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The Responses of Rising or Falling Spherical Wind Sensors to Atmospheric Wind Perturbations

George H. Fichtl

Abstract

The responses of rising or falling spherical wind sensors to atmospheric wind perturbations are analyzed using Fourier transform techniques. The linearized equations of motion of a sensor that is subject to drag and gravitational body forces are developed by perturbing a sensor about an equilibrium uniform motion with wind fluctuations which have vertical variations. The wind environment and sensor velocities are decomposed with stochastic Fourier-Stieltjes integrals, and the linearized equations of motion are used to derive the response functions and phase angles of the sensor motions. The results of the analysis are used to analyze the response properties of the Jimsphere balloon wind sensor.

The transfer functions and phase angles are functions of the wind perturbation frequency Ï‰(=Îº|w|, Îº and wÌ„ being the wavenumber of the perturbation and the mean rise or fall rate of the sensor) and two parameters T and Î± which are functions of sensor mass, apparent mass of the sensor, the mass of air displaced by the sensor, the mean ascent or descent rate of the sensor as the case may be, and the acceleration of gravity. The quantity T is a time constant of the system and Î± is the ratio of the apparent mass to the mass of the system. Once these parameters are specified, the response properties of the sensor are completely specified in a linear context. In general, the system becomes more responsive as T and Î± approach zero and unity, respectively.

As Ï‰T â†’ 0, the transfer functions and phase angles approach unity and zero, respectively; so that at sufficiently low frequencies, the sensor essentially measures 100% of the wind perturbation Fourier amplitudes with no phase shifts. As Ï‰T â†’ âˆž, the transfer functions and phase angles approach Î±2 and zero, respectively; so that at sufficiently large frequencies, the sensor is capable of detecting 100a% of the wind perturbation Fourier amplitudes, again with no phase shifts. If apparent mass effects were not present (Î± = 0), the sensor transfer functions would approach zero as Ï‰Tâ†’âˆž. Thus, the apparent mass effects make the sensor more responsive. At Ï‰T = Î±âˆ’Â½ for the horizontal fluctuations and Ï‰T=2Î±âˆ’Â½ for the vertical velocity fluctuations, the phase angles of the sensor Fourier amplitudes take on minimum values. In general, the transfer functions associated with the horizontal sensor motions are smaller than the transfer functions associated with the vertical sensor motions in the domain 0& lt; Ï‰T < âˆž; thus, the sensor is more responsive to vertical than to horizontal air motions.

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Probability Distribution of Vertical Longitudinal Shear Fluctuations

George H. Fichtl

Abstract

In order to properly design aerospace systems (conventional airplanes, V/STOL vehicles, space vehicles, etc.) the engineer must consider the vertical structure of the horizontal wind during the launch and landing phases of flight. One way do this is with vertical two-point wind differences (wind shear). In turbulent flows like those found near the ground, wind shear is composed a steady-state part associated with the mean wind profile and a fluctuating part produced by atmospheric turbulence. Mean wind profile theory can be used to specify the steady-state wind shear; however, the fluctuating part is a stochastic process which can only be specified statistically. This paper discusses some recent measurements of third and fourth moments of vertical differences (shears) of longitudinal velocity fluctuations obtained in unstable air at the NASA 150 m meteorological lower site at Cape Kennedy, Fla. Each set of measurements consisted of longitudinal velocity fluctuation time histories obtained at the 18, 30, 60, 90, 120 and 150 m levels, so that 15 wind-shear time histories were obtained from each set of measurements.

It appears that standardized third and forth moments S and K of wind shear are universal functions of Î”z/zÌ„ and zÌ„/L 0, where Î”zÌ„ is the vertical distance between the two points over which the wind difference is calculated, zÌ„ the height of the mid-point of Î”z above natural grade, and L 0 the surface Monin-Obukhov stability length. As Î”z/zÌ„ â†’ 2, K â†’ 3, S â†’ 0, and S > 0, K > 3 for Î”z/zÌ„ < 2. Thus, it appears that the joint distribution function of the longitudinal wind fluctuations at two levels is not bivariate Gaussian and that it can only be approximated with a Gaussian distribution for sufficiently large value of Î”z/zÌ„. The kurtosis K appears to be independent of zÌ„/L 0. However, the skewness S seems to experience a rather abrupt transition at zÌ„/L 0âˆ¼O(âˆ’1). The implications of these and other results relative to the design and operation of aerospace vehicles are discussed.

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Characteristics of Turbulence Observed at the NASA 150-m Meteorological Tower

George H. Fichtl

Abstract

Recent observations of turbulence obtained at the NASA 150-m meteorological tower located at Kennedy Space Center, Fla., are presented. The wind data were obtained at the 18-, 30-, 60-, 90-, 120- and 150-m levels, while temperature data were obtained at the 3-, 18-, 60-, 120- and 150-m levels. Most of the tests were made during the daylight hours, and the duration time of each test ranged between 30â€“60 min.

A survey of the surface roughness length associated with the tower site is presented. Estimates of the roughness length were calculated with wind profile laws consistent with the Monin-Obukhov similarity hypothesis. In the case of those wind directions Î¸ in the ranges 0Â° â‰¦ Î¸ < 150Â°, 180Â° â‰¦ Î¸ < 240Â°, and 300Â° â‰¦ Î¸ < 360Â°, the roughness length is 0.23 m, while for those wind directions in the ranges 150Â° â‰¦ Î¸ < 180Â° and 240Â° â‰¦ Î¸ < 300Â°, the roughness length has the values 0.51 m and 0.65 m, respectively. Longitudinal turbulence spectra calculated from recent observations are also presented. It is shown that the nondimensional frequency nz/u, associated with the peak of the logarithmic spectrum is proportional to z â…˜, where u is the wind speed at height s and n the frequency. Based upon an analysis of the logarithmic spectrum in the inertial subrange, it is implied that the local mechanical production of turbulent energy is balanced by the local viscous dissipation at the 18-m level.

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Longitudinal and Lateral Spectra of Turbulence in the Atmospheric Boundary Layer at the Kennedy Space Center

George H. Fichtl
and
George E. McVehil

Abstract

An engineering spectral model of turbulence is developed with horizontal wind observations obtained at the NASA 150-m meteorological tower at Cape Kennedy, Fla. Spectra, measured at six levels, are collapsed at each level with [nS(n)/u *0 2,f]-coordinates, where S(n) is the longitudinal or lateral spectral energy density at frequency n(Hz), u *0 the surface friction velocity, and f = nz/Å«, Å« being the mean wind speed at height z. A vertical collapse of the dimensionless spectra is produced by assuming they are shape-invariant in the vertical.

An analysis of the longitudinal spectrum in the inertial subrange, at the 18-m level, implies that the local mechanical and buoyant production rates of turbulent kinetic energy are balanced by the local dissipation and energy flux divergence, respectively.

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The Role of Latent Heat Release in Baroclinic Waves-Without Î²-Effect

Chung-Muh Tang
and
George H. Fichtl

Abstract

In this paper we develop the analytical theory of two-level quasi-geostrophic baroclinic waves without Î²-effect aimed at understanding the role of latent heat release on the development of baroclinic waves.

When the release of latent heat is introduced with pseudo-adiabatic ascent and dry adiabatic descent the width a of the ascending region is different from the width b of the descending region and, furthermore, a static stability-vertical velocity correlation results in the mean state thickness increasing with time. However, the basic state shell is defined a priori, independent of the perturbations, in the formulation of the stability problem. Integro-differential equations for the perturbations are developed. Due to, the mass continuity constraint, the unstable waves in the dry and moist regions are stationary in a frame of reference which translates with the mean zonal wind at the middle level, and the growth rate in the moist region is equal to that in the dry region, the same as in the dry model. We define the parameter F = 2f 2/Sdp 2 2 kd 2, where f is the Coriolis parameter, Sd the static stability in the dry region, p 2 the pressure at the middle level, and kd = Ï€/b. The ratio a/b is a function of F. For F > 1, two unstable modes appear. The first mode has a narrow region of strong ascending motion and a wide region of weak descending motion (a/b < 1), and the second mode has a narrow region of strong descending motion and a wide region of weak ascending motion (a/b > 1). As F â†’ 1, the modes become steady and neutral and are characterized by 1) a/b = (Sm /Sd )Â½ (Sm is static stability in the moist region), and 2) a/b â†’ âˆž. As F â†’ âˆž, the modes are steady and neutral and are characterized by 1) a/b â†’ 0, and 2) a/b â†’ 1. In comparison with the dry model, the structure of the first unstable mode shows that the ridge and trough of the streamlines shift slightly toward the region of sinking motion, and the warm advection occurs at the node of the vertical motion, while the structure of the second unstable mode shows that the ridge and trough of the streamlines shift slightly toward the region of rising motion, and the cold advection occurs at the node of the vertical motion.

An analysis of the energetics shows the presence of a latent heat release term which directly contributes to the generation of eddy available potential energy. Although this term is small compared to the vertical and horizontal heat transports, latent heat release causes a significant change in the structure of the waves such that large departure in the horizontal heat transport from dry atmospheric values can occur.

The multicomponent solution is also discussed. It is stressed that the first harmonic must be present and even harmonics are allowed provided the vertical motion is upward everywhere in the moist region of the width a and downward everywhere in the dry region of the width b. The solution is not Fourier decomposition in the normal sense because except for the first harmonic odd modes are not allowed.

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Non-Quasi-Geostrophic Effects in Baroclinic Waves with Latent Heat Release

Chung-Muh Tang
and
George H. Fichtl

Abstract

A second-order theory of baroclinic waves is developed to investigate non-quasi-geostrophic behavior in disturbances in which latent heat release associated with condensation is permitted to occur in an atmosphere saturated with water vapor. A two-level formulation without Î²-effect is used to analyze these disturbances. The analysis involves an expansion of the flow into a basic state zonal flow with superimposed perturbation which is assumed to be independent of the meridional direction. The superimposed perturbation consists of linear combination of a quasi-geostrophic contribution and a non-quasi-geostrophic departure. The basic state flow with the superimposed quasi-geostrophic perturbation has been investigated by the authors in a previous paper. The governing equations for the non-quasi-geostrophic contribution consist of a nonlinear (thermodynamic) integro-differential equation and a nonhomogenous (vorticity) differential equation. The nonlinearity is a direct result of latent heat release associated with pseudo-adiabatic assent; i.e., saturated ascending air parcels and dry descending air parcels. The nonhomogeneity arises from the non-quasi-geostrophic terms in the vorticity equation. In this theory we use the quasi-geostrophic contribution to calculate the non-quasi-geostrophic terms which generate the second-order solution.

The problem is characterized by two parameters, namely a rotational Froude number F = 2f 2(S dP2 2 k d 2)âˆ’l (where f is the Coriolis parameter, S d the static stability in descending portion of the wave, p 2 the pressure at the middle level, and kd = Ï€/b, b being the horizontal extent of the descending or dry portion of the wave) and a moisture parameter Îµ which is proportional to the midlevel vertical gradient of mean flow specific humidity. For Îµ â‰  0 the disturbances are characterized by a/b â‰  1, where a is the horizontal extent of the ascending or wet portion of the wave. The quasi-geostrophic contribution to the disturbance is characterized by two modes for F > 1. The first mode has a narrow region of strong ascending motion and a wide region of weak descending motion (a/b < 1), with the reverse for the second mode (a/b > 1). Thew solutions, developed by the authors in an earlier paper, are used to calculate the non-quasi-geostrophic solution terms mentioned above.

For the first moist mode, due to the non-quasi-geostrophic effects, both the trough and ridge are intensified at the upper level with stronger intensification of the trough and are weakened at the lower level with considerable weakening of the ridge. The formation of the frontal zone on the east side of the descending region is a feature similar to that in the dry model with non-quasi-geostrophic effects. For the second moist mode, due to the non-quasi-geostrophic effects, both trough and ridge are weakened at the upper level, but they am intensified at the lower level. The temperature profile in each region is nearly symmetric. The total vertical motion field is asymmetric in each region for both the first and second moist modes.

The main characteristics of the energetics are described by the transports due to the first-order eddy. The transports due to the second-order eddy have only minor influence except for large F such as F â‰¥ 10 for the first mode and except for Îµ near unity for the second mode.

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Approximate Equations of Motion for Gases and Liquids

John A. Dutton
and
George H. Fichtl

Abstract

A set of conditions which justify the application of the Boussinesq approximation to compressible fluids is developed. Two cases are found and compared. In the first, in which the vertical scale of the motion can be of the same order of magnitude as the scale height of the medium, the perturbation momentum must be nondivergent and the effects of perturbations of pressure appear in several places. In the other case, where the vertical scale of the motion is much less than the scale height, the perturbation velocities are non-divergent and the perturbation pressure appears only in the pressure gradient force.

The approximate equations lead to linearized equations controlling the stability of wave motion which are formally equivalent to those for the same problem in the flow of a stratified medium which is incompressible in the sense that the flow is solenoidal. Thus, a variety of results about such motions are made applicable to the problems of convection and gravity wave motion in the atmosphere.

Various properties of the approximate equations are investigated; it is shown that acoustic modes are not permitted; quadratic forms which can serve as energies in various cases are developed; and integral methods of determining stability criteria are reviewed and applied.

In order to give the results wider applicability than to ideal gases, an ideal liquid is defined (cp and the coefficients of expansion all being constant). The thermodynamic functions of this ideal liquid, including the entropy, internal energy and potential temperature, are determined explicitly.

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Aerodynamic Properties of Spherical Balloon Wind Sensors

George H. Fichtl
,
R. E. DeMandel
, and
S. J. Krivo

Abstract

A first-order theory of the fluctuating lift and drag coefficients associated with the aerodynamically induced motions of rising and falling spherical wind sensors is developed. The equations of motion of a sensor are perturbed about an equilibrium state in which the buoyancy force balances the mean vertical drag force. It is shown that, to within first order in perturbation quantities, the aerodynamic lift force is confined to the horizontal, and the fluctuating drag force associated with fluctuations in the drag coefficient acts along the vertical. The perturbation equations are transformed with Fourier-Stieltjes integrals and the resulting equations lead to relationships between the power spectra of the aerodynamically induced velocity components and the spectra of the fluctuating lift and drag coefficients.

Experimental evidence shows that the aerodynamically induced motions of the Jimsphere balloon occur predominantly in the horizontal plane. This implies that the root-mean-square (rms) horizontal lift coefficient is much larger than the rms vertical drag coefficient. The aerodynamically induced motion of the Jimsphere is found to be sinusoidal in nature. The dimenionless frequency (Strouhal number) and nondimensional variance of the induced zonal and meridional velocity components are given as functions of the Reynolds number. The experimental range of the Reynolds number is 1.4 Ã— 105 to 6.6 Ã— 105 The ratio between the rms lift coefficient and the mean, or zero-order, drag coefficient is found to be approximately 0.36.

The theory shows that the Fourier components of the first-order fluctuating horizontal lift coefficient vector lead those of the induced horizontal velocity vector, and that the fluctuating part of the drag coefficient lags the induced vertical velocity for rising balloons and leads the induced vertical velocity in the case of falling balloons. The phase angles of the induced lift and drag associated with the characteristic frequency of oscillation of the Jimsphere are given as functions of the Reynolds number.

The rms lift coefficient of smooth 2 m ROSE balloons operating at supercritical Reynolds numbers is found to be approximately twice the value obtained from wind tunnel data. This result suggests that caution should be exercised when wind tunnel data of constrained bodies are applied to free balloons.

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Analysis of In-Flight Winds for Shuttle Mission STS 51-L

George H. Fichtl
,
Nathaniel D. Reynolds
,
Alan E. Johnston
,
,
,
Larry Lott
,
Paul J. Meyer
,
Orvel E. Smith
,
Marion S. Swint
, and
Otha H. Vaughan Jr.

Abstract

Television photos of smoke plumes an analyzed to estimate meridional wind shear on the space shuttle Challenger associated with the accident of Mission 51-L. Gust velocities were obtained by detailed examination of the debris trails. The shuttle exhaust trail was used to establish altitudes of significant features in the photographs. Wind data obtained from the photographs compare favorably with data obtained from a rawinsonde released 9 min after the launch of the shuttle.

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