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@begin(titlepage)
Advanced Technology Research Center | Siengg Division | Samani International Enterprises
A New Correlation of the Universal Logarithmic Turbulent Fluid Velocity Distribution
VASOS-PETER JOHN PANAGIOTOPOULOS II [President, Samani International Enterprises]
@value(date)
@begin(researchcredit)
@center[@b(ABSTRACT)]
The correlations of Prandtl and Deissler, as they appear, modified,
in a recent textbook, are recorrelated as a single continuous function.
This was originally developed while trying to determine dimensional
variables which involved graphical iteration and complex calculations. With
this method numerical analysis can simplify calculations on a simple
programmable calulator.
@end(researchcredit)
@end(titlepage)
@pageheading(left "Logarithmic Turbulence Correlation",center"@value(page)",
right"Vasos-Peter John Panagiotopoulos II")
@pagefooting(left"@value(sectiontitle)", CENTER "@value(date)",
RIGHT "Samani Siengg ATRC")
@chapter(Previous Work)
The logarithmic correlation was initially proposed by Prandtl in
1933@foot{Prandtl, L, @i<VDI Z.>, @b<77>:105(1933).} Deissler had
determined an experimental velocity distribution @foot{ Deissler,
R.G., "Analysis of Turbulent Heat Transfer, Mass Transfer, and
Friction in Smooth Tubes at High Prandtl and Schmidt Numbers",
@i<United States National Advisory Committee on Aeronautics
Report> 1210, (1955), p.3.} Based on data of Laufer @foot{Laufer,
J., "The Structure of Turbulence in Fully Developed Pipe Flow",
@i<United States National Advisory Committee on Aeronautics
Technical Note> 2954(1953).} and himself @foot{Deissler, R.G.,
"Analytical and Experimental Investigations of Adiabatic Turbulent
Flow in Smooth Tubes", @i<United States National Advisory Committee
on Aeronautics Technical Note> 2138(1950).} His correlation was
@equation{
u@+<+>= [H(y@+<+>)-H(y@+<+>-26)]@ovp<(>,y@-<0>@+[+]dy@+<+>/(1+.015u@+<+>y@+<+>[1-exp<-.015u@+"+"y@+"+">]) + (2.8 ln y@+<+> +3.8
with
@equation{u@+<+>=u(4@ovp(o),Lg@-"c"@+"-1"D@+"-1"@u</\>P@+"-1")@+".5", and y@+<+>=y(.25g@-"c"D@u</\>P@ovp<o>,@+"-1"L@+"-1"@ovp(u
u@+<+> and y@+<+> are dimensionless velocity and displacement,
respectively, as functions of density, tube length, tube diameter,
pressure drop, and viscosity. H is Heaviside's function, often
appelated the unit step function. Ocular inspection yields the
accuracy measurement @equation{u@+<+>:=u@+<+> @u(+) 1.}
This has recently been simplified via further
discretisation into three zones, the viscous sublayer, the buffer
zone and the turbulent core. @foot{ As appeared in, without direct
reference, @i<Unit Operations of Chemical Engineering>, McCabe,
W.L., and Smith, J.C., 3rd.ed., NY:McGrawHill, 1976, p.95.}
@equation{u@+<+>=y@+<+>[H(y@+<+>)-H(y@+<+>-5)] + [5lny@+<+> -3.05][H(y@+<+>-5)-H(y@+<+>-30)] + [2.457lny@+<+>+5.67]H(y@+<+>-30)
indicated, however, that the discontinuities lack physical
correspondence. Since no data or refernce is provided for the
latter correlation and since ocular inspection indicates
sufficient similarity, the accuracy of the former shall be assumed
for the latter.
@comment( Schlichting; Knudsen&Katz p 97,101-5,158-71)
@Chap(The Proposed Correlation)
@sec(Preliminary Correlation)
A function was assumed and its constants were obtained
numerically @foot{Marquardt-Levenberg iterative curve fitting
algorithm, in: Knott, G.D., & Reece, D.K., MLAB:A Civilised Curve
Fitting System, @i<ONLINE'72 International Conference>, @b<1>, p.
497-526, Brunel,UK(1972); Knott, G.D., @i< Computer Programs in
Biomedicine>, @b<10>(1979)271-280; Knott, G.D., & Reece, D.K.,
@i<MLAB(Modeling LABoratory)>, interactive program in SAIL, for DEC
PDP-10 systems, Laboratory of Statistical and Mathematical
Methodology, DCRT, NIH, Bethesda, Md.(1980). The curve fitting
performed later in this document were obtained via MLAB's
Marquardt-Levenberg error optimisation capabilities in which: (1) The
standard errors are standard deviations when the model is linear in
its coefficients; (2) Dependency values positively measure
intercoefficional dependency and uniqueness of fit, i.e., the
sharpness of the stationary point; (3) Sum of squares are a
chi-squared statistic with degrees of freedom being the number of
observations less the number of coefficients; (4) RMS error is a
dimensional goodness of fit measure equal to the square root of the
chi-square/degrees-of-freedom; (5) t-statistic may be calculated by
dividing the coefficient by the standard error; (6) r-squared may be
proxied via 1-RMS-error/Best-data-value. }, yielding,
@equation{u@+<+>=y@+<+>(-.452-.813x10@+<-4>y@+<+>)+(8.44+.0721y@+<+>)lny@+<+>-9.15+11.5/y@+<+> t=[-40.6,-18.5,111,35.6,-66.7,57
@u(+)(.23=1.645*.142) |95% confidence | 1<y<600 (theoretical)
@u(+).5|2<y<600 or @u(+)3|1<y<800 (actual extrema) } @foot{The RMS error
is the standard deviation of the residuals and the t-statistic for so many
degrees of freedom approaches 1.645 at 95% confidence level. Error for
y+=(1,2,632.31,806) is (.9,.35,.5,3). Error peaks at ends and
discontinuities.}
Although the error seems to be a subtractive exponential, when such
a function was attempted, the power's coefficient proved insignificant,
with t-statistic of .09, a very low confidence level.
@begin(figure)
@begin(verbatim)
{MLAB OUTPUT EDITED FOR BREVITY}
*FCT UX(Y)=IF Y<5 THEN Y ELSE IF Y<30 THEN 5*LOG(Y)-3.05 ELSE 2.457*LOG(Y)+5.67
*UQ=POINTS(UX,1:30:.1&31:120:1&170:600:50);FCT U(Y)=Y*(A+B*Y)+(C+D*Y)*LOG(Y)+E
*FIT(A,B,C,D,E),U TO UQ
FINAL NORMAL ERROR DEPENDENCY
PARAMETER VALUES: STANDARD ERRORS: VALUES:
.453416@@-1 .213130@@-1 .999795 A
.711906@@-4 .107396@@-4 .994506 B
4.42922 .912667@@-1 .994221 C
-.146479@@-1 .410606@@-2 .999822 D
-1.57767 .105343 .955864 E
RMS ERROR= .437056
FINAL SUM OF SQUARES= 73.5418
*FCT U(Y)=Y*(A+B*Y)+(C+D*Y)*LOG(Y)+E+F/Y;*F=.001
*FIT(A,B,C,D,E,F),U TO UQ
FINAL NORMAL ERROR DEPENDENCY
PARAMETER VALUES: STANDARD ERRORS: VALUES:
-.452000 .111451@@-1 .999920 A
-.812952@@-4 .440195@@-5 .996528 B
8.43652 .763184@@-1 .999123 C
.721114@@-1 .202527@@-2 .999922 D
-9.14607 .137165 .997236 E
11.4980 .201788 .951213 F
RMS ERROR= .142405
FINAL SUM OF SQUARES= 7.78725
{?U:=U-.0435EXP(.00467Y):}
*FCT U(Y)=Y*(A+B*Y)+(C+D*Y)*LOG(Y)+E+F/Y-G*EXP(H*Y);F=11.5;G=.0435;H=.00467
*FIT(A,B,C,D,E,F,G,H),U TO UQ
FINAL NORMAL ERROR DEPENDENCY
PARAMETER VALUES: STANDARD ERRORS: VALUES:
.462015@@-1 .186014 .999988 A
.724636@@-4 .165689@@-3 .999894 B
4.42368 .834649 .999683 C
-.144936@@-1 .333804@@-1 .999988 D
-1.83068 8.19489 .999967 E
6.80982 1.72133 .971030 F
.123637 7.61536 .999981 G
.405598@@-2 .465814@@-1 .999640 H {TSTAT=.09=>NOT SIGNIF}
RMS ERROR= .936093
FINAL SUM OF SQUARES= 334.735
@end(verbatim)
@caption(Correlations with MLAB)
@end(figure)