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Summary details of an aperiodic time asymptotic numerical solution for the square drive cavity at Re = 10000 are presented. The data presented is for 6100 less than t less than or equal to 7100, and is representative of the data that characterizes the aperiodic asymptotic state.
The present consideration of the flight trajectory of hypersonic aerospace vehicles subject to a class of path constraints notes the constrained dynamics to constitute a natural two-timescale system, so that problems of trajectory optimization and guidance can be dramatically simplified by means of the asymptotic analytical solutions thus obtained. An illustrative application in ascent trajectory optimization for an aerospace vehicle is presented.
A detailed evaluation is performed of a guidance algorithm for aeroassisted orbit transfer that was developed earlier based on the method of matched asymptotic expansions. It is shown that, by exploiting the structure of the matched asymptotic expansion solution procedure, the original problem which requires the solution of a set of 20 implicit algebraic equations can be reduced to a problem of six implicit equations in six unknowns. The main contribution here is that it was possible to obtain a solution that is near optimal, requires a minimum of computation, and thus can be implemented in real time and on-board the vehicle. Guidance law implementation entails treating the current state as a new initial state and repetitively solving the matched asymptotic expansion problem to obtain the feedback controls.
The asymptotic description of the interaction between a normal shock wave and a turbulent boundary layer is reviewed. The layers necessary in a rational analysis of the interaction are discussed with emphasis on the differences from an interaction with a laminar boundary layer, the uncoupling of solutions for the distribution of pressure and skin friction at the wall, and the role of the Reynolds shear stress in these solutions. The accuracy of asymptotic solutions in flows at Reynolds numbers of technical interest is discussed. Solutions for the distribution of pressure and skin friction at the wall and the shape of the shock are considered for the case where the flow is near separation. For the pressure and skin friction, it is possible to write two simplified partial solutions, one valid at the beginning of the interaction and one valid somewhat downstream of the shock wave. A solution composed of these two parts and a linear interpolation between them appears to give good comparison with experiment; one unknown constant, independent of the parameters of the interaction, must be found from experiment. The simplified relations are presented. Comparison of numerical computations with experimental data indicates a possible value for the constant and shows quite satisfactory results.
The Burnett equations in the cylindrical coordinate system have been applied to the hypersonic flow over a cylinder in low density environments. Through a local similarity transformation, the Burnett equations are degenerated into nonlinear third-order ordinary differential equations which are only valid along the stagnation line. These equations are solved by the multi-staged Runge-Kutta integration technique. Steady state solutions are asymptotically reached by the time-marching procedure. A newly-developed computer code is used to solve both the Navier-Stokes and the Burnett equations within the shock layer. The computed heat transfer coefficients are consistent with wind-tunnel and STS flight data. Comparisons with the DSMC solutions and the Boltzmann solutions show that the Burnett equations are more appropriate as the governing equations than the Navier-Stokes equations in the transitional flow regime.
We study the conditions which lead to the conversion of most Poynting flux into kinetic energy flux in cold, relativistic hydromagnetic winds. It is shown that plasma acceleration along a precisely radial flow is extremely inefficient due to the near cancellation of the toroidal magnetic pressure and tension forces. However, if the flux tubes in a flow diverge even slightly faster than radially, the fast magnetosonic point moves inward from infinity to a few times the light cylinder radius. Once the flow becomes supermagnetosonic, further divergence of the flux tubes beyond the fast point can accelerate the flow via the 'magnetic nozzle' effect, thereby further converting Poynting flux to kinetic energy flux. We show that the Grad-Shafranov equation admits a generic family of kinetic energy-dominated asymptotic wind solutions with finite total magnetic flux. The Poynting flux in these solutions vanishes logarithmically with distance. The way in which the flux surfaces are nested within the flow depends only on the ratio of angular velocity to poliodal 4-velocity as a function of magnetic flux. Radial variations in flow structure can be expressed in terms of a pressure boundary condition on the outermost flux surface, provided that no external toriodal field surrounds the flow. For a special case, we show explicitly how the flux surfaces merge gradually to their asymptotes. For flows confined by an external medium of pressure decreasing to zero at infinity we show that, depending on how fast the ambient pressure declines, the final flow state could be either a collimated jet or a wind that fills the entire space. We discuss the astrophysical implications of our results for jets from active galactic nuclei and for free pulsar winds such as that believed to power the Crab Nebula.
Numerical solution of boundary layer differential equations
The application of the nonsteady Lax-Wendroff technique to problems with asymptotically periodic solution which offers a potentially powerful method for the investigation of the interaction of rotating and stationary blade rows in turbomachinery is reported. A technique for specifying boundary conditions with phase lag was developed to accomplish this. A complete nonlinear analysis is carried out numerically to determine the entire flow field without recourse to the assumption of small disturbances of linear equations which underlie the previous acoustic theories. The result, obtained for the case of equal number of rotor and stator blades shows that transonic flow can be handled without difficulty. In addition, the program is not limited with regard to blade thickness, camber or loading. Extension of this method to incorporate viscous wakes and to analysis of fully three dimensional configuration is feasible, and would greatly expand its utility in practical applications.
The status of an investigation of four numerical techniques for the time-dependent compressible Navier-Stokes equations is presented. Results for free shear layer calculations in the Reynolds number range from 1000 to 81000 indicate that a sequential alternating-direction implicit (ADI) finite-difference procedure requires longer computing times to reach steady state than a low-storage hopscotch finite-difference procedure. A finite-element method with cubic approximating functions was found to require excessive computer storage and computation times. A fourth method, an alternating-direction cubic spline technique which is still being tested, is also described.
Fluctuations in intensity due to wave propagation in a wide range of turbulent media, are discussed on the basis of a previously developed theory of strong scintillations. An asymptotic analytic solution to the fourth-moment equation is presented, and its range of validity is discussed. Various parameters characterizing the intensity fluctuations, such as scintillation index and correlation scale, are expressed in terms of turbulence parameters for both Gaussian and power-law irregularity spectra as well as for thick and thin turbulent media.
Lifting line theory is applied to describe the flow about a lifting wing at transonic speeds. The method extends that of Van Dyke (1975), in which lifting line theory is viewed as a singular perturbation problem, to transonic flows. Inner and outer expansions as the aspect ratio approaches infinity of the transonic small disturbance equations are found. It is shown that the solutions match asymptotically. A boundary value problem is formulated which describes the first aspect ratio correction to the two dimensional cross sectional transonic flow. The theory is especially applicable to wings of similar cross-sections.
A method for predicting the strong interaction between the viscous and inviscid flows which occurs in flow separation is reviewed. An inverse boundary-layer procedure approximately accounts for normal pressure gradients that may be important in strongly interacting flows. Transformed boundary-layer equations are written in which the pressure gradient is set equal to the inviscid pressure gradient. As the boundary-layer edge is approached and the viscous shear and heat conduction terms vanish, the viscous flow solution is required to asymptotically approach the inviscid solution over the generalized displacement body. Attention is then focused on viscous-inviscid interacting flows with a first-order viscous formulation and constant pressure across the boundary layer. Results obtained with this procedure are presented for: (1) transitional separation bubbles near an airfoil leading edge, (2) subsonic boattail separated turbulent flow, and (3) transonic turbulent shock wave boundary-layer interaction on an axisymmetric bump configuration.
Two recently developed aerodynamic prediction techniques based on the steady full potential equation and the unsteady Euler equations have been applied to a variety of three-dimensional supersonic flow problems exhibiting embedded subsonic regions. Both techniques utilize planar Gauss-Seidel relaxation in the marching direction and approximate factorization in the cross-flow plane. A conservative switching scheme and flux bias technique are employed in the full potential method to transition from the supersonic marching procedure to a subsonic relaxation algorithm and vice versa. A new unified approach with finite volume, high accuracy (up to third order) Total Variation Diminishing formulation (based on Roe's scheme) is used in the Euler solver. In the supersonic regions of the flow an 'infinitely large' time step is employed, and a finite time step is applied in the subsonic regions of the flow to reach the steady-state as a time-asymptote. Numerical solutions are obtained for a number of complex configurations, including: (1) an elliptic waverider, (2) a realistic fighter configuration, (3) the Space Shuttle, and (4) a Shuttle-like configuration. Both the Full Potential and Euler numerical results are in good agreement with available experimental data.
The development of Gortler vortices in boundary layers over curved walls in the nonlinear regime is investigated. The growth of the boundary layer makes a parallel flow analysis impossible except in the high wavenumber regime so in general the instability equations must be integrated numerically. Here the spanwise dependence of the basic flow is described using Fourier series expansion whilst the normal and streamwise variations are taken into account using finite differences. The calculations suggest that a given disturbance imposed at some position along the wall will eventually reach a local equilibrium state essentially independent of the initial conditions. In fact, the equilibrium state reached is qualitatively similar to the large amplitude high wave-number solution described asymptotically by Hall (1982). In general, it is found that the nonlinear interactions are dominated by a mean field type of interaction between the mean flow and the fundamental. Thus, even though higher harmonics of the fundamental are necessarily generated, most of the disturbance energy is confined to the mean flow correction and the fundamental. A major result of the calculations is finding that the downstream velocity field develops a strongly inflection character as the flow moves downstream. The latter result suggests that the major effect of Gortler vortices on boundary layers of practical importance might be to make them highly receptive to rapidly growing Rayleigh modes of instability.
The development of Goertler vortices in boundary layers over curved walls in the nonlinear regime is investigated. The growth of the boundary layer makes a parallel flow analysis impossible except in the high wavenumber regime so in general the instability equations must be integrated numerically. Here the spanwise dependence of the basic flow is described using Fourier series expansion while the normal and streamwise variations are taken into account using finite differences. The calculations suggest that a given disturbance imposed at some position along the wall will eventually reach a local equilibrium state essentially independent of the initial conditions. In fact, the equilibrium state reached is qualitatively similar to the large amplitude high wave-number solution described asymptotically by Hall (1982). In general, it is found that the nonlinear interactions are dominated by a mean field type of interaction between the mean flow and the fundamental. Thus, even though higher harmonics of the fundamental are necessarily generated, most of the disturbance energy is confined to the mean flow correction and the fundamental. A major result of the calculations is finding that the downstream velocity field develops a strongly inflectional character as the flow moves downstream. The latter result suggests that the major effect of Goertler vortices on boundary layers of practical importance might be to make them highly receptive to rapidly growing Rayleigh modes of instability.
An asymptotic theory is presented for the determination of velocity and linear stability of a steady symmetric bubble in a Hele-Shaw cell for small surface tension. In the first part, the bubble velocity U relative to the fluid velocity at infinity is determined for small surface tension T by determining transcendentally small correction to the asymptotic series solution. It is found that for any relative bubble velocity U in the interval (U(c),2), solutions exist at a countably infinite set of values of T (which has zero as its limit point) corresponding to the different branches of bubble solutions. U(c) decreases monotonically from 2 to 1 as the bubble area increases from 0 to infinity. However, for a bubble of arbitrarily given size, as T approaches 0, solution exists on any given branch with relative bubble velocity U satisfying the relation 2-U = cT to the 2/3 power, where c depends on the branch but is independent of the bubble area. The analytical evidence further suggests that there are no solutions for U greater than 2. These results are in agreement with earlier analytical results for a finger. In Part 2, an analytic theory is presented for the determination of the linear stability of the bubble in the limit of zero surface tension. Only the solution branch corresponding to the largest possible U for given surface tension is found to be stable, while all the others are unstable, in accordance with earlier numerical results.
The Roe flux-difference splitting method has been extended to treat two-dimensional viscous flows with nonequilibrium chemistry. The derivations have avoided unnecessary assumptions or approximations. For spatial discretization, the second-order Roe upwind differencing is used for the convective terms and central differencing for the viscous terms. An upwind-based TVD scheme is applied to eliminate oscillations and obtain a sharp representation of discontinuities. A two-stage Runge-Kutta method is used to time integrate the discretized Navier-Stokes and species transport equations for the asymptotic steady solutions. The present method is then applied to two types of flows: the shock wave/boundary layer interaction problems and the jet in cross flows.
An asymptotic theory is presented for the determination of velocity and linear stability of a steady symmetric bubble in a Hele-Shaw cell for small surface tension. First the bubble velocity relative to the fluid velocity at infinity is determined for small surface tension by means of a transcendentally small correction to the asymptotic series solution. In addition, a linear stability analysis shows that only the solution branch corresponding to the largest possible bubble velocity for given surface tension is stable, while all the others are unstable.