213 research outputs found
Stability of Inverse Problems for Steady Supersonic Flows Past Lipschitz Perturbed Cones
We are concerned with inverse problems for supersonic potential flows past
infinite axisymmetric Lipschitz cones. The supersonic flows under consideration
are governed by the steady isentropic Euler equations for axisymmetric
potential flows, which involve a singular geometric source term. We first study
the inverse problem for the stability of an oblique conical shock as an
initial-boundary value problem with both the generating curve of the cone
surface and the leading conical shock front as free boundaries. We then
establish the existence and asymptotic behavior of global entropy solutions
with bounded BV norm of this problem, when the Mach number of the incoming flow
is sufficiently large and the total variation of the pressure distribution on
the cone is sufficiently small. To this end, we first develop a modified
Glimm-type scheme to construct approximate solutions by self-similar solutions
as building blocks to balance the influence of the geometric source term. Then
we define a Glimm-type functional, based on the local interaction estimates
between weak waves, the strong leading conical shock, and self-similar
solutions, along with the construction of the approximate generating curves of
the cone surface. Next, when the Mach number of the incoming flow is
sufficiently large, by asymptotic analysis of the reflection coefficients in
those interaction estimates, we prove that appropriate weights can be chosen so
that the corresponding Glimm-type functional decreases in the flow direction.
Finally, we determine the generating curves of the cone surface and establish
the existence of global entropy solutions containing a strong leading conical
shock, besides weak waves. Moreover, the entropy solution is proved to approach
asymptotically the self-similar solution determined by the incoming flow and
the asymptotic pressure on the cone surface at infinity.Comment: 41 pages, 5 figures. arXiv admin note: text overlap with
arXiv:2008.0240
Stability of Conical Shocks in the Three-Dimensional Steady Supersonic Isothermal Flows past Lipschitz Perturbed Cones
We are concerned with the structural stability of conical shocks in the
three-dimensional steady supersonic flows past Lipschitz perturbed cones whose
vertex angles are less than the critical angle. The flows under consideration
are governed by the steady isothermal Euler equations for potential flow with
axisymmetry so that the equations contain a singular geometric source term. We
first formulate the shock stability problem as an initial-boundary value
problem with the leading conical shock-front as a free boundary, and then
establish the existence and asymptotic behavior of global entropy solutions in
of the problem. To achieve this, we first develop a modified Glimm scheme
to construct approximate solutions via self-similar solutions as building
blocks in order to incorporate with the geometric source term. Then we
introduce the Glimm-type functional, based on the local interaction estimates
between weak waves, the strong leading conical shock, and self-similar
solutions, as well as the estimates of the center changes of the self-similar
solutions. To make sure the decreasing of the Glimm-type functional, we choose
appropriate weights by careful asymptotic analysis of the reflection
coefficients in the interaction estimates, when the Mach number of the incoming
flow is sufficiently large. Finally, we establish the existence of global
entropy solutions involving a strong leading conical shock-front, besides weak
waves, under the conditions that the Mach number of the incoming flow is
sufficiently large and the weighted total variation of the slopes of the
generating curve of the Lipschitz perturbed cone is sufficiently small.
Furthermore, the entropy solution is shown to approach asymptotically the
self-similar solution that is determined by the incoming flow and the
asymptotic tangent of the cone boundary at infinity.Comment: 50 pages; 7 figue
Multidimensional Conservation Laws: Overview, Problems, and Perspective
Some of recent important developments are overviewed, several longstanding
open problems are discussed, and a perspective is presented for the
mathematical theory of multidimensional conservation laws. Some basic features
and phenomena of multidimensional hyperbolic conservation laws are revealed,
and some samples of multidimensional systems/models and related important
problems are presented and analyzed with emphasis on the prototypes that have
been solved or may be expected to be solved rigorously at least for some cases.
In particular, multidimensional steady supersonic problems and transonic
problems, shock reflection-diffraction problems, and related effective
nonlinear approaches are analyzed. A theory of divergence-measure vector fields
and related analytical frameworks for the analysis of entropy solutions are
discussed.Comment: 43 pages, 3 figure
Two-Dimensional Steady Supersonic Exothermically Reacting Euler Flow past Lipschitz Bending Walls
We are concerned with the two-dimensional steady supersonic reacting Euler
flow past Lipschitz bending walls that are small perturbations of a convex one,
and establish the existence of global entropy solutions when the total
variation of both the initial data and the slope of the boundary is
sufficiently small. The flow is governed by an ideal polytropic gas and
undergoes a one-step exothermic chemical reaction under the reaction rate
function that is Lipschtiz and has a positive lower bound. The heat released by
the reaction may cause the total variation of the solution to increase along
the flow direction. We employ the modified wave-front tracking scheme to
construct approximate solutions and develop a Glimm-type functional by
incorporating the approximate strong rarefaction waves and Lipschitz bending
walls to obtain the uniform bound on the total variation of the approximate
solutions. Then we employ this bound to prove the convergence of the
approximate solutions to a global entropy solution that contains a strong
rarefaction wave generated by the Lipschitz bending wall. In addition, the
asymptotic behavior of the entropy solution in the flow direction is also
analyzed.Comment: 58 pages, 16 figures; SIAM J. Math. Anal. (accepted on November 1,
2016
Exploratory Measurement of Recession Rates of Low Temperature Ablators Subjected to Mach 6 Flow
The high speed/high-temperature effect of heat shield ablation was simulated in the low-enthalpy AFRL Mach 6 Ludwieg Tube using solid dry ice as a low-temperature sublimator. The experiments utilized both 21 half-angle cones and bi-conic models with a 7 ° half-angle leading edge followed by a 26° half-angle base contained within a cryogenic-cooled stainless steel holder. A method of fabricating dry ice test articles was developed using commercially procured dry ice and custom-made aluminum molds. Tests were performed at Mach 6.1 with a stagnation temperature of 490 K and stagnation pressures ranging from 40 - 500 psi. Unit Reynolds number ranged from 2.6 x 106 to 23 x 106 m−1 . High-speed Schlieren photography with a frame rate of 20 kHz was used for visualization and data analysis. The observed ablation rates compared favorably to previous research and were analyzed using the Fay-Riddell stagnation point heating correlation. This exploratory effort demonstrated the potential for other uses of dry ice test models in the facility, including for store separation experiments and localized particle-based flow visualization
Development of a computational model for predicting solar wind flows past nonmagnetic terrestrial planets
A computational model for the determination of the detailed plasma and magnetic field properties of the global interaction of the solar wind with nonmagnetic terrestrial planetary obstacles is described. The theoretical method is based on an established single fluid, steady, dissipationless, magnetohydrodynamic continuum model, and is appropriate for the calculation of supersonic, super-Alfvenic solar wind flow past terrestrial ionospheres
Multidimensional Riemann Problems: Transonic Shock Waves and Free Boundary Problems
We are concerned with global solutions of multidimensional Riemann problems
for nonlinear hyperbolic systems of conservation laws, focusing on their global
configurations and structures. We present some recent developments in the
rigorous analysis of two-dimensional Riemann problems involving transonic shock
waves through several prototypes of hyperbolic systems of conservation laws and
discuss some further multidimensional Riemann problems and related problems for
nonlinear partial differential equations. In particular, we present four
different two-dimensional Riemann problems through these prototypes of
hyperbolic systems and show how these Riemann problems can be
reformulated/solved as free boundary problems with transonic shock waves as
free boundaries for the corresponding nonlinear conservation laws of mixed
elliptic-hyperbolic type and related nonlinear partial differential equations.Comment: 46 pages; 10 figures. arXiv admin note: substantial text overlap with
arXiv:2109.1024
Multidimensional transonic shock waves and free boundary problems
We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for potential flow, and related nonlinear PDEs of mixed elliptic-hyperbolic type. The transonic shock problems include the problem of steady transonic flow past solid wedges, the von Neumann problem for shock reflection-diffraction, and the Prandtl-Meyer problem for unsteady supersonic flow onto solid wedges. We first show how these longstanding multidimensional transonic shock problems can be formulated as free boundary problems for the compressible Euler equations and related nonlinear PDEs of mixed type. Then we present an effective nonlinear method and related ideas and techniques to solve these free boundary problems. The method, ideas, and techniques should be useful to analyze other longstanding and newly emerging free boundary problems for nonlinear PDEs
Two-dimensional riemann problems: transonic shock waves and free boundary problems
We are concerned with global solutions of multidimensional (M-D) Riemann problems for nonlinear hyperbolic systems of conservation laws, focusing on their global configurations and structures. We present some recent developments in the rigorous analysis of two-dimensional (2-D) Riemann problems involving transonic shock waves through several prototypes of hyperbolic systems of conservation laws and discuss some further M-D Riemann problems and related problems for nonlinear partial differential equations. In particular, we present four different 2-D Riemann problems through these prototypes of hyperbolic systems and show how these Riemann problems can be reformulated/solved as free boundary problems with transonic shock waves as free boundaries for the corresponding nonlinear conservation laws of mixed elliptic-hyperbolic type and related nonlinear partial differential equations
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