How to break the density-anisotropy degeneracy in spherical stellar systems

We present a new non-parametric Jeans code, GravSphere, that recovers the density $\rho(r)$ and velocity anisotropy $\beta(r)$ of spherical stellar systems, assuming only that they are in a steady-state. Using a large suite of mock data, we confirm that with only line-of-sight velocity data, GravSphere provides a good estimate of the density at the projected stellar half mass radius, $\rho(R_{1/2})$, but is not able to measure $\rho(r)$ or $\beta(r)$, even with 10,000 tracer stars. We then test three popular methods for breaking this $\rho-\beta$ degeneracy: using multiple populations with different $R_{1/2}$; using higher order `Virial Shape Parameters' (VSPs); and including proper motion data. We find that two populations provide an excellent recovery of $\rho(r)$ in-between their respective $R_{1/2}$. However, even with a total of $\sim 7,000$ tracers, we are not able to well-constrain $\beta(r)$ for either population. By contrast, using 1000 tracers with higher order VSPs we are able to measure $\rho(r)$ over the range $0.5 < r/R_{1/2} < 2$ and broadly constrain $\beta(r)$. Including proper motion data for all stars gives an even better performance, with $\rho$ and $\beta$ well-measured over the range $0.25 < r/R_{1/2} < 4$. Finally, we test GravSphere on a triaxial mock galaxy that has axis ratios typical of a merger remnant, $[1:0.8:0.6]$. In this case, GravSphere can become slightly biased. However, we find that when this occurs the data are poorly fit, allowing us to detect when such departures from spherical symmetry become problematic.Comment: 19 pages; 1 table; 11 Figures. Version accepted for publication in MNRAS. (Minor changes from previously. Appendix B added showing decreasing bias of VSP estimators with increasing sampling.

An efficient approximate factorization implicit scheme for the equations of gasdynamics

An efficient implicit finite-difference algorithm for the gas dynamic equations utilizing matrix reduction techniques is presented. A significant reduction in arithmetic operations is achieved while maintaining the same favorable stability characteristics and generality found in the Beam and Warming approximate factorization algorithm. Steady-state solutions to the conservative Euler equations in generalized coordinates are obtained for transonic flows about a NACA 0012 airfoil. The theoretical extension of the matrix reduction technique to the full Navier-Stokes equations in Cartesian coordinates is presented in detail. Linear stability, using a Fourier stability analysis, is demonstrated and discussed for the one-dimensional Euler equations. It is shown that the method offers advantages over the conventional Beam and Warming scheme and can retrofit existing Beam and Warming codes with minimal effort

The numerical calculation of laminar boundary-layer separation

Iterative finite-difference techniques are developed for integrating the boundary-layer equations, without approximation, through a region of reversed flow. The numerical procedures are used to calculate incompressible laminar separated flows and to investigate the conditions for regular behavior at the point of separation. Regular flows are shown to be characterized by an integrable saddle-type singularity that makes it difficult to obtain numerical solutions which pass continuously into the separated region. The singularity is removed and continuous solutions ensured by specifying the wall shear distribution and computing the pressure gradient as part of the solution. Calculated results are presented for several separated flows and the accuracy of the method is verified. A computer program listing and complete solution case are included

Generation of three-dimensional body-fitted coordinates using hyperbolic partial differential equations

An efficient numerical mesh generation scheme capable of creating orthogonal or nearly orthogonal grids about moderately complex three dimensional configurations is described. The mesh is obtained by marching outward from a user specified grid on the body surface. Using spherical grid topology, grids have been generated about full span rectangular wings and a simplified space shuttle orbiter

Developments in the simulation of compressible inviscid and viscous flow on supercomputers

In anticipation of future supercomputers, finite difference codes are rapidly being extended to simulate three-dimensional compressible flow about complex configurations. Some of these developments are reviewed. The importance of computational flow visualization and diagnostic methods to three-dimensional flow simulation is also briefly discussed

Flux vector splitting of the inviscid equations with application to finite difference methods

The conservation-law form of the inviscid gasdynamic equations has the remarkable property that the nonlinear flux vectors are homogeneous functions of degree one. This property readily permits the splitting of flux vectors into subvectors by similarity transformations so that each subvector has associated with it a specified eigenvalue spectrum. As a consequence of flux vector splitting, new explicit and implicit dissipative finite-difference schemes are developed for first-order hyperbolic systems of equations. Appropriate one-sided spatial differences for each split flux vector are used throughout the computational field even if the flow is locally subsonic. The results of some preliminary numerical computations are included

Numerical simulation of steady supersonic flow

A noniterative, implicit, space-marching, finite-difference algorithm was developed for the steady thin-layer Navier-Stokes equations in conservation-law form. The numerical algorithm is applicable to steady supersonic viscous flow over bodies of arbitrary shape. In addition, the same code can be used to compute supersonic inviscid flow or three-dimensional boundary layers. Computed results from two-dimensional and three-dimensional versions of the numerical algorithm are in good agreement with those obtained from more costly time-marching techniques

A conservative implicit finite difference algorithm for the unsteady transonic full potential equation

An implicit finite difference procedure is developed to solve the unsteady full potential equation in conservation law form. Computational efficiency is maintained by use of approximate factorization techniques. The numerical algorithm is first order in time and second order in space. A circulation model and difference equations are developed for lifting airfoils in unsteady flow; however, thin airfoil body boundary conditions have been used with stretching functions to simplify the development of the numerical algorithm

Shock waves and drag in the numerical calculation of isentropic transonic flow

Properties of the shock relations for steady, irrotational, transonic flow are discussed and compared for the full and approximate governing potential in common use. Results from numerical experiments are presented to show that the use of proper finite difference schemes provide realistic solutions and do not introduce spurious shock waves. Analysis also shows that realistic drags can be computed from shock waves that occur in isentropic flow. In analogy to the Oswatitsch drag equation, which relates the drag to entropy production in shock waves, a formula is derived for isentropic flow that relates drag to the momentum gain through an isentropic shock. A more accurate formula for drag, based on entropy production, is also derived, and examples of wave drag evaluation based on these formulas are given and comparisons are made with experimental results

Implicit approximate-factorization schemes for the low-frequency transonic equation

Two- and three-level implicit finite-difference algorithms for the low-frequency transonic small disturbance-equation are constructed using approximate factorization techniques. The schemes are unconditionally stable for the model linear problem. For nonlinear mixed flows, the schemes maintain stability by the use of conservatively switched difference operators for which stability is maintained only if shock propagation is restricted to be less than one spatial grid point per time step. The shock-capturing properties of the schemes were studied for various shock motions that might be encountered in problems of engineering interest. Computed results for a model airfoil problem that produces a flow field similar to that about a helicopter rotor in forward flight show the development of a shock wave and its subsequent propagation upstream off the front of the airfoil