Slender Body Of Revolution Research Articles

Theoretical analysis of convective transport over a slender body of revolution

The study of thermal transport in the laminar boundary layer had been a great area of research since long ago because of its practical necessity in different fields of engineering such as aerodynamics, electronics cooling, designing of turbine blades, etc. The primary goal of the research has always been to look into geometries or shapes that have higher heat transfer rates but a less viscous drag. This is also a topic of fundamental interest to the aerospace industry. For this purpose, a cone-shaped body of revolution has been considered. Such geometries are chosen due to their wider acceptance in modern aerodynamics, with their enhanced heat transfer rates and lower aerodynamic drag. In this connection, axisymmetric bodies of revolution having different body contours and involving surface transverse curvature are usually considered. The selection of such a curved body and the imposed particular type of momentum and thermal boundary conditions make the problem of non-similar nature. Such kinds of problems are normally dealt with by an implicit finite difference scheme due to the accuracy of the solution. The current problem has also been solved with the same numerical scheme. It is concluded that a more streamlined body of revolution has a less spread of velocity and temperature within the boundary layer with a large heat transfer rate and wall skin-friction. On the other hand, the parabolic body of revolution has a large distribution of velocity and temperature but less heat transfer rate and wall skin-friction. Moreover, the assisting role of surface transverse curvature on temperature distribution and wall skin-friction has also been observed in this study.

Development of nonstationary side forces along a slender body of revolution at incidence

The flow field around an axisymmetric slender body at an angle of attack is complex and typically contains extensive regions of three-dimensional crossflow separation; although the body is axisymmetric the side force induced by the flow on the body is significant and may change greatly along the body. This study provides a new perspective on the development of flow unsteadiness and the side-force distribution along the body for a wide range of angles of attack.

Experimental investigations of the asymmetric vortex formation over a slender body of revolution at high angles of attack

This paper describes an experimental investigation of the initial growth of flow asymmetries over a slender body of revolution at high angles of attack with natural and disturbed noses. Time-resolved particle image velocimetry (PIV) is used to investigate the flow field around the body. Flow visualization clearly shows the formation of the asymmetric vortices. Instantaneous PIV shows that the amplified asymmetric disturbances lead to Kelvin–Helmholtz instability appearing first on one side, which increases the momentum exchange crossing the layer. As a result, the separation region shrinks which creates the initial vortex asymmetry.

Study on the manoeuvring characteristics of axisymmetric underwater vehicles with different slenderness and control fin aspect ratios

This paper presents the effect on the manoeuvring characteristics of an axisymmetric underwater vehicle due to its body slenderness ratio and aft fins. The vehicle is a slender body of revolution with elliptical nose, tapered tail and cylindrical middle body. Five vehicle lengths with slenderness ratio from 10.8 to 14.4 with two sets of aft cruciform fins with different aspect ratios are employed for this study. The effect of manoeuvrability is evaluated by simulation of standard manoeuvres using 6 degrees-of-freedom (6DOF) equations of motion. The variation of normal force and pitch moment at oblique flows/pitch rates due to slenderness ratio and aft fin are evaluated by experimental and numerical methods. It is seen that the straight line stability in horizontal plane improves with slenderness ratio and aft fins size.

Зависимость звукового удара от взаимного расположения тел в сверхзвуковом потоке

Results of calculating the sonic boom generated in a supersonic air flow by two bodies (disk and slender body of revolution) are presented. The bodies are arranged one behind the other. The slender body is aerodynamically shaded by the disk. The free-stream Mach number is 2. The calculations are performed by a combined method of “phantom bodies.” By changing the disk position and its size, it is possible to reduce the sonic boom level. Based on the calculation results, the gas-dynamic factors affecting the sonic boom level are described.

Gas-dynamic factors controlling the level of the sonic boom generated by two bodies in a supersonic flow

The results of calculation of the sonic boom generated in a supersonic air flow by two bodies (disk and slender body of revolution) are presented. The bodies are arranged one behind the other. The slender body is aerodynamically shaded by the disk. The free-stream Mach number is 2. The calculations are performed by a combined “phantom bodies” method. The sonic boom level can be reduced by more than 12% via the disk position and size changing, as compared to the sonic boom generated by the slender body. the gas-dynamic factors affecting the sonic boom level are described on the base of the calculation results.

Flow Characteristics Over a Missile at Higher Angle of Attacks using Experiments and Simulation

The need to determine forces and moments acting on bodies of revolution at high angles of attack originally arose in connection with airships. In general missile aerodynamics is dominated with vertical and separated flows. The advent of highly manoeuvrable missile design concepts has required a major effort in understanding the problem of slender bodies of revolution at high angles of attack. An extensive experimental investigation was conducted to determine the aerodynamic forces and moments. The aerodynamic data are needed to establish the structural and control system requirements. Tests were conducted on a smooth missile model with several interchangeable nose parts in a wind tunnel at subsonic level at high angle of attack. The yaw moments and side forces are determined using force measuring systems by which maximum induced side force was found. Using oil flow visualization technique, vortex shedding location and vortex strength were traced at various angles of attack. Computational analysis was carried out for the cone, elliptical and ogive nose shapes. It is then compared with the experimental results to know where the least flow separation occurs and proper reasons for the side forces.

Sonic boom generated by a slender body aerodynamically shaded by a disk spike

The sonic boom generated by a slender body of revolution aerodynamically shaded by another body is numerically investigated. The aerodynamic shadow is created by a disk placed upstream of the slender body across a supersonic free-stream flow. The disk size and its position upstream of the body are chosen in such a way that the aerodynamically shaded flow is quasi-stationary. A combined method of phantom bodies is used for sonic boom calculations. The method is tested by calculating the sonic boom generated by a blunted body and comparing the results with experimental investigations of the sonic boom generated by spheres of various diameters in ballistic ranges and wind tunnels. The test calculations show that the method of phantom bodies is applicable for calculating far-field parameters of shock waves generated by both slender and blunted bodies. A possibility of reducing the shock wave intensity in the far field by means of the formation of the aerodynamic shadow behind the disk placed upstream of the body is estimated. The calculations are performed for the incoming flow with the Mach number equal to 2. The effect of the disk size on the sonic boom level is calculated.

Behaviour of Mean Velocity in the Turbulent Axisymmetric Outer Near Wake

An asymptotic study of the outer near wake of a long slender body of revolution is carried out. The long slender body is a cylinder which is kept parallel to the flow and takes the shape of a simplified geometry such as that of an underwater vehicle, a rocket or hull form of a ship model. The wake flow is axisymmetric and the analysis has been carried out without any assumption on the eddy viscosity but utilizing the general behaviour of turbulent shear stress in the near wake. The governing equations are solved with appropriate boundary conditions. Similarity analysis for the mean velocity characteristics is carried out which shows the existence of a logarithmic region in the normal direction in the overlap region between the outer near wake and the inner near wake. Also shown is the exponential decay of the mean velocity defect as freestream velocity is reached. Validation of the results of the analysis is done using available experimental data.

Flow separation from submarine shaped bodies of revolution in steady turning

Flow separation from 3 slender bodies of revolution in steady turning is studied using Computational Fluid Dynamics RANS simulations. The body profiles include the DRDC-STR, DARPA SUBOFF, and the Series-58 Model 4621. The flow conditions consider varying turning radii, 1.0≤R/l≤10 (where R is the radius of the turn and l is the body length) with a fixed pivot point located 25% aft of the nose. The Reynolds numbers considered were 6.5, 11.7 and 23 million. Scaling laws deduced for the development of separation along the midbody and tail were used to develop an analytical model for predicting the separation line along a body of revolution of arbitrary profile.

Experimental investigations of the initial growth of flow asymmetries over a slender body of revolution at high angles of attack

This paper describes an experimental investigation of the initial growth of flow asymmetries over a slender body of revolution at high angles of attack with natural and disturbed noses. Time-resolved particle image velocimetry was used to investigate the flow field around the body. The experimental results show that initially different amplitudes of unsteady disturbances near the tip are established owing to the tip imperfections. These unsteady disturbances experience a super-exponential growth near the tip and continue to grow exponentially due to linear instabilities. Attachment of a piece to the tip brings a larger initial difference and extends the super-exponential growth region. Thus, the disturbance amplitudes and their differences are larger for the disturbed case than for the natural case before reaching the neutral point of linear instability. The amplified disturbances lead to different instability vortex strengths in the separated shear layers, which feed continuously into the two primary concentrated vortices. As a result, the primary vortex strengths differ, which result in the initial vortex asymmetry. The experiment results demonstrate that the initial flow asymmetry arises from an asymmetric development of the boundary layer instability.

Slender-body approximations for advection–diffusion problems

We consider steady advection–diffusion about a slender body of revolution at arbitrary $O(1)$ Péclet numbers ($\mathit{Pe}$). The transported scalar attenuates at large distances and is governed by axisymmetric (either Dirichlet or Neumann) data prescribed at the body boundary. The advecting field is assumed to be an axisymmetric Stokes flow approaching a uniform stream at large distances and satisfying impermeability at the boundary; otherwise, the interfacial distribution of tangential velocity is assumed to be arbitrary, irrotational and no-slip Stokes flows being particular cases. Employing the method of matched asymptotic expansions, we develop a systematic scheme for the calculation of the scalar concentration in increasing powers of $\ln ^{-1}(1/{\it\epsilon})$, ${\it\epsilon}\ll 1$ being the body’s characteristic thickness to length ratio. The leading term in the inner expansion coincides with the pure diffusion case, the second term depends nonlinearly on the magnitude of the far-field stream and higher-order terms depend on the boundary distribution of tangential velocity. In the special case of irrotational flow and Neumann boundary conditions the logarithmic expansion terminates, leaving an algebraic error in ${\it\epsilon}$. The general formulae developed can be directly applied to numerous physical scenarios. We here consider the classical problem of forced heat convection from an isothermal body, finding a two-term expansion for $\mathit{Nu}({\it\epsilon},\mathit{Pe})/\mathit{Nu}({\it\epsilon},0)$, the ratio of the Nusselt number to its value at $\mathit{Pe}=0$. This ratio is insensitive to the particle shape at the asymptotic orders considered; at moderately large $\mathit{Pe}$ ($\ll {\it\epsilon}^{-1}$) its deviation from unity is $O[\ln (\mathit{Pe})/\text{ln}\,(1/{\it\epsilon})]$, marking the poor effectiveness of advection about slender bodies. The expansion is compared with a numerical computation in the case of a prolate spheroid in both irrotational and no-slip Stokes flows.

Experimental study on side force alleviation of conical forebody with a fluttering flag

Slender bodies of revolution placed at high angles of attack will produce a steady asymmetric vortex pair. This behavior is determined by the spatial instability of the flow. The asymmetric vortex pair is very sensitive to small disturbances near the tip. This paper presents a simple side force alleviation method without a power unit for slender bodies at high angles of attack. The method uses the self-excited oscillation (flutter) of a small cantilever flag as the excitation source on the cone tip. An experimental study is performed to investigate the characteristics of the asymmetric side forces of a 20° cone-cylinder slender body with and without a small flutter flag. Pressure distribution at eight cross sections of the cone forebody is measured at different free-stream velocities and different angles of attack. The process demonstrates that the excitation provided by the fluttering flag can change the asymmetric bistable flow state to a well symmetric one at high angles of attack, and that the corresponding side force and yaw moment will significantly decrease. The flutter strength of the flag will be enhanced by increasing the free-stream velocity. The effectiveness of the side force alleviation method will also be increased.

Turbulent skin-friction drag on a slender body of revolution and Gray's Paradox

The boundary layer on a slender body of revolution differs considerably from that on a flat plate, but these two cases can be connected by the Mangler-Stepanov transformations. The presented analysis shows that turbulent frictional drag on a slender rotationally symmetric body is much smaller than the flat-plate concept gives and the flow can remain laminar at larger Reynolds numbers. Both facts are valid for an unseparated flow pattern and enable us to revise the turbulent drag estimation of a dolphin, presented by Gray 74 years ago, and to resolve his paradox, since experimental data testify that dolphins can achieve flow without separation. The small values of turbulent skin-friction drag on slender bodies of revolution have additional interest for further experimental investigations and for applications of shapes without boundary-layer separation to diminish the total drag and noise of air- and hydrodynamic hulls.

The aerodynamics of a slender body moving very close to a water wave surface

A slender body is assumed to translate horizontally above and in very close proximity to a water surfaceon which there is a progressive harmonic wave. The air flow induced by the water wave surface and thebody is studied by developing classical slender body theory. The outer flow far away from the body issolved analytically, while the inner flow near the body reduces to a two-dimensional flow problem inthe transverse cross-flow plane. An analytical inner solution is obtained for an arbitrary slender body ofrevolution for different angles of attack and yaw. The attraction and side forces, and pitching and yawingmoments, acting on the body, are obtained in terms of integrals along the body length. Numerical analysesreveal the variation trends of the force and moment in terms of the angles of attack and yaw of the body,the wavelength, amplitude, phase speed and direction of the water wave and the clearance between them.Significant dynamic effects of the water wave surface are noted, even when the wave phase speed is oneorder of magnitude smaller than the translation speed of the body.Keywords: slender body theory; ground effect; matched asymptotic expansions.

Study of transonic effect of flows around slender bodies of revolution and plane profiles

Transonic effect with presence of local supersonic terminated by shock wave is not wished. In spite of wishes or not, the appearance of shock waves is not avoided. Strong shock waves cause losses, even a grand decrease of aerodynamic quality in fluid-solid interaction. However, under certain conditions, transonic effect increases the drag and at the same time increases the lift leading the amelioration of aerodynamic quality. This work uses numerical methods and established programs for solving differential equations of the axisymmetric flow and the plane flow to study the influence of transonic effects for the two cases.

A source term model of perturbation in a numerical study on flows around a slender body of revolution at a high angle of attack

A numerical study on flows around a slender body of revolution at a high angle of attack is conducted to investigate the influence of strength and circumferential angle of perturbation on flow asymmetry. A source term model is applied to simulate a real geometrical perturbation near the tip of the slender body. It can greatly facilitate the adjustment of perturbation strength. The results show that the bistable phenomenon does not appear any more at a small perturbation strength. For different perturbation strengths, the energy of the asymmetric perturbation flow experiences a similar spatial exponential growth regime with the same growth rate. The appearance of the bistable phenomenon is closely related to nonlinear saturation of the perturbation flow as perturbation strength increases.

Influence of nose-perturbation location on behavior of vortical flow around slender body at high incidence

Response of the vortical flow around a slender body of revolution at high incidence to the shift of a single nose perturbation was investigated systematically using numerical methods. A minute geometric bump was employed to act as the nose perturbation, and all computations were performed for subsonic flows at incidence of 50°. The computational results show that the vortical flow is more sensitive to the perturbation located axially closer to the nose apex of a slender body. With perturbation shifting axially downstream away from nose apex, there is a critical axial location appearing. The vortical flow is less sensitive to the perturbation located axially closer to the critical axial location; when perturbation traverses axially around the critical axial location, the vortical flow switches between opposite asymmetric patterns. The eventual influence of perturbation axial location on the vortical flow lies on both its relative locations to nose apex and the critical axial location. The vortical flow is more sensitive to the perturbation located circumferentially farther away from the fore-and-aft symmetric plane of a slender body, and just the asymmetrically-located perturbation can provoke the vortical flow to asymmetry. With perturbation shifting circumferentially in sequence, the vortical flow varies by degrees in manner of a single periodicity. A convective-type of instability existing in the flow field is responsible for the influence of nose perturbation on the vortical flow.

Supersonic scattering of a wing-induced incident shock by a slender body of revolution

The lift force acting on a slender body of revolution that separates from a thin wing in supersonic flow is analysed using Prandtl–Glauert linearized theory, scattering theory and asymptotic methods. It is shown that this lift is associated with multi-scattering of the wing-induced shock wave by the body surface. The local and global lift coefficients are obtained in simple analytical forms. It is shown that the total lift is mainly induced by the first scattering. Contributions from second, third and higher scatterings are zero in the leading-order approximation. This greatly simplifies calculations of the lift force. The theoretical solution for the flow field is compared with numerical solutions of three-dimensional Euler equations and experimental data at free-stream Mach number 2. There is agreement between the theory and the computations for a wide range of shock-wave strength, demonstrating high elasticity of the leading-order asymptotic approximation. Theoretical and experimental distributions of the cross-sectional normal force coefficient agree satisfactorily, showing robustness of the analytical solution. This solution can be applied to the moderate supersonic (Mach numbers from 1.2 to 3) multi-body interaction problem for crosschecking with other computational or engineering methods.

An analytical solution for two slender bodies of revolution translating in very close proximity

The irrotational flow past two slender bodies of revolution at angles of yaw, translating in parallel paths in very close proximity, is analysed by extending the classical slender body theory. The flow far away from the two bodies is shown to be a direct problem, which is represented in terms of two line sources along their longitudinal axes, at the strengths of the variation rates of their cross-section areas. The inner flow near the two bodies is reduced to the plane flow problem of the expanding (contracting) and lateral translations of two parallel circular cylinders with different radii, which is then solved analytically using conformal mapping. Consequently, an analytical flow solution has been obtained for two arbitrary slender bodies of revolution at angles of yaw translating in close proximity. The lateral forces and yaw moments acting on the two bodies are obtained in terms of integrals along the body lengths. A comparison is made among the present model for two slender bodies in close proximity, Tuck & Newman's (1974) model for two slender bodies far apart, and VSAERO (AMI)–commercial software based on potential flow theory and the boundary element method (BEM). The attraction force of the present model agrees well with the BEM result, when the clearance,h0, is within 20% of the body length, whereas the attraction force of Tuck & Newman is much smaller than the BEM result whenh0is within 30% of the body length, but approaches the latter whenh0is about half the body length. Numerical simulations are performed for the three typical manoeuvres of two bodies: (i) a body passing a stationary body, (ii) two bodies in a meeting manoeuvre (translating in opposite directions), and (iii) two bodies in a passing manoeuvre (translating in the same direction). The analysis reveals the orders of the lateral forces and yaw moments, as well as their variation trends in terms of the manoeuvre type, velocities, sizes, angles of yaw of the two bodies, and their proximity, etc. These irrotational dynamic features are expected to provide a basic understanding of this problem and will be beneficial to further numerical and experimental studies involving additional physical effects.