- Email: [email protected]
The resolution of turbulent pressures at the wall of a boundary layer
J. Sound V& (1967) 6
(I),
59-70
THE RESOLUTION WALL
OF TURBULENT
PRESSURES
OF A BOUNDARY
AT THE
LAYER
G. M. CORCOS College of Engineering, University
of California, Berkeley, California,
U.S.A.
(Received 19 September 1966) This paper is a discussion of the resolution by a piezo-electric transducer of the local pressure fluctuations at the wall of a turbulent boundary layer. The first part applies to transducers whose sensing area is large in comparison with the boundary layer thickness, and the second part to transducers whose sensing area is either of the order of, or smaller than, the boundary layer thickness. For large transducers, the main result, derived for a sensing surface of arbitrary shape and with an arbitrary spatial sensitivity distribution, is that under circumstances which are perhaps unrealistically stringent, the attenuation of a given frequency component will increase as the cube of the transducer face linear dimension. In general, it should be expected to increase only as the square of the dimension. In the second part, the probable dependence of the resolution on frequency and on transducer size is indicated by using both recent experimental evidence and physical reasoning bearing on the nature of the pressure cross-spectral density. Both approaches suggest that in a co-ordinate system translated downstream at an appropriate velocity, as the streamwise wave number increases, the spectral density in wave number-frequency space is associated with proportionately increasing spanwise wave numbers and almost proportionately increasing frequencies. Experimental evidence suggests departure from this finding if the fixed axis longitudinal wave length is very large (of the order of the boundary layer thickness) in which case both the characteristic time in convected axes and the characteristic lateral coherence length appear to reach asymptotic finite values. For shorter waves, the strong relationship between the two space scales and the convective time scale necessarily implies that the resolution of the frequency spectral density by a finite transducer continues to deterioriate as the frequency increases. It is concluded that the attenuation predictions previously published by the author are not likely to be seriously in error. I.
INTRODUCTION
When a pressure transducer of non-zero sensing area is mounted as part of the solid boundary surface of a turbulent boundary layer it responds to some area integral of the instantaneous pressure found on the boundary. The manner in which the output signal thus fails to measure pressure at a point has been discussed by Corcos [I] (henceforth referred to as [I]), Gardner [2], White [3] and Willmarth and Roos [4]. In what follows we first consider again the case of large transducers by removing assumptions which are not strictly necessary and by focusing on a theoretical constraint which is provided by the equation of motion and which was overlooked in [I]. The notation is that in [I]. For two identical linear transducers with a perfect time response, mounted as part of (but uncoupled from) a solid surface along which the turbulent pressure field is statistically stationary and homogeneous, it has been shown (see, e.g., [I]) that the measured crossspectral density is given in terms of the actual cross-spectral density by
60
G. M. CORCOS
where c is the distance vector between the two transducer Thus, the measured frequency spectral density is Q&J)
= rm(o, W) = ,I
r(e,
positions,
and w is the frequency.
CO)O(r) dA(s),
(a)
where O(E) = j- K(s)K(s+ m
E) dA(s).
Here K(s) is the unit response of the transducer, i.e. the contribution to the transducer output signal due to a unit pressure at point s on its face; i.e. the transducer located at x measures a pressure p, given by P,(x,
t) = j” P(S, 4 K(s - 4 dA(s).
Hence O(E) is a function wholly defined by the transducer shape and characteristics. E is the two-dimensional vector (E%,E~)defining the co-ordinates on the transducer face, c = (4,~) is the separation between two identical transducers used for the measurements, w is the frequency and i- 00 r&CO)
= ,‘,
W,
I --co
7) exp [ - i(+]
d7,
where R(<,T) is the space-time covariance or correlation of the pressure at two points separated by the distance vector < and at two times separated by the delay 7. 2. ASYMPTOTIC DEPENDENCE SPECTRAL DENSITY
OF THE MEASURED FREQUENCY ON TRANSDUCER SIZE
The behavior of transducers of large sensing face dimensions is of interest in connection with sonar design. It was briefly considered in [I]. For transducers of given shape and characteristics, with a typical face dimension L, of L, is a complex we have O= @(e/L), whereas r(<, w), which is evidently independent function whose modulus at a given w is a maximum at < = o and which decreases to zero for any w. as \
‘1
l
c3 /
,--4
&
i
I.
\--
?
‘L_+’
Figure
1’
g
The area of integration.
We consider the open set ~2 of points in the plane whose boundary is a perimeter V described by the locus of the end of the distance vector between the origin of two identical transducers of identical orientation if one is held fixed and the other is translated to assume all possible contacts with the first (i.e. if the area of the transducer defines the set of points a, then the set &‘=g--a). The characteristic length for the perimeter $5’is
TURBULENT
PRESSURES
ON A BOUNDARY
61
LAYER
called L. We assume that 0 is a continuous function of (e/L), non-stationary at the origin, with continuous first derivatives on & and identically zero on and outside V. We also assume that the transducer is sensitive to pressure rather than to force, so that 0 is of the form 0 = if(E/L), where A is the sensing surface area of order L2 andf(s/L) Then we expand 0 in a partial series
wheref,=--*f
af
ax’ y
=--
af
ay
ando
is a non-dimensional
function.
Substitutingin(
The two integrals on the right-hand side can each be written as the difference between an integral over an infinite plane and an integral over the part of that plane which excludes the domain &. However, the second integral vanishes in the second integration because for points outside the contour %7all derivatives off vanish. We thus have
Now we assume that for ]$ and ]c~~$Q, I~(E~,E~,w)/=O{I/(E,J~~~~~~~}, where m and n > 2. As L + m, the average values of ccand j3 in the third integral tend to zero and
(3) where KI = j-1 F(E, w) icxfx(o,
0)
=yfy(o,
0))
de,
dcy
a function of w only. The first integral on the right-hand side of (3), which is the coefficient of (r/A) - ( 1/L2), can be shown [s] to vanish exactly along a plane and rigid surface for a turbulent pressure field which obeys a strictly incompressible Navier-Stokes equation, while in general the second integral, ICI, the coefficient of (I/AL) - ( 1/L3) does not. Thus it might appear from the foregoing result that the leading term of the asymptotic attenuation is proportional to 1/L3.i But in a laboratory experiment or, say, on the hull of a ship, several factors will inevitably conspire to prevent the first integral from vanishing, in general. They are, is
-f This
TN-65-2).
result was also quoted by D. M. Chase, Technical
Research Group Technical
Note (TRG-01
l-
62
G. M. CORCOS
for instance, the radiation field caused either by the finite Mach number characterizing the boundary layer or by vibrations of the boundary induced by the turbulence, or the slight departure from statistical (planar) homogeneity which is inherent in the pressure field of a boundary layer. For none of these pressure perturbations does the area integral of the space correlation vanish. Under nearly ideal conditions+, the coefficient of the first or l/_Cz term may well be smaller than I/L times that of the next, up to fairly large values of L. In such cases the asymptote of @,(w,L) as L increases may appear to behave as 1/L3 first and then as 1/L2. It is thus apparent that the values given in equation (22) of [I] for the coefficients of 1/L2 (square transducer) and 1/L2 (circular transducer) have no general significance and should be ignored. They were obtained from an integration of experimental values of r(<,q,w) at a given frequency and are characteristic only of those experimental conditions. It should be noticed that the foregoing result depends neither on a convection hypothesis nor on similarity properties. It assumes onIy that the unit response of the transducer (the contribution to the signal due to unit pressure at any point of the transducer face), which may be an arbitrary spatial function, is linearly dependent upon the pressure.
3. THE CASE OF SMALLER
TRANSDUCERS
When one attempts to measure the frequency spectral density of turbulent pressure fluctuations instead of suppressing it, smaller transducers are used. It is, nevertheless, difficult to avoid some attenuation. We have seen that the measured frequency spectral density is given by
This expression, by itself, provides a means of computing the attenuation, @,/@, only for transducers whose characteristic dimensions are large enough : in the integral above, 0 is non-zero only for separation distances (~1which are smaller than the greatest dimension of the transducer. For instance for a circular face, 0 = o for E = 1st 3, 21 where T is the transducer radius and 0 is a monotonically decreasing function of E for E < 27. Thus, the required values of @(a, w) are those for which o < E < 2~. On the other hand, cross-spectral density measurements are always necessarily made for a non-zero separation distance (the minimum distance being the sum of the radii for two circular transducers). If we take the minimum value of E for which r/0 must be known empirically as E = L/4 [i.e. if we assume that interpolation errors are tolerable for o < E 6 L/4, in view of the fact that r/Q (0, w) = I], wh ere L is the maximum dimension of the transducer whose attenuation is queried, we see that equation (2) can be used directly only provided .P/@ has previously been measured(i) over a range E 2 L/4; (ii) with two transducers whose dimension Lo was small enough so that
This does not necessarily require Lo << L, since r/Q, is a function of non-dimensional variables in which scaling factors are flow parameters. Hence, the same requirement may be met by increasing the flow dimensi0ns.f As we shall see, the presently available data can be applied directly to the computation of the frequency spectral density by transt These might be visualized as the flow over the spillway of a dam, for instance. 1 This assumes either that the Reynolds’ number is kept the same or that its influence on I’/@is negligible.
TURBULENT
PRESSURES
ON A BOUNDARY
LAYER
63
L*> o~z8” and lateral dimension L,> 1.46~ ducers with streamwise total dimension according to the requirements set forth above. Here 6” is the boundary layer displacement thickness. An alternative, and on the surface a more direct course, consists in computing the Fourier transform E(k, co) with respect to E of r from the ratio of Em (the Fourier transform of r,,,) and S(k) the Fourier transform of O(E) (Corcos, Cuthbert and von Winkle [6]). But the resulting expression for @ is an integral involving a singular kernel (Willmarth and Roos [4]), which requires that Em be known for the transducer used and this with impossible accuracy in the neighborhood of the singularities. For transducers whose radius r/S* is small enough so that the two conditions set above are not satisfied, the most hopeful approach seems to attempt to infer, from general physical reasoning and from undistorted measurements of this quantity at larger values of c, how r(Z;, w) depends on <. 3. I.
SIMILARITY
In [I] it was suggested
that r’ could be approximated
in a convenient
manner
as
A and B as well as the dependence of Uc on w were inferred from the The functions experiment of Willmarth and Wooldridge [7]. The similarity form (4) for r, if it is suitable, obviates the need to measure cross-spectral density with transducers non-dimensionally smaller than the one for which a resolution correction is desired and in fact allows the substitution of measured for actual spectral-density in the integrand: according to equation (4) equation (2) is of the form
As we have seen, in equation (s), the range of integration of E is that for which O(E) f o, i.e. it is of order L. But if C is a function of the similarity variable WE/ Uc only, the required values of C may be determined for larger values of E and correspondingly lower values of W. Hence C may be obtained from measurements of C,, for which the distortion by the finite transducer size is small, i.e. for which r,,, z r. UJU, is assumed to be a function rather than of &,/Urn. This introduces an uncertainty in the comof w, i.e. of &P/U,, putations in that the dependence of UJU, upon &P/U, is not known for sufficiently large values of the frequency. But this uncertainty is numerically limited because it is found experimentally that the convection speed L’, varies little with frequency for large values of w8*ci C; m. 3.2.
VlLIDITY
OF THE SIMILARITY
ASSUMPTION
One may write in all generality r(t,?,w) where
= ~(W,~0,p,U,,U7,V)C(5,?1,~,CTCo,21~,~,~M)exp
C is a real function,
U,=
-w.$(tan-rri/r~)-‘,
ri and r,
[
-i
, 2 c c11
are the imaginary
and
real parts of r, U, = the free stream velocity, u, = the friction velocity = .\/i-,/p, 7,‘= wall shear stress, p = the density, and v is the kinematic viscosity. r has been cast as the product of the power-spectral density Q(w) and another function of which the argument corresponds to convection at a speed UCand the modulus describes the coherence in space and time in a frame of reference moving downstream with the velocity L;.
G. M. CORCOS
64
We note that for an incompressible two-dimensional boundary layer on a smooth wall, the ratio Uoolu, depends solely upon the local Reynolds’ number Rg*=S*Um~v,sothat we may write
(6) where U,/ U, depends on the same variables as C. Two questions may be asked concerning equation (6). (a) H ow does r/@ depend on Rp, i.e. how similar are the normalized cross-spectral densities for two flows at different Reynolds’ numbers? We still possess little definitive information on this point. We shall see later that the data of Priestley [S], the only cross-spectral density data gathered at Reynolds’ numbers much larger than the usual laboratory ones, suggest that Reynolds’ number effects are moderate, i.e. of the order of the effect of Rp upon u,lU,.We shall not pursue this point, because experimental difficulties may also account for these differences and we shall assume as is often done in shear flow turbulence (e.g. Townsend [12]) that the features of the flow, including the function r are not affected by the flow Reynolds’ number unless their time and length scales are of order v/u: and V/U?respectively, or smaller. (b) Are the non-dimensionalizing lengths Lg. and L, and the non-dimensionalizing frequency w. external dimensions, i.e. flow parameters such as lengths 6” or v/U,and frequency Urn/P,or internal or “local” variables such as length UC/w,and times f/U,or, say, G-‘12Kr3/2 [where G(k,) is the ordinary one-dimensional wave number spectrum of a velocity component within the boundary layer] ? The validity of the dependence of C upon two similarity variables as assumed in [I] hinges on the answer, It should be pointed out that similarity variables, which necessarily appear if external parameters are found irrelevant, need not take the form assumed in [I]. For instance, if the characteristic time of evolution, or memory time in a convected frame, for wave number component ki is kr3j2 G-‘12(Ki), the dependences of C in the streamwise direction would be expressed approximately as c(&o,w)
= C($!;jG112).
But, physically and numerically, the distinction between internal and external scaling appears to be more important than the choice of the proper internal scale. We shall see that of the two available external length scales, 6” is not suitable except for very long wave lengths, so that for the bulk of the wave lengths for which approximately
C must be a function of similarity or internal variables. 3.3.
EXPERIMENTAL
EVIDENCE
The experimental evidence offered in support of the similarity form (4) in [I] was rather incomplete and, as has already been pointed out, it applied to r,, not necessarily to r.t Two sets of measurements of the pressure cross-spectral density have been published recently, one by Bull [IO] and the other by Priestley [8]. It is possible (Corcos [I I]) to determine that for the frequency range investigated, the resolution errors were either small (Bull) or completely negligible (Priestley). t The suggestion in [I], p. 197, that I‘,/@,=
r/Q
ISerroneous in general (Corcos [ll]).
TURBULENT
PRESSURES
ON A BOUNDARY
LAYER
65
According to Bull, for C&*/U, >o.s, the normalized cross-spectral density is well scaled by the similarity variables C&/U, and wv/Ijo for a frequency ratio of about ten distances in this experiment were 0.82
3
Figure 2. The magnitude of the cross-spectral density according to Priestley. cross-stream. Boundary layer turbulent but stably stratified. (-) Bull [IO] ; (--) and Wooldridge [7]; ~/6”valuesasfollows: (0)0.36; (0) 0.54; (A) 0.90; (c) 1.67; (v) 0.48 < wS*/lJ < 60 for all points.
Separation
is
Willmarth
1.98; (+) 2.j2.
The cross-spectral density was measured over a range of separation distances (0.35 < q/S* < 2.5 ; 0.05 < t/S* < 2.5) which includes the smallest separations achieved so far. Yet the separation was always very large compared with the transducers’ effective radius (t/r > IOO; r/r> 600) so that the resolution of the measured wave lengths was unimpaired. The frequency range was approximately+ 0.16 < (wS”/U) < 60, where U is the wind velocity measured at some arbitrarily fixed point, and a good approximation to the convection velocity UC at high frequencies. According to Priestley’s data the lateral cross-spectral density is given within experimental scatter as a function of wq/UC over the separation range. This is shown on Figure 2 where the inclusion of several runs, in which the separation distances and frequencies were the same, gives an idea of the effect of various wind and weather conditions and/or random errors. The data of Bull and Willmarth are summarized on the same plot by two curves. These curves are faired mostly through data for larger values of q/S* and correspondingly lower values of US*/ UC than Priestley ‘s points. Priestley’s longitudinal cross-spectral density measurements were taken over a wide range of atmospheric conditions, in particular, when the density stratification near the ground was either statically stable (no solar radiation) or unstable (considerable solar radiation and hence heating by the ground). As a result the turbulence levels and the mean profiles must have varied a great deal. When the solar radiations were intense, the 1_The
value of 6” in Priestley’s experiment is not known accurately.
66
G. M. CORCOS
pressure intensity at the ground was high and the convection velocity was almost independent of frequency, indicating vigorous mixing and a very flat velocity profile. When there were no solar radiations, the pressure intensity was lower and the convection velocity was a fairly strongly decreasing function of frequency. Surprisingly, the characteristic time scale of decay for the function A was very insensitive to these changing conditions. The data are shown on Figure 3 in terms of the similarity variable &/UC. One notes that the value of A in these experiments falls more rapidly with WZJ/U,than in
Figure 3. The magnitude of the cross-spectral density according to Priestley. Separation is nominally streamwise. Standard deviation of wind direction z 0.2 radians. For E/S+ < 0.36, boundary layer unstably stratified. For all g/8 * > 0.36 points are shown for both stably and unstably ) Bull [I o] ; (--) Willmarth and Wooldridge [7]. t/S* values as follows : stratified conditions. ((0) 0.055; (x) 0.083; (@) 0.14; (A) 0.22; (0) 0.36; (0) 0.54; (v) 0.90; (A) 1.2; (+) 2.52. For points 26.4; for those given by (e), (O), (v), (A) and (+), given by (o), (y), (8) and (A), 0.7 < w@/U< 0.5 < w@/U<60.
either Bull’s or Willmarth and Wooldridge’s case. This may be a Reynolds’ number effect. But a more likely explanation is that in an atmospheric boundary layer which has no governing characteristic dimension, spanwise velocity fluctuations can be expected to have considerable energy in the very low frequency range, or equivalently that in the course of a run the wind direction changes slowly all the time, so that the alignment of the two transducers is only nominally in the downstream direction. It typically must involve an appreciable slowly fluctuating lateral separation as well. The standard deviation of the wind direction was given by Priestley as approximately 0.2 radians in most cases. According rto Priestley, A can be approximated empirically by a function of (&/ UC)(wL/ uC)“.2B,where the (constant) length I, was not specified. This representation and the data of Figure 3 imply that A has somewhat smaller values at a given value of wf/U, for large values of w and small values of 5 than for the opposite case. In other words, not only does the coherence time e/U, not scale with an external (fixed) time such as Se/U but it decreases more than linearly with I / w. This trend is more noticeable for highly unstable conditions than for stable ones. Priestley’s cross-spectral measurements at small separations do not bear out Willmarth and Roos’ [4] expectation that marked departures from similarity occur for spatial separations of the order of or smaller than o-76* and frequencies w greater than 3 UC/S*. The departure from the kind of similarity assumed in equation (4) which does seem to be present in Priestley’s longitudinal data is easily incorporated in the evaluation of transducer resolution by the
TURBULENT
PRESSURES ON A BOUNDARY
LAYER
67
method of [I]. For high frequencies it yields attenuations which are somewhat larger than those computed in [I]. This is just the opposite of Willmarth and ROOS’conclusion. Priestley, like Bull, found that the method of scaling separation distances, according to which the coherence lengths are roughly proportional to the period of the cross-spectral density, does not apply for large eddies (those for which a typical longitudinal wave length is of the order of the boundary layer thickness or more). Both Bull and Priestley made a limited number of cross-spectral density measurements for which both [ and 7 were non-zero and found that the simplifying assumption
C(w, E,7) = 4%
E,0)B(w o,rl)
made in [I] is acceptably accurate.
3.4.
THE SENSITIVITY
OF THE ATTENUATION
TO THE VALUES OF THE CROSS-SPECTRAL
DENSITY
It can be verified that the substitution of either Bull’s or Priestley’s values for the crossspectral density into equation (2) yields transducer attenuation values which differ relatively little from those published in [I] for most transducer sizes, especially for nondimensional small ones. The discrepancy does not exceed zo%, and only for larger transducers at high frequencies. This should be expected since the data of Willmarth and Wooldridge and those of Bull and of Priestley do not differ a great deal. The attenuation of the frequency spectrum is due to three effects :
4)I
(i) the convection of the pressure field which translates longitudinal wave numbers into frequencies
and is expressed in equation (4) by the term exp - i ; [ ! UC (ii) the finite lateral coherence of a component of frequency w (i.e. of wave number k, ~oJ/UJ and; (iii) the fact th at the pressure field is not frozen, so that more than one longitudinal wave number really contribute pressure energy at a given frequency. The general character of the attenuation at a given frequency by a finite transducer which fails to resolve longitudinal and lateral wave lengths of the order of, or smaller than, the transducer diameter depends on : (a) whether the typical wave length in the lateral direction is tied to the frequency chosen (i.e. approximately to the longitudinal wave length) or independent of the frequency ; and (b) whether the typical convective coherence time is also tied to longitudinal wave length, or independent of it. Similarity variables for r (either those of Priestley or those of [I]) imply that both are so tied. Hence the attenuation is compounded and will increase rapidly with frequency, although precisely how rapidly depends somewhat on the type of similarity variable used.
3. j.
MODELS OF THE CROSS-SPECTRAL
DENSITY
Several authors have computed the function I’by making special assumptions. White [3] integrated the Poisson equation for the pressure by assuming a simple model for the velocity field. Landhal [ 131 computed the longitudinal cross-spectral density from a quasi-linear solution of the hydrodynamic equations for a disturbance in a turbulent boundary layer. Bradshaw [14] used broad dimensional argument based on the generally accepted idea that the inner part of a turbulent boundary layer is characterized primarily by one length scale, the distance y from the wall, and one velocity scale, the friction velocity U, = (TJ~J) 1/2. All these approaches which differ very substantially in their underlying assumptions but which make at least partial use of the equations of motion yield
68
G.
M. CORCOS
results which are reasonably similar. The cross-spectral density is found to be a function of similarity variables either precisely (Bradshaw) or approximately (Landhal, White). On the other hand, Chandirami [I 51 has postulated several models of the cross-spectral density which are functionally very different and which yield very different results when used to compute transducer attenuation. The difference introduced by the models is simply that the time and the length scales or the frequency and wave number scales are tied to external parameters of the flow and not to each other. If, as Chandirami assumes, the dependence of the cross-spectral density upon lateral separation is little affected by the selected frequency, the attenuation by the transducer of lateral wave lengths is almost the same at all frequencies. Predictably, this type of model tends to yield transducer attenuations which become asymptotically independent of frequency and which for high frequencies are much smaller than those inferred in [I]. But Chandirami offers no empirical or theoretical justification for his models and they are hard to reconcile with the experimental evidence which we have cited. 3.6. VERY SMALL TRANSDUCERS AT VERY HIGH
FREQUENCY
Transducers whose dimensions are a small fraction of the displacement thickness can be expected to attenuate the power spectral density at high enough frequencies. Consequently we may wonder whether there are any lower bounds to the separation distances for which the similarity variables are applicable. As we have seen, the data of Priestley and Bull imply that for components whose longitudinal wave length is smaller than the boundary layer thickness, 6, the two coherence length scales of the cross-spectral density become independent of 6 or 6”. It seems very unlikely that for much smaller separation distances, a dependence on 6” would reappear. The only other external length, V/U,, should be expected to become relevant for wave lengths of the order of the sub-layer thickness. In other words, the physical separation of the wall from the turbulent velocity fluctuations should confer upon the wall pressures a minimum length scale of the order of the sub-layer thickness. But, except at low Reynolds’ numbers, this limiting value of k, 6” should be very high, at least of the order of 50 to IOO in Willmarth and Wooldridge’s experiment, for instance, and correspondingly higher at higher Reynolds’ numbers. Thus for intermediate values of ki 6” or (wS*/U,) one should expect the function C(<, w) to be governed by internal length scales. The lateral length scale for cross-spectral density is likely to be r/k, as supposed by Bradshaw [13], and computed by White [3] because the lateral and the longitudinal scales of the covariance of the pressure at the wall are closely related by the Poisson equation for the pressure. Thus the scaling C(o, 71,w) = C(o,qw/ U,), while not inevitable, is highly probable. For the longitudinal cross-spectral density, the apparent length scale is U,t, where t is the characteristic time scale in a convected frame of reference. Such a time scale cannot be obtained uniquely from dimensional reasoning without additional assumptions. It can be given either by the differential convection of eddies spread over a height of order 1/k, or by the intrinsic excursions of a convected eddy with time scale of order ky3/2G-1/2 [where G(k,) is a one-dimensional velocity spectrum] or by fluctuations in the convection velocity which would involve the energy of eddies of different sizes. All these mechanisms no doubt contribute? to the convective lack of coherence of the pressure and they yield characteristic lengths which differ somewhat in their dependence on W. But they all couple the convective length scale at a given frequency to a positive power of the frequency which is not very different from unity for all of the mechanisms investigated. t Priestley’s data suggest that more than one of these effects must control A. Otherwise the behavior of A would be very different when differential convection is almost absent and the turbulence level is high and when differential convection is strongly present and the turbulence level is low.
TURBULENT
3.7.
DIRECT
MEASUREMENTS
PRESSURES
OF TRANSDUCER
ON A BOUNDARY
LAYER
69
ATTENUATION
The spectral density measurements of Bull [IO] were taken at various ratios of r/6* and they provide some check on the accuracy of the correction method of [I]. Bull himself used the method to his satisfaction. But the check is not decisive because the frequency range is limited on the upper end. Willmarth and Roos [4] have also attempted to check They inferred the attenuation by the accuracy of the method by direct measurements. obtaining the frequency spectral density with transducers of four different sizes and extrapolating the result at each frequency to zero transducer size. They concluded from their data that the transducer attenuation depends not only on the variable For large values of this parameter their results WT/ 0; (as in [I]) but also upon &*/U,. differ considerably from those of [I]. A re-examination of the data of Willmarth and Roos (Corcos [I I]) has led us to conclude that their data are incompatible, not only with the assumptions underlying the computation of [I], but also with the properties of any physically plausible model of the pressure field. Specifically, the data (which indicate that the resolution actually improves with frequency when the latter exceeds a certain value) imply that as the longitudinal wave number increases, the cross-spectral density either regains a longer convective memory or is contributed by decreasing lateral wave numbers. This seems impossible to reconcile even dimensionally with our ideas of turbulence or with Priestley’s measurements. On the other hand, a modest amount of high frequency radiated noise could easily have led to this result. We conclude that for w6*/ U, > 0.5 a similarity variable of the form WV/U, is likely to exist for the lateral cross-spectral density. For the longitudinal cross-spectral density, the coherence length appears to be also an internal scale under the same restriction. Its form is not uniquely given by unaided dimensional considerations, but they and the experimental evidence together suggest that this scale is proportional to a power of the frequency which is close to - I. The power-spectral density for the pressure at the wall should have an upper frequency cut-off around-f-
and C should
be a function
of similarity
variables up to the cut-off. The approximation seems to be empirically justified. C(@$/ u,, w?ll Cic) = A(wQ UC) B(wrll L7J From the standpoint of transducer resolution, it would thus appear that, while the estimates published in [I] are not necessarily numerically very accurate, they should lead to no gross errors of the kind inferred by Willmarth and Roos. In particular, for w~*/U~ > 0.5, one should expect the dependences of the spectral resolution on any parameter other than WY/C’,, to be weak. REFERENCES G. M. CORCOS 1963J. acoust. Sot. Am. 35, No. z, 192. Resolution of pressure in turbulence. S. GARDNER 1963 Tech. Res. Grp Rep. TRG rqz-TN 63-s. Surface pressure fluctuations produced by boundary layer turbulence. 3. F. M. WHITE 1964 U.S. Navy Underwater Sound Laboratory Rept USL No. 639. A unified theory of turbulent pressure fluctuations. 4. W. W. WILIMARTH and F. W. Roos 1965 J. Fluid Mech. zz, 81. Resolution and structure of the wall pressure field beneath a turbulent boundary layer. 5. R. N. KRAICHMAN 1956J. acoust. Sot. Am. 28, 378. Pressure fluctuations in the turbulent flow over a flat plate. I.
2.
f Here the sub-layer thickness has been taken as 10(,/p,),
the ratio C’,/.~‘X as 0.5 and k, as w/LTC.
70
G. M. CORCOS
6. G. M. CORCOS,J. W. CUTHBERTand W. A. VONWINKLE 1959 Univ. Calif Inst. Engng Res. Rep. Series 82, No. 12. On the measurement of turbulent pressure fluctuations with a transducer of finite size. 7. W. W. WILLMARTHand C. E. WOOLDRIDGE1962J. Fluid Mech. 14, 187. Measurements of the fluctuating pressure at the wall beneath a thick turbulent boundary layer. 8. J. T. PRIESTLEY 1965 N. B. S. Rep. 8942. Correlation studies of pressure fluctuations on the ground beneath a turbulent boundary layer. 9. W. W. WILMARTHand C. E. WOOLDRIDGE1963 AGARD Rept 456. Measurements of the correlation between the fluctuating velocities and the fluctuating wall pressure in a thick turbulent boundary layer. IO. M. K. BULL 1963 AGARD Rept 455. Properties of the fluctuating wall pressure field of a turbulent boundary layer. I I. G. M. CORCOS1966 Univ. CaZif. Inst. Engng lies. Rep. No. AS-66-13. Turbulent pressure cross-spectral density and transducer resolution. 12. A. A. TOWNSEND1956 The Structure of Turbulent Shear Flow. p. 37. Cambridge University Press. r3. T. M. LANDHAL1966 Unnumbered report under NASA Contract Cr-317. A wave-guide model for turbulent shear flow. Propagation characteristics for a flat-plate boundary layer. (Also to appear in Physics Fluids.) 14. P. BRADSHAW 1965 N. P. L., Aero. Rep. 1172. “ Inactive ” motion and pressure fluctuations in turbulent boundary layers. 15. K. L. CHANDIRAMANI 1965 Bolt, Beranek &Newman Rep. No. IgIo. Interpretation of wall pressure measurements under a turbulent boundary layer.
(I),
59-70
THE RESOLUTION WALL
OF TURBULENT
PRESSURES
OF A BOUNDARY
AT THE
LAYER
G. M. CORCOS College of Engineering, University
of California, Berkeley, California,
U.S.A.
(Received 19 September 1966) This paper is a discussion of the resolution by a piezo-electric transducer of the local pressure fluctuations at the wall of a turbulent boundary layer. The first part applies to transducers whose sensing area is large in comparison with the boundary layer thickness, and the second part to transducers whose sensing area is either of the order of, or smaller than, the boundary layer thickness. For large transducers, the main result, derived for a sensing surface of arbitrary shape and with an arbitrary spatial sensitivity distribution, is that under circumstances which are perhaps unrealistically stringent, the attenuation of a given frequency component will increase as the cube of the transducer face linear dimension. In general, it should be expected to increase only as the square of the dimension. In the second part, the probable dependence of the resolution on frequency and on transducer size is indicated by using both recent experimental evidence and physical reasoning bearing on the nature of the pressure cross-spectral density. Both approaches suggest that in a co-ordinate system translated downstream at an appropriate velocity, as the streamwise wave number increases, the spectral density in wave number-frequency space is associated with proportionately increasing spanwise wave numbers and almost proportionately increasing frequencies. Experimental evidence suggests departure from this finding if the fixed axis longitudinal wave length is very large (of the order of the boundary layer thickness) in which case both the characteristic time in convected axes and the characteristic lateral coherence length appear to reach asymptotic finite values. For shorter waves, the strong relationship between the two space scales and the convective time scale necessarily implies that the resolution of the frequency spectral density by a finite transducer continues to deterioriate as the frequency increases. It is concluded that the attenuation predictions previously published by the author are not likely to be seriously in error. I.
INTRODUCTION
When a pressure transducer of non-zero sensing area is mounted as part of the solid boundary surface of a turbulent boundary layer it responds to some area integral of the instantaneous pressure found on the boundary. The manner in which the output signal thus fails to measure pressure at a point has been discussed by Corcos [I] (henceforth referred to as [I]), Gardner [2], White [3] and Willmarth and Roos [4]. In what follows we first consider again the case of large transducers by removing assumptions which are not strictly necessary and by focusing on a theoretical constraint which is provided by the equation of motion and which was overlooked in [I]. The notation is that in [I]. For two identical linear transducers with a perfect time response, mounted as part of (but uncoupled from) a solid surface along which the turbulent pressure field is statistically stationary and homogeneous, it has been shown (see, e.g., [I]) that the measured crossspectral density is given in terms of the actual cross-spectral density by
60
G. M. CORCOS
where c is the distance vector between the two transducer Thus, the measured frequency spectral density is Q&J)
= rm(o, W) = ,I
r(e,
positions,
and w is the frequency.
CO)O(r) dA(s),
(a)
where O(E) = j- K(s)K(s+ m
E) dA(s).
Here K(s) is the unit response of the transducer, i.e. the contribution to the transducer output signal due to a unit pressure at point s on its face; i.e. the transducer located at x measures a pressure p, given by P,(x,
t) = j” P(S, 4 K(s - 4 dA(s).
Hence O(E) is a function wholly defined by the transducer shape and characteristics. E is the two-dimensional vector (E%,E~)defining the co-ordinates on the transducer face, c = (4,~) is the separation between two identical transducers used for the measurements, w is the frequency and i- 00 r&CO)
= ,‘,
W,
I --co
7) exp [ - i(+]
d7,
where R(<,T) is the space-time covariance or correlation of the pressure at two points separated by the distance vector < and at two times separated by the delay 7. 2. ASYMPTOTIC DEPENDENCE SPECTRAL DENSITY
OF THE MEASURED FREQUENCY ON TRANSDUCER SIZE
The behavior of transducers of large sensing face dimensions is of interest in connection with sonar design. It was briefly considered in [I]. For transducers of given shape and characteristics, with a typical face dimension L, of L, is a complex we have O= @(e/L), whereas r(<, w), which is evidently independent function whose modulus at a given w is a maximum at < = o and which decreases to zero for any w. as \
‘1
l
c3 /
,--4
&
i
I.
\--
?
‘L_+’
Figure
1’
g
The area of integration.
We consider the open set ~2 of points in the plane whose boundary is a perimeter V described by the locus of the end of the distance vector between the origin of two identical transducers of identical orientation if one is held fixed and the other is translated to assume all possible contacts with the first (i.e. if the area of the transducer defines the set of points a, then the set &‘=g--a). The characteristic length for the perimeter $5’is
TURBULENT
PRESSURES
ON A BOUNDARY
61
LAYER
called L. We assume that 0 is a continuous function of (e/L), non-stationary at the origin, with continuous first derivatives on & and identically zero on and outside V. We also assume that the transducer is sensitive to pressure rather than to force, so that 0 is of the form 0 = if(E/L), where A is the sensing surface area of order L2 andf(s/L) Then we expand 0 in a partial series
wheref,=--*f
af
ax’ y
=--
af
ay
ando
is a non-dimensional
function.
Substitutingin(
The two integrals on the right-hand side can each be written as the difference between an integral over an infinite plane and an integral over the part of that plane which excludes the domain &. However, the second integral vanishes in the second integration because for points outside the contour %7all derivatives off vanish. We thus have
Now we assume that for ]$ and ]c~~$Q, I~(E~,E~,w)/=O{I/(E,J~~~~~~~}, where m and n > 2. As L + m, the average values of ccand j3 in the third integral tend to zero and
(3) where KI = j-1 F(E, w) icxfx(o,
0)
=yfy(o,
0))
de,
dcy
a function of w only. The first integral on the right-hand side of (3), which is the coefficient of (r/A) - ( 1/L2), can be shown [s] to vanish exactly along a plane and rigid surface for a turbulent pressure field which obeys a strictly incompressible Navier-Stokes equation, while in general the second integral, ICI, the coefficient of (I/AL) - ( 1/L3) does not. Thus it might appear from the foregoing result that the leading term of the asymptotic attenuation is proportional to 1/L3.i But in a laboratory experiment or, say, on the hull of a ship, several factors will inevitably conspire to prevent the first integral from vanishing, in general. They are, is
-f This
TN-65-2).
result was also quoted by D. M. Chase, Technical
Research Group Technical
Note (TRG-01
l-
62
G. M. CORCOS
for instance, the radiation field caused either by the finite Mach number characterizing the boundary layer or by vibrations of the boundary induced by the turbulence, or the slight departure from statistical (planar) homogeneity which is inherent in the pressure field of a boundary layer. For none of these pressure perturbations does the area integral of the space correlation vanish. Under nearly ideal conditions+, the coefficient of the first or l/_Cz term may well be smaller than I/L times that of the next, up to fairly large values of L. In such cases the asymptote of @,(w,L) as L increases may appear to behave as 1/L3 first and then as 1/L2. It is thus apparent that the values given in equation (22) of [I] for the coefficients of 1/L2 (square transducer) and 1/L2 (circular transducer) have no general significance and should be ignored. They were obtained from an integration of experimental values of r(<,q,w) at a given frequency and are characteristic only of those experimental conditions. It should be noticed that the foregoing result depends neither on a convection hypothesis nor on similarity properties. It assumes onIy that the unit response of the transducer (the contribution to the signal due to unit pressure at any point of the transducer face), which may be an arbitrary spatial function, is linearly dependent upon the pressure.
3. THE CASE OF SMALLER
TRANSDUCERS
When one attempts to measure the frequency spectral density of turbulent pressure fluctuations instead of suppressing it, smaller transducers are used. It is, nevertheless, difficult to avoid some attenuation. We have seen that the measured frequency spectral density is given by
This expression, by itself, provides a means of computing the attenuation, @,/@, only for transducers whose characteristic dimensions are large enough : in the integral above, 0 is non-zero only for separation distances (~1which are smaller than the greatest dimension of the transducer. For instance for a circular face, 0 = o for E = 1st 3, 21 where T is the transducer radius and 0 is a monotonically decreasing function of E for E < 27. Thus, the required values of @(a, w) are those for which o < E < 2~. On the other hand, cross-spectral density measurements are always necessarily made for a non-zero separation distance (the minimum distance being the sum of the radii for two circular transducers). If we take the minimum value of E for which r/0 must be known empirically as E = L/4 [i.e. if we assume that interpolation errors are tolerable for o < E 6 L/4, in view of the fact that r/Q (0, w) = I], wh ere L is the maximum dimension of the transducer whose attenuation is queried, we see that equation (2) can be used directly only provided .P/@ has previously been measured(i) over a range E 2 L/4; (ii) with two transducers whose dimension Lo was small enough so that
This does not necessarily require Lo << L, since r/Q, is a function of non-dimensional variables in which scaling factors are flow parameters. Hence, the same requirement may be met by increasing the flow dimensi0ns.f As we shall see, the presently available data can be applied directly to the computation of the frequency spectral density by transt These might be visualized as the flow over the spillway of a dam, for instance. 1 This assumes either that the Reynolds’ number is kept the same or that its influence on I’/@is negligible.
TURBULENT
PRESSURES
ON A BOUNDARY
LAYER
63
L*> o~z8” and lateral dimension L,> 1.46~ ducers with streamwise total dimension according to the requirements set forth above. Here 6” is the boundary layer displacement thickness. An alternative, and on the surface a more direct course, consists in computing the Fourier transform E(k, co) with respect to E of r from the ratio of Em (the Fourier transform of r,,,) and S(k) the Fourier transform of O(E) (Corcos, Cuthbert and von Winkle [6]). But the resulting expression for @ is an integral involving a singular kernel (Willmarth and Roos [4]), which requires that Em be known for the transducer used and this with impossible accuracy in the neighborhood of the singularities. For transducers whose radius r/S* is small enough so that the two conditions set above are not satisfied, the most hopeful approach seems to attempt to infer, from general physical reasoning and from undistorted measurements of this quantity at larger values of c, how r(Z;, w) depends on <. 3. I.
SIMILARITY
In [I] it was suggested
that r’ could be approximated
in a convenient
manner
as
A and B as well as the dependence of Uc on w were inferred from the The functions experiment of Willmarth and Wooldridge [7]. The similarity form (4) for r, if it is suitable, obviates the need to measure cross-spectral density with transducers non-dimensionally smaller than the one for which a resolution correction is desired and in fact allows the substitution of measured for actual spectral-density in the integrand: according to equation (4) equation (2) is of the form
As we have seen, in equation (s), the range of integration of E is that for which O(E) f o, i.e. it is of order L. But if C is a function of the similarity variable WE/ Uc only, the required values of C may be determined for larger values of E and correspondingly lower values of W. Hence C may be obtained from measurements of C,, for which the distortion by the finite transducer size is small, i.e. for which r,,, z r. UJU, is assumed to be a function rather than of &,/Urn. This introduces an uncertainty in the comof w, i.e. of &P/U,, putations in that the dependence of UJU, upon &P/U, is not known for sufficiently large values of the frequency. But this uncertainty is numerically limited because it is found experimentally that the convection speed L’, varies little with frequency for large values of w8*ci C; m. 3.2.
VlLIDITY
OF THE SIMILARITY
ASSUMPTION
One may write in all generality r(t,?,w) where
= ~(W,~0,p,U,,U7,V)C(5,?1,~,CTCo,21~,~,~M)exp
C is a real function,
U,=
-w.$(tan-rri/r~)-‘,
ri and r,
[
-i
, 2 c c11
are the imaginary
and
real parts of r, U, = the free stream velocity, u, = the friction velocity = .\/i-,/p, 7,‘= wall shear stress, p = the density, and v is the kinematic viscosity. r has been cast as the product of the power-spectral density Q(w) and another function of which the argument corresponds to convection at a speed UCand the modulus describes the coherence in space and time in a frame of reference moving downstream with the velocity L;.
G. M. CORCOS
64
We note that for an incompressible two-dimensional boundary layer on a smooth wall, the ratio Uoolu, depends solely upon the local Reynolds’ number Rg*=S*Um~v,sothat we may write
(6) where U,/ U, depends on the same variables as C. Two questions may be asked concerning equation (6). (a) H ow does r/@ depend on Rp, i.e. how similar are the normalized cross-spectral densities for two flows at different Reynolds’ numbers? We still possess little definitive information on this point. We shall see later that the data of Priestley [S], the only cross-spectral density data gathered at Reynolds’ numbers much larger than the usual laboratory ones, suggest that Reynolds’ number effects are moderate, i.e. of the order of the effect of Rp upon u,lU,.We shall not pursue this point, because experimental difficulties may also account for these differences and we shall assume as is often done in shear flow turbulence (e.g. Townsend [12]) that the features of the flow, including the function r are not affected by the flow Reynolds’ number unless their time and length scales are of order v/u: and V/U?respectively, or smaller. (b) Are the non-dimensionalizing lengths Lg. and L, and the non-dimensionalizing frequency w. external dimensions, i.e. flow parameters such as lengths 6” or v/U,and frequency Urn/P,or internal or “local” variables such as length UC/w,and times f/U,or, say, G-‘12Kr3/2 [where G(k,) is the ordinary one-dimensional wave number spectrum of a velocity component within the boundary layer] ? The validity of the dependence of C upon two similarity variables as assumed in [I] hinges on the answer, It should be pointed out that similarity variables, which necessarily appear if external parameters are found irrelevant, need not take the form assumed in [I]. For instance, if the characteristic time of evolution, or memory time in a convected frame, for wave number component ki is kr3j2 G-‘12(Ki), the dependences of C in the streamwise direction would be expressed approximately as c(&o,w)
= C($!;jG112).
But, physically and numerically, the distinction between internal and external scaling appears to be more important than the choice of the proper internal scale. We shall see that of the two available external length scales, 6” is not suitable except for very long wave lengths, so that for the bulk of the wave lengths for which approximately
C must be a function of similarity or internal variables. 3.3.
EXPERIMENTAL
EVIDENCE
The experimental evidence offered in support of the similarity form (4) in [I] was rather incomplete and, as has already been pointed out, it applied to r,, not necessarily to r.t Two sets of measurements of the pressure cross-spectral density have been published recently, one by Bull [IO] and the other by Priestley [8]. It is possible (Corcos [I I]) to determine that for the frequency range investigated, the resolution errors were either small (Bull) or completely negligible (Priestley). t The suggestion in [I], p. 197, that I‘,/@,=
r/Q
ISerroneous in general (Corcos [ll]).
TURBULENT
PRESSURES
ON A BOUNDARY
LAYER
65
According to Bull, for C&*/U, >o.s, the normalized cross-spectral density is well scaled by the similarity variables C&/U, and wv/Ijo for a frequency ratio of about ten distances in this experiment were 0.82
3
Figure 2. The magnitude of the cross-spectral density according to Priestley. cross-stream. Boundary layer turbulent but stably stratified. (-) Bull [IO] ; (--) and Wooldridge [7]; ~/6”valuesasfollows: (0)0.36; (0) 0.54; (A) 0.90; (c) 1.67; (v) 0.48 < wS*/lJ < 60 for all points.
Separation
is
Willmarth
1.98; (+) 2.j2.
The cross-spectral density was measured over a range of separation distances (0.35 < q/S* < 2.5 ; 0.05 < t/S* < 2.5) which includes the smallest separations achieved so far. Yet the separation was always very large compared with the transducers’ effective radius (t/r > IOO; r/r> 600) so that the resolution of the measured wave lengths was unimpaired. The frequency range was approximately+ 0.16 < (wS”/U) < 60, where U is the wind velocity measured at some arbitrarily fixed point, and a good approximation to the convection velocity UC at high frequencies. According to Priestley’s data the lateral cross-spectral density is given within experimental scatter as a function of wq/UC over the separation range. This is shown on Figure 2 where the inclusion of several runs, in which the separation distances and frequencies were the same, gives an idea of the effect of various wind and weather conditions and/or random errors. The data of Bull and Willmarth are summarized on the same plot by two curves. These curves are faired mostly through data for larger values of q/S* and correspondingly lower values of US*/ UC than Priestley ‘s points. Priestley’s longitudinal cross-spectral density measurements were taken over a wide range of atmospheric conditions, in particular, when the density stratification near the ground was either statically stable (no solar radiation) or unstable (considerable solar radiation and hence heating by the ground). As a result the turbulence levels and the mean profiles must have varied a great deal. When the solar radiations were intense, the 1_The
value of 6” in Priestley’s experiment is not known accurately.
66
G. M. CORCOS
pressure intensity at the ground was high and the convection velocity was almost independent of frequency, indicating vigorous mixing and a very flat velocity profile. When there were no solar radiations, the pressure intensity was lower and the convection velocity was a fairly strongly decreasing function of frequency. Surprisingly, the characteristic time scale of decay for the function A was very insensitive to these changing conditions. The data are shown on Figure 3 in terms of the similarity variable &/UC. One notes that the value of A in these experiments falls more rapidly with WZJ/U,than in
Figure 3. The magnitude of the cross-spectral density according to Priestley. Separation is nominally streamwise. Standard deviation of wind direction z 0.2 radians. For E/S+ < 0.36, boundary layer unstably stratified. For all g/8 * > 0.36 points are shown for both stably and unstably ) Bull [I o] ; (--) Willmarth and Wooldridge [7]. t/S* values as follows : stratified conditions. ((0) 0.055; (x) 0.083; (@) 0.14; (A) 0.22; (0) 0.36; (0) 0.54; (v) 0.90; (A) 1.2; (+) 2.52. For points 26.4; for those given by (e), (O), (v), (A) and (+), given by (o), (y), (8) and (A), 0.7 < w@/U< 0.5 < w@/U<60.
either Bull’s or Willmarth and Wooldridge’s case. This may be a Reynolds’ number effect. But a more likely explanation is that in an atmospheric boundary layer which has no governing characteristic dimension, spanwise velocity fluctuations can be expected to have considerable energy in the very low frequency range, or equivalently that in the course of a run the wind direction changes slowly all the time, so that the alignment of the two transducers is only nominally in the downstream direction. It typically must involve an appreciable slowly fluctuating lateral separation as well. The standard deviation of the wind direction was given by Priestley as approximately 0.2 radians in most cases. According rto Priestley, A can be approximated empirically by a function of (&/ UC)(wL/ uC)“.2B,where the (constant) length I, was not specified. This representation and the data of Figure 3 imply that A has somewhat smaller values at a given value of wf/U, for large values of w and small values of 5 than for the opposite case. In other words, not only does the coherence time e/U, not scale with an external (fixed) time such as Se/U but it decreases more than linearly with I / w. This trend is more noticeable for highly unstable conditions than for stable ones. Priestley’s cross-spectral measurements at small separations do not bear out Willmarth and Roos’ [4] expectation that marked departures from similarity occur for spatial separations of the order of or smaller than o-76* and frequencies w greater than 3 UC/S*. The departure from the kind of similarity assumed in equation (4) which does seem to be present in Priestley’s longitudinal data is easily incorporated in the evaluation of transducer resolution by the
TURBULENT
PRESSURES ON A BOUNDARY
LAYER
67
method of [I]. For high frequencies it yields attenuations which are somewhat larger than those computed in [I]. This is just the opposite of Willmarth and ROOS’conclusion. Priestley, like Bull, found that the method of scaling separation distances, according to which the coherence lengths are roughly proportional to the period of the cross-spectral density, does not apply for large eddies (those for which a typical longitudinal wave length is of the order of the boundary layer thickness or more). Both Bull and Priestley made a limited number of cross-spectral density measurements for which both [ and 7 were non-zero and found that the simplifying assumption
C(w, E,7) = 4%
E,0)B(w o,rl)
made in [I] is acceptably accurate.
3.4.
THE SENSITIVITY
OF THE ATTENUATION
TO THE VALUES OF THE CROSS-SPECTRAL
DENSITY
It can be verified that the substitution of either Bull’s or Priestley’s values for the crossspectral density into equation (2) yields transducer attenuation values which differ relatively little from those published in [I] for most transducer sizes, especially for nondimensional small ones. The discrepancy does not exceed zo%, and only for larger transducers at high frequencies. This should be expected since the data of Willmarth and Wooldridge and those of Bull and of Priestley do not differ a great deal. The attenuation of the frequency spectrum is due to three effects :
4)I
(i) the convection of the pressure field which translates longitudinal wave numbers into frequencies
and is expressed in equation (4) by the term exp - i ; [ ! UC (ii) the finite lateral coherence of a component of frequency w (i.e. of wave number k, ~oJ/UJ and; (iii) the fact th at the pressure field is not frozen, so that more than one longitudinal wave number really contribute pressure energy at a given frequency. The general character of the attenuation at a given frequency by a finite transducer which fails to resolve longitudinal and lateral wave lengths of the order of, or smaller than, the transducer diameter depends on : (a) whether the typical wave length in the lateral direction is tied to the frequency chosen (i.e. approximately to the longitudinal wave length) or independent of the frequency ; and (b) whether the typical convective coherence time is also tied to longitudinal wave length, or independent of it. Similarity variables for r (either those of Priestley or those of [I]) imply that both are so tied. Hence the attenuation is compounded and will increase rapidly with frequency, although precisely how rapidly depends somewhat on the type of similarity variable used.
3. j.
MODELS OF THE CROSS-SPECTRAL
DENSITY
Several authors have computed the function I’by making special assumptions. White [3] integrated the Poisson equation for the pressure by assuming a simple model for the velocity field. Landhal [ 131 computed the longitudinal cross-spectral density from a quasi-linear solution of the hydrodynamic equations for a disturbance in a turbulent boundary layer. Bradshaw [14] used broad dimensional argument based on the generally accepted idea that the inner part of a turbulent boundary layer is characterized primarily by one length scale, the distance y from the wall, and one velocity scale, the friction velocity U, = (TJ~J) 1/2. All these approaches which differ very substantially in their underlying assumptions but which make at least partial use of the equations of motion yield
68
G.
M. CORCOS
results which are reasonably similar. The cross-spectral density is found to be a function of similarity variables either precisely (Bradshaw) or approximately (Landhal, White). On the other hand, Chandirami [I 51 has postulated several models of the cross-spectral density which are functionally very different and which yield very different results when used to compute transducer attenuation. The difference introduced by the models is simply that the time and the length scales or the frequency and wave number scales are tied to external parameters of the flow and not to each other. If, as Chandirami assumes, the dependence of the cross-spectral density upon lateral separation is little affected by the selected frequency, the attenuation by the transducer of lateral wave lengths is almost the same at all frequencies. Predictably, this type of model tends to yield transducer attenuations which become asymptotically independent of frequency and which for high frequencies are much smaller than those inferred in [I]. But Chandirami offers no empirical or theoretical justification for his models and they are hard to reconcile with the experimental evidence which we have cited. 3.6. VERY SMALL TRANSDUCERS AT VERY HIGH
FREQUENCY
Transducers whose dimensions are a small fraction of the displacement thickness can be expected to attenuate the power spectral density at high enough frequencies. Consequently we may wonder whether there are any lower bounds to the separation distances for which the similarity variables are applicable. As we have seen, the data of Priestley and Bull imply that for components whose longitudinal wave length is smaller than the boundary layer thickness, 6, the two coherence length scales of the cross-spectral density become independent of 6 or 6”. It seems very unlikely that for much smaller separation distances, a dependence on 6” would reappear. The only other external length, V/U,, should be expected to become relevant for wave lengths of the order of the sub-layer thickness. In other words, the physical separation of the wall from the turbulent velocity fluctuations should confer upon the wall pressures a minimum length scale of the order of the sub-layer thickness. But, except at low Reynolds’ numbers, this limiting value of k, 6” should be very high, at least of the order of 50 to IOO in Willmarth and Wooldridge’s experiment, for instance, and correspondingly higher at higher Reynolds’ numbers. Thus for intermediate values of ki 6” or (wS*/U,) one should expect the function C(<, w) to be governed by internal length scales. The lateral length scale for cross-spectral density is likely to be r/k, as supposed by Bradshaw [13], and computed by White [3] because the lateral and the longitudinal scales of the covariance of the pressure at the wall are closely related by the Poisson equation for the pressure. Thus the scaling C(o, 71,w) = C(o,qw/ U,), while not inevitable, is highly probable. For the longitudinal cross-spectral density, the apparent length scale is U,t, where t is the characteristic time scale in a convected frame of reference. Such a time scale cannot be obtained uniquely from dimensional reasoning without additional assumptions. It can be given either by the differential convection of eddies spread over a height of order 1/k, or by the intrinsic excursions of a convected eddy with time scale of order ky3/2G-1/2 [where G(k,) is a one-dimensional velocity spectrum] or by fluctuations in the convection velocity which would involve the energy of eddies of different sizes. All these mechanisms no doubt contribute? to the convective lack of coherence of the pressure and they yield characteristic lengths which differ somewhat in their dependence on W. But they all couple the convective length scale at a given frequency to a positive power of the frequency which is not very different from unity for all of the mechanisms investigated. t Priestley’s data suggest that more than one of these effects must control A. Otherwise the behavior of A would be very different when differential convection is almost absent and the turbulence level is high and when differential convection is strongly present and the turbulence level is low.
TURBULENT
3.7.
DIRECT
MEASUREMENTS
PRESSURES
OF TRANSDUCER
ON A BOUNDARY
LAYER
69
ATTENUATION
The spectral density measurements of Bull [IO] were taken at various ratios of r/6* and they provide some check on the accuracy of the correction method of [I]. Bull himself used the method to his satisfaction. But the check is not decisive because the frequency range is limited on the upper end. Willmarth and Roos [4] have also attempted to check They inferred the attenuation by the accuracy of the method by direct measurements. obtaining the frequency spectral density with transducers of four different sizes and extrapolating the result at each frequency to zero transducer size. They concluded from their data that the transducer attenuation depends not only on the variable For large values of this parameter their results WT/ 0; (as in [I]) but also upon &*/U,. differ considerably from those of [I]. A re-examination of the data of Willmarth and Roos (Corcos [I I]) has led us to conclude that their data are incompatible, not only with the assumptions underlying the computation of [I], but also with the properties of any physically plausible model of the pressure field. Specifically, the data (which indicate that the resolution actually improves with frequency when the latter exceeds a certain value) imply that as the longitudinal wave number increases, the cross-spectral density either regains a longer convective memory or is contributed by decreasing lateral wave numbers. This seems impossible to reconcile even dimensionally with our ideas of turbulence or with Priestley’s measurements. On the other hand, a modest amount of high frequency radiated noise could easily have led to this result. We conclude that for w6*/ U, > 0.5 a similarity variable of the form WV/U, is likely to exist for the lateral cross-spectral density. For the longitudinal cross-spectral density, the coherence length appears to be also an internal scale under the same restriction. Its form is not uniquely given by unaided dimensional considerations, but they and the experimental evidence together suggest that this scale is proportional to a power of the frequency which is close to - I. The power-spectral density for the pressure at the wall should have an upper frequency cut-off around-f-
and C should
be a function
of similarity
variables up to the cut-off. The approximation seems to be empirically justified. C(@$/ u,, w?ll Cic) = A(wQ UC) B(wrll L7J From the standpoint of transducer resolution, it would thus appear that, while the estimates published in [I] are not necessarily numerically very accurate, they should lead to no gross errors of the kind inferred by Willmarth and Roos. In particular, for w~*/U~ > 0.5, one should expect the dependences of the spectral resolution on any parameter other than WY/C’,, to be weak. REFERENCES G. M. CORCOS 1963J. acoust. Sot. Am. 35, No. z, 192. Resolution of pressure in turbulence. S. GARDNER 1963 Tech. Res. Grp Rep. TRG rqz-TN 63-s. Surface pressure fluctuations produced by boundary layer turbulence. 3. F. M. WHITE 1964 U.S. Navy Underwater Sound Laboratory Rept USL No. 639. A unified theory of turbulent pressure fluctuations. 4. W. W. WILIMARTH and F. W. Roos 1965 J. Fluid Mech. zz, 81. Resolution and structure of the wall pressure field beneath a turbulent boundary layer. 5. R. N. KRAICHMAN 1956J. acoust. Sot. Am. 28, 378. Pressure fluctuations in the turbulent flow over a flat plate. I.
2.
f Here the sub-layer thickness has been taken as 10(,/p,),
the ratio C’,/.~‘X as 0.5 and k, as w/LTC.
70
G. M. CORCOS
6. G. M. CORCOS,J. W. CUTHBERTand W. A. VONWINKLE 1959 Univ. Calif Inst. Engng Res. Rep. Series 82, No. 12. On the measurement of turbulent pressure fluctuations with a transducer of finite size. 7. W. W. WILLMARTHand C. E. WOOLDRIDGE1962J. Fluid Mech. 14, 187. Measurements of the fluctuating pressure at the wall beneath a thick turbulent boundary layer. 8. J. T. PRIESTLEY 1965 N. B. S. Rep. 8942. Correlation studies of pressure fluctuations on the ground beneath a turbulent boundary layer. 9. W. W. WILMARTHand C. E. WOOLDRIDGE1963 AGARD Rept 456. Measurements of the correlation between the fluctuating velocities and the fluctuating wall pressure in a thick turbulent boundary layer. IO. M. K. BULL 1963 AGARD Rept 455. Properties of the fluctuating wall pressure field of a turbulent boundary layer. I I. G. M. CORCOS1966 Univ. CaZif. Inst. Engng lies. Rep. No. AS-66-13. Turbulent pressure cross-spectral density and transducer resolution. 12. A. A. TOWNSEND1956 The Structure of Turbulent Shear Flow. p. 37. Cambridge University Press. r3. T. M. LANDHAL1966 Unnumbered report under NASA Contract Cr-317. A wave-guide model for turbulent shear flow. Propagation characteristics for a flat-plate boundary layer. (Also to appear in Physics Fluids.) 14. P. BRADSHAW 1965 N. P. L., Aero. Rep. 1172. “ Inactive ” motion and pressure fluctuations in turbulent boundary layers. 15. K. L. CHANDIRAMANI 1965 Bolt, Beranek &Newman Rep. No. IgIo. Interpretation of wall pressure measurements under a turbulent boundary layer.