On the recording of turbulent longitudinal and transverse fluctuations

'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 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 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. 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 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
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 rp and its intensity by Lu, and if there exists a
Jul Lu2
velocity gradient --- in y-direction 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 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 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 c-term 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 three-wire probe, b = con-
stant and thus K = 0. The iy-terms 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 c-term.

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. Thus the
measuring points obtained with the three-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






NACA TM 1313






















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