ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Final Report RESEARCH STUDY PERTAINING TO LOW-LEVEL WIND STRUCTURE May 1, 1955 - November 30, 1956 E. Wendell Hewson Max A. Woodbury Project 2377 LABORATORY PROCUREMENT OFFICE SIGNAL CORPS SUPPLY AGENCY CONTRACT NO. DA-36-039, SC-64676 DA PROJECT NO. 3-17-02-001, SC PROJECT NOo 1052A July 1957

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The University of Michigan * Engineering Research Institute TABLE OF CONTENTS Page LIST OF FIGURES iv ABSTRACT v OBJECTIVE vi PURPOSE 1 PUBLICATIONS, LECTURES, REPORTS, AND CONFERENCES 1 FACTUAL DATA TASK A: STATISTICAL DESIGNS-ROCKET RESPONSE TO ATMOSPHERIC TURBULENCE 2 1. Development of Rocket Equations 2 2. Estimation of Terms in Response Equations 7 3. Improved Estimates for Homogeneous Field 10 4. Application to Rocket Fire Systems 14 5. Conclusions and Recommendations 17 TASK A: STATISTICAL DESIGNS-ROCKET TRAJECTORY DISPERSION BY ATMOSPHERIC DISTURBANCES 19 1. Structure of Wind in the Lower Layers of the Atmosphere 20 (a) Variation of wind with height based on theories of momentum transfer 20 (b) Analysis of the atmospheric surface layer 21 2. General Remarks on Turbulence 22 35 Response of the Rocket 22 4o Proposed Methods of Attack on the Problem 23 APPENDIX A. VARIATION OF WIND WITH HEIGHT BASED ON CONSIDERATIONS OF MOMENTUM TRANSFER 25 APPENDIX B. ANALYSIS OF THE ATMOSPHERIC SURFACE LAYER 28 A. The Adiabatic Atmosphere 30 Bo The Diabatic Atmosphere 30 TASK A: STATISTICAL DESIGNS-EVALUATION OF WIND DATA 33 OVERALL CONCLUSIONS 33 RECOMMENDATIONS 33 REFERENCES 35 iii

The University of Michigan * Engineering Research Institute LIST OF FIGURES No, Page 1 Wind effects on a rocket for two influence functions-one constant and the other inversely proportional to distance from point of launching 4 2 Broken line. typical wind influence function as given by Hunter, Shef, and Black. Solid lines. typical mean windspeed ratios for day and night conditions at OVNeill, Neb. 7 3 Space spectrum for u = 5 m/sec. 13 4 Filters corresponding to a simple correction scheme. 16 A-1 Relations among the geostrophic wind 0, the surface wind Vo, and the arbitrary constant A, whereby A is expressed in terms of Vo. 27 -V

The University of Michigan ~ Engineering Research Institute ABSTRACT TASK A. STATISTICAL DESIGNS A system of equations is developed to calculate the atmospheric effect on a rocket along its flight trajectory. It is found that the major effect is contributed by low-frequency turbulence components. The rocketinfluence function is defined which acts as a filter on the turbulent eddies operating along the rocket trajectory, The magnitudes of many terms in the rocket equations are estimatedn It is found that the total of turbulent effects is almost constant for any trajectory, whereas the mean wind effect is a definite function of a given rocket trajectory. Through a Fourier transform of the influence function, a correlation function is determined that would be observed *by an object moving along a mean wind path, The binormal dispersion of the rocket is calculated by transforming frequency spectra into space spectra according to rTaylorTs hypothesis, A prediction scheme is developed to reduce dispersion whereby'the rocket launcher will be adjusted to the effects of low-level winds, The scheme is adjudged inferior to a least-squares technique'which, ho'wevrer is much more difficult to evaluate. The mean wind is less applicable to the- actual mechanics of a rocket firing system than is the whole wind, The quality of prediction schemes may be improved by the use of Taylor's hypothesis at levels below 150 ft. The combination of a least-squares prediction technique with Taylor's hypothesis for heights of interest holds promise of development in future work., The hypothesis could be tested below 1050 ft by analysis of data: obtained from instrumentation on and between two television towers withlin a mile of each other. The structure of low-tropospheric winds is considered from momentumtransfer theories and by analysis of the surface layers and turbulence regimes. The atmospheric parameters that affect the dispersion of the rocket trajectory after launching are reviewed. TASKS B, C, AND D Detailed investigations of anemometry, data-reduction systems, and wind-tunnel studies appropriate to this research were deferred in the interest of developing the statistical design. v

The University of Michigan * Engineering Research Institute OBJECTIVE The object of the research is to analyze low-level wind structure as it pertains to dynamic wind loading. vi

The University of Michigan * Engineering Research Institute PURPOSE The purpose of this investigation is to analyze the problem of the determination of the structure of the wind in the lower layers of the atmosphere as it pertains to dynamic wind loading of objects. The resources of mathematical physics, meteorology, aerodynamics, and statistics are employed in the analysis The research program may be considered as consisting of four tasks, as follows: TASK A: To produce one or more statistical designs for field experiments which will reveal the wind-flow features that are significant for dynamic loading problems. TASK B: To evaluate existing or possible wind-measuring instruments, such as anemometers, gustometers, and bivanes, to determine their suitability for field use in measuring the three-dimensional largescale structure of the atmosphere. TASK CO To recommend one or more systems to reduce the data obtained by the sensing elements of the instruments to usable form, TASK D: To assess the suitability of the wind tunnel as a device for simulating eddy structure over specified terrain features. No continuous analysis of wind instruments of suitable response characteristics was made during the project, It was felt that detailed studies of appropriate anemometry should be deferred until adequate statistical designs had been developed to serve as a guide for such studies. Anemometers are mentioned briefly, however, in the Recommendationso The analysis of statistical designs has not advanced far enough to warrant proceeding with studies of data-reduction systems and with wind-tunnel studies. PUBLICATIONS, LECTURES, REPORTS, AND CONFERENCES There have been no publications or lectures during the reporting periodo 1 1

The University of Michigan * Engineering Research Institute A number of informal meetings have been held since the Third Progress Report to elaborate the ideas developed in the summer conference of 1956 and to prepare this material in a form suitable for the Final ReportO FACTUAL DATA TASK A: STATISTICAL DESIGNS-ROCKET RESPONSE TO ATMOSPHERIC TURBULENCE by Ben Davidson and Leo J. Tick 1o DEVELOPMENT OF ROCKET EQUATIONS It is instructive to consider the dynamic wind loading problem in coordinate systems which are standard for atmospheric turbulence work and then to rotate these standard coordinates to one which is convenient for calculating wind effects on rocketso The standard meteorological system defines the x axis in the direction of the mean wind, u, the y axis in the cross-wind direction, and the z axis vertically upward. The turbulent velocities in these directions are ul, v', and wl, respectively. The results of almost all atmospheric turbulence work are expressed in terms of u~, v, and wt as described above. For this reason, it is desirable to maintain these familiar velocity components The natural coordinate system for a straight line ideal trajectory prior to burn-out is one where the x' axis is along the flight trajectory, the z" axis is normal to the flight trajectory, and the y" axis is in the binormal direction. The wind relationships between the tTo systems are: u" = [ (u + u) cos A - v~ sin A] cos E - w sin E; vv' = ( + u' ) sin A + v2 cos A; and (1) w = [ ( + u') cos A - v~ sin A] sin E + wS cos E, where u", v", w" are components of the wind in the x", y", and z" directions. The angle A represents the angle between the x and x" axis and E is the elevation angle of the rocket trajectory, We are assuming that the rocket trajectory is a straight line in space for distances of interest to uso The quantity u may vary with height. We also assume that below 150 meters there is no direction shear, so that V = 0, and we are dealing with two sets of axes fixed in space. *The notion of a mean wind is, at best, an elusive one and most attempts to formalize it have been, in the main, unsuccessfulo We leave it here as mostly an intuitive notion, to be dealt with in each individual case, 2 ---

The University of Michigan * Engineering Research Institute The u" component of wind affects the rocket by slight variations in drago The v" and w" components of wind, the binormal and normal wind components, affect the rocket motion by introducing variations in angle of attacko We shall assume that the angle-of-attack effects are more important than the drag effects. Let us now calculate atmospheric effects on the object along its flight trajectory. The integrated wind along the flight trajectory to burn-out sb is nb sb Ob S b v' ds = sin A u (s sin E)ds + sin A uds+cos A vds so so so so w"ds = cos A sin E I i (s sin E)ds + ulds (2) 0 0 S 0 o O - sin A sin E J vSds + cos E / w ds Over a long series of trials the average value of the integrals of the turbulent velocities is zero, by the nature of the mean, so that the only nonzero quantities are the integrals of the mean velocity along the trajectory. For individual trials, integrals along the trajectory are not zero. If we express the turbulent velocities in a sine series,* v"(s) = Z ak sin (I 2s + 8) (3) The integrated wind along the path is then v ds = Z ak b sin ---- + k) ds J o S'b - so (4) =, ak (Sb-So) sin ek in2 ik 2ikk 5b s i sin k sS +~+ 5skin L - ~ k (Sb,-So)) k 2 irk Sb - so _j 2 qk for 6k = Oo Aside from the phase relationship implicit in the oscillating term in brackets we see that if the amplitudes are constant (constant energy spectrum), contributions to nonzero values of the turbulent integrals are inversely proportional to wave number. The major contribution comes from relatively low wave number (long wave length) components. ~This is not a very meaningful representation for the larger wave lengths, but we only use it to discuss the high-frequency behavior, 5

1.0 -( 1.~~~~~~~~~~~~'0 \ sInfluence function 0 0.8,o 0.6 —| 0 3 IoO. w Iii z Z -l 0.4 Fig Wind effets on a rockeonstant for two influence functionone constant and ~~~~~~0.2 ~s. -— I 0 o 2 3 4 5 6 7 8 9 k SPACE- FREQUENCY Fig. 1, Wind effects on a rockett for tio Lnfluence functions-one constant and the other inderfecy proportional to distance from point of launching.

The University of Michigan * Engineering Research Institute The previous result assumes that a turbulent wind impulse, no matter where applied, has an identical effect on the objecto This will not usually be the case. The aerodynamics of the rocket and. the rocket-velocity history along the trajectory will determine the relative effects of a gust at different points along the trajectory. Suppose we introduce a function G(s) in Eq. (2) which expresses this dependenceo G(s) is usually called the "rocket influence functiono" We now have to integrate terms like G I Sb S(bs \ G(s) u(s)ds = G(s) ak sin (2 + s o (5) so \sb - S o Now the wave numbers which contribute to nonzero values of the integral are not so obviouso The function G(s) acts like a filter. As an example we take G(s) inversely proportional to s, which is saying that the turbulent wind effects are greatest in the beginning of the trajectory. The integral of Eq. (5) with G(s) = 1/s is E ak (Sb - So) Sr i 2k 5b -Si( 2r skl cos Sk ak - S o) i Sb S ok k 2it SL \b -so / Sb - so + CL fi b Sb) - Ci ( ^ so)] sin 6k, (6 L sb - so ~ _.,' when Si and Ci are the so-called sine-integrals and. cosine-integrals. For illustrative purposes we have plotted the integrals given by Eqs. (4) and (6) (assuming 6k = 0) in Fig. 1o It is evident that substantial contributions to nonzero values of the integral occur at higher wave numbers (shorter wave lengths) for the curve corresponding to Eq. (6) (influence function like 1/2) than for the curve corresponding to Eq. (4) (influence function like 1). From this discussion it appears that one cannot fix the range of frequencies which should be measured or estimated without first knowing the character of the G(s) function which will differ for various types of rockets. We now calculate the mean square dispersion of the rocket at burn-out, assuming that the influence function G(s) is symmetrical: 2 =F b G(s)v"ds = si2 A b G(s)i(s sin E)ds + sin2 A[ G(s)ulds] + cos2 A G(s)vds + 2 sin A cos A / G(s)u'ds / G(s)v'ds (7a) so Suso ------------------------— 5 -------------

The University of Michigan * Engineering Research Institute _-___ _ i So _ 2_ _ n2 = [J r G(s)w"ds = cos2 A sin2 E G(s)u(s sin E)ds Lno - fo S Sb Sb + cos2 A sin2 E G(s)u'ds + sin2 A sin2 E G(s)w'ds L~o J o + 2 cos A sin E cos E s G(s)w'ds G(s)v'ds - 2 cos A sin A sin2 E G(s)u'ds G(s)v'ds So Jo - 2 sin A sin E cos E G(s)wds G(s)uds. (7a) o 0 Here G represents the angular deviation from the windless trajectory of the rocket at burn-out in the binormal (b) and normal (n), directions, respectively2 2 __ G2 script 13 2e- wb en G(s)(s sin Ed sin2d A cos2 A sin2 E o G(s) G(s') RB11[(s) x(s')] sin2 A cos A sine E J G(s) G(s') R22[X(S) C(s')i cos2 A sine A sin2 E /G(s) G(s') Rs3[(s) (s) cos2 E (7b) G(s) G(s8) R2[ x(s) (s8)1 2 sin A cos A -2 cos A sin A sin2 E s2[~(S)'(SI)] G(s) G(sv?) R1[X(s) X(s)] -2 sin A sin E cos E /G(s) G(s') R23s[(s)'(s')] 2 cos A sin E cos E where the R terms represent the correlation of eddy wind components at two points s and s' along the line defined by the angles A and E. The subscripts 1,2,3 refer to the r, v, and w components of the turbulent wind, while H is the mean wind assumed to be a function of z (or s sin E) onlyo ----------------------- 6 —----------

The University of Michigan * Engineering Research Institute With respect to a coordinate system fixed to the earth, the rocket trajectory is arbitrary, all values of A(O < A < 2r) and E(O < A < rt) are permissible; the influence function G(s) is a function of the aerodynamics of the particular rocket under consideration, A complete solution of the problem thus involves knowledge of appropriate correlation and influence functions along an infinite number of lines within the atmospheric boundary layer. Such an observational task is clearly impossible, We therefore turn to methods of characterizing and modeling the atmosphere. 2. ESTIMATION OF TERMS IN RESPONSE EQUATIONS We shall attempt to do this by first summarizing all that is preently known concerning the structure of atmospheric mean and turbulent wind. fields, Using this information coupled with an increasing set of minimum assumptions, we extract information as to the relative importance of the various terms in Eqo (7b) and suggest possible methods of improving the present rocketlaunch systemo It is convenient at this point to introduce a realistic influence function, and we take as a typical wind influence function that given by Hunter, Shef, and Blacko1 The dotted curve of Figo 2 is a graph of this function, and for the moment all that concerns us is that G(s) is a decreasing monotonic function of distance along the trajectory-~ We nowi cite some obs.erva30' Trajectory (day) 1.0- 0.8 0 I 0 r i < 30~ Trajectory (night) 0 0.8- 0.68.6 o x z'o+ 0.6 -- 0.4 0.4 - 0.2 - \ G(s) 0 1.2 2.4 3.6 LENGTH ALONG PATH IN YAW WAVE LENGTHS Figo 20 Broken line: typical wind influence function as given by Hunter, Shef, and Blacko1 Solid lines: typical mean wind-speed ratios for day and night conditions at O0Neill, Neb, (For further details on the latter, see the discussion near the end of Section 4 ) oec ------ 7-o

The University of Michigan * Engineering Research Institute tional evidence concerning the intensity of turbulence and the correlation structure of the turbulence at a fixed point. These data represent the results of observations at Brookhaven, Long Island, No YO, Buchanan, N. Y., Round Hill, Mass., and OeNeill, Nebo.2 3 It is important to note that the values of gustiness vary with the averaging time used to define the mean wind; the appropriate averaging time for most of the data cited below is one hour. TABLE I TYPICAL VALUES OF GUSTINESS RATIOS AND ZERO SEPARATION CORRELATIONS FOR TYPICAL LAPSE CONDITIONS Condition (uN2)1/2/R (vR2)1/2/ (w92)1/2/i Ri2(0) R23() R13(O) Lapse 0.2 5 005 02 - 05 0.07 - 0,3 0 (03 to O6) cau The lower values of the gustiness ratios are associated with relatively smooth terrain (O'Neill, Neb.), while the higher value has been observed at Buchanan, N, Yo, under strong wind flow, over a ridge about 1.5 miles upwind It is worth noting that under these conditions an anemometer, located 1/2 mile downwind of the ridge, registers almost pure turbulence; it is impossible to define a mean wind from the trace. We now assume that the rocket is aimed to hit a target under zero wind conditions, i.eo, that no wind correction is made. Not correcting for the mean wind introduces a systematic bias in the results so that the center of burst will not be around the target. Since -G(s) is a monotonic decreasing function of s, the integrals involving the hourly mean wind are In a similar fashion, the integrals involving identical subscript — ~2 [c —(s ).b Using the observed R.i (0) correlations, the integrals involving cross correlation functions are << CiCj [G(so)]2 (sb - so)2 The reason for the double inequality in the last relationship is because of the zero contribution from Rij(O) where G(s) is largest. It is extremely unreasonable to expect Rij(s) terms to reach high values for 0 < s < Sb, and 8

The University of Michigan * Engineering Research Institute even if they did, the character of the G(s) function would minimize them, We now compare [UI(sb)]2 with a? > ajaj or ra? Crij 1 with > T2 cr Using the values in Table I, we are comparing 1 with numbers ranging from 0.04 to 0025. If for the moment we disregard the trigonometric terms in Eqo (7b), the above indicates that if no mean wind correction is made, the squared error due to mean wind speed is about 4 to 25 times that due to turbulence. The bias error due to mean wind (a more realistic number) is about 2 to 5 times that due to the standard deviation of the turbulent windo If we consider the trigonometric terms in Eq. (7b), we emerge with quite different results, depending on the angles A and Eo For example, the binormal comparisons would then involve U with Rl + R22 > R12 sin A wh c2 2 sin A cos A sin2 A with > If the rocket is aimed in the direction of the mean wind, then the only contribution to the scatter in the binormal direction is that due to turbulent wind componentso For the normal direction (assuming A = 0), we have u R11 R R3 23.2 a 2 EO2 2 avw ~~sin2 E usin2 E CaYu CosE 2 sin E cos E v o s2 si _ _2 If we take a typical value of E, say 50~, then we find for the normal component R11 R33 R 23 0.25 0o25 [0.04 + 0o25] + 0.75 + 0.86 The leading term in the squared error in the normal direction is the mean wind termn Comparing the normal and binormal scatter for this example, we find the squared error due to the mean wind in the normal direction is about 1 to 6 time that due to turbulence in the binormal direction, The realistic standard error would range from 1 to about 2,5, For elevation angles of 20~ or less, and an azimuth angle of 0, the contribution of turbulence to the scatter in the bi----------------------- 9

The University of Michigan * Engineering Research Institute normal direction will exceed the error due to mean wind in the normal directiono From the foregoing, it is apparent that generalizations about the error due to the relative importance of mean and turbulent wind are difficult to make without considering the path of the rocket with respect to the vertical and mean wind axis. Perhaps a more realistic criterion would be to accept a certain wind error and then to re-examine the problem in view of the acceptable wind erroro For example, despite the fact that turbulent wind effects are of the same order as mean wind effects for the special trajectory A = O, E f 2C, it may be that the mean wind effect is itself negligible for this trajectoryo In other words, Eqo (7b) implies that the total turbulent effects are almost constant for any trajectory, while the mean wind effect is very much a function of specified rocket trajectory. 30 IMPROVED ESTIMATES FOR HOMOGENEOUS FIELD We may get a better estimate of the dispersion due to turbulence alone by introducing the notion of homogeneous and isotropic turbulence. In an homogeneous turbulent field, the mean quantities characterizing the field are independent of translations It follows that the c'or:relation tensors are functions only of the vector separation between points, and the spatial gradients of all mean quantities specifying the turbulence are -.er o For exampleo Rij() = ui (u + r) uj(X) __'(8) V u- = 0 In an isotropic field the mean quantities characterizing the -turbulence ares independent of all rigid body rotations. It follo'w- that (using the equation of continuity), Rijr) = f(r) [rirj + 6i = 1 i. j (9) =0 i j and that uT2 -= v2 = w'22 Now atmospheric turbulence is neither isotropic nor even homogeneous. In some restricted aspects, however, the atmosphere does behave like an homogeneous, isotropic medium, and any practical solution of the total problem must take advantage of the homogeneous or isotropic features of the atmosphere. We summarize now some of the knowledge which has been accumulated concerning the eddy structure in the lowest 100 m of the. atmosphereo Most of these data are 10

The University of Michigan * Engineering Research Institute analyses in the frequency domain of three-component wind data obtained from meteorological towers located at Brookhaven, Long Island, No Yo, OVNeill, Nebo, and at Round Hill, Mass. Summarizing the data available at the present time and speaking in an average sense only, the following seems to represent the facts roughly: A. Under unstable conditions: 1o u'2 X v'2 20 Sz(ua) q Sz(v ) 3~ Sz(u) Sz(v ) ( O0 4o [uSvj(o)] = [vw'(O)] = 0 50;u w12 60 (uwv )z / 0 7~ Sz(w) 0 3Z Bo Under stable conditions: lo u- X 4v92 20 Not enough spectra are available to warrant further statements Here the Sz is the Eulerian time spectrum, calculated from the readings of a fixed anemometer located at height z During unstable conditions (clear sunlight hours), the atmosphere exhibits some features of isotropy (.Al-4) or homogeneity (Al-5) in the horizontal components, but is clearly neither isotropic or homogeneous with respect to the vertical velocity (A5-7)~ If we refer back to Eqo (7), we see that the vertical velocity does not enter into the expression for the binormal at all, but does enter into the expression for the wind that acts normal to the trajectoryo Moreover, we have already shown that terms involving cross correlations are likely to be quite a bit smaller than the terms involving identical subscripts in the correlation tensor. The conclusion clearly is that we may regard the turbulence entering into the binormal equation as at least homogeneous in the x and y directions while A3 implies some sort of homogeneity in the z direction for horizontal velocity components Assuming that homogeneity for the horizontal velocity components exists in the x, y, and z directions, we may then write the correlation functions in the binormal component of Eqo (7b) as Rij(s) = Rij {[(s) - x(s )] [y(s) -y(sB)] [z(s) - z(s)]} l 11

The University of Michigan * Engineering Research Institute In this form the Fourier Transform of Rij is the spectrum, and the binormal component of Eq. (7b) can be written as G = / sin2 A 0il + cos2 A 022 + 2 sin A cos A 012 lr(k,k2,k3) 12 dkidk2 dkX Sb = 2 + sin2 A [ " (s sin E)d, (10) where 0ij is the space-spectrum tensor of the turbulent velocity components, and r is the Fourier Transform of the influence function G(s). For a rocket fired in the direction of the mean wind - = f 022 lr(kl,o,o)|2 dkldk2dkk3 = 0*(kl) I r(k1,,O) 12 dkl, (11) where 0 = 022 (kl) dk2 d.k3 is the Fourier Transform of the one-dimensional lateral correlation function along a line parallel to the mean wind. This is the correlation function that would be observed by an aircraft flying along a mean wind path. As was pointed out previously, the meteorological spectra available at the present time are in the frequency domain and are computed from the time history of the wind going by a fixed pointo It is customary to employ Taylor's hypothesis to transform the time spectra into space spectra. The hypothesis states that x = (12) where U is the mean wind, k1 the wave number parallel to the mean wind, and cX the time frequency. A frequency spectrum can be transformed into a space spectrum by dividing the abscissa and multiplying the ordinate by u. Considerable activity at Pennsylvania State University, Cornell Aeronautical Laboratory, New York University, Brookhaven National Laboratory, and Massachusetts Institute of Technology is now going on to determine the validity of Taylor s hypothesis for atmospheric levels of turbulence0 The one report5 which is available does not contradict the hypothesis0 We now transform a frequency spectrum into a space spectrum by use 12

The University of Michigan * Engineering Research Institute of Taylor's hypothesis. Using the Fourier Transform of the influence function given in Refo 1, we calculate the binormal dispersion of the rocket. The relatively high wave-number portion of the spectrum was take from Refo 2 while the low wave-number portion of the spectrum was obtained from Refo 35 We have plotted the space-equivalent spectrum obtained through use of Taylorgs hypothesis in Fig~ 3 for a typical wind speed of 5 m/sec together with the square of the admittance function given as a function of rocket-yaw wave length, which we assume to be 600 fto The dispersion due to turbulence is the integral of the product of the two functionso I Z w 7.5- 2 -\ r _(k) = Square of normalized admittance function WIZ I.0 8-Io 2.5 - 0 0 0.1 0.2 0.3 k CYCLES/ YAW WAVE LENGTH Figo 3. Space spectrum for u = 5 m/sec. Figure 3 is extremely instructive. It will be noted that, for the influence function used, the cutoff point for high wave numbers is determined not by the admittance function, but by the character of the turbulence spectrum, In other words for relatively high wave numbers the spectrum goes to negligible values much sooner than. does the admittance function. The other point of major interest is that the value of the dispersion will be very much a function of the lower limit of integration of the product of the spectrum and the square of the admittance function. For example, if the lower limit of integration is k = 0o01 (corresponding to a mean wind averaging time of 60 min), the dispersion is about twice that for a lower limit k = 0~04 (corresponding to a mean wind averaging time of 15 min) 13

The University of Michigan * Engineering Research Institute 40 APPLICATION TO ROCKET FIRE SYSTEMS The above suggests that the notion of a mean wind, while useful for a general approach to the problem and for estimates of orders of magnitude of possible dispersion, is of marginal utility in the actual mechanics of a rocket fire system. For example, the dispersion due to turbulence is a function of the averaging time used to define the mean windo In Sections 2 and 3 the dispersion is estimated in terms of an averaging time of one houro Application to a rocket fire system would assume that an hourly mean wind is predicted, that this wind is inserted into the rocket aiming system, and that after an hour of firing, the dispersion around the mean point of impact will be as given. From an applied point of view, the mean wind inserted into the system may not be the true mean wind, and the dispersion will be centered not around the target but at some point distant from the targeto Experience indicates roughly that- the error in forecasting hourly mean winds is of the same magnitude as the turbulent variations about the hourly meano For this reason, although the predicted dispersion may be correct, the actual point target may never be hito What is desirable in a rocket fire system is first of all a system which, after a series of firings, will insure that the center of scatter is around the target, and secondly that the dispersion around the target be as small as possibleo Restricting our discussion now to a horizontal trajectory along the mean wind, the first requirement is obviously satisfied by any system which continuously feeds a new aiming correction on the basis of current winds into the system as the rounds are firedo This has the virtue of closely approximating the true mean wind existing while the rounds are being firedo The second requirement is more difficult to satisfyo To illustrate the possibilities of reducing the dispersion by a simple prediction scheme, we will assume that it takes a minimum of two minutes to make an adjustment in the launchero We further assume that the turbulence is stationary and homogeneous in a horizontal plane and. that we are justified in invoking Taylor s hypotheses If no wind correction is made, the rocket deviation is given by r5 gn = G(x) V(x) dx, (13) o where V (x) is the whole wind and where the trajectory is assumed to be horizontal into the windo A simple prediction of Gn, say 9I, could be given by averaging the wind at launcher height over some time interval T and assuming this to hold, along the entire path 2 minutes later~ 1.4

The University of Michigan * Engineering Research Institute Then G = Sb G(x) Vp(x) dx and (14) -2 (T+2)u V (x) = T T T/ V(t) dt= V() dx vP T (XT-2 T J2u Therefore the mean square deviation of the difference between the predicted impact and the true impact is given by...... 2 ps^ r i p(T+2)U 2 Sb, 1 (n Gp)2 = bG(x) [(x) - T T+2) V(x- )dx- dXJ = [sb G(x) V (x) dx rs t r JsO J -= G(x) G(xa) V*(x) V*(xv) dx dx.() Now *z 1 p(T+2)u V (x) V*(x ) = R (x - x ) + Tu 2 J R(y - y ) dy dy' 2 p(T+2)u 2- - fT2 R(x - y) dy T- T T- sin2 ku eik A 2- sin - uk = Ti(x-x')k si 2 _ u J iu s(k) dk (16) 4T 4 u T Thus T2 j (Gn - p)2 = F r(k) 2 + (0) sin 2 -pT2 2 F(k) () -iUk[2+T/2] sin 2 s(k) (17) I —— r(k) -- (O)e s 2k ) dko157 15

The University of Michigan * Engineering Research Institute We make the further assumption that the imaginary part of r(k) is zeroo This is a fairly good approximation for rough estimation purposes as the sine is small when G(x) is large. The imaginary part of r(k) will grow with increasing k, but the spectrum decreases rapidly with increasing k. Hence Eq. (15) becomes r2(0) sin2 T 2itku (Gn - Gp)= rI(k) 2 +- 2 s (2t)2 ^T2T sin 2x T -k ) - 2 r(k) r'(0) cos 2n uk [2+T/2]- s(k) dk. (8) 2) Here k is cycles per meter if u is in m/sec, and T is in seconds. For T = 1 and 2 sec, and U = 5 m/sec, the results of this correcting scheme are given in Fig. 4, together with the corresponding wind velocity. As the filters are almost zero for long wave lengths, we see that these methods will reduce the bias, i.e., the center of impact can be made to coincide with the target. However, the filters rise extensively compared to the filter for an uncorrected rocket (the squared admittance function) which is given in Fig. 3. If we perform the integration, the square dispersion we get is almost 50% larger for the corrected case. A more meaningful measure is the square root, which makes the dispersion about 20% higher. Whether this is an improvement over no correction at all is dependent on the relative importance of scatter about the mean value as against the mean value not being at the target. I0 z I Minute averages I w 75 \ 0 -3 N Yr I/ U H - W 3a / x 50 -\ 250 Fl \on 2 Minute averages co - -1 H0~~~0 uj o / / q^ \1a 0 O.I 0.2 0.3 k CYCLES/YAW WAVE LENGTH Fig. 4. Filters corresponding to a simple correction scheme. 16

The University of Michigan * Engineering Research Institute It should be recognized that this is a simple and crude schemeo The best predictor would be the "least squares" one, whose characteristics are considerably harder to evaluate. The major trouble with the simple scheme given above is that it weighs all the past observations equally in determining the predictor. If one used a weighted estimate of the mean so that the weights were decreasing for decreasing time, the cosine term in Eqo (18) would be damped for increasing k, and these would reduce the height of the peak of the dispersion filter, hence reducing the dispersiono It is possible to construct a wide variety of weighted means which would be good for different spectrum formso A more involved correction scheme for the horizontal trajectoryhomogeneous turbulence case would attempt to use the observed wind 2 minutes upwind of the launcher as the aiming correction. In a wind speed of 5 m/sec this would mean a separation of 600 mo It is not known how much of an improvement this would give for anemometers located at So, about 50 ft above the ground. It is, however, fairly easy to devise such experiments and to estimate the reduction in scatter using such a simple system. The results would depend on the adequacy of Taylor's hypothesis for horizontal turbulence components 50 ft above the ground, For a nonhorizontal trajectory, the variation of the mean or whole wind with height must be taken into account. The solid lines in Fig. 2 are plots of typical mean wind ratios for day and night conditions as observed at O'Neill, Nebo The horizontal scale for the dashed lines is in terms of s (s assumed to be 50 ft, sb = 2000 ft)o The original wind ratios are in terms of z, and we have used z = s sin F(E = 30~) in the ploto The relative mean wind shear is quite sensitive to E and the graphs should be replotted as E changes A rough correction scheme for this case is to find a value of u* so that u G(s) ds = S (s sin E) G(&) d.s Then the ratio u /u(so) can be used to correct the prediction anemometer at launcher height soo Observed values of wind shear may be incorporated into the system as indicatedo 5o CONCLUSIONS AND RECOMMENDATIONS We should like to emphasize that all statements made previously are based on the assumed influence function given in Figo 2o Our conclusions are that high-frequency turbulence components contribute very little to the dispersion, mostly because there is so little energy in horizontal turbulence components at high frequencies. The major contribution to rocket dispersion comes 17

The University of Michigan * Engineering Research Institute from low-frequency turbulent componentse Aside from attempting to correct for vertical variation of mean wind, the notion of mean wind is difficult to apply to the actual mechanics of a rocket fire system. More meaningful results are obtained if one deals with whole windso (This confirms the views in Refo 6.) In terms of standard meteorological notation, the mean deviation of the rocket from the target can be ascribed to mean wind. The dispersion around the point of mean deviation can be ascribed to turbulent windso Because of the nature of the velocity fluctuations in atmospheric turbulence, the dispersion is due mainly to the Rii components of the correlation tensor, while the contributions of the cross-correlation terms are considerably lesso This is especially true for the influence function with which we have been working. A rocket launching system should ensure that the center of the statistical scatter is on the target, and that the dispersion around the target is minimizedo The simple prediction schemes used here satisfy the first requirement, but the dispersion is increased by 20% over what it would have been if the hourly mean wind had been known in advanceo An optimum technique would be a least-squares technique such as described in Refo 7. This would require a lengthy series of computations to evaluate and it is doubtful that sufficient meteorological information is available to evaluate this technique generally. Application of Taylor's hypothesis to a prediction scheme would improve estimates to the extent that Taylorgs hypothesis is correct for levels from 50 to 150 ft above the groundo Our recommendations for future work in order of expected contribution to reduction in scatter are: 1) Investigation of the adequacy of Taylor's hypothesis for heights of interesto 2) Development of a model according to Taylor's hypothesis that also incorporates the vertical variation of the mean windo 3) Development of more general models incorporating the vertical correlation structure of atmospheric turbulence 4) Experimentation to evaluate these modelso The experimental setup for 1) is simple and would require stringing anemometers upwind of a potential firing site. Experimentation connected with more complicated models would involve more complicated experimental schemes such as those used in the Santa Barbara, Calif., projecto Since towers are fixed on the earth's surface and therefore limit separation distances, we suggest a study of the potentialities of properly instrumented aircraft in conjunction with meteorological tower observations in verifying models under 2) or 53) Some data are available to check 1) and. 2) from Santa Barbara, Brookhaven, and Cornell Aeronautical Laboratory; moreover, the Santa Barbara data can be used partially to construct models under 3)~ __________________________ 18

The University of Michigan * Engineering Research Institute We should like to emphasize that, in our opinion, the major return in reducing dispersion is likely to come from 1) coupled with a least-squares prediction technique, and that the improvements resulting from 2) and/or 3) are likely to be of secondary importance as long as the influence function in Figo 2 is typicalo Acknowledgments This analysis is an outgrowth of the conference held at The University of Michigan in Ann Arbor during July, 1956. Undoubtedly each participant has contributed in some form or other to this report. The following were present at that conference: E Wendell Hewson, R J. Deland, H. E Reinhardt, G. Keller, M, Ao Woodbury, and V. C. Liuo The responsibility for the contents of this section of the report, however, rests solely with the two authors. TASK A: STATISTICAL DESIGNS-ROCKET TRAJECTORY DISPERSION BY ATMOSPHERIC DISTURBANCES by Tse-Sun Chow List of Symbols a Proportional constant G Geostrophic wind. g Acceleration of gravity K Coefficient of eddy diffusivity; Km coefficient of diffusivity for momentum; Kh coefficient of diffusivity for heat k Von Karmanns constant I Mixing length p Pressure Qa Heat flux at earth s surface R Gas constant; also = K/u zo Sn Defined to be (g Zo/u*2T) (Qa/u* Cp ~ ) T Temperature t Time V Mean wind velocity; also, the complex wind velocity u + iv uvw Components of velocity u* Defined to be 4NTo/ xyz Position coordinates z Height above the earth surface;z*, a length proportional to the mean height of surface irregularities cu Angular speed of the rotation of earth Geocentric latitude 19

The University of Michigan * Engineering Research Institute p Density p. Proportional constant aC Angle between G and Vo (see figure) X\ Coefficient of heat conductivity v Kinematic viscosity G Potential temperature ] ~Viscosity ~T Shearing stress; ~ (z + Zo)/Zo 1o STRUCTURE OF WIND IN THE LOWER LAYERS OF THE ATMOSPHERE (a) Variation of wind with height based on theories of momentum transfer o-When the wind varies with the height, there is a transfer of momentum when one eddy travels from one level at one velocity to another level at a different velocity. Early investigators such as Schmidt8 and Taylor9 introduced the coefficient of eddy diffusivity K and established the net rate of gain of momentum per unit volume due to the eddying motion to be where p is the density and V is the average wind velocity at level z. According to Prandtl~ s development of these concepts, the coefficient of eddy diffusivity K may be expressed in terms of the vertical gradient of the wind velocity and a length' Y, which he defines to be the mean vertical distance traversed by an eddy before it mixes with its environment. Because an eddy is a somewhat undefined entity, and the dynamical processes involved in the motion of an individual eddy traveling from one level to another are not clear, it is not possible to derive an expression for the mixing length & based on theoretical considerationso Observations from experiments, however, tend to indicate that i does not depend on the velocity but only on the distance from, and the nature of, the boundary surface. Various assumptions have been made regarding the dependence of & on the distance from, and the nature of, the boundary surfaces Rossby and Montgomery10 assume that over a rough surface and in an adiabatic atmosphere it can be written, = a (z + z*), where z* is a length proportional to the mean height of the surface irregularities and a is a nondimensional constant. It has been observed by experiment, however, that a is not actually constant. In 1946, Frost11 put forward the hypothesis that & = zl-mz*m, where m is a nondimensional constant and lies between 0 and 1, if ~ is to increase with both height above and the roughness of the surface. When this assumption is used in conjunction with the Prandtl's development, it is found that the coefficient of eddy diffusivity K is proportional to zl-m. This expression is similar to that derived by Suttonl2 and Calderl based on different ----------------------- 2 0

The University of Michigan * Engineering Research Institute assumptions. The proportional constant can be evaluated numerically from meteorological measurements Based on the formulation that K is proportional to zl-m, 0 < M < we have found it possible to solve the problem of the variation of wind with height in the atmospheric boundary layer. The geostrophic wind can be taken to be constant in the atmospheric boundary layer and the motion is assumed to be steady and two-dimensional. The boundary conditions are such that the wind should approach the geostrophic wind at a great height and also that the wind direction should coincide with its vertical derivative at the earth's surfaceo This is similar to the approach adopted by Taylor in solving the same problem with K assumed to be constant, a reproduction of which is given by Brunto14 The details are given in Appendix A, and the final solution by Eqo (A-14). Numerical computation has to be carried out using different values of the parameters, and it may be possible to extend the consideration to threedimensional turbulence, using an approach recently suggested by Davies.15 (b) Analysis of the atmospheric surface layer.-For the atmospheric surface layer which constitutes the lower part of the atmospheric boundary layer, the analysis is much more complicated due to the presence of convective currents resulting from the heat exchange at the earth's surface. As a result of these convective currents there are two kinds of turbulence in the atmospheric surface layer: convective turbulence and frictional turbulence produ.ced by the friction at the earth's surface. Besides employing the dynamical equations of motion, the analysis calls for the use of the heat convection equation| By starting with the four basic equations, the continuity equation, the equation of state, the Navier-Stokes equations, and the Fourier equation of heat convection, Businger,16 in a recent study of the influence of the earth's surface on the atmosphere, is able to derive, after making various approximations, two transfer equations relating the vertical gradients of the mean'velocity and the mean potential temperature with the shearing stress at the earthps surface, the flux of heat at the earth's surface, and the coefficient of eddy transfer for heat and. momentum. For an adiabatic atmosphere in which the potential temperature is everywhere the same, the velocity profile is logarithmico This is in good agreement with the experimental work of many investigators,17 For the diabatic atmosphere further assumptions are required, Businger makes the formal distinction between convective turbulence and mechanical frictional turbulence and assumes that the total turbulence is the sum of the two partso With these assumptions certain relations can be derived in terms of several nondimensional parameters showing the variation of the coefficient of eddy diffusivity with height, With these relations the velocity and the temperature profiles in the atmospheric surface layer can be determined. A summary of such an analysis is presented in Appendix Be The experimental results regarding the variation of wind profile for the adiabatic and diabatic atmosphere have been summarized in Refo 17, 2.1

The University of Michigan * Engineering Research Institute Note: The atmospheric boundary layer extends to a height of about 3000 ft and the atmospheric surface layer to about 80 ft above the earth's surface.. 2o GENERAL REMARKS ON TURBULENCE The statistical part of the problem has already been presented by Hewson and Woodbury,7 who derive the mean and second moment of the rocket dispersion in terms of velocity correlation coefficients. To visualize the simplest possible model, it is assumed that the turbulence is homogeneous and isotropic. Extension to the higher moments is immediate| In the theory of turbulence one usually confines the discussion to homogeneous turbulence, which is a random motion whose statistical properties are independent of position in the fluid. To simplify the problem further, one ignores the directional preference of the statistical properties of the turbulent motion. This is the simplest case possible, and the turbulence is said to be isotropic. It should be remembered, however, that such cases are highly idealized, and can only be realized or approximated in an unbounded fluid extending theoretically to infinity in all directions. Thus, Kolmogoroff's theory of local isotropy asserts the existence of such a statistically steady, homogeneous, and isotropic turbulence of an unbounded fluid for a certain range of wave numbers, provided the Reynolds number of the flow is sufficiently higho In practical problems such as this one where in particular the lowlevel wind structure is to be studied, the presence of a rigid boundary will probably make the turbulence nonhomogeneous. It is known in meteorology that the presence of the earthts surface will not affect the motion of the air at sufficient heights (above about 3000 ft) from the ground, i.eo, outside the atmospheric boundary layer6 The motion of the air in the low-level wind layer under consideration will lie well within the atmospheric boundary layer. Thus, because of the presence of the ground, there will be a variation of the mean velocity and the turbulent fluctuating motion with height, and there is a lack of homogeneity. There will also be effects of radiation, heat transfer to and from the soil, etc., which will further complicate the problem. These various aspects have already been discussed in Section 1. 3. RESPONSE OF THE ROCKET In Ref. 18, the basic equations for the motion of the rocket have been derived in terms of the various aerodynamic coefficients, Within the framework of the linearized theory, it is possible to express the rocket dispersion due to a particular wind profile in the form of an integral over the time or space domain in terms of the rocket response functions due to gusts of unit impulse. This is the approach adopted by Hewson and Woodbury7 in presenting the statistical aspect of the problem. 22

The University of Michigan * Engineering Research Institute 4. PROPOSED METHODS OF ATTACK ON THE PROBLEM The pertinent problem is to determine the dispersion of the rocket trajectory after launching due to atmospheric disturbances, If we assume the statistical uniformity of such disturbances in the horizontal plane, the description of atmospheric disturbances can be given by Io the variation of the average wind with height, and II. the variation of the correlation functions with height (second order and higher moments) or their spectra by carrying out a statistical analysis, When the average wind-velocity profile and the vertical variation of the correlation functions of the atmospheric disturbances are known, the dispersion of the rocket trajectory can be calculated (in I, the average value, in II, the higher moments, so that I and II give the complete probabilistic value) by making use of the rocket response function to gusts of unit impulse (Section 3). By proceeding from the studies and correlating the results of Section 1 and 2 (also, Appendix A and B), the average wind-velocity profile in the lower layers of the atmosphere can be determined in terms of certain parameters which can be measured close to the groundo The relative significance of each parameter on the rocket dispersion can then be ascertained after numerical calculations showing the effect of eacho On the other hand, the variation of the correlation functions with height has yet to be investigated. This can be carried out in a manner similar to that described in Appendix Bo If this is not carried out one has to limit the analysis to homogeneous turbulence, which is probably too idealized for the case under consideration (Section 2). Our proposed methods of attack on this problem are as follows: Il Continuation and completion of the investigation of the variation of the average wind with height by proceeding from the studies in Section 1 and 2 (Appendix A and B), IIo Investigation of the vertical variation of the correlation functions with height by conducting an analysis similar to that in Appendix B. IIIo Comparison of the results obtained in Tasks I and II with the experimental data already in existenceo On the basis of this comparison, models embodying essential wind structure in the lower atmosphere pertinent to dynamic wind loading'will be devisedo IVo Selection of a design for a field experiment or experiments to yield maximum information on rocket dispersion, to be followed by extensive 25

The University of Michigan * Engineering Research Institute numerical calculations. The design will include data-reduction reccommendations V. Determination of anemometer types which have response characteristics suitable for field use. 24~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The University of Michigan * Engineering Research Institute APPENDIX A VARIATION OF WIND WITH HEIGHT BASED ON CONSIDERATIONS OF MOMENTUM TRANSFER It has been shown by Taylor that if K denotes the eddy diffusivity, then the rate of eddy transfer of x- and y-momentum per unit volume is a ( a) and z (Kp.) and the equations of motion will be, assuming the vertical component is zero, -2s sin v = -- P +, (A-l) p bx p z \ z 1 6p 1 3 v 2w sin u u =) o (A-2) \.' p dy p dz \^' dv(A-2) Writing V = u + iv, we can combine the above two equations into d2V 1 dK dV (, i2 ) = in0 (Ai + - - - (li) - (V —G = G 0 (A-3) dz2 K dz dz K 14 1-n 1-m where G is the geostrophic wind~ An appropriate form of K is aVi z where V1 is the value of the mean wind speed at some standard height zl, and n,m are constants for a given turbulent state and may be evaluated numerically 1-m from meteorological measurementso We therefore write K = z o By substituting this expression of K into (A-3), we get d2V l-m dV + 2(l+i)2 e s z- (V-G)= (A-4) dz z dz m m+1 We now put V - G = uxl+m and z - = x and the differential equation (A-4) in the new variables becomes d2U 1 U 2 )2 c sin -_ u = O (A-(5) dx2 x dx (m+1)2 l+ x 25

The University of Michigan * Engineering Research Institute which is the standard Bessel differential equation. The solution of Eqo (A-4) is therefore V G =m/2 A J 2i(l+i) sin z 2 + BJ mL 2i(l+i) sinz 2~ ^ m+l - m+l m+ (A-6) assuming that m+ is not an integer. Observations of many investigatorsl have shown that the value of m lies between 0 and 1o As stated by Frost,11 m =1/7 approximately under a condition of thermal equilibrium in wind tunnels and the same value may be taken for the atmosphere. Thus, using m = 1/7 as an illustration, we write (A-6) as T- G = z aA -__ 4 V sin zI + B JZ I( v~G~TT A27K 7(L- ai =8 sin (A-7) To determine A and B, we first impose the condition that V-G remain finite as z + Oo Using the asymptotic formulae of Bessel functions and making use of this condition, we get A e + B 0 (A-8) We use this relation to eliminate B. in (A-7) and obtain the solution of (A-6) as so that at great heights we have 4 4 / -__ G A _ _ 7ss~a),ie e \ / \ (A-10) 5 — ~ —~-.. - (. z7z7 u sin \ 11t' We use the subscript o to denote the values of V, etco,' as z + O o As A e 52 / o zs +C, wO'e hasve 33~i V - G (A )4 2J. (A-ll) 26

The University of Michigan * Engineering Research Institute Also, as z - O, we have 7 1 ~/oV = 4A 28 a7 sin 4~ 7 W(ov4A 28 (T __f __ z _ (A-12) Following Taylor, we assume as z + O, V and 6V/kz are in the same direction, i.e,, the slip is in the direction of the strain, Let the direction G be in the direction of the x-axis, and let a be the angle between G and Vo. Then Fig. A-1 shows that 3n 1 s in s Cx A! = V sin ( a r 4 )8 (A-1) This determines the arbitrary constant A in terms of Voo The solution can now be expressed as V - G = e v3 VO si nc C (8) (Z){l[i w lh/ e 4 zj _3__X~3i 4 717 sin 0.1 7 -e 8 2 e 4 z7 (A-14) > \ t \ > /716Fig. A-lo Relations among the geostrophic wind G, the surface wind VO, and the arbitrary constant A, whereby A is expressed in terms of VOO 27 8 -O' * -------— I —--------— 2 7

The University of Michigan * Engineering Research Institute APPENDIX B ANALYSIS OF THE ATMOSPHERIC SURFACE LAYER The equations that govern the structure of the atmospheric surface layer consist of the following: (a) the equation of continuity at + y (pu) + y (v) + (pw) = 0 (B-l) (b) the equation of state p = pRT, (B-2) (c) the Navier-Stokes equations au au au au - + u -- + v - + - 1 ap u S u u a v - -V + 6y2 + - + + 1 I (B-3) X p ax )ax2 ad arm 3 a x Vy plus two similar equations in the v,w components, and (d) the Fourier equation a((pTa)(p ) (ppT) ap) / T aT + T (B4) t + u + a + W ip ayZ 2 (B-4) 6t + x;y z y z2 Assuming that the velocity components, pressure, density, etc., can be split into a fluctuating part (indicated by a prime symbol) superimposed on the mean value (indicated by an upper bar), we can split each of the above equations, one for the mean state and the other for the fluctuating part. In many cases the equation for the fluctuating part can be left out of consideration. It will be assumed that the mean state motion is steady, and that the mass flow takes place in the x direction, with the z axis perpendicular to the horizontal plane, so that pv = pw = 0 and pu t Oo The equation of continuity is seen to be reduced to, after integration, pu + p'ut = f(z) o (B-5) 28

The University of Michigan * Engineering Research Institute Calder13 has shown that p / << p /p and T/T, so that the equation of state becomes ps/_p = -Tv/T. In the atmospheric surface layer the difference between the temperature T and the potential temperature G, which is defined to be the temperature the air would attain if brought adiabatically to a standard pressure, is small, so that the equation of state becomes P - -- e (B-6) P Q To simplify the Navier-Stokes equations, Businger assumed that in the atmospheric surface layer, p-w" remains practically constant and p u"w' > w7 p-u1, puw, wI p P pow2. The Navier-Stokes equations are then reduced to 1 p ~x +f / uVw^ =- 1 ~ +v —v (s, (B-7) p ax az (B-7) pw2 = p(O) - (z) - g d z, (B-8) O 0 after integration with respect to z and assuming that ip/6x is constant, where the subscript o indicates the value at z = 0, Now v(u/kz)o is the shearing stress at the earth's surface and is equal to To/p' In the atmospheric surface layer, p x is small, and v(a/az) is also small compared with u-w- as soon as there is any turbulence. Thus for a first approximation we have VUWV = 12 o (B-9) Equation (B-9) implies that the vertical turbulent motion causes a pressure rise with regard to the static pressure in the free flowo Finally, the Fourier equation is reduced, after integration, to pwT = x_ a ( (B-10) CP 6z CP \z/0 Here k(T/kz)o = Qa is the heat flux at the surface of the earth. Also, (x/Cp)(7T/6z) is the heat flux by pure conduction and is therefore negligible at the earth's surface. Furthermore, pwT w p w'TT, so that we have Qa w T- = --- - (B-ll) Cp P 29

The University of Michigan * Engineering Research Institute Thus from (B-6) we have w -Ig ^= (B-12) The velocity profile in the atmospheric surface layer can not be determined. A. THE ADIABATIC ATMOSPHERE -/ /1 By means of the assumption of von Karman, which has also been proved by Hamel,20 ulwT = -k2 - (B-13) k being von Karman's constant, we write (B-9) as k (5 = ) (B-34) and upon integration we get au 1 (.-15 Let C = kzo/u* where u* = o/ then further integration gives u 1 logZ + ^o (B-16+ klog --, (B-16) u* kk zo which is a logarithmic profile. B. THE DIABATIC ATMOSPHERE When the atmosphere is diabatic, von Karman's assumption (B-1) is no longer valid and we cannot solve (B-9). Instead, the two equations (B-9) and (B-12) have to be solved simultaneously. Following Taylor,1 we write, in place of (B-9) and (B-12), 30

The University of Michigan * Engineering Research Institute Km = u*2 (B-17) Qa Kh = -cp (B-18) Such a formulation is similar to that for the flux of momentum and heat in laminar flow. The unknowns are now Km, Kh, u, and Go From the kinetic theory of gases, Held22 pointed out that the two coefficients Kmi Kh are equal if the Prandtl number is 1 for an infinite number of degrees of freedom of the molecules. By writing Km = Kh = K we have three unknowns K, u5, G but only two equations so that further assumptions have to be madeo In an adiabatic atmosphere the turbulence is entirely due to friction at the earth's surface. In the diabatic atmosphere the turbulence is due to mechanical friction and convective currentso We make a formal distinction between these two kinds of turbulence and assume that the total turbulence is the sum of frictional turbulence and convective turbulence. Based on this statement, Busingerl6 formulates the following relation K2 Kf2 -g 3- - - (B-19) from a consideration of the acceleration exercised on an eddy and the relation ao K2 fa -K g - (B-20) from a consideration of energy per unit mass. In these equations ~, If are the mixing lengths of total turbulence and frictional turbulence, respectivelyo Furthermore, Businger formulates the relation Kf = ( (B-21) Now by combining (B-15) and (B-17) we get Kf = k u* (z + Zo) o (B-22) The solution of (B-17), (B-18), (B-19), and (B-21) gives R = k252 Sn + 1 k+ [1 + (1 + 4k Sn)1/] (B-23) 31

The University of Michigan * Engineering Research Institute and the solution of (B-17), (B-18), (B-20), and (B-21) gives R = kS + Sn k22, (B-24) where R = K/u*zo, a dimensionless coefficient of eddy transfer, k is. von Karmanvs constant,' = (z + zo)/o, the dimensionless height, and Zog Qa Sn = U*2 T u Cp p Thus, from the relations of R and, Sn it is possible to find the velocity and temperature profileso The practical significance of these results is that, if the profiles have once been determined for one given set of conditions involving a given value of Sn, then they are also determined for any other set of conditions which yield the same value of S, 52

The University of Michigan * Engineering Research Institute TASK A: STATISTICAL DESIGNS-EVALUATION OF WIND DATA The preliminary evaluation of the wind data obtained by Professor Ro Ho Sherlock and his associates was completed. It was subsequently decided not to use these data in the present research, so that the evaluation will not be presented here. OVERALL CONCLUSIONS Briefly stated, the overall conclusions are as follows: 1, Low-frequency turbulence makes the major contribution to rocket dispersiono 2o The concept of the whole wind is more meaningful and easier to apply in this problem than that of the mean windo 35 A least-squares technique of prediction would be valuable but atmospheric information adequate to evaluate it may not be available yeto 4. Application of Taylorts hypothesis should improve estimates obtained by a prediction method. RECOMMENDATIONS Progress in this area is seriously hampered by the lack of information on the degree of applicability of Taylor-s hypothesis. This hypothesis suggests that the structure of turbulence passing an area normal to the mean wind specifies the structure of the turbulence downwind from that area. There is obviously some degree of validity to this assumption, but precise information of the type needed for the present problem is lacking. RECOMMENDATION lo It is therefore recommended that field studies be conducted to determine the range of validity of Taylor s hypothesis for the large-scale turbulence which is of primary importance in the present problemo 33

The University of Michigan * Engineering Research Institute It is proposed that several TV and FM antenna towers closely grouped in an area of flat horizontal terrain on the northwest outskirts of Detroit be instrumented with anemometers and bivanes at several levels. The tower of WJBK-TV is 1050 ft high and has platforms at 300, 600, and 870 ft, which are reached by a small elevator. Temperature lapse-rate measurements are already being made at these levels on a routine basis. An almost identical tower, with platforms at the same heights, and owned by WWJ-TV, lies about one mile to the north-northeast of WJBK-TV. A shorter tower, 468 ft high, and operated by WLDM-FM, lies about one mile to the northeast of WJBK-TV; only the top 80 ft of this tower is energized, so that the portion up to 388 ft could be instrumented. Tentative approval for the instrumenting of each tower has been obtainedo The distance between the two high towers is sufficiently great to permit aliasing in spectra in the significant ranges unless intermediate observations are obtained. It is proposed to fly one or more kytoons between the two high towers when winds are NNE and SSWo A number of zero-lift balloons would be attached at intervals to the kytoon cable by elastic cords to form one or more arrays of gustometers of the type described by Hewson.2e After the validity of Taylor~s hypothesis had been assessed by use of the tower instrumentation in conjunction with arrays of balloon gustometers, and the characteristics of the latter determined more fullty, then the gustometers supported by kytoons would be used in rough terrain of various types and in various climatic regimes to determine how Taylores hypothesis stands up under topographic and climatic conditions very different from those near Detroit. A final step would be to devise, calibrate, and test simplified field apparatus that would provide more readily the basic information supplied by the balloon gustometers. RECOMMENDATION 2 It is further recommendedtly that concuretly ith the above program the theoretical studies described in this report be continued along the lines which follow directly from the progress and findings to date, ---------------------- 4 —----------

The University of Michigan * Engineering Research Institute REFERENCES 1o Hunter, Mo W., Shef, Ao, and Black, Do V., "Some Aerodynamic Techniques in Design of Fin-Stabilized Free-Flight Missiles for Minimum Dispersion," Jo of Aero, Scio, 23 (1956, 571-5770 2o Panofsky, Ho, Statistical Properties of the Vertical Flux and Kinetic Energy at 100 Meters, Scientific Report No. 2, AF 19(604)l166, The Penno State Univ., Univ. Park, Pao, July, 19535 3o Van der Haven, I,, "Power Spectrum of Horizontal Wind Speed in the Frequency Range from 0~0007 to 900 Cycles per Hour," J. Met,, 14 (April, 1957, 160-164o 4. "Great Plains Turbulence Field Program," to be published in 1957o 5. Panofsky, Ho, and Van der Haven, Io, Structure of Small Scale and Middle Scale Turbulence at Brookhaven, AFCRC-TN-56-254, The Penn. State Univ., Univ. Park, Pao. 1955. 60 Shaffer, P. A,, Jr, Wind Structure Study for Rockets, Final Report, Contract DA-36-039-SC-52599, North American Instruments, Santa Barbara, Calif. 1956o 7. Hewson, Eo W., and Woodbury, Mo A., Research Study Pertaining to Low-Level Wind Structure, The Univ. of Micho, Eng. Reso Inst. Report No, 2377-15-P, Ann Arbor, October, 1956. 8. Schmidt, Wo "'Der Massenaustausch bei der ungeordneten Stromung in freier Luft und seine Folgen," Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Abo IIa, 126 (1917), 757-804. 9. Taylor, G. I., "Eddy Motion in the Atmosphere," Phil o Trans, A, 21.5 (1915), 1-26o 10. Montgomery, R. Bo, "Vertical Eddy Flux of Heat in the Atmosphere," J. Met,, 5 (1948), 265-274. 11o Frost, Ro, "Turbulence and Diffusion in the Lower Atmosphere," Proc. Roy. Soc., A, 186 (1946), 20-355 12o Sutton, 0o G,'"Wind Structure and Evaporation in a Turbulent Atmosphere," Proc. Royo Soco, A, 1.46 (1934), 701-722. 55

The University of Michigan * Engineering Research Institute REFERENCES (Concluded) 135 Calder, K. L., "Eddy Diffusion and Evaporation in Flow over Aerodynamically Smooth and Rough Surfaces: A Treatment Based on Laboratory Laws of Turbulent Flow with Special Reference to Conditions in the Lower Atmosphere," Quart, Journ. Mech. and Appl. Math., 2 (1949), 153-176. 14, Brunt, Do, Physical and Dynamical Meteorology, Cambridge Univ. Press, N. Y. 1944. 15o Davies, Do Ro, "Three Dimensional Turbulence and Evaporation in the Lower Atmosphere, I, II," Quart. Journ. of Mech. and Applo Matho, 3 (1950), 5173. 16, Businger, J. Ao, Some Aspects of the Influence of the Earth's Surface on the Atmosphere, Koninklijk Nederland Meteorologisch Institut, is Gravenhage, 1954. 17. Ellison, To H., "Atmospheric Turbulence," Surveys in Mechanics, eds. Batchelor, G. K.,, and Davies, Ro Mo, Cambridge Univo Press, N.Y., 1956. 18o Rosser, Jo B., Newton, Ro R., and Gross, Go Lo, Mathematical Theory of Rocket Flight, McGraw-Hill Book Company, N. Y,, 19470 19. Von Karman, Tho., Mechanische Ahnlichkeit -nd Turbulenz," Gesell. Wiss. Gbttingen, Matho-Phys. Klo Nachr., No. 8, 1930o 20. Hamel, Go, "Streifenmethode und Ahnlichkeitsbetrachtungen zur turbulenten Bewegung " Akad. Wisso Berlin, Klo Mth. nat urwiss., Abhand. No. 8 (1943), 1-25, 21. Taylor, G. I., "Effect of Variation in Density on the Stability of Superposed Streams of Fluid," Proc. Roy. Soc, A, 135 (1931), 499-5235 22. Held, E. E. M., Standaarddictaat Gelijkvormigheidsleer, Utrecht, 1947. 23. Hewson, Eo W., "A Method for Determining Atmospheric Turbulent Flow Parameters Throughout the Vertical Extent of the Boundary Layer," Geophysical Research Papers, No 19 (1952), 165-170. ----------- 56. —------—. —--------