HYDRO PAGE


Hydro Equations and Boundary Conditions


Initial Conditions


Cartesian (x,y) or cylindrical (z,r) coordinates, depending on the geometry of the rigid body, are introduced to treat the case of a planar shock interacting with a spherical or cylindrical obstacle centered in x (or z)=2 and y (or r)=0 and a radius R normalized to 1. The rigid body is located in front of the incident shock and its radius R is a free parameter that selects the length scale of every single model. Our units are such that the density and the sound velocity in the ambient medium are set equal to 1 so that pressure becomes p=1/gamma. The mass and time scales can be easily deduced from the physical quantities previously defined.
The incident planar shock (e.g. in cartesian coordinates), is perpendicular to the x axis and is moving from (x=0, y=0) (at the initial time tini=0) towards the positive x direction. For all cases, we actually compute using different Mach numbers. We stop the computation after a time tfin=f\tau, where tau=2*R/us is the crossing time of the rigid body by the incident shock, and us is the shock velocity. We chose, for the factor f, f=20, so that the final time is reached for all computations after approximately 18 crossing times.
The Mach number of the incident shock is defined as M=us/ca, where us is the shock velocity. The shocked quantities are fixed by the Rankine-Hugoniot conditions: and the y-component of the shock velocity is equal to zero.

Numerical Grid


The calculation around the rigid body is made using a uniform grid. this approach has several advantages: a) the mesh generation is straightforward and easily computed , b) the grid refinement is easy to do and c) the grid effects arising from non-uniformities in the mesh are eliminated.
The numerical grid is set up as a rectangular domain with a uniform "active" part, surrounded by an "expanded" one, in which the mesh points are not equidistant, but designed to permit the free development of the bow shock through the upper side of the active grid. The active part of the grid extends from x=0 to x=9 and from y=0 to y=3, and selects the region of the numerical computation with uniform mesh of size Dy=Dx, depending on the resolution of the problem. Non uniform expanding zones, from -10 to 0 in x and from 3 to 20 in y, define the expanded region.

Boundary Conditions


The primary difficulty with this kind of grid is, of course, represented by the boundary conditions. Flow-in (x=-10) boundary conditions are fixed on the left side of the x-y grid,in order to make sure that the shocked values are incoming. Flow-out (x=9) and (y=20) boundary conditions are imposed on the right and top sides of the x-y grid, in order to allow the outflow of the fluid quantities. An outflow condition is accomplished by maintaining a "zero gradient" in all fluid variables . In the next sections, we will take care only of the numerical results in the active (uniform) part of the grid.
We set reflecting boundary conditions at the bottom (y=0) of the x-y grid, enforcing an axial symmetry with respect to the plane containing the x axis and perpendicular to the y axis. We should note here, that the fact of adopting an axial symmetry, while loosing -of course- some spatial detail and forcing some geometrical "unnatural" behaviour in the numerically simulated turbulence, has the advantage of using a remarkably smaller amount of computing time. Moreover, as we will show later, the actual real flows are almost axisymmetric, except along the axis of the motion, and the latter fact does not affect the flow macroscopically.
In this regard, we note that the choice of axial symmetry has been used also in numerical experiments for viscous flows, using a deterministic vortex method (e.g. Chang and Chern 1991), in order to reproduce the experimental flow visualization performed by Ta Phouc Loc and Bouard (1985). However, the time evolution of viscous flow around a rigid body (see also Bouard and Coutanceau (1980), BG61 and Van Dyke (1988)) shows a strong symmetry around the main axis. Therefore, the reflecting condition does not seem to limit the development of the vortex structures behind the object even for Mach number >= 1.

Boundary Wall on the Surface of the Object


We paid particular attention to the calculations on the boundary wall between the object and the gas (see figure 1), represented as an impenetrable series of "stair steps". As the resolution increases, the uniform grid approaches very well the "natural" boundary of the body. For the cases of low resolution, a new treatment has been used in order to determine better approximated fluxes at the zone interfaces. In order to do this, Like in Edgar and Woodward (1992) we allow "fractional" steps, producing a smoother boundary for a given computational mesh. This correction technique yields a sensible improvement in the comparison with the experimental data, especially for the case of the sphere.