The conditional sampling technique is analyzed as a weighted time average for turbulent flow. The various conditional averages are obtained by using different types of weighting functions. A second averaging relation is obtained between the conventional averages and the conditional averages. A few examples are given in which simplified expressions are used

Wang, F. C.

English

NASA Technical Reports Server

NASA TECHNICAL 
MEMORANDUM 
NASA TM X-64886 
ON CONDITIONAL SAMPLING FOR TURBULENT FLOW STUDIES 
By F. C. Wang 
Systems Dynamics Laboratory 
(NASA-TH-X-64886) ON CONDITIONAL SAHPLING 
FOR TURB'JLENT I L O Y  STUDIES (NASA) 20 p CSCL 20D 
NASA 
George C. 
~ 7 4 -  34696 
Unclas 
Marshall Space Flight Center 
Space Fh'gh t Center, A h  bdmd 
https://ntrs.nasa.gov/search.jsp?R=19740026583 2020-03-23T02:27:02+00:00Z
T E C H N I C A L  R E P O R T  S T A N D A R D  1 l T I  E P A G (  
George C. Marshall Space Flight Center 
Marshall Space Flight Center, Alabama 35812 
1. REPORT NO. 2. GOVERNMNT ACCESSION NO. 
NASA TM X- 64886 
4. TITLE AND SUBTITLE 
ON CONDlTlONAL SAMYLlNG FOR TURBULENT FLOW STUDIES 
7. AUTIIMI(S) 
F. C. Wang* 
9. PERFORMING ORGANIZATION NAWL AN0 ADDRESS 
1 1. CCNTRACT OR GRANT NO. 
1. RECIPIENT'S CATALOG NO. 
I. REPORT O N E  
September 16, 1974 
.,. ,AT, 
6234 
8. PERFORMING ORGANIZATION REPCU1 t 1 
10. WORK UNIT NO. 
National Aeronautics and Space Administration 
Washington, D.C. 20546 
12. SPONSORING AGENCY NAME AN0 AOORESS 
Technical Memorandum 
- 
14. SPONSORING AGENCY CODE t 
C 
13. TYPE O f  REPOR; & PERIOD COVERPI 
I 
IS. SUPPLEMENTARY NOTES 
Prepared by Systems Dynamics Laboratory, Science and Engineering 
*National Research Council - National Aeronautics and Svace Administration Postdoctoral Research 
Associate 
16. ABSTRACT 
The conditional sampling technique is analyzed as a weighted time average. The various 
conditional averages are obtained by using different types of weighting functions. A "second averaging" 
relation is obtained between the conventional averages and the conditional averages. A few examples 
are given in which simplified expressions are used. 
7. KEV WORDS 18. OlSTRlBUTlON STATEMENT 
Unclassified-unlimited 
I 
9 .  SECURITY CLASSIF. (d thb n p a m  
Unclassified 
ISEC - Form 8 I B Y  (Rev Dwrmbor 1#11) For u l e  by National Technical Inforntation 9crlcr, Springfield, Virginia 21 15 1 
20. SECURITY CLASSIF. (of thh  VS*) 
Unclassified 
21. NO. OF PAGES 
20 
22. PRICE 
NTlS 
This work was supported by the National Research Council while the author was 
a Postdoctoral Research Associate at NASA-Marshall Space Flight Center. 
TABLE OF CONTENTS 
Page 
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 
11. THE WEIGHTED TIME AVERAGES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 
111. CORRELATIONS IN CONDITIONAL AVERAGES . . . . . , . . . . . . . . . . . . . . . 8 
IV. SIMPLE EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 
V. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 
LIST OF ILLUSTRATIONS 
Figure Title Page 
1. Turbulent boundary layer near the wall . . . . . . . . . . ... . . . . . . . . . . . . . . . 2 
2. Fluctuating interface in an infinite domain . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 
3. Linear stretching of two fluid regions between two solid boundaries . . . . . 13 
iii 
TECHNICAL MEMORANDUM X-64886 
ON CONDITIONAL SAMPLING FOR TURBULENT FLOW STUDIES 
I. INTRCDUCTION 
A conditional sampling technique has been used recently by many researchers in 
studying turbulent shear flows [ 1-91 . Under the conditional sampling, a discriminating 
criterion is chosen and flow measurements are made using two or more probes. One of the 
probes is the triggering probe and the rest are sampling probes. Averaging at the sampling 
probe is made only when the discriminating criterion is satisfied at the triggering probe. 
By so doing, the flow field is divided into different regions and average flow quantities in 
each region are obtained. A tremendous amount of data may be generated and by using 
the various forms of correlations, the detailed structure gf the turbulent flow field is 
obtained. This is usually shielded from us in a conventional measurement. 
Any detectable flow quantity may be used to provide the discriminating criterion. 
The location of the turbulent-nonturbulent interface at the outer edge of a turbulent 
boundary layer has been used by Kovasznay et al. [ I ] ,  by Imaki [2] ,  and by Antonia [4].  
The same has been used by Wagnanski and Fielder [S] in a two-dimensional mixing layer, 
and by Thomas [6] in a two-dimensional wake. The location of the viscous sublayer was 
used in studying turbulent "bursting" phenomena by Narahari et al. [7 1 and by Blackwelder 
and Kovasznay [8].  Other quantities that may be used include the flow temperature used 
by Dean [9] ,  the species concentration, and the location of a fluctuation shock front. 
Mathematically speaking, the conditional average is just a form of weighted average, 
with the weighting function generated from the reading at the triggering probe. By using 
different types of weighting functions, different types of conditional averages can be 
obtained. The conventional average, for which no discriminating criterion is used, becomes 
a special case. 
In the following sections, the concept of conditional sampling is demonstrated by 
considering the turbulent boundary layer over a flat plate. The flow field consists of two 
distinctive regions: the turbulent region near the flat plate, and the nonturbulent region of 
the ambient flow. These regions are separated by a thin, sharp interface that is highly 
contoured, and its position fluctuates in time. By using the location of the interface as a 
discriminating criterion, the following types of conditional averages can be obtained: the 
turbulent zone average, the nonturbulent zone average, and the point conditional average. 
A "second averaging" relation is then obtained which relates the conditional 
averages to the conventional average through the statistical distribution of the interface 
position. With this relation it is thus possible to incorporate the experimental data of 
conditional measurement into the theoretical modeling of turbulent shear flow. Simplified 
relations for the "second averaging" are obtained under special assumptions. 
II. THE WEIGHTED TIME AVERAGES 
Consider the turbulent boundary layer over a flat plate as shown in Figure 1. 
Figure 1. Turbulent boundary layer near the wall. 
Let the instantaneous location of the turbulent interface, assumed to  be single- 
valued, be given by 7 = h(x,t). Let ~ ( f , t )  be a flow quantity measured at a point 
P(x,y). Then at any time t ;  one of the following conditions must be satisfied: 
where the point P is said to be either inside the turbulent region, inside the nonturbulent 
region, or at the interface, depending on whether condition (la), ( lb) ,  or (lc) is satisfied. 
-+ 
If a weighting function w( r ,t) can be defined that takes different values for P in 
different regions, then a weighted average of Q is defined as 
-C 
According to  the weighting function w(r ,t), the types of averages listed as follows 
can be obtained. 
1. Conventional Averages, If a constant weighting function is used for P in all 
the three regions in (I), then the conventional average, or the "long time" average, is 
obtained, 
The fluctuation of Q (f .t) about its average Q (t) is given as 
which has a "long time" average of zero, 
The standard deviation of Q(r,t) is given by 
which is a measure of the intensity of the fluctuation. 
2. Zone Avera If the functional for the weighting function to be used is the 4
turbulent s@al I ( r  ,t), which is unity in the turbulent zone and zero elsewhere, i.e. 
then the following turbulent zone average is obtained: 
A nonturbulent zone average can be obtained similarly by I 
function, giving 
using 1-1 ( f , t )  as the weighting 
Introducing an intermittency factor, y(r), as the conventional average of the 
turbulent signal, 
then the turbulent and nonturbulent zone averages can be rewritten as 
and 
Obviously, the conventional average can be obtained from the zone averages u'sing the 
relation 
which was first obtained by Townsend [ 101 
3. Multiple Zone Averages. We shall now deduce from the zone averages the point 
conditional averages. First let us introduce the multiple zone averages. Assume that the 
range of the possible values of the interface position h(x,t) is given by the open interval 
(a,b), where a may be minus infinity and b may be plus infkity. Assuming that a 
sequence of ordered points po < p1 < pZ < ... < p, exists such thut po ='a and 
pJ = b , then the interval (a,b) is divided into J smaller intervals by the sequence of 
pgints. If the following weighting function for each interval is defined 
( 0 , otherwise 
for j = 1 , 2 , 3  ,..., J 
then a multiple zone average at the jth interval is obtained as 
Similar to equation (13), the conventional average is related to the multiple zone 
averages by the summation relation 
where 
j wj(f,t) dt 
+ - lim 0 f j ( r )  = T- T 
is the probability that the interface position h(x f )  falls in the interval bctwesn pj-1 
and pj . A normalization condition has to be satisfied by these probability functions, 
which yields 
4. Point Conditional Averages. Let us now take the limit such that the largest 
interval in the multiple zone average oC item 3 becomes infinitesimally small, so that the 
weighting function in (15) becomes the Dirac delta functions, 
Then the following point condition average is obtained: 
I 
In the same limit, as the largest interval goes to  zero, a probability density function 
Pj(h) for the turbulent interface position can be introduced in the expression 
-* 
wj(r ,t) = S(h - pj) = 
Substituting (21 ) into equation (16) and taking the same limit, the following 
integral expression is obtained: 
1 , h = p j  
0 , otherwise 
\ 
Here integration is taken wer all possible values of h ,  assumed to be from minus 
infinity to plus infinity. This can be rewritten as 
by using the normalization relation 
Noticing the analogy between equations (2) and (221, the relation between the con- 
ventional averages and the conditional averages can thus be interpreted as another weighted 
A + 
averaging process. Since the conditional average Q ( r  ) is already an averaged quantiiy, 
this relation is thus 1~:rned a "second averaging" relation. The weighting function for this 
"second averaging" is :ne probability density function P (h) of the turbulent interface 
position, which may take any value from 0 to .. 
Ill. CORRELATIONS IN  CONDITIONAL AVERAGES 
In this section, we shall derive the relations between the fluctuatio- :elations 
in the conditional and conventional averages using the "second averagir ion, equation 
(22). 
Let the flow quantity to be averaged, ~( ; f , t ) ,  be the product o ro measurable 
-b 
fluctuating quantities, ~ ( f t )  and S( r .t); i.e., 
Hem tho dependency of Q, R, and S on the space cc -dinate f is well under- 
stood and shall be omitted from the following discussions. 
Substituting equatibn (24) into equation (22) gives the relation, 
Separating R(t) and S(t) into their conventional mean and fluctuating parts gives 
~ ( t )  ~ ( t )  = [R t r(t)l [S t s ( t ) ~  = R S + ii s(t) + i r(t) + r(t) s(t) . 
(26)  
Defining fluctuating parts in the point conditional average as 
gives similarly 
A A A 
R(t) S(t) = R S + R sA (t) + S rA (t) + r, (t) sA (t) . (28) 
Substituting (26) and (28) into (25) and using the relations 
gives 
which can be rewritten as 
Thus the conventional cordation E can be obtained from the point conditional 
A h  
averages R, S and the point conditional correlation G A o n c e  the probability density 
distribution function Pj(h) is known. Contracting equation (3 1 ) gives the fluctuating 
level, which is 
00 
- A A 
r2 = 1 Pj(h) (R2 + < ). dh - J (' Pj(h) kj dh) 
-00 
Higher order correlations can also be obtained in a similar manner. The expression 
always includes correlation of the same order in the ;oint conditional average and conela- 
tions of lower order. A third order correlation is given here as an example: 
m 
- 
A A A  A A  A A  A A  A 
qrs = $ Pj(h) (QRS + QrAsA + R q A s A  + Sq,,rA + q,r,,~,,)~dh 
-m 
The above expressions can also be applied to two-point correlations like 
q(x,t) r(x + dx,t) . Similar relations for time-delayed correlations like ( q(x,t) r(x, t+dt)) 
can be obtained using ensemble averages. 
IV. SIMPLE EXAMPLES 
Applicatian of the "second averaging" relations, equations (22) and (3 1) through 
(33), in a real flow situation may become impractical when extensive ionditional measure- 
ments are needed. Under certain conditions, however, approximations for the conditional 
averages can be made, and simplified relations for the second averaging can be obtained. 
In the following examples, similarity assumptions are used to draw up sirnplificd relations. 
Example 1. Fluctuations in an Infhite Domain. Consider the motion of a 
fluctuating interface in an infinite domain as shown in Figure 2. This interface may be a 
material surface which separates two different flow regions, or a nonmaterial surface such 
as a shock front or a shear front. It is reasonable to assume that the motion of the fluid 
very near the interface is always in phase with the motion of the interface; and the flow 
properties at a fixed distance from the interface position always remain constants, so that 
they can be given a single-valued function F as 

which reduces to  
by using new variables 
A particular example of this case is the averaging of the turbulent signal, which is 
The average from (35b) gives 
which gives the intermittency factor. 
Example 2. Linear Fluctuation Between Nodal Points. Consider two layers of 
mutually immiscible fluid bounded on both sides by solid walls as shown in Figure 3. 
Assume a one-dimensional fluctuating motion is produced in the x-direction, normal to  
the interface, and no reflective waves are produced so that the motion of the fluid is in 
phase with the motion of the interface. Furthermore, assume each fluid layer, is stretched 
and compressed linearly by the motion of the interface. Then the point conditional 
averages in the fluid region can be represented by one of the two similar profiles given 
below : 
The functions F, and Fa are both smooth and singlwalucd, and they ut normrlizad 
with the distance between the interface and the walls. 
Figure 3. Linear stretching of two fluid regions between two solid boundaries. 
Substituting equation (38) into equation (22) gives 
which, after changing variables, becomes 
Thus, by knowing the distributions of P , F, , and Fa, the simple integration will give the 
conventional averages. 
V. CONCLUSION 
The conditional sampling averages have been analyzed as weighted time averages. 
The various forms of cnn4t i - , i i  a;:mges are obtained by using different types of 
weighting functk,!s. A relation is derived which relates the conventional averages to the 
conditional averages through a "second averaging" process. This "second averaging" 
process is dso a weighted averaging over possible values of the discriminating quantity. 
A,few exam~4cs are given in which simple expressions for the point conditional averages 
can be used. 
The "second averaging" relations, (22b) and (31) through (33), relate the condi- 
tional averages to the conventional averages. Application of relation (22b) in the mathe- 
matical modeling of turbulent shear flow has been done by Nee and Kovasznay [ 111 and 
by Wang [ 121. Using a simplified form, encouraging improvements in the mean flow 
fields were obtained. The justification for these relations lies in the assumptions made on 
the conditional averages. Detailed conditional measurements are thus needed for this 
purpose. Recently, Paizis and Schwan [ 131 used a rack of 20 probes in studying the 
structure of the turbulent interface. From the simultaneous measurements at the 20 
locations, detailed structure and motion of the interface were obtained. This information 
is valuable in the constructing of a better model for the turbulent shear flows. 
REFERENCES 
1. Kovasznay, L. S. G . ;  Kiben, V.; and Blackwelder, R.: Large-scale motion in the 
intermittent region of a turbulent boundary layer. J. Fluid Mech., vol. 41, 1970, 
p. 283. 
2. Imaki, K.: Bull. Inst. Space Aeronaut. Sci. University of Tokyo, vol. 4,  1968, 
p. 448 (In Japanese 3nly). 
3. Kaplan, R. E.; and Laufer, J.: Proc. 12th Int. Cong. Appl. Mech. Springer, 1969, 
p. 236. 
4. Antonia, R. A.: Conditionally sampled measurements near the outer edge of a 
turbulent boundary layer. J. Fluid Mech., v d .  56, 1972, p. 1. 
5. Wygnanski, I.; and Fielder, H. E.: The two-dientional mixing region. J. Fluid 
Mech., vol. 41, 1970, p. 327. 
6 .  Thomas, R. H.: Conditional sampling and other measurements in a plane turbulent 
wake. J. Fluid Mech., vol. 57, 1973, p. 549. 
7. Narahari Rao, K.; Narasimha, R.; and Badri Narayanan, M. A.: The "bursting" 
phenomenon in a turbulent boundary layer. J. Fluid Mech., vol. 48, 1971, p. 339. 
8. Blackwelder, R. F.; and Kovasznay, L. S. C.: Large-scale motion of a turbulent 
boundary layer during laminarization. J. Fluid Mech., vol. 53, 1972, p. 61. 
9. Dean, R. C.: Interaction of Turbulent Shear' Layers in Ducts and Diffusers. Ph.D. 
Thesis, Imperial College, London, 1972. 
10. Townsend, A. A.: The Structure of Turbulent Shear Flow. Cambridge Univ. 
Press, 1956. 
11. Nee, V. W.; and Kovasznay, L. S. G.: Simple phenominological theory of turbulent 
shear flow. Phys. Fluids, vol. 12, 1969: p. 473. 
12. Wang, F. C.: Structure of Turbulent Jets and Wakes Using Eddy Viscosity Models. 
Ph.D. Thesis, University of Minnesota, Minneapolis, 1972. 
13. Paizis, S. T.; and Schwarz, W. H.: An investigation of the topography and motion 
of the turbulent interface. J. Fluid Mech., vol. 63, 1974, p. 315. 
APPROVAL 
ON CONDITIONAL SAMPLING FOR TURBULENT FLOW STUDIES 
By F. C. Wang 
The information in this report has been reviewed for security classification. 
Review of any information concerning Department of Defense or Atomic Energy 
Commission programs has been made by the MSFC Security Classification Officer. This 
report, in its entirety, has been determined to be unclassified. 
This document has also been reviewed and approved for technical accuracy. 
HOMER B. WILSON, JR. 
Chief, Thermal Environment Branch 
WERNER K. DAHM 
Chief, Aerophysics Division 
firector, systems Dynamics Laboratory 
MICC-RSA, Alb 


On conditional sampling for turbulent flow studies

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