An approach to the Riemann problem in the light of a reformulation of the state equation for SPH inviscid ideal flows: a highlight on spiral hydrodynamics in accretion discs
In physically inviscid fluid dynamics, "shock capturing" methods adopt either an artificial viscosity contribution or an appropriate Riemann solver algorithm. These techniques are necessary to solve the strictly hyperbolic Euler equations if flow discontinuities (the Riemann problem) are to be solved. A necessary dissipation is normally used in such cases. An explicit artificial viscosity contribution is normally adopted to smooth out spurious heating and to treat transport phenomena. Such a treatment of inviscid flows is also widely adopted in the Smooth Particle Hydrodynamics (SPH) finite volume free Lagrangian scheme. In other cases, the intrinsic dissipation of Godunov-type methods is implicitly useful. Instead "shock tracking" methods normally use the Rankine-Hugoniot jump conditions to solve such problems. A simple, effective solution of the Riemann problem in inviscid ideal gases is here proposed, based on an empirical reformulation of the equation of state (EoS) in the Euler equations in fluid dynamics, whose limit for a motionless gas coincides with the classical EoS of ideal gases. The application of such an effective solution to the Riemann problem excludes any dependence, in the transport phenomena, on particle smoothing resolution length $h$ in non viscous SPH flows. Results on 1D shock tube tests, as well as examples of application for 2D turbulence and 2D shear flows are here shown. As an astrophysical application, a much better identification of spiral structures in accretion discs in a close binary (CB), as a result of this reformulation is also shown here.
A conservative primitive variable discrete exterior calculus (DEC) discretization of the Navier-Stokes equations is performed. An existing DEC method (Mohamed, M. S., Hirani, A. N., Samtaney, R. (2016). Discrete exterior calculus discretization of incompressible Navier-Stokes equations over surface simplicial meshes. Journal of Computational Physics, 312, 175-191) is modified to this end, and is extended to include the energy-preserving time integration and the Coriolis force to enhance its applicability to investigate the late time behavior of flows on rotating surfaces, i.e., that of the pla…
研究了非旋转黑洞周围一维准球形跨音速吸积流中由于角动量转移而引起的平流吸积流的不稳定性。该代码的设计包括粘性的影响，流体动力学组件保持严格的拉格朗日和重映射方法在没有粘性的角动量，而粘性组件通过隐式方法更新粘性角动量传递。我们进行了两个测试，以证明适合的增长研究的代码。首先，我们模拟了黑洞周围的无粘、低角动量、带激波的跨音速吸积流，然后模拟了牛顿体周围的亚音速、自相似的ADAF解。两种模拟都非常好地拟合了相应的分析曲线。然后，我们模拟了旋转，粘性，跨音速流体与冲击。我们表明，对于低粘度参数，稳定的冲击在更大的距离是可能的。当粘性系数较大时，激波后盘中角动量的传递效率较高，激波结构振荡。此外，当激波漂移到更大的距离时，会产生二次内激波。我们表明，内激波是外激波膨胀的直接结果，也是由于内音速点附近更有效的角动量转移而产生的具有$\partial l / \partial r < 0$的区域。我们发现，所有的磁盘参数，包括发射率，振荡与相同的周期的冲击振荡。我们的模拟可能对低频QPO有影响，例如，GRO J1655-40和XTE J1550-564。
We use spherical coordinates to devise a new exact solution to the governing equations of geophysical fluid dynamics for an inviscid and incompressible fluid with a general density distribution and subjected to forcing terms. The latter are of paramount importance for the modeling of realistic flows-that is, flows that are observed in some averaged sense in the ocean. Owing to the employment of spherical coordinates we do not need to resort to approximations (e.g. of $f$- and $β$-plane type) that simplify the geometry in the governing equations. Our explicit solution represents a steady purely…
In this paper, Beltrami vector fields in several orthogonal coordinate systems are obtained analytically and numerically. Specifically, axisymmetric incompressible inviscid steady state Beltrami (Trkalian) fluid flows are obtained with the motivation to model flows that have been hypothesized to occur in tornadic flows. The studied coordinate systems include those that appear amenable to modeling such flows: the cylindrical, spherical, paraboloidal, and prolate and oblate spheroidal systems. The usual Euler equations are reformulated using the Bragg--Hawthorne equation for the stream function …
We derive the general solution of the unsteady Stokes equations for an unbounded fluid in spherical polar coordinates, in both time and frequency domains. The solution is an expansion in vector spherical harmonics and given as a sum of a particular solution, proportional to pressure gradient exhibiting power-law spatial dependence, and a solution of vector Helmholtz equation decaying exponentially in far field, the decomposition originally introduced by Lamb. The solution can be applied to construct the transient exterior flow induced by an arbitrary velocity distribution at the spherical boun…
