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 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1313 
 
 ON THE RECORDING OF TURBULENT LONGITUDINAL 
 
 AND TRANSVERSE FLUCTUATIONS 
 
 By H. Reichardt 
 
 Translation of "Uber das Messen turbulenter Langs- und 
 
 Querschwankungen." Zeitschrift fiir angewandte Mathsmatik und 
 
 Mechanik, Band 18, Heft 6, December 1938 
 
 Washington ^Y OF FLORIDA 
 
 P.O. BOX 11 7011 
 GAINESVILLE. FU 3261 > 
 
 1 \i9J, 
 
^M f (^ 16 Off n 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM I313 
 
 ON THE RECORDING OF TURBULENT LONGITUDINAL 
 
 AND TRANSVERSE FLUCTUATIONS* 
 
 By H. Reichardt 
 
 1. ON THE SIGNIFICANCE OF FLUCTUATION MEASUREMENTS 
 
 A thorough understanding of the turbulent flow movements cannot be 
 arrived at from investigations of the temporal mean values of the flows 
 alone. Study of the fluctuation phenomena themselves is indispensable. 
 Thus turbulence research entered a new promising stage when investigators 
 started performing fluctuation measurements and basing theories on those 
 measurements. 
 
 The investigations of fluctuations carried out so far refer almost 
 exclusively to the so-called isotropic turbulence. It represents a 
 damping phenomenon; it is the simplest type of turbulence where merely 
 the longitudinal fluctuations need to be measured. However, it is pre- 
 cisely nonisotropic turbulence which involves the most essential problem 
 of the turbulent exchange movement and the turbulent apparent friction. 
 Investigation of nonisotropic turbulence requires measurement of longi- 
 tudinal and transverse fluctuations. 
 
 2. DIRECTIONAL PROBES FOR FLUCTUATION MEASUREMENTS 
 
 Reliable measurements of quadratic mean values (as are required for 
 fluctuation research) are practically feasible only with hot wires. Any 
 hot-wire directional probe that can be manufactured in sufficiently 
 small dimensions is usuable for simultaneous recording of longitudinal 
 and transverse fluctuations. 
 
 Directional probes consist of at least two hot wires of the same 
 type in an arrangement which is sensitive to angle changes. The differ- 
 ence in voltages over these wires is a measure for the angle deviation 
 and the transverse velocity v' of the flow; the variation of the volt- 
 age sum is a measure for the varia^ion u' of the longitudinal velocity. 
 
 *"Uber das Messen turbulenter Langs- und Querschwankungen. " 
 Zeitschrift fii'r angewandte Mathematik und Mechanik, Band I8, Heft G, 
 December I938, pp. 358-361. 
 
NACA TM 1313 
 
 In the arrangement of Simmons and Bailey (fig. la), the directional 
 sensitivity of the hot wires is based on their being placed in oblique 
 flow. In case of the two parallel hot wires of Burgers (cross section 
 represented in fig. lb), the directional sensitivity depends on the 
 mutual influencing of the temperature and velocity fields of the wires. 
 
 In the three-wire arrangement of the author (fig. Ic), the velocity 
 and temperature gradients in the wake of the front (third) hot wire are 
 made use of. The front wire may serve for the measurement of u* but 
 also for the measurement of temperature fluctuations T'. Thus i t is 
 possible to determine not only the turbulent apparent friction pu'v', 
 but also the turbulent heat transfer pc T'v* directly from the fluctu- 
 ations (p = air density, c = specific heat). 
 
 3. GENERAL RELATIONS 
 
 Quite independently of the manner in which the directional sensi- 
 tivity is attained, the following general considerations apply to direc- 
 tional probes used for the measurement of fluctuations. 
 
 The "effective cooling velocity" -t^ on a hot wire is a function 
 of the angle a (fig. l) and of the velocity u of the undisturbed 
 flow. The w(a) -curves of a hot wire for the separate velocities are 
 symmetrical to the axis a = on which lie the minimum values of w. 
 The maximum values of w are generally identical with the undisturbed 
 velocity u. 
 
 For fluctuation measurements, the probe must be set up in such a 
 manner that the hot wires form fixed angles a, or -a, respectively, 
 with the main flow direction x. Then the flow flows against both 
 hot wires with the same velocity w if the flow vector lies in the 
 x-direction and the velocity of the oncoming flow u is locally 
 constant. 
 
 If the flow varies its direction with respect to the x-axis by 
 the small angle cp and its intensity by Au, and if there exists a 
 
 Auq^ - Au2 
 velocity gradient — in y-direction vertically to x, one 
 
 obtains for the modifications of the cooling velocities Aw, and AWp 
 approximately 
 
 ■""According to definition, w includes also the effects of an 
 altered temperature of the air. 
 
NACA TM 1313 
 
 5w „ . S' 
 
 
 Aw2 = - ^ cp + ^ AU2 (2) 
 
 The voltage E over a hot wire is a function of w. This function 
 f (w) depends in the individual case on the chosen connection. Generally 
 one may state that f decreases with increasing w to the asymptotic 
 value E of the cold wire. 
 
 For small variations. Aw, one may write 
 
 E = f (w) + — Aw (3) 
 
 dw 
 
 or, respectively, for the voltage difference measured in the Wheatstone 
 bridge 
 
 E, - E2 = e^ = y 2 |ii cp + ^ ^ Ay) (k) 
 
 -L ^ ^ dw \ da ou dy / 
 
 (Ay = distance of th3 hot wires). If one further introduces the 
 
 transverse fluctuation v' = cpu and denotes t^ = — , one obtains 
 
 u 
 
 e^ = b(v' + cPqU + c -^ j (5) 
 
 Here cPq signifies a small angle of deviation which may very easily 
 appear in the setting up of the probe 
 
 b = 2 ^ |a (6) 
 
 dw da 
 
NACA TM 1313 
 
 ^ 
 
 Tl 
 
 da da 
 
 The quantity b is a measure for the sensitivity of the v' -measurement 
 while c represents the influence of the velocity gradient on the 
 v' -measurement . 
 
 The quantity b is, in general, dependent on the velocity. In 
 case of the three-wire probe, however, b may be made constant within 
 the range of small velocities since for the small Reynolds numbers of 
 the front wire (of about 0.01-mm diameter) two effects act against one 
 another: While df/dw decreases with increasing w, the dent in the 
 velocity curve behind the front wire gradually deepens whereby hri/ha 
 increases; however, when the velocity is increased far beyond 1 m/s, 
 the dent variation no longer progresses so rapidly, and the damping of 
 df/dw predominates. 
 
 Since the turbulent fluctuations in velocity are not always small, 
 the dependence of b on the velocity makes a correction necessary. One 
 may write as an approximation 
 
 b = b(u) + ^ u' (8) 
 
 du 
 
 where u signifies the mean velocity and u' the longitudinal fluc- 
 tuation. After Introducing the dimensionless number 
 
 b du 
 
 one obtains, neglecting the correction terms of second degree, instead 
 of equation (5) 
 
 e > = V' + cpou + '^ ^^ + c fi (10) 
 
 ' u dy 
 
NACA TM 1313 5 
 
 k. FORMATION OF VARIOUS MEAN VALUES 
 
 In general, one does not measure the bridge voltage itself but a 
 current caused by the bridge voltage. If this current is proportional 
 to the bridge voltage, the current mean values also are proportional to 
 the corresponding voltage mean values. 
 
 For the linear voltage mean, one obtains 
 
 _ /, - u'v' du , , 
 
 ey/h = cpQU + K -z — + c — 11) 
 
 u dy 
 
 This equation is to be used for the experimental determination of c. 
 In order to find the quadratic mean value, the equation 
 
 must be squared. One obtains 
 
 e,/. = V . .00. . c 3^ JH^ - S^ (l.) 
 
 e 
 v 
 
 — / — — 2 
 
 '^b2 = v'2 + 2cPqu'v^ + 2c ^(uMF^) + 2k ^'^' (13) 
 
 )z 
 is taken into account in the c-term of this equation 
 
 The assumption that u' -v;;— vanishes because of lacking correlation 
 
 If the equation 
 
 e^a = u' (1^] 
 
NACA TM 1313 
 
 exists for the longitudinal fluctuat ions u', one obtains according to 
 equation (12) for the mixed product u'v' which is proportional to the 
 turbulent apparent friction 
 
 ^n^\r r\ r^ r\^^ ^ ^ 
 
 'u "^v , , ,0 c du' u' v' /,_N 
 
 = u'v' + CpQU'^ + 3 -T— + K = (15) 
 
 ab <^ dy u 
 
 The terms with k in the equations (13) and (15) are small and 
 vanish, except in case k = 0, even for symmetrical u'v' distribution. 
 
 5. EXAMPLE 
 
 \/v'2 
 
 Figures 2 and 3 show Vv'2 and u'v' measurements with the three- 
 wire probe in a fully developed tunnel flow as functions of the wall 
 distance y. The distances of the hot wires from one another were of 
 the order of magnitudes 0.1 millimeter. The distance of the tunnel walls 
 was 2k-. k centimeters, and the mean maximum velocity was Ujji = 100 centi- 
 meters per second. 
 
 In the present velocity range, for the three-wire probe, b = con- 
 stant and thus k = 0. The K-terms would have been eliminated anyway 
 since the u'v' distribution had been found to be symmetrical. 2 
 
 The dashed curve in figure 2 co nnects the \/v'^ measuring points 
 
 \/v'2 
 
 The solid curve represents the y v'^ values improved with consideration 
 of the c-term. 
 
 The u'v' -curve of figure 3 passing through the zero point corre- 
 sponds to the turbulent apparent friction 
 
 P^^ = -tot - ^ f (16) 
 
 The total shearing stress t^q^ corresponds to the straight line passing 
 
 through the points y = 12.2 and 10^ ^^-^ = 25.8. The position of 
 
 Um'^ 
 _ 
 
 Reichardt, H. : Messungen turbulenter Schwankungen (Measurements 
 of turbulent fluctuations). Naturwissenschaften 26, 1938^ P- ^0^+. 
 
NACA TM 1313 7 
 
 this straight line was determined from the pressure drop. Thus the 
 measuring points obtained with the thr ee-wire probe lie, on the average, 
 with satisfactory accuracy on the u'v'-curve required by the pressure 
 gradient and the velocity gradient. Without consideration of the 
 c-term, the mean measured values would come to lie on the lightly drawn 
 curve underneath. 
 
 Translated -by Mary L. Mahler 
 National Advisory Committee 
 for Aeronautics 
 
8 
 
 MCA TM 1313 
 
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