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OPEN BOUNDARY CONDITIONS FOR THESTREAMFUNCTION -VORTICITY FORMULATION OFUNSTEADY LAMINAR CONVECTIONG. Comini a , M. Manzan a & G. Cortella aa Departmento di Energetica e Macchine , Universit degli Studi di Udine , Via delleScienze 208, Udine, 33100, ItalyPublished online: 17 Apr 2007.

To cite this article: G. Comini , M. Manzan & G. Cortella (1997) OPEN BOUNDARY CONDITIONS FOR THE STREAMFUNCTION-VORTICITY FORMULATION OF UNSTEADY LAMINAR CONVECTION, Numerical Heat Transfer, Part B: Fundamentals: AnInternational Journal of Computation and Methodology, 31:2, 217-234, DOI: 10.1080/10407799708915106

To link to this article: http://dx.doi.org/10.1080/10407799708915106

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OPEN BOUNDARY CONDITIONS FOR THESTREAMFUNCTION -VORTICITY FORMULATIONOF UNSTEADY LAMINAR CONVECTION

G. Comini, M. Manzan, and G. CorteUaUnioersita degli Studi di Udine, Departmento di Energetica e Macchine,Vw delle Scienze 208, 33100 Udine, Italy

The streamj'wu:tion-vortiJ:ily formulation is used to tmIllyze unsteady laminar-conoeaionproblems in two-dimensional incompressible flows. The Bubnov-Gakrlcin jinile..lemetumethod and a sequential procedure are employed to discretize and sollJe the governingdifferential eqUllliDns. Very accurate resulss are obtained by employing "advedivetkrivaJive condmons" at the ouJjlowfor 00 the variables invohJed. The boundJuy conditionsfor the streamfunction at internal woHs are imposed during the assembly pTOC1!SS, and thevortiJ:ily at inflow and woO boundaries is eva1UlJled in theframework ofthe streamfunctionequation. The accuracy of the approach is tkmonstrated by the solution of two weU-knownbenchmark problems concerningforced convedion over a circular

218 G. COMINl ET AL

NOMENCLATUREa thermal diffusivity (= klpCp) T dimensionless temperatureA area projected perpendicular to the I vector of nodal values of temperature

free-stream velocity u,w velocity components in theA advection matrix [Eq. (22) (x, z) directionsb vector of buoyancy contributions U,W velocity components in the (X, Z)

[Eq. (23)] directions, dimensionlessB body force per unit volume V volumecp specific heat at constant pressure x,z Cartesian coordinatesCx drag coefficient, dimensionless X,Z Cartesian coordinates, dimensionlessCz lift coefficient, dimensionless /3 coefficient of thermal expansionD diameter it timep force 6 time, dimensionless (= it u IL)r vector of tangential velocity contributions 8 period, dimensionless

[Eq. (24)] A wavelength, dimensionlessFr Froude number I" dynamic viscosity

[= Re 2/Gr = u2/g /3(lh - le)L) P densitys modulus of the gravity vector (T normal component of the stress vector,g gravity vector dimensionlessGr Grashof number[ = p2g /3(lh - le)L) I 1"'] r tangential component of the stress vector,H height of a channel dimensionlessk thermal conductivity

'"

streamfunctionK Laplacian matrix [Eq. (21)]

'"

vector of nodal values of theKo Laplacian matrix referred to the outflow streamfunction

boundary [Eq. (27) 'I' streamfunction, dimensionless (= "'luL)L reference length w vorticityM mass matrix [Eq. (20) ... vector of nodal values of vorticityMo mass matrix referred to the outflow n vorticity, dimensionless (= wLjU)

boundary [Eq. (25)n number of nodes Subscripts

~ shape and weighting functions related tothe node i c cold

Nu Nusselt number, equal to the h hotdimensionless temperature gradient at I inflowthe surface i, j indices or node numbers

Nu space-averaged Nusselt number n normal component, positive when(Nul time-averaged Nusselt number outward oriented(Nul space- and time-averaged Nusselt number 0 outflowp pressure p prescribedp pressure, dimensionless [ = pI( pu 2)] s tangential component, or at the localPo outflow boundary vector surface coordinate s

[Eqs. (26) and (28)] S synunetryPe Peclet number (= Re Pr = uLla) X,Z in the (X, Z) directionsPr Prandtl number (= cpI"lk) 0 reference valueRe Reynolds number (= puLI 1")s linear coordinate in the direction of s Superscriptss unit vector tangent to a boundary surfaceS surface I m at the time step mSt Strouhal number C1/8) - average value

.

expended matrix or vector,I temperature

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OPEN BOUNDARY CONDmONS FOR UNSTEADY C0NVECI10N 219

After having obtained a temperature solution, we compute local and averageNusselt numbers from the resulting nodal rates of heat transfer. Similarly, from thestreamfunction and vorticity solutions we evaluate the normal and tangentialcomponents of the stress vector at wall boundaries, in order to compute suchderived quantities as drag, lift, and pressure distributions.

In the finite-element discretization we employ the Bubnov-Galerkin method;i.e., we use the same functions for interpolating the variables and for weighting theresiduals. In this way we do not rely on the effects of numerical diffusionintroduced by upwinding techniques related to Petrov-Galerkin methods. Finally,we demonstrate the accuracy of the approach by the comparisons with well-estab-lished literature results.

STATEMENT OF THE PROBLEMFor a constant-property Boussinesq fluid, when B, = - pg is the only contri-

bution to the body force per unit volume, in the absence of volumetric heating andneglecting the effects of viscous dissipation, the streamfunction, vorticity, andenergy equations can be written in dimensionless form as [1]

(1)

(2)

and

In the above equations, the space coordinates, the velocities, and the time arenormalized with respect to a reference length L, to the average flow velocity ii,and to a characteristic time 110 = L Iii, respectively, while the dimensionlessvariable T = (r - t)I(th - tc ) is referred to suitably defined "hot" and "cold"temperatures.

In the problems considered here, we can have inflow boundaries, internal andexternal wall boundaries, symmetry boundaries, and outflow boundaries. At inflowboundaries, external walI boundaries, and symmetry boundaries, we assume thatthe streamfunction is known and its distribution can be prescribed as a boundarycondition of the first kind:

(4)

At internal wall boundaries belonging to multiply connected domains, the stream-function does not change with. the space coordinate, but has an unknown valuewhich can be time-dependent and must be determined as part of the solution

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220 G. COMINI ET AI..

process. Thus, for the time being, we will express this condition as

'If = const

while, in a later section, we will show how to enforce Eq. (5) during the assemblyprocess.

At inflow, wall, and symmetry boundaries, the normal derivative of thestreamfunction is also known. At inflow and wall boundaries we have

a'Ifu = -- =0

s an

while at symmetry boundaries we have

a'Ifu = -- * 0

s an

(6)

In a sequential procedure, we cannot use the additional conditions (6) and (7) forthe solution of the streamfunction equation because, in this way, we wouldoverspecify the problem. However, the information on the normal derivative of thestreamfunction can be incorporated in the boundary condition for the vorticity [1,9]. Thus at inflow, wall, and symmetry boundaries, we can obtain the values of thevorticity from a solution of Eq. (1) where conditions (6) and (7) are taken intoaccount. At this point, we must remark that, by using Eq. (6) also at inflowboundaries (instead of specifying the inlet vorticity directly, as we did in [1] and [9]),we have followed the mathematically elegant approach suggested by Gresho [17].Numerical experiments have shown that, in this way, we obtain slightly moreaccurate results at the expense of a decrease in the stability limits for the timeintegration algorithms [15, 16].

To complete the specification of inflow, wall, and symmetry boundary condi-tions, we must deal with the temperature variable. However, this presents nodifficulty, since at inflow and wall boundaries we have

while at symmetry boundaries we can confidently assume that

aT-=0an

(8)

(9)

In transient problems, we suggest using the outflow boundary conditionsidentified as "advective derivative conditions." By using the average normal veloc-ity lJ" as an estimate of the constant average phase speed [18], these conditions canbe written directly as

an _ an-+u-=oao n an (10)

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OPEN BOUNDARY CONDmONS FOR UNSTEADY CONVECTION 221

and

aT _ aT-+u-=oa(J n an (11)

for the vorticity and the temperature, respectively. The advective derivative condi-tion for the streamfunction can be obtained from the corresponding advectivecondition for the tangential component of the velocity,

au. - au.-+u-=oa(J n an (12)

Taking into account the definition U. = -(a'ltjan) and the auxiliary expression

(13)

derived from Eq. (1), we can first write Eq. (12) as a second-order differentialequation,

a (a'lt) _(a2'1t )- - =U --+0a(J an n as2 (14)

defined on the outflow boundary and subjected to the conditions of 'It = const atthe two endpoints. Then we can discretize Eq. (14) with respect to the timevariable, arriving at the desired final expression of the outflow boundary conditionfor the streamfunction,

( a'lt )m+! _ (a2'1t )m- :;; U 1i(J -- + 0 +an n as2 (15)

In stationary problems, Eqs. (10) and (11) yield again the usual zero normalderivative conditions, while Eq. (14) leads to the expression

(16)

which represents the "least constraining" condition for the streamfunction [19].At this point it must be remarked that, in the context of integrated solution

procedures, a set of alternative downstream boundary conditions has been derivedin [20, 21] as the streamfunction-vorticity equivalent of the traction-free conditionsin primitive variables [21, 22]. However, these alternative conditions are not easilyimplemented and cannot be employed in completely sequential procedures ofsolution.

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2ZZ G. COMINI ET AL

FINITE-ELEMENT FORMULATIONGoverning differential equations (1)-(3), with boundary conditions (4)-(11)

and (15), are first discretized in space using a Bubnov-Galerkin procedure. Then, aCrank-Nicolson time stepping scheme is applied to the partially discretized ver-sions of Eqs. (1)-(3), subjected to the boundary conditions (10) and (11). In matrixform, the final algebraic equations are [I, 9, 12, 15, 16]

for the streamfunction,

Kt/I = Mw - Po - r (17)

[~ (M +.2-Mo ) = ~ (_1 K+A)]wm+ 1se u; Re 2 Re= [~(M +.2-M )- ~(~K+ A)]Wm +~b (18)so u; Re 0 2 Re Fr

for the vorticity, and

[~ (M +.2-Mo ) +~(~K+A)]tm + 1/i.O u; Pe 2 Pe= [_1 (M +.2-Mo ) - ~ (~K+A)]tm (19)s u; Pe 2 Pe

for the temperature. In the above equations t/I, w, t, r, b, and Po are vectors oflength n (the number of node points), while M, K, A, and Mo are (n x n) matrices.

The mass matrices M, the Laplacian matrix K, the advection matrix A, thebody force vector b, and the load vector r are evaluated from the usual definitions:

Mj j = 1N;~dVv

1(aN; a~ aN; a~)K .. = --+-- dV'I v ax ax az az(20)

(21)

1 ( aN aN) 1 [(a'l') aN (a'l') aN]A .. = N U _I + V _I dV = N _ _I - - -r-_I dV (22)II v I ax aY v I az ax ax ozb. = ( N( aT) dV (23)

I lv I ax

and

(24)

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OPEN BOUNDARY CONDmONS FOR UNSTEADY CONVEcrlON 223

where (N;,~) are shape/weighting functions, with i, j = 1, n. The derivativesa'I'/ ax, a'I'/ az, and aT/aX are computed at the integration points using themost recent values of '" and t. The expanded matrix Mo and the expanded vectorPo have nonzero entries only on the outflow boundary. The nonzero entries in Mocorrespond to the entries in the matrix M o ' evaluated from the definition

(25)

with i, j = 1, no' The nonzero entries in Po correspond to the entries in the vectorPo' and Po is computed, at each new time step m + 1, from the discretized versionof Eq. (15),

(26)

using the null vector p~ = 0 as a starting value at tm = 1). The additionaldefinitions

and

f. aN aN(K ) .. = -' _J dSo IJ So as as

a'I'(P)i = - f. N;-dS

So an

(27)

(28)

hold good for the outflow boundary calculations, again with i,j = 1, no' It must bepointed out, however, that we do not form Eq. (28) explicitly, since the vector Po isevaluated directly from Eq. (26).

At this point, we must still impose the condition (5) in Eq. (17) on the nodesthat belong to internal wall boundaries in multiply connected domains. This can bedone by following an assembly procedure which can be better explained byreferring to a practical situation, such as the one illustrated in Figure 1. Let usassume, for example, that we look for a solution of Eq. (17) where the streamfunc-tion values at nodes 3, 5, and 7 must coincide with the streamfunction value at .

4Figure 1. Internal boundary 1357 sur-rounded by elements (1), (2), (3), and (4).

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224 G. COMINI ET AI..

Table I. Connectivity lists for the internal boundary represented in Figure 1

Element Node Numbers Connections

1 1 2 4 3 Real1 1 2 4 1 Auxiliary2 3 4 6 5 Real2 1 4 6 1 Auxiliary3 5 6 8 7 Real3 1 6 8 1 Auxiliary4 7 8 2 1 Real4 1 8 2 1 Auxiliary

node 1. Then, in addition to the "real" connectivity lists for the element nodes, wecan define "auxiliary" lists, where the "master" node 1 substitutes the "slave"nodes 3, 5, and 7. These lists are reported in Table 1 for the elements (1)-(4) thatsurround the internal wall boundary 1357. Obviously, we use the real lists forcomputing the element contributions to the global matrices, while we refer to theauxiliary lists in the assembly process. In this way, the algebraic equations corre-sponding to the slave nodes are effectively eliminated from the global system ofalgebraic equations (17). The only penalty is the bandwidth of the system increases,but this can be a minor difficulty if one uses, as we do, an iterative solver.Alternative methods for dealing with multiply connected domains, in the context ofstreamfunction-vorticity formulations, are illustrated in [4], but we believe that theprocedure suggested here has the advantage of simplicity.

Equations (17)-(19) and (26) can be decoupled and solved in sequence,provided that special care is taken to represent properly the vorticity. boundaryconditions at inflow and wall boundaries. The calculation procedure, to advancefrom the step m ;;. 0 to the step m + 1, can be described as follows.

Once we have determined the contributions of the outflow boundary condi-tion from Eq. (26), we can impose the boundary conditions (4) and (5) at theappropriate nodes. Therefore, we can eliminate from the system (17) the nodalequations that correspond to "slave" nodes or to points where the streamfunctionis known. Afterward, the new values of the streamfunction at the remaining nodescan be computed from the reduced system (17) as

(29)where w m is the vorticity solution at the previous step, while pm+ 1 has beenobtained from the solution of Eq. (26). Once Eq. (29) has been solved, we can goback to the original equation (17) and, using the most recent values of thestreamfunction '" m + 1, we can calculate the vorticity at inflow boundaries, wallboundaries, and symmetry boundaries as

(30)Actually, in Eq. (30), we do not have to consider a complete new solution, since thecomputations involve only a strip of elements adjacent to the boundaries consid-ered. Finally, having obtained the boundary values of vorticity from Eq. (30), wecan solve in sequence Eqs. (18) and Eq. (19), advancing to the next time step.

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OPEN BOUNDARY CONDmONS FOR UNSTEADY CONVECTION 225

CALCULATION OF STRESS COMPONENTSIn the solution of fluid flow problems, a very important issue is the evaluation

or pressure distributions along internal or external wall boundaries. To thispurpose, in the streamfunction-vorticity formulation, we can utilize the projectionof the momentum equation in the direction s tangential to the wall boundary [221,

au. 1 a(u.2 + U}) ap 1 an 1 g- + - - un + - + - - + - T- s = 0 (31)ao 2 as n as Re an Fr g

At no-slip walls we assume U; = U. = 0 and thus, by integrating Eq. (31) from areference point So to the point of interest s, we obtain

,( 1 an 1 g )P - Po = - f - - + - T- . s tis

'0 Re an Fr s(32)

where the unit gravity vector is defined as gig = (0, -1). After having computedthe pressure distribution from Eq. (32) we can calculate, if required, the normaland tangential components of the stress vector, using the dimensionless expressions

u= -P

and1

r > --nRe

which are normalized with respect to pu2

SOLUTION OF BENCHMARK PROBLEMS

(34)

The examples presented here concern transient solutions of two well-knownlaminar benchmark problems: forced convection over a circular cylinder in crossflow and mixed convection in a plane channel heated from below. In the computa-tions, we have used eight-node isoparametric elements. The space-discretizationerrors and the sensitivity of the computed fields to the location of the outflowboundary have been discussed in [IS, 161. At each time step, the discretizedequations have been iterated in sequence until convergence has been reached,employing underrelaxation both for the vorticity and the energy equation. Thecalculations were continued in time until the spatially averaged Nusselt numbersdiffered less than 0.1% at two successive extremum points.

The space-averaged and the time-averaged Nusselt numbers have beendefined as

- IfNu = S s Nus dS1 f92(Nus) = 8 Nus ae

O2 - I 9,

(35)

(36)

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226

and

G. COMINI ET AL.

- 1 f6 2 -(Nu) = Nud882 - 81 6,

(37)

In the above equations, the time interval (82 - 81) is large compared to the periodof oscillations, and the local Nusselt numbers Nu, are computed directly from thedimensionless temperature gradients, that is, from the "reactions" (1).

Forced Convection over a Circular Cylinder in Cross FlowIn the present test problem we have followed [13] by taking Re = 100 and an

average normal velocity equal to the total inlet velocity. We have also assumed avalue of the Prandtl number Pr = 0.71, and we have disregarded the naturalconvection contributions by taking llFr = O. The boundary conditions and thefinite-element mesh utilized are shown in Figure 2, where the flow domain hasbeen divided into 952 eight-node, isoparametric elements, yielding a total of 3,002nodes.

,,-0

1-0

d"~-- --I .-0(a)

dl--0iI.

(b)

1.(illf).iIIJ iI.

(8.8) 12$.8)

(e)(25.-8)

Figure 2. Forced convection over a circu-lar cylinder in cross flow: (a) boundaryconditions for the flow problem; (b)boundary conditions for the thermalproblem; (e) finite-element mesh.

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OPEN BOUNDARY CONDmONS FOR UNSTEADY CONVECfION 227

The basic characters of the flow are illustrated in Figures 3 and 4, where weshow the periodic evolution of streamlines and temperature contours starting froma reference time (J = 0, which corresponds to the instant when a minimum isreached in the temperature at point (X, Z) = (4,0). The qualitative agreementbetween our streamline plots and those of [13] is very good. Moreover, thesefigures demonstrate that we have very little reflection and distortion at the outflow,thanks to the advective derivative conditions. We can also note that, by allowingthe value of the streamfunction at the cylinder wall to be unknown, we have arrivedat a solution that changes periodically with time in an interval of amplitude 0.0256around 'It = 8. Since, as pointed out in [4], solutions obtained by assuming 'It = 8compare well with the benchmark solution, we can infer that the agreementbetween out results and such a resonable assumption implicitly validates ourprocedure.

By taking into account the projections of the stress vector components in thedirection of the main flow, we have evaluated the drag coefficient,

and the lift coefficient,

FzC = -----,,--z A pu 2j 2

(38)

(39)

EllEllI!!IEiII Figure 3. Forced convection over a circu-lar cylinder in cross flow at Re = l()():periodic evolution of streamlines at timeintervals of tEl.

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228 G. COMINI ET AI..

Figure 4. Forced convection over a circu-lar cylinder in cross flow at Re = 100:periodic evolution of temperature con-tours at time intervals of teo

where A is the cylinder frontal area. The time histories of these coefficients,represented in Figure 5, start from the previously defined reference time and arefound again to be in good agreement with the benchmark results [13]. From theperiodic evolution of the lift coefficient, we estimate the Strouhal number, definedas the inverse of the dimensionless period, to be St = 1.70. This value is in closeagreement with the best estimate (St = 1.73) obtained in (13) using a 14,000-nodemesh.

The variation of the local, time-averaged Nusselt number with the angularcoordinate t/> is shown in Figure 6. From the periodic evolution of the space-aver-aged Nusselt number (not re~rted here), we estimate the time- and space-aver-aged Nusselt number to be (Nu) = 5.14. The empirical Hilpert correlation

(Nu) = 0.683 ReO.466 Pr 1/ 3 (40)

suggested in (23), yields (Nu) = 5.21 for Re = 100 and Pr = 0.71. This kind ofagreement must be considered good, since experimental results are undoubtedlyinfluenced by the finite length of the cylinder.

Mixed Convection in a Plane Channel Heated from BelowIn the present test problem we have followed (5) by assuming that Re = 10,

Pe = 20/3, and Fr = 1/150. The boundary conditions and the finite-element mesh

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OPEN BOUNDARY CONDmONS FOR UNSTEADY C0NVECI10N 229

, f\ , 1\ ' ,

,-

I-

t- -

f- V V V V V, , I I I I I

1.342

1.334o 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

8(a)

1.338

1.350

0.3 r--r-.--,--,---,-,---,,-,

Cz02

0.1

0.0

-0.1

-0.2

02 0.4 0.6 0.8 1.0 12 1.4 1.6

8(b)

Figure 5. Forced convection over a circu-lar cylinder in cross flow at Re = 100:periodic evolution of drag (a) and lift(b) coefficients.

Figure 6. Forced convection over a circu-lar cylinder in cross flow at Re = 100:circumferential variation of the localtime-averaged Nusselt number.

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230 G. COMINI ET AI..

utilized are shown in Figure 7, where the flow domain has been divided into 8 X 20eight-node isoparametric elements, yielding a total of 537 nodes.

The basic characters of the flow after are illustrated in Figures 8 and 9, wherewe represent the periodic behavior of streamlines and temperature contoursstarting from a reference time 8 = 0, which corresponds to the instant when aminimum is reached in the temperature at point (X, Z) = (5,0.5). As can be seen,the advective derivative conditions lead to very little reflection and distortion at theoutflow, even with a very short mesh.

More quantitative comparisons, taken from [16], concern the period e, thetime- and space-averaged Nusselt number (Nu), the wavelength A, and thethermal wave speed A/e. These results, reported in Table 2, demonstrate goodaccuracy, even if we always utilized relatively coarse meshes.

From Eq. (33), we have evaluated the pressure at the Z = 0 wall boundary,finding a pressure distribution that is not spatially periodic. However, as demon-strated in Figure 10, where we plot the auxiliary variable P + 12/Re X, we findthat a periodic component is superimposed on the overall linear pressure distribu-tion,

12P= --X

Re(41)

which corresponds to the Poiseuille flow. These results are, once again, in goodagreement with those of [14].

(a)

(b)roo

L"{al_'(c)

(0.1) (S.I)

-(0.0)

(d)(S.O)

Figure 7. Mixed convection in a planechannel at Re = 10, Pe = 20/3, andFr = 1/150: (a) boundary conditions forvelocitycomponents; (b) boundary condi-tions for streamfunction and vorticity;(c) boundary conditions for temperature;(d) finite-element mesh.

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OPEN BOUNDARY CONDmONS FOR UNSTEADY CONVECTION 231

Figure 8. Mixed convection in a planechannel at Re = 10, Pe = 20/3, andFr = 1/150: periodic evolution ofstreamlines at time intervals of teo

Figure 9. Mixed convection in a planechannel at Re = 10, Pe = 20/3, andFr = 1/150: periodic evolution of tem-perature contours at time intervals ofteo

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232 G. COMINI ET AL.

Table 2. Mixed convection in a plane channel at Re = 10, Pe = 20/3, and Fr = 1/150

Period (8) (Nul Wavelength (Al Aje Reference1.3319 2.5583 1.4465 1.09 Evans and Paolucci [I4]

8,000 nodes1.273 2.574 1.45 1.14 Our results [161

2,097 nodes

Finally, in Figure 11, we show the time evolution of the spatially averagedNusselt number, starting from a Poiseuille velocity distribution and a lineartemperature field. As we can see, at first heat transfer takes place only byconduction (Nu = 1), while advection is the dominating effect after the formationof the vortices. When the flow becomes unstable, the space-averaged Nusseltnumber Nu suddenly increases and starts oscillating around the final space- andtime-averaged value (Nu). It must be pointed out, however, that the value(Nu) = 2.34, obtained in Figure 11, is referred to the short domain of Figure 7d,and thus cannot be compared directly with the results of Table 2, which arereferred to a domain four times longer.

CONCLUSIONS

A finite-element method has been presented to solve transient laminar-con-vection problems in two-dimensional incompressible flows, using the streamfunc-tion-vorticity formulation. Advective derivative conditions have been used at theoutflow for all the variables involved. In multiply connected domains, the boundaryconditions for the streamfunction at internal walls have been imposed during theassembly process. The vorticity at inflow and wall boundaries has always beenevaluated in the framework of the streamfunction equation. The accuracy of theapproach has been demonstrated by the solution of two well-known benchmarkproblems, concerning forced convection over a circular cylinder in cross flow andmixed convection in a plane channel heated from below.

x

Figure 10. Mixed convection in a planechannel at Re = 10, Pe = 20/3, andFr = 1/150: pressure distribution atZ = 0 with the Poiseuille pressure fieldsuperimposed.

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OPEN BOUNDARY CONDmONS FOR UNSTEADY C0NVECI10N 233

01234.5678910

9

Figu", 11. Mixed convection in a planechannel at Re = 10, Pe = 20/3, andFr = 1/150: time history of the spatiallyaveraged Nusselt number.

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