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'A f 13
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1313 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 socalled 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 hotwire directional probe that can be manufactured iu 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 variation u' of the longitudinal velocity. *"Uber das Messen turbulenter Lgngs und Querschwarnkungen." Zeitschrift fur angewandte Mathematik und Mechanik, Band 18, Heft 6, December 1938, pp. 358361. 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 threewire 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 it is possible to determine not only the turbulent apparent friction pu'v', but also the turbulent heat transfer pc pT'v' directly from the fluctu ations (p = air density, cp = 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" 'w on a hot wire is a function of the angle a (fig. 1) 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 = 0 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 xdirection and the velocity of the oncoming flow u is locally constant. If the flow varies its direction with respect to the xaxis by the small angle rp and its intensity by Lu, and if there exists a Jul Lu2 velocity gradient  in ydirection vertically to x, one btains for the modifications of the cooling velocities Awl and Aw2 approximately According to definition, w includes also the effects of an altered temperature of the air. NACA TM 1313 'wl w ul (1) .'2 = 3 + E Au2 (2) 6a. )U The voltage E over a hot wire is a function of v. 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, w, one may write df E = f(w) + Aw (3) dw or, respectively, for the voltage difference measured in the Wheatstone bridge E E = ev = (2 6 (, + v du*y (4) 1 2 v dw a 6u dy ({y = distance of th3 hot wires). If one further introduces the transverse fluctuation v' = pu and denotes = one obtains u ev = b(v' + p0u + c d (5) Here cp0 signifies a small angle of deviation which may very easily appear in the setting up of the probe b = 2 f (6) dw da 4 NACA TM 1313 U u C a Ay v 7 The quantity b is a measure for the sensitivity of the v'measurement while c represents the influence of the velocity gradient on the vY measurement. The quantity b is, in general, dependent on the velocity. In case of the threewire 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 0.01mm 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 TI/3a 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(a) + db ut (8) du where U signifies the mean velocity and ut the longitudinal fluc tuation. After introducing the dimensionless number K fdb (9) b dii one obtains, neglecting the correction terms of second degree, instead of equation (5) ev/b = v' + pOu + K U + c (10) NACA TM 1313 4. 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 E'v/b = (P0U + K  + 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 I du' utvt =VT ev/b = v' + r0u' + c d (12) '*/ u ~dy \ must be squared. One obtains 2 =v2 + U, ,2 e 'Yb2 = v' + 2cpo0u'v' + 2c d(u'v') + 2K uv (13) av' The assumption that u' vanishes because of lacking correlation is taken into account in the cterm of this equation. If the equation eu/a = u( (14) o NACA TM 1313 exists for tic longitudinal fluctuations u', one obtains according to equation (12) for the mixed product u'v' which is proportional to the turbulent apparent friction eucv' c 2 c du'2 u'2v' (15) u'v' 4 ':POu'2 + + K (15) ab 2 dy u The terms with K in the equations (13) and (15) are small and vanish, except in case r = 0, even for symmetrical u'v' distribution. 5. EXAMPLE Figures 2 and 3 show V/v,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 24.4 centimeters, and the mean maximum velocity was Um = 100 centi meters per second. In the present velocity range, for the threewire probe, b = con stant and thus K = 0. The iyterms would have been eliminated anyway since the u'v' distribution had been found to be symmetrical.2 The dashed curve in figure 2 connects the v'2 measuring points. The solid curve represents the v12 values improved with consideration of the cterm. The u'v'curve of figure 3 passing through the zero point corre sponds to the turbulent apparent friction pu'v' = 7tot t i (16) dy The total shearing stress itot corresponds to the straight line passing through the points y = 12.2 and 104 u = 25.8. The position of Um2 Urn 2Reichardt, H.: Messungen turbulenter Schwankungen (Measurements of turbulent fluctuations). Naturwissenschaften 26, 1938, p. 4o04. NACA TM 1313 7 this straight line was determined from the pressure drop. 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