Geometrical shock dynamics for magnetohydrodynamic fast shocks

2016 ◽  
Vol 811 ◽  
Author(s):  
W. Mostert ◽  
D. I. Pullin ◽  
R. Samtaney ◽  
V. Wheatley
Keyword(s):  
Mach Number ◽  
Two Dimensions ◽  
Mhd Equations ◽  
Relative Pressure ◽  
Gas Dynamic ◽  
Shock Dynamics ◽  
Geometrical Shock Dynamics ◽  
Ideal Magnetohydrodynamic ◽  
The Stability ◽  
Spatially Varying

We describe a formulation of two-dimensional geometrical shock dynamics (GSD) suitable for ideal magnetohydrodynamic (MHD) fast shocks under magnetic fields of general strength and orientation. The resulting area–Mach-number–shock-angle relation is then incorporated into a numerical method using pseudospectral differentiation. The MHD-GSD model is verified by comparison with results from nonlinear finite-volume solution of the complete ideal MHD equations applied to a shock implosion flow in the presence of an oblique and spatially varying magnetic field ahead of the shock. Results from application of the MHD-GSD equations to the stability of fast MHD shocks in two dimensions are presented. It is shown that the time to formation of triple points for both perturbed MHD and gas-dynamic shocks increases as $\unicode[STIX]{x1D716}^{-1}$, where $\unicode[STIX]{x1D716}$ is a measure of the initial Mach-number perturbation. Symmetry breaking in the MHD case is demonstrated. In cylindrical converging geometry, in the presence of an azimuthal field produced by a line current, the MHD shock behaves in the mean as in Pullin et al. (Phys. Fluids, vol. 26, 2014, 097103), but suffers a greater relative pressure fluctuation along the shock than the gas-dynamic shock.

Accounting for transverse flow in the theory of geometrical shock dynamics

1993 ◽  
Vol 442 (1916) ◽  
pp. 585-598 ◽  
Keyword(s):  
Order Approximation ◽  
Wave Structure ◽  
Two Dimensions ◽  
Successive Approximations ◽  
Shock Propagation ◽  
Truncation Scheme ◽  
Shock Dynamics ◽  
Geometrical Shock Dynamics ◽  
First Order Approximation ◽  
Original Derivation

The theory of geometrical shock dynamics is deduced as a formal approximation to the equations of gas dynamics in two dimensions, thus overcoming the approximation of quasi-one-dimensional flow utilized in the original derivation. Successive approximations may be obtained by using the truncation scheme developed by Best (1991) for considering shock propagation down a tube of slowly varying cross section. The zero-order approximation yields the theory as developed by Whitham (1957). The first-order approximation includes a term which is additional to those appearing in the equations derived by Best (1991). The characteristic speeds for disturbances propagating on the shock are computed and a hierarchical wave structure is evident. In view of this structure, comment is made upon the problem of blast diffraction by a convex corner.

k-t scattering formulation of the absorptive acoustic wave equation: Wraparound and edge‐effect elimination

Geophysics ◽  
1986 ◽  
Vol 51 (12) ◽  
pp. 2185-2192 ◽  
Author(s):  
B. Compani‐Tabrizi
Keyword(s):  
Wave Equation ◽  
Numerical Solutions ◽  
Two Dimensions ◽  
Time Step ◽  
Absorbing Boundary ◽  
Temporal Derivative ◽  
Time Domain Scattering ◽  
The Stability ◽  
Spatially Varying ◽  
And Storage

The solution algorithm to the absorptive acoustic scalar wave equation with spatially varying velocity and absorptive fields is numerically examined in the context of the k-space time‐domain scattering formalism to construct an absorbing boundary potential which eliminates wraparound and edge effects. The absorptive potential is constructed by using the absorptive coefficient, i.e., the coefficient of the first temporal derivative in the differential equation. Numerical solutions, in two dimensions, show the stability of the algorithm and the elimination of wraparound and edge reflections through use of the constructed absorptive potential. The numbers of calculations and storage requirements per time step are on the order of [Formula: see text] and N, respectively, where N is the number of points into which the problem is discretized.

On generation and convergence of polygonal-shaped shock waves

1996 ◽  
Vol 309 ◽  
pp. 301-319 ◽  
Author(s):  
N. Apazidis ◽  
M. B. Lesser
Keyword(s):  
Mach Number ◽  
Shock Waves ◽  
Wave Front ◽  
Numerical Scheme ◽  
Numerical Procedure ◽  
Arbitrary Form ◽  
Shock Focusing ◽  
Shock Dynamics ◽  
Mach Number Distribution ◽  
Geometrical Shock Dynamics

A process of generation and convergence of shock waves of arbitrary form and strength in a confined chamber is investigated theoretically. The chamber is a cylinder with a specifically chosen form of boundary. Numerical calculations of reflection and convergence of cylindrical shock waves in such a chamber filled with fluid are performed. The numerical scheme is similar to the numerical procedure introduced by Henshaw et al. (1986) and is based on a modified form of Whitham's theory of geometrical shock dynamics (1957, 1959). The technique used in Whitham (1968) for treating a shock advancing into a uniform flow is modified to account for non-uniform conditions ahead of the advancing wave front. A new result, that shocks of arbitrary polygonal shapes may be generated by reflection of cylindrical shocks off a suitably chosen reflecting boundary, is shown. A study is performed showing the evolution of the shock front's shape and Mach number distribution. Comparisons are made with a theory which does not account for the non-uniform conditions in front of the shock. The calculations provide details of both the reflection process and the shock focusing.

Modeling of External Pressure Gradient Influence on the Stability of Disturbances in the Boundary Layers of Compressed Gas

2013 ◽  
Vol 8 (4) ◽  
pp. 64-75
Author(s):  
Sergey Gaponov ◽  
Natalya Terekhova
Keyword(s):  
Mach Number ◽  
Pressure Gradient ◽  
Boundary Layers ◽  
Leading Edge ◽  
External Pressure ◽  
Transition To Turbulence ◽  
Compressed Gas ◽  
High Mach ◽  
The Stability ◽  
Very High

This work continues the research on modeling of passive methods of management of flow regimes in the boundary layers of compressed gas. Authors consider the influence of pressure gradient on the evolution of perturbations of different nature. For low Mach number M = 2 increase in pressure contributes to an earlier transition of laminar to turbulent flow, and, on the contrary, drop in the pressure leads to a prolongation of the transition to turbulence. For high Mach number M = 5.35 found that the acoustic disturbances exhibit a very high dependence on the sign and magnitude of the external gradient, with a favorable gradient of the critical Reynolds number becomes smaller than the vortex disturbances, and at worst – boundary layer is destabilized directly on the leading edge

scholarly journals Levitation of non-magnetizable droplet inside ferrofluid

2018 ◽  
Vol 857 ◽  
pp. 398-448 ◽  
Author(s):  
Chamkor Singh ◽  
Arup K. Das ◽  
Prasanta K. Das
Keyword(s):  
Magnetic Field ◽  
Steady State ◽  
Viscosity Ratio ◽  
Temporal Dynamics ◽  
Liquid Droplet ◽  
Droplet Surface ◽  
Two Dimensions ◽  
Projection Algorithm ◽  
The Stability ◽  
The Individual

The central theme of this work is that a stable levitation of a denser non-magnetizable liquid droplet, against gravity, inside a relatively lighter ferrofluid – a system barely considered in ferrohydrodynamics – is possible, and exhibits unique interfacial features; the stability of the levitation trajectory, however, is subject to an appropriate magnetic field modulation. We explore the shapes and the temporal dynamics of a plane non-magnetizable droplet levitating inside a ferrofluid against gravity due to a spatially complex, but systematically generated, magnetic field in two dimensions. The coupled set of Maxwell’s magnetostatic equations and the flow dynamic equations is integrated computationally, utilizing a conservative finite-volume-based second-order pressure projection algorithm combined with the front-tracking algorithm for the advection of the interface of the droplet. The dynamics of the droplet is studied under both the constant ferrofluid magnetic permeability assumption as well as for more realistic field-dependent permeability described by Langevin’s nonlinear magnetization model. Due to the non-homogeneous nature of the magnetic field, unique shapes of the droplet during its levitation, and at its steady state, are realized. The complete spatio-temporal response of the droplet is a function of the Laplace number $La$ , the magnetic Laplace number $La_{m}$ and the Galilei number $Ga$ ; through detailed simulations we separate out the individual roles played by these non-dimensional parameters. The effect of the viscosity ratio, the stability of the levitation path and the possibility of existence of multiple stable equilibrium states is investigated. We find, for certain conditions on the viscosity ratio, that there can be developments of cusps and singularities at the droplet surface; we also observe this phenomenon experimentally and compare with the simulations. Our simulations closely replicate the singular projection on the surface of the levitating droplet. Finally, we present a dynamical model for the vertical trajectory of the droplet. This model reveals a condition for the onset of levitation and the relation for the equilibrium levitation height. The linearization of the model around the steady state captures that the nature of the equilibrium point goes under a transition from being a spiral to a node depending upon the control parameters, which essentially means that the temporal route to the equilibrium can be either monotonic or undulating. The analytical model for the droplet trajectory is in close agreement with the detailed simulations.

On the Combined Effect of Mean Flow and Acoustic Excitation on Structural Response and Radiation

1997 ◽  
Vol 119 (3) ◽  
pp. 448-456 ◽  
Author(s):  
A. Frendi ◽  
L. Maestrello
Keyword(s):  
Mach Number ◽  
Response Spectrum ◽  
Structural Response ◽  
Harmonic Excitation ◽  
Two Dimensions ◽  
Physical Problem ◽  
Acoustic Excitation ◽  
Mean Flow ◽  
Power Spectral ◽  
The Mean

Numerical experiments in two dimensions are carried out in order to investigate the response of a typical aircraft structure to a mean flow and an acoustic excitation. Two physical problems are considered; one in which the acoustic excitation is applied on one side of the flexible structure and the mean flow is on the other side while in the second problem both the mean flow and acoustic excitation are on the same side. Subsonic and supersonic mean flows are considered together with a random and harmonic acoustic excitation. In the first physical problem and using a random acoustic excitation, the results show that at low excitation levels the response is unaffected by the mean flow Mach number. However, at high excitation levels the structural response is significantly reduced by increasing the Mach number. In particular, both the shift in the frequency response spectrum and the broadening of the peaks are reduced. In the second physical problem, the results show that the response spectrum is dominated by the lower modes (1 and 3) for the subsonic mean flow case and by the higher modes (5 and 7) in the supersonic case. When a harmonic excitation is used, it is found that in the subsonic case the power spectral density of the structural response shows a subharmonic (f/4) while in the supersonic case no subharmonic is obtained.

scholarly journals Exact solution and the multidimensional Godunov scheme for the acoustic equations

2022 ◽  
Author(s):  
Wasilij Barsukow ◽  
Christian Klingenberg
Keyword(s):  
Exact Solution ◽  
Mach Number ◽  
Euler Equations ◽  
Two Dimensions ◽  
Finite Volume Scheme ◽  
Godunov Scheme ◽  
Low Mach Number ◽  
Low Mach Number Limit ◽  
Acoustic Equations ◽  
Spatial Dimensions

The acoustic equations derived as a linearization of the Euler equations are a valuable system for studies of multi-dimensional solutions. Additionally they possess a low Mach number limit analogous to that of the Euler equations. Aiming at understanding the behaviour of the multi-dimensional Godunov scheme in this limit, first the exact solution of the corresponding Cauchy problem in three spatial dimensions is derived. The appearance of logarithmic singularities in the exact solution of the 4-quadrant Riemann Problem in two dimensions is discussed. The solution formulae are then used to obtain the multidimensional Godunov finite volume scheme in two dimensions. It is shown to be superior to the dimensionally split upwind/Roe scheme concerning its domain of stability and ability to resolve multi-dimensional Riemann problems. It is shown experimentally and theoretically that despite taking into account multi-dimensional information it is, however, not able to resolve the low Mach number limit.

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