given these conditions then the solution of the system (6)(9) is unique provided that lin-ear constraints K (U) = 0, i = 1, . . . , N are also specied. We x both U and U at a point and either U at aions, see [6,21]. In terms of these variables the enclosed ow conditions (3) are

U 3 g2x; y; U 4 g1x; y on C; 10Appropriate boundary conditions for this system of equations are as for the equivalent system for the Stokesoy ox

2moU 3oy

2m oU 4ox

f4. 9oU 1 oU 2 f3; 8 oU 1ox

oU 2oy

2m oU 3oy

2m oU 4ox

U 23 U 24 f1; 6oU 1oy

oU 2ox

2m oU 3ox

2m oU 4oy

2U 3U 4 f2; 7So the NavierStokes equations (1) and (2) can be written in rst-order form asoy

ox

2moy

ox

2U 3U 4.oU 1 oU 2 oU 4 oU 3 U 1 /x; U 2 /y ; U 3 wx; U 4 wytain

oU 1ox

oU 2oy

4m oU 3oy

U 23 U 24;We eliminate the pressure p and make the substitutionsu1 wy ;u2 wx.an write (5) in terms of / and w as

/yy p 2mwxy w2y ; /xy mwyy wxx wxwy ;/xx p 2mwxy w2x ./xy /xxand a stream function w dened asrR /yy /xyThen introduce a tensor rR such that

rR pI 2md R; 5where I is the identity matrix. Given that (1) and (2) hold then

r rR 0.We introduce a stress function / dened so that !ive technique, with an updated solution U = (U1, U2, U3, U4) to be obtained from an estimate or

f 3 ; f

with

~U U 2 H X jU g ; U g on C; K U 0; i 1; . . . ;N ;such that the functional in Eq. (15) is a minimum. Given that the function U minimises the functional in Eq.

lim X 0 8V 2 ~V

and t

LULV dX LVF dX 8V 2 ~V .

nite-dimensional subset ~V h of the test space ~V . The nite element solution Uh satises the relation

In [5,equatmassAlthoweighothervation term or terms in these reformulations has been considered elsewhere, see for instance [8].

P. Bolton, R.W. Thatcher / Journal of Computational Physics 213 (2006) 174183 177We have found that the terms corresponding to Eqs. (8) and (9) in the least-squares functional which arisesfrom the NavierStokes system (6)(9) also have to be weighted. Here, we present solutions obtained by mini-

misinZXLUhLV h dX

ZXLV hF dX 8V h 2 ~V h.

6], we showed that mass is not conserved well in the solutions of the equivalent system for the Stokesions for incompressible ow with enclosed ow boundary conditions. We saw in [5,6] that much moreis conserved if we weight the terms in the least-squares functional corresponding to Eqs. (8) and (9).ugh conservation of mass is enforced by Eq. (9), Eq. (8) is of the same form and it seems natural tot this equation by the same factor. We note that loss of mass is observed in least-squares solutions ofrst-order reformulations of the Stokes and NavierStokes equations and weighting of the mass conser-X X

In obtaining nite element solutions we work with a nite-dimensional subset ~Uh of the trial space ~U and at!0 dt

hereforeZ Z(15) then

dR LU tLV F 2dX3 2 4 1 i cWe seek the function U, where U = (U1, U2, U3, U4), from an appropriate trial space U , namely

1 4n o~V fV 2 H X jV 3 0; V 4 0 on C; KiV 0; i 1; . . . ;N cg.T ~VT = (V1, V2, V3, V4), namely

1 4i1

X the domain on which Eqs. (11)(14) hold. We introduce a test space ~V with elements V such thatXkLi U F i k20 154 . The corresponding least-squares functional is

4We write this system of equations as L*U = F*, with L* a linear operator, U as above and F T f 1 ; f 2 ;2moU 3oy

2m oU 4ox

f4 f 4 . 14previous iterate U^T U^ 1; U^ 2; U^ 3; U^ 4. Applying Newtons linearisation method to the system (6)(9) we

obtain the equations

oU 1ox

oU 2oy

2m oU 3oy

2m oU 4ox

2U 3U^ 3 2U 4U^ 4 f1 U^ 32 U^ 42 f 1 ; 11oU 1oy

oU 2ox

2m oU 3ox

2m oU 4oy

2U 3U^ 4 2U 4U^ 3 f2 2U^ 3U^ 4 f 2 ; 12oU 1oy

oU 2ox

f3 f 3 ; 13g both the unweighted functional

highly dependent on the form of the grid, see [6]. In approximating the NavierStokes equations, we use so-calledcomp

On

whils

The lparam

WFor i

178 P. Bolton, R.W. Thatcher / Journal of Computational Physics 213 (2006) 174183U 3 0; U 4 0.inear constraints are that U1 = 0 and U2 = 0 at the point B and that U2 = 0 at the point D. The viscosityeter m is set equal to 102.

e see from Table 1 that there is substantial loss of mass in the solution of the unweighted SN functional.On the walls BC, DE and AO, we apply the no-slip boundary conditionsthe inlet AB, we apply the boundary conditions

U 3 0; U 4 y 1 y ;t on the outlet CD, we apply the conditions

U 3 0; U 4 0:125 1 y2

.not necessarily hold for other forms of grid, see [20]. In particular they satisfy the grid decomposition prop-erty, see [1,9,10].

As a particular nite-dimensional space, we chose the set of piecewise continuous linear functions denedon a triangulation of X. In [7], convergence rates of least squares solutions of the rst-order reformulation ofthe Stokes equations in terms of the velocity, vorticity and pressure and with enclosed ow boundary condi-tions are analysed. It is shown there that if linear elements are used to approximate all the variables then con-vergence is suboptimal in the vorticity and pressure. But the equivalent system to Eqs. (6)(9) for the Stokesequations is the one in terms of the stress and stream functions presented in [21]. In [22], convergence rates ofsolutions of this system are analysed. Optimal convergence in H1, and in L2 given suitably smooth analyticalsolutions, is proved for a number of dierent boundary conditions including enclosed ow (10).

Local and global stiness matrices and right-hand side vectors can be generated and assembled in the usualway to give a linear system for the unknown nodal values. As they arise from a least-squares functional, thelinear systems are symmetric and positive-denite at each iteration.

The stream function can be recovered from the solution in the velocity variables by minimising thefunctional

Ss wx U 3k k20 kwy U 4k20.

2. Simulation of ow over a backward facing step

Our region is [2, 0] [1, 0] [ [0, 6] [1, 1] as illustrated in Fig. 1.Union Jack grids or grids which are topologically equivalent, in preference to other grids like thoseosed of unidirectional triangles. It is known that Union Jack grids possess special properties which dooy ox 0 oy ox 0In this paper, we set W = 103 which is large enough so that there is relatively little ow lost but not so largethat the linear algebra systems arising in the nite element solution are ill-conditioned, see [5]. Experience insolving the corresponding Stokes system suggests that the quality of solutions when weighting is employed isW oU 1oU 2 f3 2W 2moU 32moU 4 f4 2.SN ;W ox oy 2m oy 2m ox U 3U 4 f1 0 oy ox 2m ox 2m oy 2U 3U 4 f2 0

oU 1 oU 2 oU 3 oU 4 2 2

2 oU 1 oU 2 oU 3 oU 4 2

and the weighted functionaloy

ox

f3 0

2moy

2mox

f4 0SN oU 1ox oU 2oy

2moU 3oy

2moU 4ox

U 23U 24 f1

2

0

oU 1oy

oU 2ox

2moU 3ox

2moU 4oy

2U 3U 4 f2

2

0

oU 1 oU 2 2 oU 3 oU 4 2nstance at ny = 4 over 86% of the mass is lost between the inlet and the line x = 0 through the re-entrant

corner. Even at ny = 32 almost 34% of the mass is lost between these two lines. Much less mass is lost in thesolution of the weighted SN, W functional, see Table 2. In this case at ny = 32 virtually all of the mass is con-served between the inlet and the line x = 0.

Figs. 2 and 3 show the contours of the stream functions obtained from the unweighted and weighted solu-tions respectively. Fig. 2 demonstrates graphically the loss of uid in the solution, particularly in the area nearthe re-entrant corner. The solution shown in Fig. 3 is much more acceptable. There is some recirculation in theweigh

3. Flo

A

B C

DE

O

Fig. 1. Planar backward facing step grid at ny = 2.

P. Bolton, R.W. Thatcher / Journal of Computational Physics 213 (2006) 174183 179Our elements are triangular and the approximations are piecewise linear. The grid for the region with param-eter ng = 1 is shown in Fig. 4 and its rst renement, for which ng = 2, is shown in Fig. 5. On the inlet line AFand the outlet line DE the boundary conditions are

U 3 0; U 4 0:16y 2:5 y .The uid is stationary on the walls AB, CD and EF as well as on the restriction itself so that

U 3 0; U 4 0.

Table 1Axial ow in the solution of the SN formulation

ny Axial ow

x = 2 x = 0 x = 34 0.15625 0.02080 0.05201

81632

TableAxial

ny

481632fx; y j x 2 0; 10; y 2 0; 2:5; x 32 y2 P 1g.We approximate ow through a channel [0, 10] [0, 2.5] with a semi-circular restriction of radius unity, asillustrated in Fig. 4. The region isw over a semi-cylindrical restrictionin this portion of the region.

ted solution close to the corner labelled E in Fig. 1. This is indicated in Fig. 3 by the closed contour lines0.16406 0.04288 0.077330.16602 0.07463 0.104800.16650 0.11067 0.13061

2ow in the solution of the SN, W formulation, W = 10

3

Axial ow

x = 2 x = 0 x = 30.15625 0.14755 0.155030.16406 0.16171 0.163590.16602 0.16536 0.165850.16650 0.16632 0.16645

180 P. Bolton, R.W. Thatcher / Journal of Computational Physics 213 (2006) 174183The viscosity parameter m is set equal to 102. In this case, we nd that we require three linear constraints. Weset U1 equal to zero at A and U1 and U2 as zero at E.

Just as in the solutions of the backward facing step problem there is a great deal of ow lost in the solutionof the unweighted functional, see Table 3.

Fig. 2. Stream function contours, SN formulation (ny = 16).

Fig. 3. Stream function contours, SN, W formulation (ny = 16).

A B C D

EF

P

Q

Y

Z

Fig. 4. Mesh at ng = 1.

Fig. 5. Mesh at ng = 2.

Table 3Axial ow in the solution of the SN formulation

ng Axial ow

AF PQ YZ DE

4 0.41594 0.12013 0.18685 0.415948 0.41649 0.17901 0.23873 0.4164916 0.41662 0.25224 0.29755 0.41662

Table 4Axial ow in the solution of the SN, W formulation, W = 10

3

ng Axial ow

AF PQ YZ DE

4 0.41594 0.38965 0.39807 0.415948 0.41649 0.41003 0.41212 0.4164916 0.41662 0.41515 0.41563 0.41662

Fig. 6. Stream function contours, SN formulation (ng = 4).

Fig. 7. Stream function contours, SN, W formulation (ng = 4).

Fig. 8. Recirculation in the solution of the SN, W formulation, at ng = 8.

P. Bolton, R.W. Thatcher / Journal of Computational Physics 213 (2006) 174183 181

182 P. Bolton, R.W. Thatcher / Journal of Computational Physics 213 (2006) 174183At ng = 4 even in the solution of the weighted functional 6.3% of the ow is lost between the inlet line AFand PQ, see Table 4. It is only in solutions of the SN, W functional on the ner grids that the ow is conserved.

The stream functions for the unweighted and weighted solutions at ng = 4 are shown in Figs. 6 and 7,respectively. The diverging contours in Fig. 6 indicate a loss of ow, whilst the plot in Fig. 7 is closer to whatwe might expect for the true stream function. At this level of renement there is no separation or recirculationvisible even in the solution of the SN, W functional. There is some recirculation in the solution of the SN, Wfunctional at ng = 8 and ng = 16 in the region to the immediate right of the cylinder, close to the cornerlabelled C in Fig. 4. A quiver plot of the velocity eld in the solution on the grid ng = 8 in this portion ofthe region is shown in Fig. 8.

4. Conclusion

The least-squares nite element method oers much promise because it gives rise to symmetric and positive-denite systems for which fast direct and indirect methods of solution exist. Although much work has beendone, ecient solution techniques for standard Galerkin formulations of mixed methods for the NavierStokesequations are still being developed. In the formulation presented here the non-linear terms are algebraic andhence the system is semi-linear. This is similar to the formulation in terms of the velocity, the vorticity andthe total pressure or head discussed for instance in [3]. We note that for these semi-linear systems the classi-cation is the same as for the corresponding linear Stokes-equivalent systems so that the equations are elliptic forall values of the Reynolds number. Furthermore boundary conditions which are appropriate for the Stokes-equivalent system will also be appropriate for the system equivalent to the NavierStokes equations. The anal-ysis of semi-linear systems may also be simpler. Other rst-order formulations of the NavierStokes equationsused in obtaining least-squares solutions do not possess this advantageous property.

We have shown here that apparent problems with lack of mass conservation in solutions of incompressibleow obtained using this formulation can be overcome by modifying the technique in a simple way, namelyweighting of particular terms in the least-squares functional. This modication preserves the symmetric posi-tive-deniteness of the linear systems that occur when using this method. Finally we note that the formulationcan also be extended into three dimensions, see [5], although this idea is still being developed.

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[18] J.N. Reddy, An Introduction to Nonlinear Finite Element Analysis, Oxford University Press, Oxford, 2004.[19] D.G. Shepherd, Elements of Fluid Mechanics, Harcourt Brace and World, Inc., New York, 1965.[20] D.J. Silvester, R.W. Thatcher, The eect of stability of mixed nite element approximations on the accuracy and rate of convergence

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P. Bolton, R.W. Thatcher / Journal of Computational Physics 213 (2006) 174183 183

A least-squares finite element method for the Navier - Stokes equationsA first-order reformulation of the Navier ndash Stokes equationsThe Navier ndash Stokes equationsThe first-order least-squares finite element methodA first-order reformulation of the Navier ndash Stokes equations in terms of stress and stream functions

Simulation of flow over a backward facing stepFlow over a semi-cylindrical restrictionConclusionReferences