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In this paper, we consider analytic initial conditions with finite energy, whose complex spatial continuation is a superposition of a smooth background flow and a singular field. Through explicit calculation in the complex plane, we show that under some assumptions, the solution to the 3-D Euler equation ceases to be analytic in the real domain in finite time.
The Korteweg-de Vries (KdV) and the modified Korteweg-de Vries (MKdV) equations are shown to have, besides the regular real solutions, exact regular complex as well as singular solutions. The singular solution for the KdV is real but for the MKdV it is pure imaginary. Implications of the complex solutions are discussed.
In its standard form, geostrophic balance uses a partial representation of the Coriolis force. The resulting formation has a singularity at the equator, and violates mass and momentum conservation. When the horizontal projection of the planetary rotation vector is considered, the singularity at the equator disappears, continuity can be preserved, and quasigeostrophy can be formulated at planetary scale.
The study of radiative heat transfer with scattering usually leads to the solution of singular Fredholm integral equations. The present paper presents an accurate and efficient numerical method to solve certain integral equations that govern radiative equilibrium problems in plane-parallel geometry for both grey and nongrey, anisotropically scattering media. In particular, the nongrey problem is represented by a spectral integral of a system of nonlinear integral equations in space, which has not been solved previously. The numerical technique is constructed to handle this unique nongrey governing equation as well as the difficulties caused by singular kernels. Example problems are solved and the method's accuracy and computational speed are analyzed.
Singular perturbation problems for partial differential equations
Einstein equations solution for empty space without singularities containing metric, with closed homogeneous space type hypersurfaces expanding anisotropically
Recent advances in FEM techniques and applications are discussed in reviews and reports presented at the ICASE/LaRC workshop held in Hampton, VA in July 1986. Topics addressed include FEM approaches for partial differential equations, mixed FEMs, singular FEMs, FEMs for hyperbolic systems, iterative methods for elliptic finite-element equations on general meshes, mathematical aspects of FEMS for incompressible viscous flows, and gradient weighted moving finite elements in two dimensions. Consideration is given to adaptive flux-corrected FEM transport techniques for CFD, mixed and singular finite elements and the field BEM, p and h-p versions of the FEM, transient analysis methods in computational dynamics, and FEMs for integrated flow/thermal/structural analysis.
Infinite pointwise stretching in a finite time for general initial conditions is found in a simulation of the Biot-Savart equation for a slender vortex tube in three dimensions. Viscosity is ineffective in limiting the divergence in the vorticity as long as it remains concentrated in tubes. Stability has not been shown.
The problem of potential flow around two-dimensional airfoils is solved by using a new singular integral method. The potential flow equations for incompressible potential flow are written in a singular integral equation. The equation is solved at N collocation points on the airfoil surface. A unique feature of this method is that the airfoil geometry is specified as an independent variable in the exact integral equation. Compared to other numerical methods, the present calculation procedure is much simpler and gives remarkable accuracy for many body shapes. An advantage of the present method is that it allows the inverse design calculation and the results are extremely accurate.
The bypass duct of an aircraft engine is a low-pass filter allowing some spinning modes to radiate outside the duct. The knowledge of the radiated modes can help in noise reduction, as well as the diagnosis of noise generation mechanisms inside the duct. We propose a nonintrusive technique using a circular microphone array outside the engine measuring the complex noise spectrum on an arc of a circle. The array is placed at various axial distances from the inlet or the exhaust of the engine. Using a model of noise radiation from the duct, an overdetermined system of linear equations is constructed for the complex amplitudes of the radial modes for a fixed circumferential mode. This system of linear equations is generally singular, indicating that the problem is illposed. Tikhonov regularization is employed to solve this system of equations for the unknown amplitudes of the radiated modes. An application of our mode detection technique using measured acoustic data from a circular microphone array is presented. We show that this technique can reliably detect radiated modes with the possible exception of modes very close to cut-off.
The vertical structure equation is a singular Sturm-Liouville problem whose eigenfunctions describe the vertical dependence of the normal modes of the primitive equations linearized about a given thermal profile. The eigenvalues give the equivalent depths of the modes. The spectrum of the vertical structure equation and the appropriateness of various upper boundary conditions, both for arbitrary thermal profiles were studied. The results depend critically upon whether or not the thermal profile is such that the basic state atmosphere is bounded. In the case of a bounded atmosphere it is shown that the spectrum is always totally discrete, regardless of details of the thermal profile. For the barotropic equivalent depth, which corresponds to the lowest eigen value, upper and lower bounds which depend only on the surface temperature and the atmosphere height were obtained. All eigenfunctions are bounded, but always have unbounded first derivatives. It was proved that the commonly invoked upper boundary condition that vertical velocity must vanish as pressure tends to zero, as well as a number of alternative conditions, is well posed. It was concluded that the vertical structure equation always has a totally discrete spectrum under the assumptions implicit in the primitive equations.
The vertical structure equation is a singular Sturm-Liouville problem whose eigenfunctions describe the vertical dependence of the normal modes of the primitive equations linearized about a given thermal profile. The eigenvalues give the equivalent depths of the modes. The spectrum of the vertical structure equation and the appropriateness of various upper boundary conditions, both for arbitrary thermal profiles were studied. The results depend critically upon whether or not the thermal profile is such that the basic state atmosphere is bounded. In the case of a bounded atmosphere it is shown that the spectrum is always totally discrete, regardless of details of the thermal profile. For the barotropic equivalent depth, which corresponds to the lowest eigen value, upper and lower bounds which depend only on the surface temperature and the atmosphere height were obtained. All eigenfunctions are bounded, but always have unbounded first derivatives. It was proved that the commonly invoked upper boundary condition that vertical velocity must vanish as pressure tends to zero, as well as a number of alternative conditions, is well posed. It was concluded that the vertical structure equation always has a totally discrete spectrum under the assumptions implicit in the primitive equations.
This thesis develops a method to model the acoustic field generated by a monopole source placed in a moving rectangular duct. The walls of the duct are assumed to be infinitesimally thin and the source is placed at the center of the duct. The total acoustic pressure is written in terms of the free-space pressure, or incident pressure, and the scattered pressure. The scattered pressure is the augmentation to the incident pressure due to the presence of the duct. It satisfies a homogeneous wave equation and is discontinuous across the duct walls. Utilizing an integral representation of the scattered pressure, a set of singular boundary integral equations governing the unknown jump in scattered pressure is derived. This equation is solved by the method of collocation after representing the jump in pressure as a double series of shape functions. The solution obtained is then substituted back into the integral representation to determine the scattered pressure, and the total acoustic pressure at any point in the field. A few examples are included to illustrate the influence of various geometric and kinematic parameters on the radiated sound field.
A meteoroid environment model must describe, at a bare minimum, the local density or flux of meteoroids. To accomplish this, many models make use of analytic equations to convert orbital distributions to number density, or to compute the effects of a planet's gravity on flux and speed. However, these equations often contain singularities that are unrealistic. For instance, the orbit-to-density conversion used by Jones (2006) is unbound at peri- and aphelion, and the gravitational focusing algorithms presented by Staubach et al. (1997) are unbound along the anti-radiant line. We present methods for removing these singularities that result in more realistic descriptions of the meteoroid environment.
Prediction of aerodynamic loads on bodies in arbitrary motion is considered from an acoustic point of view, i.e., in a frame of reference fixed in the undisturbed medium. An inhomogeneous wave equation which governs the disturbance pressure is constructed and solved formally using generalized function theory. When the observer is located on the moving body surface there results a singular linear integral equation for surface pressure. Two different methods for obtaining such equations are discussed. Both steady and unsteady aerodynamic calculations are considered. Two examples are presented, the more important being an application to propeller aerodynamics. Of particular interest for numerical applications is the analytical behavior of the kernel functions in the various integral equations.
Prediction of aerodynamic loads on bodies in arbitrary motion is considered from an acoustic point of view, i.e., in a frame of reference fixed in the undisturbed medium. An inhomogeneous wave equation which governs the disturbance pressure is constructed and solved formally using generalized function theory. When the observer is located on the moving body surface there results a singular linear integral equation for surface pressure. Two different methods for obtaining such equations are discussed. Both steady and unsteady aerodynamic calculations are considered. Two examples are presented, the more important being an application to propeller aerodynamics. Of particular interest for numerical applications is the analytical behavior of the kernel functions in the various integral equations.
The singular points in steady state, field-aligned plasma transport models based on velocity moment theory are examined. In particular, two separate singular points in the equations obtained from the 16-moment approximation are identified. These equations are presented in a form that makes the singularities apparent, and they are solved in a simple illustrative case. The singular points, one occurring at the sonic point and the other at a critical value of the parallel heat flux, give rise to different outflow regimes, characterized generically by different asymptotic behavior. The existence of the different outflow regimes separated by the heat flux critical point has been only hinted at in previous discussions of numerical simulation of the polar wind.
A new method for calculating the three-dimensional inviscid surface streamlines and streamline metrics using Cartesian coordinates and time as the independent variable of integration has been developed. The technique calculates the streamline from a specified point on the body to a point near the stagnation point by using a prescribed pressure distribution in the Euler equations. The differential equations, which are singular at the stagnation point, are of the two point boundary value problem type. Laminar heating rates are calculated using the axisymmetric analog concept for three-dimensional boundary layers and approximate solutions to the axisymmetric boundary layer equations. Results for elliptic conic forebody geometries show that location of the point of maximum heating depends on the type of conic in the plane of symmetry and the angle of attack, and that this location is in general different from the stagnation point. The new method was found to give smooth predictions of heat transfer in the nose region where previous methods gave oscillatory results.