The effect of compressible flowon the perceived temperature fluctuations measured by a moving sensor
George Y. Jumper*, Robert R. Beland?, John R. Roadcap*, and Owen R. Cote'§Air Force Research LaboratoryHanscom AFB, MA 01731-3010
ABSTRACT
Equations are developed for the effect of velocity fluctuations on the perceived temperature fluctuationsdetected by a moving temperature sensor. The impact of these fluctuations on perceived fluctuations ofindex of refraction are computed. The magnitude of this effect is estimated using known ranges ofatmospheric fluctuations in temperature and velocity. Using a method to estimate velocity fluctuations fromtemperature fluctuations and concurrently measured atmospheric data, balloon derived fluctuation data isused to estimate what would be perceived by a faster moving sensor. We conclude that normally observedfluctuations can have a significant impact on the perceived optical properties.
NOMENCLATUREa = Constantb = ConstantBr(x) = Spatial Covariance (units of r squared)Cx
2 = Structure constant of the quantity x (units arethe units of x^m273)
CP = Specific heat at constant pressure (J/kg/K)Dx(r) = Structure function of quantity x (units are the
units of x squared)d = Distance (m)i,j,k = with caret, unit vectorsK = Spatial wave number (m"1)L0 = Outer scale of turbulence (m)n = Index of refraction (n. d.)P = Pressure (mbars)r = Recovery factor (n. d.)(see Eq. 2)SX(K) = One-sided, one-dimensional spatial power
spectrum of xT = Temperature (K)U = True Air Speed Vector (m/s)u,v,w = components of wind velocity (m/s)V = Velocity (m/s)x = Distance (m)e = Turbulent kinetic energy dissipation rate (W kg"
1 commonly shown as mV3)^ = Dissipation rate of temperature variance(K2s~')0 = Potential Temperature (K)
Subscripts and Subscripts:GS = Ground speedn = Index of refractionr = RecoveryT = TemperatureU or u = Velocityw = Wind1,2 = Locations 1 or 2Bar = Time averaged or mean component' = Fluctuating component
INTRODUCTION
Optical turbulence in the atmosphere is most commonlyknown as the phenomena that causes stars to twinkle.The performance of optical systems is degraded by thephenomena. For instance, in the presence of opticalturbulence, a projected beam appears to wander orscintillate, effectively reducing the average power thatarrives at a spot. Optical turbulence is caused by thepresence of adjacent parcels of air, which are at slightlydifferent index of refraction moving about in the field ofview of the instrument.
The index of refraction, n, in the atmosphere can bewritten as
(D
where P is the partial pressure of either the dry air (d) orthe water vapor (w) and the a coefficients are thewavelength dependent coefficients for either the dry airor the water vapor. Fluctuations in the index ofrefraction are caused by fluctuations of either thedensity of the air or the concentration of water vapor inthe air or both.2 For radiation in the visible and near DR.,the water vapor term is rarely significant, except,perhaps, right over a body of water. Fluctuations inindex of refraction can be measured optically over apath with instruments such as a scintillometer. Radarsdetect fluctuations in index of refraction in the viewvolume by Bragg Scatter3. At microwave frequencies,*Sr. Aerospace Engineer, Battle Space Environment Division, Assoc.Fellow AIAAtSr. Research Physicist, Battle Space Environment Division•^Research Physicist, Battle Space Environment Division^Research Physicist, Battle Space Environment Division
1
This paper is declared a work of the U.S. Government and is not subject to copyright protection in theUnited States.
the index of refraction is much more sensitive to watervapor and the results must be corrected when applied tooptical and near IR systems. While both pressure andtemperature appear in Equation 1, fluctuations inpressure dissipate rapidly through acoustic processes.Therefore, in situ measurement of optical turbulence is,in practice, reduced to measurement of fluctuations intemperature.In situ temperature fluctuations have been measured byfine scale wire temperature sensor carried aloft byballoons and aircraft. These platforms complementeach other to provide a more complete picture of theatmosphere. Each balloon-borne sensor passes throughonly a small portion of the atmosphere at any altitude,while a plane can probe the horizontal dimension withease. Furthermore, the balloon is passing through thelayers at about 5m/s, so there are statistical samplingquestions to be addressed. The time that an airborneplatform can sample is only limited by the aircraft speed(typically from lOOm/s to over 200m/s) and thehorizontal dimension of a homogeneous portion of theatmosphere. While the aircraft is limited to a maximumaltitude of around 14km, the balloon normally ascendsto around 30km, which is where the probes beginloosing sensitivity due to heat transfer considerations.While aircraft are typically restricted to certain discretealtitudes, a balloon provides continuous profiles ofturbulence with height. The probes currently used onballoons are sensitive to sunlight and can only be usedin the dark; aircraft probes have shown no suchsensitivity. Furthermore, since balloon payloads are notrecovered, economics influenced the current practice ofaveraging the temperature fluctuations and transmittingrms averages at normal meteorological radiosonde datarate of about O.SHz. Temperature fluctuation data fromthe aircraft are collected and stored at high rates,allowing a more detailed analysis. Since a balloon-borne payload is attached to a meteorologicalradiosonde, concurrent wind, temperature, and humidityinformation is also available.
While both aircraft and balloon-borne temperaturesensors are often fine wire resistance probes, theapproaches to the determination of optical turbulencefrom these instruments are distinctly different. Balloon-borne measurements are made at a vertical ascent rateof approximately 5m/s, while traveling horizontally atessentially the horizontal wind velocity4. The balloonuses two wire probes spaced one meter apart in thehorizontal plane to determine a Im-structure function,from which the temperature structure constant, Cj?, is
deduced, using the Kolmogorov hypothesis5. Thetemperature difference is sensed with wires that are3.45jim in diameter, which results in a time constant ofless than 1ms. The temperature difference is averagedwith an onboard rms integrated circuit. The output ofthe rms chip is transmitted back to the ground station at1.2s intervals.
Otten and other aerothermal researchers at Kirtland AirForce Base6'7'8 have used a 5(im diameter wire probesmounted on a sting at a forward location on the aircraftwhere the local pressure is approximately the freestream pressure. The probe signal is divided into dcand ac components and recorded at 12kHz. After theflight, turbulence levels are determined by analysis ofthe spectrum of short segments of the flight. Airbornemeasurements are made at nearly constant altitude overpaths of tens to hundreds of kilometers.
Since the desired state variable is the variation in freestream air temperature, one very importantconsideration is the ability of any temperature sensor tomeasure fluid temperature while moving through thefluid. This is a classic heat transfer problem faced byanyone trying to measure temperature in a movingstream. To achieve maximum sensitivity totemperature, the wire probes are used in the "cold wire"mode or, more precisely, the constant current, lowoverheat mode. The wires used on the aircraft and onthe balloons are only slightly heated by the currentflowing through the wire. Consequently, the wire seeksan equilibrium temperature determined by conservationof energy. At low velocity, the probe temperature isonly slightly higher than the ambient air temperature,also called the static temperature. At higher velocities,the probe, or in fact any temperature sensor, responds tothe combination of the thermal energy and theorganized kinetic energy of the flow relative to themoving sensor.
This problem is usually analyzed in terms of theequilibrium temperature that an adiabatic surface willachieve. This "adiabatic wall temperature", or"recovery temperature"9 is defined by the equation:
T=T + r-2c_
(2)
Where Tr is the recovery temperature, T is the ambientair temperature, U is the magnitude of the air velocityrelative to the probe, r is the recovery factor, and Cp isthe heat capacity at constant pressure of the air. Therecovery factor is equal to unity at the stagnation point,
American Institute of Aeronautics and Astronautics
where all of the kinetic energy is converted to heat, andthe recovery temperature becomes the "stagnationtemperature" or the "total temperature." As one movesaway from the stagnation point, the recovery factor getssmaller, it can become zero or even a slight negativenumber on the leeward side of a body. Thisphenomenon is sometimes known as the compressibilityeffect and is sometimes referred to as dynamic heating.
The wire on a sensing probe is very long compared toits diameter, so conduction along the wire length isslight except for near the ends. The wire does conductheat through the cross section, so one can define anaverage recover factor for a given cross sectionalgeometry. Measured values of the average value of rfor a thin cylindrical wire depend on Mach number.10
The values tend to be greater than or equal to therecovery factor for a flat plate, which is the square rootof the Prandtl Number (for air, Pr = 0.7, so r = 0.84).As we will show below, when the flow is fast enough,fluctuations of either ambient temperature or velocitycan cause a fluctuation in recovery temperature. Ifvelocity fluctuations are significant, then the velocityfluctuation contribution must be removed to determinethe actual air temperature fluctuation so thatfluctuations in index of refraction, the "opticalturbulence," can be computed. The methods forobtaining the fluctuations in velocity in a compressibleflow were derived by Morkovin1' and have continued tobe refined (for a recent example, see Nagabushana12).Typically at least two wires are used, with different wiretemperature "overheat" conditions, and, typically, twoor more equations must be solved simultaneously todetermine the fluctuations in air temperature andvelocity. Clearly, the correction for velocity is notrequired for low Mach numbers, but, at some point, it isnecessary. The addition of another wire at a differentoverheat condition, the additional calibration and dataacquisition requirements, and the coupled solution ofequations for very large data sets, make the decision tocorrect for velocity fluctuation measurements a seriousmatter.This paper examines the consequences of ignoring thevelocity fluctuations on the apparent optical turbulenceas a function of increasing velocity, and estimates theresulting distortion to the optical turbulence profile andcalculated optical performance. We hope to approachthe topic in a way that is comprehensible to atmosphericscientists interested in optical turbulence as well asaeronautical engineers.
THEORY
COORDINATES
The U in Equation 2 is the velocity of the air relative tothe temperature-sensing instrument. Refer to Figure 1for the vectors used in the analysis. Consider theaircraft moving with a velocity VG$ relative to theground. The plane is moving through a wind field ofvelocity, Vw, characterized by the mean and fluctuatingcomponents shown below:
Vw = (u \v')k (3)
where u denotes velocity from West to East, v fromSouth to North, and w upward, all defined relative tothe ground. For convenience, the wind vector is shownas the sum of mean and fluctuating parts:
V =Vrw ' +V~ 'w (4)
The true air speed velocity vector, U, is determined bysubtracting the aircraft velocity, assumed constant, fromthe wind velocity.
(5)
Performing the indicated operations we have:
U = V -u "w +V' '
Next we define the mean true air speed vector, U, asthe mean wind velocity minus the ground speed vector:
U = V — V (7)W GS *
The result is that the true air speed vector is the meantrue air speed vector, t7, plus the fluctuatingcomponents of the wind velocity vector. In the lowerpart of Figure 1, the fluctuating components have beenmoved over to the end of the U vector. Theinstantaneous true air speed (not shown) is a vectorfrom the tail of the U vector to the head of thefluctuating part of the wind vector.VELOCITY IN THE RECOVERY TEMPERATUREEQUATION:
The recovery temperature equation, Equation 2, is anoutgrowth of the conservation of energy equation. Theinstantaneous velocity, U, in that equation is themagnitude of the instantaneous velocity vector, so thesquare of that velocity is just the instantaneous velocityvector, U, doted with itself. Therefore:
American Institute of Aeronautics and Astronautics
where U' is the component of the fluctuating windvelocity in the mean true air speed direction, defined bythe equation:
U' = (8)
where time averages for this equation are denoted bythe symbols ( /.
For a stationary, homogeneous random process, themeans are a constant which we can write as (r) and thecovariance depends only on the magnitude of x = (x2-Xj). To simplify the notation, we shall use the subscript1 and 2 to denote the location xi and x2.
U
FLUCTUATIONS IN RECOVERY TEMPERATURE: Now substituting Equations 10 and 11 yields:
The primary concern of the majority of thecompressible flow anemometry literature is the Br(x} =determination of velocity fluctuation in the face of thecontaminating temperature fluctuations. The emphasishere is just the opposite, the contamination of thetemperature fluctuation measurement by velocityfluctuations.
(14)
, U2
Assuming that the measured temperature is essentiallythe recovery temperature, we proceed to compute themean of the recovery temperature. Assume that we arein a homogeneous region where the aforementionedmean values of the velocities are steady and the mean ofthe fluctuating parts is zero. Further, air temperature isexpressed as the sum of a mean and fluctuating part:
T = T + T' (9)Inserting the definitions into Equation 2, we have
2c(10)
Taking the mean of the above equation, we get the meanrecovery temperature:
(11)
Subtracting Equation 11 from 10, we obtain thefluctuating part of the recovery temperature:
,' , r t / £ / ' , r .(V'2-V'2} (12)\ w v l
Next we calculate the spatial covariance of themeasured temperature, B^x), which is defined as:
= (Tr(Xl)Tr(X2))-(Tr(Xl))(Tr(X2))(13)
4c; 4C.(15)
We can identify these various terms with covariancesand cross-covariances, namely:
Br(x) = BT(x) + -
17rr2
r2U'2C2
(16)This is the final, formal expression that also explicitlyincludes and specifies the higher order terms andmoments. Note that in SI units, Cp is on the order of103 for air, so terms with U/Cp can become significantas U becomes large. Terms without velocity probablyremain quite small, except for the temperature term.Note also that the dot product of the mean true air speedvector and the fluctuating wind velocity vector reversesign if the true air speed vector reverses sign. Thishappens when an aircraft flies back over the sameground track. Equation 16 can immediately be writtenin terms of a structure function:
-U'T' + r' U -U' + H.O.T.
(17)
American Institute of Aeronautics and Astronautics
Using the Fourier transform relation betweencovariances and power spectra, this expression can beimmediately written in terms of spectra of recoverytemperature (STr), temperature (5r), and velocity (Su),and the cospectra of velocity and temperature (SUT) asfollows:
2U V
(18)
where K is the spatial wave number (rn'). Note that wehave dropped the primes in the spectral expression,since it is clear that we are looking at fluctuations over arange of wave numbers.
The next task at hand is to determine the relativemagnitudes of the terms. At this point, it is worthwhileto discuss the origin of temperature fluctuations*. Theessence of turbulence is fluctuations in flow velocity.In the presence of temperature gradients, thefluctuations in flow velocity drive fluctuations in fluidtemperature. In the atmosphere, it is the presence ofgradients in potential temperature which must be usedto adjust for the adiabatic temperature changes whichoccur with changes in altitude. It is very possible tohave velocity turbulence with very little fluctuation intemperature. In fact, turbulence itself tends to mix fluidparcels of different potential temperature, and,eventually, reduce the temperature fluctuations whilevelocity fluctuation persist at high levels.
For purposes of comparison, we will focus onturbulence which has matured to an equilibrium state.This ignores the times when turbulence is just starting,which may not be a large portion of the time, and whenit is dying out, at which times the turbulence levels aretoo small to matter. At equilibrium, the Kolmogorovhypothesis is that turbulent kinetic energy is ducteddown from the largest eddies, called the outer scale ofthe turbulence, to the smallest eddies, the inner scale, byinertial processes only. At the inner scale, kineticenergy is converted to thermal energy by viscousdissipation. When the velocity spectrum is within theinertial range (between the outer and inner scale), aresult of the Kolmogorov hypothesis is that the one
* See any fluid mechanics text which treats turbulence. Forinstance, Boundary-Layer Theory, Hermann Schlichting,McGraw-Hill Book Company, "Fundamentals of turbulentflow" chapter.
way, one dimensional power spectral density of velocityhas the following functional relationship13:
c CR'") = f)25/"2 ff""5'3 (19)
Where Cv2 is the spectral constant of the velocity
fluctuations. Obhukov showed that spectra forconservative passive scalars such as temperature shouldfollow the same relationship14, i.e.:
ST(K) = 0.25 C2T K'5' (20)
The cross spectrum may not enjoy the same functionalrelationship with K, so it will be left as an unspecifiedfunction of wave length:
5TT2
STi = (0.25)Cf K2 rr-5/3
c.(21)
Now, combining the temperature and velocity structureconstant terms, the total temperature spectra is:
The terms in the bracket show that the velocity structureconstant, which is never negative, adds to thetemperature structure constant throughout the inertialrange in a way that is indistinguishable fromtemperature fluctuations if only the temperature sensoris used.
Next, we consider the term which contains the varianceof the UT' product in Equation 17 or the cospectra ofvelocity and temperature fluctuations in Equation 22.As mentioned above, this term can be either positive ornegative. Some observations can be made about thecontribution of this term: 1) If we are examining acollection of many data runs flown in oppositedirections, the contribution of this term should notaffect the mean of the collected data, even though it willincrease the variance of the data set. 2) Whilecorrelation of velocity and temperature fluctuations isexpected when caused by gravity waves at large scales,there may be little to no correlation of the two inisotropic turbulence. The term contains the horizontalturbulent heat flux which would be significant whenthere are strong temperature gradients in the positive ornegative U' direction (which is usually in the horizontalplane). Horizontal temperature gradients are usuallyvery weak compared to vertical gradients. 3) Finally,by the Schwartz inequality, the magnitude of the term is
American Institute of Aeronautics and Astronautics
bounded by the magnitude of the product of theindividual components U" and T. Therefore in times oflow temperature fluctuation, the cross term must alsohave a low value. So, for these reasons, we concludethat this term is not important when looking atdifferences in the mean of collections of data fromdifferent directions, nor can it be important when thetemperature variance is small.
RELATIVE MAGNITUDES
Setting aside the cross term, the problem now is todetermine the relative magnitudes of C-2 and C2.Temperature Structure Constant. Consider first thetemperature structure constant. The procedure used toconvert balloon-borne instrument data to thetemperature structure constant assumes the Kolmogorovresult15 that:
n { A\ — r^A™ (23)
where d is the distance between the points ofmeasurement and Dj(d) is the structure function16. Atthe relatively low ascent velocity of 5m/s, thetemperatures measured by the balloon-borne sensors donot suffer from compressibility effects. The balloonacquires temperature differences at a 1m separation;therefore the magnitude of the 1m structure function,DT(lm), is the magnitude of the structure constant. Theamplitude of balloon acquired temperature structureconstant (CT
2) typically varies from the thermosondenoise floor, IxlO"6 K2/mm to a maximum of around0.001 K2/mM, with values above lxlO-5K2/mM deemedsignificant. Conversion of the temperature structureconstant to the refractive index constant, Cn
2, dependson local pressure and temperature and the wavelength ofthe radiation which is being propagated. For radiationnear the visible spectrum, it is customary to use theformula17:
C2 = (79.XKT6 PIT2}2 C2 (24)
Velocity structure constant. Velocity fluctuations havebeen measured over the years and are often expressed interms of the turbulent kinetic energy dissipation rate, E,which, by virtue of the Kolmogorov hypothesis, is18:
,2/3 (25)
Where a is a dimensionless constant which wasdetermined to be approximately 2 to 3 frommeasurements in the free atmosphere. Some earlyaircraft measurements19 in quiet air resulted in 8 valuesof about IxlO^nWs3. The transition from negligible
turbulence to low turbulence is roughly 3xlO"3m2/s3.We have estimated e from balloon data, using arelationship of Cn
2 and the Brunt Vaisala frequency20, inthe range from 10"5 to lO'W/s3 at aircraft samplingaltitudes. Equation 25, which relates e to the velocityvariance or structure constant, is the result of anassumption of small scale isotropy. That is, thedirection of the measurement is not important.Therefore, the value of e is assumed to apply in anydirection including the true air speed direction.
RESULTS
PERCEIVED OPTICAL TURBULENCE
In Figure 2, we compute the contribution of the Qj2term which would be added to CT
2 to result in the CT2
that would be perceived by a a sensor traveling atlOOm/s and 200m/s. The velocity fluctuations areexpressed in terms of a range of dissipation rates, e.Also shown in Figure 2 is the perceived Cn
2 that themoving sensor would see above the actual Q,2 assumingstandard conditions at 12km, since at 12km theconversion factor from CT
2 to Cn2 is a simple power of
10"13. The figure shows that the additive factor canapproach significant levels, especially for the highervalue of fluid velocity. The contribution spans ordersof magnitude, ranging from negligible to significant.Clearly the impact on the final result depends on theactual value of the fluctuations of temperature in theatmosphere.
In Figure 3, we show the effect of the velocitycontribution over a range of e values, combined withtemperature fluctuations, CT
2, to obtain the value of CT2
perceived by a probe moving at 200m/s. The mostdramatic effect occurs when the temperaturefluctuations are low, but velocity fluctuations are strong.For instance if the actual level of Cn
2 is 10"1S, themoving probe would perceive values an order ofmagnitude higher if e exceeded IxlO^n^/s3. If thatsame e level occurs when the actual Q2 is an order ofmagnitude higher, 10"17, then the effect of that 6 value isthat the perceived value of the moving probe is doublethe actual value.
Estimating Velocity Fluctuations
It is hard to make a realistic estimate of the impact ofthe effect, with the hypothetical comparisons shown inFigure 3. In an attempt to show a more realisticestimate of the effect, we have adapted a commonmethod of obtaining e from radar data20, to obtain anestimated profile of e from balloon data. Using the
American Institute of Aeronautics and Astronautics
Kolmogorov hypothesis and assuming that the verticalcomponent of potential temperature is the primarycontribution to the total gradient of potentialtemperature, a common relationship of the ratio of Qjto CT
2 is the following:
Cl _ £ _ g (26)
*T7*;The new symbol, %, is the dissipation rate oftemperature variance, analogous to 8 for velocityvariance. Applying Equation 25, we arrive at theformula:
£ =
-.3/2
(27)
where 0 is the potential temperature, g is thegravitational constant, a and b are atmosphericparameters where the ab product is approximately unityfor a Richardson Number greater than 1A, which is mostof the flight. If the Richardson Number drops below 1A,the product gets smaller, so e is under-estimated if avalue of unity is used for the entire flight. A problemfor radar people is insuring that they have reasonabletemperature and temperature gradient data, which isusually obtained from a nearby balloon flight. Weenjoy a higher confidence in our calculation since ourtemperature data is gathered concurrently with thetemperature fluctuation information.
To illustrate the phenomena, we present the results ofcalculations using data from 5 balloon flights over theWhite Sands Missile Range flown in September of1997. These flights are used since they had good datacoverage throughout most of the flight. For each flight,we show the ratio of Qj2 to CT
2 ( or e/%) for the entirerange of altitude. Also shown is our estimate of theperceived Cn
2 along with the balloon derived Cn2. The
solid line on the plot on the right is Cn2 determined by
the balloon, the dotted line is our estimate of theperceived Cn
2 for a probe traveling at 200m/s. Thelatter results are shown from 10km to 15km, bracketingthe typical altitudes of airborne sampling. Finally, wesuperimpose
The actual amount of velocity amplification is quitedependent on the altitude. A mean amplification overthe region of 10 to 15km is less than 2, but some localspikes are observed. As expected, the largestamplifications are in the regions where C,,2 is low. Insome regions of low Q2, the amplification is a low
value. These must be interpreted as regions of very lowvelocity turbulence, i.e., laminar regions. In others, theperception is quite a bit higher. These are regions werethe vertical gradient of potential temperature is quitelow, often bounded by peaks of high optical turbulence.
Also shown in Figure 4 is the ratio of perceived toactual Cn
2 is shown for the entire altitude regime. Inthese charts it is quite evident that amplification can bequite large at lower altitudes, but in the stratosphere,above the 17km tropopause, the ratio approaches unity,the dotted line. Mathematically, this is due to the factthat in the stratosphere the vertical gradient of potentialtemperature is high, suppressing the value of ecomputed in Equation 27. The implication is that itmay not be necessary to correct for velocity when in thestratosphere.
Comparison to other research
Is it common for there to be regions of high velocityturbulence and low temperature turbulence? There havebeen only a few reported concurrent measurements ofboth velocity and temperature fluctuations. Barat andBertin21 reported ratios of Qj2 to CT
2of 10, often, and inone instance, of 40 in the center of a large billow. Themeasurements were made in the stratosphere. While afactor of 40 would only double the actual opticalturbulence predictions, the strongest enhancements areexpected in the troposphere.
Recent direct numerical simulation (DNS) of breakinggravity waves22 have a least qualitatively shown thisphenomena. The simulated wave shows strong velocityand temperature variance at early stages of thedevelopment of the turbulent region (or billow). Astime goes on, the temperature differences in the centerof the billow diminish, suppressing temperaturefluctuations, while velocity fluctuations remain strongthroughout a turbulent layer. Actual quantitativeestimates from their work is risky, but the ratio of Qj2 toCT
2is 6 to 8 times higher in the center of the billow thanthe edges.OPTICAL PERFORMANCE CALCULATIONS
The bottom line of optical turbulence is the effect thatCn
2 has on the computation of optical performance.There are many different measures of opticalperformance. One simple measure of the turbulenceeffects on imaging is the line integral of Q2, whichappears in the calculation of the transverse coherencelength and Fried's coherence length for plane waves.
American Institute of Aeronautics and Astronautics
The higher the value of the line integral, the more thedegradation of a beam due to optical turbulence23.The comparison of the integrals of actual and perceivedCn
2 from 10 to 15 km for the 5 WSMR profiles is shownin Table 1. The results are that one might expect anincrease in the predicted optical degradation of from50% to 80% above the predicted degradation from theactual optical turbulence.
Table 1. Comparison of the integral of the profile ofthe actual Cn
2 to the integral of the Cn2 profile as
perceived by a temperature probe moving at 200m/s.The integral is performed with limits of 10km to15km.FlightHOL74037404740574077410
Actual Cn2
(m1/3x!014)1.292.730.671.271.90
Perceived Cn2
(m1/3 x 1014)2.254.271.042.312.87
Ratio Per/Act
1.741.561.551.811.51
The ratio, above can also be used to estimate a meanvalue that one might expect in the overestimation ofoptical degradation with uncorrected data.
CONCLUSION
An analysis of the fluctuation of temperature givenfluctuations in both fluid temperature and fluid velocityshows that velocity fluctuations are in two terms whichcan become significant as fluid velocity increases. Thefirst term is always a positive contribution to theperceived temperature fluctuations. The results of thisanalysis show that even moderate levels of velocityturbulence can contribute significantly to the perceivedlevels of temperature fluctuations and consequentlyindex of refraction fluctuations. The contribution ismost significant in regions that have low temperaturefluctuations and high velocity fluctuations. Thecontribution from the second term, the velocity -temperature cross correlation, can be either positive ornegative, depending on direction of flight, so for acollection of samples flown back and forth over aground track, the term should not contributesignificantly to the mean of the perceived opticalturbulence, but would tend to spread the measuredvalues. The contributions from the cross term are notexpected to be large when the actual temperatureturbulence fluctuations are small.
When there is low temperature turbulence, it can be dueto either an absence of any turbulence, or a strong
velocity turbulence which has mixed the fluid wellenough to reduce temperature gradients. In the lattercase, the moving sensor will perceive high temperaturefluctuations when, in fact, there are very little. Thiscase is not unusual, but the natural result of turbulencein the atmosphere. In these cases, aircraft derivedoptical turbulence levels are higher than actual levels.
The analysis indicates that the effect of velocityfluctuations is stronger in the troposphere than thestratosphere. Some balloon derived data was used toestimate the value of perceived optical turbulence froman aircraft in the high troposphere flying at 200m/s.The result was that the predicted optical degradationwould be 50% to 80% too high.
We predict that there should be differences in the levelsof optical turbulence deduced from airborne and theballoon-borne temperature sensing instruments,especially in the troposphere. We recommend seriousconsideration of the addition of a velocity fluctuationsensor to the instrument suite of aircraft performingoptical turbulence measurements so that the wiretemperature fluctuations can be properly adjusted forfluctuations in velocity.
As a final caveat, we want to clarify a point about ourunderstanding of the atmosphere. We can not, at thistime, state that our values of Cn
2 is the correct value forany airborne instrument flying within some proximity tothe balloon, and that any deviation from that value mustbe attributed to some level of Cu2. Our results apply toa locus of points through the atmosphere for the timethat the balloon passed through. Depending on the localcause of the turbulence and the extent of the turbulentlayer, we might expect the turbulence environment toapply for some horizontal extent. We still do not knowthe full spatial and temporal extent of the turbulence.This is one of the objectives of an Air Force ResearchLaboratory basic research program in turbulence.Balloons, aircraft and radar are integral parts of theprogram, and the addition of velocity fluctuationsensors will be essential in providing a full scientificcharacterization of the turbulence.
ACKNOWLEDGEMENTThe authors are indebted to Demos Kyrazis and John B.Wissler, who have spent long hours quantifying velocityeffects, and who helped us work out the correct form ofthe equations and offered many valuable suggestions forthis work.
REFERENCES
American Institute of Aeronautics and Astronautics
Owens, J. C., "Optical refractive index of air: Dependenceon pressure, temperature and compositions," Appl. Opt. 6, SI-59, 1967.2 Beland, R.R, "Propagation through Atmospheric OpticalTurbulence," The Infrared & Electro-Optical SystemsHandbook, J. Acetta and D. Schumaker, ex. eds., Vol. 2,Atmospheric Propagation of Radiation, F.G. Smith, ed.,Infrared Information Analysis Center, Ann Arbor, MI andSPIE Optical Engineering Press, Bellingham, WA, 1993.3 Tatarski, V.I., Wave Propagation in a Turbulent Medium,McGraw-Hill, New York, 1961.4 Brown, J.H., R. E. Good, P.M. Bench, and G. Faucher,"Sonde measurements for comparative measurements ofoptical turbulence," Air Force Geophysics Laboratory,AFGL-TR-82-0079, ADA118740, NHS, 1982.5 Jumper, G.Y., H.M. Polchlopek, R.R. Beland, E.A. Murphy,P. Tracy, K. Robinson, "Balloon-borne measurements ofatmospheric temperature fluctuations", AIAA-97-2353, 28th
Plasmadynamics and Lasers Conference, June 23-25, 1997,Atlanta, GA.6 Otten, L.J., "Airborne observations of tropopausalturbulence", AIAA-85-0342, AIAA 23rd Aerospace SciencesMeeting, Reno, NV, Jan 14-17,1985.7 Otten, L.J., A. L. Pavel, W. E. Finley, and W. C. Rose, "ASurvey of recent atmospheric turbulence measurements froma subsonic aircraft, AIAA-81-0298, AIAA AerospaceSciences Meeting, St. Louis, MO, January, 1981.8 Finley, W. E., L. J. Otten, A. L. Pavel, and W. C. Rose,"Fine wire anemometry measurements through a tropopausewith vertically propagating gravity waves", Third Conferenceon the Meteorology of the Upper Atmosphere, San Diego,C A, January 1981.9 White, F. M., Heat and Mass Transfer, Addison-WesleyPublishing Company, Reading, MA, 1988, p. 358ff10 Laufer J. and McClellan, "Measurement of heat transferfrom fine wires in supersonic flows", J. Fluid Mech., 1(1956), p. 276-289. (Caution: this paper does not presentrecovery factor, but the ratio of equilibrium (recovery)temperature to total temperature.)11 Morkovin, M.V., "Fluctuations and Hot-wire Anemometryin Compressible Flows," AGARDograph 24,1956.12 Nagabushana, K.A., P.C. Stainback, and G.S. Jones, "ARational Technique for Calibrating Hot-Wire Probes inSubsonic to Supersonic Speeds, AIAA 94-2536, 28* AIAAAerospace Ground Testing Conference, Colorado Springs,CO, June 20-23,1994.13 Beland, R. R., 1993, op cit., p. 167ff.
15 Ibid, p 170.16 Ibid, p 167.17 Ibid, p 173.l* Ibid, p 116.19 Trout, D. and H. A. Panofsky, "Energy dissipation near thetropopause", TeUus, Vol. XXI (1969), No. 3, pp. 355-358.20 Hocking, W.K. and P.K.L. Mu, "Upper and middletropospheric kinetic energy dissipation rates frommeasurements of Q,2 - review of theories, in-situinvestigations, and experimental studies using the BucklandPark atmospheric radar in Australia", J. Atmospheric andSolar-Terrestrial Physics, Vol. 59, No. 14, pp.' 1779-1803,1997.21 Barat, J. and F. Bertin, "Simultaneous Measurements ofTemperature and Velocity Fluctuations within Clear AirTurbulence Layers: Analysis of the Estimate of DissipationRate by Remote Sensing Techniques", J. of the AtmosphericSciences., Vol. 41, May 1984, p.1613.22 Werne, J and D. C. Fritts, 'Turbulence in Stratified andSheared Fluids: T3E Simulations", Colorado ResearchAssociates, Boulder, CO, Conference Proceedings, 8th DODHigh Performance Computing User Group Conference,Houston, TX, 1998.23 Beland, R. R., 1993, op cit., p!91f.
14 Ibid, p. 169ff
Figure 1. Vectors used in the equations. Vgs is thevelocity vector of the aircraft relative to the ground.Vw is the wind velocity vector, composed of a meanwind speed with the bar, and fluctuatingcomponents u', v', and w'. The true air speed vectoris U (not shown) composed of the mean value withthe bar and the fluctuating wind components.
American Institute of Aeronautics and Astronautics
Figure 2. The contribution of the term, r U Cn2/Cp on the perceived CT
2 and Cn2 at 12km for a cold wire
probe moving at 200m/s through the air, using a recovery factor of 0.837. Velocity fluctuations areexpressed in terms of the dissipation rate, e. The relationship to Cu
2 is shown in Equation 25. While thereis no universal agreement on the turbulence classification, values shown typically range from none tonegligible turbulence at the minimum values to low turbulence at 3xlO"3m2/s3.
Approximateperceived Perceived
Cn2 at 12km Of
1.E-15 1.E-02
1.E-16 1.E-03--
1.E-17 1.E-04'r
1.E-18 1.E-05--
1.E-19 1.E-06
TurbulentEnergy
DissipationRate
(Epsilon)(m2/s3)
1.E-051.E-18
1.E-041.E-17
1.E-03 Actual CT2(K2/mM)
1 E-16 ApproximateCn
2 at 12km (1/m20)
Figure 3. Given an actual value of CT on the X axis, the effect of the term, r U Cu /Cp for various levels ofe is added to produce the perceived CT
2 on the Y axis for a probe moving at 200m/s with a recovery factorof 0.837. Also shown are the approximate actual and perceived Cn
Figure 4. The graphs on the left side are the ratio of 8 to % which is equal to the ratio of Cu2 to CT2 for the
entire flight of 5 balloon flights over White Sands Missile Range in September of 1997. In the middle isshown the actual Q,2 sensed by the balloon payload (solid line) and the estimate of the perceived Cn
2 for aninstrument traveling at 200m/s for the 10km to 15km altitude range. On the right is shown the ratio of theperceived Cn2 to the actual Cn2 for the entire 30km altitude range.
![Page 1: [American Institute of Aeronautics and Astronautics 29th AIAA, Plasmadynamics and Lasers Conference - Albuquerque,NM,U.S.A. (15 June 1998 - 18 June 1998)] 29th AIAA, Plasmadynamics](https://reader035.documents.pub/reader035/viewer/2022080406/575095201a28abbf6bbf1d8c/html5/thumbnails/1.jpg)
![Page 2: [American Institute of Aeronautics and Astronautics 29th AIAA, Plasmadynamics and Lasers Conference - Albuquerque,NM,U.S.A. (15 June 1998 - 18 June 1998)] 29th AIAA, Plasmadynamics](https://reader035.documents.pub/reader035/viewer/2022080406/575095201a28abbf6bbf1d8c/html5/thumbnails/2.jpg)
![Page 3: [American Institute of Aeronautics and Astronautics 29th AIAA, Plasmadynamics and Lasers Conference - Albuquerque,NM,U.S.A. (15 June 1998 - 18 June 1998)] 29th AIAA, Plasmadynamics](https://reader035.documents.pub/reader035/viewer/2022080406/575095201a28abbf6bbf1d8c/html5/thumbnails/3.jpg)
![Page 4: [American Institute of Aeronautics and Astronautics 29th AIAA, Plasmadynamics and Lasers Conference - Albuquerque,NM,U.S.A. (15 June 1998 - 18 June 1998)] 29th AIAA, Plasmadynamics](https://reader035.documents.pub/reader035/viewer/2022080406/575095201a28abbf6bbf1d8c/html5/thumbnails/4.jpg)
![Page 5: [American Institute of Aeronautics and Astronautics 29th AIAA, Plasmadynamics and Lasers Conference - Albuquerque,NM,U.S.A. (15 June 1998 - 18 June 1998)] 29th AIAA, Plasmadynamics](https://reader035.documents.pub/reader035/viewer/2022080406/575095201a28abbf6bbf1d8c/html5/thumbnails/5.jpg)
![Page 6: [American Institute of Aeronautics and Astronautics 29th AIAA, Plasmadynamics and Lasers Conference - Albuquerque,NM,U.S.A. (15 June 1998 - 18 June 1998)] 29th AIAA, Plasmadynamics](https://reader035.documents.pub/reader035/viewer/2022080406/575095201a28abbf6bbf1d8c/html5/thumbnails/6.jpg)
![Page 7: [American Institute of Aeronautics and Astronautics 29th AIAA, Plasmadynamics and Lasers Conference - Albuquerque,NM,U.S.A. (15 June 1998 - 18 June 1998)] 29th AIAA, Plasmadynamics](https://reader035.documents.pub/reader035/viewer/2022080406/575095201a28abbf6bbf1d8c/html5/thumbnails/7.jpg)
![Page 8: [American Institute of Aeronautics and Astronautics 29th AIAA, Plasmadynamics and Lasers Conference - Albuquerque,NM,U.S.A. (15 June 1998 - 18 June 1998)] 29th AIAA, Plasmadynamics](https://reader035.documents.pub/reader035/viewer/2022080406/575095201a28abbf6bbf1d8c/html5/thumbnails/8.jpg)
![Page 9: [American Institute of Aeronautics and Astronautics 29th AIAA, Plasmadynamics and Lasers Conference - Albuquerque,NM,U.S.A. (15 June 1998 - 18 June 1998)] 29th AIAA, Plasmadynamics](https://reader035.documents.pub/reader035/viewer/2022080406/575095201a28abbf6bbf1d8c/html5/thumbnails/9.jpg)
![Page 10: [American Institute of Aeronautics and Astronautics 29th AIAA, Plasmadynamics and Lasers Conference - Albuquerque,NM,U.S.A. (15 June 1998 - 18 June 1998)] 29th AIAA, Plasmadynamics](https://reader035.documents.pub/reader035/viewer/2022080406/575095201a28abbf6bbf1d8c/html5/thumbnails/10.jpg)
![Page 11: [American Institute of Aeronautics and Astronautics 29th AIAA, Plasmadynamics and Lasers Conference - Albuquerque,NM,U.S.A. (15 June 1998 - 18 June 1998)] 29th AIAA, Plasmadynamics](https://reader035.documents.pub/reader035/viewer/2022080406/575095201a28abbf6bbf1d8c/html5/thumbnails/11.jpg)
