IN HFATANDMASS TRANSFER 0094-4548/82/040291-08503.00/0 Vol. 9, pp. 291-298, 1982 Pergamon Press Inc. Printed in the United States

MIXED CONVECTION - A COMPARISON BETWEEN EXPERIMENT AND AN EXACT NUMERICAL SOLUTION OF THE BOUNDARY LAYER EQUATIONS

Roland Hunt and Graham Wilks University of Strathclyde Department of Mathematics

Glasgow, Scotland

(C~LLnicated by D.B. Spalding)

Introduction

Although there have been a significant number of theoretical papers on

mixed convection flows adjacent to vertical surfaces, experimental corrobora-

tion of their conclusions have been limited. This is particularly true of

flow regions in which buoyancy forces have been dominant over external

streaming effects. Recnetly, Carey and Gebhart [i] have provided some experi-

mental evidence for the flow associated with the uniform heat flux constraint.

In a supporting analysis of the full Navier Stokes equations they adopt a

matched asymptotic expansion technique to obtain downstream perturbation

series solutions, about the pure free convection state, which include both

higher-order boundary layer and external streaming effects. A very satis-

factory reconciliation between theory and experiment is displayed for

Pr = 0.733. It is noted that a boundary layer formulation of the problem is

available in Wilks [2] which incorporates external streaming effects only.

The results of [i] are accordingly ideally suited for examining the relevance

of associated boundary layer solutions. Consequently, in this paper, we

present a full numerical solution for the particular value Pr = 0.733 which

specifically incorporates nodal points along the surface that are commensurate

with the experimental parameter values of [I].

291

292 R. Hunt and G. Wilks Vol. 9, No. 4

THE GOVERNING EQUATIONS

The flow envisaged is that of a uniform stream past a vertically

aligned semi-infinite plate. Buoyancy forces occur as a result of

a uniform heat flux from the plate. As a consequence, the boundary

layer flow adjacent to the plate evolves between the Blasius similarity

state at the leading edge and the pure free convection similarity

state downstream of Sparrow and Gregg [3]. The governing boundary

layer equations representing conservation of mass, momentum and energy

respectively are

a--Eu + a-Iv = 0 (1) ~x ~y

~u ~u ~2u u ~xx + v -~y = gB(T-T o) +.-- (2)

~y2

~T ~T ~2T U~fx +v~ = K (3)

~y2

to be solved subject to boundary conditions

~T - q on y = O u=v=0; ~-~= (4)

u + U, T + T as y + ~. o

Here T is temperature and (u,v) are velocity components associated

with increasing (x,y), measured from the leading edge and normal to the

plate respectively. Conditions in the free stream are represented by

the uniform velocity U and constant temperature T . The kinematic o

viscosity v, the acceleration due to gravity g, the coefficient of

thermal expansion 8 and the thermal diffusivity r are taken as

constant. The specified uniform heat flux at the plate is represented

by q and k is the thermal conductivity of the fluid.

TRANSFORMATIONS AND NUMERICAL SOLUTION

The unifying basis for numerical solution over the semi-infinite

flow field is the characterising non-dimenslonal coordinate

(23g282q2 v I 1/3

= \ 52k2U5 / x . (5)

Vol. 9, No. 4 MIXED ODNVECTION 293

Accordingly ~ reflects the local relative magnitudes of buoyancy and

inertia forces at stations along the plate.

Transformations appropriate to perturbations about the Blasius

similarity state (small ~) and the Sparrow and Gregg similarity state

(large ~) are

BLASIUS SPARROW and GREGG

4 = (2 vUx) ~ f (~, rl) ~=C2x4/5~ (~, b

T c(2vx~ T_To qxl/5 8(~,~) 6(b) T . . . . . ~8(~,~) 6(a) o k U ) = kC I -

U ~ ClY z-r/'s x

1/5, .245~gBqv3. I/5 C, =( gBq ) " C2=t k )

2.5.kv2

Each set of transformations yields an associated form of the governing

boundary layer equations. An algorithm for complete numerical

integration over all ~ may be initiated using the transformed form

of equations according to 6(a) and subsequently switching at ~ = i

to the transformed form of equations according to 6(b). However,

Hunt and Wilks [4] have recently introduced a single set of continuous

transformations in ~ which accommodate the evolution of the flow

between the appropriate similarity states. These read

3/ln~ ~ = - q(2vxl } (1+~)-3110~(E,~) = (2vUx)~(I+C) "~f(~'q); T-To k" U"

( U )~(I+C)3/I O _ (7) n =Y~vx

Integration is then achieved in the context of the single set of

equations

s(z+~) -~- @n 3 On

6~ .O~. 2 E 3 / 2 5(1+~) t---~~ - 5 ( i -~ - ~) o

On

e {02 }

OqO~ 3n On 2

(8)

294 R. Hunt and G. Wilks Vol. 9, No. 4

i ~2~

Pr a~ 2 5+8~ ~ ~f (5+2~) ~ ~ 5(1+%) ~ 5(I+~)

3n an

~n ~n

(9)

Boundary conditions on equations (8) and (9) are

~ ( ~ , o ) = ~ , . ( ~ , o ) = e ~ ( ~ , o ) - 1 = o n (10)

~ (i+~)-3/5, and f~(~,q) + ~(~,~) ÷ 0 as n ÷ ~. q

The boundary condition on ~ at large ~ reflects the external streaming n

effects on the basic pure free convection flow.

The algorithm used for the solution of (IO) is based upon a

Keller [5] box scheme and has been fully reported in [4]. In

particular the range of ~ is represented by a number of non-unlform

nodal values starting with step length 0.2 and ending with step length

0(102). As the experimental parameter e M of [i] may be correlated

with ~ as 3/5 o

= (~) (ii) £m

it was possible to include nodal points in the ~ range which coincided

with the reported values of E M. A direct comparison between

experimental and theoretical boundary layer velocity profiles could

then be established.

RESULTS

Experimental evidence in [~ is presented in the context of

two parameters (CM,CH). The former is associated with external

streaming effects and the latter with second order boundary layer

contributions to the flow field. The parameter pairings reported in

[I] are (EM,EH) = (0.75, 0.028), (0.60, 0.026), (0.45, 0.022),

(0.27, 0.020) and (0.IO, O.016). Accordingly nodal values of

= (3.23, 4.69, 7.57, 17.73, 92.83) were specifically incorporated in

the numerical scheme. Complete solution was obtained for Pr = 0.733.

FIG.I displays velocity profiles at various locations of ~ in terms

of the computational velocity variable ~-~f In FIG. 2 the results ~ "

have been recast in terms of the variables used in [i]. Velocity

295

0 2 5

3 4 5

FIG. 1 .

Velocity profiles in computational variables at appropriate values

0.

F

0.

0

I

3-

,

2-

.I -

1- I I I I

( L ) C” \ 0 (0.15, 0.028) 1 = 3.23

v (0.60, 0.026) 3 =4.69

0 (0.45, 0.022) J = 1.51

&ji&

P\ 3 ‘Y B

i

J

k ~. \\ %. \ -.a- - ^_ _o_ __A_ - -

I I I I I

I 2 37 4

5

FIG. 2

of 5.

Comparison of boundary layer velocity profiles and experimental profiles.

296 R. Hunt and G. Wilks Vol. 9, No. 4

profiles at the appropriate values of ~ have been superimposed on

the experimental observations. Theoretical predictions of [I]

incorporating hlgher-order boundary layer effects are included as

dashed line plots.

For completeness FIG. 3 presents the skin friction and heat

transfer coefficients prescribed by

I 52k2~2 I I/6 (~u.

W \23U492B2q2] ~Y)y=O

and Q =

= (I+~)7/I0 [a2f 1

\1/6 52k2~2u21 I ~T (i+~) 3/10

3g2 2q2/ (rY)Y:° (i3)

FIG. 3

Skin friction and heat transfer coefficlents, Pr = 0.733.

DISCUSSION AND CONCLUDING REMARKS

The appropriateness of the numerical algorithm is displayed in

FIG. I. The evolution between forced and free convection velocity

profiles is apparent. FIG. 2 indicates that certain features of

the numerical solution of the boundary layer equations are entirely

consistent with the experimental results, particularly in the

immediate vicinity of the plate. Here correlations of velocity

Vol. 9, No. 4 MIXED OClqVEL-TION 297

profiles and maximum velocities are excellent. However at large

distances from the plate boundary layer solutions consistently

underestimate the experimental evidence. The extent of this under-

estimation at large q may be quantified from [I~ in terms of the most

signiflcant higher order boundary layer theory contribution as an

amount 0.559 e H.

Accordingly we have confirmed the significance of h~gher order

boundary layer effects in correlating theoretical and experimental

flow fields at a distance from the plate. The magnitude by which

numerical solutlons of the boundary layer equations may be expected

to underestimate the flow away from the plate may be readily assessed

in terms of the parameter ~H' On the other hand it would appear

that boundary layer solutions remain adequate, for practical purposes,

for establishing the appropriate physical parameters at the plate,

namely the skin friction and heat transfer coefficients as displayed

in FIG. 3.

C1 = ( gBq )11_ 15~

2.5.k.~ 2

3 C 2 = 2.5 ~ C I

f,f,~

g

k

Pr = -- K

q

Q

T

T o

(u,v)

U

(x,y)

NOb~NCLATURE

downstream transform constants.

non-dimensional stream function for leading edge,

downstream and numerical formulations respectively.

acceleration due to gravity.

thermal conductivity.

Prandtl number.

surface heat flux constant.

heat transfer coefficient at the plate.

temperature.

temperature of ambient fluid.

dimensional velocity components associated with

(x,y).

free stream velocity.

distances along and normal to the plate.

298 R. Hunt and G. Wilks Vol. 9, No. 4

GREEK SYMBOLS

el'I' eH rU

~,, r~ rl

6,6,8

T w

coefficient of thermal expansion.

expansion parameters.

Don-dimensional normal variable for leading edge,

downstream and numerical formulations respectively.

non-dimensional temperature for leading edge,

dok~stream and numerical formulations respectively.

thermal diffusivity.

kinematic viscosity.

characterising non-dimenslonal coordinate along

the plate.

skin friction coefficient at the plate.

stream function.

REFERENCES

I. V. P. Carey and B. Gebhart, Int. J. Heat Mass Transfer 25 (2),255 (1982).

2. G. Wilks, Int. J. Heat Mass Transfer 17, 549 (1974).

3. E. M. Sparrow and J. L. Gregg, Trans. A.S.M.E. 78, 435 (1956).

4. R. Hunt and G. Wilks, J. Comp. Phys. 40, (1981).

5. H. B. Keller, Ann. Rev. Fluid Mech. IO, 417 (1978).