Showing papers in "Journal of Aircraft in 1995"

Finite state induced flow models. I - Two-dimensional thin airfoil

Abstract: A new finite state aerodynamic theory is presented for incompressible, two-dimensional flow around thin airfoils. The theory is derived directly from potential flow theory with no assumptions on the time history of airfoil motions. The aerodynamic states are the coefficients of a set of induced-flow expansions. As a result, the finite state equations are hierarchical in nature and have closed-form coefficients. Therefore, the model can be taken to as many states as are dictated by the spatial texture and frequency range of interest with no intermediate numerical analysis. The set of first-order state equations is easily coupled with structure and control equations and can be exercised in the frequency or Laplace domain as well as in the time domain. Comparisons are given with Theodorsen theory, Wagner theory, and other methods. Excellent results are found with only a few states.

Finite state induced flow models. II - Three-dimensional rotor disk

Abstract: In Part I of this two-part article, we developed a finite state induced flow model for a two-dimensional airfoil. In this second part, we develop a finite state induced flow model for the three-dimensional induced flow for a rotor. The coefficients of this model are found in a compact closed form. Although the model does not presuppose anything about the source of lift on the rotating blades, applications are given in which the Prandtl assumption is invoked. That is, the two-dimensional lift equations are used at each radial station, but with the inflow from the three-dimensional model. The results are shown to reduce (in several special cases) to Prandtl-Golds tein theory, Theodorsen theory, Loewy theory, dynamic inflow, and blade-element momentum theory. Comparisons with vortex-filament models and with experimental data in hover and forward flight also show excellent correlation.

Natural-laminar-flow airfoil for general-aviation applications

Abstract: A natural-Iamina r-flow airfoil, the NLF(1)-0115, has been recently designed for general-aviation aircraft at the NASA Langley Research Center. During the design of this airfoil, special emphasis was placed on experiences and observations gleaned from other successful general-aviation airfoils. For example, the flight lift-coefficient range is the same as that of the turbulent-flow NACA 23015 airfoil. Also, although beneficial for reducing drag and producing high lift, the NLF(1)-0115 airfoil avoids the use of aft loading, which can lead to large stick forces if utilized on portions of the wing having ailerons. Furthermore, not using aft loading eliminates the concern that the high pitching-moment coefficient generated by such airfoils can result in large trim drag if cruise flaps are not employed. The NASA NLF(1)-0115 airfoil has a thickness of 15% chord. It is designed primarily for general-aviation aircraft with wing loadings of 720-960 N/m2 (15-20 lb/ft2). Low-profile drag as a result of laminar flow is obtained over the range from c, = 0.1 and R = 9 x 106 (the cruise condition) to c, = 0.6 and R = 4 x 106 (the climb condition). While this airfoil can be used with flaps, it is designed to achieve a c,,max of 1.5 at R = 2.6 x 10 6 without flaps. The zero-lift pitching moment is held to c,H,0 = -0.055. The hinge moment for a 20% chord aileron is fixed at a value equal to that of the NACA 632-215 airfoil, CH = — 0.0022. The loss in cAmax due to leading-edge roughness at R = 2.6 x 10 6 is 11% as compared with 14% for the NACA 23015.

Summary of an Active Flexible Wing program

Abstract: This paper presents a summary of the NASA/ Rockwell Active Flexible Wing program. Major elements of the program are presented. Key program accomplishmerits included singleand multiple-mode flutter suppression, load alleviation and load control during rapid roll maneuvers, and multi-input/multi-output multiple-function active controls tests above the open-loop flutter boundary.

Wing flutter boundary prediction using unsteady Euler aerodynamic method

Abstract: Modifications to an existing three-dimensional, implicit, upwind Euler/Navier-Stokes code (CFL3D Version 2.1) for the aeroelastic analysis of wings are described. These modifications, which were previously added to CFL3D Version 1.0, include the incorporation of a deforming mesh algorithm and the addition of the structural equations of motion for their simultaneous time-integration with the government flow equations. The paper gives a brief description of these modifications and presents unsteady calculations which check the modifications to the code. Euler flutter results for an isolated 45 degree swept-back wing are compared with experimental data for seven freestream Mach numbers which define the flutter boundary over a range of Mach number from 0.499 to 1.14. These comparisons show good agreement in flutter characteristics for freestream Mach numbers below unity. For freestream Mach numbers above unity, the computed aeroelastic results predict a premature rise in the flutter boundary as compared with the experimental boundary. Steady and unsteady contours of surface Mach number and pressure are included to illustrate the basic flow characteristics of the time-marching flutter calculations and to aid in identifying possible causes for the premature rise in the computational flutter boundary.

Subsonic wing planform design using multidisciplinary optimization

Abstract: This article presents basic results from wing planform optimization for minimum drag with constraints on structural weight and maximum lift. Analyses in each of these disciplines are developed and integrated to yield successful optimization of wing planform shape. Results demonstrate the importance of weight constraints, compressibility drag, maximum lift, and static aeroelasticity on wing shape, and the necessity of modeling these effects to achieve realistic optimized planforms.

Real-time estimation of atmospheric turbulence severity from in-situ aircraft measurements

Abstract: The quality of atmospheric turbulence detection and forecast information for the operational meteorology and aviation communities is directly linked to the quality of real-time measurements. Currently, the only direct data are subjective, qualitative, and intermittent pilot reports. This article describes techniques, suitable for real-time application on commercial transport aircraft, to generate quantitative and comprehensive turbulence measurements. These algorithms build on standard methods used in the analysis of aircraft response to turbulence, but are specifically designed to address the limitations of the available on-board data and computational

Free vibration of composite beams - An exact method using symbolic computation

Abstract: An exact dynamic stiffness matrix method has been developed to predict the free vibration characteristics of composite beams (or simple structures assembled from them) for which the bending and torsional displacements are (materially) coupled. To achieve this, an explicit expression is presented for each of the elements of the dynamic stiffness matrix of a bending-torsion coupled composite beam. This was made possible by performing symbolic computing with the help of the package Reduce. Programming the stiffness expressions in Fortran on a SUN SPARC station indicates about 75% savings in computer time when compared with the matrix inversion method normally adopted in the absence of such expressions. The derived dynamic stiffness matrix is then used in conjunction with the Wittrick-Williams algorithm to compute the natural frequencies and mode shapes of composite beams with substantial coupling between bending and torsional displacements. The results obtained from the present theory are compared with those available in the literature and discussed.

Flutter suppression control law design and testing for the active flexible wing

Abstract: Design of a control law for simultaneously suppressing the symmetric and antisymmetric flutter modes of a sting-mounted, fixed-in-roll aeroelastic wind-tunnel model is described. The flutter suppression control law was designed using linear quadratic Gaussian theory, and involved control law order reduction, a gain root-locus study, and use of previous experimental results. A 23% increase in the open-loop flutter dynamic pressure was demonstrated during the wind-tunnel test. Rapid roll maneuvers at 11% above the symmetric flutter boundary were also performed when the model was in a free-to-roll configuration.

Efficient parametrization of generic aircraft geometry

Abstract: A new method is presented for the parametrization of aircraft geometry; it is efficient in the sense that a relatively small number of "design" parameters are required to describe a complex surface geometry. The method views surface generation as a boundary-value problem and produces surfaces as the solutions to elliptic partial differential equations, hence its name, the PDE method. The use of the PDE method will be illustrated in this article by the parametrization of double delta geometries; it will be shown that it is possible to capture the basic features of the large-scale geometry of the aircraft in terms of a small set of design variables.

Rotor free-wake modeling using a pseudoimplicit relaxation algorithm

Abstract: A pseudoimplicit predictor-cor rector relaxation algorithm with five-point central differencing in space has been developed for the solution of the governing differential equations of the helicopter rotor free-wake problem. This new approach is compared and contrasted with more conventional explicit-type free-wake algorithms. A convergence analysis shows that the new algorithm provides for much more rapid convergence characteristics compared to explicit methods, with improvements in numerical efficiency and predictive accuracy. Nomenclature be = spatial boundary condition vector CT = rotor thrust coefficient, Tlp7rR2([lR)2 c = blade chord, m E — shift operator i,/, k = unit vectors in the x, y, and z directions, respectively L = spatial discretization operator / = length of discretized vortex element, m Nh = number of blades N.t. = number of vortex filaments P = iteration scheme operator R = rotor radius, m r(. = vortex core radius, m r, = spanwise location from which vortex filaments are trailed, m r = position vector of a point on a vortex filament, m S = source vector T = rotor thrust, N / = time, s V.,_ — freestream velocity, m/s V = time invariant flowfield velocity, m/s Vind = induced velocity, m/s Vloc = local velocity at a point in space, m/s VH = tangential velocity, m/s Cartesian coordinate system, origin at hub center rotor shaft angle (negative forward), deg )3() = blade coning angle, deg r = circulation, m2/s Ar" = nth iteration position-vector correction f = distance along trailed wake filament (wake age), rad A.- = uniform-induced inflow ratio

Computation of Delta-Wing Roll Maneuvers

Abstract: This article presents computations of delta-wing roll maneuvers for an 80-deg sweep delta-wing at 30-deg angle of attack. Three constant roll-rate maneuvers are considered. Two of the maneuvers consist of a roll from 0 to 45 deg at nondimensiona l roll rates of = 0.0233 and 0.0467. The third roll maneuver computed starts at a 45-deg roll angle and rolls back to a -45-deg roll angle at a roll rate 4> = -0.0467. The governing equations are the unsteady, three-dimensional Navier-Stokes equations. The equations are solved using the implicit, approximately-factored, diagonal form of the Beam-Warming algorithm. Subiterations are used to provide a more accurate means of implementing the diagonal form of the algorithm for unsteady flows. The effects of roll-rate and differing initial roll angles on the dynamical behavior of the vortices positions and strengths as well as their corresponding effect on surface pressure and roll moment coefficient are described.

Aeroelastic analysis of a flexible control surface with structural nonlinearity

Abstract: This article is concerned with a time domain approach to the flutter analysis of a flight vehicle control surface with concentrated nonlinearities. In this study, an elastic model of a control surface with root freeplay nonlinearity in pitch is considered. A finite element structural model is used for structural analysis and a doublet lattice unsteady aerodynamic model is used for the calculation of aerodynamic loads. In approximating the frequency domain aerodynamic forces, the least-square rational function approximating method is used with an optimizing algorithm. To transform the frequency domain aerodynamic forces to the time domain forces, the method of Brace and Eversman is used. To reduce the problem size and the computation time, the fictitious mass modal approach is used, which can afford the possible local change of structural properties. The effects of the initial conditions and the magnitude of nonlinearity on the aeroelastic characteristics are examined. The aeroelastic responses are sensitive to initial conditions. Limit cycle oscillation and chaotic motion are observed in this study. The presence of freeplay makes the divergent flutter speeds larger than those of a linear case.

Structural Dynamic Loads in Response to Impulsive Excitation

Abstract: The modal approach is used to analyze the dynamic loads on a flexible structure due to local impulsive excitations such as that caused by store ejection from a flight vehicle. First-order, time-domain equations of motion in generalized coordinates are constructed for restrained and free-free structures, without and with unsteady aerodynamic effects. The dynamic loads associated with the structural response are expressed by the mode displacement (MD) and by the summation-of-forces methods. The MD approach is simpler and easier to apply, but requires the inclusion of more modes for obtaining results of acceptable accuracy. A rigorous comparison between the resulting loads shows that the performance of the MD method is especially poor when the excitation is local and impulsive. A dramatic improvement is obtained when the generalized coordinates are based on normal modes calculated with fictitious masses at the excitation points. Fictitious masses are also used to generate artificial load modes that yield simple and efficient expressions for integrated shear forces, bending moments, and torsion moments at various structural sections.

Design Analysis Methodology for Solar-Powered Aircraft

Abstract: A methodology for conceptual design of solar-powered aircraft is described. The method is based on traditional design methodologies adapted to address the peculiar characteristics of solar-powered aircraft. The solar propulsion notation is applied to the analysis to create constraint diagrams that have takeoff wing loading ft = Wto/S as their independent variable, and the ratio of solar collector area to reference planform area R — Sce,,/S as their dependent variable. Constraints determined by the propulsion requirements for the design mission of the aircraft define a solution space on the constraint diagram. A design point is selected from within the solution space. Parametric estimates for component weights are then represented as wing loading portions. When these component wing loading portions are summed along with an expression for the required payload wing loading portion, they are equated to the design point wing loading. The resulting equation is solved for the required reference planform area, thus sizing the conceptual design. Three typical conceptual designs are described and analyzed, demonstrating the utility of the methodology.

Flutter suppression for the active flexible wing - A classical design

Abstract: The synthesis and experimental validation of a control law for an active flutter suppression system for the active flexible wing wind-tunnel model is presented. The design was accomplished with traditional root locus methods making extensive use of interactive computer graphics tools and simulation-based analysis. The design approach relied on a fundamental understanding of the flutter mechanism to formulate a very simple control law structure resulting in a filter with a "inverted notch" characteristic. This unusual filter characteristic was required to compensate for adverse zero locations in the frequency range near flutter. Wind-tunnel tests of the flutter suppression controller demonstrated simultaneous suppression of two flutter modes, significantly increasing the flutter dynamic pressure. The flutter suppression controller was also successfully operated in combination with a rolling maneuver controller to perform flutter suppression during rapid rolling maneuvers.

Flow developments above 50-deg sweep delta wings with different leading-edge profiles

Abstract: The characteristics of flow developments above 50-deg sweep delta wings with different leading-edge profiles are shown by flow visualizations and velocity measurements. The Reynolds number based on freestream velocity and root chord is about 7 x 103. The leading-edge profiles studied include the shapes of square, round, windward surface beveling, leeward surface beveling, and wedge. Based on the velocity data obtained along the leading edges of the delta wings it is noted that the flow angles associated with the separated shear layers vary with the leading-edge profiles studied. This finding infers that varying the leading-edge profile has an impact on the initial development of the separated shear layer, consequently, the formation of leading-edge vortex. Furthermore, it is shown that the leading edge of windward beveling causes the largest leading-edge flow angle and produces the most organized leading-edge vortex.

Navier-stokes applications to high-lift airfoil analysis

Abstract: This article presents applications of a compressible Reynolds-averaged Navier-Stokes method to the calculation of flows about a transport-type multielement airfoil. The unstructured-mesh method used utilizes multigrid techniques for computational efficiency and includes a selection of turbulence models. The airfoil used to benchmark the computational capability is a three-element airfoil configured for landing for which extensive experimental data have been acquired both on and off the airfoil surface at high Reynolds numbers. Comparisons of computational results vs experimental data shown here include traditional airfoil performance calculations due to configuration changes. Also discussed are detailed comparisons of computational results and experimental data obtained in the flap well region to assess the applicability of existing turbulence models to the flap slot flow. Performance comparisons are conducted for configurations tested at chord Reynolds numbers of 5 x 10 6 and 9 x 106 and the flap well study is based on data obtained on a similar airfoil at a chord Reynolds number of 5 x 10 5.

Lift-Enhancing Tabs on Multielement Airfoils

Abstract: The use of flat-plate tabs (similar to Gurney flaps) to enhance the lift of multielement airfoils is extended here by placing them on the pressure side and near the trailing edge of the main element rather than just on the furthest downstream wing element. The tabs studied range in height from 0.125 to 1.25% of the airfoil reference chord. In practice, such tabs would be retracted when the high-lift system is stowed. The effectiveness of the concept was demonstrated experimentally and computationally on a two-dimensional NACA 63(sub 2)-215 Mod B airfoil with a single-slotted, 30%-chord flap. Both the experiments and computations showed that the tabs significantly increase the lift at a given angle of attack and the maximum lift coefficient of the airfoil. The computational results showed that the increased lift was a result of additional turning of the flow by the tab that reduced or eliminated now separation on the flap. The best configuration tested, a 0.5%-chord tab placed 0.5% chord upstream of the trailing edge of the main element, increased the maximum lift coefficient of the airfoil by 12% and the maximum lift-to-drag ratio by 40%.

Estimating noise characteristics from flight test data using optimal Fourier smoothing

Abstract: A technique based on Fourier series analysis was developed to separate signal from noise for flight test data. This was done with an optimal filter designed in the frequency domain. The method is general, and separates signal and noise based on the spectral content of the measurement time history. Smoothed time histories with no time lag were computed, and noise characteristics were accurately estimated. The technique can be used independently of other procedures, and does not require assumptions about the independence of the noise processes or the frequency content of the measurements. Simulated data was used to demonstrate the technique and to evaluate the accuracy of estimated noise characteristics. For 20 simulation cases, noise standard errors were estimated within 5% of the true values. Flight test data from a lateral maneuver of the F-18 High Alpha Research Vehicle was then analyzed. The theoretical analysis was shown to be sound for both simulated data and flight test data.

Aerodynamics and Acoustics of Rotor Blade-Vortex Interactions

Abstract: Helicopter rotor simulation codes of different organizations are validated against experimental data obtained in the DNW. The comparison addresses rotor blade loading, wake geometry, blade motion and noise radiation. Although specific differences exist the general prediciton of rotor noise is reasonably well.

Nonlinear Aerodynamic Analysis of Grid Fin Configurations

Abstract: The aerodynamic analysis of grid fin configurations has been extended to generic cruciform configurations oriented at any azimuthal angle. The generality of grid fin designs has been retained providing orientations for up to four fins as mounted on or near the surface of a missile airframe. The theoretical analysis is based on a vortex lattice overlay of the lifting elements and includes appropriate body upwash terms as well as wing-body carry over load prediction. This basic approach produces adequate modeling for small angles of attack and small fin incidence angles for any grid fin located at any azimuthal position around the body. For higher angles of attack, empirical equations have been developed for fin coefficients and for the body aerodynamic coefficients. Entire four fin configurations are generically modeled and solutions are obtained in an iterative manner. Excellent agreement between experimental data and theoretical predictions have been obtained for the configuration considered up to angles of attack of twenty degrees and fin incidence angles of thirty degrees. For other grid fin designs other than the one considered in the wind tunnel test and for higher angles of attack, the agreement between the theory and experimental results has not been established.

Computation of compressible, laminar boundary layers on swept, tapered wings

Abstract: Cp = pressure coefficient, 2(p — px)/pxUl, c = chord length cfx = local skin friction coefficient for circumferential flow, 2(/jLdu/dz)w/peul f = transformed circumferential component of vector potential, i^(0, r, z)/(2£r/r0) G = transformed radial component of vector potential (alternative), (ve/qe)g g = transformed radial component of vector potential, w>2(0, r, z)/r(2gr/r0)ve H = total enthalpy Mx = Mach number of undisturbed stream, UJa p = static pressure qe = resultant external velocity, r = radial distance along generator, measured from cone apex, O r() = radius of sphere centered at O, intersecting leading edge of streamwise section S = enthalpy parameter, HIHe — 1 T = static temperature Ux = velocity of undisturbed stream (w, v, w) = velocity components in (0, r, z) directions v = spanwise velocity, — v x = arc distance around surface, r00 y = spanwise distance, r() r z = distance normal to surface y = ratio of principal specific heat capacities of gas A = transformed boundary-layer thickness 8 * = displacement thickness of circumferential flow, S'o (1 pulpeue) dz 77 = similarity variable used in present work, ue /f, p dz/V2fr/r0 VK-C = similarity variable used by Kaups and Cebeci, Vve/pe/zer0 Jg p dz 0 = polar angle in developed plane, measured from stagnation line AH A 2 = sweep angles of leading and trailing edges fji = coefficient of viscosity of gas f = transformed circumferential coordinate, So Pet*>euer() d0 p = mass density of gas o= Prandtl number of gas

Inlet acoustic mode measurements using a continuously rotating rake

Abstract: Comprehensive measurements of the spinning acoustic mode structure in the inlet of the advanced ducted propeller were obtained using a unique method that was first proposed by Sofrin. A continuously rotating microphone system was employed. Three inlet configurations with cut-on as well as cut-off stator vane sets were tested. The cut-off stator was designed to suppress all modes at the blade passing frequency. Rotating rake measurements indicate that several extraneous circumferential modes, possibly due to the interaction between the rotor and small interruptions in the casing tip treatment, were present. The cut-on stator produced the expected circumferential modes plus higher levels of the unexpected modes seen with the cut-off stator. HE next generation of fan engines will likely employ a marriage of turbofan and propeller technologies to achieve significant noise and fuel consumption reductions. The ad- vanced ducted propeller (ADP) model used in this investi- gation was designed and built by Pratt and Whitney, a Di- vision of United Technologies, and tested in the NASA Lewis 9- by 15-ft Anechoic Wind Tunnel. Typical of propeller tech- nology, the ADP allows for the in-flight adjustment of the blade pitch angle. This provides reverse thrust and optimum performance over a wide range of conditions. The duct pro- vides the noise suppression advantage of a conventional fan engine. Since future engines are expected to use still higher bypass ratios, fan noise is likely to be the dominant engine source. One of the most important features of fan tone noise is its modal structure. Knowledge of these spinning modes helps to identify the generation mechanism, control duct propa- gation (thus, mode knowledge is needed for acoustic treat- ment design) and control far-field radiation. Previous at- tempts at direct mode measurements1'3 have faced formidable practical difficulties such as: very large axial and circumfer- ential arrays of wall microphones that are not practical for the short ducts of ultrahigh bypass engines, and radial mea- surements upstream of the fan that introduce a wake that interacts with the rotor, thus causing extraneous modes. A continuously rotating microphone technique first proposed by Sofrin4 overcomes the problem of wake-generate d modes, reduces the number of microphones and the duct length re- quired. This technique has been implemented for the first time in this investigation. Two important features of this tech- nique are as follows:

Spanwise variations in profile drag for airfoils at low Reynolds numbers

Abstract: In a nominally two-dimensional flow, extensive wake surveys were performed on two airfoils at low Reynolds numbers to quantify the profile drag variation along the airfoil model span. Wake profile measurements were made at 57 spanwise stations spaced 2% chord apart and 1.25 chord lengths downstream of the trailing edge. Results at a Reynolds number (Re) of 2 X 10' show that variations on the order of 5-40% are typical. In an extreme case, however, over a spanwise distance of less than 12% chord, the profile drag coefficient changed from approximately 0.006 to 0.016, which illustrates the sometimes rather dramatic three-dimensional nature of the flow. Measurements taken at higher Reynolds numbers and closer to the trailing edge showed significant reductions in the spanwise drag variation, which suggests that the laminar separation bubble and the developing wake play an important role. A zigzag boundary-layer trip and an isolated roughness element were also investigated to examine their effects as compared with their respective undisturbed cases. Finally, the results suggest that the lack of good agreement in profile drag measurements between different wind-tunnel facilities can partly be traced to the associated measurement techniques used as weD as the flowfield three dimensionality. Differences in agreement between facilities can be expected when only one spanwise wake profile is taken per angle of attack, as has often been done in the past.

Neural network approach to fatigue-crack-growth predictions under aircraft spectrum loadings

Abstract: An artificial neural network (NN) method is developed to represent the fatigue-crack-growth and cycle relationships under spectrum loadings of the Mirage aircraft operated by the Royal Australian Air Force. This method utilizes load cycle spectrum using available flight and experimental data for crack growth vs cycles as input. The trained network is able to predict the relationship between the crack-growth and the loading cycles. The neural network is able to predict the crack-growth cycle behavior for different variations in the original loading spectrums. The results predicted by the NN model seem reasonable and the model is capable of representing crack-growth behavior for various arbitrary aircraft spectrum loadings with certain limitations. In addition, an attempt is made to predict the material parameters for Walker's fatigue-crack-growth relationship using a different neural network. Because of the demonstrated performance, it is possible that the proposed NN approach can be extended with more research effort to estimate the fatigue life of arbitrary cracked structural components under complex loadings in real time.

Evolution of airplane gust loads design requirements

Abstract: Nomenclature a = lift curve slope, per rad #„, bn — Fourier series coefficients for e.g. acceleration c = wing mean geometric chord, ft FK — flight profile alleviation factor FKm = flight profile factor due to airplane weight FKZ = flight profile factor due to altitude g = acceleration due to gravity, 32.2 ft/s g;j, hn = Fourier series coefficients for true gust velocity H = gust gradient distance, ft K, KK = gust load alleviation factors L = scale of turbulence, ft M = Mach number q = dynamic pressure, psf /?, = maximum landing weight/maximum takeoff weight R2 = maximum zero fuel weight/maximum takeoff weight S = wing area, ft s = distance along flight path, ft T = local atmospheric temperature, °R 7,,, /„ = real and imaginary parts of the e.g. acceleration transfer function T() = ambient atmospheric temperature, °R U = gust velocity, fps, true airspeed (7dc = derived equivalent gust velocity, fps, equivalent airspeed (7ds = design gust velocity, fps, equivalent airspeed £/dt = derived true gust velocity, fps, true airspeed £/rct = design reference gust velocity, fps, equivalent airspeed Utr = power spectral scale factor V = aircraft velocity, fps, true airspeed VB = rough air penetration speed, Kt, equivalent airspeed Vc = design cruise speed, Kt, equivalent airspeed VD = design dive speed, Kt, equivalent airspeed W = aircraft weight, Ib Zmo = maximum operating altitude, ft y = ratio of specific heats Aft = incremental load factor, g fjif, = airplane mass parameter p = air density, slugs/ft Introduction S UBSONIC aircraft respond to atmospheric turbulent air motions or eddies of, approximately, 30-2000 ft in extent. Smaller eddies will generally be averaged out over the surface of the aircraft, larger eddies typically will not cause sharp or excessive aircraft accelerations or structural loads on the airplane. Aircraft in supersonic flight respond to ever longer wavelengths as the flight speed is increased. From the aircraft design standpoint, atmospheric turbulence may be separated into two categories: 1) turbulence, which contributes to aircraft structural fatigue and passenger inconvenience and discomfort, is generally associated with the less intense, smaller scales (small eddy size or higher spatial frequencies as characterized by the turbulence power spectrum); and 2) turbulence that can cause aircraft upset, passenger injury, and possibly structural damage or failure is associated with the more intense larger scales of the turbulence spectrum. Whereas the first category may be in the inertial subrange of the turbulence power spectrum where local homogeneity and stationarity may be assumed, the second category is definitely associated with the larger energy-containing scales of turbulence that are definitely nonstationary and inhomogeneous. In fact, the second type of atmospheric turbulence may not be turbulence at all, but may be a part of, or derive directly from, the ordered convective or geostrophic motions of the atmosphere. Frictionally induced turbulence in the planetary boundary layer is dependent on wind speed near the ground and the surface roughness. Convective turbulence in the planetary boundary layer is dependent on the lapse rate (the rate of temperature change with altitude) and the depth of the mixing layer. Thus, the wind speed and surface roughness and the lapse rate and mixing layer thickness largely determine the intensity and scale of the boundary-layer turbulence. This low altitude boundary-layer turbulence will affect landing and takeoff operations for the larger commercial aircraft that normally cruise at high altitude, and it will be the primary cause of disturbance for small aircraft that operate at low altitudes. Usually, it can be characterized as random, locally stationary, and Gaussian, and the turbulence scale at the surface is of the order of 1000 ft and increases with altitude. It is only natural that pilots avoid flying into areas of obviously rough air such as thunderstorms or even cumulus clouds;