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National Advisory Commitiee for Aeronautics NATIONAL ADVISORY COMMITIEE FOR AERONAUTICS TECHNICAL NOTE 2438 HEAT TRANSFER TO BODIES IN A HIGH-SPEED RAREFIED- GAS STREAM By Jackson R . Stalder , Glen Goodwin, and Marcus O. Creager Ames Aeronautical Laboratory Moffett Field , Calif. Teel :1'I 1"Al LI ("I RA V AIRESEARCH W ':C r C~"RlI' G C 9851-9951 S: p,o, ~~A BL'lD. II'.GLE WOOD, CALI FOR r~ IA Was hington August 1951 1 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL NOTE 2438 HEAT TRANSFER TO BODIES IN A HIGH-SPEED RAREFIED - GAS STREAM By Jackson R. Stalder) Glen Goo dwin) and Marcus O. Creager SUMMARY The equilibrium temperature and heat-transfer coefficients for transverse cylinders in a high- speed stream of rarefied gas were measured over a range of Knudsen numb ers (ratio of molecular-mean-free p ath to cylinder diameter) from 0 .025 to 11.8 and for Mach numbers from 2.0 to 3.3. The range of free - stream Reynolds numbers was from 0 .28 to 203 · The models tested were 0 .0010-) 0 .0050- ) 0 . 030-) 0 .051-) 0 .080 -) and 0 .126- inch-diameter cylinders held normal t o t he stream. For the case of high- speed flow of a rarefied gas about cylinders) the data indicate that fully developed free-molecule flow first occurs at Knudsen numbers of approximately 2.0 and the temperature-recovery factor depends primarily upon the Knudsen number . For Knudsen numbers greater than 0.2) the temperature-recovery factor exceeds a value of unity even though free -mol ecule flow may not be fully developed . Over t he range of conditions cover ed in the present tests) the Nusselt number is a function only of t he Reynolds number if the vis­ cosity and thermal conductivity are ba sed upon stagnation temperature and the density based upon free - stream conditions. For free -molecule f low) the heat- transfer data are well cor related by the theory with the a ccommodation coeffi cient equal t o 0 .90. INTRODUCTI ON The sub j ect of heat transfer to or f r om bodies in a high- speed rarefied gas stream has become of importance in aeronautics . The recent p ractical realization of missil e flight at very high altitudes has s timulated interest in the subject . Also) in certain aspects of 2 NACA TN 2438 wind- tunnel testing, notably turbulence studies in supersonic wind tunnels by means of hot-wire anemometers, the effects of slip may not be negligibl e . (See, e . g . , reference 1.) The various phenomena encountered in high- speed flow of gases are often described in terms of the Reynolds number and the Mach number of the flow . The former p arameter may be considered to be a measure of the effect of viscosity and the l atter a measure of the effect of compressibility on the flow field. When considering the f low of rarefied gases, however, an additional parameter, the Knudsen number, becomes important . The Knudsen number is defined as the ratio of the mean- free molecular path to a character­ istic body dimension . This pa:rameter, which is a measure of the degree of gas rarefaction, may a l so be construed as defining the importance of the effect of molecular motion:3 on the flow field . When the Knudsen number is small (less than, sa;f, 0 . 001), the effect of molecular motions on the flow is negligible and, in this regime, the gas can be treated as a continuous medium. On the other hand, for large values of the Knudsen number (of the order 0:[ 10 ), the effect of molecular motions is all - important and the phenomena which occur can be completely described in terms of the motions of ind:Lvidual molecules - in a statistical sense , of course . By virtue of the well- established laws of molecular motions, this so- call ed "free-mol ecule" regime is readily susceptible to analysis and has been thor011ghly examined (references 2 through 8) wi th respect to both aerodynam:~c and thermodynamic phenomena . Experi­ mental work in this regime with respect to external flows (as opposed to flow in tubes) is, however, quite meager. The pioneer work of Epstein in 1924 (reference 3) .ras concerned with the measurement of the drag of spheres; the tests were confined to very low-speed motions. Recently, the drag and equilibrium temperature of a cylinder in high­ speed free -molecule flow were reported in reference 8. The regime of intermediate Knudsen numbers (0.001 < K < 10) is difficult to deal with analytical ly and, at this time, little success has b een attained in attempts to analyze this portion of the field of fluid mechanics . Some progresE, has been made, however, by Chapman and Cowling (reference 9), Schamberg (refer ence 10), and Grad (reference 11). The two general types of a,pp roach to the problem used by the above authors have been described by Uhlenbeck (reference 12). Briefly, the first a:p:proach consists of a:pplying correction terms to the Navier­ Stokes continuum equations and boundary conditions, the first approxi­ mati on resulting in a set of higher- order equations - the Burnett equations . The other approach is to attempt to solve the Boltzmann equation for t he molecular velccity distribution, 'which, in effect, results in obtaining corr ection terms to the Maxwell velocity distri­ bution equation . Both of these methods are essentially perturbation methods . In the first approacb solutions are obtained in terms of NACA TN 2438 3 ~ ower series of the Knudsen number, while in the second method solutions are obtained in terms of the reci~rocal of the Knudsen number. Both methods will fail for Knudsen numbers of t he order of unity. As in the case of free-molecule flows , little experimental work has been done in the intermediate regime f or the case of high-speed external flows . However, Kane (reference 13) has re~orted results of drag tests on s~heres in this regime. The general purpose of the resear ch described in this ~aper was to s tudy the heat-transfer processes that occur between a body and a high­ speed rarefied gas stream. More parti cularly, it was hoped that the experimental data would yield two important results: first, the proper gr ouping of variables t o obtain correlation of heat-transfer data in the analytically intractable "slip-flow" r egime intermediate between the continuum and free -molecule regimesj and second, to define ap~roxi­ mat e values of the parameters which del ineate the three regimes for the p articular configuration used in thes e t ests. Experimental data were obtained on t he heat transfer frcm circular cylinders in the three regimes of free -mol ecule, slip, and continuum f lows. The data are compared at small values of Knudsen number wi th the data of reference 1 and, f or large values of Knudsen number, wi th the free-molecule analysis of r eference 8. NOTATI ON cp specific heat at constant pressure , foot-pounds per slug, Of d cylinder diameter, feet f (s) dimensionless function of s def i ned by f(s) = Zl (S 2 + 3) + Z2 (S2 + ~) g (s) dimensionless function of s def i ned by g(s) = 3(ZI + Z2) g gravitational acceleration, 32 .2 feet per second squared h heat-transfer coefficient, foot-pounds per square foot, second, ~ 10 modified Bessel function of f irst kind and zero order II modified Bessel function of first kind and first order K Knudsen number (~), dimensionl ess k Boltzmann constant, 5.66 X 10-24 foot-pounds per Op per molecule I _J 4 NACA TN 2438 1 mean- free molecular path, feet M Ma ch number ( v: ), dimensionl ess Nu Nusselt number ( h:), dimensionless p static pressure, pounds per square foot P r Prandtl number (Cp ~) , dimensionl ess Q heat transferred from b ody, foot -pounds per square foot, second r temperature- recovery factor, defined by Te = T (1 + ,;1 r M2), dimensionless R gas constant, foot-poilllds per pound, Op absolute Fe Reynolds number ( u:p), dimensionless s molecular speed ratio ( .;r ), dimensionless \ m T free - stream static tern,perature, Op absolute Tc temperature of t he cylinder, Op ab solute Te equilibrium temperature, ~ absolute U stream mass velocity, feet per second va acoustic speed ( J , gRT), f e et per second vm most prob able molecular speed (J 2gRT) , feet per second v mean mol ecular speed (~~ 8 ~T ), feet per second 2 Z l dimensionless function of s defined by Z l = reI o( ~ )exp ( _ ~ 2) Z 2 d imensionl ess function of s defined by a, accommodation coeff icie:J.t , di mensionless , ratio of specific heats , di mensi onl e ss NACA TN 2438 5 thermal conductivity, f oot -pounds per second, square foot, Op per f oot ~ viscosity, pound-seconds p er square foot '\) kinematic viscosity ( ~ ), square feet per second p gas density, slugs per cubic f oot Subs cript 10 r efers to the fact that the viscosity and thermal conductivity were evaluated at stagnation t emperature while the density was evaluated at free - stream conditions ANALYSIS Some General Relations Between Variables Before proceeding with the analysis, it is desirable to set down a few relations which are used throughout. The Knudsen number, K = 7. Id, can be r elated to the Mach and Reynolds numbers as follows: U Va M ( 1) Re VdP ~ From reference 14, page 147, the visc osi~y of a rarefied gas composed of hard elastic spheres and having a Maxwel lian velocity distribution can be expressed in terms of the denSity, mean molecular velocity, and mean ­ free molecul ar path, l-L = o .
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