scholarly journals Analytic growth rate of gravitational instability in self-gravitating planar polytropes

2018 ◽  
Vol 859 ◽  
pp. 362-399 ◽  
Author(s):  
Jean-Baptiste Durrive ◽  
Mathieu Langer
Keyword(s):  
Growth Rate ◽  
Numerical Solutions ◽  
Gravitational Instability ◽  
External Pressure ◽  
Second Order ◽  
Stratified Medium ◽  
Valuable Insight ◽  
Rigid Boundary ◽  
Fourth Order Eigenvalue Problem ◽  
Polytropic Exponent

Gravitational instability is a key process that may lead to fragmentation of gaseous structures (sheets, filaments, haloes) in astrophysics and cosmology. We introduce here a method to derive analytic expressions for the growth rate of gravitational instability in a plane stratified medium. First, the main strength of our approach is to reduce this intrinsically fourth-order eigenvalue problem to a sequence of second-order problems. Second, an interesting by-product is that the unstable part of the spectrum is computed by making use of its stable part. Third, as an example, we consider a pressure-confined, static, self-gravitating slab of a fluid with an arbitrary polytropic exponent, with either free or rigid boundary conditions. The method can naturally be generalised to analyse the stability of richer, more complex systems. Finally, our analytical results are in excellent agreement with numerical solutions. Their second-order expansions provide a valuable insight into how the rate and wavenumber of maximal instability behave as functions of the polytropic exponent and the external pressure (or, equivalently, the column density of the slab).

Saturation mechanism and generated viscosity in gravito-turbulent accretion disks

2020 ◽  
Vol 640 ◽  
pp. A53
Author(s):  
L. Löhnert ◽  
S. Krätschmer ◽  
A. G. Peeters
Keyword(s):  
Growth Rate ◽  
Numerical Simulations ◽  
Accretion Disks ◽  
Gravitational Instability ◽  
Turbulent Viscosity ◽  
Radial Distance ◽  
Maximum Growth ◽  
Wave Number ◽  
Radiative Cooling ◽  
Mixing Length

Here, we address the turbulent dynamics of the gravitational instability in accretion disks, retaining both radiative cooling and irradiation. Due to radiative cooling, the disk is unstable for all values of the Toomre parameter, and an accurate estimate of the maximum growth rate is derived analytically. A detailed study of the turbulent spectra shows a rapid decay with an azimuthal wave number stronger than ky−3, whereas the spectrum is more broad in the radial direction and shows a scaling in the range kx−3 to kx−2. The radial component of the radial velocity profile consists of a superposition of shocks of different heights, and is similar to that found in Burgers’ turbulence. Assuming saturation occurs through nonlinear wave steepening leading to shock formation, we developed a mixing-length model in which the typical length scale is related to the average radial distance between shocks. Furthermore, since the numerical simulations show that linear drive is necessary in order to sustain turbulence, we used the growth rate of the most unstable mode to estimate the typical timescale. The mixing-length model that was obtained agrees well with numerical simulations. The model gives an analytic expression for the turbulent viscosity as a function of the Toomre parameter and cooling time. It predicts that relevant values of α = 10−3 can be obtained in disks that have a Toomre parameter as high as Q ≈ 10.

scholarly journals Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Darae Jeong ◽  
Yibao Li ◽  
Chaeyoung Lee ◽  
Junxiang Yang ◽  
Yongho Choi ◽  
...  
Keyword(s):  
Parabolic Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Second Order ◽  
Order Convergence ◽  
Numerical Convergence ◽  
Verification Method ◽  
First Order ◽  
Convergence Test ◽  
Second Order Convergence

In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

Mass transport under partially reflected waves in a rectangular channel

1994 ◽  
Vol 266 ◽  
pp. 121-145 ◽  
Author(s):  
Jiangang Wen ◽  
Philip L.-F. Liu
Keyword(s):  
Mass Transport ◽  
Vector Potential ◽  
Numerical Solutions ◽  
Rectangular Channel ◽  
Scalar Potential ◽  
Second Order ◽  
Nonlinear Transport ◽  
Reflected Waves ◽  
Transport Velocity ◽  
Mass Transport Velocity

Mass transport under partially reflected waves in a rectangular channel is studied. The effects of sidewalls on the mass transport velocity pattern are the focus of this paper. The mass transport velocity is governed by a nonlinear transport equation for the second-order mean vorticity and the continuity equation of the Eulerian mean velocity. The wave slope, ka, and the Stokes boundary-layer thickness, k (ν/σ)½, are assumed to be of the same order of magnitude. Therefore convection and diffusion are equally important. For the three-dimensional problem, the generation of second-order vorticity due to stretching and rotation of a vorticity line is also included. With appropriate boundary conditions derived from the Stokes boundary layers adjacent to the free surface, the sidewalls and the bottom, the boundary value problem is solved by a vorticity-vector potential formulation; the mass transport is, in gneral, represented by the sum of the gradient of a scalar potential and the curl of a vector potential. In the present case, however, the scalar potential is trivial and is set equal to zero. Because the physical problem is periodic in the streamwise direction (the direction of wave propagation), a Fourier spectral method is used to solve for the vorticity, the scalar potential and the vector potential. Numerical solutions are obtained for different reflection coefficients, wave slopes, and channel cross-sectional geometry.

Hybrid Nodal Integral / Finite Element Method for Time-Dependent Convection Diffusion Equation

2021 ◽  
Author(s):  
Sundar Namala ◽  
Rizwan Uddin
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Numerical Solutions ◽  
Second Order ◽  
Time Dependent ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Integral Methods ◽  
Element Method

Abstract Nodal integral methods (NIM) are a class of efficient coarse mesh methods that use transverse averaging to reduce the governing partial differential equation(s) (PDE) into a set of ordinary differential equations (ODE). The standard application of NIM is restricted to domains that have boundaries parallel to one of the coordinate axes/palnes (in 2D/3D). The hybrid nodal-integral/finite-element method (NI-FEM) reported here has been developed to extend the application of NIM to arbitrary domains. NI-FEM is based on the idea that the interior region and the regions with boundaries parallel to the coordinate axes (2D) or coordinate planes (3D) can be solved using NIM, and the rest of the domain can be discretized and solved using FEM. The crux of the hybrid NI-FEM is in developing interfacial conditions at the common interfaces between the NIM regions and FEM regions. We here report the development of hybrid NI-FEM for the time-dependent convection-diffusion equation (CDE) in arbitrary domains. Resulting hybrid numerical scheme is implemented in a parallel framework in Fortran and solved using PETSc. The preliminary approach to domain decomposition is also discussed. Numerical solutions are compared with exact solutions, and the scheme is shown to be second order accurate in both space and time. The order of approximations used for the development of the scheme are also shown to be second order. The hybrid method is more efficient compared to standalone conventional numerical schemes like FEM.

Lobar Buckling of Thin-Walled Cylindrical and Truncated Conical Shells Under External Pressure

1974 ◽  
Vol 18 (04) ◽  
pp. 272-277
Author(s):  
C. T. F. Ross
Keyword(s):  
Finite Element ◽  
Circular Cylinder ◽  
Numerical Solutions ◽  
External Pressure ◽  
Element Solution ◽  
Conical Shells ◽  
Varying Thickness ◽  
Complex Boundary ◽  
Thin Walled ◽  
Unsymmetrical Loading

Numerical solutions have been produced for the asymmetric instability of thin-walled circular cylindrical and truncated conical shells under external pressure. The solutions for the circular cylinder have shown that the assumed buckling configurations of Nash [l]2 and Kaminsky [2] were quite reasonable for fixed ends. Comparison was also made of the finite-element solution of conical shells with other analyses. From these calculations, it was shown that the numerical solutions were superior to the analytical ones, as the former could be readily applied to vessels of varying thickness or those subjected to unsymmetrical loading or with complex boundary conditions.

scholarly journals Galactic Dynamos and Density Wave Theory

1990 ◽  
Vol 140 ◽  
pp. 135-135
Author(s):  
L Mestel ◽  
K Subramanian
Keyword(s):  
Growth Rate ◽  
Angular Velocity ◽  
Numerical Solutions ◽  
Density Wave ◽  
Wave Theory ◽  
Interstellar Gas ◽  
Global Magnetic Field ◽  
Dynamo Equation ◽  
Density Wave Theory ◽  
Image Position

A steady density wave in the stellar background of a disk–like galaxy is supposed to force a spiral shock wave in the interstellar gas. The jump in vorticity across the shock leads to a locally enhanced helicity, and so to an α–effect which is steady but azimuth–dependent in the frame rotating with the angular velocity ω of the density wave. This is simulated by the adoption of the form for the local dynamo growth rate arising when the standard kinematic dynamo equation is treated by the thin–disk approximation (Ruzmaikin et al 1988). The global magnetic field is proportional to the function Q satisfying where η is the turbulent resistivity (for simplicity assumed uniform) and is the laminar angular velocity of the gas in the inertial frame. We look for solutions of the form where is a global eigen-value, and the non-vanishing of couples all odd or all the even m-values. Anticipating that the strong differential rotation will ensure that in the modes with the largest growth-rate the higher-m parts are weak, the equations are truncated, leaving just a pair in q1, q-1, to describe a basically bisymmetric (m = 1) mode. Approximate treatment by the WKBJ technique suggests that a corotating growing mode (with Γ real and positive) will differ significantly from zero over the range between the points where Numerical solutions have been found for a set of illustrative parameters with corotation occurring at 6.67 kpc, and the turbulence parameters close to those in the M51 mode studied by Ruzmaikin et al which extends over = 1 kpc. Three growing corotating modes were found, the fastest extending for ~ 3 kpc, the other two for over 4 kpc. The first two grow 2-3 times faster, the third somewhat slower, than the M51 mode.

Improved Approach to the Streamline Curvature Method in Turbomachinery

1987 ◽  
Vol 109 (3) ◽  
pp. 213-217 ◽  
Author(s):  
S. Abdallah ◽  
R. E. Henderson
Keyword(s):  
Flow Field ◽  
Velocity Gradient ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Second Order ◽  
Curvilinear Coordinates ◽  
First Order ◽  
Streamline Curvature ◽  
Stators And Rotors ◽  
Streamline Curvature Method

Quasi three dimensional blade-to-blade solutions for stators and rotors of turbomachines are obtained using the Streamline Curvature Method (SLCM). The first-order velocity gradient equation of the SLCM, traditionally solved for the velocity field, is reformulated as a second-order elliptic differential equation and employed in tracing the streamtubes throughout the flow field. The equation of continuity is then used to calculate the velocity. The present method has the following advantages. First, it preserves the ellipticity of the flow field in the solution of the second-order velocity gradient equation. Second, it eliminates the need for curve fitting and strong smoothing under-relaxation in the classical SLCM. Third, the prediction of the stagnation streamlines is a straightforward matter which does not complicate the present procedure. Finally, body-fitted curvilinear coordinates (streamlines and orthogonals or quasi-orthogonals) are naturally generated in the method. Numerical solutions are obtained for inviscid incompressible flow in rotating and non-rotating passages and the results are compared with experimental data.

scholarly journals A Partial Lagrangian Approach to Mathematical Models of Epidemiology

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
R. Naz ◽  
I. Naeem ◽  
F. M. Mahomed
Keyword(s):  
Mathematical Models ◽  
Exact Solutions ◽  
Numerical Solutions ◽  
First Integrals ◽  
Order Equation ◽  
Second Order ◽  
Lagrangian Approach ◽  
Hiv Model ◽  
First Order ◽  
Partial Lagrangian

This paper analyzes the first integrals and exact solutions of mathematical models of epidemiology via the partial Lagrangian approach by replacing the three first-order nonlinear ordinary differential equations by an equivalent system containing one second-order equation and a first-order equation. The partial Lagrangian approach is then utilized for the second-order ODE to construct the first integrals of the underlying system. We investigate the SIR and HIV models. We obtain two first integrals for the SIR model with and without demographic growth. For the HIV model without demography, five first integrals are established and two first integrals are deduced for the HIV model with demography. Then we utilize the derived first integrals to construct exact solutions to the models under investigation. The dynamic properties of these models are studied too. Numerical solutions are derived for SIR models by finite difference method and are compared with exact solutions.

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