Interferometric investigations of convection around cylinders at small Grashof numbers Luca Crescentini and Giorgio Fiocco Experiments with a Twyman-Green interferometer have been carried out to determine the temperature distribution around a horizontal cylinder in the presence of natural convection for Grashof numbers around 10-4 and a Prandtl number of 0.7, a region of parameters poorly covered by previous investigations. Temperature differences are recovered by digital analysis of the fringe pattern perturbation. The results indicate significant discrepancies in the temperature profile, when compared with numerical models. 1. Introduction Due to its engineering interest, natural convection from horizontal circular cylinders and from line heat sources has been extensively studied in the past, both numerically and experimentally. For a survey, see Refs. 1, 2, and 3. The process is governed by three adimensional pa- rameters, namely the Grashof, Nusselt and Prandtl numbers, defined as follows: g6a 3 (To-T T.) NQ - cpv , (1) 2rkv 2 27rk(T-TJ kp where g is the gravity acceleration, ,3 the expansion coefficient of air, a the cylinder radius, k the thermal conductivity of air, v the kinematic viscosity, Q the power dissipated per unit length, cp the specific heat of air at constant pressure, p the air density, To the temperature of the cylinder, and T the ambient tem- perature. The Grashof number indicates the relative impor- tance of buoyancy with respect to viscosity; the Nus- selt number represents the ratio of the total heat trans- port to the diffusive heat transport when the temperature radial gradient is given by (To - T)Ia; the Prandtl number describes the relative importance of the two diffusive processes which tend to retard the motion, i.e., viscosity and molecular diffusion. Luca Crescentini is with University of LAquila, Physics Depart- ment, I-67100,L'Aquila, Italy, and G. Fiocco is with University of Rome, Physics Department, -00100 Rome, Italy. Received 28 February 1989. 0003-6935/90/101490-06$02.00/0. i) 1990 Optical Society of America. Numerical analyses have been carried out by several authors covering a very large range of Grashof num- bers, 10-10 < Gr < 108. Limiting our attention to Gr < 10-2, the problem of pure convection was analyzed in Ref. 4, by joining the circumferential average tempera- ture in the conduction-dominated region around the cylinder to that in the far field, governed mainly by convection. In another paper, 5 a finite-difference method was used for integrating the simplified equa- tions in the thick boundary layer, for large values of r, defined as the ratio of the enclosure to the cylinder. More recently, the method of series truncation has been introduced, 3 where the temperature distribution and flow variables are expanded as Fourier series with variable coefficients. Despite the large amount of available numerical analysis, to our knowledge, the existing experimental verifications for small Grashof numbers have been lim- ited to establishing a correlation between Grashof and Nusselt numbers. 6 - 8 Measured temperature distribu- tions are in the range of moderate to large Grashof numbers. In this paper, an experimental investigation of pure convection around a horizontal conducting wire for Gr _ 10-4 is presented. The wire is set along the axis of one arm of a Twyman-Green interferometer and the temperature distribution around the cylinder is stud- ied by analyzing the interference fringe pattern. This technique has been widely used in the past 9 but has been improved here by introducing a high-sensitivity interpolation method. II. Experimental The experimental setup and the image analysis technique have been described in detail in Refs. 10 and 11. Only a brief summary will be given here. 1490 APPLIED OPTICS / Vol. 29, No. 10 / 1 April 1990
Transcript
Interferometric investigations of convection aroundcylinders at small Grashof numbers
Luca Crescentini and Giorgio Fiocco
Experiments with a Twyman-Green interferometer have been carried out to determine the temperaturedistribution around a horizontal cylinder in the presence of natural convection for Grashof numbers around10-4 and a Prandtl number of 0.7, a region of parameters poorly covered by previous investigations.Temperature differences are recovered by digital analysis of the fringe pattern perturbation. The resultsindicate significant discrepancies in the temperature profile, when compared with numerical models.
1. Introduction
Due to its engineering interest, natural convectionfrom horizontal circular cylinders and from line heatsources has been extensively studied in the past, bothnumerically and experimentally. For a survey, seeRefs. 1, 2, and 3.
The process is governed by three adimensional pa-rameters, namely the Grashof, Nusselt and Prandtlnumbers, defined as follows:
g6a3(To-T T.) NQ - cpv , (1)2rkv 2 27rk(T-TJ kp
where g is the gravity acceleration, ,3 the expansioncoefficient of air, a the cylinder radius, k the thermalconductivity of air, v the kinematic viscosity, Q thepower dissipated per unit length, cp the specific heat ofair at constant pressure, p the air density, To thetemperature of the cylinder, and T the ambient tem-perature.
The Grashof number indicates the relative impor-tance of buoyancy with respect to viscosity; the Nus-selt number represents the ratio of the total heat trans-port to the diffusive heat transport when thetemperature radial gradient is given by (To - T)Ia;the Prandtl number describes the relative importanceof the two diffusive processes which tend to retard themotion, i.e., viscosity and molecular diffusion.
Luca Crescentini is with University of LAquila, Physics Depart-ment, I-67100, L'Aquila, Italy, and G. Fiocco is with University ofRome, Physics Department, -00100 Rome, Italy.
Received 28 February 1989.0003-6935/90/101490-06$02.00/0.
i) 1990 Optical Society of America.
Numerical analyses have been carried out by severalauthors covering a very large range of Grashof num-bers, 10-10 < Gr < 108. Limiting our attention to Gr <10-2, the problem of pure convection was analyzed inRef. 4, by joining the circumferential average tempera-ture in the conduction-dominated region around thecylinder to that in the far field, governed mainly byconvection. In another paper,5 a finite-differencemethod was used for integrating the simplified equa-tions in the thick boundary layer, for large values of r,defined as the ratio of the enclosure to the cylinder.More recently, the method of series truncation hasbeen introduced,3 where the temperature distributionand flow variables are expanded as Fourier series withvariable coefficients.
Despite the large amount of available numericalanalysis, to our knowledge, the existing experimentalverifications for small Grashof numbers have been lim-ited to establishing a correlation between Grashof andNusselt numbers.6 -8 Measured temperature distribu-tions are in the range of moderate to large Grashofnumbers.
In this paper, an experimental investigation of pureconvection around a horizontal conducting wire for Gr_ 10-4 is presented. The wire is set along the axis ofone arm of a Twyman-Green interferometer and thetemperature distribution around the cylinder is stud-ied by analyzing the interference fringe pattern. Thistechnique has been widely used in the past9 but hasbeen improved here by introducing a high-sensitivityinterpolation method.
II. Experimental
The experimental setup and the image analysistechnique have been described in detail in Refs. 10 and11. Only a brief summary will be given here.
The Twyman-Green interferometer is shown in Fig.1. A He-Ne laser beam, expanded to a -2-cm diam,after dividing through a beam splitter, travels throughtwo horizontal 90-cm long, 12-cm diam pipes consti-tuting the two arms of the interferometer. The sepa-rate beams are then reflected by 6-cm diam mirrors atboth ends of the arms and recombine at the beamsplitter, giving rise to interference fringes on a screen.The fringe pattern is observed by a commercial CCDvideo camera, digitized at 6 bits by a Data TranslationDT2803 256 X 256 frame grabber and stored in apersonal computer memory.
The horizontal cylinder is a 76-cm long, 0.32-mmdiam constantane wire, set along the axis of one arm.The wire is heated by a small electric current flowingthrough it.
The temperature distribution around the wire isobtained by comparing the fringe patterns in the pres-ence and absence of the current.
The refractive index of dry air h, at the wavelength X= 0.6328 Atm, depends on local temperature and pres-sure through well-known relations 2 and, in the regionof the parameters of interest, the dependence can bewritten as:
dT , 1.21 X 10-9 po C-l, (2)aT
where p is expressed in mm Hg.Thus, if L is the wire length, neglecting edge affects,
the phase perturbation experienced by the beam in itstwo-way propagation along one arm of the interferom-eter, due to a homogeneous change in temperature, isgiven by
6sp= 4rL ah 6T = 0.0182 p5T [rad]. (3)X 072
Since the fringe pattern analysis is able to detect 2-Dphase perturbations of about 0.1 rad, a sensitivity ofthe order of 10-20C is expected; this is achieved with aspatial resolution of few pixels taking into account thenecessity for smoothing.
Although the CCD video camera contains 512 X 512active elements, the frame grabber can only digitize a256 X 256 image. To study the transition region be-tween the inner conduction-dominated layer and theouter region, the focusing optics were chosen so that100 pixels correspond to -1 cm. Spatial resolution is,then, of the order of a few 10-2 cm, that is of the wiredimensions.
III. Inversion Method
In principle, the acquisition of the temperature fieldin the region surrounding the wire should be done bythe differential technique based on comparing the per-turbed and the unperturbed fringe patterns. In prac-tice, at the level of sensitivity implied by the smallvalues of temperature differences actually present,several causes of error may be altering the measure-ments. This difficulty becomes particularly severewhen trying to obtain the temperature differences atvery close distances from the wire, or at the surface of
mirrorIII
I I
IijIIjII
camera Iinterfcace
Fig. 1. Schematic diagram of the experimental setup.
the wire, since the presence of the holders precludesstudying the process in this region. Installing a ther-mometer on the wire could alter the process itself. Onthe other hand, the value of the temperature of thewire is an essential element in a normalization scheme.
We have resorted to an indirect procedure just toobtain the temperature of the wire surface. The pro-cedure is based on the analytical dependence of thecircumferential average adimensional temperature (t)on the adimensional distance r from the wire axis, thatcan be expanded in the term (r - 1) to yield4:
r a(t) = -Nu + D(r-1)3 + DI(r-1)4 + . . .,Or
(4)
where2,
(t) = [(T(r,O) - T.)I(To - T.,I)]dO,
r is expressed in units of the wire radius, and D and Diare appropriate coefficients.
After integrating in r from 1 to a general value andintroducing the actual circumferential average tem-perature (T), we get
(T) (r)-T-To-T<.- Q nr+C J1(p-1)'dp27rk fr p
+ Cl 1 (p - 1)4 dp, (5)
where C = Di(To-T).This result can be used to obtain (To - T), Q, C,
and C by least-squares fitting expression (5) to theexperimental values of (T) (r) - T_ The values of Qinferred from electrical measurements can be used forcomparison.
Once the scaling factor (To - T.) has been deter-mined, it is possible to generate 2-D maps of t(r,6;Nu,Gr).
In practice, the limited extension of the analyzedregion, and the presence of long-term modifications inthe interferometer, make it hard to measure the tem-perature distribution with respect to the unperturbedambient temperature (T(r,O) - T.) and, consequently,the term (To - T). Consequently, the temperaturedistribution has been retrieved by assuming that theheating in the region below the wire was negligible for r>70. The computation of Q, C, and Cl is not affectedby this assumption.
A few correlations between Gr and Nu have beenproposed in the past, some based on experimentalresults and other ones on numerical simulations.4 -8For Gr 10-3/10-4, Nu values generally agree inside10%. This permits us to obtain an independent evalu-ation of (To - To), since the cited correlation, togetherwith the definitions of Gr and Nu [Eq. (1)], form aclosed set of equations when Q is known.
The former method of directly computing (T(r,O) -T.) from the phase perturbation map, the direct tech-nique (d.t.), and the latter method of using a correla-tion between Gr and Nu obtained from the literatureto compute (To - T), were used independently andthen compared.
IV. Results and Discussion
The circumferential average of the adimensionaltemperature (t) (r) can be directly compared with theresults obtained in Ref. 4, hereafter referred to as NO.
Nakai and Okazaky4 included the terms up to thesecond in the r.h.s. of Eq. (4) and determined D withthe conditions obtained in the outer region, obtainingD = CB(NUGr)2IGr; CB depends on the Prandtl num-ber and is approximately equal to 0.068 for air. Theexpansion is considered valid at least for r < rmax =CA(NuGr)- 113 , where CA 1.6 for air. The depen-dence of D on the parameters can be considered as anindication of the closeness of the theory with the ex-periments.
For computing (To - To), we used two differentcorrelations:
= lnE- ln(NuGr) (6)Nu 3 3
Nu = 0.466(PrGr)1/1 5, (7)
where E 20.4 for air.The two correlations were respectively proposed by
Nakai and Okazaky4 and by Tsubouchi et al.6 hereaf-ter referred to as NO and TSN, respectively. Theywere chosen as representative of the range of variationof Nu(Gr) existing in the literature.
Introducing the definitions of Gr and Nu, the follow-ing relations can be obtained:(a) NO correlation:
N = 3/(15.97 - nQ),Gr = 8.03 X 10- 7Q(15.97 - nQ),
To-T = 2.49 X 103Gr;(b) TSN correlation:
Nu = 0.213Q1/ 16,Gr = 1.13 X O-5Q15/1',
To-TX = 2.49 X 103Gr;where Q is measured in mW/m.
Each quantity can be measured only approximately,because of three main sources of errors:
(1) uncertainty in the position of the wire axis;(2) overall error in the detection of the phase per-
turbations by the image analysis; and(3) misalignment of the wire with respect to the
laser beam.The effects of misalignment can be considered negli-
gible when compared with the other sources of errors.Uncertainty about the position of the wire axis can
be estimated in +2 pixels in the x direction and 1pixel in the y direction. Tests have shown that therelated uncertainty in the determination of the param-eters is always <10%.
Errors in the detection of the phase perturbationshave been discussed in Refs. 10 and 11; the overalluncertainty, taken as the full width at half maximumof the b8o distribution, is 0.3 rad, corresponding to±0.02°C in temperature: its effect has been computedby applying the error propagation laws to the linearbest fit mentioned previously.
Five sets of three experiments have been analyzed.In each set, characterized by a value of the currentflowing in the wire, the final heating map has beenobtained by averaging the three maps corresponding tosingle experiments.
Table I shows the experimental values of Gr, Nu,(To - T.), D, and Di for both correlations and for thedirect technique. The column before the last lists thecorresponding values of D as predicted by NO. Thelast column shows the standard deviation betweenthe experimental data and the theoretical curve, bothexpressed as (t) (r). Figure 2 plots (t) (r) obtained inthe fourth set (solid lines), together with the best-fitcurves from Eq. (5) (dashed lines), using the NO corre-lation, the TSN correlation, and the direct technique.The corresponding curves obtained using the NO pre-diction of D are also shown (dotted lines); these lattercurves have been drawn for r rmax-
Figure 3 shows the experimentally determined val-ues of D vs the Grashof number, compared with the NOprediction, with the errors induced by the uncertain-ties in wire position and heating detection.
The following considerations arise:(1) the values of D predicted by NO are smaller thanthe experimental ones by a factor up to an order ofmagnitude; moreover, the predicted dependence on Grand Nu is not observed;(2) the discrepancy in (t) (r) between theory and ex-periment becomes as large as 30% near the transitionregion; and(3) the values of (To - T) inferred from the heatingmap generally corresponds to those computed usingthe TSN correlation within 0.020 C, while the dis-agreement with those based on the NO correlationranges from 0.04-0.150C.
For the sake of comparison, the preceding calcula-tions have been made again, including only the firsttwo terms in the r.h.s. of expression (4), as done in Ref.
Fig. 2. Circumferential average adimensional temperature (t) vs
adimensional distance r, after using the NO correlation, the TSNcorrelation, and the direct technique: experimental results (solidlines); best-fit three-term expansions (dashed lines); NO predictions
(dotted lines) (fourth set of experiments).
4. Even if the qualitative agreement with experimen-tal data was still good, the standard deviation a- in-creased by 50% and Q was underestimated by 20%,regardless of the correlation used. Consequently, atleast the third term has to be introduced in expression(4) to achieve a good accuracy.
We turn now to consider the temperature profilealong the vertical direction.
The plots in Fig. 4 show the experimental values ofthe plume center line temperature t, vs the distance zfrom the wire axis for the fourth set of experiments(solid lines); the dashed lines represent the best-fitcurves of the family t(z) = az-0, computed by apply-ing the least-squares fit method to the points z 2 rmax.Distance is measured in units of the wire radius.
Figure 5 shows the estimated values of a and # for allthe sets of experiments. The factor a is approximately
-612. 12C
10
0.. 8.I1
L_
.:_
0
a0
)80
:5
0 2 4 6 8GR( 0-4 )
Fig. 3. Experimental values of the expansion coefficient D vs the
Grashof number Gr, compared with theoretical predictions: NO,NO correlation; TSN,,TSN correlation; direct technique, d.t.
equal to one, independently of the Grashof numberand of the kind of correlation introduced. On theother hand, the exponent /3 appears to be independentof the Grashof number, but is equal to 2/5 using the NOcorrelation and to 1/2 using the TSN correlation or thedirect technique.
This indicates that the plume can not be consideredfully developed in the present experiments. In fact,when a buoyant plume above a horizontal line heatsource is fully developed, the center line temperaturefollows a dependence on the distance from the wireaxis according to the -3/5 power law. This depen-dence has been experimentally verified at higher Gra-shof numbers,13 and has been assumed by NO in theouter region.
We could mention that both a and /3 have beennumerically computed in Ref. 2, obtaining a = 1.1 and
zFig. 4. Center line temperature t vs the vertical distance z; thedashed lines represent the best-fit curves t4(z) = az- (fourth set of
experiments).
1.2
1.0
0.8
0.6
0.4
0.2
0.00 2 4
GR(1 0-4 )6
Fig. 7. Same as Fig. 6, but using the TSN correlation.
100
Fig. 8. Same as Fig. 6, but using the direct technique.
8
Fig. 5. Values of a and Al obtained by fitting tz) = az- to theobserved distributions of the center line temperature t,.
Fig. 6. Two-dimensional distribution of the adimensional tem-perature t obtained in the fourth set of experiments using the NOcorrelation. The scale segment corresponds to ten adimensional
radii.
, = 1/2. Unfortunately, the computations have beenmade for Gr 10-1 and a direct comparison is notpossible.
Figures 6 and 7 show the 2-D temperature distribu-tions respectively in the case of the NO and TSNcorrelations. Again, Fig. 7 gives a description of thetemperature field which agrees more closely at largedistances with the one obtained using the direct tech-nique, shown in Fig. 8.
V. Conclusions
Using a Twyman-Green interferometer and a com-puterized procedure of fringe pattern analysis, naturalconvection around cylinders in air has been investigat-ed at Gr 10-4.
The 2-D temperature distribution around the wirehas been derived from the phase perturbation mapobtained by analyzing different interferograms andintroducing various proposed correlations between theGrashof number and the Nusselt number.
The experiments indicate that the temperature dis-tribution is always strongly affected by convection butthe plume can not be considered fully developed, evenin the outer region, where previous investigations ex-pected the plume to be fully developed.
Also the circumferential average temperature is indisagreement with some numerical simulations, being-30% higher then predicted at the fringe of the outerregion.
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