Hydrodynamic Computation of the Flow around a Cylinder

R. Bedogni

Osservatorio Astronomico di Bologna

A. Di Fazio

Osservatorio Astronomico di Roma

Abstract

Il flusso
idrodinamico, attorno ad un cilindro rigido, è stato calcolato utilizzando un
codice numerico basato sullo schema PPM alle differenze finite. Il calcolo è
stato effettuato nel limite "euleriano" utilizzando gli effetti di dissipazione
insiti nel metodo numerico. Il programma di calcolo fornisce anche il valore
della forza di resistività sul cilindro e quello della energia di turbolenza
nella sua scia. Si tratta di una simulazione nel complesso più che accurata
anche se l'utillizo della "viscosità numerica" è limitato, nella parte di grandi
numeri di Reynold, dall'aumento della risoluzione e quindi dei tempi di
calcolo.

A 2-dim
PPM finite difference scheme is used in a numerical program to evaluate the
hydrodynamic flow around a cylinder. The problem has been investigated in the
Eulerian limit including the numerical dissipation effects of the PPM method.
The program is designed to be able to compute also the resistivity force, the
turbulent energy in the fluid and the main scale of turbulence in the wake. This
is a fairly good simulation, considering also that the Reynolds numbers that can
be simulated using the PPM numerical viscosity are partly limited by the
computing times and the costs induced by the chosen grid
zoning.

Introduction

The problem of evaluating the resistivity force exerted on an
object moving in a fluid in a turbulent regime, and sometimes even in a
supersonic turbulent regime, is very fierce, and it challenges theoretical
physics and engineering already from the second decade of this century. The most
known applied cases of this scientific problem are the predictions of the flight
speeds of airplanes and jets, the cruising speeds of ocean liners and cargo
ships, the maximization of the speeds of racing cars, the design of the foils of
rockets and re-entry vehicles from space, the design of the most efficient sail
plans and hulls of the modern ocean racers and America's cup racers, and many
others.

The problematic that motivated us to build a suitable
instrument to calculate the resistivity force are of astrophysical nature. In
fact, we work in the general line of investigation of the evolution of
self-gravitating objects in their primordial state, such as protogalaxies and
protoglobular clusters, as well as the formation of open clusters, of stars
inside them, and of molecular clouds, but also of the great scales: primordial
clusters of galaxies. In all these cases, although with different (and
non-linearly scaling) physical and chemical conditions, a self-gravitating cloud
undergoes fragmentation via gravitational, thermal, and Rayleigh-Taylor
instabilities, while moving under the effect of its own
gravity.

Results

In this work, we test our hydrodynamical tool, by reproducing
the experiments made on cylinders. Of course, this will not guarantee that the
method is valid for the compressible case, but at least it will be valid in the
limiting cases of the experiments that are available. The main parameters of the
flow that determine the flow around the cylinder and the resistivity force are
the Reynolds number and the Mach number.

The compressible flow around a full circular cylinder has been
investigated, by Ishii at al.1, at Mach number 0.3, using the
Beam-Warming-Steger scheme for the Navier-Stokes equations. They compute the
drag coefficients for four Reynolds numbers in the subcritical, supercritical
and transcritical regimes. In a previous paper Ishii and Kuwahara2,
calculate the transonic flow past a circular cylinder with Mach numbers 0.8 and
0.95. The same problem, but in the context of inviscid Eulerian flow, has been
approached recently also by Botta3 in order to study the transition
from a quasi-periodical to a chaotic, turbulent regime as the Mach number is
increased from 0.6 to 0.98.

Figure 1

The our numerical method solves the Euler equations of inviscid
flow. Therefore, the method's viscosity is actually due to the approximation of
calculating the equations as finite differences performed on a computational
grid. In particular, the behaviour of the numerical viscosity is very similar to
a true viscosity and dissipates the kinetic energy of small-scale fluid motions
into heat. This dissipation is discussed by Porter and Woodward4 in
the contest of turbulent motions produced by thermal convection.

For some well representative case, the effective kinematic
viscosity n eff was found to be
approximately :

n eff
/(csD) = (D/dx)–3 (Ml + 0.25)3.15

where c s is the local sound speed, M
l is the local Mach number of the flow, dx the resolution of
the computational mesh and D the scale length –i.e. the diameter
D = 2·R – of the cylinder.

The Reynolds number Re is defined as :

Re = (u·D) /(n eff)

where n eff is the
kinematic viscosity, while in our case it is the numerical viscosity.

Figure 2

We actually compute the flow around the cylinder using
different Mach numbers and with different resolutions and Reynold's number. We
stop the computation after a time tfin = ft , where t
= 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 = 40,so that the final time is reached, for all
computations, after approximately 66 crossing times.

Here we present, using the same inviscid approximation by Botta
but with a different numerical technique and numerical grid around the full
cylinder, the results of the numerical calculations with "local" Mach numbers
Ml = 0.288, 1.55 while the incident shock has Mach
numbers M = 1.2 and M = 4.

The results of the subsonic case M = 1.2, with a grid
resolution dx = 0.0185 and corresponding "numerical" Reynold's number
Re = 2.7·105, have been displayed in the Figure 1.
The four snapshots describe the flow represented by the density that reaches
relative minima in the center of the vortexes. The main conspicuous feature is
the progressive formation of a series of Karman vortexes and a large number of
secondary eddies. The "full" computation around the cylinder breaks the symmetry
on the flow after 10 crossing times.

For the supersonic case M =  4 see Figure 2 the
grid resolution is dx = 0.02 and the corresponding "numerical"
Reynold's number is Re = 3.3·104, Here we can see
how the bow and tail shocks are farther away from the cylinder and the wake
oscillates slightly around the symmetry axis of the solid object. The higher
Mach number of the incident shock preserve the symmetry of the flow even for
time t = 15.

Acknowlegment

This work was supported by a Grant n. 92-3-102-6 of the CINECA
center in Casalecchio (Bologna). The same calculation, but for the hydrodynamic
flux around a sphere instead on a cylinder, are in [1]

References

1Ishii,K., Kuwahara K., Ogawa, S., Chyu, W.J. and Kawamura, T.:1985
Computation of flow around a circular cylinder in a supercritical regime AIAA
paper 85-1660, AIAA 18th Fluid Dynamics, Plasma Dynamics an Lasers
Conference.

2Ishii,K., and Kuwahara K.: 1984 Computation of compressible flow
around a circular cylinder AIAA paper 84-1631, AIAA 17th Fluid Dynamics,
Plasma Dynamics an Lasers Conference.

3Botta, N.: 1995 The inviscid transonic flow about a cylinder J.
Fluid Mech. 301 225-250.

4Porter, D.H., Woodward, P.R.: 1994, Astrophys. J. Suppl.
Series, vol. 93,309

Publications

[1] R. Bedogni, A. Di Fazio
A numerical method for the calculation of
the resistivity coefficient in the flow around a sphere
submitted to
"Nuovo Cimento"