Aerodynamic Equations Research Articles

The Distribution of Surface Fluxes and Boundary Layer Divergence in Midlatitude Ocean Storms

Abstract Surface meteorological observations have been used to calculate the distributions of surface fluxes of momentum, sensible heat, and latent heat and the distributions of surface divergence and curl of surface stress which are characteristic of ocean storms in the Gulf of Alaska. Flux calculations were based on aerodynamic equations which include the effects of high wind speed and stratification of the surface layer. Distributions of divergence determined directly from surface winds were compared with those calculated using the curl of the surface stress. Results indicate that heat fluxes were weak or oven negative for the most intense storms in a band about 300 km wide immediately east of strong cold or occluded fronts. Maximum upward latent heat fluxes occurred to the west of the front, with secondary maxima along the eastern edge of the storm, beyond the 300 km band of weak flux. Surface convergence occurred to the east and divergence to the west of cold or occluded fronts. Vertical velocities c...

A Simple Method for Solving Three-Dimensional Inverse Problems of Turbomachine Flow and the Annular Constraint Condition

In 1950, Wu [1] suggested that the three-dimensional inverse (design) problem of turbomachine flow may be solved approximately by a Taylor series expansion in the circumferential direction based on the known flow variation over an S2 stream surface in the midchannel of the blade passage. This idea has been realized recently. A new coordinate transformation has been developed. The coordinate surfaces are coincident with the S2 stream surfaces. This transformation leads to a new method to calculate the aerodynamic equations of three-dimensional flow. By the use of this transformation, a high-order expansion is realized to determine the shape of the blade surfaces from the fluid state on the S2m stream surface directly. Computation in this manner soon leads to the discovery that theoretically the distribution of flow parameter (usually Vθr) on S2m prescribed by the designer should satisfy a constraint condition, which guarantees that the S1 stream surfaces along the hub and shroud obtained from circumferential extension of the S2m surface are surfaces of revolution. An approximate method is suggested to meet this condition.

Design Considerations for Aerodynamically Quenching Gas Sampling Probes

An aerodynamic quench is the most rapid method for quenching temperature and pressure-dependent chemical reactions. Attempts have been made to quench gas samples aerodynamically, but many of these attempts have been unsuccessful because of a lack of understanding of the internal aerodynamics of sampling probes. A one-dimensional model developed previously by the authors has been used for the design and analysis of aerodynamically quenching probes. This paper presents in detail the important aerodynamic and heat transfer equations used in the model, a description of the method of solution, and the results of a sensitivity study. These calculations demonstrate the limitations and important trade-offs in design and operating conditions of probes using an aerodynamic quench.

Aeroelastic time response analysis of thin airfoils by transonic code LTRAN2

A computer based numerical procedure is used to perform aeroelastic time response analysis of thin airfoils oscillating with single and two d.o.f.'s in two-dimensional, unsteady, low-frequency, small-disturbance, transonic potential flow. A computational transonic code LTRAN2 based on fully implicit time integration scheme is employed to obtain the aerodynamic forces. Structural equations of motion and the aerodynamic equations are integrated simultaneously by a numerical method. Results are first obtained for a time response example for a flat plate oscillating at M ∞ = 0.7 with (1) a single pitching d.o.f.; (2) a single plunging d.o.f.; and (3) two d.o.f.s—plunging and pitching. A parallel set of results are also obtained based on subsonic aerodynamic theory for comparison. The agreement is good. Results are then obtained for a time response example for a NACA 64A006 airfoil pitching and plunging at M ∞ = 0.85 which was found to give the lowest flutter speed for the case considered. The parameters that result in neutrally stable response agree with those found in a separate flutter analysis. It was found through comparison between flutter results and response analysis that the principle of superposition of airloads is valid for the present example cases. The results also include the effect of altitude on aeroelastic response.

Estimating Monthly Averaged Air-Sea Transfers of Heat and Momentum Using the Bulk Aerodynamic Method

Abstract Air-sea transfers of sensible heat, latent heat and momentum are computed from 25 years of middle-latitude and subtropical ocean weather ship data in the North Atlantic and North Pacific using the bulk aerodynamic method. The results show that monthly averaged wind speeds, temperatures and humidities can be used to estimate the monthly averaged sensible and latent heat fluxes from the bulk aero-dynamic equations to within a relative error of ∼10%. The estimates of monthly averaged wind stress under the assumption of neutral stability are shown to be within ∼5% of the monthly averaged non-neutral values.

Time-Dependent Response of a Two-Dimensional Airfoil in Transonic Flow

A procedure is developed for the aeroelastic analysis of a two-dimensional airfoil in transonic flow. The fluid is assumed to be described by the unsteady low-frequency small-disturbance transonic potential equation for which a fully time-implicit integration scheme exists. Structural equations of motion are integrated in time simultaneously with the potential equation in order to predict the unsteady airfoil motion. As a computational example, a three-degree-o f-freedom NACA 64A010 airfoil is considered using representative values of the structural parameters. The method is shown to be both stable and accurate, and the time response for several choices of initial conditions and reduced freest ream density is presented. Oscillations with either growing or decaying amplitudes are indicated depending upon the prescribed initial conditions. OR the case of flow over an airfoil in a freestream at Mach numbers near 1, small amplitude motions of the body surface can produce large variations in the aerodynamic forces and moments acting on the structure. In addition, phase differences between the flow variables and the resultant forces may be great. These characteristics tend to enhance the probability of encountering aeroelastic instabilities in the transonic flow regime, and thus evidence a need for tech- niques of analyzing the coupled unsteady flowfield and resultant structural response in such situations. In the subsonic and supersonic cases, the governing flow equations may be linearized such that the aerodynamic forces depend upon the body motion in a linear fashion. Moreover, the resultant forces acting on the airfoil may be obtained through superposition by summing the contributions due to each of the various types of body motion permitted. This allows the linear structural equations of motion to be solved independent of the governing aerodynamic equations which provide only the force coefficients. Uncoupling of the fluid and structural equations is not, in general, possible for the transonic regime due to its inherent nonlinear nature. Recent advances in computational methods have made several approaches available for computing unsteady tran- sonic flows. While a number of different techniques have evolved and various physical problems have been con- sidered, M6 the unsteady body motion was generally prescribed as a known function of time, thereby precluding the simulation of true aeroelastic behavior. Only more recently have these procedures been applied to actual aeroelastic problems.17'18 It is the intent here to describe a method for obtaining the time-dependent response of a two- dimensional airfoil in transonic flow and to provide a computational example by applying this technique to a physical situation of practical interest. The governing aerodynamic equation of motion is assumed to be the unsteady low-frequency small-disturbance transonic equation for the velocity potential function which is capable of simulating nonlinear flow phenomena including irregular shock wave motions. Solutions to this equation have corn-

Aerodynamics of slender lifting surface in accelerated flight

HERE are many researches into the aerodynamics of lifting surfaces in unsteady motion. Most of these, however, concern out-of-plane motions at a uniform flight speed, with few studies for nonuniform flight speeds.1'2'3 These problems have become increasingly more important, because recent airplanes, such as STOL or supersonic, can experience considerable acceleration during the take-off climb or landing approach. More precise estimation of take-off or landing distance would require some aerodynamic theories for accelerated flight. In Ref. 4, a fundamental formulation was made for wings in nonuniform motion in an inviscid incompressible flow. A Fourier transform for the in-plane coordinate variables was used. In Ref. 5, application was made to two-dimension al airfoils with a sinusoidally pulsating speed. A significant result was the fact that the difference between unsteady and quasi-steady lifts is significant, as was also shown by the previous writers.2'3 In general, the lift of a wing depends on the complete history of its motion, and not just on the instantaneous acceleration. In this paper, we treat the problem of slender wings in accelerated motion. This problem also is solvable and represents the low aspect ratio limit of wing theory opposite that of the two-dimensional problem. An interesting result has been obtained in that the lift force for slender wings is shown to depend only on the instantaneous acceleration and not on the history of its motion. An acceleration (deceleration) increases (decreases) the lift from that for the uniform flight speed. II. General Formulation In this section, a general formulation is given for the case of a wing in nonuniform motion (flight speed) in an inviscid fluid. The analysis employs a moving axes system fixed to the wing. According to Reissner, 6 the unsteady linear equations of aerodynamics expressed in a moving axes system can be written as follows:

Computations of Surface Energy Flux and Annual Air–Sea Interaction Cycles of the North Atlantic Ocean

Abstract Energy fluxes through the surface of the North Atlantic Ocean have been calculated using bulk aerodynamic equations with exchange coefficients which vary with wind speed and stability, and radiation equations. Values of the coefficients were determined from published results of many experiments made at sea, coastal installations and in laboratories. Eight million ship weather reports were used for flux calculations. Reliabilities of monthly meteorological and flux averages are found to be good in areas having SW or move observations per average. A chart of the net annual heat gain by the ocean shows regions with characteristic gains or losses such as the Gulf Stream, Norwegian Sea, Labrador Current, trade winds and upwelling areas. Annual cycles of fluxes, winds, temperatures, clouds and rainfall in eight regions are presented and discussed from the point of view of predominating air–sea interaction processes. Contributions of meteorological variables, oceanic currents, atmospheric circulations a...

Energy Exchange Charts of the North Atlantic Ocean

Abstract Charts of calculated energy exchange across the surface of the North Atlantic Ocean have been constructed. Wind and temperature observations obtained from 8 million ship weather reports were entered individually into the bulk aerodynamic equations with exchange coefficients that varied with wind speed and stability. The individual fluxes were averaged to obtain monthly and annual means of latent and sensible heat momentum. Net radiation fluxes were calculated using Budyko's (1963) formulas. Monthly and annual averages for 32 years have been formed for 500 subdivisions of the ocean. Averages for each month from 1941 through 1972 were computed for 66 10° squares to study the variations and anomalies of the fluxes. meteorological variables, and sea temperature. Charts giving annual averages of the net heat gain by the ocean, evaporation, sensible and radiational heat exchange, wind stress components, and meteorological variables are presented. A graph of the monthly variations for Marsden Square 116...

Rapid Finite-Difference Computation of Subsonic and Slightly Supercritical Aerodynamic Flows

Rapid iterative (or semidirect) computation methods are developed for highly efficient finite-difference solution of the nonlinear equations of subsonic and transonic aerodynamics. Applicability is restricted to subsonic through slightly supercritical conditions. At each iteration, a fast, direct elliptic algorithm solves the entire computation field. A special acceleration device for the iterative convergence is developed. The methods are applied, using a recently developed direct Cauchy-Riemann solver, to the nonlinear transonic small-disturbance equations for a biconvex airfoil. At M = 0.7, t/c = 0.1 (subcritical), three iterations on a 39 x 32 mesh (totalling 2.45 sec on an IBM 360/67 computer) obtain convergence within 0.1%. A slightly supercritical case requires seven iterations (6.75 sec) for convergence within 1%. A very recent revision of the Cauchy-Riemann solver reduces these computing times by more than 30%. Strong supercritical cases do not converge, but the results, including the small computation times and the computational accuracy, demonstrate the usefulness of these techniques in subsonic flow computations through the critical condition and suggest the potential usefulness of further developments for transonic flow.

Modeling implications of measurements on excised larynxes

Results of experiments with excised canine larynxes are examined within the framework of a two-mass model of the larynx, and the implications of the results for laryngeal modeling in general are considered. Sketches of glottal shapes were constructed on the basis of detailed measurements during a glottal cycle [J. Acoust. Soc. Am. 55, S79 (1974)] and during static experiments. A two-section model was fit to these sketches, and the resulting fits were used to estimate values for some parameters of the two-mass model. To the extent that analysis was possible, the aerodynamic equations of the model were found to be adequate. However, the design of the mechanical system, consisting of two coupled masses capable of only horizontal excursions, was not able to account for some experimental results. A description of the vibrations in terms of a propagating membrane or surface wave on the vocal-fold walls as well as string vibrations of the vocal ligament is proposed. Representation of these phenomena is discussed.

Comparison of aerodynamic and energy budget estimates of fluxes over a pine forest

AbstractValues of the total vertical flux of sensible and latent heat over a level forested region, obtained from aerodynamic (profile‐gradient) formulae appropriate to airflow over relatively smooth surfaces, are found to fall consistently short of independent energy‐balance estimates by a factor of 2 to 3 in unstable and near‐neutral conditions (Richardson number, Ri, in the range −0·4 to +0·01), whereas for Ri > + 0·02 no similar discrepancy is detected. These results, based on tangents drawn to (semi‐logarithmic) profiles at a height of about nine aerodynamic roughness parameters (z0) above the zero plane displacement level (d) of the forest, rely on the basic assumption that the value of d established in very nearly neutral conditions (|Ri| < 0·003), namely 0·76 mean tree heights, holds under all conditions of thermal stability.Wake diffusion and thermal seeding effects are discussed as possible additional transfer mechanisms acting to reduce profile gradients immediately over aerodynamically rough surfaces. In terms of the former mechanism (assumed to operate below d + 25z0, or so), approximate empirical formulae are derived which attempt to quantify the observed discrepancy in terms of Ri and the proximity of the surface.It is concluded that aerodynamic equations ought not to be used to give independent flux estimates close to aerodynamically rough surfaces.

Comparison of aerodynamic and energy budget estimates of fluxes over a pine forest

Values of the total vertical flux of sensible and latent heat over a level forested region, obtained from aerodynamic (profile-gradient) formulae appropriate to airflow over relatively smooth surfaces, are found to fall consistently short of independent energy-balance estimates by a factor of 2 to 3 in unstable and near-neutral conditions (Richardson number, Ri, in the range −0·4 to +0·01), whereas for Ri > + 0·02 no similar discrepancy is detected. These results, based on tangents drawn to (semi-logarithmic) profiles at a height of about nine aerodynamic roughness parameters (z0) above the zero plane displacement level (d) of the forest, rely on the basic assumption that the value of d established in very nearly neutral conditions (|Ri| < 0·003), namely 0·76 mean tree heights, holds under all conditions of thermal stability. Wake diffusion and thermal seeding effects are discussed as possible additional transfer mechanisms acting to reduce profile gradients immediately over aerodynamically rough surfaces. In terms of the former mechanism (assumed to operate below d + 25z0, or so), approximate empirical formulae are derived which attempt to quantify the observed discrepancy in terms of Ri and the proximity of the surface. It is concluded that aerodynamic equations ought not to be used to give independent flux estimates close to aerodynamically rough surfaces.

A quasi-empirical model of the hurricane boundary layer

Solutions have been obtained for the temperature and moisture distributions within the atmospheric boundary layer of an axisymmetric hurricane model. The intensity of the hurricane is related to the equivalent potential temperature gradient produced by a balance of heat sources and advection within the boundary layer. Solutions are obtained by using the bulk aerodynamic transport equations or applying a two-layer baroclinic boundary layer model by Cardone (1969). Equilibrium maximum wind speeds vary from minimal hurricane force to about 80 m/s for fixed ocean temperatures between 27° and 31°C. Observations of the oceanic heat loss in actual storms are necessary to establish the ratio of heat flux to momentum transfer in the high wind speeds treated by the model. Various applications of the model are proposed with an ultimate goal of a time-dependent simulation of hurricane-ocean coupling.

Use computers to demonstrate the differential equations of aerodynamics

A majority of engineering students are apprehensive about the first courses they take in which they are expected to solve differential equations. Fluid mechanics in aeronautical engineering is perhaps the most difficult engineering specialty from the point of view of differential equations. For this reason, aeronautics instructors should become familiar with and use techniques which make the solutions to differential equations less frightening to the students. Versatile computer techniques to demonstrate the characteristics of solutions to differential equations and the effect of boundary conditions on their solutions are currently available for student use. Several of the computational techniques not requiring an extensive computer background are discussed.

Generalization of basic equations of aerodynamics and electrodynamics

Generalization of basic equations of aerodynamics and electrodynamics

Generalization of basic equations of aerodynamics and electrodynamics

Generalization of basic equations of aerodynamics and electrodynamics

Hypersonic lifting body optimization.

Lifting hypersonic configurations were optimized to produce maximum lift-drag ratios. Simplified aerodynamic equations (with Newtonian pressure and laminar skin friction proportional to the inverse root of length) were written for a generalized shape with ten degrees of geometric freedom; a numerical searching routine was employed to ascertain the optimum vehicle. Constraints of volumetric efficiency, nose-heating limit, and skin friction were imposed singly and in combination. In general, the derived configurations tended towards delta planforms when the leading-edge bluntness drag exceeded the skin-friction drag, and for the alternate situation, the planforms were trapezoidal. Geometric details of the optimal configurations are tabulated for each set of constraints. Experimental correlation of the mathematical model for the aerodynamic characteristics was obtained by conducting a wind-tunnel test at a Mach number of 14 to obtain the lift-drag ratio. The results provided confidence in the adequacy of the formulation.

Approximate methods for solving nonlinear differential equations in hypersonic aerodynamics

Approximate methods for solving nonlinear differential equations in hypersonic aerodynamics

風の影響を受けない気圧計感部の開発

Wind tunnel experiments have been carried out to determine the design of a barometric sensor insensitive to high winds, particularly from necessity for cosmic ray observations. Four different types of models are tested in the wind tunnel of Japan Meteorological Agency for a wind speed range up to 40m/sec, One of them is found to keep the pressure deviation coefficient C in an aerodynamic equation ΔP=C1/2ρV2 always within 0.04 in absolute value, corresponding to a pressure deviation of 0.4mb at 40m/sec, for a zenith angle range of ±40°and for an azimuth angle range of ±20°.