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A program to compute three-dimensional subsonic unsteady aerodynamic characteristics using the doublet lattic method, L216 (DUBFLX). Volume 1: Engineering and usage

The program input presented consists of configuration geometry, aerodynamic parameters, and modal data; output includes element geometry, pressure difference distributions, integrated aerodynamic coefficients, stability derivatives, generalized aerodynamic forces, and aerodynamic influence coefficient matrices. Optionally, modal data may be input on magnetic file (tape or disk), and certain geometric and aerodynamic output may be saved for subsequent use.

Richard, M.↗

Dynamic Aeroelastic Flight Dynamic Modeling of Mach 0.745 Transonic Truss-Braced Wing

This paper presents a modeling approach for dynamic aeroelastic flight dynamic analysis of the Mach0.745 Transonic Truss-Braced Wing. The modeling approach is based on a transonic correction method to correct the Theodorsen’s theory for transonic flow. CFD unsteady Reynolds-averaged Navier-Stokes equations (RANS) simulations are conducted using FUN3D solver for a series of wing sections from the Mach 0.745Transonic Truss-Braced Wing in pitch and plunge oscillations. Unsteady lift and pitching moment coefficients are obtained and used to develop the correction terms in the Theodorsen’s theory to account for transonic aerodynamics. The unsteady lift and pitching moment derivatives with respect to the unsteady angle of attack are obtained as functions of the reduced frequency. These derivatives are used to compute the unsteady lift and pitching moment contributions by the angle of attack, pitch rate, and roll rate. They are then approximated using a frequency-domain regression to obtain the dynamic stability derivatives for the Mach 0.745 Transonic Truss-Braced Wing. The structural dynamic mode shapes of the Mach 0.745 TTBW are extracted from a NAS-TRAN finite-element model. These mode shapes are used to compute the generalized unsteady aerodynamic forces. The aerodynamic mass, damping, and stiffness and the aerodynamic lag states are constructed to couple the dynamic aeroelastic contribution to the flight dynamic model of the Mach 0.745 Transonic Truss-Braced Wing. The coupled dynamic aeroelastic flight dynamic equations of motion are formulated. The eigenvalues of the coupled system are computed. All the flight dynamic modes and structural dynamic modes are stable at Match 0.745. The effect of transonic aerodynamics generally causes all the dynamic modes to have lower damping values

Flight Dynamics↗

Unsteady incompressible aerodynamics and forced response of detuned blade rows

A mathematical model is developed and utilized to demonstrate the enhanced forced response behavior associated with aerodynamic, structural, and combined aerodynamic-structural detuning of a loaded rotor operating in an incompressible flow field. The unsteady aerodynamic gust response and oscillating cascade aerodynamics are determined by developing both a complete first-order unsteady aerodynamic analysis and a locally analytical solution in individual grid elements of a body fitted computational grid. The aerodynamic detuning is accomplished by means of alternate circumferential airfoil spacing, with alternate blade structural detuning also considered. The beneficial forced response effects of these detuning techniques are then demonstrated by applying this model to various detuned rotor configurations.

Chiang, Hsiao-Wei D.↗

Thrust law effects on the longitudinal stability of hypersonic cruise

In the present analytical treatment of the cruising flight stability of an aerospacecraft in nearly circular orbit, on the basis of a thrust law which arbitrarily depends on altitude, speed, and angle of attack, attention is given to thrust law effects on the translational (height and phugoid) and rotational (angle of attack) modes. The partial derivatives of the propulsive forces in conjunction with the aerodynamic forces, with respect to speed and altitude, are noted to exert a major influence on the stability of translational dynamics; the partial derivative of the component of propulsive and aerodynamic forces which is perpendicular to vehicle velocity (with respect to the angle of attack) is a primary determinant of the damping of angle-of-attack oscillations, while the partial derivative of the sum about the center of mass of aerodynamic and propulsive pitching moments, with respect to the angle of attack, determines the corresponding period.

Markopoulos, Nikos↗

Flutter and Forced Response Analyses of Cascades using a Two-Dimensional Linearized Euler Solver

Flutter and forced response analyses for a cascade of blades in subsonic and transonic flow is presented. The structural model for each blade is a typical section with bending and torsion degrees of freedom. The unsteady aerodynamic forces due to bending and torsion motions. and due to a vortical gust disturbance are obtained by solving unsteady linearized Euler equations. The unsteady linearized equations are obtained by linearizing the unsteady nonlinear equations about the steady flow. The predicted unsteady aerodynamic forces include the effect of steady aerodynamic loading due to airfoil shape, thickness and angle of attack. The aeroelastic equations are solved in the frequency domain by coupling the un- steady aerodynamic forces to the aeroelastic solver MISER. The present unsteady aerodynamic solver showed good correlation with published results for both flutter and forced response predictions. Further improvements are required to use the unsteady aerodynamic solver in a design cycle.

Reddy, T. S. R.↗

Aerocapture: An Enabling Technology for Flagship-Class Uranus Orbiter and Probe Mission

Introduction: The current decadal survey published by the National Academies of Sciences has informed National Aeronautics and Space Administration (NASA) to prioritize the study of the Ice Giants, especially Uranus. To gather the required data that addresses the science questions raised in this survey, a mission to Uranus with an orbiter and atmospheric probe must be designed. The Uranus Orbiter and Probe (UOP) study, which the survey identified as the flagship mission of this decade, proposes a 2031 launch to take advantage of a Jupiter fly-by and utilizes a fully propulsive orbit insertion design with an Earth-to-Uranus transit times ranging from 13 to 15 years. This fully propulsive orbit insertion at Uranus will be very fuel expensive (wet mass percentages of around 60-70\%) thereby leaving less mass for the scientific payload and additional planetary probes. In addition, scientists are more interested in visiting Uranus before 2049, when the Spring Equinox will occur, as it allows studying Uranus seasons not seen during Voyager 2's flyby in 1987. A NASA Flagship-Class mission would require at least 10 years of lead time prior to launch thereby making the 2031 launch to take advantage of the Jupiter fly-by extremely challenging. The consequence of missing the Jupiter fly-by and launching in the late 2030s is the challenge of a fully-propulsive mission like UOP to have a feasible alternative interplanetary trajectory that reaches Uranus before 2050. As an alternative, to address the shortcomings of the fully propulsive mission, one can design a mission to Uranus using aerocapture. What is aerocapture: Aerocapture is an atmospheric maneuver that uses aerodynamic forces generated by flight within a planetary atmosphere to decelerate and achieve orbit insertion. Using aerocapture allows one to provide the change in velocity (Delta V) needed to slow down from the approach hyperbolic trajectory to achieve the desired captured orbit around the target planet using aerodynamic forces of the vehicle (lift and drag). Using aerodynamic forces instead of fully-propulsive maneuvers results in significant savings in the fuel. Furthermore, aerocapture can also allow one to consider interplanetary trajectories with higher approach hyperbolic velocities, thus reducing the mission transit times. Aerocapture as an enabling technology: To use aerocapture as an enabling technology for the Uranus exploration mission, one would require an integrated system-level design, including a Thermal Protection System (TPS), hardware needed for aerodynamic modulation, and autonomous Guidance, Navigation, and Control (GNC) systems. Aerocapture has yet to be demonstrated, despite considering it for several past missions. Recent advancements in TPS and GNC capabilities show the potential for using rigid, heritage entry vehicle configurations already flown at other planetary bodies for Uranus aerocapture. Aerocapture can be a robust technology that can deliver spacecraft to Uranus science orbits while substantially increasing on-orbit payload mass (more than 40\%) that can enable robust atmospheric entry probes. Additionally, the aerocapture maneuver would reduce the interplanetary transit time by 2–5 years (15-30\%) relative to fully-propulsive orbit insertion. Recent work has shown that one can conduct a flagship-class mission in a shorter period than fully-propulsive missions if using aerocapture. What does aerocapture bring in for a Uranus Mission: Using aerocapture for a Uranus orbiter and atmospheric probe mission can provide one with considerable propellant savings. Spacecraft in interplanetary trajectories to Uranus typically need an Delta V in orders of kilometers per second to insert into science orbit. One would require thousands of kilograms of fuel to achieve such a Delta V using a traditional fully propulsive maneuver, leaving less mass for payload during the mission launch. Aerocapture can reduce the propulsion needs by dissipating energy in the sizable atmosphere of Uranus without a significant mass increase due to the need for an aeroshell. One can use the mass savings achieved using the aerocapture to reduce the launch vehicle requirements. In addition, one can also have additional science instruments on the orbiter or create a robust instrumentation suite on an atmospheric probe that can significantly increase the science outcome of the Uranus exploration mission. Furthermore, since aerocapture performance is relatively insensitive to increases in hyperbolic excess velocity, one can design the interplanetary trajectory to arrive at Uranus faster, reducing the interplanetary transit time and operations cost. All these savings, achieved using aerocapture, could help fit a larger class mission, such as the Uranus mission within a smaller capital, e.g., a Flagship-class orbiter mission in a New Frontiers class capital. Summary: This talk will provide an overview of how aerocapture can enable the Uranus exploration mission. Specifically, this talk will discuss the latest advancements made in the Uranus aerocapture study, such as investigating interplanetary trajectories with higher hyperbolic approach velocities and their implications on the aero heating and the TPS design, incorporating FNPAG (an advanced numerical-predictor guidance) and comparison of multiple navigation approaches. In addition, this talk will focus on mechanical design that can house more than one atmospheric probe and the six degrees of freedom simulation of aerocapture at Uranus. Findings from a recent NASA Space Technology Mission Directorate (STMD)-funded activity studying the aerocapture as an enabling technology for a Uranus orbiter will be presented. Using the science payload recommended by the Decadal Survey for Uranus exploration, this work shows many improvements over the baseline fully-propulsive mission. These improvements include a shorter cruise phase, flexibility in launch opportunities late into the 2030s while reaching Uranus before the 2050 equinox for the desired science opportunities, and lower propellant mass needs. This talk will highlight how aerocapture can be utilized for Uranus science orbit insertion using a lower-risk, heritage entry vehicle configuration used extensively as a Mars entry, descent, and landing vehicle. Furthermore, this talk will explore how the demonstration of aerocapture at Earth can benefit the aerocapture-enabled Uranus mission.

Aerocapture↗

Study of flutter related computational procedures for minimum weight structural sizing of advanced aircraft, supplemental data

Computational aspects of (1) flutter optimization (minimization of structural mass subject to specified flutter requirements), (2) methods for solving the flutter equation, and (3) efficient methods for computing generalized aerodynamic force coefficients in the repetitive analysis environment of computer-aided structural design are discussed. Specific areas included: a two-dimensional Regula Falsi approach to solving the generalized flutter equation; method of incremented flutter analysis and its applications; the use of velocity potential influence coefficients in a five-matrix product formulation of the generalized aerodynamic force coefficients; options for computational operations required to generate generalized aerodynamic force coefficients; theoretical considerations related to optimization with one or more flutter constraints; and expressions for derivatives of flutter-related quantities with respect to design variables.

Oconnell, R. F.↗

Calculation of AGARD Wing 445.6 Flutter Using Navier-Stokes Aerodynamics

The flutter characteristics of the first AGARD standard aeroelastic configuration for dynamic response, Wing 445.6, are studied using an unsteady Navier-Stokes algorithm in order to investigate a previously noted discrepancy between Euler flutter characteristics and the experimental data. The algorithm, which is a three-dimensional, implicit, upwind Euler/Navier-Stokes code (CFL3D Version 2.1), was previously modified for the time-marching, aeroelastic analysis of wings using the unsteady Euler equations. These modifications include the incorporation of a deforming mesh algorithm and the addition of the structural equations of motion for their simultaneous time integration with the governing flow equations. In this paper, the aeroelastic method is extended and evaluated for applications that use the Navier- Stokes aerodynamics. The paper presents a brief description of the aeroelastic method and presents unsteady calculations which verify this method for Navier-Stokes calculations. A linear stability analysis and a time-marching aeroelastic analysis are used to determine the flutter characteristics of the isolated 45 deg. swept-back wing. Effects of fluid viscosity, structural damping, and number of modes in the structural model are investigated. For the linear stability analysis, the unsteady generalized aerodynamic forces of the wing are computed for a range of reduced frequencies using the pulse transfer-function approach. The flutter characteristics of the wing are determined using these unsteady generalized aerodynamic forces in a traditional V-g analysis. This stability analysis is used to determine the flutter characteristics of the wing at free-stream Mach numbers of 0.96 and 1.141 using the generalized aerodynamic forces generated by solving the Euler equations and the Navier-Stokes equations. Time-marching aeroelastic calculations are performed at a free-stream Mach number of 1.141 using the Euler and Navier-Stokes equations to compare with the linear V-g flutter analysis method. The V-g analysis, which is used in conjunction with the time-marching analysis, indicates that the fluid viscosity has a significant effect on the supersonic flutter boundary for this wing while the structural damping and number of modes in the structural model have a lesser effect.

Lee-Rausch, Elizabeth M.↗

Investigation of a Technique for Measuring Dynamic Ground Effect in a Subsonic Wind Tunnel

To better understand the ground effect encountered by slender wing supersonic transport aircraft, a test was conducted at NASA Langley Research Center's 14 x 22 foot Subsonic Wind Tunnel in October, 1997. Emphasis was placed on improving the accuracy of the ground effect data by using a "dynamic" technique in which the model's vertical motion was varied automatically during wind-on testing. This report describes and evaluates different aspects of the dynamic method utilized for obtaining ground effect data in this test. The method for acquiring and processing time data from a dynamic ground effect wind tunnel test is outlined with details of the overall data acquisition system and software used for the data analysis. The removal of inertial loads due to sting motion and the support dynamics in the balance force and moment data measurements of the aerodynamic forces on the model is described. An evaluation of the results identifies problem areas providing recommendations for future experiments. Test results are validated by comparing test data for an elliptical wing planform with an Elliptical wing planform section with a NACA 0012 airfoil to results found in current literature. Major aerodynamic forces acting on the model in terms of lift curves for determining ground effect are presented. Comparisons of flight and wind tunnel data for the TU-144 are presented.

Graves, Sharon S.↗

Flight Dynamics of an Aeroshell Using an Attached Inflatable Aerodynamic Decelerator

An aeroelastic analysis of the behavior of an entry vehicle utilizing an attached inflatable aerodynamic decelerator during supersonic flight is presented. The analysis consists of a planar, four degree of freedom simulation. The aeroshell and the IAD are assumed to be separate, rigid bodies connected with a spring-damper at an interface point constraining the relative motion of the two bodies. Aerodynamic forces and moments are modeled using modified Newtonian aerodynamics. The analysis includes the contribution of static aerodynamic forces and moments as well as pitch damping. Two cases are considered in the analysis: constant velocity flight and planar free flight. For the constant velocity and free flight cases with neutral pitch damping, configurations with highly-stiff interfaces exhibit statically stable but dynamically unstable aeroshell angle of attack. Moderately stiff interfaces exhibit static and dynamic stability of aeroshell angle of attack due to damping induced by the pitch angle rate lag between the aeroshell and IAD. For the free-flight case, low values of both the interface stiffness and damping cause divergence of the aeroshell angle of attack due to the offset of the IAD drag force with respect to the aeroshell center of mass. The presence of dynamic aerodynamic moments was found to influence the stability characteristics of the vehicle. The effect of gravity on the aeroshell angle of attack stability characteristics was determined to be negligible for the cases investigated.

Cruz, Juan R.↗

Flutter analysis of supersonic axial flow cascades using a high resolution Euler solver. Part 1: Formulation and validation

This report presents, in two parts, a dynamic aeroelastic stability (flutter) analysis of a cascade of blades in supersonic axial flow. Each blade of the cascade is modeled as a typical section having pitching and plunging degrees of freedom. Aerodynamic forces are obtained from a time accurate, unsteady, two-dimensional cascade solver based on the Euler equations. The solver uses a time marching flux-difference splitting (FDS) scheme. Flutter stability is analyzed in the frequency domain. The unsteady force coefficients required in the analysis are obtained by harmonically oscillating (HO) the blades for a given flow condition, oscillation frequency, and interblade phase angle. The calculated time history of the forces is then Fourier decomposed to give the required unsteady force coefficients. An influence coefficient (IC) method and a pulse response (PR) method are also implemented to reduce the computational time for the calculation of the unsteady force coefficients for any phase angle and oscillation frequency. Part 1, this report, presents these analysis methods and their validation by comparison with results obtained from linear theory for a selected flat plate cascade geometry. A typical calculation for a rotor airfoil is also included to show the applicability of the present solver for airfoil configurations. The predicted unsteady aerodynamic forces for a selected flat plate cascade geometry and flow conditions correlated well with those obtained from linear theory for different interblade phase angles and oscillation frequencies. All the three methods of predicting unsteady force coefficients, namely, HO, IC, and PR, showed good correlations with each other. It was established that only a single calculation with four blade passages is required to calculate the aerodynamic forces for any phase angle for a cascade consisting of any number of blades, for any value of the oscillation frequency. Flutter results, including mistuning effects, for a cascade of stator airfoils are presented in Part 2 of the report.

Reddy, T. S. R.↗

An analytical technique for approximating unsteady aerodynamics in the time domain

An analytical technique is presented for approximating unsteady aerodynamic forces in the time domain. The order of elements of a matrix Pade approximation was postulated, and the resulting polynomial coefficients were determined through a combination of least squares estimates for the numerator coefficients and a constrained gradient search for the denominator coefficients which insures stable approximating functions. The number of differential equations required to represent the aerodynamic forces to a given accuracy tends to be smaller than that employed in certain existing techniques where the denominator coefficients are chosen a priori. Results are shown for an aeroelastic, cantilevered, semispan wing which indicate a good fit to the aerodynamic forces for oscillatory motion can be achieved with a matrix Pade approximation having fourth order numerator and second order denominator polynomials.

Dunn, H. J.↗

High angle of attack hypersonic aerodynamics

A new aerodynamics force model is presented which is based on modified Newtonian theory and empirical correlations. The algebraic model was developed for complete vehicles from take off to orbital speeds and for large angles of attack. Predictions are compared to results for a wind tunnel model at a Mach number of 20, and the full scale Shuttle Orbiter for Mach numbers from 0.25 to 20 for angles of attack from 0 to 50 deg. The maximum shuttle orbiter lift/drag at Mach 10 and 20 is 1.85 at 20-deg angle-of-attack. Aerodynamic force predictions are made for a transatmospheric vehicle, which is a derivative of the Shuttle Orbiter, for Mach numbers from 4 to 27 at angles of attack from 5 to 40 deg. Predicted aerodynamic force data indicate that lift/drag ratios of 5.2 at Mach number 10 and 3.6 at Mach number 26 are obtainable. Changes in force coefficients with changes in: nose angle, sweep angle, and (volume exp 2/3)/planform area are quantified for Mach numbers of 10 and 26. Lift/drag ratios increase with decreasing nose angle and (volume exp 2/3)/planform area and increasing wing sweep angle. Lift/drag ratios are independent of these variables for angles of attack in excess of 20 deg at Mach 10 and 30 deg at Mach 26.

Harloff, Gary J.↗

Space Shuttle flutter as affected by wing-body aerodynamic interaction

In the NASA Langley Research Center 26-inch transonic blowdown wind-tunnel, flutter speeds were measured on 1/80-th scale semispan models of the orbiter wing, the complete Space Shuttle, and intermediate component combinations. Using the doublet lattice method combined with slender body theory to calculate unsteady aerodynamic forces, subsonic flutter speeds were computed for comparison. Aerodynamic interaction was found by test and analysis to raise the flutter speed in some configurations while lowering it in others. Although at Mach number less than 0.7, predicted speeds correlated to within 6% of those measured, rapid deterioration of the agreement occurred at higher subsonic Mach numbers, especially on the more complicated configurations. Additional analysis showed that aerodynamic forces arising from body flexibility potentially can have a large effect on flutter speed, but that the current shuttle design is not so affected.

Chipman, R. R.↗

Linearized Aeroelastic Solver Applied to the Flutter Prediction of Real Configurations

A fast-running unsteady aerodynamics code, LINFLUX, was previously developed for predicting turbomachinery flutter. This linearized code, based on a frequency domain method, models the effects of steady blade loading through a nonlinear steady flow field. The LINFLUX code, which is 6 to 7 times faster than the corresponding nonlinear time domain code, is suitable for use in the initial design phase. Earlier, this code was verified through application to a research fan, and it was shown that the predictions of work per cycle and flutter compared well with those from a nonlinear time-marching aeroelastic code, TURBO-AE. Now, the LINFLUX code has been applied to real configurations: fans developed under the Energy Efficient Engine (E-cubed) Program and the Quiet Aircraft Technology (QAT) project. The LINFLUX code starts with a steady nonlinear aerodynamic flow field and solves the unsteady linearized Euler equations to calculate the unsteady aerodynamic forces on the turbomachinery blades. First, a steady aerodynamic solution is computed for given operating conditions using the nonlinear unsteady aerodynamic code TURBO-AE. A blade vibration analysis is done to determine the frequencies and mode shapes of the vibrating blades, and an interface code is used to convert the steady aerodynamic solution to a form required by LINFLUX. A preprocessor is used to interpolate the mode shapes from the structural dynamics mesh onto the computational fluid dynamics mesh. Then, LINFLUX is used to calculate the unsteady aerodynamic pressure distribution for a given vibration mode, frequency, and interblade phase angle. Finally, a post-processor uses the unsteady pressures to calculate the generalized aerodynamic forces, eigenvalues, an esponse amplitudes. The eigenvalues determine the flutter frequency and damping. Results of flutter calculations from the LINFLUX code are presented for (1) the E-cubed fan developed under the E-cubed program and (2) the Quiet High Speed Fan (QHSF) developed under the Quiet Aircraft Technology project. The results are compared with those obtained from the TURBO-AE code. A graph of the work done per vibration cycle for the first vibration mode of the E-cubed fan is shown. It can be seen that the LINFLUX results show a very good comparison with TURBO-AE results over the entire range of interblade phase angle. The work done per vibration cycle for the first vibration mode of the QHSF fan is shown. Once again, the LINFLUX results compare very well with the results from the TURBOAE code.

Reddy, Tondapu S.↗

Improvements to the SADSAM computer program for aeroelasticity analysis

A computer program for calculating the elastic element forces and strain energy distribution for normal mode shapes is presented. The subjects discussed are: (1) revised aerodynamic force representation, (2) flap aerodynamics, (3) chordwise motions and aerodynamic forces, (4) wing-tail interaction, and (5) gust analysis. An aircraft configuration used in sample gust analysis is included.

Schwendler, R. G.↗

Aeolian Sand Transport in the Planetary Context: Respective Roles of Aerodynamic and Bed-Dilatancy Thresholds

The traditional view of aeolian sand transport generally estimates flux from the perspective of aerodynamic forces creating the airborne grain population, although it has been recognized that "reptation" causes a significant part of the total airborne flux; reptation involves both ballistic injection of grains into the air stream by the impact of saltating grains as well as the "nudging" of surface grains into a creeping motion. Whilst aerodynamic forces may initiate sand motion, it is proposed here that within a fully-matured grain cloud, flux is actually governed by two thresholds: an aerodynamic threshold, and a bed-dilatancy threshold. It is the latter which controls the reptation population, and its significance increases proportionally with transport energy. Because we only have experience with terrestrial sand transport, extrapolations of aeolian theory to Mars and Venus have adjusted only the aerodynamic factor, taking gravitational forces and atmospheric density as the prime variables in the aerodynamic equations, but neglecting reptation. The basis for our perspective on the importance of reptation and bed dilatancy is a set of experiments that were designed to simulate sand transport across the surface of a martian dune. Using a modified sporting crossbow in which a sand-impelling sabot replaced the bolt-firing mechanism, individual grains of sand were fired at loose sand targets with glancing angles typical of saltation impact; grains were projected at about 80 m/s to simulate velocities commensurate with those predicted for extreme martian aeolian conditions. The sabot impelling method permitted study of individual impacts without the masking effect of bed mobilization encountered in wind-tunnel studies. At these martian impact velocities, grains produced small craters formed by the ejection of several hundred grains from the bed. Unexpectedly, the craters were not elongated, despite glancing impact; the craters were very close to circular in planform. High-speed photography showed them to grow in both diameter and depth after the impactor had ricochetted from the crater site. The delayed response of the bed was "explosive" in nature, and created a miniature ejecta curtain spreading upward and outward for many centimeters for impact of 100-300 micron-diameter grains into similar material. Elastic energy deposited in the bed by the impacting grain creates a subsurface stress regime or "quasi-Boussinesq" compression field. Elastic recovery of the bed occurs by dilatancy; shear stresses suddenly convert the grains from closed to open packing, and grains are consequently able to eject themselves forcefully from the impact site. Random jostling of the grains causes radial homogenization of stress vectors and a resulting circular crater. There is a great temptation to draw parallels with cratering produced by meteorite impacts, but a rigorous search for common modelling ground between the two phenomena has not been conducted at this time. For every impact of an aerodynamically energized grain, there are several hundred grains ejected into the wind for the high-energy transport that might occur on Mars. Many of these grains will themselves become subject to the boundary layer's aerodynamic lift forces (their motion will not immediately die and add to the creep population), and these grains will become indistinguishable from those lifted entirely by aerodynamic forces. As each grain impacts the bed, it will eject even more grains into the flow. A cascading effect will take place, but because it must be finite in its growth, damping will occur as the number of grains set in motion causes mid-air collisions that prevent much of the impact energy from reaching the surface of the bed -thus creating a dynamic equilibrium in a high-density saltation cloud. It is apparent that for a given impact energy, the stress field permits a smaller volume of grains to convert to open packing as the size of the bed grains increases, or as the energy of the "percussive" grain decreases (by decrease in velocity or mass). Thus, the mass of the "repercussive" grain population that is ejected from the impact site becomes a function of the scale of the stress field in relation to the scale of the bed material (self-similarity being applicable if both bed size and energy are simultaneously adjusted). In other words, in a very high energy aeolian system where an aerodynamically raised grain can ballistically raise many more grains, the amount of material lifted into the wind becomes largely a function of a dilatancy threshold. If this threshold is exceeded, grains are repercussively injected into the saltation cloud. The "dilatancy threshold" may be defined in terms of the saltation percussive force required to convert the bed, through elastic response, from a closed to an open packing system. If open packing cannot be created, the grains cannot escape from the impact site, even though the elastic deformation and percussive force may be able to reorganize the grains with respect to one another. As the crossbow experiments showed, for an ever-increasing bed grain size, a point is reached when no material can be moved because the energy of the percussive grain is insufficient to dilate the relatively coarse bed. Although this seems to be stating the obvious -- that too little energy will not cause the bed to splash -- the consequences of exceeding the "splash threshold" by dilatancy are not so obvious for high-energy aeolian transport. It is noted that the force required to elastically dilate the bed has to overcome Coulombic grain attractions such as dipole-dipole coupling, dielectric, monopole, contact-induced dipole attractions, van der Waals forces, molecular monolayer capillary forces, as well as the mechanical interlocking frictional resistance of the grains. On Mars, it is predicted that the dilatancy threshold may be the prime control of grain flux. Additional information is contained in the original.

Marshall, J. R.↗

Semi-empirical model for prediction of unsteady forces on an airfoil with application to flutter

A semi-empirical model is described for predicting unsteady aerodynamic forces on arbitrary airfoils under mildly stalled and unstalled conditions. Aerodynamic forces are modeled using second order ordinary differential equations for lift and moment with airfoil motion as the input. This model is simultaneously integrated with structural dynamics equations to determine flutter characteristics for a two degrees-of-freedom system. Results for a number of cases are presented to demonstrate the suitability of this model to predict flutter. Comparison is made to the flutter characteristics determined by a Navier-Stokes solver and also the classical incompressible potential flow theory.

Mahajan, Aparajit J.↗