Normal Force Coefficient Research Articles

Predicting the supersonic aerodynamics of very-low-aspect-ratio lifting surfaces

Improved methods for predicting aerodynamics of lifting surfaces are needed at the very low aspect ratios found in wing (housing) designs of span limited missiles and at moderate supersonic to hypersonic Mach numbers. A successful empirical method for estimating the normal force coefficient, CN, and center of pressure location, Xcp, for this class of surfaces has been developed for unbanked ( = 0 deg) cruciform wing configurations and for wing configurations banked at = 45 deg. The values of CN and X^ obtained using these empirical curves provide good estimates at Mach numbers above 2.5, and for angles-of-attack to 20 deg for thick and thin surfaces mounted in cruciform on cylindrical bodies.

A study on the behavior of separated flow around elliptic airfoils and the unsteady aerodynamic forces under dynamic stall. (2nd report, Numerical analysis. Part 2)

The stalling characteristics of 15% thickness elliptic airfoils in incompressible turbulent flows at a Reynolds number of 5×105 are investigated numerically by solving the Reynolds equations and the boundary-layer equations. Calculations were performed on the airfoils impulsively started at high angles of attack and oscillating in pitch with large amplitude. The development of the separated flows and the behaviour of the forces are studied in terms of patterns of streamlines, vectors of velocity and lines of constant vorticity, and normal force and moment coefficients it is observed from the results that the separated flows over impulsively started airfoils at high angles of attack show oscillatory behaviour.

Effect of edge-tone noise on supercritical airfoil data

A N investigation has recently been carried out in the National Aeronautical Establishment (NAE) 0.38x1.5 m two-dimensional test section1 to determine: 1) The possibility of edge-tone noise suppression by overlaying the perforated floor and ceiling with a fine gauze or screen. 2) The change in wall interference characteristics due to the application of point 1. 3) The effect of edge-tone noise suppression on twodimensional airfoil test data. The investigation was performed with a 0.254 m chord model of the BGK No. 1 airfoil and covered the Mach number range of 0.3-0.8 at chord Reynolds numbers of 10-21 x 106. A full account of the investigation is given in Ref. 2. This Note deals primarily with selected data related to point 3. However, first some brief comments on points 1 and 2. The fine gauze technique, first demonstrated by Vaucheret,3 was found to be both a practical and an effective means for edge-tone suppression, as demonstrated in Figs. 1 and 2, respectively. It was also found that the presence of the gauze, although reducing the geometric porosity from 20.5 to 8%, had a very minor effect on the wall interference characteristics. Measurements on the BGK No. 1 airfoil comprised balance, wake, and surface pressure. Only the latter two are discussed here. The data depicted in Fig. 2 were obtained by a sidewallmounted differential pressure transducer, with the reference side detuned and connected to the plenum chamber. The frequency response of the active side was flat up to 16 kHz. The data are presented in the form of the root mean square of the fluctuating wall static pressure coefficient, Cpj^, referenced to the mean value of the freestream dynamic pressure. The significant reduction in noise level due to the edge-tone suppression is clearly evident. In fact, at the higher Mach number the measured noise level is reduced to a level comparable to that inherent in the wall boundary layer itself. For a well-developed turbulent boundary layer, the wall CPRMS (referenced to the dynamic pressure at the edge of the boundary layer) is known to be about 0.6-0.7%. To what degree then are model test data influenced by this improvement in flow quality? The obvious thought that comes to mind is that the edge tones could excite the TollmienSchlichting waves and thus cause early transition. A cursory investigation reveals that, at the Reynolds numbers here of interest, the dangerous frequencies are an order of magnitude higher than the edge-tone frequencies. Thus the edge tones are not expected to influence the boundary-layer transition. However, we may speculate that they could interfere with, for instance, a turbulent boundary layer close to separation, although the mechanism in such a case would be far from clear. We shall first examine drag data obtained from wake pressure measurements. In Fig. 3 we have plotted the drag coefficient vs freestream Mach number, both corrected for wall interference effects, for two values of the normal force coefficient CN, 0.3 and 0.6. The data points are obtained by a

Equivalent angle-of-attack method for estimating nonlinear aerodynamics of missile fins

A method has been developed for estimating the nonlinear aerodynamic characteristics of missile wing and control surfaces. The method is based on the assumption that if a fin on a body has the same normal-force coefficient as a wing alone composed of two of the same fins joined together at their root chords, then the other force and moment coefficients of the fin and the wing alone are the same, including the nonlinearities. The method can be used for deflected fins at arbitrary bank angles and at high angles of attack. In this paper a full derivation of the method is given, its accuracy is demonstrated, and its use in extending missile data bases is shown.

On the Nonlinear Lift Characteristics of Slender Bodies at Incidence

Nonlinearity in normal force for a slender body with small incidence to flow is discussed by the asymptotic expansion method under the usual assumptions, 1>>e>>1/√Rn and k=β/e=O (1), where e is a slenderness parameter, β an angle of attack, k relative angle of attack, and Rn Reynolds number. The main purpose of the present paper is to develop the method to determine the separation line and the strength of the vortex sheet separating from there. The near field is composed of continuous vortex sheet whose complex potential is expressed with the circulation as a parameter, which is measured from the end point of vortex sheet to the point considered. Kutta's condition to be used to determine the strength density of the vortex sheet near the separation line is proposed by investigating the characteristics of the governing equation of vortex sheet shape. It is shown that this condition is consistent with the vortex conservation law. A method is proposed to solve simultaneously potential stream-lines and boundry layer parameters including the effects of the vortex sheet by only one marching downstream. The shape of the vortex sheet and the separation line are calculated and the normal force coefficient is compared with experimental results.

Three-dimensional supersonic flow of ideal gas past a body with tail fins

The results are given of numerical and experimental investigations of the steady supersonic flow of an ideal gas past a model of a body with + and X fins. The angle of attack varied from 0 to 20 °. A study is made of the physical structure of the flow, and the pressure distributions, the coefficients of the normal force and the pitching moment of the complete body and the individual fins, and also the increment of the coefficient of the normal force acting on the body due to the influence of the fins are found. A comparison with linear theory and the experimental data is made. The agreement between the calculated results and the experimental data is satisfactory.

Improved Potential Gradient Method to Calculate Airloads on Oscillating Supersonic Interfering Surfaces

sectional normal force coefficient in AGARD notation, Eq. (7) = sectional pitching moment about quarter-chord point coefficient in AGARD notation, Eq. (8), C*mi is moment about midchord = variation of the potential jump behind the leading edge of an element on which a constant potential gradient in streamwise direction is assumed (Fig. Ib) = amplitude of the normal displacement of the oscillating wing = reduced frequency = ul/ U supersonic pressure dipole kernel = reference length = Mach number = normal directions of receiving wings = total number of unknowns = dynamic pressure = /2pU generalized aerodynamic coefficients , generalized forces, Eq. (6)

The differences between a wing and a planing plate in two-dimensional flow

The normal force coefficient on a flat planing surface having arbitrary heave and pitch motion in two-dimensional flow is compared with the lift coefficient of a thin wing in an infinite fluid. Despite the totally different derivations, they are found to be identical (at large Froude numbers and low trim angles and allowing for the wing's interaction with twice as much fluid) at low reduced frequencies. For higher frequency motions, the wing's angle of attack induced lift and its pitch and heave damping are less than those of a planing surface, but the acceleration terms remain identical. The differences at the higher reduced frequencies are due to the fact that, in invisad irrotational flow, the planning plate cannot leave a vortex wake, whereas a wing does. It seems to follow that the “virtual mass” planing hull analysis can be applied to “quasi-static” problems involving wings and bodies in an infinite fluid without the slenderness restriction originally imposed by Jones (1946). Certainly, it is remarkable that the so called “quasi-steady” forces on a two-dimensional wing can be obtained in a few lines of elementary analysis. On the other hand, the method fails entirely when used to compute the pitching moment on a two-dimensional plate, even though it has been found to give good results for the three-dimensional case (Payne, 1981c). This work is offered as a very incomplete study of an intriguing relationship between two very different bodies of analysis. Much more work will need to be done before the relationship between the two approaches will be fully understood.

Nonlinearity in Lift of Slender Bodies at Incidenc

Nonlinearity in normal (lateral) force for a slender body with small incidence to flow is discussed by the asymptotic expansion method under the assumptions, 1>>β>>ε>>1/√Rn and (ε/β) 1/2>>β, where β is an angle of attack, ε a slenderness parameter, and Rn the Reynolds number. The theory is developed utilizing a slender body theory technique. The near field is composed of two vortex layers extending downstream infinitely in cross flow direction. Kirchhoff's dead water theory is applied to the cross flow with the inclusion of three-dimensional effect which increases the velocity between the vortex layers in a lateral section. The first term of the normal force coefficient is derived only from the near field consideration and expressed by lateral drag coefficient of the Kirchhoff's theory. The second term is derived by considering the increase of lateral flow velocity by the effect of vortex layers in the vortex field, which is a flow field existing between the near and far fields and consists of vortex layers of finite extension. The second term of the coefficient is obtained analytically by assuming the form of vortex layers and the strength of vorticity. The obtained results are compared with experimental results.

On the skin friction and resulting wake of a two-dimensional planing plate

The skin friction of a two-dimensional planing flat plate is made up of two opposing components; a drag force from the flow aft of the stagnation line and an opposing thrust force from the jet flow. This paper is concerned only with the drag term and the wake velocity defect which it causes in the water behind the transom. It is concluded that the skin friction is less than would be expected from a flat plate at ambient static pressure ( D fo say) and is approximately equal to D fo (1 − C R ), where C R is the normal force coefficient based on wetted area. The wake velocity decrement due to this drag is found to be significant, particularly for surface piercing propellers.

The Hydrodynamic Force Acting on the Ship in a Following Sea (1 st Report)

Broaching phenomena are most likely to occur in a following sea to relative small and fast craft under the condition that the wavelength/the ship length ratio is about 1. 5 to 2. 0. There are many studies and explanations about it, and it is thought that the forces acting on the ship are almost consist of the hydrostatic force and steady state lift.Towing flat plate model in following waves without attack angle it become obvious that side force and heading moment are of such magnitude as to a marked extend. These changed or additional disturbing forces depend upon the wave height and upon the relative position of the ship to waves.In this paper two attempt are made in order to investigate the hydrodynamic force. Firstly, transforming vortex core model is used under the assumption of the free surface is flat. Slender-rectangular thin wing in the constant stream, 3-dimensional separation occurs at the end of the side edges, which form a symmetrical spiral vortex sheet.In a following sea, these spiral vortex sheet may be affected by the wave orbital velocity. A simple flow model, consists of core vortices and feed vortex sheets which emerge from side edges to the core vortices, is used.And core vortices are transformed by the wave orbital velocity and the induced velocities of vortices. By the method, reasonable results are obtained for normal force coefficient, centre of pressure and location of the core vortex in the constant flow. But in a following sea, calculated normal force coefficient is not so good.Secondly, high speed slender body theory is used introducing the data of incident wave as free surface.These method, calculated normal force and heading moment coefficients show quite good agreement to the data obtained from towing flat plate model test.

Comparisons of Aerodynamic Data Obtained by Static and Dynamic Techniques

Introduction T HE conventional internal strain gage balance system has been a standard tool for wind tunnel aerodynamics for many years. Although widely used to obtain static aerodynamic data, experimentalists have been well aware of the difficulties in obtaining accurate force and moment data at small angles of attack and of the inaccuracies which result from differentiating experimental moment data to obtain static stability derivatives. Free oscillation techniques have likewise been widely used to obtain dynamic stability data. Although persons familar with dynamic testing techniques are certainly cognizant, the technical community in general may not be aware that static aerodynamic data can readily be obtained from dynamic tests. The purpose of this paper is to make a comparison of static aerodynamic data obtained with an internal strain gage balance and a small amplitude, free oscillation technique. Normal force coefficients, pitching moment coefficients, static stability derivatives, and center of pressure locations are compared for a biconic configuration with a 15% spherically blunted nose. Also included are the results of numerical calculations, using a shock capturing code developed by Solomon.

Experimentally Determined Stability Parameters of a Subsonic Cascade Oscillating Near Stall

Tests were performed on a linear cascade of airfoils oscillating in pitch about their midchords at frequencies up to 17 cps, at free-stream velocities up to 200 ft/s, and at interblade phase angles of 0 deg and 45 deg, under conditions of high aerodynamic loading. The measured data included unsteady time histories from chordwise pressure transducers and from chordwise hot films. Unsteady normal force coefficient, moment coefficient, and aerodynamic work per cycle of oscillation were obtained from integrals of the pressure data, and indications of the nature and extent of the separation phenomenon were obtained from an analysis of the hot-film response data. The most significant finding of this investigation is that a change in interblade phase angle from 0 deg to 45 deg radically alters the character of the unsteady blade loading (which governs its motion in a free system) from stable to unstable. Furthermore, the stability or instability is governed primarily by the phase angle of the pressure distribution (relative to the blade motion) over the forward 10–15 percent of the blade chord. Reduced frequency and mean incidence angle changes were found to have a relatively minor effect on stability for the range of parameters tested.

Subsonic Flow Past a Slender Body of Revolution Placed Eccentrically in a Tube

For a subsonic flow, the surface pressure coefficients of a slender body of revolution, which is placed eccentrically in a tube, have been calculated by means of M2∞-expansion-method and as slender body in slender tube theory. The assumptions used in this calculation are as follows : (1) The fluid is inviscid. (2) The fluid is irrotational, as the eccentricity is small. The results obtained here are as follows : (1) With this method, analytic solutions can be easily obtained to a required higher order approximation. (2) When the uniform stream Mach number is 0.3 and the blockage ratio is 0.4, the absolute values of the surface pressure coefficient in the neighbourhood of the center cross section of the body are about 20 percent larger than those of incompressible flow. (3) As for the body shapes, that of parabolic spindle takes a larger value for the normal force coefficient than other body shapes.

An inviscid model of two-dimensional vortex shedding for transient and asymptotically steady separated flow over an inclined plate

A potential flow model of two-dimensional vortex shedding behind an inclined plate is developed. The free shear layers which emanate from the sides of the plate are represented by discrete vortices through the use of the appropriate complex-velocity potential, the Kutta condition and the Joukowsky transformation between a circle and the plate cross-section. The analysis is then applied to predict the kinematic and dynamic characteristics of the flow for various angles of attack. The results compare favourably with the available experimental data as far as the form of vortex shedding and the Strouhal number are concerned. The calculated normal-force coefficients are 20−25% yo larger than those measured by Fage & Johansen (1927).

A new device for simultaneous measurement of friction force, normal force and friction coefficient

To meet the demands for higher accuracy in the measurement of forces in tribological experiments a force transducer has been developed which allows the simultaneous measurement of friction force, normal force, and, via an electronic divider, the friction coefficient. The device consists essentially of four bending rods notched at four points to localize strains monitored at these points by strain gauges. Tilting and bending of the probe is avoided by this construction. The details, performance and formulae for dimensioning are given.

Calculation of slender wing with rolling-up vortex sheet along side edges

The Report deals with a characteristic flow field around a slender rectangular thin wing, where free vortices, which separate from side edges, roll up and form a symmetrical spiral vortex sheet. The aims are to obtain lift distribution and to grasp the macroscopic flow field around the wing.Referring to the observed flow pattern around the wing by means of a hydrogen bubble method in a water tunnel, a simple flow model is assumed. The model consists of a pair of core vortices and feed vortex sheets which emerge from side edges to the core vortices.With the simplification of the flow field, boundary conditions need to be made less detailed. That is, the condition on the wing surface is satisfied only along the center line of it, and the condition for the free vortices is simplified to the condition that total forces acting on the free vortices are zero.By the method, reasonable results are obtained for normal force coefficient, center of pressure and average location of core vortices.

A four-level technique for estimation of tactical missile aerodynamic parameters

A four-level technique is described for estimation of aerodynamic moment coefficient M <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</inf> , normal acceleration force coefficient Z <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</inf> , and velocity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</tex> . These parameters are associated with a simplified model of pitch plane dynamics for a tactical missile. Contrary to past studies where aerodynamic parameters were estimated from measurements which are very difficult to obtain (e.g., angle of attack), it is shown herein how estimates of the above parameters can be realized from measurements which are more easily obtained. Specifically, parameters are estimated from measurements of rate, normal acceleration, and gimbal angle deflection.

Hydrodynamic Forces Acting on a Ship in Oblique Towing

The objective of this paper is to afford a more rational method of calculation for hydrodynamic forces on a ship in manoeuvering motion. Applying the slender body theory and extending the Barche's method to a ship form associated with free vortex layers which represent 3-dimensional separation vortices, the problem can be reduced to some 2-dimensional problems. More precise hull form or ship condition can then be taken into consideration for hydrodynamic force calculation. The author calculates by this method normal force coefficient CN and moment coefficient Cm_??_ for various ship models, and compares the calculated results with tank test results. These comparisons affirm the validity of this method. He also measures the pressure distribution on a ship hull in steady oblique towing, the results of which show a good agreement with those of potential flow calculation except in small region located at lee side and aft part of the ship.

Optimal Aerodynamic Attitude Stabilization of Near-Earth Satellites

A near-Earth satellite orbiting in the altitude range of 150 km to 450 km encounters small but non-negligible aerodynamic forces. An analysis of a system which generates sufficient aerodynamic torques to achieve active attitude control from all-moving control surfaces is presented. The satellite configuration considered has four control surfaces and the dominant gravity gradient and aerodynamic torques are included in the analysis. The equations of motion are linearized and optimal control theory concepts are applied. A feedback control system is synthesized subject to the constraint of minimum drag. Numerical studies indicate that aerodynamic surfaces of reasonable size are achievable. Damping times of the order of from a few orbits to a fraction of an orbit are possible. Nomenclature A, jB, C = moments of inertia of satellite about roll, pitch, yaw axes At = area of control surface i bi = center of pressure of 8th control surface with respect to satellite mass center d =cos9i(i= 1,2, 3) CPi, Ctj, Cot = normal force, shear force, and drag coefficients for control surface / CmA