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REPORT NO. HE-1O-136

STECH1•ICAL REJPORT

UNIVERSITY CALIFORNIASIJINSTITUTE OF EN ERING RESEARCH

BERKELE L LIFORNIA

FREE M0LECULE FLOW FORCE5 AND HEAT TRANSFM FOR ý11 INF'INITE

CIRCULAR CYLINDM AT AN ANGLE OF ATThCi

by

L. Talbot

SERIES N.. ....... .................. .......

IssuE NO 10.. 7.! .........................

DATE ............ M.!aroh. 2.6, .1.9$56. I.............

CONTRACT N7-onr-295-Task 3 JOINTLY SPOIDORED BYPROJECT NR 061-003 OFFICE OF NAVAL RESEARCH ANDREPORT N0. HF-150-136 OFFICE OF SCIENTIFIC RESEARCHSERIES NO. 20-107MARCH 26, 1956 Reproduction in whole or in

part is permitted for anypurpose of the United StatesGovernment

FLUID FLOW ANDHEAT TRANSFER

AT LOW PRESSURESAND TEMPERATURES

FREE IMOLECULE FLOW FORCES AND HEAT TRANFER FOR AN INFINITE

CIRCULAR CILINDER AT AN ANGLE OF ATTACK

by

L. Talbot

FACULTY INVESTIGATORS-.

S. A. Schaaf, Associate Professor of Engineering Science

L. Talbot, Assistant Professor o± Mechanical Engineering

Approved by. _____

ABSTRACT2

Expressions are derived for the recovery factor and heat transfer

and the normal and tangential force characteristics of an infinite circu-

lar cylinder at arbitrary angle of attack in free molecule flow. The

results are expressed in closed form for a cylinder at arbitrary uniform

temperature in terms of three gross molecular surface interaction coef-

ficients, the normal and tengential momentum transfer coefficients and

the thermal accommodation coefficient.

1

Free Molecule Flow Forces and Heat Transfer for an Infinite

Circular Cylinder at an Angle of Attack

The calculations of Stalder, et al. (Refs. I and 2) for the force

and heat transfer charicteristics of an infinite cylinder transverse to a

free molecule flow can be extended to include the case of arbitrary angle

of attack at zero yaw.

I Heat Transfer to a Surface Element

Assuming that the undisturbed flow is in Maxwellian equilibrium, the

distribution function for the incident molecules is given by

.2.T

where u, v, w are molecular velocities in the x, y, z directions of Fig. 1,

U is the gas velocity, p the gas density (p = RT), m the mass of a mole-,

cule, and 6 the local angle of attack of the surface element. The flux

of energy incident on the surface element dA is comprised of translational

molecular energy, dE itrans, and internal energy, dgiint. The first is

given by

tr 4- U,92 7 /RTi

-~ -~O0

2

where s = U/ 2RT is the molecular speed ratio and erf(x) is the error

function A/-•'T • -J + The number of molecules striking dA per

u n it t im e is '0

i'dr 7rSAii 41 3

According to the principle of equipartition of energy, each molecule

carries, on the average, jRT units of internal energy per unit mass, where

j is the equivalent number of fully excited internal degrees of freedom

of the molecule. In terms of the specific heat ratio _) j

Then the flux of internal energy to dA is

5- £3 r (/.7 PTY'[ e- (S ^" -- (4)

We now define a thermal accommodation coefficient a. for the surface inter-

action

(5)

Where dEi dEi,trans + dEi,int r dE is the total energ.y- flux of the mole-

cules re-emitted from the surface; and dE the energ-y flux which would bew<

3

re-emitted if all molecules left in Maxellian equilibrium at the surface

temperature Tw. It is easily shown that

(4 R Ty

The total convective heat transfer to the surface dA is

dQ ~ (d 5,, s (din+ d Fz dew

Then, in terms of the appropriate integrals

- 4 . ) 7 S -'6!Q AZT a~j'-e-)$

- C e-(sA•40)7dA (8)

II Strmsses on a Surface Element

The stress calculations require the introduction of two additional

surface interaction parameters

(9)

A V (10)

where Ci and •r are the incident and re--emitted tangential momentum

flushes, and pi., Pr P, are respectively the normal fluxes of momentum

incident, re-emitted, and that which wouild be re-emitted if all molecules

left in Ramxellian equ1L.brium with the surface. We have then,

_ ~00 ~

5a 4-ý0 e+ St

- V~q [e- r4r (S2 -(12)

(The minus sign indicates that ri acts in the direction of negative y

in Fig. 1)

The net normal and shear stresses on dA are

C A'.Y- C- ýW(13)

and) with

,a (jCS)

(see Refs. 3, 4)) the final, expressions are

V(16)

7.1772 .JL P

The expressions, Eqs. 8 and 17, are identical with those obtained in

Ref. l, 2, except that they are presented in the more convenient form first

used by Schaaf and Chambre (Ref. 3). Eq. 16 differs from the equivalent

result of Ref. 2 in that the additional surface interaction parameter 0

(Ref. h) is included.

ITI Integration of the Stress aid Heat Flux Equations Over the Cylinder

The coordinate system on the cylinder and the appropriate angles are

shomn in Fig. 2. The angle of attack is p . It is assumed that the para-

meters C , Cr', and o/ , and the surface temperature Tw are constant over

the Pntire cylindrical surface.

For the heat transfer per unit length of cylinder, with d the cylinder

diameter, we have 7/

in wihich , .,,, sinfi . The results of the integrations may be convenient-

ly expressed in terms of a Stanton number St and recovery factor r defined

by

V(20)

z 1 7* 1(2o-

v~here C1" is the constant pressure specific hc~at of the gam, T is tLhe•i

adiabatic cylinder temperature (q u) s anf T the aiiabatic s tay na bionv

otemperature of the raso We find tioSit o

6

4 fYVT 5y~rz (21)i- 7,

~. (22)

where a-daT, aro modioied Besse!.. unctions of the

first kind.

The force calculations are most conveniently expressed in terms OL'

the non-nal and tangential coefficients

c - (23)

(-7 =(24)

where N and T are the forces per unit length of cylinder normal to and

parallel to the cylinder axis. We haveIV/L

Al Irz 0" ' 7 VW

)(25)

+ s + 0aoCOS 0

'~ 7-

7

and

j' r _ •f__ / +"(26)

These integrals, when evaluated, give

L

4. F/_ _F_?__ _ T 7_ w-( 2 7 )

+ I(28)

The final results, Eqs. 21, 22, 27, 28, agree with those obtained in

refs. 1 and 2 for = 900 , and for 0, for the two particular cases

of molecule-surface interaction that the authors have considered, namely,

perfectly diffuse reflection (T.- a'= . =:I) and perfectly specular

reflection o= 0I ' -t).

REFERENCES

1. Stalder, J. R., Goodwin, G., and Creager, M. 0., "A Comparison ofTheory and Experiment for High Speed Free Molecule Flow", NAGA TN 2244,1950.

2. Stalder, J. R., and Zurick, V. J., "Theoretical Aerodynamic Character-istics of Bodies in a Free Molecule Flow Field, INACA TN 2h23, 1951.

3. Schaaf, S. A. and dhambre, P. L., " Flow of Rarefied Gases', PrincetonSeries on High Speed Aerodynamics and Jet Propuls.ion, Vol. IV, Part G,(in press).

h. Bell, S., and Schaaf, S. A., "Aerodynamic Forces on a Cylinder for theFree Molecule Flow of a Non-Uniform GaS J. American Rocket Society,v. 23, Sept.-Oct. 1953, pp. 3-14-317.

U

P~ x

z

FIG. I LOCAL COORDINATE SYSTEM

HYD 6792-150-0

U

FIG. 2 COORDINATE SYSTEM ONSURFACE OF CYLINDER

H YD 6793- 150-0

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