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W9 da7 12 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1566 HEAT TRANSFER BY FREE CONVECTION FROM HORIZONTAL CYLINDERS IN DIATOMIC GASES By R. Hermann I. DETERMINATION OF HEATTRANSFER LAW FOR HORIZONTAL CYLINDER FROM TESTS WITH PARTICULAR ACCOUNT TAKEN OF THE TEMPERATURE CHARACTERISTIC Tel 1. Introductory Remarks The case of the horizontal cylinder is of particular importance in the study of heat transfer by free convection for the following reasons: in the first place, next to the rectangular plate it represents the simplest twodimensional case; and second, a very wide range of measure ments is possible, from the finest electrically heated glow lamp wires to pipes heated by liquids or gases flowing through them. To investigate freeconvection flow from the point of view of similarity considerations, it is convenient to consider the case of small temperature differences between the heated body and the surround ings; in this case all the properties of the medium, even the density (ref. 1, p. 429), in the entire temperature field may be assumed constant. The case of large temperature differences, for which the variation in properties over the temperature range can no longer be "Warmeubergang bei freier Stromung am wagrechten Zylinder in zweiatomigen Gasen." VDI Forschungsheft, No. 379, 1936, pp. 124. 1Accepted as dissertation by the Technical High School at Aachen with the approval of Prof. Dr.Ing. C. Wieselsberger and Prof. Dr. W. Muller. The tests were carried out in the division for applied mechanics and thermodynamics of the Physical Institute at the University of Leipzig. The author takes this occasion to thank Prof. Dr. L. Schiller for the interest which he took in this work and for his valua ble advice. NACA TM 1366 neglected, is treated separately. As first pointed out by W. Nusselt (ref. 2), it follows from the differential equations of free convection, together with the uniquely determining boundary values (ref. 1, p. 428) for the case of small temperature differences, that the nondimensional heattransfer parameter Nu (Nusselt number2) is determined by the lift coefficient Gr (Grashof number) and the characteristic Pr (Prandtl number) for the molecular constitution of the gas3: Nu = F(Gr, Pr) (1) where Nu = ad/X, Gr = d3 g p /v2 Pr = v/a (2) a is the heattransfer coefficient (cal/cm2)(sec)(oC), d is the cylin der diameter (cm), g is acceleration of gravity (cm/sec2), X is the heat conductivity (cal/(cm)(sec)(oC)), v is the kinematic viscosity (cm2/sec), a is the temperature conductivity (cm2/sec), 0 is the coef ficient of expansion (0C1), and 8 is the temperature difference (oC) between body (ty) and medium (t.). The case of large temperature difference between the body and the surroundings, for which the gas properties vary over the field, was similarly first considered by W. Nusselt (ref. 2). The following ex pression is obtained for the heat transfer under the restricting con dition that the temperature dependence on v and a may be repre sented by exponential laws in the absolute temperature, the exponents of which for gases of the same substance must be equal4; Nu = F(Gr, Pr, Te) (3) The form here chosen for the temperature characteristic5 2In the notation of the characteristic numbers, the proposals of the Heat Conference at Koln in 1931 are followed. See Forschg. Ing.Wes., Bd. 2, p. 380, 1931. 3See eq. (5). 4This is approximately the case, for example, for air, oxygen, and hydrogen; see reference 1, p. 433. 5The abbreviation Te is chosen to conform to the other abbrevia tions for the characteristic numbers and to remind the reader that it refers to a temperature relation. NACA TM 1366 Te = 8/T, (4) (T,, T.,(K) = absolute temperatures of the body and the surroundings, respectively) differs inessentially from the nondimensional ratio T,/T. chosen by Nusselt, namely by the constant 1. Equality of Gr, Pr, and Te in two different cases means complete similarity, that is, similarity of the velocity field, the temperature field, and the field of all the coefficients characteristic of the substance (ref. 1, p. 434), and therefore also equality of the Nu number. The possibility still remains, however, that similarity may exist at the same time that the previously mentioned condition of the power law is not satisfied. In the present report the characteristics represented in equa tion (3) will be computed for the heat transfer from heated wires and pipes in air, hydrogen, and oxygen; and from these values the most probable form of the heattransfer law, obtainable at present, will be determined for the case of the horizontal cylinder in diatomic gases (Pr = 0.74).5a The dependence of Nu on the temperature characteris tic Te, which follows from the similarity theory and is illustrated by an example (ref. 1, fig. 1) as a third independent variable, will be ver ified for several tests in the range of Gr from 104 to 10. For the smallest values of Gr, the effect of Te considerably outweighs that of Gr. In the region of large Gr from 104 to 107, on the contrary, Te is practically without effect. The large scatter of the test points in the nondimensional representation of the test results of Nusselt (ref. 2) and Davis (ref. 3) (32 percent in the case of Nusselt) is due to the fact that the parameter Te is not taken into account in accord ance with equation (1). The dependence of Nu on Gr, Pr, and Te will be theoretically clarified hereinafter. The effect of Te for small Gr may also be correctly estimated quantitatively. From the tests conducted by various investigators for different finite Te values, the limiting law of small temperature difference is determined in each case by extrapolation to Te = 0. This is of primary theoretical and practical significance since it is identical with the theoretically simpler case (1) and is thus free from the pre viously mentioned restrictive assumption with regard to the temperature dependence of the constants defining the properties of the substance. baFor the values of X and v required for computing the char acteristics, values are assumed for the temperature t. (in agreement with E. Schmidt). The similarity consideration leaves the choice of the reference temperature free. NACA TM 1366 2. Radiation From the measured overall heat losses directly determined in the tests, the amount due to radiation must be subtracted. For this pur pose, a well known equation (ref. 4, eq. 38, p. 232) was employed; the best mean values at present known were substituted as radiation con stants for the metal cylinder surfaces, as follows: for platinum, nickel, silver, and tantalum, the values given by H. Schmidt (ref. 5); and for copper and iron, the values given by Gr6ber (ref. 6, p. 196). Small deviations as compared with the values used by the investigators them selves are without significance in the range of small values of Gr because the radiation component in the case of thin wires amounts to only a few percent of the heat of convection. For the tests of Wamsler and of Koch on thick pipes, however, for which the radiation losses are up to 60 percent, an accurate knowledge of the radiation coefficients is required. These coefficients were obtained by Wamsler (ref. 13) through his own radiation tests. After correction6 of computations of reference 13, the values obtained herein for the radiation heat are 0.2 to 2 percent larger. 3. PropertyDetermining Constants of Gases In order to compute Nu, Gr, and Pr, values of the following properties are required: density p, dynamic viscosity u, heat con ductivity X, and specific heat c for air, hydrogen, and oxygen. (a) Constants for air. The density was computed throughout the temperature and pressure range according tc the ideal gas law7. The true specific heat for constant pressure was obtained from an equation given by Holborn and Jakob8 for the mean specific heat. The temperature dependence of the dynamic viscosity was determined by graphical adjustment of the values given by Erk9. These values 6Wamsler (ref. 13) computes incorrectly with the previously men tioned radiation equation without the factor of the area ratio in the denominator. 7LandoltB6rnstein: 5th ed., vol. I, p. 43. 8LandoltBornstein: 5th ed., vol. II, p. 1274. 9LandoltB6rnstein: 5th ed., vol. I, pp. 177181, Eg. I, pp. 143 144, Eg. IIa, pp. 138141. NACA TM 1366 differ only slightly (at most, 4 percent) from the graphical mean values given by Erk from 0 to 7000 C10. The heat conductivity at high temperatures was obtained by a gas kinetic extrapolation, since the test values collected by Jakobl1 lie only between 0 and 2120 C. The Nusselt formula (ref. 1, p. 491; ref. 6, p. 192), which deviates from the available test values by at most 2.8 percent, is employed herein. From the gas kinetic relation12 Pr = i c/X = (n + 2)/(n + 4.5) (5) where n is the number of degrees of freedom of the molecular motion according to which the Pr number of a gas is independent of the tem perature (and moreover is the same for all gases with the same number of atoms); there is obtained for air a small decrease of Pr = 0.739 for 0 C to 0.721 at 2000 C and 0.679 at 10000 C (value obtained from formula for n = 5 is Pr = 0.737), which, however, is of no signifi cance because it lies within the uncertainty limits of the p and X values. The same holds for the differences in the Pr numbers among air, hydrogen, and oxygen. (b) Constants for hydrogen and oxygen. For the kinematic vis cosity and heat conductivity the values given by Davis (ref. 3) are employed, which also for air are in good agreement (deviation, at most 2.7 percent) with the chosen values. A knowledge of the specific heat is unnecessary if the Pr value for 00 C is satisfactory; the values are 0.717 for hydrogen and 0.731 for oxygen13. (c) Gas constants for high pressures. From the tests of Petavel, which extend up to 160 atmospheres, only those up to 40 atmospheres were computed in order that reliable gas constants would be available. In this pressure range and up to 10000 C, the computation was conducted 10WienHarms: Handb. d. Exp. Phys., IV, 4, p. 531. llLandoltBornstein: 5th ed., vol. II, p. 1304. 12 1A. Busemann in WienHarms: Handb. d. Exp. Phys., IV, 1, p. 359. This relation may be obtained from an equation given by Eucken (Physik Z., vol. 14 (1913), p. 324. 13A. Busemann in WienHarms: Handb. d. Exp. Phys., IV, 1, p. 362. NACA TM 1366 with the normal values since the dependence on the pressure of X and p and the deviations of p from the ideal gas law are either not known at all or are only partly known, and the known deviations amount to only a few percent14. 4. Individual Investigations15 (a) General. The nondimensional representation of the results is effected according to equation (3) by logarithmic graphs with Nu as a function of Gr and with Te as the only parameter, since Pr (according to sec. 3a) may be considered as constant. Even if the slight variation of Pr there indicated is considered as real, there would, according to a test result of Davis (ref. 3) on the effect of the Pr number, be obtained only displacements of the curves of the order of magnitude of 1/100 of those which can actually be observed. In table I are summarized the test objects and test conditions of the individual investigators and the range of Te used. In order to judge to what extent the condition of infinite extension of the fluid was satisfied, the ratio of the height H of the surrounding space to the cylinder diameter d is shown. In figures 1 to 7, approximately equal Te values are indicated by the same symbols. The decrease in Nu with increasing Te is particularly evident in the individual point groups (because of the same wire), whereas the values for different wires naturally show larger scattering as compared with one another. The curves Te = constant were obtained by a graphical adjustment pro cess and, similarly, the extrapolation points and curves for Te = 0 and the dependence of Nu on Te (relative decrease in Nu for ATe = 1). The indicated minimum and maximum values hold for the indi vidual wires (point groups); the indicated mean values, for the individ ual investigators. (b) Results of Ayrton and Kilgour (ref. 1), figure 1. The authors of reference 1 give in the form of diagrams for each wire the total heat transfer coefficient as a function of the wire temperature; two wires (0.0206 and 0.0282 cm diam.) because of the very large scattering and a third wire (0.0102 cm diam.) because of its strong deviation in position and inclination were excluded from the evaluation of the tests. The 14LandoltB6rnstein: 5th ed., Eg. I, p. 64 ff, Eg. IIa, p. 144. 15Acknowledgement is made to Dr. K. Winkler, Leipzig, for his assistance with the computations. NACA TM 1366 points computed from the low temperatures of 400 to 800 C were simi larly not used on account of the large scattering. (c) Results of Langmuir (ref. 8), figure 2. Langmuir gives in a table the adjusted values of the total energy losses as a function of the absolute wire temperature increasing in stages of 2000 C. Only the tests up to 13000 K were computed herein in order to have reliable values available for v and I. The convergence of the curves Te = 1.33 and Te = 3.35 shows that the dependence of Nu on Te decreases with increasing Gr, the decrease independently determined for the individual wires being: log Gr 0.14 0.84 0.63 0.52 0.41 Decrease, percent 7.2 7.5 5.4 3.2 2.2 This experimental confirmation of the results following from theore tical considerations (sec. 6b) acquires significance in that the tests of Langmuir are characterized by a large change of the Te number and smallest scattering. The extrapolation from Te = 1 to Te = O, on account of the absence of test points for this value, was effected with the mean value of 13.2 percent taken from reference 7. Langmuir (ref. 9) also discusses his own tests at various air pressures (10 to 760 mm Hg) and variable room temperature (t. = 1900 C to 6000 C) which on account of the strong change in the expansion coef ficient (P = 1/T,) would be of great interest and the only tests of their kind. Unfortunately, these tests have not as yet been published, and the test data are not available in a form in which they may be evaluatedl. (d) Results of Bijlevelt (ref. 10), figure 3. Since the tests of Bijievelt were primarily for the purpose of investigating forced con vection, the experimental setup was designed for the sensitivity re quirements of this flow. This explains the greater scatter of the test points of reference 10 as compared with those of other investigators. For each individual wire (with the exception of the Ta and Ni wire), a grouping is nevertheless observed with respect to Te in the usual sense. (e) Results of Kennelly, Wright, and Bijlevelt (ref. 11), fig ure 4. The investigation of reference 11 is noteworthy in that through the pressure (p,v) changes, even for constant wire diameter and constant surrounding temperature, a change is effected in the Gr number so that 16For these data the author is indebted to a personal communication from Dr. I. Langmuir. NACA TM 1366 curves of Te = constant are here obtained from tests on a single wire. The nondimensional representation of these tests (fig. 4) has been given previously (ref. 1). The authors give for each wire a dia gram with the heating current as a function of the pressure for three or four different temperature differences, as parameter of which the values for five different pressures were computed in terms of the characteristics and were connected by the curves Te = constant. The extrapolation to Te = 0 shows extremely good agreement for both wires. The values of the wire of medium diameter (d = 0.02616 cm) were not plotted because of the irregularities obtained. (f) Results of Petavel (ref. 12), figure 5. From the overall heattransfer coefficients given for constant pressure as a function of the temperature difference, values were computed for 2000, 4000, 6000, 8000, and 10000 C temperature differences up to pressures of 40 atmospheres. For the sake of clarity, however, only the curves for the extreme Te values are plotted in figure 5. Since the heat con ductivity and density of hydrogen differ from the values of air by the factors 6 and 14, respectively, these tests would have been particu larly suitable for checking the similarity law. Unfortunately, however, these tests are evidently unreliable because of the far too small jacket pipe (l/d = 18.6 as the smallest value of all investigators). The similarity law for the three gases is confirmed only for Te = 0.69 and Gr > 102. In the case of air and oxygen, there is a splitting with respect to the dependence on Te, but in the reverse sense from that observed for all wires of the other investigators. In the case of hydrogen, a dependence on Te in the correct sense is observable only at the smallest values of Gr. Finally, the Nu values also for small Gr lie too high as compared with the values of all other in vestigators and the rise in the Nu with increasing Gr has too great a lag. For these reasons, the results of Petavel for determining the heattransfer law were not used. (g) Results of Wamsler (ref. 13), figure 6. Wamsler's curve, which was obtained on the basis of correctly computed radiation loss (sec. 2), shows a dependence on Te such that the mean values for Te = 0.60 lie about 5 percent higher than those for Te = 0.20. This small splitting of the effect of Te which occurs in the opposite sense to that in the case of small Gr, however, in all probability does not contradict the facts but may be explained by the consideration that the radiation coefficients in the entire temperature range from 500 to 2700 C were assumed constant, whereas in general for metals there has been established an increase of the radiation coefficient with the temperature. If in the previously mentioned temperature range a rise in the radiation coefficients of about 5 percent is assumed, as follows for cast iron from Wamsler's determination of the radiation coefficient and for wrought iron from the determination of Nusselt (ref. 14), the observed splitting of the effect of Te is neutralized and the curve shown, valid for all Te values, is obtained. The test values of the NACA TM 1366 copper pipe and of a wrought iron pipe (of 5.9cm diam.) were not em ployed in figure 6 because of the obvious irregularities indicated by the scattering. (h) Results of Koch (ref. 15), figure 7. The nondimensional characteristics were redetermined herein with the values X and v for wall temperature, since the characteristics given by Koch were com puted for a mean temperature of the substances. The test results show an unusually small scattering which is to be ascribed a careful account ing for various factors (e.g., determination of the temperature of the air in the room and at the walls, and also the end effect of the pipes). For the two intermediate pipes, after graphical adjustment, a decrease of Nu for Te = 0.07 to 0.52 by about 2.5 percent is observed, where as in the case of the smallest pipe no ordering is observed and in the case of the largest pipe there is a partial decrease and a partial scattering. In agreement with the results of Wamsler, the dependence on Te is thus practically zero also for large Gr (104 to 107). 5. Summary of Test Results The results of the different investigators as regards their agree ment may be compared in two groups. In the region of small Gr (104 to 10), the tests of Ayrton and Kilgour, Langmuir, Bijlevelt, and Kennelly and coworkers are in agreement. From these investigations are determined, on the one hand, the curves Te = 0 for the limiting law of small temperature difference (fig. 8) and, on the other hand, the curves Te = 0.65 corresponding to an intermediate test range (fig. 9). For Te = 0 the maximum scatter of the results of the fourinvestigators is 10.5 percent, the minimum 5 percent, the corresponding values for Te = 0.65 being 22 and 8.5 percent, respectively. The middle curves of figures 8 and 9 obtained for Te = 0 and Te = 0.65 give for this change in Te an average change in Nu of 14.7 percent (fig. 10). A relative decrease in the Nu number by 22 percent with increase of Te from 0 to i is thus obtained as the experimental mean value of the effect of Te for small Gr. In the region of large Gr (l04 to 107), the test values of Wamsler and Koch from energy measurements and the values obtained from the temperature field measurements of Jodlbauer (ref. 16) by integrat ing over the cylinder perimeter are shown in figure 10. As follows from the discussion of the results of Wamsler and Koch, the values of Koch must be considered as more reliable. There is nevertheless an uncertainty in the values due to the uncertainty in the radiation com ponents which lie between 40 and 60 percent. On the other hand, the temperature field measurements of Jodlbauer are free from this source of error. Since, moreover, the results of reference 16 are in good NACA TM 1366 agreement (mean deviation, 4 percent) with the theoretical solution given in Part II they were assigned a greater weight in the evaluation. Jodlbauer himself, however, considers his energy measurements, which are about 10 percent higher and lie near the values of Koch, to be correct and his field measurements in error because of the introduction of the thermocouple. The test points of Wamsler are, on the average, 18 percent higher than the theoretical values, and those of Koch, 11 percent higher. The interpolation curve is not everywhere satis factory as regards its slope and curvature and could be determined only with a partial deviation from the test values. The differences still existing, after extensive adjustment, among the individual investigators and the difficulty of interpolation show that the heattransfer curve is still not conclusively determined with the desirable accuracy of several percent, but that, on the contrary, further tests are required. The given interpolation curve nevertheless represents the most probable curve of the heattransfer law, obtainable at the present time, for diatomic gases with account taken of the de pendence on Te in the range of Gr from 104 to 107. The numerical values of this curve are given in table II. 6. Qualitative Theoretical Interpretation of HeatTransfer Law (a) Streamlines. It is necessary to explain theoretically the dependence of the Nu number on Gr and Te, determined experimentally in sections 4 and 5 (fig. 10), and also the dependence on Pr as ob tained by A. H. Davis (ref. 3) for liquids. A solution of the differ ential equations under the assumption of a thin (as compared with the cylinder diam.) heated laminar layer is given in Part II for diatomic gases (Pr = 0.74) and small temperature differences (i.e., approximately constant properties, Te 0) and is valid for mean values of Gr of about 104 to 3x108. On the basis of these theoretical results and with the aid of suitable assumptions, the dependence of Nu on small and large Gr and also on Pr and Te will be discussed theoretically without any necessary implications of finality in the conclusions. The streamline plot (fig. 11) obtained from the solution and the isotherms of figure 12 show that cold air streams from the bottom and sides of the cylinder with increasing velocity in laminar flow upwards along its surface, is deflected away from the surface in the region of the upper stagnation point, and forms a rising current of warm air. With increasing Gr of the cylinder, the thickness of the streaming layer relative to the cylinder diameter decreases. Since the velocities increase at a higher rate, however, the Reynolds number Re of the boundary layer increases so that the boundary layer finally becomes turbulent. According to the schlieren photographs of figure 23, this occurs in the neighborhood of the upper stagnation point at Gr = 3.5x108. NACA TM 1366 With further increasing Gr, the point of turbulence transition travels upstream (fig. 26) and reaches the equatorial region at Gr = 3x109. With decreasing Gr of the cylinder, the Re of the boundary layer likewise decreases, and at the same time the thickness of the layer in relation to the cylinder diameter increases so that the given solution of the differential equations finally (below Gr = 104) becomes in valid. At very small Gr (very small Re), the isothermal picture of the free flow in the neighborhood of the body becomes increasingly similar to the concentric isothermal picture of the pure heat con ductionl7 so that almost static heatconduction relations may.here be supposed. An estimate of the Te effect made under this assumption at small Gr likewise gives good agreement with the measurements (see following section b). The following fundamental difference is never theless to be observed. For initially given cylinder temperature and heat transfer, the solution of the heatconduction equation has as the potential function at infinity a singular point (negatively infinite temperature). That is, even at a very large distance from the cylinder the room temperature continues to decrease, whereas actually in the case of free flow the room temperature at large distance from the body very soon becomes constant. (b) Heat transfer. Dependence on Gr. For the dependence of Nu on Gr, the solution of the differential equations under the familiar assumptions of the Prandtl boundarylayer theory gives Nu Grl/4 (6) as is also approximately shown by the tests of Wamsler and Koch (fig. 10). At very large Gr with at least partial turbulent flow, the heat trans fer must rise more strongly on account of the increased mixing. With increasing turbulent mixing the dependence of the heattransfer coef ficient on the position must also decrease. The assumption that a is independent of the position (or on the cylinder diam.), as is obtained from the tests of Griffiths and Davis (ref. 17) for a vertical plate, gives on the basis of dimensional considerations Nu Grl/3 (7) Above Gr = 3.5x108, the proportionality of the Nu number to the fourth root of Gr corresponding to the upstream travel of the point of transition to turbulence must gradually become a proportionality to the cube root. Experiments of W. King (ref. 18) on vertical cylinders, 17See photographs by R. B. Kennard; Bur. Stand. J. Res., vol. 8 (1932), p. 787. Report on this by M. Jakob: Forschg. Ing.Wes., vol. 4 (1933), p. 45. NACA TM 1366 plates, and blocks at Gr up to 1012 confirm the cuberoot law as already pointed out by M. Jakob and W. Like (ref. 19). For very small Gr, for which the amount of heat transferred by convection is always below that transferred by conduction, the Nu number in accordance with its definition must gradually become constant, as may be seen from the discussion of data below Gr = 103 (fig. 10). Dependence on Pr. The dependence of Nu on Pr for Gr = constant and Te = constant may be understood by assuming a change in X with all the remaining magnitudes kept constant. The increase in Pr, through a decrease in X, gives in the region of predominant heat conduction (very small Gr) a proportional decrease in the quantity of heat transferred, that is, Nu = constant. In the region of large Gr, a decrease in )X (on account of the smaller con duction component as compared with the convection component) gives a relatively smaller decrease in a, that is, an increase in Nu. In agreement, investigations by Davis refss. 3 and 20) on liquids of very different Pr (Pr = 0.74 to 7940) for small Gr give the limiting value of Nu which is independent of Gr and also of Pr. For medium and large values of Gr, an increase of Nu with Pr was obtained. Dependence on Te. The dependence of Nu on Te for Gr = constant and Pr = constant for diatomic gases is determined by the temperature dependence on X, v, a, and also by the temperature chosen in computing the characteristics Gr and Nu (Pr is independ ent of the temperature) for the fluid properties X and v (0 = 1/T. is constant in the entire field, ref. 1); the wall temperature tw or T, was chosen herein. An increase of Te = (Tw/T,)l is considered to be due to a de crease in T. with Tw constant so that the values of X and v employed for the computation of Nu and Gr remain unchanged. The associated increase of p and 0 is compensated by a decrease in the gravity field g so that gp8, that is, the lift acceleration and Gr remain constant. Hence colder outer layers with smaller X and v but larger p now take part in the heat transfer. In the region of very small Gr with predominating heat conduction X, the magnitude a, and hence Nu, must therefore decrease with increasing Te. This is also shown by the data discussed previously between Gr = 104 and 10 (see fig. 10). For the region of very large Gr with smaller vis cosity and heatconduction effect as compared with the turbulent momen tum and heat mixing, it also follows18 from the momentum equation, 18Under the reasonable assumption that with the change of Te the profiles of the velocity and temperature to a first approximation under go affine variations. The 'standard velocity W' may thus be taken for example as the maximum value of the tangential velocity at the equa torial region of the cylinder. NACA TM 1366 because of the constancy of the lift acceleration, that the standard velocity W is constant. The participation of colder layers with in creased value of p in the heat transfer here means an increase in the heattransfer coefficient a, which is proportional to cpW, and there fore in Nu. The decrease of Nu with increasing Te for very small Gr contrasts, therefore, with an increase for very large Gr. The effect of Te for small Gr values must therefore, in agreement with the test results of Langmuir, first decrease with increasing Gr values and finally, somewhere in the range of medium Gr, must practically vanish, as is evidently the case according to the experiments between Gr = 104 and 107 (fig. 10). For the range of very small Gr, the order of magnitude of the dependence of Nu on Te under the approximating assumption of purely static heat conduction may be estimated when a mean conductivity is used, m 1 T T X(T)dT (8) the following expression is obtained for the Nu numbers corresponding to two different Te values Tel and Te2 for equal wall temperature T = Tw2 Nu(Tel) X(Tel) Nu(Te2) = m(Te2) (9) The value Tel = 0 is chosen, that is, vanishing temperature difference or T1w = T,; and Te2 = 1, that is, Tw2/T.2 = 2. For the temperature dependence of X, the previously (ref. 1) given exponential form is used9 X/O0 = (T/To)n (10) with n = 0.73820 190 is the value of X for the arbitrarily chosen temperature TO. 2Determined from the previously given exponent for a of 1.738 neglecting the temperature dependence on c. NACA TM 1366 From 4 Xm(Te2) = T 1 fW2 X dT m T 2 T T 2 with TO = T = T2 in equation (10) and by integration: X(Twl) Tw2 T, T. n + km(Te2) = n+ T,' (I) Since Xm(Tel) = X(Tw ), the following expression results after sub stitution of the numerical values: m(Te2) 2 J 1 7 1= T = 0.81 (1) n(Tel) = 1.738 G The decrease of Nu for Gr = constant with increase from Te = 0 to Te = 1 is thus theoretically obtained as 19 percent. The pre viously discussed tests of the four investigators between Gr = 104 and 10 gave values between 13 percent and 26 percent with a mean value of 22 percent (see fig. 5). This good agreement justifies the assumption of approximately static conducting conditions at small Gr. II. THEORETICAL SOLUTION OF THE BOUNDARY EQUATIONS FOR THE HORIZONTAL CYLINDER (CASE OF STEADY MOTION) 1. Abstract The large number of experimental investigations of heat transfer in free convection correspond to only a single physically satisfactory theoretical solution of the differential equations, namely the solution for the vertical plate (twodimensional steadyflow case) for moderate temperature differences, which was given by Schmidt and Beckmann (ref. 21) with the help of Pohlhausen. The solution is based on the approximation of the Prandtl boundarylayer theory laminarr flow in a layer which is thin as compared with the distance from the lower edge; velocity normal to the wall small as compared with that in the direction of the principal flow along the wall). The results of the theory as regards the velocity and temperature fields and therefore the heat transfer agree very well with the corresponding measurements. NACA TM 1566 For the case of the horizontal cylinder, schlieren photographs in air (see ref. 22 and figs. 23 and 25) show that, for Grashof character istics (Gr = d3gpe/v2) of about 104 and above, the heating of the air extends to only a relatively thin layer around the cylinder, a fact which permits the mathematical simplification of the problem. On the assumption that the heattransfer and the flow processes are restricted to a thin film, as compared with the cylinder diameter, with laminar flow (boundarylayer assumption) the differential equations (ref. 23) for moderate temperature differences can be solved approximately for the velocity and temperature fields; and therefore the heat transfer can be computed which, in the range of Gr 104 to 35108, within which the initial assumptions are satisfied, is in good agreement with experiment. 2. Setting up of Differential Equations To start, the hydrodynamic differential equations are used in a form previously given (ref. 1)21 which shows a "lift term", charac terizing the free convection, that arises from the combining of the gravity term and the hydrostatic component of the pressure drop, so that only the gradient of the dynamic pressure p* remains as the pressure force. This dynamic pressure is equal to zero where there is no motion and no temperature difference with respect to the medium at a large distance. is was indicated in detail in reference 1 (p. 429), the density and other characteristics of the substance in the entire field may be considered as constant (case of small temperature difference) and the equations will continue to describe the free convection problem. The equations in vector form are then22 w grad w = 1/p grad p* v rot rot w gp8. div w = 0 (12) w grad 0 = aA e with the boundary conditions: w = 0, 0 =9 on the cylinder surface; and w = 0, 8 = 0, p* = 0 at infinity. (w, velocity; 0, temperature differ ence with respect to temperature at infinity; p*, dynamic pressure; p, density; v, cinematic viscosity; P, expansion coefficient; a, temper ature conductivity; g, gravity acceleration vector; g, its absolute value; r, cylinder radius.) 21In equation (3c) in reference 1 a minus sign mistakenly appears on the right side. 22In place of Aw the invariant form is written grad div w  rot rot w. 16 NACA TM 1366 For treating the twodimensional flow about the cylinder, these equations are conveniently written in arc length coordinates with the arc length s on the cylinder perimeter (taken in clockwise direction with the lower stagnation point as origin) and the normal distance n from the cylinder surface, and the velocity components u and v parallel and normal to the wall, respectively. There is then obtained, without neglecting any terms23: r ou ou u*v 1 op* r s u + V 6 + r + = + gp sin + r + n rs /n +n sr n r 1.1 6.1/6 1.6 f2u 1 ou Sn2 r + n n 62 1/62 1/6 r 6v ov u2  u  + v _  r + n +s 1n r + n 1. 6. 1 1 1 6v r2 62v + n (r + n)2 s2 r + n an (r" + n)2 ;82 r2 (r + n)2 u 2r (r + n)2 + (r + n)2 6v 3s 2  1 bp* s J gpo cos + r v 2r (r + n)2 (r + n)2 uls 3s 2 1 r u v+ v +n + r+ r + n s n r + n 1 1 6 r o0 a0 f 9  u + v = a  r + n 7s 6n 2 1.1 S.1/6 E2 1 E62 1 0 + + r + n Tn 1/e The approximations made according to the boundarylayer theory are now introduced. The cylinder radius r and hence also s (with the ex ception of the lower stagnation point) and u are assumed to be of the order of magnitude 1; n is assumed to be of the order of magnitude of 23The hydrodynamic equations agree, except for the dynamic pres sure and lift terms, with the equations (10), (11), and (12) for r = constant of W. Tollmien: in WienHarms, Handb. d. Exp. Phys., IV, 1, p. 248. 62V VaIn2 62 1/6 (13) r2 (r + n)2 1Ye 3s2 1 NACA TM 1366 the boundarylayer thickness s <<1, so that n + r may be re placed by r. The estimate of the order of magnitude of the individual terms according to the method given by Prandtl and his coworkers then gives the results written underneath the individual terms. In order that the friction and inertia forces may be of equal effect, it is necessary that v (and correspondingly a) be of the order of magnitude of C2. In order that the lift forces for the motion in the direction of the main stream be of significance as compared with the inertia and friction forces, that is, in order that 'free convection' exist at all, it is necessary that gpAe 1. From the two momentum equations it then follows that the pressure terms are at most. 1. Since p* is zero at the outer edge of the boundary layer, it follows from 1 )p* by integration from the outer edge of the boundary layer to the cylinder surface over the length e that I*/p in the boundary layer can at most be of the order of magnitude 6, that is, so that 1/p p 6/os = 6. Since in what follows only the terms of the order 1 are retained, the following result, important for simplifying the mathematical treat ment, is obtained: The tangential drop of the dynamic pressure is to be neglected as compared with the tangential lift, inertia, and friction forces. The term p* alone is contained in the equation of the normal momentum. Hence, u, v, and 0 are to be computed from the equation of the tangential momentum, the continuity equation, and the heattransfer equation (eq. (14)). Thereafter p* may, if desired, be computed from equation (15) of the normal momentum. Hence, the boundarylayer equa tions for the horizontal cylinder for free convection are 2 )u Ou u .+ S u s + v = t + ge sin s n2 r 6u 6v + = 0 (14) u + v w = a , as on 2 oan u 1 3p* (15) r = 5T g cos r 18 NACA TM 1366 A corresponding estimate of equations (13), first rendered non dimensional, gives the two corresponding conditions: 1 2 Gr 1  and  1, i.e. r (b) Re Re2 4 For a maximum e, in the sense of the approximations, of 0.1 (approxi mation to about 10 percent), this gives Re = 100 and Gr = 104 as lower limits of the computations. The approximation is therefore better the larger Gr, that is, the smaller the boundarylayer thickness e. The upper limit is then attained if the boundary layer becomes turbu lent, which (according to part IV) is the case for Gr = 3.5xl08 at the upper stagnation point, for Gr = 3x109 at the equator. The range of applicability of the laminarboundarylayer computations therefore lies between Gr = 104 and Gr = 109. 3. Reduction to Ordinary Differential Equations The usual introduction of the stream function U V (16) on 3s eliminates the continuity equation. By introducing nondimensional magnitudes with Gr' = r3gg3/v2 by means of the transformation24: x= n Gr1l/4 = Gr'1/4 (17) r r V g there remain of the five constants (v, gp, a; 0, r) of differential equations (14) only two in the nondimensional combination of the Prandtl characteristic Pr = v/a. y 82y 32 3 S + sin x dydx dy d y2 y3 (18) BYi J Yt br 1 o22 by bx bx by Pr dy2 Equation (17) is not an arbitrary transformation which carries equation (14) over into equation (18), but a definitely determined transformation with the aid of which it is possible to eliminate Gr 24Gr' referred to r is more convenient for the computation than Gr referred to d. In the final result, the more usual Gr is again introduced. NACA TM 1366 so that it no longer occurs in equation (18). The solution of equa tion (18) therefore likewise no longer contains Gr. It is therefore unnecessary to know the solution of equation (18) if it is of interest to know only how the solution (e.g., velocity, temperature, boundary layer thickness, heattransfer coefficient) of the initial equation (14) depends on the Gr number. This dependence of Gr is already com pletely represented by the transformation (17), whereas equation (18) contains the further dependence on Pr and on the space coordinates. In other words, the solution of equation (18), obtained for a definite Pr value24a gives, by means of equation (17), at the same time. the velocity and temperature fields for all Gr values. On the other hand, the transformation (17) without the solution of equation (18) already gives important general information on the flow condition and the heat transfer for free convection with respect to the dependence on Gr. From equation (17) there follows, if the following expression is set up, U (19) that n yGrl1/4 = U Gr'1/2 vr = V Gr1l/2 r = Gr1,/4 (20) That is, the relative boundarylayer thicknesses decrease with Gr1/4 the nondimensional velocities increase with Grl/2; and the non dimensional temperature drop increases with Grl/4, from which there follows directly the wellknown 1/4power law of heat transfer in free convection: Nu m Grl/4 (21) These theoretical results agree with those for the rectangular plate and are not restricted to the latter and the horizontal cylinder alone, but depend on more general assumptions (ref. 19). They are valid, in general, for all twodimensional cases of free flow with a boundarylayer character (ref. 23), that is, where the thickness of the heated laminar layer is small compared with the distance from the inci dence edge and the radius of curvature of the wall, and the latter does not change discontinuously in the flow direction. The existence of sharp edges about which the flow occurs is therefore excluded. In this more general case, there are on the right side of equation (13) addi tional terms which are determined by the radius of curvature and which 24alt is here assumed as self evident from physical considerations that one and one only such solution exists. On the mathematical side of these existence and uniqueness proofs, see the discussion in refer ence 1 (p. 428). NACA TM 1366 vary with the arc length (or/6s) but which drop out in the estimate made under the initial assumptions. Furthermore, in place of the sine and cosine of the lift term there appear functions which determine the direction of the surface element with respect to the horizontal and vertical but which similarly do not change anything in the transfor mations (17) and the conclusions from equations (20) and (21) based on these transformations. In agreement with these theoretical conclusions equation (21) has already been confirmed for vertical plates (ref. 21), horizontal cyl inders25, and cylindrical layers (ref. 24), and even for the heat transfer from the two sides of a square horizontal thick plate with edges (ref. 25) and for the heat transfer with evaporation at a verti cal cylinder in water and carbon tetrachloride (ref. 26) for which the previously mentioned assumptions are not all satisfied. A reduction of the partial differential system (18) to systems of ordinary differential equations is attempted by setting q = y g(x) (22a) Y(x,y) = p(q)f(x) (22b) :(x,y) = t(q) (22c) for example, U = = f(x)g(x)p'(q) (22d) based on the notion that all the profiles of the stream function (y), tangential velocity U(y), and temperature T(y) for the different azimuths x of the cylinder are produce by affine distortions by means of the azimuth functions f(x) and g(x) from a single "base profile" of the stream function p(q) or the velocity p'(q) and one base profile of the temperature t(q) (q = normal coordinate of the base profile). Substitution of equation (22) in equation (18) gives (primes of p and t denote differentiation with respect to q; primes of f and g denote differentiation with respect to x): p,2(f2gg, + ff'g2) pp"ff'g2 = p"'fg3 + t sin x /Prgt" + f't'(23) 1/Pr gt" + f'pt' = 0 25See Part I, section 5, figure 10. NACA TM 1366 A separation into functions of the two independent variables x and q, that is, a successful application of the assumed expression (eq. (22)) for solving the equations, is possible only if the two func tions f(x) and g(x) can be uniquely determined from the four ordinary differential equations f'(x) = ag(x) (24a) f(x)f'(x)g2(x) = c sin x (24b) f2(x)g(x)g'(x) = b sin x (24c) f(x)g3(x) = d sin x (24d) (where a,b,c,d are initially undetermined constants) in spite of the redundancy of the equations26. The possibility of such unique determi nation, which is the necessary assumption for the applicability of equation (22), will be discussed in more detail after equation (29) is considered. If such determination is possible, there remain for p(q) and t(q) two ordinary differential equations: (b + c)p'2(q) cp(q)p"(q) = dp"'(q) + t(q) (25) t"(q) + a Pr p(q)t'(q) = 0 j In section 4, equation (24) will be solved and in section 5 equation (25) will be discussed. 4. Determination of the Azimuth Functions F(x) and G(x) (a) Setting up of the two differential equations for' F(x). The boundary values of f(x) and g(x) which are required for solving equa tion (24) are determined from physical considerations. From = dt(q) x =y dq x) it follows that g(x = 0) = go o 0 (26a) 26The author is indebted to Dr. A. J[aumann, Leipzig, for several mathematical suggestions. NACA TM 1366 Otherwise, the temperature of the cylinder surface would also be the temperature in the field along the entire normal at the lower stagna tion point, which cannot be the case. From equation (24a) there is then obtained f'(x = 0) = agg (26b) Since the tangential velocity at the lower stagnation point must, for reasons of symmetry, be different, there fellows from equation (22d): f(x = 0) = 0 (26c) The system of equation (24) with the boundary conditions (26a), (26b), and (26c) thus contains five free available constants a,b,c,d, and go, which are determined in subsequent computation. By introducing the normed functions27: F(x) = f(x) G(x) g(x) (27) ago g0 equation (24) may be transformed to: F' = G (28a) F2 G G' = a sin x (28b) a2g04 F F' G2 = sin x (28c) a2g04 F G3 = sin x (28d) ago4 with the boundary conditions F(x = 0) = 0 F'(x = 0) = 1 G(x = 0) = 1 (29) A necessary assumption for the applicability of the expressions of equation (22) for solving the equations is that the two functions F and G can be uniquely determined from the four equations (28a) to (28d). This would be the case if it could be proven that the four 27By normed is meant fully determined also with respect to a factor. NACA TM 1366 equations are not independent of one another, but that two equations follow from the other two. This proof is partly possible only inso far as one equation (for suitable choice of the constants) is a con sequence of the remaining equations. Thus, elimination of G with the aid of equation (28a) gives F2 F' F" = b sin x (30a) a2g04 F F'3 = C sin x (30b) a2g04 a9 / F F'3 d sin x (30c) ago4 Equation (30c) is thus a consequence of equation (30b) in the case that the following first condition for the constants is satisfied: d = c/a (31) Setting for briefness c/a2g04 = a (32a) b/a2g04 = ( (32b) there then remain for the determination of F the two equations F F' = a sin x (331) F2 F' F" = 0 sin x (3311) with freely available values of a and P. As a necessary assumption for the applicability of equation (22), there remains the further con dition that the two equations I and II possess two equal (or at least approximately equal) sDlutions. This proof will be obtained by solving equations I and II by a powerseries development above and below the equator x = 4/2. For this purpose, with p = x n/2, I and II are transformed into F(q)F'3(p) = a cos p (341) F2(p)F'(()F"(g) = P cos 9 (3411) where the primes now denote differentiation with respect to 9. NACA TM 1366 (b) Solution of differential equation (I) for F. The value of a, by the boundary values of equation (29), is first determined as S= 1 (32c) as is seen from the first term of the powerseries development of equation (331) about the point x = 0. From the expression for solving equation (341): FI()) = a0(1 + Cl ( + c2 92 + .) there follows FI ( = 0) = ao F '(p = 0) = a0 c1 (36a) and by substituting in equation (341) a04 c13 = a = 1 (36b) Substitution of expression (35) in differential equation (341) gives the unknown coefficients, which are taken up to the fifth, in clusive, and which can be successively determined in terms of Cl: 1 c2 = c1 1 5 c3 = 18 cic8 1 5 1 c5 = 6 c 13 + 5 3_601 324 1] 2 = 1 2 5 .4 S(37) 3 . 1L The boundary condition FI(P = r/2) = 0 according to equation (29) gives for the value of cl the solution of an algebraic equation of the fifth degree which is solved by trial CI = 0.581 (38) Of the five possible real roots it is necessary to choose which no further zero of FI is obtained between cp = 0 The coefficients then assume, according to equation (37), c = + 0.581 c2 = 0.05626 that one for and 9 = n/2. the values c3 = 0.01412 c4 = 0.00165 (39) C5 = 0.00066 According to equations (36a), and (36b), there is obtained FI'( = 0) = a0 c1 = 0.874 (35) FI(9 = 0) = ao = 1.504 (40) NACA TM 1366 and the numerical solution of FI(cp) is thereby completely given (see table III). It may be remarked27a that for equation (331) a solution was also obtained by a powerseries expansion at the point x = 0 In powers of sin (x/2) which, because of the fact that the curvature of F at that point is zero, converges with particular rapidity: F*(x) = 2 sin + 1 sin2 + sin4 + (4 2 10 2 100 2 This solution assumes at the equator the values F*(x = r/2) = 1.500 F*'(x = 4/2) = 0.874 (e) which are in good agreement with the values of equation (40) of the series expansion at the equator. In this way, the sufficient conver gence of FI(() with the five computed terms is assured, and a good check is obtained against computation errors. The solution F* will not be used in what follows; this is because of the singular behavior of equation (3311) at the point x = 0, and the identity proof must therefore be carried out by the series expansion at the equator. (c) Solution of differential equation II for F. For equa tion (3411), corresponding to equation (35), the following expression is assumed: FI(,) = b0(1 + dP + d2 + .) (42) Substitution in the differential equation gives the following values for the constants: bO and di are arbitrary d2= 1 2 2b04dI d d = 2 (d2d + d22) (43) 13 3 1 where, on account of the length of the expressions, d4 and d5 are not given as functions of di and d2. The function FII thus depends 27aAcknowledgment is made to Prof. W. Muller, Aachen, for the suggestion that the solution of equation (331) may be reduced to two quadratures by separation of the variables. One of the quadratures is not in closed form, however, but can be carried out only by a series expansion so that this method would not be essentially different from the one employed. NACA TM 1366 on the as yet freely available values of bo, dl, and B. The solu tions F and FII are equal if the corresponding coefficients of the series are equal. Therefore, the as yet undetermined coefficients are equated as follows: bo = a0 di = cl (44) The condition d2 = c2 gives L.= I 2 2bo4d = 6 or with equations (36b) and (44) n = 1/3 a = 1/3 All the coefficients may now be expressed in terms of dl: (f) (45) d3= 5 dl3 d5 7 1 + L 5 540 di 216 d = d2 d 4 4 =+ 21 ~T2_ 1 (46) Comparison of the coefficients of the two series for gives b0 ao = 0 di cI = 0 d2 c2 0 F and FII 1I d3 c3 1 '1 Id ~ T 3=7Bc1 d41 2 4 d4 c4 = 1T S1 1 3 d C5 = 60 1 + 405 This result states that the solutions of the two different differen tial equations (341) and (34II) agree at the equator in the value of the function, in the tangent, and in the curvature, but not in the higher derivatives in which they differ fundamentally to a small degree. FI and FII are approximately, but not rigorously, equal. The system (24) is, strictly speaking, overdetermined and has only approximately unique solution. This solution of expression (22) is not rigorously, but only approximately, valid. Because of the agreement of F and FI at the equator up to and including the curvature, however, these differ ences become more appreciable in the neighborhood of the lower and upper stagnation points where the initially assumed boundarylayer assumptions are not satisfied. The method that has here been employed thus gives an approximate solution of the boundarylayer equations with in the accuracy limits determined by the physical approximations of the boundarylayer assumptions. S 12 d2 = 6 1d (47) NACA TM 1366 27 The numerical computation of equations (44) and (46) with a0 and cl according to equations (40) and (39) gives: bo = 1.504 d1 = 0.581 d2 = 0.05626 d3 = + 0.01816 d4 = 0.00321 d5 = + 0.00144 (48) Table III shows the values of the functions FI(q) and FII(p) computed according to equations (39), (40), and (48), with sufficient agreement between x = 30, 1500 and even at 1650 (graphically interpolated for F I). The negative value of FII for x = 0 has no significance since, on account of the singular behavior of equation (3411) for F = 0, the series (42) beyond this singular point no longer represents the solution of the differential equation. For the further equations, FI is employed because only this function possesses the required property of vanishing at the lower stagnation point. (d) Determination of C. The term G is determined according to equation (28d) where, according to equation (49), d = ag04. Table III shows the computed values based on F . 5. Determination of Basic Profiles p(q) and t(q) The constants b, c, d which occur in equation (25) can now, with the aid of equations (31), (32a), to (32c), and (45) be expressed in terms of a and go: b = 1/5 a2 g04 c = a2 go4 d = a g04 (49) Equation (25) then becomes 2/3 a2 0g4 p'2 a2 g04 p p = a gO4 p"' + t (50) t" + a Pr p t' = 0 with the boundary conditions: q = 0: p = 0 p' = 0 t =1 q = : p' = 0 t = 0 (g) Equation (50) for a = 1 and go = 1 may now be numerically integrated. The following values were chosen for the as yet free remaining constants a and gg NACA TM 1366 a = 3 g = 31/4 (51) equation (50) becomes p"' + 3 p p" 2 p2+t = (52) (52) t" + 3 Pr p t' = 0 This system with the same boundary conditions and for Pr = 0.733 (diatomic gases) has already previously been numerically integrated by E. Pohlhausen for the vertical plate so that, because of the introduc tion of the normed functions F and G and the consequent availability of the constants, a new numerical integration is unnecessary. The stream function p(q) and temperature function t(q) correspond to the functions t(() and 0e() of Pohlhausen. Their values are given in reference 21 in tabular form (ref. 4, p. 187). Graphical representa tions of the velocity profile P'(q) and of the temperature profile t(q) are given in figures 13 to 16 (continuous curves). 6. Complete Solution for Velocity and Temperature Fields With the values of equations (51) for a and go, the azimuth functions f(x) and g(x) according to equation (27) are now uniquely determined. For a g = 33/4 = 2.280 g = 31/4 = 0.760 (53) there is obtained f(x) = 2.280 F(x) g(x) = 0.760 G(x) (54) Table IV shows the values of f(x), g(x), and f(x)g(x) computed from FI and G of table III. Figure 17 shows these values graphically. With the values f(x) and g(x) from table IV, the values p(q) and t(q) from the table of Pohlhausen and the transformations (16), (17), and (22) there is obtained the complete solution of the boundary layer equations (14) for the velocity (u and v) and temperature fields (8) obtained without any empirical value. Collecting results yields (see also eq. (i ) and (20) for simultaneous passage from Gr' to Gr (Gr = 8Gr'): NACA TM 1366 29 u Gr_/2 u(s,n) = F 2 f(x)g(x) p'(q) v(sn) = r l4ff) ) + ()g'(x) P) (55a v'sn =' 77 f '(x)p(q) + l ^q 5(q)) (sn) Gr1/4 O(s,n) = @t(q) (srn) =  g(x)t'(q) s n Gr1/4' with x = and q = 7 g(x) r r For the determination of the normal velocity v it is required to know f'(x) and g'(x). The term f'(x) is, according to equation (24), determined by means of g(x); g' (x) was determined by graphical differ entiation of g(x)28. Here the qualitative description of the normal velocity obtained from the streamline picture is sufficient. It can be shown that the boundary conditions of u, v, e for q = 0 are cor rectly assumed. If the assumption n are obtained for u and 0 the correct value zero; but for v, in contradiction to the boundary conditions (12), a finite value is ob tained which is related to the fact that the cylinder curvature was neglected. The boundarylayer thickness, tangential velocity, temperature, and heattransfer coefficients according to equation (55) behave as follows. With regard to the dependence of these magnitudes on Gr, there hold first of all the considerations of section 3 in connection with equations (20) and (21). The dependence on the cylinder azimuth is such that all profiles of the tangential velocity and temperature are obtained by affine distortions from the basic profiles p'(q) and t(q). The value l/g(x) (fig. 17) represents the 'extension' of the tem perature and velocity profiles normal to the surface with increasing azimuth, that is, the development of any characteristic distance from the wall, for example, of the place of maximum velocity (q = 0.95) or of the boundarylayer thickness to be defined later (q = 2.18). In accordance with this, the boundarylayer thickness at the lower stag nation point possesses a finite approximately constant value over a 28The values are not given here. NACA TM 1366 large azimuth range but increases at first gradually then more and more rapidly and attains at the upper stagnation point a theoretically in finitely large value (upward current of warm air). The function g(x), since it is proportional to the temperature drop, likewise represents the variation of the local heattransfer coefficient with the cylinder azimuth. At the lower stagnation point, g(x) is constant and with constant'slcpe connects with the values of the other side of the cyl inder. At the upper stagnation point (x = i) this is no longer the case; g(x) has a sharp peak which is associated with the fact that the boundarylayer assumptions at this point strongly deviate from the actual conditions (normal velocity large as compared with the tangen tial velocity). Under actual conditions this peak is balanced out by a very sharp minimum value as is shown by the value of Nu along the cylinder perimeter computed from the schlieren pictures of E. Schmidt (ref. 22) No quantitative comparison of this experimental curve of g(x) with the theoretical curve given herein can be given because the conditions of the twodimensional problem are so little satisfied for the schlieren pictures (pipe length = 2 times cylinder diam.) that the computed heattransfer coefficient is 40 percent greater than that of the tests of Koch (ref. 22). The development of the tangential velocity with the azimuth, that is, the increase of any characterizing velocity, for example the max imum velocity, is described by f(x) g(x) (see fig. 17). The velocity increases from the value zero at the lower stagnation point (up to 600 approximately linearly) to a maximum at about 1280 and, on approaching the upper stagnation point (convergence of the flow on either side) rapidly decreases to zero. This physically correct behavior of the solution at the upper stagnation point is not introduced as a boundary condition but is obtained as a necessary consequence of the theory. An overall picture of the velocity and temperature fields is given by the streamlines and isotherms (figs. 11 and 12). They are computed from the equations (x,n/r) = f(x)Grl1/4 p [1 Gr,1/4 g(x)] = const O(x,n/r) n Gr'1/4 g(x) = cnst (55e,f) for various values of the constants30. The scale of the representation 2See figure 13 for maximum Gr = 16xl06. 30The streamline picture for azimuth intervals of 150 and in addi tion for 50, 100, 1700, 1750; the isotherm picture for azimuth intervals of 300 and in addition for 165, 1700, 1750. NACA TM 1366 holds for Gr = 104; for Gr = 106 the distances from the cylinder surface should be reduced by a factor of 3.16 and for Gr = 108, by 10. At the lower and equatorial neighborhood of the cylinder up to an azi muth of about 1050, the flow near the wall is directed toward the sur face, while above it is directed away from the surface upwards. In this respect the theoretical solution for the cylinder differs funda mentally from that for the vertical plate, which gives only a flow toward the plate. The isotherms at the lower part form almost con centric circles. At the upper stagnation point (upward warm current) the isotherms theoretically go toward infinity, whereas actually (corre sponding to the gradual dissolution and spreading out of the warm up streaming air) they close at a relatively large distance over the cylinder. The isotherm 0.53 gives at the same time the place of the maximum tangential velocity. 7. HeatTransfer Law From the expression for the quantity of heat dQ passing from an element of area df, dQ = ]) df (56) dI = 0 ) the value of the temperature gradient at the wall obtained from equation (55c) a (n = 0) = Gr'l/4 g(x)t'0 (57) (t'0 = t'(q = 0) = 0.508), and the defining equation for locallyy variable heattransfer coefficient a(x) a(x) = d (58) there is obtained a()r =g(x)(to') Cr'1/4 (59) From this there is obtained for the dependence of the nondimensional local heattransfer coefficient on the cylinder azimuth x and the Gr number (Grd = 8Gr')30a: 30aSince several formulas of this section will be used later in Part III, and in order to avoid misunderstandings, indices (e.g., d,h,H) for the characteristics Gr and Nu are written which indicate the lengths with which these magnitudes are formed. NACA TM 1366 a x = Nud(x) = 0.604 g(x) Grdl/4 (60) see figure 18. To compute the mean heattransfer coefficient averaged over the cylinder perimeter, am = (x) cx (61) there is required the mean value 1 JA g = g(x) dx (62) which, by planimetering g(x) (fig. 17), is found to be g = 0.616 (63) It may also be computed by the following relation from equation (24a) with a = 3 according to equation (51) 3 it g = f(it) (64) which gives g = 0.620. However, the value 0.616 is preferable because this value takes into account the total curve g(x) and does not depend on the end point of the series expansion f(x). From equation (60) the following is then obtained for the mean heattransfer coefficient as a function of Gr for free flow at a horizontal cylinder am = Nud = 0.372 Grdl/4 (65) A comparison of this theoretical heattransfer coefficient for diatomic gases (Pr = 0.74) with test results is given in Part I, section 5; see also figure 10. 8. Comparison of Theoretical Velocity and Temperature Fields with Available Measurements The comparison of the computed velocity and temperature fields is carried out with the measurements of Jodlbauer (ref. 16) which are the NACA TM 1366 only ones thus far available3. Of the fields measured for six differ ent Gr values, the following ones were computed: 2r = 9 cm, tw = 99.2, t = 18.1, Gr = 3.76x106 (figs. 13, 14) 2r = 5 cm, tw = 104.6, t = 18.10, Gr = 6.54x105 (figs. 15, 16) The best agreement with theory is to be expected for the maximum Gr. On the basis of the theory it would be desirable to have measurements for the largest possible laminar Gr, that is (see Part IV) values of about 5.5x10 which could be realized in air at 200 C at a surface temperature of 100 C with cylinders of 42 centimeters diameter. The comparison with the theory was carried out in such manner that all velocity profiles u(s,n) and temperature profiles e(s,n) corre sponding to a value of Gr were, with the aid of equation (55), re computed for the theoretical basic profiles p'(q) and t(q) in the representation of which they must all coincide. In the case of the velocity profiles (figs. 13 and 14), there is a regular deviation in that the measured velocities are greater than the computed ones, the deviation from the theory in the maximum velocity amounting on the average to 22 percent for the smaller and 17 percent for the larger value of Gr. A part of the deviations occurring for large distances from the wall beyond the maximum value should be ascribed to the uncertainties in the difficult measurements of such small air velocities, as can be seen from the following discrepancies. The measured velocities at the azimuth 600 for Gr = 5.76x106 lie considerably above the theoretical curve, whereas for Gr = 6.54x105 they lie on the theoretical curve. On the other hand, the velocities measured at x = 30:' for Gr = 6.54x105 show strong deviations upward, whereas for Gr = 3.76x10l they are in agreement with thetheory. The measured distance from the wall of the maximum value of the velocity is everywhere in very good agreement with the theory. The measured temperature profiles show very good agreement with the theory even for the azimuth of 1650. A small regular deviation is noted for all azimuths and Gr numbers for medium distances from the wall. The small deviations at small distance from the wall can be ascribed partly to the fact that the condition for the isothermal sur face was not quite satisfied in the tests. The measured temperature 31 1Acknowledgment is made to Prof. E. Schmidt and Dr. K. Jodelbauer for their aid in providing the data. NACA TM 1366 drops at the wall for Gr = 6.54xi05 are on the average 3 percent smaller than the values given by the theory, while for Gr = 3.76X06 they are 1 percent higher32. It is likewise of interest to compare the solution based on the boundarylayer theory for the vertical plate with the corresponding test results. Reference 21 (figs. 20 to 23) also shows the velocity fields to have stronger deviations than the temperature fields. For the small plate with smaller deviations, the measured velocities also lie somewhat higher than the theoretical values; but for the larger plate with larger deviations the contrary is true. For the temperature fields, particularly for the large plate, no regular deviation can be established between theory and experiment. The deviations occurring for the plate are clearly smaller in comparison with those for the cylinder. This is to be ascribed to the fact that in the case of the plate only the boundarylayer terms of the order 2 << 1 (rectilinear flow) were neglected, whereas for the case of the cylinder there were, in addition, neglected terms of the order el << 1 that were related to the curvature. III. HYDRODYNAMIC AND THERMAL COCMPARISCOI BETWEEN VERTICAL PLATE AND HORIZONTAL CYLINDER FOR FREE CONVECTION AND FOR PLATE IN PARALLEL FLOW 1. Abstract Now that the theoretical boundarylayer solutions for the vertical plate and the horizontal cylinder for free convection have been obtained, it is of advantage to compare these two standard bodies of two dimensional free flow with regard to the shape of the stream and the heat transfer. Similarly, a comparison of the boundarylayer develop ment between the vertical plate for free convection and for the plate in parallel flow (Blasius solution of the bourdary;layer equations) is of interest. The following features are characteristic for the three types of flow. 32The fact that the values of the heattransfer coefficient ob tained from the field measurements of J.dlbsuer through integration of the temperature gradients at the wall over the yliinder perimeter lie, on the average, 4 percent higher than the thecretical values (see fig. 10) is due to the deviation between the theoretical and the actual conditions in the neighborhood of the upper stagnation point. NACA TM 1366 The boundarylayer thickness of the plate in parallel flow in creases with the square root of the distance from the incidence edge; that of the vertical plate in free convection increases, however, as the fourth root of the height above the lower edge (cf. eqs. (69) and (71)). Both start with the thickness zero, which for the plate in parallel flow results in infinitely large velocity gradients, and for the free plate results in an infinitely large temperature drop and an infinitely large local heattransfer coefficient at the lower edge. In the case of the cylinder, on the contrary, the boundary layer starts at the lower stagnation point of the horizontal cylinder with a finite thickness and thereafter with finite velocity and temperature gradients and a finite local heattransfer coefficient (fig. 17). It increases according to a complex law (l/g(x), see eqs. (70) and (82)) and at the upper stagnation point reaches a theoretically infinite thickness, with vanishing velocity and temperature drops normal to the surface (upward stream of warm air) and with a vanishing heattransfer coefficient. In spite of these fundamental differences it is possible to set up a number of relations between the three cases. These show on the one hand a close hydrodynamic and thermal kinship which is due to the boundarylayer character of the differential equations underlying the theoretical solutions. On the other hand, they give practical view points with regard to the application of rectangular plates or hori zontal cylinders for the heat transfer. For this purpose it is first of all necessary to represent the Reynolds numbers of the characteristic length, of the boundarylayer thickness, and of the nondimensional local and average heattransfer coefficients as functions of the Grashof numbers of the plate and cylinder, which in turn requires the introduc tion of a boundarylayer thickness (flow discharge thickness) for the velocity profile of the free convection. A knowledge of the dependence of Re of the boundarylayer thickness on Gr of the plate, as well as of the azimuth and Gr of the cylinder, is in addition required for the later evaluation of the tests as regards the occurrence of turbulence (Part IV) for free convection at the plate and the cylinder. 2. BoundaryLayer Thickness for Free Convection It is first necessary to determine the magnitude to be associated with the boundarylayer thickness. For the velocity profile u(y) of the plate in parallel flow (U = maximum velocity), Prandtl and his coworkers, as is known, introduced the displacement thickness 8*, defined as r 5 U = (U = u) dy (66) which is a measure of the decrease in the flow discharge as a result of the friction. NACA TM 1366 For the velocity profile of the free convection u(y), a flow discharge thickness 6 is introduced which is hydrodynamically equiv alent to the displacement thickness, being like the latter character istic of the development of the friction layer, which in this case takes up the entire flow: SU = u dy (67) or expressed in words: Through the flow discharge thickness 6 would flow the same fluid mass with the maximum velocity as actually flows with the total stream. In the nondimensional velocity profile p'(q) (q = ncrndimensional distance from the wall, p = stream function) which occurs in the theo retical solution, the discharge flow thickness is denoted by q., the maximum velocity by p',. From equation (67) there is then obtained: qmax p (q)dq = p() p(0) and with the numerical values: p' = 0.275 max p(m) = 0.6033 p(0) = 0 the value of qc is obtained as qg = 2.18 (68) For orientation purposes34, it is assumed that the velocity maximum lies at q = 0.95 and the point of inflection of the profile at q = 1.85. With the values of q_ for the plate and cylinder from equa tions (72) and (82), respectively, there is obtained for the vertical plate, where h is the distance from the lower edge: 33Extrapolation from last computed value p(6.0) = 0.5928. 34See continuous curves in figures 13 and 15. NACA TM 1366 5 21/2 r/4 = 2.18 h 2 and for the horizontal cylinder of diameter d at azimuth x Sg(x)21/4 Crl/4 = 2.18 .g = (70) 3. Nondimensional Representation of Theory for Vertical Plate (a) Reynolds numbers. From the theoretical solution for the vertical plate given by Schmidt and Pohlhausen (ref. 21) the following expression is obtained according to their equations (23), (24), and (31)35, since c4 = Grh/4h3, for the velocity u and the distance from the wall y: u = p' (q) Grhl/2 S= qif, Grh1/4 For the flow discharge thickness y = 8 and the maximum velocity u = U, there is then obtained U 2v Grhl/2 U 8 pm 1/4 V 9v2 Pmax q5 Grh h = q4,/" Grh1/4 Uh = 2 Pax Grh1/2 V and using the abbreviations Rh = Uh/V and R= = U6/v tuting the numerical values for pma and q (72) (73) and substi Rg = 1.695 Grh1/4 Rh = 0.550 Grhl/2 (74) (75) Elimination of Grh from the last two equations gives the following expression as the relation between Re of the flow thickness and the height of the plate for a vertical plate in free flow: 35The distance from the lower edge is denoted by h instead of x, the nondimensional distance from the wall 4 by q, the nondimensional stream function C by p. (69) (71) NACA TM 1366 RB = 2.29 Rh1/2 (76) (b) Nusselt characteristics. For the local heattransfer coef ficient a(h) at the height h above the lower edge, the following equation holds (ref. 21)*: a(h) = X( to) h) 1/ (77) (tO = 0.508 is the value of the derivative of the nondimensional temperature function t(q) for q = O) or, in nondimensional form, the numerical values may be substituted: a(h).h = 0.359 Crhl/4 (78) The variation of the heattransfer coefficient along the variable height h of a plate of total height H is most conveniently obtained by introducing a local heattransfer coefficient which has been made nondimensional by dividing through by H a.(H = Nu(h) = 0.359 Grll/4 )1/4 (79) The heattransfer coefficient (fig. 18) at the lower edge of the plate with zero boundarylayer thickness is theoretically infinitely large and then continuously drops with the reciprocal of the fourth root of the height. The mean heattransfer coefficient corresponding to a plate of height H a 1 = ft a(h)dh (j) is obtained from equation (77) as .4 L (to0) 1 (80) a,=: 3 2 t) or, nondimensionalized and with numerical values substituted (fig. 18), as =H =0.479 r1/4 X = NuH = 0.479 GrH1/4 (81) *Extrapolation. NACA TM 1366 4. Nondimensional Representation of Theory for Horizontal Cylinder (a) Reynolds numbers. The theoretical solution given previously for the tangential velocity u (eq. (55a)) and the distance n from the wall (eq. (55d)) for the azimuth x of a horizontal cylinder, with the values n = 8 and the maximum velocity u = U, may be trans formed to U = f(x)g(x)pa Grl/2 6 = q Grr1/4 (82) For the upper stagnation point x = t, because g(t) = 0 and f(n) = 5.84, the boundarylayer thickness becomes infinite and the tangential velocity zero. The quantity flowing through U6 and there fore the Re of the boundary layer U8/v, however, remain finite. To introduce Re for the characteristic length results in no simplication for the cylinder because, in addition to the characteristic length, the cylinder radius occurs as an additional characteristic length of the system. From equation (82), U&/v = f(x)Pax q Grrl/4 (83) Substituting the numerical values and using Grd result in the follow ing expression for Re of the boundary layer at azimuth x: Rg(x) = 0.357 f(x) Ord1/4 (84) and for Re of the boundary layer at the upper stagnation point (with f(t) = 5.84): R5 = 2.08 Grdl/4 (85) where R8 is written for R8(r). Although the theoretical solution for x = i is no longer valid, equation (85) because of the small change of f(x) gives the correct relations in the region of the upper stagnation point (up to about 1650), so that for the sake of simplic ity the formula will be used in the following discussion. (b) Nusselt characteristics. Figure 18 shows the variation of the nondimensional local heattransfer coefficient according to equa tion (60) over the developed semicircumference of the cylinder. In the region of the lower stagnation point, it has a finite, practically constant value; and at the upper stagnation point, it drops rapidly to zero in accordance with the upstreaming warm air at that point. In the same figure the mean heattransfer coefficient Nud for a cylinder is plotted according to equation (65). NACA TM 1366 5. Hydrodynamic Comparison (a) Between vertical plate and horizontal cylinder in free con vection. Although the boundarylayer growth for a vertical plate and a horizontal cylinder, as already remarked (Part III, sec. 1) and as seen from equations (72) and (82), differ fundamentally in character, it is nevertheless of interest to compare the flow condition (always characterized in what follows by Rg) at the upper edge of the plate and in the region of the upper stagnation point of the cylinder. The following three questions will here be considered: 1. The behavior of the flow at the upper edge of the plate and at the upper stagnation point of the cylinder for the case that the height of the plate is equal to the cylinder diameter. 2. The behavior of the flow at the upper edge of the plate and at the upper stagnation point of the cylinder for the case that the height of the plate is equal to the developed semicircumference of the cyl inder, that is, the characteristic lengths are equal. 3. The relation of the height of a plate whose flow condition at the upper edge is equal to that of the upper stagnation point of a cylinder to the diameter of the cylinder. Before each of these questions is considered, the ratio of the boundarylayer Re numbers at the upper stagnation point of the cyl inder RB(Z) and at the upper edge of the plate RB(P) are written, with the aid of equations (74) and (85): R5(Z)/R5(P) = 1.226 Grd1/4/Grhl/4 (86) where R5(Z) and Grd refer to the cylinder, and R%(P) and Grh refer to the plate. The coefficient 1.226 has the following signifi cance (cf. eqs. (85) and (83) for the cylinder, eqs. (74) and (73) for the plate): 1.22 2.08 23/4pmax'q8f() f() (8 1.226 = 165= ax = (87) 1.695 23/2p'. q 421/4 2 Pmax 4.21/4 1. For equal Gr of plate and cylinder, Grd = Grh, equation (86) gives R5(Z) = 1.226.R5(P) (88) NACA TM 1366 that is, for equal Cr of plate and cylinder, for example, for a plate of the height of the cylinder diameter h = d under otherwise equal conditions (with regard to temperatures and materials), Re at the upper stagnation point of the cylinder is 22 percent greater than that of the upper edge of the plate (fig. 19). This result states nothing about the hydrodynamic relation between the two, since the comparison refers to different characteristic lengths. It is useful only for the rapid comparison of plate and cylinder for the usual values of Gr. It is then known that for equal Gr the cylinder possesses the more developed flow, so that the flow at the cylinder may be turbulent while that of the plate is still laminar. 2. For equal characteristic length h = A/2d there is obtained from equation (86) Rg(Z)/Rg(P) = 1.226 (2/t)3/4 = 0.874 (89) The boundarylayer Re at the upper stagnation point of a cylinder is thus 13 percent smaller than that of a plate of the height of the developed semicircunf'erence of the cylinder (fig. 19). This retar dation of the boundarylayer development as compared with the vertical plate is due to the fact that the boundary layer at the lower stagnation point of the cylinder already has a finite thickness, whereas in the case of the plate the thickness must increase from zero. 3. For equal boundarylayer Re values for the plate and cylinder, equation (86) gives Grd/Grh = 1.2264 = 0.441 (90) At the upper stagnation point of the cylinder, there is therefore the same flow condition as at the upper edge of the plate if Gr of the cylinder is 0.14 times that of the plate. This permits a rapid con version of the flow data for the plate into those for the cylinder and vice versa. If, for example, it is known from heattransfer experiments on a plate that the departure from the laminar Gr1/4 law occurs at (Grh),r = Sxl18, then it can be concluded without experiment from equa tion (90) that this must be the case for the horizontal cylinder for (Grd)kr = 3.5x0I. If the plate and cylinder are under otherwise equal conditions, it is possible from equation (90) to derive a simple rela tion for the height of a plate ha which is hydrodynamically equivalent to a cylinder: d3:ha3 = 0.441 hg:d = 1.311 (91) NACA TM 1366 In the neighborhood of the upper stagnation point of a horizontal cyl inder, the same flow condition prevails (Rg) as under otherwise equal conditions at a vertical plate of height 1.31 times that of the cylin der diameter (fig. 19). As was to be concluded from the answers to questions 1 and 2, this height must lie between the cylinder diameter and the developed semicircumference. The answer to the three questions on the flow condition at a cylinder and at plates of various heights is shoum in figure 19 (Rg of the cylinder set equal to i). (b) Hydirodynamic comparison between a vertical plate in free con vection and a plate in a parallel flow. The relation (76) between Re of the boundarylayer thickness and that of the height of the plate (characteristic length) Rh for the vertical plate in free convection is strongly analogous to the corresponding relation (92) between Re of the boundarylayer thickness RB*(referred to the displacement thickness 8*) and the characteristic length Rx for a plate in a stream parallel to its plane (Blasius solution of the boundarylayer equations): R = 2.29 Rhl/2 (76)II R* = 1.73 Rxl/2 (92) In spite of the already mentioned entirely different rate of growth of the boundarylayer thickness, in the case of the parallel flow as the square root of the distance from the incident edge ani in the case of the free convection as the fourth root of distance from the lower edge, the development of the Reynolds number of the boundarylayer thickness as a function of the Reynolds number of the characteristic length in the case of these two very different types of flow follows the same exponential law and with approximately equal magnitude. 6. Thermal Comparison between Vertical Plate and Horizontal Cylinder (a) Mean heattransfer coefficients. For equal Gr of plate and cylinder (GrH = Grd), there is obtained from equations (81) and (65) the following ratio of the nondrimensional heattransfer coefficients of the horizontal cylinder Nud and of the vertical plate INuH NuJdNuH = 0.777 (93) Under otherwise equal conditions, therefore, the mean heittransfer cofficienL of a cylinder is only 78 percent of that of a plate of the NACA TM 1366 height of the cylinder diameter (fig. 18). In agreement with this, the comparison given by M. Jakob and W. Linke (ref. 19) of convection tests on vertical plates and horizontal cylinders in a single NuGr diagram (with R and d denoting the length dimension) gives a lower position of the cylinder tests. The modified determination of the critical num ber, which follows from these results, will be discussed later (Part IV, sec. 5). For equal mean heattransfer coefficient am of plate and cylinder under otherwise equal conditions, there is obtained from equations (81) and (65) 0.479 H1/4 = 0.372 d1/4 H/d = 2.76 (94) The mean heattransfer coefficient of a cylinder is therefore, under otherwise equal conditions, equal to that of a plate of height 2.76 times that of the cylinder diameter. On account of the small decrease of the local heattransfer coefficient at large plate heights, this factor of 2.76 is not very sharply determined. Thus Jodlbauer (ref. 29) found it possible, without considering the previously given theoretical solution for the cylinder, to represent the mean heattransfer coef ficients for cylinders from tests by Koch by the formulas for the ver tical plate if H is replaced by 2d (instead of theoretically by 2.76d). As a matter of fact, the am values thus determined for the cylinder lie only 8.3 percent above the values given by the theory according to equation (65), as can be seen by replacing H by 2d in equation (81). (b) Total heat transfer. The total heat transfer Q(Z) of a cylinder of diameter d along the entire circumference and the total heat transfer Q(P) along the two sides of a plate of the height H, under otherwise equal conditions, are in the ratio Q(z) am(Z)id = Nud(Z) Q(P am(P)2H 2 NuH(P) and from equations (81) and (65), Q(Z)/Q(P) = 1.221 Grdl/4/GrHl/4 (95) where the significance of the factor 1.221 is as follows (cf. eqs. (81), (80), (65), and (59)): S0.372 21/4g(to') 3_ g ) 1.221 = (96) 2 0.479 24 f 421/4 5 2 (t NACA TM 1366 A comparison of this factor 1.221 with the factor 1.226 occurring in the ratio of the boundarylayer Re of cylinder and plate (eq. (86)) according to equation (87) shows that both are identical, since accord ing to the theoretical solution, the following relation holds as an immediate consequence of the validity of the heattransfer equation: 3i5g = f(:K) (97) The small difference of 1/2 percent is explained by the fact that (as mentioned in connection with eq. (64)) g was computed not according to equation (97) but was obtained by planimetering. For equal Grd and GrH values in equations (86) and (95), the following is thus obtained: Q(Z)/Q(P) = R8(Z)/RB(P) (98) that is, the total heat transfer of a horizontal cylinder and that of the two sides of a vertical plate under otherwise equal conditions are in the same ratio as the boundarylayer Re values at the upper stag nation point of the cylinder and at the upper edge of the plate, respectively. On the basis of relation (98) it is now possible, without further computation, to answer the three questions on the total heat transfer of a cylinder and plates of different height, analogous to the questions referring to the hydrodynamic comparison (see fig. 19; the heat transfer of the cylinder has been set equal to 1). Corresponding to equation (88), Q(Z) = 1.22 Q(P) (99) The total heat transfer of a cylinder is thus 22 percent greater than that for the two sides of a plate of a height equal to the cylinder diameter under otherwise equal conditions (fig. 18). Corresponding to equation (89), Q(Z) = 0.87 Q(P) (100) The total heat transfer of a cylinder is 13 percent smaller than that of the two sides of a plate of height equal to the developed semi circumference of the cylinder (equal characteristic length) under other wise equal conditions. NACA TM 1366 Corresponding to equation (91) for Q(Z) = Q(P) there is obtained the height of a plate Hg thermally equivalent to a cylinder: Eg : d = 1.31 (101) The total heat transfer of a cylinder under otherwise equal conditions is therefore as great as for the two sides of a vertical plate of height 1.51 times that of the cylinder diameter. In summary it may therefore be stated that a horizontal cylinder and a vertical plate, which transfers heat on both its sides, of a height equal 1.31 times the cylinder diameter are equivalent both thermally (with reference to the total heat transfer) and hydrodynamically (with reference to the boundarylayer development at the upper stagnation point and at the upper edge, respectively). IV. THE OCCURRENCE OF TURBULENCE IN FREE FLOW ABOUT A HORIZONTAL CYLINDER AND ALONG A VERTICAL PLATE 1. Introductory Observations For the determination i:f the transition from laminar to turbulent flow in the case of free convection, tests were carried out on a verti cal plate and a horizontal cylinder of sufficient size. Two main reasons underlay the investigation. In the first place, it was desired to determine the upper limit of the validity of the laminar boundary layer theories for plate and cylinder and therefore the upper limit of applicability of the heattransfer formulas developed from these theo ries for practical application. The second reason was of a more theoretical nature. The numerous turbulence investigations, both ex perimental and theoretical, undertaken in recent decades have been concerned almost exclusively either with the flow between two parallel walls with linear velocity distribution (Couette flow) or with the flow in pipes, channels, about rotating cylinders, along plates, cylinders, or other resistance bodies which have in common a velocity profile which rises uniformly from the value zero at the wall up to a maximum value. Differing essentially from these profiles are evidently velocity pro files for which the velocity rises from the value zero at the wall to a maximum value and then, after passing through a point of inflection, again drops to the value zero at a large distance from the wall. Such profiles are typical in the case of heat transfer by free convection. NACA TM 1366 Some further observations on the setting up of turbulence in free con vection will be made later in connection with the discussion of results (sec. 5). 2. Test Procedure For the investigation, the schlieren method developed b, E. Schmidt was applied, which as an optical method provides the possibility of an instantaneous view of the entire field, and is therefore of advantage for determining the critical number. The schlieren method has been described in detail by E. Schmidt (ref. 22) so that only the essential points will be given here. It is based on the deflection suffered by a light ray in passing through a field with density stratification (i.e., variation of the index of refraction) normal to its direction of propagation toward the colder layer. In the test, a parallel light beam is allowed to pass tangen tially along the surface of the heated body. The rays in the neighbor hood of the wall, because of the maximum temperature drop at that place, are most strongly deflected from their initial direction; those farther away from the wall are less deflected, while those rays which are out side the temperature field undergo no deflection. On a screen which is set up at a sufficient distance behind the heated body, the rays near the wall appear the farthest from the wall. The following picture is obtained: the heated body throws a completely dark shadow which, as compared with that of the cold body, is increased by the thickness of the temperature boundary layer. This shadow is bounded by a first "interior caustic curve" adjacent to which is formed a mediumbright region (rays which have passed through the temperature field), which in turn is bounded toward the outside by a second "external caustic curve" originating from the tangential rays in the neighborhood of the heated surface. Still farther toward the outside is the uniformly bright field of the undisturbed illumination. For an accurate evaluation, the method is suitable only for bodies having one of its length dimensions (twodimensional problem), so that the effect of the temperature fields at the ends on the path of the light ray is small compared with the deflection undergone in passing through the distance along the length. On the other hand, the length should be chosen only large enough so that the light ray at the end of the body still is approximately at the same distance from the wall as at the initial point of the body. NACA TM 1366 3. Test Setup and Procedure36 (a) Plate. The plate was 100 by 100 centimeters and 1.0 centi meter thick. It consisted of two zinc sheet plates 1 millimeter thick which by interposing a frame of 8millimeter Ubrass were soldered onto each other. In the interspace there was placed a heating coil, insu lated by asbestos boards, which consisted of 36 meters of chromenickel wire of 1millimeter diameter with a total resistance of 51 ohms. In air of room temperature, the plate could be kept at 1000 C surface tem perature with 5.6 amperes. The plate was suspended on two bicycle wheel spokes which were attached to the upper strip of the frame. Through the latter the current was also conducted to the interior. For measuring the surface temperature a silverconstantan thermocouple was used which was flatly soldered on about 5 centimeters from the upper rim of the plate and the wires of which extended a few more centimeters quite close to the plate. This simple temperature measurement could be applied because extreme accuracy was not required. Figure 20 shows the plate suspended with the currentsupplying wires, and the thermo couple and two pendulums each with two spheres which served to determine the vertical and made possible the mutual comparison of the different photographs. (b) Cylinder. The cylinder consisted of two layers of 0.7 millimeterthick zinc sheet which was rolled over two bicycle wheel rims and soldered to their upper sheathing strip. Between the sheets there was placed the heating coil insulated by asbestos board, the coil being of 54 meters of chromenickel wire of 1millimeter diameter with a total resistance cf 77 ohms, which brought the cylinder in air of room temperature to 1000 C surface temperature with a current of 4.1 amperes. The cylinder was suspended on two bicycle wheel spokes. The current leads and thermocouple were similarly located on the upper sheathing strip. The mean length of the cylinder was 100.2 centimeters, and its mean outer diameter 58.45 centimeters (fluctuating between 58.35 in the horizontal and 58.75 over the soldered seam). The frontal surfaces were closed off with asbestos board. Figure 21 shows a view with the suspension, current leads, and the two pendulums. (c) Setup. Since it is very difficult to produce a parallel bundle of rays of the diameter of the body here employed by means of lenses, such a beam was replaced by the light from a strongly screened (as a rule of 3mm aperture) arc lamp 32 meters from the heated body. The distance between the body and the screen was 8 meters. The appa ratus was set up on the largest floor, of 42meter length, of the 36Acknowledgment is made to Herr Dr. R. Weise and Dr. H. Kurzweg, Leipzig. For the test setup, acknowledgment is made to Master Mechanic C. Hentsch, Leipzig. NACA TM 1366 Institute. Because of the window in one of the .alls and the connec tion to the staircase halls, it was not possible to exclude entirely currents of the room air so that only at high temperature differences (above 500 C) could useful schlieren photographs be obtained. The existing dissymmetry of the critical points on the two sides of the bodies, both for the plate and cylinder (see table V) was due to a unilateral convection flow from the window wall. The horizontal adjustment of the test bodies was effected by means of a water balance; the adjustment parallel to the light cone, by measuring the shadow of the cold body. (d) Photography37. The schlieren pictures were obtained by two methods. In one method the shadow pictures were taken on the screen by means of a lens and camera (13x18 cm, ZeissTessar 1:2.7). In spite of highly sensitive plates (AgfaSuperpan 16/1C DIm), exposure times of 5 to 25 seconds were required on account of the low brightness, be cause of the strong screening and large distance from the arc lamp. For determining a time mean value of the fluctuating critical points, these time pictures were, however, particularly well suited. The photograph was taken either somewhat from the side (plate) or, in order to avoid distortions, exactly central (partly in the case of the cyl inder); the camera was set up in the shadow of the cylinder so that the legs of the tripod threw an additional shadow (fig. 23). The second method consisted of a direct illumination of one or several photoplates simultaneously (AgfaIsochron 13/10 DIN, 10x15 cm), which were brought to the position of the screen. The greater brightness obtained in this way made possible instantaneous photographs (through light flashes from the arc lamp), which in connection with their greater scale pro.'ided a view in instantaneous detail of the transition f'r:m laminar to turbu lent flow. 4. Evaluation of the Schlieren Pictures Figures 22 and 23 show time exposures of the plate and cylinder taken by means of lens and camera. In the center the magnified shadow of the body by the thickness of the temperature boundary layer, the "inner caustic curve" surrounding it, and further toward the outside the "outer caustic curve" arising from the rays near the wall. The dissymmetry of the lower edge of the plate is due to the disturbances of the room air that were mentioned previously. The upper edge of the plate is immediately below the frame of the picture in the "necking" of the central shadow. In the upper stagnation point of the cylinder are seen the shadows of the suspension wires, current leads, and thermo couple wires; below are seen the three tripod legs for the central 37For valuable advice in connection with the photography, ac knowledgment is made to Herr Dr. A. Naumann, Leipzig. NACA TM 1366 photograph. Figure 24 shows for simultaneously obtained instantaneous photographs of the left38 side of the plate (in the case of direct illumination about 25 cm from the lower edge starting with small inter spaces of about 1 cm), a region of about 50centimeter height: at 24(a) and the lower part of 24(b), laminar; in the upper part of, 25(b), a transition; 24(c) and 24(d), turbulent. Figure 25 shows three simul taneously taken instantaneous pictures of the cylinder with direct illumination: 25(a) at the lower stagnation point, laminar; 25(b) at the left azimuth of 120, turbulent; 25(c) at the upper stagnation point, warm air stream. As shown by figures 22 to 25, the outer caustic curve of the rays near the wall is particularly suitable for determining the critical location. If this line is broken up, then turbulence certainly exists; whereas the flow regions farther from the wall are more easily disrupted through external disturbances. In the laminar (lower) region, the caustic curve is sharp and is completely stationary; in the turbulent (upper) region it is torn into several pieces and in disordered motion. In the time photographs, this means complete washing out. The critical position itself fluctuates upwards and downwards, at one time very rapidly and at another slowly. For evaluating the time photographs there was taken as the "time mean value of the critical position" the end of the region in which the existence of an outer caustic curve through differences in brightness could still be determined outwards and inwards arrowss). Above the critical position the heated layer rises as a whole outwards as broad, uniformly bright strips. Comparison of the photographic observation thus determined with the mean critical position estimated from naked eye observation over a period of time showed good agreement and therefore the justification of this procedure. The critical positions obtained by this procedure are'collected together with the remaining test values in table V. Each critical point is the mean value of four determinations two of which were made with direct illumination, and two with a magnified projection picture. The Gr values are formed in the case of the cylinder with the diameter, in the case of the plate with the critical height as characteristic length. By means of equation (74) for the plate and, with f(xkr), equation (84) for the cylinder, the critical Reynolds numbers of the boundary appear ing in the last column were computed. Their mean values of 303 for the plate and 285 for the cylinder are weighted according to the different illimination times of the individual pictures. From the mean value R5kr = 303 the value 1.0x109 is obtained for the mean critical Gr for the plate. 380n the right of the figure; the pictures with direct illumination are laterally interchanged. NACA TM 1366 5. Discussion of Results The mean value of 303 for the critical Reynolds number of the boundary later for the plate and 285 for the cylinder can be regarded in agreement, in view of the accuracy of determination of a critical number in general and the small number of the schiieren pictures and disturbances of the room air in particular. This is understandable from the fact that the velocity profiles of both flow processes at any distance from the lower edge or lower stagnation point are affine to each other. The development along the initial stretch occurs in a different manner, but this should have only a higherorder effect. As was to be expected, the critical number of the free profile Bkr 300 lies considerably below the critical number of the profile in the case of forced flow (plate), which, referred tc the displacement thickness of Burgers and Hansen, amounts to 950 in the wind tunnel, and accord ing to more recent tests in the GCttingen water tunnel for very quiet approach, it amounts to about 1400. It would be of interest to know whether a theoretical stability investigation for the free profile, the analog of the familiar computations in the forcedflow case, would have to be conducted in order to give this increased instability. With the determination of the critical number for the vertical plate and the horizontal cylinder and reduction to the critical Reynolds number of the boundary layer, the data of other investigators will be compared with regard to the transition from laminar to turbulent flow. E. Schmidt (ref. 26) notes that according to tests in air, the flow at about 1000 C plate temperature remains laminar up to a plate depth of about 50 centimeters, corresponding to a critical Gr of 8x10, in good agreement with the values obtained herein. M. Jakob and W. Linke (ref. 19) determined from a combination of the measurements of various investigators on vertical plates, vertical and horizontal cylinders, block, and sphere, as the transition point from the Gr1/4 to the Grl/3 law, the critical number of (Gr.Pr)kr 3.0xl07, that is, for diatomic gases Grkr m 4.0x107. The difference as compared with the present determinations should in large part be ascribed to the fact that hyvirodynamically unequivalent bodies (Reynolds number of the boundary, see Part III, sec. 5a, eq. (88)) were here treated together. If account is taken only of the tests on the vertical plates and vertical cylinders (Part III, sec. 6a), which for large cylinder diameters and large Gr (thin boundary layer) are at least approximately equivalent hydrodynamically, there is obtained a critical Gr of 4x108, which is therefore quite close to the present determinations. NACA TM 1366 E. Griffith and A. H. Davis (ref. 17) investigated the local distribution of the heat transfer on a 270centimeterhigh wall, which consisted of 25 individual elements. They obtained a minimum value of the heat transfer at about 38centimeter height (their own determina tion) which they regarded as the transition from the laminar to the turbulent condition. This agrees in order of magnitude with the pres ent values (columns between the individual elements), where it should be further clarified which point of their curve is to be regarded as the "critical point" proper. 6. Travel of the Critical Azimuth for the Horizontal Cylinder On the basis of the given theoretical solution of the flow about the horizontal cylinder and the point of transition experimentally determined for a Gr = i09, the travel of the critical azimuth for a variable Gr can be computed if it is assumed that the turbulence always occurs if the boundarylayer Re has the same value of 285. This reasonable assumption will correspond all the more to fact since, as has just been shown, even for the vertical plate with a quite differ ent boundarylayer development, the turbulence is set up at the same critical number of the boundary layer. From equation (84). the following relation is obtained, with the value Rkr = 285: (Grd) =4 (102) kr f xkrT which is evaluated in table VI and plotted on semilogarithmic grid in figure 26. In accordance with the basic considerations, for a Gr = 109 the start of turbulence occurs for an azimuth of about 1200. Complete ly laminar fl.j about the cylinder, for extension of the theory up to the upper stagnation point, occurs for Gr<3X108. The transition point reaches the equator for Gr = 3x109. For greater clarity, the Gr will be expressed for cylinders of varying diameters, which are in air of 20 C with a surface temperature of 1000 C. From the definition of Gr, for 6 = 80 (C), P = 1/293 (OC1), v(l00o) = 0.231 (cm2/sec), the following relation obtains: (Grd 3\ d[c = 15 X 103)1/3 (103) which with the aid of the critical Grd values from equation (102) has similarly been evaluated in table VI and plotted in figure 27. Corre sponding to the basic considerations, the critical azimuth for the NACA TM 1366 relations considered for a cylinder of 60centimeter diameter is 1200. For cylinders below 41 centimeters, no turbulence occurs at all. The transition point reaches the equator for a cylinder of 84centimeter diameter. For practical applications, the result is thus obtained that for heated horizontal conducting pipes up to 40 centimeters in diameter and surface temperatures of 1000 C in roomtemperature air the previously given formulas are valid for the heat transfer in air (in particular, eqs. (60) and (65)). For accurate computaticn, the nondimensional formulation of figure 26 is naturally to be applied. The formulas can, in the absence of other suitable data, still be applied up diameters of 60 centimeters, because two thirds of the cylinder periphery is still in laminar flow. There should not likewise he much of a change for considerably higher surface temperatures (2000 or 3000 C). The experimental determination of the variation of the critical azimuth for the cylinder with the Gr, through varying the temperature, remained unsuccessful. For temperature difference up to about 500 C the outer caustic curve is still not sufficiently far from the central shadow in order to be perceived separately from the latter, as is re quired for determining the critical number. For this, the distance between the cylinder and the screen would have to be considerably in creased, which is not possible on account of space requirements. Going beyond the usually employed temperature differences of 800 C, because of the strong increase in v, no longer gives an increase in Gr and moreover gives increasingly stronger deviations from the assumption of moderate temperature differences, which is at the basis of the theory. The travel of the critical azimuth with Gr is best determined by varying the cylinder diameter, as follows from figure 27. V. SUMMARY From the numerous tests already available on the heat transfer from horizontal pipes and wires in diatomic gases, the dependence of the nondimensional heattransfer number Nu on the Grashof number Gr and the temperature coefficient Te, which enters as a further non dimensional factor at large temperature differences, is determined. The effect of Te on the heat transfer is quantitatively determined for the first time. It is particularly large in the region of small Gr (104 to 10), where on the average a decrease in Nu by 22 percent for in crease of Te from Te = 0 to Te = 1 is obtained. The experimentally found dependence of Nu on Gr, Pr, and Te can then obtain a qualita tive theoretical explanation. NACA TM 1366 For the region of Gr (about 104 to 3xlO8) in which the heat trans fer is limited to a thin (in comparison with the cylinder diameter), heated layer with laminar flow, the velocity and temperature fields and the heat transfer are quantitatively computed from the boundarylayer differential equations without any additional empirical values; these computations are found in good agreement with the available measure ments. In particular, the onefourth power law of the heat transfer is obtained theoretically as Nu = 0.37 Grl/4. The flow and heattransfer relations thus computed (Re of the boundarylayer flow, variation of the local heattransfer coefficient along the cylinder periphery, mean heattransfer coefficient, and total heat given off by the cylinder) are compared with the already known relations for the vertical plate. Among other results, the depth of a vertical plate is determined which for free convection shows equal flow condition and equal total heat transfer at a given cylinder, so that a simple computation is made possible for converting the heattransfer data for a plate to those of a cylinder and conversely. In order to know the upper limit of validity of the laminarflow and heattransfer computations and the laminar heattransfer formulas, the start of turbulence was determined by schlieren photographs on a vertical plate and a horizontal cylinder of sufficient size. This occurs for different values of Gr, namely, Gr = l.OX109 for the plate and Gr = 3.5xi0O for the cylinder, but for equal Re of the boundary layer flow, namely, for Re = 300. In this way, the critical Reynolds number of the velocity profile of the free flow was determined for the first time; as compared with other velocity profiles, this profile is marked by the presence of a maximum value and a point of inflection. Likewise computed with the Grashof number was the 'critical azimuth' on the cylinder at which the laminar flow passes into the turbulent flow. REFERENCES 1. Hermann, E.: Phys. Z., Bd. 33, 1932, p. 425. 2. Nusselt, W.: Gesundh. Ing., Bd. 38, 1915, p. 477. 3. Davis, A. H.: Coll. Res. Nat. Phys. Lab., vol. 19, 1926, p. 193. (See also Phil. Mag., vol. 43, 1922, p. 329.) 4. Gr6ber, H., und Erk, S.: Die Grundgesetze der Warmeiibertragung. Julius Springer Berlin, 1933. 5. Schmidt, H.: Ergebn. exakt. Naturwiss., Bd. 7, 1928, p. 342. NACA TM 1366 6. Gr6ber, H.: Warmeibertragung, Julius Springer (Berlin), 1926. 7. Ayrton, W. E., and Kilgour, H.: Phil. Trans. Roy. Soc. (London), vol. A183, 1892, p. 371. 8. Langmuir, I.: Phys. Rev., vol. 54, 1912, p. 401. 9. Langmuir, I.: Trans. Am. Inst. Elec. Eng., vol. 31, 1912, p. 1229. (See also Proc. Am. Inst. Elec. Eng., vol. 32, 1913, p. 407.) 10. v. Bijlevelt, J. S.: Die kiinstliche Konvektion an elektrischen Hitzdrahte. Diss., T. H. Dresden, 1915. 11. Kennelly, A. E., Wright, C. A., and v. Bijlevelt, J. S.: Trans. Am. Inst. Elec. Eng., vol. 28, 1909, p. 365. 12. Petavel, J. E.: Phil. Trans., Bd. (A)197, 1901, p. 229. 13. Wamsler, F.: VDI Forschungsheft 98/99 (Berlin), 1911. 14. Nusselt, W.: Diss. T. H. Minchen, 1907. 15. Koch, W.: Beih. z. Gesundh.Ing. Reihe 1, Heft 22 (Berlin und Munchen), Oldenbourg 1927. 16. Jodlbauer, K.: Forsch. Ing.Wes., Bd. 4, 1933, p. 157. 17. Griffiths, E., and Davis, A. H.: Transmission of heat by radiation and convection. (London), 1931. 18. King, W. J.: Mech. Eng., vol. 54, 1932, p. 347. 19. Jakob, M., and Linke, W.: Forsch. Ing.Wes., Bd. 4, 1933, p. 75. 20. Davis, A. H.: Phil. Mag., vol. 44, 1922, p. 920. 21. Schmidt, Ernst, und Beckmann, Wilhelm: Das Temperatur und Geschwindigkeitfeld vor einer Warme abgebenden senkrechter Platte bei naturlicher Konvektion. Tech. Mech. u. Thermodynamik, Bd. 1, Nr. 10, Okt. 1930, pp. 341349; cont., Bd. 1, Nr. 11, Nov. 1930, pp. 391406. 22. Schmidt, E.: Forsch. Ing.Wes., Bd. 3, 1932, p. 181. 23. Hermann, R.: Phys. Z., Bd. 34, 1933, p. 211. NACA TM 1366 55 24. Beckmann, W.: Forsch. Ing.Wes., Bd. 2, 1931, p. 165. 25. Weise, R.: Forsch. Ing.Wes., Bd. 6, 1935, p. 281. 26. Jakob, M., und Linke, W.: Phys. Z., Bd. 36, 1935, p. 275. 27. Schmidt, E.: 1. VDI, Bd. 76, 1932, p. 1025. Translated by S. Reiss National Advisory Committee for Aeronautics NACA TM 1366 TABLE I. SUMMARY OF TESTS BY DIFFERENT INVESTIGATORS Invest igator Ayrxton, Langmuir BiJlevelt Kennelly, Petavel WamEler Koch KLigour Wright, ErBlevelt Substance 9 Pt wiree 5 Pt wires 6 wires: 3 Cu wires 1 Pt wre Pipes: 4 steel Ta, Pt, Fe, 6 wrought pipes Cu, Ag, Ni Fe; 6 Cu, 1 cast Fe Cylinder aiam., 0.0031 0.00404 0.0043 0.01143 0.1106 2.058.9 1.4 Scm 0.0556 0.0510 0.1000 0.06907 5.9 10.05 5.9 temperature of 4000 2271027 46239    27.6188.6 body, ty. . Excess tempera   6 0151 2001000 36243 13174.4 ture, 6, C Room tempera 10. 515.9 i?22 20 (approx) 16 1029 14.222 tare, t o: Medium Air Air Air Air Air, 2, Air Air 0, Preeure 760 750 50 70 1201 900 0.1i 715 715.7722.3 mm fig 1 : atm Cylinder lerntn, 32.5 100 35 150 (approx) 45 300 138198 cm Srround ing BoraiHintal Bas", ,10C': Height, Vertical Horizontal Ba e, ".35 Height, space pipe 15 im:; 300 cm tanr: pipe: zq meters; 400 cm lengthr, beigrit, length, Length, height, 2.5 cm; 30 cm 152 cm; 45 cm, 210 cm d am. height, diam. 5.O cm 660 cm .0 cm Ratio of ;pace 1640:143 7450:5886 0,000:j30.0 58,0C0:9640 18.6 102:24 286:40 neieht to ylirider diam. le number 0.701.00 0.673..3 0.090..58 0.C.'0.Ei 0.693.4r. 0.130.82 0.045 0.606 Relative 7.616.6 7.22.2 9.445.6 25..2 Irreguiar None None decrease of Nu for z'e = 1, percent Mean value S'temat i: 25.6 25.2    of ru for '', = 1i, percent Fige re 1 2 4 6 7 NACA TM 1366 57 co 0 aj O 0 0 w .0 3 0 ,i H LO r. ')  O C.I C Wo 0 j ED rx S 0 Lb   CJ 0 o O O N F) 0 CO 0 p o Om H n r ci w a j 49 o1 .o 'di.. ur OO H L LO 00 u H H r LO S 1 *) r J C 0 o 0 1O II r C ) 1r 0 11 Lj ftj c r N C I) n CDO T S" '1 O I O o C to Oi *'" a at  t 1 H V) ( * 0 H ,.u t o S "] ,: 0 ,O '4 0 0 U 4 ," ,:D ,9I., C, o 0 1 0 O 0 Oa4LO O o a0 5 x o Co a o x 0 S '1' rj al 0 if) b 0 00 C D S, r. , .l n1 ,I H ,. LiL. ,o ,O 0 D 0 r4 O H .rH ) v l CO H . .i W H 0 0. S t 'O 'C0 D uo Oo O O o Fri 7o ,, ZH oD ,' 0 E. uC. P Pq 0.. ^ f, v * H" E H ,i u p "rL 0 I o0 H uW . r W "' H> 0 , 4C l C: H HQ H D) H 'C U. 0 L, G) 0 0 H r Cf o 0 . .H * 0_ P4 Z T. '44 .4 ,, i : S3 . , . m > B < .I . f I HI X C b i II Li' '1' NACA TM 1366 TABLE V. EVALUATION OF SCHLIEREI] PHOTOGRAPHS FOR OCCURRENCE .JF TURBULENCE (a) Vertical plate. Number b, t,, Exposure tw, Critical height, h cm Grhkr R5kr mm oC time, 0C Left Pight Middle se.id 24 74d 16.5 20 78 76.0 55.9 F6.0 13.42 x10 324 8 25 748 16.5 20 113 53.8 47.3 50.h 6.83 x10 274 26 748 16.5 25 100 66.2 60.8 633.5 13.13 x1O0 323 27 748 16.5 20 100 60.0 47.3 54.1 8.12 x108 286 Mean: 303 (b) Horizontal cylinder. Diameter, d, 58.45 centimeters. lumber b, t. Exposure ty, Critical azimuth, X Gr kr f(xkr) RBkr am oC time, 0oC Left Ri, ht Middle sec 6 760 19 10 102 125.50 123.5. 124.5 10.26 x 10 4.55 291 10 760 19 20 102 125.8 110.00 117.90 10.26 w 108 4.35 278 11 760 19 5 102 120.90 116.0 11,.50 10.26 x 10' 4.37 279 12 760 19' 210 102 126.7 117. 'H 122.1' 10.261 x 10 4.48 286 17 752 18 20. 102 13. 1.0 11.40 10.23x X 10 4.39 281 1i 752 1. 5 102 144.5'' 12.5 37 .00I' 10.23 x 1o 4.90 513 Mean: 285 TABLE VI. TRAVEL OF TURBULENCE IRAI'JSITIOI] POC1nT (CRITICAL AZIMUTH xkr) WITH GRASHOF I UMBER OF C YLTDER Crd OR WITH CYLINDER DIAMETER d [ty, 100i'' C; te, 200 C; b, 760 mm.] x. 1 0 1 65 150 120r' 90' '. 3 50C 150 10 5 Grd 3.52 4.30 5.42 1.07 2.96 1.37 2.10 3.35 1.69 2.72 xl08 xO18 x108 x109 x109 xl'0 xio11 xi012 x1013 xl014 d, cm 41.3 44.1 47.7 59.9 84.0 140 347 890 1500 3790 NACA TM 1366 59 0.fPr oor rP i 00 Te ___ ____ .3 /./ S.... ,; J, (Extrapolated) > /" S Adjutment  I O , 0 IC 0 ^7.7    j^ ^ ^    o +' eO, a 1,3 , Tests x 1 1 ',0 7 0 4 19 Grf A JJ3 . v Te0 (Extrapolated) " Adjustment . f,33 G3  2 log Gr Figure 2. Convection tests of Langmuir on five platinum wires. 60 NACA TM 1566 prLv r at + le 'k to r " e =isIto. / \ / Tests x 0oZto2 Z' / N =Cito.e5 / S_ 4"Zto47F ____ ____ rTep (Extrapolated) Adjustmeati 0.J .65 7,/* , a,71 log Gr Figure 3. Convection tests of Bijlevelt on six wires o.f tantalum, platinum iron, copper, silver, and nickel. 4z I r I , Ci A4. A f0 .Wire diaom, I, Tests / " Wire diam,.. c. em .Tes 0o 0 1xt iupw li O,./ ^   ,  4791 '4  _______ _____ log Gr Figure 4. Convection tests of Kennelly, Wrsht, ar.d Bijlevelt on three copper wires. NACA TM 1366 log Gr Figure 5. Convection tests of Petavel on one platinum wire. Pa0 Te S3 ~ aJ to 026 Tests : 0 .X to JS9 S +z to 0.50 9 2 054O to O. _ Adjustment : All values of'Te/ y ,i Figure 6. Convection tests of Wamsler on one copper, one cast iron, and six wrought iron pipes. 62 NACA TM 1366 7.4 Pr 0. Te 7.3 OfO ro .10 I0 0 to 025 Te tS 7.2 0. 5 to 0.6 AdjuEtment :Al v values of Te _ / 1.0 I I    9 / rO 5I .5 4 5 6 7 log 'r Fivpure 7. ic.rn.e.tion tests of FKocr orn four teel pipes. 1 Ay,Tt.:,n arina Kliloour 2 Bijleve 3 Kennelly et ai. 7 O longmuir  .,_^  rJ _ O j7^ ^ J log ,or 0 Figure 8. CompariZon iof teit result of, different investigators for Te = 0 (extrapolster) Limitminr la of smill temperature differences. NACA TM 1366 63 I I Pr=0.74 S Ayrton and Kilgour ____ 2 Bijlevelt J Kenrell:, et al. 0 Languruir .91 a  b D f  >  Figure 9. Comparison of test results of various investigators for Te = 0.65. 4 3 2 1 0 f 2 3 4 5 6 7 DAo" log Gr Figure 10. Summary of all test results with diatomic gases (Pr = 0.74) with the adjustment curve \viing the most probable heattransfer law based on present irU'ormation. 64 NACA TM 1566 co S I( 0 U ** N 0O S r4 _0 r4 o  a) ci) 0c. o \ ... I Sm  > l  r *M U) C m 0 A rd u E,\ i / yn u a r /1 1 1 O o +2 Ar PO ?D CD il1 03 +) O LQ V4 TIACA TM 1366 ___ " .1 o1 a 0 tSS9,I C, 18.10 C O G OrJ3. 6 o UO 0 ., , r0 + O & 1/4 S .5 11e,11 en .,),lt uer ir ir = : r ,:ntimeters; t, = 99.20 C; to= 18.10 C; A.z uatr t deg 0 JO IZO 7 0 Or3. 6 VF J 1 ... Fijurc L3. '..'Vloc it: Frofi'1,r r 'r .yrirer referred to nondimensional coordinates. ;:.' nruj 'e :r''6 r'ipre.eriT. th4i.:.r.frci le :ls.:.1in. Points according to ri ,,:miieT.; :i ...lts'je.r icr r *:TrITimetiers; tW, = 99.20 C; ty,= 18.10 C; SA;i ut n / deg .8 90 t 9S2! t 18.1 c  o 30 Grj376x10' 5 __ 0 Lr*r'l nGri/4 "r. .: i l ; d i; .;n.c : m r ill, " $ 1 gl ) r8 6 1/4 ,t re 14. i.,pe'ture Tjro frilc. near "lin.er referred to nondimensional oI'd I;Ljtes. t 'rnt inuor, 'ur.e rieprient: heoretical solution. Points accrirric rt me's.uremerint of .TdlbIsu!er f:', 2r = 9 centimeters; tw = 99.20 C; ':, = i i.1 ,I ; ,r = ,'.. xi,?' . NACA TM 1366 S2r 5cm 00 o 00 o 0 0 ./ I I*" .,^ \  ,4 0 0 0 W^ n Gr4 Dimensionless distance from wall, rL . Figure 15. Velocity profiles near cylinder. Continuou: :ar. c.'e theoretical solution. Points according to measurement o: .*._1. i.er _:r 2r = 5 centimeters; t, = 104.60 C; to3= 18.10 C G = r Y . de, Zr Scmn t, 6.54, tl. Grb.54.i0 5i .N. i 0 2 3 .* 'A n6r% Dimensionless distance from wall, 8 gtl Figure 16. Temperature profiles near cylinder. ConTu;,. cr. v ..ve theoretical solution. Points according to measureme. :.: .o.icauer for 2r = 5 centimeters; t, = 1:4.. C; to = 18.10 C; Gr : .4xi'. TIACA TM 1366 Iv EWISW Azimuth, x, deg .9 .6 .2 0 Figure 17. Azimuth function g(x) giving variation of local heattransfer coefficient or of a reciprocal distance from the wall which is characteristic for profile (for example, boundarylayer thickness) along cylinder perimeter.  Local heattransfer coefficient   .lean heattransfer coefficient, as 1:0.777 ///. Total heattransfer coefficient, as 1:1.22 . / t i. ' 0 .2 .6 .8 0 Bei.ht H of vertical plate 0 M0 6M0 0 x a 5 fII W *Azimuth, x deg Diameter d = 2r of horizontal cylinder Figure 18. Cornparis.uon of heat trianEier for rectangular plate and horizontal cylinder. For H = 1, under otheriize equal conditions, the two diagrams give dimeni.anal hesttarinsfer coefficients a to scale. NACA TM 1366 157 1Z00 : : 1.00 1 Z5 Figure 19. Ratio of total heat transfer of horizontal cylinder and vertical plates (on both sides) of different heights under otherwise equal conditions. Ratio of boundarylayer Re at upper stagnation point of cylinder and upper edge of plate. Abscissa, heat transfer and Re; ordinate, plate height. Figure 20. Vertical plate 100x100x1 centimeters for determining occurrence of turbulence for free convection. IACA TM 1366 Figure 21. Borizontal cylinder (58.5cm diam., 100cm length) for determining occurrence .:,f turt.ulenc.e for free *:n.'.ection. E ,ir'e L'. :rlterer ph t.3grap h ji re.r transfer at the vertical plate. Time prct.,gapi E i ' .: .Iti L:;. t r.. .I: .. Surface temperature, t, = 1000 C; t = 1 .:. '.; L. T4 milli.eter.; .rit ial height (arrows), left 66.2 center rep ri,rr ... :Ertmet r: critical Grashof number, 13.1x108; 'rit i :'j1 '_. uijn' r':,' ." : 'e; ] e, _" .. NACA TM 1366 .' ." 253. Selie:e:: 7. e nee sser  he ariz aal cylider. b : '__e.r ; .':" he 7:yLiie. '. " s~_i l  arows) " _ NACA TM 1366 7] b d  .. ' 2 .er . 07 NACA TM 1366 Figure 25. Schlieren photographs of heat transfer Simultaneous instantaneous photographs (1/20 sec) tw = 1020 C; tc= 180 C; b = 752 millimeters; Gr, nation point, laminar; (b) azimuth of about 1200, point, upflow of warm air with current lead wires at the horizontal cylinder. with indirect illumination. 10.2x108. (a) lower stag turbulent; (c) upper stagnation and so forth. NACA TM 1366 a, I " J 0 I15 JO W0 3 iZO 15 W 10 (1 Critical cylinder azimuth, deg Figijre 26. Variation of position o.f ,tart of turbulence with Gr number of cylinder. Computed for critical boundarylayer Re, 285. 3) '~ L   I SI\ 0 O 80 2# 720 1_ 0 ___ _ iM jCriticil cylinder a:Lmuth. deg Figure V7. r, nation of Ionition of tart of turbulence Vith the cylinder diameter for surface temperature of l,'C0 iLn air at 200 C and 760 millimeters of mercur,'. Computed for critical boundarylayer Re, 285. 0) \ s \ ^ ' n < a ,** * 15 ^ ^ 0  \  so U 'i.^ of mrercur,'. Computed I'.^r critical sijoundarylay:er Re, 285. NACALingley 11354 1000 j ~ C:c U~i L: CS : L. ;  43 F. r b :.5 a ,c z > 0n C, M 0 d = < c eOi . *z5 E rZC) 30 ' u 60 u C, E<> L e. i 'n z5Ec U i C S!LM3 " u 2.C M u 5 B 203 2 zw0 o> < z z = : tNzo ^ or'* ~ "' ^*1.5ii^ Z Z B NZ N U u a > ip S u N c< c: 0 c 43" > o c c w lwEL iS'.t E 4" Ei oa~ N.o 545g5> Cu. o d : i CW 06 5 L C' 0 041N. cca ZE LC ME 5 M G 05.( , r& . V 43 ss'S C.'^ 43a0 >' wl cd Q f> C us t Sa t i a54a C w VCu4 C . Z1 Ea ;a 0639 c d c c cbBal S~t 2 ? nl a M'a w m a M= E2 ;QO bdc it . 0 V. 0 z 0O g c u 4;4 U Zb 4 E 0s0Q0 a t^ P4 bDg *Og cd e4 44 G w L) b aM >4 (Z:S k E 4 >s .4 u iil l P4 0 Z 2a Bilz N S << o l ilg" ll zzaaSMaoa I 3 I ^^^ 003 1 0 0 > k 43, j'g 9 S ti0  006 4 " vo c "45 aos .0 X 43* F44)tsV RI w 0 a)Y~' E 0,81 3 tr a > r,~ I I 0 s .. z c. u C0 2 z *~ > ^ e 1.03 0 < u E .5 c . rs 6  00 L ..  i aE 2 it  E 0 w  ZOz=tjy^" <a *1Sz 2zaaM z ,z " G WC 0 0. e > > Cu 060 " EL 0 0 . 0 0 4 00 ia c ec a C u  '0 0 * PC 3 C u Iliillll ?" ArJ% g". 4., iM rs c 3 54 r; i.4B IS.. witn sino Si i i.0 9 0. 0) >J 'Q .lin s. in fc in 0 inSl in CU 0 .0 .~ in s>...s in >( 0 i 00 tw 4 > a* i 00 eso QCP S > in. in  o P ins S..i..00 U > U,* in& *" in 0.0> 4 3. in w' o u '0 2~ '0 S '0 ?* i n1 .0 a .'0 ' in in f.* 01 0 5.. e . 0. " a 0 0 0a U Sa) 001 UCd s2 ,cs 2 E. LW0) . 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