Supersonic Cone Research Articles

Analytical Solution to the Steady Navier-Stokes Equation for a Supersonic Cone in the Area of Boundary Layer Behind the Shock Wave by Means Tunnel Mathematics

The means of tunnel mathematics (the theory of functions of a spatial complex variable) allow for an analytical solution to the problem of supersonic flow around a cone in the area of the boundary layer and beyond. The peculiar feature of the Navier-Stokes equations is that they allow for the determination of analytical velocity fields of fluids only for a small number of simple problems. Of course, the problem of the supersonic motion of fluid around a cone is not included in this number. Tunnel mathematics is a method for finding analytical vector velocity fields for steady flows of fluids with axial symmetry. The Navier-Stokes equations are then used to determine pressure and temperature distributions. The main theorem of tunnel mathematics allows for the determination of these distributions for planes z = const (similar to constructing slices of a brain in an MRI procedure). By collecting these “slices,” we can obtain full space distributions of pressure and temperature around a supersonic cone. At this stage of investigation, the conclusions obtained through tunnel mathematics make it possible to qualitatively assess the thickness of the boundary layer on the cone's surface, the shape of the shock wave, and whether the shock wave intersects the boundary layer. First of all, we focused on ensuring that the resulting solutions corresponded to the physical pattern of phenomena. No doubt, solutions obtained through tunnel mathematics must be confirmed experimentally.

Estimation of Drag on Supersonic Cone Varying Nose Radius at Different AOA

Estimation of Drag on Supersonic Cone Varying Nose Radius at Different AOA

The computation of the pitch damping stability derivatives of supersonic blunt cones using unsteady sensitivity equations

The numerical methods for computing the stability derivatives of the aircraft by solving unsteady sensitivity equations which was proposed in our previous papers was extended to solve three-dimensional problems in this paper. Both the static and dynamic derivatives of the hypersonic blunt cone undergoing pitching oscillation around a fixed point were computed using the new methods. The predicted static derivative and dynamic derivative were found to be in reasonable agreement with the experimental data. For the present method, it is possible to distinguish the components of dynamic derivatives caused by different state parameters. It is found that {C}_{m_{dot{alpha}}} and {C}_{m_q} are usually of opposite signs and tend to eliminate each other, which makes {C}_{m_{dot{alpha}}}+{C}_{m_q} being much smaller than its individual components. Another feature of this method is that the moment of pressure derivatives proposed in the present paper can be used to predict the contribution of each part of the blunt cone to the overall stability quantitatively. It is found that the head region is crucial for the static stability and the body region contributes most to the dynamic stability.

The improvement of turbulence modeling for the aerothermal computation of hypersonic turbulent boundary layers

The engineering computation of turbulent flows is mainly based on turbulence modeling, however, accurate aerothermal computation of hypersonic turbulent boundary layers is still a not well-solved problem. Aerothermal computation for turbulent boundary layers on a supersonic or hypersonic blunt cone with small bluntness is done firstly by using both direct numerical simulation and BL model, and seven different cases are investigated. Then the results obtained by the two methods are compared, and the reason causing the differences is found to be the incorrect assumption in the turbulence modeling that the ratio between eddy heat conductivity and eddy viscosity is constant throughout the whole boundary layer. Based on certain theoretical arguments, a method of modifying the expression of eddy heat conductivity in the region surrounding the peak location of the turbulent kinetic energy is proposed, which is verified to be effective, at least for the seven cases investigated.

Inflow boundary condition for DNS of turbulent boundary layers on supersonic blunt cones

For direct numerical simulation (DNS) of turbulent boundary layers, generation of an appropriate inflow condition needs to be considered. This paper proposes a method, with which the inflow condition for spatial-mode DNS of turbulent boundary layers on supersonic blunt cones with different Mach numbers, Reynolds numbers and wall temperature conditions can be generated. This is based only on a given instant flow field obtained by a temporal-mode DNS of a turbulent boundary layer on a flat plate. Effectiveness of the method is shown in three typical examples by comparing the results with those obtained by other methods.

Variation of the temperature factor of models on a ballistic range

A technique for variation of the temperature factor of free-flight models by varying their initial temperature is described. An experiment on a ballistic range is carried out with a free-flying supersonic blunt cone with a half-angle of 15° at a Mach number of 2.3. The flow at the cone base is studied in the transition range (from the laminar to turbulent flow) of Reynolds numbers. The base flow pattern is determined from the shadowgraphs of the flow about the models. The drag coefficient of the blunt cone at a zero angle of attack is found by processing trajectory data. It is found that the near wake geometries and the drag coefficients of the models tested at the laboratory temperature and a temperature of 120 K differ. Explanations of this effect are given.

Numerical investigation of evolution of disturbances in supersonic sharp cone boundary layers

The spatial evolution of 2-D disturbances in supersonic sharp cone boundary layers was investigated by direct numerical simulation (DNS) in high order compact difference scheme. The results suggested that, although the normal velocity in the sharp cone boundary layer was not small, the evolution of amplitude and phase for small amplitude disturbances would be well in accordance with the results obtained by the linear stability theory (LST) which supposes the flow was parallel. The evolution of some finite amplitude disturbances was also investigated, and the characteristic of the evolution was shown. Shocklets were also found when the amplitude of disturbances increased over some value.

Features of the Application of Slender-Body Theory to the Calculation of Supersonic Water-Stream Cavitation Flow Past Cones

Features of the application of slender-body theory to the calculation of supersonic water-stream cavitation flows past cones are described. It is shown that the boundary condition on the cone edge in whose neighborhood Prandtl-Meyer flow develops cannot be satisfied within the framework of slender-body theory. The cone drag coefficients determined on the basis of slender-body theory are compared with the results of numerical calculations of conical flows.

Numerical simulation of asymmetric non-steady-state turbulent flows around supersonic sharp cones

Results are presented of a numerical simulation of turbulent flows around pointed circular supersonic cones at angles of attack α≥3θc, where θc is the cone half-angle. The problem is solved in the framework of a locally conical approximation of a set of Reynolds’ equations written in terms of mass-averaged quantities with the use of a differential one-parametric turbulence model. The numerical solutions are obtained with the help of an implicit finite-difference unidirectional scheme, with the focus on violation of the symmetry and stationarity of the separated cross flow.

Quantitative comparison between interferometric measurements and Euler computations for supersonic cone flows

Dual-reference-beam, plane-wave digital holographic interferometry has been applied to obtain quantitative interferometric data in the three-dimensional supersonic flow over circular cones. The interferometric data are compared quantitatively on a two-dimensional grid with the postprocessed results of an Euler code that simulates three-dimensional inviscid compressible flows. The comparison involves two different combinations of cone angle and angle of incidence. The maximum deviations between the interferometric data and the numerical data are found to lie in the error interval [-2.0%, +2.0%]. O validate numerical codes for the simulation of threedimensional compressible flows, extensive experimental flowfield data are required. The nonintrusiveness and the high resolution of the data provided by a single measurement (512 x 512 data points) make digital holographic interferometry (DHI) in some respects a good alternative for conventional measuring techniques like pressure probe measurements. DHI is the optical measuring technique that combines holographic interferometry to visualize the flowfield in an interferogram, phase stepping of the interferogram and digital image processing to compute the phase map from the digitized interferograms. The phase map reflects the deformation of the wave fronts of a laser light beam that has passed through the flowfield inside a test section. The phase is proportional to the mean density over the width of the test section along the direction of light propagation. At the same time, this explains the inherent disadvantage of DHI. In three-dimensional flows it provides only quantitative information about the average density along light rays through the flowfield. In this paper the experimental data provided by DHI will be compared with the numerical data obtained by a computer code for inviscid compressible flow (three-dimensional Euler code). The supersonic flow around circular cones will serve as the test flowfield in the comparison. Several good reasons exist for choosing this flowfield as the test flowfield. First, if the bow shock is attached, it represents a relatively simple flowfield because of its conical symmetry. In this case no dominant viscous interaction effects are present and the assumption of an inviscid fluid holds. Furthermore, extensive results of inviscid flow computations are available for several Mach numbers M^, angles of incidence a, and cone half-angles Oc.l~3 Finally, the applicability of holographic interferometry in the investigation of the supersonic flow around circular cones has been successfully demonstrated before.4-5

Particle Impaction on a Supersonic Cone

The theoretical collection efficiency has been determined for particles impinging on a supersonic cone with an attached shock. By defining a moving coordinate system such that the gas is stationary on the cone surface behind the shock and scaling the particle Stokes number with a shock-to-cone distance, a similarity parameter has been denned that properly accounts for the effects of freestream Mach number, cone angle, and large particle Reynolds number. The effective Stokes number, which represents the ratio of the particle stopping distance in the moving frame to the appropriate shock-to-cone length, is shown to correlate the total collection efficiency.

Linear stability of supersonic cone boundary layers

The effect of the variable surface geometry of a cone on the linear stability of a supersonic boundary layer flowing over it is investigated subject to different quasiparallel flow approximations. It is shown that, if a suitable set of disturbance state variables is chosen for the normal mode analysis, these effects can accurately be accounted for. In fact, a planar coordinate system can be used for the stability analysis of the cone boundarylayer profiles and a simple correction can subsequently be applied to obtain an accurate approximation to the spatial growth rates.

Influence of the angle of attack on the transition from a laminar to a turbulent boundary layer in the case of supersonic flow past sharp circular cones

Some results are given of an experimental investigation into the influence of the angle of attack on the transition in the symmetry plane of a sharp cone at Mach number M = 6.1. These results and available experimental data are used to establish the dependence of the transition Reynolds number on the angle of attack on the flow division line of sharp circular cones.

Supersonic Cone with Surface Blowing

SummaryThe velocity field around a circular cone in supersonic flow is determined by considering fluid injection and suction through the porous wall of the body. By using a new method of linearisation of the equations of motion, analytical formulae are obtained which yield almost exact results, as compared with numerical calculations. In addition, the method proposed suggests new problems related to fluid injection and suction which are important for practical applications.

Comment on "Boundary-Layer Transition on Supersonic Cones in an Aeroballistic Range"

Comment on "Boundary-Layer Transition on Supersonic Cones in an Aeroballistic Range"

Super-Alfvénic supersonic flow over a cone in aligned fields

Super-Alfvénic supersonic aligned magnetogasdynamic flow over a cone of finite semi-apex angle, with an attached fast shock wave, is solved numerically. We obtain ‘almond curves’ in the plane of magnetic induction vector variation, analogous to Busemann's ‘apple curves’ for supersonic cone flows, to describe the flow field near the cone. Total surface pressure coefficients, current and vorticity distributions are presented. A closed-form solution of the flow is obtained when a switch-on shock occurs.

Experimental investigation of the stability of the laminar supersonic cone wake

An instability investigation was carried out in the wake of a 7° half angle, sharp cone suspended magnetically. Hot-wire fluctuation measurements were made in the wake for a range of Reynolds numbers and at a Mach number 4.3. These measurements indicated a completely stable near wake at Remd = 51,600 and large amplification of small disturbances at Remd = 61,900. The amplified waves were highly concentrated with respect to frequency, with a number of pronounced harmonics being present. Amplitude and phase measurements of the spectral components indicated that the instability process fits within the framework of linear stability theory as formulated by Gold, with each mode having a wavefront shaped like a circular helix. It appears that each succeeding mode has an additional thread in the helical wavefront. However, the measured fundamental oscillation has a larger amplification rate and a much lower frequency than predicted by Gold's theory. In addition, weak nonlinear interactions were observed, which grow stronger with downstream position.

Experimental investigation of the mean flow of the laminar supersonic cone wake

Experimental investigation of the mean flow of the laminar supersonic cone wake

Laser Planogram Measurements of Turbulent Mixing Statistics in the Near Wake of a Supersonic Cone

The laser planogram technique is a new method for the study of turbulent mixing. It utilizes a pulsed laser and a particulate tracer to determine the spatial mixing field of tagged and untagged fluids. The laser planogram technique is described along with design considerations. As an example of its implementation, laser planogram measurements have been obtained and analyzed to provide turbulent mixing statistics in the wake of a cone at a Mach number of 2.5 and a Reynolds number of 3 × 106. The mean radial concentration profile of tagged material is shown to be a Gaussian in agreement with theory. A k−5/3 spectral dependence on wavenumber is observed which is characteristic of high turbulent Reynolds number flows. Experimental limitations prevent the resolution of the turbulent scale lengths in the present experiment although the integral scale should be easily resolved with increased data length.

Effect of nonequilibrium dissociation on supersonic three-dimensional flow past inverted cones

Effect of nonequilibrium dissociation on supersonic three-dimensional flow past inverted cones