C. H. Su

KORTEWEG--DE VRIES EQUATION AND GENERALIZATIONS. III. DERIVATION OF THE KORTEWEG--DE VRIES EQUATION AND BURGERS EQUATION.

The Korteweg‐de Vries equation and the Burgers equation are derived for a wide class of nonlinear Galilean‐invariant systems under the weak‐nonlinearity and long‐wavelength approximations. The former equation is shown to be a limiting form for nonlinear dispersive systems while the latter is a limiting form for nonlinear dissipative systems.

On head-on collisions between two solitary waves

We consider a head-on collision between two solitary waves on the surface of an inviscid homogeneous fluid. A perturbation method which in principle can generate an asymptotic series of all orders, is used to calculate the effects of the collision. We find that the waves emerging from (i.e. long after) the collision preserve their original identities to the third order of accuracy we have calculated. However a collision does leave imprints on the colliding waves with phase shifts and shedding of secondary waves. Each secondary wave group trails behind its primary, a solitary wave. The amplitude of the wave group diminishes in time because of dispersion. We have also calculated the maximum run-up amplitude of two colliding waves. The result checks with existing experiments.

Continuum Theory of Spherical Electrostatic Probes

A continuum theory for spherical electrostatic probes in a slightly ionized plasma is developed. The density of the plasma is taken to be sufficiently high such that both ions and electrons suffer numerous collisions with the neutrals before being collected by an absorbing probe. A general discussion of probes at an arbitrary potential is given. It is found that for very negative probe potentials the sheath thickness can be comparable to the probe radius. Two explicit forms of current‐voltage characteristics are given; one for very negative probes, the other for probes at nearly plasma potential. Both of these are based on the assumption that the probe radius is large compared with the Debye length. Numerical computation is also given for negative probes of a wider range of probe sizes.

Stochastic Modeling of Flow-Structure Interactions using Generalized Polynomial Chaos

We present a generalized polynomial chaos algorithm to model the input uncertainty and its propagation in flow-structure interactions. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as the trial basis in the random space. A standard Galerkin projection is applied in the random dimension to obtain the equations in the weak form. The resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode. This approach is a generalization of the original polynomial chaos expansion, which was first introduced by N. Wiener and employs the Hermite polynomials (a subset of the Askey scheme) as the basis in random space. The algorithm is first applied to second-order oscillators to demonstrate convergence, and subsequently is coupled to incompressible Navier-Stokes equations. Error bars are obtained, similar to laboratory experiments, for the pressure distribution on the surface of a cylinder subject to vortex-induced vibrations

Spectral Polynomial Chaos Solutions of the Stochastic Advection Equation

We present a new algorithm based on Wiener–Hermite functionals combined with Fourier collocation to solve the advection equation with stochastic transport velocity. We develop different stategies of representing the stochastic input, and demonstrate that this approach is orders of magnitude more efficient than Monte Carlo simulations for comparable accuracy.

Collisions between two solitary waves. Part 2. A numerical study

Collisions between two solitary waves are investigated using a numerical scheme. The phase shifts and maximum amplitude of a collision are checked with a corresponding perturbation calculation and compared with the available experiments. We found a wave train trailing behind each of the emerging solitary waves from a head-on collision. The properties of the wave train are in agreement with those of the perturbation solution. After the collision, the solitary waves recover almost all of their original amplitude for the length of time in our calculation. However, the difference (less than 2 % of their original value) persists and accounts for the energy residing in the wave train.

Kinetic Theory Approach to Electrostatic Probes

A spherical electrostatic (Langmuir) probe in a slightly ionized plasma is studied from a kinetic theory point of view. The two‐sided distribution function method of Lees, which embodies the Mott‐Smith approach, is used. The velocity space is divided into two regions along the straight cone tangent to the spherical probe, and different distribution functions are defined in the two regions. On satisfying the two relevant moments of the distribution function (continuity and number density flux) three simultaneous ordinary nonlinear differential equations, which are appropriate to all values of the Debye length, collision mean free path and probe potential, are obtained for determining the ion and electron number densities, and the potential. These equations reduce to the usual linear flux equations when the mean free path is much shorter than the probe radius and the Debye length. The equations are first linearized and solved for the case of small probe potential. Explicit solutions are given for the curren...

Continuum Theory of Electrostatic Probes

Electrostatic probes of general ellipsoidal configurations are analyzed from a continuum viewpoint for the conditions of a quiescent slightly ionized plasma. Approximate analytic current—voltage characteristics are obtained for arbitrary ellipsoidal shapes. The cylindrical probe and circular disk probe are discussed in detail as two special cases.

Predicting shock dynamics in the presence of uncertainties

We revisit the classical aerodynamics problem of supersonic flow past a wedge but subject to random inflow fluctuations or random wedge oscillations around its apex. We first obtain analytical solutions for the inviscid flow, and subsequently we perform stochastic simulations treating randomness both as a steady as well as a time-dependent process. We use a multi-element generalized polynomial chaos (ME-gPC) method to solve the two-dimensional stochastic Euler equations. A Galerkin projection is employed in the random space while WENO discretization is used in physical space. A key issue is the characteristic flux decomposition in the stochastic framework for which we propose different approaches. The results we present show that the variance of the location of perturbed shock grows quadratically with the distance from the wedge apex for steady randomness. However, for a time-dependent random process the dependence is quadratic only close to the apex and linear for larger distances. The multi-element version of polynomial chaos seems to be more effective and more efficient in stochastic simulations of supersonic flows compared to the global polynomial chaos method.

COMPRESSIBLE PLASMA FLOW OVER A BIASED BODY

Abstract : The flow field of a compressible plasma over a biased body is discussed with special emphasis on the electrical characteristics. The governing equations in various asymptotic regions are investigated. A stagnation point probe theory, because of its great practical interest, is given in detail. An analytic current-voltage characteristic is obtained for this case under the assumption of a very thin electrical sheath.