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AlAA 89-0756 /

Use of the Median Volume Droplet Diameter in the Characterization of Cloud Droplet Spectra K. Finstad, E. Lozowski, University of Alberta, CANADA; L. Makkonen, Technical Research Centre of Finland, Espoo, FINLAND

27th Aerospace Sciences Meeting January 9-12, 19891Ren0, Nevada

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 370 L'Enfant Promenade, S.W., Washington, D.C. 20024

USE OF THE MEDIAN VOLUME DROPLET DIAMETER IN THE

CHARACTERIZATION OF CLOUD DROPLET SPECTRA . / / 0

Karen J. Finstad ,&+d ,LC- Edward P. Lozowski ' )

I

Division of Meteorology The University of Alberta Edmonton, Alberta, Canada

Lasse Makkonen Laboratory of Structural Engineering Technical Research Centre of Finland

Espoo, Finland

Abstract

An approximation often used in icing applica- tions for calculating the collision efficiency or liquid water content of a spectrum of cloud drop- let sizes, is that of the median volume diameter, or mvd. In this paper we present a mathematical justification for using the mvd by deriving it from a single-point numerical integration formula, which can be extended to derive new approximation formulae using 2, 3, or 4 droplet sizes. Then, by using a selection of droplet size spectra collected from many different sources, we calcu- late the spectrum-weighted average collision efficiencies for circular cylinders and NACA 0015 airfoils in order to empirically demonstrate the good agreement given by the mvd. We also show that the new schemes can significantly improve the accuracy, with only a small increase in computational costs, and with additional numerical errors much smaller than those already present due to measurement and sampling procedures.

I. INTRODUCTION

In the study of atmospheric ice accretion onto airfoils and other objects, perhaps the most difficult quantities to measure are the sizes and number concentration of the cloud water droplets which are the source of the ice. Two parameters which are crucial to the icing process depend strongly on the droplet spectrum; they are the liquid water content of the cloud, and the collis- ion efficiency of the droplets with the collecting object. How accurately we can measure the spect- rum, and what parameters we use to characterise it, are thus important factors in determining how well we can describe or predict icing.

A very common practice is to characterise the complete spectrum of droplet sizes by a monodis- perse spectrum of droplets at the median volume diameter (mvd) of the original spectrum. This approximation is made in calculations of both liquid water content and collision efficiency. The advantage is in computational savings. A collision efficiency may then be calculated for one droplet size, the mvd, rather than for many

sizes or size categories, which are needed for the formation of a volume-weighted mean collision efficiency.

Similarly, the same approximation may be used to calculate the liquid water content (lwc) using the total droplet concentration, rather than a breakdown of the droplet concentrations within different droplet size categories.

For over forty years the mvd has been known to provide a good characterization of the full spect- rum, Langmuir (1944) being one of the first to use it. However, the reasons for this generally good agreement have been obscure. We felt that we ought to try to give some justification for it. In so doing, we also hope to give some idea as to how the approximation may be improved without having to calculate the spectrum weighted collis- ion efficiency in detail.

11. A Mathematical Justification for the mvd Approximation

In the discussion below we use the notation E(D) to denote the collision efficiency of a droplet with diameter D on some larger object. The collision efficiency could be an overall or a local value, and the object could be a circular cylinder or an airfoil of any type. The details of the collision efficiency function are unimport- ant for the formal mathematical analysis, and we could equally well substitute a function LWC(D), the liquid water content contained in all droplets of diameter D, for E(D) .

Let g(D) be the volumetric density function of the droplet spectrum, so that g(D) is the fraction of the total droplet volume (or mass) contained in droplets of size D to D+dD. If Espec is the exact spectrum-weighted mean overal-1

collision efficiency for a droplet distribution with diameters less than or equal to Dmax, then:

Copyright O American institute of Aeronautics and Astronautics, Inc., 1989. All r~ghts rererved.

Espec = jDmax g(D) E (D) dD (1)

0

Let f(D) be the volumetric distribution function defined so that:

D

f ( D ) = j g(D) dD

0

Then in terms of f, (1) may be re-written as:

1

Now let D* be a single droplet size which yields a collision efficiency equal to E that is: spec'

When Espec is already known, it is always possible

to find D*. What we would like to know is whether there is a rule for finding D*, a priori. One reason for wishing to find such a rule is to yield an algorithm for finding E which is

spec faster than numerically integrating (1) over the entire spectrum, keeping in mind that E(D) must be determined for each size category D in the integration. If E(D) is found through the inte- gration of individual droplet trajectories, this procedure could be very computationally intensive.

Another reason is that the detailed droplet spectrum may be unmeasured or at least unavail- able, but that certain spectral statistics such as the median volume diameter or total droplet concentration, may be available or otherwise determinable from the measurements. For some types of measurements, such as those made with a rotating multicylinder, for example, D* itself is one of the quantities determined. In this case, it is useful to know the relationship between D* and the mvd or other spectrum para- meters.

In the following we will use the notation D X

for the xth percentile droplet diameter where Dx is defined so that f(D,) = x/100, where 0 < x < 100. In this notation mvd = Dso. Why should DsO = D*?

We can begin to answer this question by numerically integrating Eqn. (3) using the well- known method of Gauss-Legendre Quadrature (Abramowitz and Stegun, 1964).' In order to do this, we must first transform the integration interval to (-1, 1). Let

Then Eqn. (3) becomes:

which has the n-point Gauss-legendre Quadrature approximation:

For n from 1 to 4, the weighting factors, w. and 1' the zeroes of the Legendre polynomials, xi, are

as listed in Table 1. This gives the following approximation formulae:

single-point,

two-point ,

three-point,

f our-point ,

Thus we see that DS0 - D* because E(D) is the first order approximation to E spec ' and that the mvd is in this sense the optimum choice for approximating the collision efficiency of the full spectrum with that of a single droplet. Further- more, by extending the Gauss-Legendre formula to 2, 3, or 4 points, one can proceed in a rational fashion to develop new and more accurate approxi- mation formulae requiring only slightly more computational effort than it takes to calculate the mvd approximation.

In the next section we will examine the per- formance of these approximations when applied to the collision efficiencies of realistic droplet spectra onto both circular cylinders and airfoils.

111. Comparisons of the Approximation Formulae

In order to compare collision efficiencies calculated according to the approximation schemes given above with spectrum-weighted collision efficiencies, we have collected examples of drop- let size spectra measured both in the field and in wind tunnels from a number of sources. The spectra include real icing fogs, stratus and cumulus clouds, sea spray, nozzle spray, and para- meterized distributions. A full list of the sources and details of the spectra have been given in Finstad et al. (1988b).

We p r e s e n t comparisons f o r t h e o v e r a l l c o l l i s i o n e f f i c i e n c y of a c i r c u l a r c y l i n d e r (F igure l ) , and f o r t h e l o c a l c o l l i s i o n e f f i c i e n c y a t t h e s t a g n a t i o n l i n e of a NACA 0015 a i r f o i l (F igure 2 ) . Fa r t h e l a t t e r comparison, a s m a l l e r

F ig . 1 . O v e r a l l c o l l i s i o n e f f i c i e n c i e s (E) of a c i r c u l a r c y l i n d e r i n p o t e n t i a l f low under t h e fo l lowing c o n d i t i o n s : a i r speed = 10 m/sec , c y l i n d e r d iamete r = 0.034 m , a i r t empera tu re = - 1 0 ' ~ . The s p e c t r a l weighted ave rage (E ) h a s

spec been c a l c u l a t e d f o r a s e l e c t i o n of 27 d r o p l e t s p e c t r a , and i s h e r e shown com- pared w i t h t h e approximat ions us ing t h e mvd, 2 -po in t , 3 -po in t , and 4-point formulae d e s c r i b e d i n t h e t e x t (Emvd,

E2, E g , and Eq, r e s p e c t i v e l y ) . The s o l i d

l i n e i n d i c a t e s a 1 : l s l o p e

number of d r o p l e t s p e c t r a were cons ide red f o r r easons of computa t iona l economy. The s p e c i f i c c o n d i t i o n s of wind speed , c y l i n d e r d i a m e t e r , chord l e n g t h , e t c . a r e l i s t e d i n t h e a p p r o p r i a t e f i g u r e c a p t i o n s .

The mvd and o t h e r d r o p l e t s i z e pa ramete r i za - t i o n s used i n Equat ions (8) through (11) were determined fo l lowing t h e method g iven by Lozowski (1978). ' For t h e c i r c u l a r c y l i n d e r we have used t h e a n a l y t i c a l approximat ion formulae f o r o v e r a l l c o l l i s i o n e f f i c i e n c y g iven by F i n s t a d e t a l . (1988.~1). For t h e NACA 0015, t h e s t a g n a t i o n l i n e c o l l i s i o n e f f i c i e n c i e s were c a l c u l a t e d from numerical d r o p l e t t r a j e c t o r y i n t e g r a t i o n s u t i l i s i n g a pane l method s o l u t i o n f o r t h e p o t e n t i a l f low ( F i n s t a d , 1986) .

I n bo th f i g u r e s i t i s c l e a r l y seen t h a t t h e mvd approximat ion does indeed g i v e a good approxi- mation t o t h e spectrum-weighted c o l l i s i o n e f f i c - iency over a wide v a r i e t y of s p e c t r a , w i t h a n average a b s o l u t e e r r o r of 0 .02 i n F i g u r e 1, and

F ig . 2. Local s t a g n a t i o n l i n e c o l l i s i o n e f f i c i e n c y (B) f o r a NACA 0015 a i r f o i l i n p o t e n t i a l f l o w under t h e f o l l o w i n g c o n d i t i o n s : a i r speed = 50 m/sec, chord = 0.5 m , a i r t empera tu re = - 1 0 " ~ . The s p e c t r a l weighted ave rage (B ) h a s been ca lcu -

spec l a t e d f o r a s e l e c t i o n of 20 d r o p l e t s p e c t r a , and i s h e r e shown compared w i t h t h e approx imat ions u s i n g t h e mvd, 2 -po in t , 3 -po in t , and 4-point formulae d e s c r i b e d i n t h e t e x t (Bmd, B 2 , B j , B 4 , r e s p e c t -

i v e l y ) . The s o l i d l i n e i n d i c a t e d a 1 : l s l o p e .

0.06 i n F i g u r e 2 . Fur thermore, use of t h e two- p o i n t formula r educes t h e ave rage e r r o r f o r t h e s e s e t s of s p e c t r a by 50 p e r c e n t o r more over t h e mvd approximat ion. Indeed , t h e ave rage e r r o r i s rough ly ha lved each time a n o t h e r p o i n t i s added t o t h e approximat ion formula . However, t h e degree of improvement v a r i e s somewhat w i t h t h e type of spectrum, being most marked f o r t h e more compli- c a t e d s p e c t r a such a s double-peaked d i s t r i b u t i o n s .

I V . Comments on Measurement E r r o r s

Up t o t h i s p o i n t , we have n o t cons ide red t h e p o s s i b i l i t y of measurement e r r o r s i n t h e d r o p l e t s i z e s o r c o n c e n t r a t i o n s , which would add u n c e r t a i n - t i e s t o t h e c a l c u l a t e d v a l u e s of Espec o r lwc.

( C o l l i s i o n e f f i c i e n c y c a l c u l a t i o n s c o n t a i n numerical e r r o r s a s w e l l . ) We have compared t h e approx imat ions , such a s Emvd, t o E a s i f t h e

s p e c l a t t e r were an e x a c t v a l u e .

However, r e a l d r o p l e t s p e c t r a a r e indeed sub- j e c t t o s i g n i f i c a n t measurement and sampling e r r o r s . Analyses such a s t h o s e of Baumgardner (1983) , Mossop (1983) , and S t a l l a b r a s s (1986) f i n d e r r o r s i n t h e range of 10 t o 20 p e r c e n t i n bo th number c o n c e n t r a t i o n and d r o p l e t s i z e s a s d e t e r -

mined by a variety of different instruments and methods. ' 9 ' y9 The resulting combined errors in lwc, for example, are estimated by Baumgardner to be greater than 30 percent for both the soot slide and optical scattering probe methods.

Given this level of uncertainty in the basic observations, the order 10 percent errors produced by use of the mvd approximation are acceptable.

V. Conclusions

The mvd approximation has been justified on mathematical grounds, by applying Gauss-Legendre Quadrature to the integral defining the spectrum- weighted mean collision efficiency. This pro- cedure has also yielded improved approximation methods which utilise two, three, or four droplet size parameterizations in lieu of the full spect- rum.

The success of these parameterizations in providing accurate estimates of spectrum-weighted collision efficiencies (at considerable compu- tational savings) has been demonstrated for circular cylinders and airfoils, and a wide variety of droplet spectra.

Acknowledgements

6. Langmuir, I., 1944: Super-cooled water droplets in rising currents of cold saturated air. Collected Works of Irving Langmuir, Vol. 10, Pergamon Press, 199-334.

7. Lozowski, E.P., 1978: Stochastic effects in spray droplet sampling with oiled slides. Laboratory Memorandum LT-172, Low Temperature Laboratory, Division of Mechanical Engineering, National Research Council of Canada, Montreal Road, Ottawa, Canada KIA OH3

8. Mossop, S.C., 1983: Intercomparison of instru- ments used for measurement of cloud drop con- centration and size distribution. J . Climate Appl. Meteor. 2, 419-428.

9. Stallabrass, J.R., 1986: Comparison of drop- let size measurements by three methods. Proceedings, Third International Workshop on Atmospheric Icing of Structures, Vancouver, Canada (in press).

Table 1. Constants for the Gauss-Legendre inte- gration formulae (Abramowitz and Stegun, 1964).

Numerous discussions with Mr. J.R. Stallabrass of NRC piqued our interest in this problem and

1 0 .o gave us some ideas as to how to examine it. This research was sponsored through a grant from the 2 k0.57735 Natural Sciences and Engineering Research Council - - of Canada. K. Finstad is grateful to the Royal Norwegian Council for Scientific and Industrial

3 0.0, k0.77459 0.88888, 0.55555 -

Research for a Post-Doctoral Fellowship, and L. Makkonen to the Finnish Broadcasting Co., 4 0.33998, +0.86113 0.65214, 0.34785 Imatran Voima Co., and the Finnish Post and Telecommunications Administration for financial support.

References

1. Abramowitz, M. and I.A. Stegun, Eds., 1964: Handbook of Mathematical Functions. National Bureau of Standards Applied Mathematics Series 55, United States Department of Commerce.

2. Baumgardner, D., 1983: An analysis and com- parison of five water droplet measuring instruments. Jour. Atmos. Sci. z, 891-910.

3. Finstad, K . J . , 1986: Numerical and Experiment- al Studies of Rime Ice Accretion. Ph.D. thesis, University of Alberta, Edmonton, Alberta, Canada.

4. Finstad, K . J . , Lozowski, E.P., and Gates, E.M., 1988a: A computational investigation of water droplet trajectories. J . Ocean. Atmos. Tech. 5, 160-170. - -

5. Finstad, K . J . , Lozowski, E.P. and L. Makkonen, 1988b: On the median volume diameter approxi- mation for droplet collision efficiency. J. Atmos. Sci., (in press).