Three-dimensional Supersonic Flow Research Articles

Blunt-fin induced interaction of wall shear layers in supersonic flow

The topic of this contribution is a numerical study of three-dimensional supersonic flow over a blunt fin mounted on a flat plate. The flow pattern is characterized by complex interaction of shock waves and the boundary layer along the plate surface. In this paper the steady-state results for two free stream Mach numbers M = 3 and 7 will be discussed and compared with experiments. For the numerical investigation a time-marching integration method has been applied based on a three-stage Runge-Kutta scheme. Several convergence acceleration techniques have been adopted. Since crisp shock fronts must be resolved for a proper computation of the interaction phenomena the mesh has been adapted in those regions where strong pressure gradients are expected. In addition, a special treatment of the artificial viscosity terms which enhances stability in regions with shocks has been introduced to achieve some upwind properties of the scheme which utilizes central differences for the spatial derivatives. For the simulation of turbulent flow algebraic models have been applied, namely, a Baldwin-Lomax model and a new turbulence model which is based on RNG-theory. The comparison with the experiments shows that for the case studied here the transitional character of the boundary layer has to be taken into account.

A numerical method for solving the three-dimensional parabolized Navier-Stokes equations

A numerical technique that solves the parabolized form of the Navier-Stokes equations is presented. Such a method makes it possible to obtain very detailed descriptions of the flowfield in a relatively modest CPU time. The approach is based on a space-marching technique, using a finite volume discretization and an upwind flux difference splitting scheme for the evaluation of the inviscid fluxes. Second order accuracy is achieved following the guidelines of the ENO schemes. At first, results concerning two-dimensional test cases are shown to make possible a comparison with other experimental and numerical data. Then, the methodology is used to investigate three-dimensional supersonic viscous flows over symmetric corners. Primary and secondary streamwise vortical structures embedded in the boundary layer and originated by the interaction with shock waves are detected and studied. For the purpose of validation, results are compared with experimental data extracted from the literature. The agreement is found to be satisfactory. In conclusion, the numerical method proposed seems to be promising, as it permits us, at reasonable computational expense, to investigate complex three-dimensional flowfields in great detail.

Information technology of mathematical modelling of a three-dimensional supersonic flow on the basis of particular solutions of hydrodynamic problems

Information technology of mathematical modelling of a three-dimensional supersonic flow on the basis of particular solutions of hydrodynamic problems

Some aspects of high-speed blunt body flow computations with Roe scheme

This paper discusses some aspects of three-dimensional supersonic and hypersonic inviscid blunt body flow computations. The method used is based on the solution of the Euler equations employing an explicit, upwind total variation diminishing formulation of the Roe Riemann solver within the cell-centered finite volume approach. A comparative study of various total variation diminishing limiters and entropy fixes is carried out to identify the most appropriate combination. The effect of using cell spacing in the total variation diminishing extrapolations is highlighted. Furthermore, a Mach number dependence and grid sensitivity study is carried out on various total variation diminishing limiters scaled with mesh spacing. Local time stepping and code parallelization has been employed to accelerate convergence in terms of effective wall clock times.

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

Three-dimensional steady supersonic duct flow using Lagrangian formulation

In this paper, numerical simulation of three-dimensional supersonic flow in a duct is presented. The flow field in the duct is complex and can find its applications in the inlet of air-breathing engines. A unique streamwise marching Lagrangian method is employed for solving the steady Euler equations. The method was first initiated by Loh and Hui (1990) for 2-D steady supersonic flow computations and then extended to 3-D computation by the present authors Loh and Liou (1992). The new scheme is shown to be capable of accurately resolving complicated shock or contact discontinuities and their interactions. In all the computations, a free stream of Mach numberM=4 is considered.

Kolmogorov-like spectra in decaying three-dimensional supersonic flows

A numerical simulation of decaying supersonic turbulence using the piecewise parabolic method (PPM) algorithm on a computational mesh of 5123 zones indicates that, once the solenoidal part of the velocity field, representing vortical motions, is fully developed and has reached a self-similar regime, a velocity spectrum compatible with that predicted by the classical theory of Kolmogorov develops. It is followed by a domain with a shallower spectrum. A convergence study is presented to support these assertions. The formation, structure, and evolution of slip surfaces and vortex tubes are presented in terms of perspective volume renderings of fields in physical space.

Numerical simulation of the flow over a body flying through a thermal in a stratified atmosphere

The three-dimensional supersonic viscous flow over a blunted body flying horizontally through a cloud of hot gas (a thermal) is studied numerically. It is assumed that the cloud was formed as the result of a powerful explosion in the Earth's atmosphere. Consideration is given to the late stage of the explosion, corresponding to the vortex ring formation time. A complete problem is unstable. However, the air flow near the body at every point of the flight trajectory can be considered as a quasi-stationary flow, in view of the fact that the ratio of the characteristic time scale in the shock layer to the thermal characteristic time scale is 10 −5. The aerodynamic and heat transfer characteristics of the flying body are studied.

Planar velocity measurement in symmetric flow fields with laser-induced iodine fluorescence

A laser-induced fluorescence technique for time-averaged planar velocity measurements in nonreacting compressible flow is reported. The technique is based on the detection of fluorescence from a Doppler-shifted absorption line of iodine molecules seeded into the flow field. We exploit the symmetry of the investigated flow field to eliminate the collisional impact shift from the measured total frequency shift of the iodine absorption spectrum. This reduces the amount of spectral data required for measuring two velocity components by a factor of 2 yet retains the accuracy of a counterpropagating laser sheets approach. The demonstration of this technique in a highly three-dimensional supersonic flow field represents what is to our knowledge the first planar velocity measurement with laser-induced fluorescence in which the contribution of the collisional shift is eliminated.

Experimental investigation of a supersonic swept ramp injector using laser-induced iodine fluorescence

Planar measurements of injectant mole fraction and temperature have been conducted in a nonreacting supersonic combustor configured with underexpanded injection in the base of a swept ramp. The temperature measurements were conducted with a Mach 2 test section inlet in streamwise planes perpendicular to the test section wall on which the ramp was mounted. Injection concentration measurements, conducted in cross flow planes with both Mach 2 and Mach 2.9 free stream conditions, dramatically illustrate the domination of the mixing process by streamwise vorticity generated by the ramp. These measurements, conducted using a nonintrusive optical technique (laser-induced iodine fluorescence), provide an accurate and extensive experimental data base for the validation of computation fluid dynamic codes for the calculation of highly three-dimensional supersonic combustor flow fields.

Prediction of asymmetric vortical flows around slender bodies using Navier-Stokes equations

Steady and unsteady asymmetric vortical flows around slender bodies at high angles of attack are solved using the unsteady, compressible, this-layer Navier-Stokes equations. An implicit, upwind-biased, flux-difference splitting, finite-volume scheme is used for the numerical computations. For supersonic flows past point cones, the locally conical flow assumption has been used for efficient computational studies of this phenomenon. Asymmetric flows past a 5° semiapex-angle circular cone at different angles of attack, free-stream Mach numbers, and Reynolds numbers has been studied in responses to different sources of disturbances. The effects of grid fineness and computational domain size have also been investigated. Next, the responses of three-dimensional supersonic asymmetric flow around a 5° circular cone at different angles of attack and Reynolds numbers to short-duration sideslip disturbances are presented. The results show that flow asymmetry becomes stronger as the Reynolds number and angles of attack are increased. The asymmetric solutions show spatial vortex shedding which is qualitatively similar to the temporal vortex shedding of the unsteady locally conical flow. A cylindrical afterbody is also added to the same cone to study the effect of a cylindrical part on the flow asymmetry. One of the cases of flow over a cone-cylinder configuration is validated fairly well by experimental data.

Variational problem of optimum side wall design for a narrow three-dimensional nozzle

The optimum design of the side walls of the supersonic section of a three-dimensional nozzle with two planes of symmetry is considered in the narrow channel model approximation, which reduces three-dimensional to two-dimensional flow. This nozzle realizes maximum thrust for given sonic or supersonic inlet flow, upper and lower walls, maximum permissible length and pressure outside the nozzle. In general, an approximate solution of the variational problem can be obtained by the indeterminate control contour method [1]. For nozzles with nonexpanding “end” sections of the upper and lower walls this is a rigorous solution. Numerical algorithms, based on the method of characteristics, for constructing the optimum, side walls and calculating the flow in narrow channels are developed in the formulation adopted using the optimality conditions found, which generalize the wellknown conditions for plane and axisymmetric configurations [1]. In addition, the three-dimensional supersonic flow in the nozzles thus designed has been calculated in accordance with a shock-capturing marching scheme [2], which for the uniform grids employed in the calculations gives a second-order approximation. A rather complex relation is established between the thrust of the optimum configurations constructed and the shape of their inlet cross sections.

Computational methods for shock waves in three-dimensional supersonic flow

A central artificial-viscosity and an upwind-biased difference method have been contrived to solve the Euler equations for the flowfield over typical spacecrafts. The spatial discretization is based on either nodal or cell vertex formulation in the domain extending from free stream to the end of the vehicle. The outer boundary is treated as a bow shock in the first method but is placed in the free stream in the second, which captures both bow and internal shocks using an approximate Riemann solver based on high-order extrapolation to the cell face. These methods were tested for the Shuttle and Hermes orbiters at wind-tunnel conditions and angles of attack ranging from 0 to 60°. The artificial-viscosity method incorporated with a shock fitting procedure has shown smeared crossflow and wing shock positions and required 15% more CPU per node than the upwind method. Greater flexibility and robustness is demonstrated by the latter on a fixed grid for all cases considered.

Multicomponent three-dimensional viscous shock layer on blunt bodies with a catalytic surface at angles of attack and yaw

The three-dimensional supersonic flow of nonequilibrium dissociating air past smooth blunt bodies on whose surface heterogeneous chemical reactions are taking place is investigated within the framework of thin viscous shock layer theory. An economical numerical method of solving the equations with an improved order of approximation with respect to the normal coordinate is employed. This method does not require the preliminary solution of the Stefan-Maxwell relations for the diffusion fluxes and makes it possible to calculate flows that do not possess a plane of symmetry. The effect of the angles of attack and yaw, the catalytic reaction model and a number of other parameters of the problem on the pressure, heat flux and equilibrium surface temperature distributions is analyzed with reference to the example of flow past a triaxial ellipsoid.

A second-order finite-difference scheme for computing three-dimensional supersonic flows of an ideal gas

An explicit second-order accurate finite-difference scheme for computing the steady three-dimensional supersonic flows of an inviscid gas is proposed. The features of the scheme are described. The results of systematical computations of conical flows and submerged jets are in good agreement with data obtained using existing second-order accurate schemes. Some samples computations for compound three-dimensional supersonic flows are presented.

Use of the curvature hypothesis for calculating supersonic flow past a rotating finned body

Three-dimensional supersonic ideal-gas flow past axisymmetric finned bodies rotating about the longitudinal axis is considered. A calculation method based on the numerical solution of the Euler equations by finite differences is described. The effect of the rotation of the body is taken into account within the framework of the curvature hypothesis [1], which provided that the dimensionless rate of rotation is small reduces the solution of the unsteady three-dimensional problem of supersonic flow past a rotating body to the solution of the steady-state problem of flow past a nonrotating body with specially curved fins. The problem of the rotation of a finned body in a free stream is solved.

On three-dimensional supersonic flow past blunt bodies when interference is present

The numerical solution of the problem of three-dimensional supersonic inviscid thermally non-conducting gas flow past bodies is considered when mutual influence occurs. The characteristic-mesh method is used to solve the equations of gas dynamics. Computational results are given for the example of flow past a sphere.

Euler solver for three-dimensional supersonic flows with subsonic pockets

A new finite-difference scheme has been developed to solve efficiently the unsteady Euler equations for three-dimensional inviscid supersonic flows with subsonic pockets. The technique utilizes planar Gauss-Seidel relaxation in the marching direction and approximate factorization n the crossflow plane. An 'infinitely large' time step is used in parts of the flowfield where the component of velocity in the marching direction is supersonic - here the Gauss-Seidel sweeps are restricted to the forward direction only, and the procedure reduces to simple space-marching; a finite time step is used in parts of the flowfield where the marching component of velocity is subsonic - here, backward and forward Gauss-Seidel sweeps are employed to allow for upstream and downstream propagation of signals, and a time-asymptotic steady state is obtained. The discretization formulas are based on finite-volume implementation of high accuracy (up to third-order) total variation diminishing formulations. Numerical solutions are obtained for an analytically defined forebody, a realistic fighter configuration, and the Space Shuttle. The results are in very good agreement with available experimental data and numerical solutions of the full-potential equation.

Investigation of supersonic three-dimensional flow past pointed axisymmetric bodies

The flow pattern near bodies of revolution with very long cylindrical and pointed nose sections is studied in the framework of an ideal gas model by means of a numerical method based on MacCormack's difference scheme. The existence of internal shock waves, oriented in both the longitudinal and the transverse directions, in the shock layer is established. The variation of the aerodynamic coefficients of the configuration with its length, angle of attack, and free stream Mach number is investigated. The calculated and experimental data are compared, and the connection between the flow parameters on the body surface and the position of the separation line of the boundary layer on its lateral face is established. A method of calculating the influence of the boundary layer on the values of the aerodynamic coefficients of bodies of revolution of large aspect ratio at small angles of attack is proposed. Axisymmetric flow near blunt bodies has been studied in detail in [1].

Dynamic stability tests on finned bodies at hypersonic Mach number

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