This book describes the fundamentals and potentials of modelling and simulation of complex engineering processes, based on, as an example, simulation of the spray forming process of metals. The spray forming process, in this context, is a typical example of a complex technical spray process. Spray forming, basically, is a metallurgical process whereby near-net shaped preforms with outstanding material properties may be produced direct from a metal melt via atomization and consolidation of droplets. For proper analysis of this process, first successive physical submodels are derived and are then implemented into an integrated coupled process model. The theoretical effects predicted by each submodel are then discussed and are compared to experimental findings, where available, and are summarized under the heading ‘spray simulation’. The book should give engineering students and practising engineers in industry and universities a detailed introduction to this rapidly growing area of research and development.

In order to develop an integral model for such technically complex processes as the spray forming of metals, it is essential that the model is broken down into a number of smaller steps. For spray forming, the key subprocesses are:

• atomization of the metal melt,

• dispersed multiphase flow in the spray,

• compaction of the spray and formation of the deposit.

These subprocesses may be further divided until a sequential (or parallel) series of unit operational tasks is derived. For these tasks, individual balances of momentum, heat and mass are to be performed to derive a fundamental model for each.

Introduction

Problems of the motion and deformation of solids are rendered amenable to mathematical analysis by introducing the concept of a continuous medium. In this idealization it is assumed that properties averaged over a very small element, for example the mean mass density and the mean displacement and stress, are continuous functions of position and time. Although it might seem that the microscopic structure of real materials is not consistent with the concept of a continuum, the idealization produces very useful results, simply because the lengths characterizing the microscopic structure of most materials are generally much smaller than any lengths arising in the deformation of the medium. Even if in certain special cases the microstructure gives rise to significant phenomena, these can be taken into account within the framework of the continuum theory by appropriate generalizations.

Continuum mechanics is a classical subject that has been discussed in great generality in numerous treatises. The theory of continuous media is built upon the basic concepts of stress, motion and deformation, upon the laws of conservation of mass, linear momentum, moment of momentum (angular momentum) and energy and on the constitutive relations. The constitutive relations characterize the mechanical and thermal response of a material while the basic conservation laws abstract the common features of mechanical phenomena irrespective of the constitutive relations.

The governing equations used in this book are for homogeneous, isotropic, linearly elastic solids.

Reciprocity relations are among the most interesting and intriguing relations in classical physics. At first acquaintance these relations promise to be a goldmine of useful information. It takes some ingenuity, however, to unearth the nuggets that are not immediately obvious from the formulation. In the theory of elasticity of solid materials the relevant reciprocity theorem emanated from the work of Maxwell, Helmholtz, Lamb, Betti and Rayleigh, towards the end of the nineteenth century, and several applications have appeared in the technical literature since that time. This writer has always believed, however, that more information than is generally assumed can be wrested from reciprocity considerations. I have wondered in particular whether reciprocity considerations could be used to actually determine by analytical means the elastodynamic fields for the high-rate loading of structural configurations. I have explored this question for a number of problems and obtained the actual fields generated by loading from a reciprocity relation in conjunction with an auxiliary solution, a free wave called the “virtual” wave. These recent results comprise an important part of the book.

To my knowledge, the topic of reciprocity in elastodynamics has not been discussed in a comprehensive manner in the technical literature. It is hoped that this book will fill that void. Various forms of the reciprocity theorem are presented, with an emphasis on those for time-harmonic fields, together with numerous applications, general and specific, old and new.

Introduction

The first reciprocity relation specifically for acoustics was stated by von Helmholtz (1859). This relation caught the attention of and was elaborated by Rayleigh (1873) and Lamb (1888). Rayleigh (1873) briefly discussed the reciprocal theorem for acoustics in his paper “Some general theorems relating to vibrations.” In The Theory of Sound (1878, Dover reprint 1945, Vol. II, pp. 145–8), Rayleigh paraphrased this theorem as follows: “If in a space filled with air which is partly bounded by finitely extended fixed bodies and is partly unbounded, sound waves may be excited at any point A, the resulting velocity potential at a second point B is the same both in magnitude and phase, as it would have been at A, had B been the source of sound.” In this statement it is implicitly assumed that sources of the same strength would be applied at both places. In Rayleigh's book (1878) the statement is accompanied by a simple proof. A similar statement of the Helmholtz reciprocity theorem for acoustics can be found in the paper by Lamb (1888). Both Rayleigh and Lamb generalized the theorem to more complicated configurations, and in time the reciprocity theorem became known as Rayleigh's reciprocity theorem.

Most books on acoustics devote attention to the reciprocity theorem; see for example Pierce (1981), Morse and Ingard (1968), Jones (1986), Dowling and Ffowcs Williams (1983) and Crighton et al. (1992). A book by Fokkema and van den Berg (1993) is exclusively concerned with acoustic reciprocity.

Laminar jets can splash!

It has been observed that a liquid jet impinging on a solid surface can produce splashing. High-speed photography has revealed that, with a turbulent jet, splashing is related to the jet surface roughness. To investigate the importance of the jet shape on splashing, perturbations of known frequency or amplitude are imposed on the surface of a smooth laminar jet.

The top picture shows the unperturbed smooth jet as it spreads radially on the solid surface. The varicose deformations imposed on the jet surface alter the flow quite dramatically (center picture). As we further increase the amplitude of the oscillations, splashing starts suddenly. The bottom picture shows the beauty and complexity of splashing.

Impacting water drops

The four photographs shown here are representative of a series which recorded the structure and evolution of the vorticity generated by a water drop impacting a free surface of water in a container. The 2.8 mm diam water drop was dyed with fluorescein and released from the tip of a hypodermic needle under specific parameters, We=26, Fr=25. The Weber number (ρU2d /γ) and Froude number (U2/gd) are based on drop diameter d, impact velocity U, and surface tension γ.

Figure 1 is photographed from the side and slightly below the free surface while Fig. 2 is shot looking directly up at the free surface via a mirror. These alternative viewing angles provide a valuable tool in visualizing the threedimensional flow structure. A “primary” vortex ring can be seen convecting away from the free surface. A convoluted secondary structure can be seen wrapped around the primary core.

A new mechanism for oblique wave resonance

Despite the large body of research concerned with the near wake of a circular cylinder, the far wake, which extends beyond about 100 diameters downstream, is relatively unexplored, especially at low Reynolds numbers. We have recently shown that the structure of the far wake is exquisitely sensitive to free-stream noise, and is precisely dependent on the frequency and scale of the near wake; indeed it is shown that the presence of extremely low-amplitude peaks in the free-stream spectrum, over a remarkably wide range of frequencies, are sufficient to trigger an “oblique wave resonance” in the far wake.

We show, in the upper photograph of Fig. 1, a nonlinear interaction between oblique shedding waves generated from upstream (to the left) and 2–D waves amplified downstream from free-stream disturbances (in the central region). We use the “smoke-wire” technique (placed 50 diameters down-stream), and the wake is viewed in planview, with flow to the right. This two-wave interaction triggers a third wave, namely an “oblique resonance wave” at a large oblique angle, to grow through nonlinear effects (in the right half of the photograph), in preference to the original two waves. If smoke is introduced 100 diameters downstream, in the lower photograph (under slightly different conditions), then all that is seen is a set of such large-angle oblique resonance waves.

This work is supported by the Office of Naval Research.

Visualization of different transition mechanisms

The sequence of photos in Figs. 1(a)-1(d) illustrates the different types of boundary-layer transitions that occur as a function of Tollmien-Schlichting (T-S) wave amplitude and fetch.

We examine the form of the free surface flows resulting from the collision of equal jets at an oblique angle. Glycerol-water solutions with viscosities of 15–50 cS were pumped at flow rates of 10–40 cc/s through circular outlets with diameter 2 mm. Characteristic flow speeds are 1–3 m/s. Figures 2–4 were obtained through strobe illumination at frequencies in the range 2.5–10 kHz.

At low flow rates, the resulting stream takes the form of a steady fluid chain, a succession of mutually orthogonal fluid links, each comprised of a thin oval sheet bound by relatively thick fluid rims (Fig. 1). The influence of viscosity serves to decrease the size of successive links, and the chain ultimately coalesces into a cylindrical stream.

As the flow rate is increased, waves are excited on the sheet, and the fluid rims become unstable (Figs. 2 and 3). Droplets form from the sheet rims but remain attached to the fluid sheet by tendrils of fluid that thin and eventually break. The resulting flow takes the form of fluid fishbones, with the fluid sheet being the fish head and the tendrils its bones. Increasing the flow rate serves to broaden the fishbones.

In the wake of the fluid fish, a regular array of drops obtains, the number and spacing of which is determined by the pinch–off of the fishbones (Fig. 4). At the highest flow rates examined, the flow is reminiscent of that arising in acoustically excited fan-spray nozzles.

Interaction of 2D wake and jet plume

Burner setup. A propane-air turbulent premixed flame is stabilized on a 30 mm Bunsen-type burner by an annular pilot. The equivalence ratio is 0.68. The flame height is about 85 mm. The mean velocity of the unburned mixture is 2.36 m/sec. Turbulence is given to the mixture by a perforated plate. The turbulence rms fluctuations at the burner exit is 0.15 m/sec. The Taylor and the Kolmogorov microscales are 1.81 and 0.22 mm, respectively, and the Reynolds number based on the Taylor microscale is 17.4.

Photographic setup. This schlieren photograph was taken by a Canon-F1 camera with a 300 mm telephoto lens of f=5.6. Two 200 mm schlieren mirrors with the focal length of 2000 mm were used for the Z-light path arrangement. The vertical knife edge is mounted at the focal point of one mirror. The light source was a xenon stroboscope with a condenser lens and a pinhole. The maximum light power is 8 J (170 1x-sec). The flash duration time is typically 15 μsec. The flash timing was synchronized with the camera shutter. The film used was the Neopan SS (ASA 100) and was developed by Fujidol.

Interpretation. With a moderate or weak turbulence of the unburned mixture, the instantaneous turbulent premixed flame zone consists of a continuous wrinkled laminar flame front. The wrinkle size seems to be irrelevant to the turbulence scale. Along the unburned mixture flow, the amplitude of wrinkles increases from bottom to top. We think that the hydrodynamic instability plays an important role in the flame wrinkling.

Interface motion in a vibrated granular layer

Granular materials are now recognized as a distinct state of matter, and studies of their behavior form a fascinating interdisciplinary branch of science. The intrinsic dissipative nature of the interactions between the constituent macroscopic particles gives rise to several basic properties specific to granular substances, setting granular matter apart from the conventional gaseous, liquid, or solid states.

Thin layers of granular materials subjected to vertical vibration exhibit a diversity of patterns. The particular pattern is determined by the interplay between driving frequency f and the acceleration amplitude Γ. Interfaces in vibrated granular layers, existing for large enough amplitude of vibration, separate large domains of flat layers oscillating with opposite phase. These two phases are related to the period-doubling character of the flat layer motion at large plate acceleration. Interfaces are either smooth or “decorated” by periodic undulations depending on parameters of vibration. An additional subharmonic driving results in a controlled displacement of the interface with respect to the center of the experimental cell. The speed and the direction of the interface motion are sensitive to the phase and amplitude of the subharmonic driving.

The image sequence above shows interface nucleation and propagation towards the center of the cell, with dimensionless time tf labeled in each image. The interface forms at the right side wall of the cell due to small-amplitude phase-shifted subharmonic driving. After the additional driving stops, the interface moves towards the center, creating small-scale localized structures in the process.

Laser-induced fluorescence (LIF) diagnostics and highspeed, real-time digital image acquisition techniques are combined to map the composition field in a water mixing layer. A fluorescent dye, which is premixed with the lowspeed freestream fluid and dilutes by mixing with the highspeed fluid, is used to monitor the relative concentration of high-speed to low-speed fluid in the layer.

The three digital LIF pictures shown here were obtained by imaging the laser-induced fluorescence originating from a collimated argon ion laser beam, extending across the transverse dimension of the shear layer, onto a 512–element linear photodiode array. Each picture represents 384 contiguous scans, each at 400 points across the layer, for a total of 153 600 point measurements of concentration. The vertical axis maps onto 40 mm of the transverse coordinate of the shear layer, and the horizontal axis is time increasing from right to left for a total flow real time of 307 msec. The pseudocolor assignment is linear in the mixture fraction (ξ) and is arranged as follows: red-unmixed fluid from the low-speed stream (ξ=0); blue-unmixed fluid from the high-speed stream (ξ=1); and the rest of the spectrum corresponds to intermediate compositions.

Figures 1 and 2, a single vortex and pairing vortices, respectively, show the composition field before the mixing transition. The Reynolds number based on the local visual thickness of the layer and the velocity difference across the layer is Re=1750 with U2/U1=0.46 and U1=13 cm/sec. Note the large excess of high-speed stream fluid in the cores of the structures.

These photographs show the vortex structures that result from the interaction of vortices that are shed from a 2D bluff body and those shed from a slot jet. The slot jet (3 mm x 150 mm) is located in the center of the rectangular face of the bluff body (15 mm x 240 mm). The photographs are positioned so that the velocity of the slot jet increases from left to right. In the first three photographs starting from the left, the velocity of the jet is smaller than the velocity of the flow around the bluff-body. In the fourth picture, the shear layer velocities of the jet and bluff body are nearly equal and a wavy structure is observed. At higher velocities, as noted by the 5th and 6th photographs, the vortex structures from the jet dominate the flow field. This is noted by the change in the direction of rotation of the vortices.

The flow is visualized by the Reactive Mie Scattering (RMS) technique in which Mie scattering is observed from micron size TiO2 particles that are formed by the spontaneous reaction of TiCl4 vapor in the slot jet air with the water in the annulus air. The technique has been shown to be more effective than smoke because it highlights the streamlines where molecular mixing is taking place. The photographs were taken in the 15ns firing of a YAG laser used to form the light sheet.

For an averaged air jet velocity of 18.5 cm/s, the alternating vortex structures shed from the 2D bluff body are evident after about 5 bluff-body widths downstream. As the jet velocity increases, the wake from the bluff body is significantly modified.

The collision of a droplet with a solid surface

The photographs displayed above show the impact, spreading, and boiling history of n-heptane droplets on a stainless steel surface. The impact velocity, Weber number, and initial droplet diameter are constant (values of 1 m/s, 43 and 1.5 mm respectively), and the view is looking down on the surface at an angle of about 30°. The photographs were taken using a spark flash method and the flash duration was 0.5 μs. The dynamic behavior illustrated in the photographs is a consequence of varying the initial surface temperature.

The effect of surface temperature on droplet shape may be seen by reading across any row; the evolution of droplet shape at various temperatures may be seen by reading down any column. An entrapped air bubble can be seen in the drop when the surface temperature is 24°C. At higher temperatures vigorous bubbling, rather like that of a droplet sizzling on a frying pan, is seen (the boiling point of n-heptane is 98°C) but the bubbles disappear as the Leidenfrost temperature of n-heptane (about 200°C) is exceeded because the droplet become levitated above a cushion of its own vapor and does not make direct contact with the surface. The droplet shape is unaffected by surface temperature in the early stage of the impact process (t≤0.8 ms) but is affected by temperature at later time (cf. t≥ 1.6 ms) because of the progressive influence of intermittent solid-liquid contact as temperature is increased.

The desire to capture images of fluids in motion for both scientific and artistic reasons dates back over 500 years, at least to the time of Leonardo de Vinci who is known to have recorded visual images of the complicated patterns traced by floating seeds on the surface of naturally flowing water. Indeed, flow visualization has played a major role in the development of the science of fluid mechanics, and has certainly been a key component in major technological advances such as the evolution of flight; in understanding natural phenomena such as the atmospheric motions that are responsible for weather-related phenomena such as hurricanes or tornadic thunderstorms; and in understanding biological systems such as the heart pump. A collection of some of the most striking photographs of fluid motions from the research literature was collected and published in 1982 in a book entitled An Album of Fluid Motion, by Milton Van Dyke.

Motivated by Van Dyke's book, the Division of Fluid Dynamics (DFD) of the American Physical Society (APS) has sponsored a “photo” contest at its annual scientific meeting each year since 1983. Researchers are invited to display visual images of fluids in motion. The entries are judged by a distinguished panel of fluid dynamics researchers to choose the most outstanding contributions based upon two criteria:

1. The artistic beauty and novelty of the visualizations;

2. The contribution to a better understanding of fluid flow phenomena.

Vortex flows paint themselves

The artistlike pictures of vortex flows presented here have been produced by the flow itself. The method of this “natural” flow visualization can be described briefly as follows: The working fluid is water mixed with some paste in order to increase the viscosity. Vortex flows are produced by pulling a stick or similar devices through the fluid or by injecting fluid through a nozzle into the working tank.

The flow visualization is performed in the following way: the surface of the fluid at rest is sparkled with oil paint of different colors diluted with some evaporating chemical. After the vortex structures have formed due to wakes or jets, a sheet of white paper is placed on the surface of the working fluid, where the oil color is attached to the paper immediately. The final results are artistlike paintings of vortex flows which exhibit a rich variety of flow structures.

Mixing in regular and chaotic flows

These photographs show the time evolution of two passive tracers in a low Reynolds number two-dimensional timeperiodic flow. The initial condition corresponds to two blobs of dye, green and orange, located below the free surface of a cavity filled with glycerine. The flow is induced by moving the top and bottom walls of the cavity while the other two walls are fixed. In this experiment the top wall moves from left to right and the bottom wall moves from right to left; both velocities are of the form Usin2(2πt/T), with the same U and the same period T, but with a phase shift of 90°.