~~~~~~~~~~~~~~~ 3 ~~~~~ O98

Transcription

1 AD A FOREISII TECHNOLOGY DIV WRIGHT PATTERSON AFO OHIO FIG 20/15 ADVECTIVE FLOW FROM A LIPEAR FEAT SOUCE LOCATED ON A LIQUID StJR ETC(U) Oft 78 0 A YEVDO$(IMOVA, V D ZIMIN UNCLASSIFIED FTD ID(RS1T 18lF5 78 Pt Ufl I I a /

2 FTD ID(RS)T FOREIGN TEC HNOLOGY DIVISION i l ADVECTIVE FLOW FROM A LINEAR HEAT SOURCE LOCATE D ON A LIQUID SURFACE by O.A. Yevdokimova, V.D. Zimin Approved for public release; distribution unlimited. 3 O98

3 FTD ID(RS ) T 1 8)45 78 EDITED TR ANSLATION FTD ID(RS ) T December 1978 MICROFICHE NR: ADVECTIVE FLOW FROM A LINEAR HEAT SOURCE LOCATED ON A LIQUID SURFACE By: O.A. Yevdokimova, V.P. Zimin En glish pa ges: 6 Source: Uchenyye Zapisk i, Permsk iy Gosudarstvennyy Universitet, im A.M. Gorkogo, No. 293, 1972, pp Country of or igin: USSR Trans late d by: Joseph E. Pearson Requester: FTD/TQTA Approved for public irelease ; distrib ution unlimited NTIS DDC U NANNOUNCED JUSTIFICATION Mute Sictic., luff Ssctioi, THIS TRANSLATION IS A RENDITION OF THE ORIGI. NAL FOREIGN TEXT WITHOUT ANY ANALYTICAL OR EDITORIAL COMMENT. STATEMENTS OR THEORIES PREPARED 8Y ADVOCATED OR IMPLIED ARE THOSE OF THE SOURC E AND DO HOT NECESSARILY REFLECT THE POSITION TRANSLATION DIVISION OR OPINION OF THE FOREIGN TECHNOLOGY DI- FOREIGN TECHNOLOGY DIVISION VI SION. WP.AFB. OHIO, ru/*yw ury D! 01st. AVAIL and/v SP(CIM. H FTD ID(RS)T_18L 5_78 Date 6 Dec 19 78

4 t U, S. BOARD ON GEOGRAPHIC NAMES TRANSLITERATION SYSTEM Block Italic Transliteration Block Italic Transliteration Aa A a A, a P p p p R, r 56 B, b Cc C c S, s BB B e V, v IT T m T, t rr r. G,g Yy y y U, u D, d F, f E e E a Ye, ye; E, e* X x A x Kh, kh >14 W c Zh, zh L Q v Ts, ts 3 3, Z, z LI Ch, ch Mu I, i W w ill i Sh, sh i t R zl Y, y Lj U4 111 q Shch, shch H 14 K x K, k b J J1 fl a L, 1 Y,y I H if M, m b b HH H i N, n 33,9, E, e Oo 0 o 0, 0 X) Yu,yu fln 11 ii P,p R i Ya,ya * Initially, after vowels, and after, ; e elsewhere. When written as in Russian, transliterate as ye or ë. RUSSIAN AND ENGLISH TRIGONOMETRIC FUNCTIONS Russian English Russian English Russian English sin sin sh sinh arc sh sinh cos cos ch cosh arc ch cash 1 tg tan th tanh arc th tanh 1 ctg cot cth coth arc cth coth 1 sec see sch sech arc sch sech 1 cosec csc csch csch arc csch csch Russian rot ig English curl l og

5 Perm Order of the Red Banner of Labor State University im. A.. Gor kiy 4 Gidrodinaxnik.a 1972 Advective Flow f rom a Linear Heat Source Located on a Liquid Surface 0. A. Yevdokiinova, V. D. Zitnin Our experimonts on the natural convection from a fine heated wire located, on a horizontal liquid surface, were published earlier (ii. The temperature fields and the rates for different values of output, of heat liberated by a source, were measured!n the work. Similarity transformations, characteristic for self-similar solutions of boundary layer equations were found empirically. This probl n is examined analytically in the present work. The free convection equations for boundary layer approximation, written for the function of flow P and temperature T, in the present case take the following form ) I M 1, 43) ()X V (ix ()y J 43 y 1J g, dt (fli () T c T (4), (1) The origin of the Cartesian coordinate system was selected on the liquid surface, the y axis was directed vertically downward, the x-axis was directed along the surface. The linear heat source was located along the s-axis

6 In accordance with the conditions of the experiments, described in work (1], the boundary conditions when y 0 we will write in the following form When q =oo U= - and = - : y=o, we require the vanishing of temperature T and the rate components. This requirement denotes, that the perturbation, created by the heat source, is concentrated close to the surface and does not penetrate deeply into the liquid. Employing the syninetry of the problem, it is easy to show, that i (x) = ( x), T(x) =T( x). (2) The conditions = - 0 are equivalent to the condition - const, since taking into account (2) the boundary conditions at infinity can be written in the following form T=0 wheny. By integrating the second equation from (1) with respect to y within the limits from 0 to cx and taking the boundary conditions into account, it is possible to obtain the integral condition - - Tdy =3, which expresses the constancy of the heat flow, passing through any vertical section. The magnitude of 2Q is the output of the heat source per unit of length. Assuming (4-) f ( ). T=*(+) 2 t( ). fx\ 2 T I t,:1 ) r 2.

7 we obtain the equations for the dimensionless functions f and t: ( if - 2f ) I + t, with the boundary conditions 1I O, f=f =o, i) t (3) Tj = oe, f=1 o, (4) to which the integral condition of the constancy of heat flow is connected I, I. By integrating the equations of (3) once and by determining from the boundary conditions the integration constant in the heat conduction equation, we find (5) = -- tf. (6) Condition t - 0 when 0 is now a consequence of (6) and should be discarded. For determining c it is necessary to know the law of the decrease in t with large. If t decreases more rapidly than, then c 0. However, when c - 0 there are no solutions for (5) and (6), which satisfy the boundary condition f 0 when, thus, c, 0, and when) Equations (5) and (6) with boundary conditions (4) are invariant relative to the following transformation: --,- 4 (7) 3.

8 By employing (7), it is possible to transform a solution with an arbitrary value I into a solution with I solutions it was assumed that c 1. In connection with this when finding 1, and the integral condition was temporarily *ot taken into account. An approximate solution of the boundary value problem for equation system (5) was obtained by the method of approximating functions. In the interval O<11<q the des ired functions were in the form of polynomials f 1 and t 1, and f and t with 11>m were replaced by asymptotic solutions f 2 and t2, val id with large F, From the structure of these expansions it is evident, that a sufficiently good approximation is already ensured by terms, proportional to fl. When T1 T1t both concepts coalesce. The following expressions were obtained for f 1 and t 1 : = - -z 2 ( 1 -z) z 2(5 _ 9z2+5z ) z= - t (1 3z ± 271) 2 ++(4z _- 3z1); I The form parameters t o and ii were de termined from the cond ition of equal ity to zero of the average values of the discrepancies, which arise with the substitution of approximate expressiosn for f and t in equation (5). (f ) d 11 I (1 ) ± S (- -- i) 0 After carrying out the integration a system of algebraic equations is obtained for the values and t. The solution of this system was found numerical ly for P, 7. (The Prandtl number for distilled water at a temperature of 4.

9 20C is equal to 7.06). The values 2.01, t o 2.04 were found for the values and t o. Now by calculating the value of I and carrying out the transformation of (7), we obtain expressions for functions f and t, which already do not contain any indeterminate constants. For comparison with the experimental results [1] we write the expres8ions (1/ for the horizontal rate component u and the vertical temperature gradient -. ii -= S ) Tfl (11) L IxI I ( (, ( _) T jxr 4 y. 0 I Fig. 1. Dimensio nless profile of the vertical temperature gradient. The broken line represents the experimental results of work [1], the solid line represents the calculations. A comparison with [1] reveals complete coincidence of the functional depend en ces of the values u, - i and on x and Q. The universal profiles t and f are given in Fig. 1 and 2 (the solid line represents the calculations, the broken line represents the average of the experimental results (1]). 5.

10 . - - I I I 0,2 j _ T Fig. 2. Dimensionless profile of the horizontal rate component.the broken line is the experiment, the solid line is the calculation. As is evident from the figures, the shapes of the curves t (ii) and V (itz) are found in satisfactory agreement with the experiment, the maximum values of the magnitudes f and t are considerably higher than the experimental values. The quantitative differences are connected, apparently, with the limitedness of the volume of liquid along the direction of motion. Experiments with a broader vessel, than in work [1), were carried out to clarify the effect of the horizontal dimension. The experiments gave a more accurate coincidence of the law of decrease with theoretical (iy ( I) y. (with large y), however, in other respects the increase in the horizontal dim ension did not lead to a more accurate realization of the conditions, convenient f or analytical investigation. Bib ].io2ranhv A. Yevdokimova, V.D. 11m m, Bulletin of the Academy of Sciences of the USSR, MZhG, 1970,6. 6.

11 - DISTRIBUTION LIST DISTRIBUTION DIRECT TO RECIPIENT ORGANIZATION MICROFICHE ORGANIZATION MICROFICHE A205 DMATC 1 E053 AF/ I NAKA 1 A210 DMAAC 2 E017 AF/RDXTR-W 1 P344 DIA /RDS-3C 9 E403 AFSC/INA 1 C043 USAMIIA 1 E404 AEDC 1 C509 BALLISTIC RES LABS 1 E4 08 AFWL 1 C510 AIR MOBILITY R&D 1 E410 ADTC 1 LAB/FlO C5l3 PICATINNY ARSENAL 1 FTD C535 AVIATION SYS COMD 1 CCN 1 C591 FSTC 5 ASD/FTD/NIIS 3 C619 MIA REDSTONE 1 NIA/PHS 1 D008 NISC 1 NIIS USAICE (USAREUR) 1 P005 DOE 1 P050 CIA/CRB/.ADD/SD 1 NAVORDSTA (50L) 1 NASA/KSI 1 AFIT/LD 1 I LL/Code r I FTD- 1D(RS)T (H