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 1 FOR AERONAUTICS 
 
 NATIONAL ADVISORY COMMITTEE 
 
 TECHNICAL MEMORANDUM 
 
 NONLINEAR THEORY OF A HOT-WIRE ANEMOMETER 
 
 By R. Betchov 
 
 Translation of "Theorie non-lineaire de l'anemometre a fil chaud." 
 
 Koninklijke Nederlandsche Akademie van Wetenschappen. 
 
 Mededeling No. 61 uit het Laboratorium voor Aero- en 
 
 Hydrodynamica der Technische Hogeschool te Delft. 
 
 Reprinted from Proceedings Vol. LII, No. 3, 1949 
 
 Washington 
 July 1952 
 
<£&o rv* 7 ~ 11 i ;• 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 13^6 
 
 NONLINEAR THEORY OF A HOT-WIRE ANEMOMETER* 
 
 By R. Betchov 
 
 We study here the properties of a hot-wire anemometer under the 
 supposition that the heat transfer from the wire to the air depends, 
 first, on the difference in temperature and, second, on the square of 
 that difference. This latter hypothesis is confirmed by experience, 
 and the consequences might be of great importance, that effect of non- 
 linearity is stronger than the effect of thermal conduction. 
 
 I. THE NONLINEAR LAW OF KING 
 
 The heat quantity Q removed per second by an air stream V from 
 a wire of the diameter d and unit length is given by King in the form 
 
 Q = (a + by/Vd)T (l) 
 
 where T denotes the temperature difference between wire and air. King's 
 calculation, approximately confirmed by experience, yields 
 
 a 
 
 = %' b = \/2nX'b'c' (2) 
 
 with x ' = thermal conductivity of the air, 6' = density of the air, 
 and c' = specific heat of the air for constant volume. 
 
 These quantities may vary with T, and experience shows that a 
 increases while b remains practically constant. Intuitively, one may 
 interpret this effect by saying that the air in contact with the wire 
 is heated which increases its conductivity. In compensation, its density 
 decreases because the pressure varies only very slightly. Obviously, 
 the effects on x' and 6' compensate one another, and only a varies. 
 
 *"Theorie non-lineaire de l'anemometre a fil chaud." Koninklijke 
 Nederlandsche Akademie van Wetenschappen. Mededeling No. 6l uit het 
 Laboratorium voor Aero- en Hydrodynamica der Technische Hoge school te 
 Delft. Reprinted from Proceedings Vol. LII, No. 3, 19^9, pp. 195-207. 
 
NACA TM 13^6 
 
 King states in his original report (ref . 1) that a increases by 
 0.114 percent per degree j he also describes there an effect of the 
 diameter on that term a which we shall not discuss here. Thus it is 
 advisable to write 
 
 Q = {a(l + yl) + b^Vd}T ,(3) 
 
 where 7 takes the nonlinearity into account. 
 
 One must not forget the hypotheses on which King bases his calcu- 
 lation: he contends that the air flow is without viscosity and that 
 the heat flow in the immediate proximity of the wire is constant. He 
 uses the specific heat at constant volume although the pressure is 
 certainly more constant than the density. For that reason, we consider 
 equation (3) as an empirical relation, valid for the wire unit length, 
 and would wish to see King's problem made the subject of a more thorough 
 investigation. 
 
 Here we intend to study the effect of the term 7 on the properties 
 of the hot wire j we simplify the notation by introducing P so that 
 
 P = 7T\/Vd (k) 
 
 Q = a(l + P + 7T)T (5) 
 
 II. GENERAL EQUATION OF THE HOT WIRE 
 
 We shall use the following symbols: 
 
 S resistance of the wire, per unit length, at operating 
 temperature 
 
 S resistance of the wire, per unit length, at ambient temperature 
 
 I intensity of the electric current heating the wire 
 
 a coefficient of the variation of S according to the 
 temperature 
 
 m weight of the wire, per unit length 
 
NACA TM 13^+6 3 
 
 c specific heat of the wire, in joule/gram and degree 
 
 % coefficient of the thermal conductivity of the wire in watt/cm 
 
 and degree 
 
 a wire cross-sectional area 
 
 Z semilength of the wire 
 
 t time 
 
 x coordinate of position, varying from I to -Z 
 
 We put 
 
 A- j^U + P) = 
 
 a + 
 
 b\/vd 
 
 aS r 
 
 (6) 
 
 The equation of the hot wire must express the equilibrium between 
 the heat supplied per second, the heat removed by the air stream, the 
 heat required to raise the temperature of the wire, and the heat trans- 
 mitted by conduction. One obtains 
 
 SI 2 - A(S - So) + -&—{S - So) 2 + 2£_ M . ±1 |!| 
 
 a2S 2 
 
 aS Q St aS Q ox 2 
 
 For the steady-state case, and introducing the parameters 
 
 ■7 ! * ■ fer^y z = ^ ls - So) 
 
 G = TT 
 
 2 a? 
 
 2/a2 
 
 If/A 
 
 2 7 1 
 
 I 2 /A 
 
 3 a 2s (1 - I2/A)2 3 a 1 + P (i _ i2/ A )2 
 
 U 
 
 (8) 
 
 one obtains the equation' (7) in the form 
 
 Z + | GZ 2 - ^| = 1 
 dy^ 
 
 (9) 
 
NACA TM 13^6 
 
 III. EXACT INTEGRATION OF THE STATIC CASE 
 
 Multiplying equation (9) by dz/dy and integrating, one obtains, 
 with a constant 
 
 — = VGZ 3 + Z 2 - 2Z + const (10) 
 
 One notes that oZ,/ay is zero for a negative value of Z and may 
 be zero for two positive values of Z. At the ends of the wire, one 
 has S = S and Z = 0; at the center, Z must be positive and oZ/cVy 
 zero. The range of interest for us lies, therefore, between Z = and 
 
 the first positive root which gives — = which we shall denote by 
 
 oj 
 Z = B. We put 
 
 Z(y) = B - X 2 (y) (11) 
 
 By virtue of the relation 
 
 GB 3 + B 2 - 2B + const = (12) 
 
 one obtains 
 
 MoX/By) 2 = -G* k + EX 2 + D (13) 
 
 with 
 
 E = 1 + 3GB D = 2(1 - B) - 3GB 2 (ik) 
 
 Following, we shall consider B as a new integration constant 
 indicating the temperature in the middle of the wire. At the center of 
 the wire, one has X = and 
 
 McX/oy) 2 = D (15) 
 
NACA TM 13U6 
 
 which shows that D is positive and generally small. At the ends of 
 the wire, Z = and X = ±/b. 
 
 The roots of equation (13) are 
 
 r 2 = -E ± \/e 2 + 1<GD 
 -2G 
 
 (16) 
 
 Introducing the parameter p so that 
 
 sinh p = sh p = 
 
 2\^D 
 E 
 
 (17) 
 
 one can write equation (13) in the form 
 
 MSx/By) : 
 
 = 4- 
 
 (ch p + 1) - X 2 
 
 2G 
 
 (ch p - 1) + X' 
 
 (18) 
 
 We define the angle cp, function of y, so that 
 
 VE ch 
 
 sin cp 
 
 ]/l - k 2 sin2cp 
 
 (19) 
 
 with 
 
 .2 _ ch p + 1 
 2 ch p 
 
 (20) 
 
 Equation (l8) then becomes 
 
 dcp 
 
 \]l - k^sin 2 ^ 
 
 |VE ch p dy 
 
 (21) 
 
6 NACA TM 1 3*4-6 
 
 and we obtain the elliptic integral of the first kind 
 
 U(kjq)) = f^ dC P = | \ZTrfTpy (22) 
 
 J \/l - k 2sin2cp 2 
 
 The variable y varies from -I to I, with 
 
 I = l/l* (23) 
 
 and we have, for y = I and X = B 
 
 t . r&max , 
 
 2 ^0 \/i 
 
 k sin cp 
 
 with 
 
 sin 2 .) EB ch 3 
 
 From equation (20) one may deduce 
 
 k 2 
 
 With 3 ranging from to +<*>, k 2 varies from 1 to 0.5. 
 
 (210 
 
 ^B .J-H. , Sin »~ (25) 
 
 This last equation may be written 
 
 = k 2 + —»— (26) 
 
 — = 1 + tanh2(p/2) (27) 
 
NACA TM 1346 
 
 With G and B known, one may calculate successively D, E, B, 
 k, and 9 max . A table of U(kyp) then permits us to calculate |. 
 
 Figure 1 gives the results obtained by this procedure and allows - 
 starting out from a prescribed wire and with 7 known - to determine 
 B from G and I . 
 
 The temperature distribution over the wire is given by Z as a 
 function of y, or by X as a function of cp and cp a function of y. 
 The relation cp(y) is given by the quotient of equations (22) and' (2k), 
 namely 
 
 y 
 
 I 
 
 f 
 
 dcp 
 
 Jo /l 
 
 - k 2 sin2cp 
 
 p^max 
 
 dcp 
 
 \J± - k2sin2cp 
 
 X 2 sin 2 cp 1 - k 2 sin 2 cp max 
 
 B sin2c Pmax 1 ~ k 2 sin 2 cp 
 The total resistance R of the wire is given by 
 
 (28) 
 
 From equations (19) and (25), one obtains a relation between cp and X, 
 namely 
 
 (29) 
 
 S dx = ~"° I* I Z dy + 2S Q Z (30) 
 
 l-l A - I 2 ^0 
 
 We introduce the cold resistance R Q and the function X 
 
 (31) 
 
8 NACA TM 13^6 
 
 We replace X according to equation (19) and dy according to equa- 
 tion (21), namely 
 
 l2 _/ 2D r 9max . 
 
 - I 2 \ l(E ch p)3/2 J (1 - k2sin^p) 3/£ 
 
 ^(b ■ —^-TT73 f C Sin29 . ,/o t 
 
 The integral is equal to 2du/c)k and we obtain 
 
 / B __2D_Mas!) (33) 
 
 I E ch U / 
 
 H-R = 
 
 Rol 2 / 
 
 A - I2\ 
 
 The values of du/dk can be deduced from a good table of U(k,cp) 
 with sufficient approximation. 
 
 If the wire were perfect, the expression in parentheses in equa- 
 tion (33) would have to be replaced by unity j therefore, we shall intro- 
 duce the quantity M so that 
 
 M = 1 - B + 2D ^ k (3M 
 
 E ch 3 U 
 
 The formula (33) then gives us 
 
 H ■ K ^ZA (35) 
 
 1 - i 2 /a 
 
 and the important relation 
 
 RI 2 
 R - R Q 
 
 = A 
 
 i 1 + r^ (1 - l2/A) } (36) 
 
 This last equation permits easy determination of the wire charac- 
 teristics because R, R Q , and I can be measured accurately and because 
 
 the curves obtained as functions of R/R , for instance, indicate directly 
 
NACA TM 13^6 
 
 the effects of conduction and of nonlineaxity. We calculated the values 
 
 M 
 
 of as a function of G and Ij figure 2 shows our results. 
 
 1 - M 
 
 A good approximation is given by 
 
 M X - 1 + t^-t (37) 
 
 1 - M B n |-1 
 
 where B corresponds to B, in the case I = 
 
 y/l + 6G - 1 
 3G 
 
 Bo = +y " T wu " x (38) 
 
 IV. A FEW USEFUL APPROXIMATIONS 
 
 In performing the calculations necessary for the plotting of fig- 
 ure 1, we have noted that one may assign to k the value unity without 
 introducing large errors. 
 
 This implies P = and equation (19) then gives 
 
 X = Jg tan cp (39) 
 
 The integral (22) becomes 
 
 U = T 9 J^- = i L 1 + sin <P = 1 /Ey (1,0) 
 
 Iq cos cp 2 1 - sin cp 2 
 whence, one deduces 
 
 :h(2U) = 1 + Sin ^ = ch(y/Ey) (kl) 
 
 1 - sin^cp 
 
10 NACA TM 13^6 
 
 At the limits, one has 
 
 which gives 
 
 X 2 = ^(ch \/Ey - 1) (k2) 
 
 B =i(ch/ES - 1) (^3) 
 
 z = § L.s^M) m 
 
 1 - l/ch /El \ ch \/E| y 
 In order to calculate M, one must put 
 
 lim cXJ/c^ - \ P* 5iS?E aq> - V^-i - u(<p)) (*5) 
 
 k — »1 ^^0 cos^ 4 \cos4p / 
 
 which gives 
 
 M = 1 * A _ tWe|\ (u6) 
 
 1 - l/ch /ES\ \/eS / 
 
 One can see that, due to the nonlinearity, I is replaced by vEs, 
 and the central temperature is lowered. 
 
 If one takes equation (hk) as solution of equation (9)> one sees 
 that the equation is satisfied for a term of approximately (ch vEy/ch \/E|) , 
 and that B is given approximately by 
 
 Vi +^6g - i\ (1 _ l/ch m) 
 
 b = r \r ~ (1 - i/ch i/ed (1.7) 
 
NACA TM 13k6 11 
 
 with 
 
 E - \Jl + 6G (1*8) 
 
 We shall take an example that represents an extreme case. We 
 choose a platinum wire with 10 percent of iridium, a diameter of 7 microns, 
 and a length of 2Z = l.lU mm. Exposed to an air stream of 5 m/sec and 
 heated with 75 mA, it gives us 
 
 P = 2.9 A = 1.2 X 10" 2 I 2 /A = 0.U7 
 Z* = 0.15 mm i = 3-75 
 
 Assuming y = 1.2 X 10 ~3 and with the aid of figures 1 and 2, we 
 determined 
 
 G = 0.25 B = 0.76 E = I.56 M = 0.1+ 
 The other parameters have the following calculated values: 
 
 k = 0.99756 cp max = 79.5° 
 sh (3 = 0.11+ D = 0.01*77 U = 2.3' 1 * 
 
 In figure 3> we show the profile of the temperatures calculated 
 exactly, the profile calculated with the approximation k = 1, and the 
 profile calculated with 7=0. It can be seen that the nonlinearity 
 offers a more uniform temperature distribution, and that the approxi- 
 mation is sufficient. 
 
 By means of equation (35) one calculates =— = 1.5* whereas the 
 
 K o 
 
 calculation with y = would give the result 1.9. The mean tempera - 
 
 ture giving — =1-5 would be 38O . 
 R o 
 
 As to the term RI 2 /R - R , it changes from the value 1.24 A when 
 the current is very weak to the value 1.35 A when I attains 75 mA. 
 
12 NACA TM 13^6 
 
 R R 
 
 It varies therefore by about 10 percent between =— = 1 and =— = 1.5 
 
 which is of the same order as the variations observed. 
 
 Thus this magnitude is constant at 10 percent and King's law is 
 verified) however, we shall see later on that the thermal inertia is 
 very different from the expected value. 
 
 V. THE DYNAMIC EQUATION OF THE HOT WIRE 
 
 In order to calculate the variation of the total resistance of the 
 wire when the current or the air stream fluctuate, one must go back to 
 the equation (7). Replacing in this equation I, S, and V (contained 
 
 in the term A) by I + ie-^, S + se^, and V + ve^, one obtains 
 after suppressing the terms of the order zero, two, and more as well as 
 the factor eJ^ 
 
 2BH + I 2 s = _S*_ I(s - So) + As + -^2_(S - S )s + mc 
 
 2aS V a 2 SQ 2 aS Q aS ^2 
 
 J03S 
 
 We introduce 
 
 z = s A " T (50) 
 
 S 1 2 
 
 and, identical to the formula (9) of our Mededeling No. 55 (ref. 5) 
 
 cjd* = — 2(A - I 2 ) (51) 
 
 mc 
 
 One then obtains, after introduction of Z 
 
 o i o i I 2 /A „ 1 P Z v ,_, . / „. S 2 z 
 
 2 t + 2 t p7" z " ? T~Tp pT" v = z ^ x + 3GZ + W"*) - T~9 
 
 1 x 1 - I 2 /A 2 - 1 + p 1 - I 2 /A v oy 2 
 
 (52) 
 
NACA TM 13^6 13 
 
 The function Z(y) appears twice in this differential equation, and we 
 must utilize the approximation k = 1 in order to avoid great compli- 
 cations. Writing the expression Z according to equation (h-k) , one 
 obtains 
 
 #* + A + 3GB / _ _JGB ch ^ \ = c ch^gy 
 
 1 - 1/ch V/E6 ' ch \/E| - 1 J J X 2 ch \/E| 
 
 > c 1 \ ± - x/cn v^^ en v£i? - x y 
 
 with 
 
 (53) 
 
 Ci = 2 
 
 1 
 
 r 2 
 
 C 2 = -2 v 
 
 I 2 /A 13 + 1 v/_P 1 B \ 
 
 1 1 - I 2 /A 1 - l/ch vfl + 2 V\l + P 1 j2/ A 1 - l/ch vfS/' 
 
 (55) 
 
 The solution without the second member of equation (53) is a 
 Mathieu function taken for a purely imaginary value of the argument, 
 but if vE£ is large enough, for instance, more than U, one may neglect 
 the term with ch \/~Ey of the member on the left side of equation (53)- 
 Actually it is important only when y is close to |j however, we shall 
 take as a limiting condition z(|) = 0, and the product z ch vEy will 
 never be important. We shall also neglect the term l/ch \/e| in the 
 denominator of one of the terms on the left side of equation (53) which 
 amounts to taking Z = B in the factor term of z. 
 
 Taking the definition of E into account, one then has 
 
 -^-§ + z(E + JcM) = Cl + C 2 ^^ (56) 
 
 ^y2 ch VE| 
 
 the solution of which - null for y = ±| - is 
 
 _ C l L ch p^E~y \ + C 2 /ch v/Iy ch p\/Ey ^ 
 " Ep2^ ' ch p\AEiy j(D/cD*\ch\AE4 '" ch pjEil 
 
 (57) 
 
Ik 
 
 NACA TM 13^6 
 
 with 
 
 p = vl + jod/Ecd* 
 
 (58) 
 
 It can be seen that due to the nonlinearity the characteristic 
 frequency cd* is replaced by a new frequency which is higher by a fac- 
 tor E. (Compare with equation (8) of ref. 5-) The amplitudes according 
 to Ci and C2 also are modified by the nonlinearity. 
 
 The variation r of the total resistance R will be 
 
 -&l 
 
 r = 
 
 s dx 
 
 R r 
 
 A - I 
 
 2 I 
 
 z dy 
 
 (59) 
 
 The integration gives 
 
 V 
 
 A - I 2 I Ep 2 
 
 \(l - tanh * m ) + % 1 / tanh m 
 
 tanh 
 
 Jaaj} (60) 
 
 We assumed above that /El is sufficiently large; also, we may 
 assign to the hyperbolic tangents the value unity (the presence of a 
 complex argument is here not of importance). 
 
 One then obtains 
 
 r = 
 
 a - r 
 
 e p ^ jo)/"* m\ P / 
 
 (61) 
 
 VI. RESPONSE TO A FLUCTUATION OF THE CURRENT 
 
 If one assumes v = in equations (5*0 and (55) > one may trans- 
 form equation (6l), suppressing the terms containing l/ch V^ 
 
 rl 
 
 2iR Q I 2 /A 
 (1 - I 2 /A) 2 
 
 (1 - I /A(l - B))(l - l/p/ES) BI /A CJD* 7p - 1^ 
 Ep 2 " ^ M P 
 
 (62) 
 
NACA TM 13^6 15 
 
 This formula gives the alternating electromotive force produced 
 at the boundaries of the hot wire by the modulation current i in 
 addition to the normal electromotive force Ri. Although this formula 
 seems to be complicated, it can be adapted to the needs of practice. 
 If the frequency to/2jt tends toward infinity, equation (62) becomes 
 
 ri 2iR I 2 /A (1 . l2/A(l _ B) _ I 2/ AB //1 (63) 
 
 (1 - I 2 /A) 2 *%>* 
 
 If one takes into account that equation (35) gives R, that equation (k6) 
 gives M, and that equation (51) gives or*, one finds 
 
 ri = -^i- iCRI 2 with C = — - (64) (65) 
 
 ai/2it ranc 
 
 The constant C corresponds to that of our former publications, 
 and equation (6h) shows that the electromotive force rl taken at high 
 frequency permits to measure C without being impeded by either con- 
 duction or nonlinearity. The method described in reference k is there- 
 fore indicated rather than the one consisting of measuring the phase 
 displacements, with to being of the order of o#. 
 
 One may immediately verify this point by assuming to in the equa- 
 tion (U9) as very large, thus making the effect of the terms of conduction 
 and of nonlinearity negligible. 
 
 When cu tends toward zero, the electromotive force becomes 
 
 ri - 21R ° l2/A i (l - I! M - 1 fl - I 2 / A (l - 1 b))\ (66) 
 
 (1 _ i2/ A )2 E\ A /bS\ \ 2 J) } 
 
 Thus the complex function ri according to equation (62) will change 
 from equation (6k) into equation (66) when cd varies from to a large 
 value. The complex trace of this function gives practically a semicircle 
 which permits to put approximately 
 
 1 + jaycD** 
 
16 NACA TM 13^6 
 
 where ay** denotes the effective characteristic frequency. In fig- 
 ure k, we plotted the semicircle and indicated a few values of a>/uy**. 
 From equation (62), and in the case of the preceding example, we calcu- 
 lated the electromotive forces rl for a few values of a>/Euy*. One 
 can see that the two functions blend, at low frequencies, if one assumes 
 ay** = l.lEoy* = 1.72ci>*. At high frequency, equations (61+) and (67) will 
 be equal which permits calculation of a satisfactory value of ay* - * when 
 the effect of thermal inertia is important. One obtains 
 
 ay*-* = ^ — (68) 
 
 r 
 
 i/Ves 
 
 Va(] 
 
 1 - I 2 /Afl - \ B 
 
 2 
 
 1 - i 2 /am 
 
 In the case treated one finds ay** = 1.23Ea>*, that is, ay** = 1.92to*: 
 the effective characteristic frequency is almost twice the expected 
 valuej therefore, the approximation (67) gives a correct plot of the 
 function, but the phases according to equation (68) will be only within 
 a 10-percent accuracy. 
 
 The denominator of equation (68) depends chiefly on I, and it 
 increases the characteristic frequency. Instead of compensating each 
 other as in the static case, the two effects reduce the thermal inertia. 
 
 Intuitively, one may say that the conduction shortens the hot part 
 of the wire and thus reduces the heat required for modifying the central 
 temperature j the nonlinearity depends on the presence of hot air around 
 the wire, and the thermal inertia of the air is negligible which improves 
 the spherical response of the anemometer. 
 
 When the wire is dusty, the quantity of immobile air is greater, 
 and the experience shows that the term a in equation (3) is increased 
 while b remains unchanged. One must therefore expect a dynamic action 
 of the dust of the wire to the extent that E is modified. The dynamic 
 effect may be more important than the static effect. 
 
 The wire in the quoted example demonstrates that, with a term 
 BI 2 /B - Bo constant at 10 percent, the characteristic frequency may be 
 almost twice the normally foreseen value. 
 
NACA TM 13^6 
 
 IT 
 
 VII. RESPONSE TO A FLUCTUATION OF THE AIR STREAM 
 
 Assuming i = in the formulas (5^0 and (55) > and maintaining v, 
 one may transform equation (6l) into 
 
 rl 
 
 1 v P 
 
 R i 3 /a 
 
 B 
 
 2 V 1 + P (i _ i2/ A )2 l _ l/ch /El 
 
 1 1 
 
 1 - l/p \/Et _ o# p_ 
 Ep 2 *» 
 
 Vei 
 
 (69) 
 
 With ao tending toward zero, one has 
 
 r 
 
 1 v P 
 rl = 
 
 R n I 3 /A 
 
 2 V 1 + P (1 _ l2/A) 2 1 _ l/ch fa 
 
 1 - ■£ 
 
 3 J_ 
 2 I/El 
 
 (70) 
 
 If a) tends toward infinity, one has 
 
 rl 
 
 1 v P 
 
 R Q l3/A 
 
 2 V 1 + P (1 _ I 2 /A) 2 x _ l/ch ^ 
 
 L 
 
 1 ' 
 
 \/i? 
 J03/03* J 
 
 (71) 
 
 and one may approximately replace equation (69) by the semicircle 
 
 rl = 
 
 rl(o3 = 0) 
 1 + J03/03** 
 
 (72) 
 
 with 
 
 03*-* = E03* 
 
 1 - i/v/es 
 
 1 - § 1//E6 
 
 (73) 
 
 which, in the case of the example treated above, gives 03** = I.8I03*. 
 
18 NACA TM 13^6 
 
 Thus there is, on principle, no equality between the dynamic reac- 
 tion to a variation i and to a variation v. 
 
 This difficulty arises due to the term \/El, namely to the conduction; 
 the nonlinearity tends toward diminishing its importance (factor \/E). 
 
 We hope to publish some empirical results, and the calculation of 
 the differences indicated by different authors, in the near future. 
 
 Translated by Mary L. Mahler 
 National Advisory Committee 
 for Aeronautics 
 
NACA TM 13^6 19 
 
 REFERENCES 
 
 1. King, L. V.: On the Convection of Heat From Small Cylinders in a 
 
 Stream of Fluid: Determination of the Convection Constants of 
 Small Platinum Wires With Applications to Hot-Wire Anemometry. 
 Phil. Trans, of the Royal Society of London. Series A, Vol. 21 U, 
 19U, PP. 373-^32. 
 
 2. Simmons, L. F. G.: Note on Errors Arising in Measurements of Turbu- 
 
 lence. Aeron. Research Committee, Technical Report, R. & M. 1919 
 
 (tote), 1939. 
 
 3. Betchov, R., and Kuyper, E.: Un Amplif icateur pour l'Etude de la 
 
 Turbulence. Med. 50, Proc. Kon. Ned. Akad. v. Wetensch., Amsterdam, 
 50, 19^7, pp. 113^-11^1. 
 
 h. Betchov, R.: L'inertie Thermique des Anemometres a Fils Chauds. 
 Med. 5k, ibid. 51, 191*8, pp. 22*+-233. 
 
 5. Betchov, R.: L'influence de la Conduction Thermique sur les 
 
 Anemometres a Fils Chauds. Med. 55, ibid. 51, 19 1+8, pp. 721-730. 
 
20 
 
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 NACA TM 13^6 
 
 
 
 
 
 
 
 
 
 
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