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Introduction to the Hydrodynamicmic Computation of Flow around Solid Object


The scientific 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. As it can be easily understood, the main difficulties arise when the predictivity of a model is very important, and when the empirical heuristic experiments cannot solve a particular problem. Unfortunately, this is very frequent when we deal with turbulence and its related phenomena, understanding which in depth can be reasonably ranked as one of the nowadays hardest problems of theoretical physics.

The scientific 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. The state of the gas is determined by the radiative and chemical processes. The rate of fragmentation, in its turn, depends on the state variables, and controls the birth of the "fragment phase" at the expense of the environment gas (see Di Fazio, 1986, hereafter DF86). The fragments undergo internal self-gravitating evolution, with collapse and subfragmentation (except those which become stars), and so on (see DF86). A part from an initial, transient collision activity, their cross sections eventually reduce themselves, to yield a collisionless system of fragments that orbits in the cloud's gravitational field, embedded in the environment gas left over from the fragmentation process (see Capuzzo-Dolcetta, Di Fazio, Menshchikov, 1990). One of the macroscopically important heating mechanisms for the gas is the dissipation of the turbulent supersonic wakes that are generated by the fragment's supersonic motions in the gas (see Battinelli et al. 1992, hereafter B92).

In making a global model of evolution of our protocloud, it is important to take into account another physical phenomenon: the generation of turbulence itself, as one of the phenomena connected to the resistivity force, due to the motions of fragments (see B92 and Di Fazio et al., 1993). In fact, before decaying into brownian motion, the turbulent motions "freeze" some amount of internal energy, and modify the thermodynamic evolution of the gas, which is essential to be able to calculate the absorption and emission functions of the gas. Moreover, the presence of diffused turbulence in the gas alters the dispersion relation for the various instabilities (in first place, the gravitational one), thus influencing the critical masses for fragmentation (e.g. the Jeans mass, see B92).

In order to complete the picture regarding the need to calculate the resistivity force in our global models of evolution, we should account for the explosion of supernovae, which exerts an action which can be, from case to case, of compression, destruction, fragmentation enhancement, or wiping out of the outer parts of the shocked fragments.

In order to build a starting model of the gas flow-cloud interaction, we chose to study the simplified case of a spherical gaseous cloud which is hit at the fly-by of an interstellar shock (Bedogni and Woodward 1990). After the passage of the shock, a more quiescent phase takes place, with the formation of a turbulent wake. In this phase we evaluate and study the resistivity force, as well as other effects, like those due to the mass-loss which the cloud undergoes. The importance and the role of the resistivity force in generating turbulence was first proposed and investigated in B92, using an approximate approach to its velocity dependence in a complex global evolution program. In this work, we set up a numerical tool suitable to calculate the resistivity force quantitatively in different environment conditions.

As discussed by McKee (1988), the resistivity force ("physical drag") plays an important role in the shock-cloud interaction. The value suggested by McKee (1988) for the drag coefficient approx= 2 is too large compared with the experimental results for rigid spheres, and is it also too large for very high Mach numbers. A numerical approach seems the only way to measure the correct drag and to understand its role in compressing and accelerating the interstellar clouds.

In this work, we test our hydrodynamical tool, by reproducing the experiments made on rigid spheres and 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. Indeed, we should not forget that absolutely no laboratory experiment will reproduce the conditions of a supernova blast or the action of a supersonic motion with a Reynolds number of 106/109 and Mach numbers (at the shock front) over 500.

The first classical experiments for the measurement of the resistivity force exerted on a rigid sphere moving in a viscous flow, from laminar to highly turbulent regime (Reynolds number Re -106,107) can be found in the literature of the first three decades of this century (see, e.g. the cumulative graph reported by Sedov 1954; Sedov 1965 on the measurements by Allan, Goettingen, Libster, Schiller-Schmidel). In the data shown by Sedov, one can see the gradual, slow deviation from the so-called Stokes regime (laminar) starting at Re >= 1. Up to that point, the resistivity force is linear with the Reynolds number. The dependence becomes more and more than linear with increasing Re , and eventually becomes approximately quadratic in Re in the interval Re [103,105]. A conspicuous feature shows up after that interval, and lasting for about a dex in Re , in that the resistivity force sharply drops and then rises again. This is what is well known as the so-called "crisis of resistivity" in the subsonic case.

The measurement of drag, nevertheless, was very important also in several industrial and military cases and thus more and more efforts were spent for the understanding of this physical phenomenon. Even the measurements for the simplest objects (rigid spheres, cylinders, cones and wedges) posed severe problems, e.g. how to hold these objects in the flow without interfering heavily and uncontrollably with the measure itself. The interaction of the formed train of waves with the wind tunnel or liquid flow conveyor also interferes with the measured force, not to speak about the train of waves formed by the device holding the object in the flow (e.g. several kinds of dynamometers). Moreover, soon it was evident that the dependence of the resistivity force and of the main parameters of turbulence in the wake of the object on the Reynolds number was also a quite variable function of the Mach number.

The visualization of the flow in the experiments also turned out to be a delicate matter, and the different experimental methods adopted showed different characteristics and parameters of the flow in the wakes of the objects. In this context, for example, consider the particle suspension methods and the electrolytic hydrogen bubble methods in liquids, the electric spark tracing via high voltage electrodes (visualizing fluid particles path lines in their flow), and the photographic "schlieren" and shadowgraph methods, being respectively sensitive to the first and second spatial derivative of the density (Japan Society of Mechanical Engineers 1988).

The main parameters of the flow that determine the resistivity force and other relevant quantities are the Reynolds number and the Mach number. The surface roughness of the chosen objects also influences the drag, and so does the tunnel blockage effect, in the cases where the motion takes place in a flow tunnel.

Among the main experiments in this problematic, we mention the attention given to the high Reynolds number regime, e.g. Flachsbart (1929) Fage and Falkner (1931), and Roshko (1961). These authors studied the dependence of the detachment of the limit layer on Reynolds number, measuring the pressure distribution (around a cylinder) versus the angular distance from the stagnation point at different Reynolds numbers ranging in the interval Re =[105,107]. The above mentioned crisis of resistivity, in fact, was shown to depend critically on where and how much the pressure distribution around the object departs from the one typical of potential flow. In particular, the pressure in the back (with respect to the flow) of the cylinder is shown to vary sensibly in the crisis of resistivity regime.

Wegener and Ashkenas (1961, hereafter WA61) measured the dependence of the resistivity force acting on a sphere at supersonic speeds on Mach number, for different Reynolds numbers. The experimental apparatus consisted in a sphere held by a wire and with a damping mechanism (see WA61, p.553). They took into account the wire ("transducer") resistivity, and they could measure an increasing resistivity force with increasing Mach numbers in a small interval M= [3.87,4.33]. The different Reynolds numbers were obtained using spheres of different diameters.

Resistivity force measurements (for spheres) using various types of transducers (or suspension systems) were compared by Bailey (1974) in the case of subsonic speeds for Reynolds numbers ranging from 400 to 107. Bailey showed that the different experimental methods affected the results from approx= 2 to more that approx= 100 (see Bailey 1974, p.408), and proposed a carefully discussed derived "standard" drag curve which is very similar to that shown by Sedov (1954, p. 49).

Achenbach (1974) set up experiments in order to determine the influence of surface roughness and tunnel blockage on the resistivity force. Achenbach shows that the crisis of resistivity moves to the left (i.e., towards lower Reynolds numbers) while the roughness ratio increases (the latter ratio being the average micro-roughness linear size divided by the diameter of the sphere). Moreover, the minimum of the drag, reached in the crisis of resistivity dip, tends to increase its attained value with increasing roughness ratio (see Achenbach 1974, p. 116). These results are of fundamental importance for our present paper, in that our calculations give for the boundary of the simulated object not a smooth sphere, of course, but a stepwise-rough sphere, due to the computational grid. This, in its turn, will induce some sort of a "numerical surface roughness effect", of which we will have to be aware in the numerical simulations. Achenbach also showed that the resistivity force increases with increasing blockage ratios (tending to equal the so called "ram pressure" force when the blockage ratio tends to 1). Moreover, with higher blockage ratios, the crisis of resistivity takes place at higher Reynolds numbers, the attainment of such crisis is smoother (the slope in the dip is lower), and the minimum of the drag reached in the crisis tends to be higher (see Achenbach, 1974, p. 121).

Miller and Bailey (1979, hereafter MB79) show the drag coefficient (i.e., the resistivity force divided by the ram pressure, and by the geometrical cross section) versus the Reynolds number across the crisis zone, for spheres at various Mach numbers, from 0.3 way up to 1.8 (see MB79, fig. 4 p.456). The drag coefficient is shown to be ever increasing with Mach number, and the crisis of resistivity moves at higher Re with increasing Mach numbers. For Mach numbers greater than 0.8, the presented graph remarkably does not show any critical dip.

Bryson and Gross (1961, hereafter BG61) studied the interaction of strong shocks with several types of objects embedded in supersonic flows. The various trains of waves formed, the contact discontinuities, the shocks and the wakes are studied, and we will use some of BG61's experimental results (see BG61, plate 4) to compare with our numerical simulations.

Several previous numerical works dealing with the calculation of the resistivity force and other quantities related to the flow past spheres and cylinders have been published. Among them we mention Ta Phuoc Loc and Bouard (1985) -and reference therein- that deals with incompressible viscous flow around cylinders. The latter work computes the vorticity in the wake at medium-low Reynolds numbers 3*103- 9*103) and compares with experiments rather well. A three dimensional numerical calculation on incompressible stratified flow past a sphere was performed at low Reynolds number approx= 200) by Hanazaki (1988). In his work, the drag coefficient is calculated, and its behaviour versus the Froude number is compared with the experiments; the pressure distribution on the surface of the sphere is also computed. Fornberg (1988) presents a study of the wake vorticity for steady incompressible flow past a sphere with Reynolds numbers >=5000. No calculations on the main parameters of turbulence or on the drag coefficient are presented. Chang and Chern (1991) perform calculations on incompressible flow past a cylinder at Reynolds numbers ranging from 300 to 106, giving the time evolution for the drag coefficient, the wake length, and the surface pressure distribution. The streamline patterns are compared with the experimental ones for Re =300,550,3000,9500. No comparisons with the experiments are given for the drag vs. Reynolds number behaviour.

The transition from a subsonic to a transonic regime ( i.e. as the Mach number is increased from 0.6 to 0.98) has been recently studied, in a context of inviscid Eulerian hydrodynamics, by Botta (1995). He computed the flow around a circular cylinder, relaxing the symmetry approximation and using a finite volume upwind method. In this paper, the attention is focused on the behaviour of the flow at large times after the breakdown of symmetry and after the onset of an oscillating solution. At Mach numbers between 0.5 and 0.6 the flow turns out to be periodic, while at higher Mach numbers the numerical solution enters a chaotic turbulent regime and reaches, at large times, an almost stationary state.