In the previous chapters we saw that if We < Q−1, the disturbance of wave numbers smaller than a cut-off wave number is unstable at the onset of instability. The cut-off wave number increases with Q. The capillary force is then shown with linear theories to be responsible for the onset of instability in the presence or absence of fluid viscosities. Subsequent to the onset, the amplitude of disturbances grows rapidly and the neglected nonlinear terms in the linear theory are no longer negligible. Thus the nonlinear evolution of disturbances that lead to the eventual pinching off of drops from a liquid jet can only be described with nonlinear theories. Similarly the pinching off of small droplets from the interface caused by interfacial pressure and shear fluctuations at the onset of instability, when We > Q−1, requires nonlinear theories to describe. Experimental observations of the nonlinear phenomena are presented first.

Experiments

Capillary Pinching

Linear theory predicts that unstable disturbances of different wavelengths grow at different rates and different natural frequencies corresponding to the different wavelengths. Figure 11.1 shows the nonlinear evolution of the disturbances when external sinusoidal forcings are introduced at three different natural frequencies. The forcing frequency for Figure 11.1(a) corresponds to k = 0.683, which is close to the Rayleigh's most amplified disturbance. Figures 11.1(b) and (c) correspond to the cases of lower forcing frequencies corresponding to k = 0.25 and k = 0.075, respectively. The disturbances of wavelengths shorter than that of the fastest growing disturbance appear to grow more slowly as they are convected downstream from the nozzle exit, as predicted by linear theory.

In the previous chapters, we investigated the fairly well studied phenomena of breakup of liquid sheets and liquid jets. The basic flows were assumed to be steady in the continuum theories. Also, they were either of infinite or of semi-infinite extent in the flow direction. Physically such infinite and semi-infinite steady jets or sheets cannot exist, as predicted by stability analysis. The analytical predictions enjoyed fairly good agreement with many known experiments. However, breakup of a liquid body into smaller parts often takes place under an unsteady situation from the beginning. The examples include the formation of satellites and subsatellites from the ligaments after detaching themselves from the main drops, the formation of drops from a dripping faucet, shaped-charge jets, the formation of micro-drops by external forcing, intermittent fuel sprays, and the phenomenon of jet branching induced by external excitation. These are the subjects to be touched upon in this last chapter.

Satellite Formation

When a stretched liquid ligament is relaxed, the capillary force associated with the large surface curvature at both ends of the ligament tends to compress and fragment the ligament into small drops. We saw the formation of the ligament during the last stage of nonlinear evolution of instability. The stretching of a liquid ligament submerged in another fluid can be achieved by pure straining or shearing or a combination of both. Figure 12.1 (Stone et al., 1986) shows how a spherical drop is stretched in two purely straining external flows with two different viscosities.

This chapter elucidates the role of interfacial shear on the onset of instability of a cylindrical viscous liquid jet in a viscous gas surrounded by a coaxial circular pipe by using an energy budget associated with the disturbance. It is shown that the shear force at the liquid-gas interface retards the Rayleigh mode instability, which leads to the breakup of the liquid jet into drops of diameter comparable to the jet diameter because of capillary force. On the other hand the interfacial shear and pressure work in concert to cause the Taylor mode instability, which leads the jet to breakup into droplets of diameter much smaller than the jet diameter. While the interfacial pressure plays a slightly more important role than the interfacial shear in amplifying the longer wave spectrum in the Taylor mode, shear stress plays the main role of generating shorter wavelength disturbances.

Basic Flow

Consider the instability of an incompressible Newtonian liquid jet of radius R1. The jet is surrounded by a viscous gas enclosed in a vertical pipe of radius R2, which is concentric with the jet. For the jet to maintain a constant radius, the dynamic pressure gradients in the steady liquid and gas flows must maintain the same constant. This will allow the pressure force difference across the liquid-gas interface to be exactly balanced by the surface tension force as required. Such coaxial flows, which satisfy exactly the Navier–Stokes equations, are given by (Lin and Ibrahim, 1990).

In the previous chapter we mentioned that fluid viscosity might alter the critical Weber number that divides the parameter space into regimes of absolute and convective instability. The effects of gas and liquid viscosities are investigated separately in this chapter, not just to understand each individual effect but also to demonstrate the coupled effect, which is unexpected. In Chapter 3, stability analysis for an inviscid liquid sheet of uniform thickness was applied locally to investigate the stability of gradually thinning liquid sheets. The thinning was either due to axial expansion or gravitational acceleration. The local application of the inviscid theory for a uniform sheet to the two different cases of nonuniform sheets was made judiciously. Likewise the viscous theories given in this chapter can be applied judiciously to a gradually thinning viscous sheet whatever the cause of the thinning. The thinning may be caused by kinematic requirements, gravitational acceleration, or viscous extrusion. The breakup of a viscous liquid sheet in an inviscid gas is expounded in Section 4.1. The effect of gas viscosity is elucidated in Section 4.2. The effects of liquid and gas viscosities on the onset of sheet breakup are summarized in Section 4.3.

A Viscous Sheet in an Inviscid Gas

The basic flow attributed to G. I. Taylor is given in Section 4.1a, and its stability is analyzed in Section 4.1b. The physical mechanism of the sheet breakup is discussed in Section 4.1c, based on energy considerations.

This chapter is concerned with the main topic of the monograph, namely, the solution of the GRP for quasi-1-D, inviscid, compressible, nonisentropic, time-dependent flow. In Section 5.1 we formulate the problem and study its solution in the Lagrangian and Eulerian frames. In particular, we state and prove the main ingredient in the GRP method, Theorem 5.7. A weaker form of this theorem leads to the “acoustic approximation” (Proposition 5.9). Summary 5.24 gives a step-by-step description of the GRP analysis. In Section 5.2 we present the GRP methodology for the construction of second-order, high-resolution finite-difference (or finite-volume) schemes. Starting out from the (first-order) Godunov scheme, we present the basic (E1) GRP scheme. It is based on the acoustic approximation and constitutes the simplest second-order extension of Godunov's scheme. This is followed by a presentation of the full array of GRP schemes (as well as MUSCL). Generally speaking, the presentation in this chapter follows closely the GRP papers [7] and [10].

The GRP for Quasi-1-D, Compressible, Inviscid Flow

In Section 4.2 we studied the Euler equations (4.45) governing the quasi-1-D flow in a duct of variable cross section. We emphasized in particular the role of the Riemann problem (“shock tube problem”), namely, the IVP subject to initial data (4.100). As we shall see in this chapter, the solution to the Riemann problem is a basic ingredient in the numerical resolution of the flow.

In Definition 2.15 we gave the most practical version of the entropy condition. It limits admissible shocks to those obtained by the intersection of “forward-moving” characteristics. These are therefore discontinuities that “cannot be avoided” or replaced by a rarefaction wave. In this Appendix we give some further insight into this concept of an “entropy satisfying” weak solution to (2.1), (2.2).

Our starting point is the physical notion of a “vanishing viscosity solution.” In general terms, an equation leading to discontinuous solutions [such as (2.1)] is supplemented by “dissipative terms” (also referred to as “viscous terms”). In analogy to the physical situation, such terms have a “smoothing effect” on solutions with large gradients, thus replacing discontinuities by “transition zones” where the solution varies smoothly, albeit rapidly. As the viscous effects are diminished, those transition zones shrink to surfaces of zero width, across which the solution has a sharp jump. Mathematically speaking, the additional viscous terms are often represented by second-order derivatives with a small (“vanishing”) coefficient.

To illustrate the situation, we consider the “moving step” problem for Burgers' equation (Example 2.12).

This chapter addresses one of the most central issues of computational fluid dynamics, namely, the simulation of flows under complex geometric settings. The diversity of these issues is briefly outlined in Section 8.1, which points out the role played by the present extensions: the (1-D) “singularity tracking” and the (2-D) “moving boundary tracking” (MBT) schemes. Section 8.2 deals with the first extension, and Section 8.3 is devoted to an outline of the second one. In the former we present the scheme methodology and refer to GRP papers for examples. In the latter, the basic principles of the method are presented, and we refer to [39] for more algorithmic details. Finally, an illustrative example of the MBT method shows how an oval disk is “kicked-off” by a shock wave.

Grids That Move in Time

In Part I of this monograph we dealt with finite-difference approximations to the quasi-1-D hydrodynamic conservation laws, where the underlying grid was fixed and equally spaced in the majority of cases. In our two-dimensional numerical extension (Section 7.3) we restricted the treatment to a Cartesian (rectangular) grid. Naturally, finite-difference approximations assume their simplest form on such grids, and the motivation for seeking geometric extensions comes primarily from physical applications.

Computational fluid dynamics (CFD) is a relatively young branch of fluid dynamics, the other two being the experimental and the theoretical disciplines. Its rapid development was enabled by the spectacular progress in high power computers, as well as by a matching progress in numerical schemes.

The starting point for the formulation of CFD schemes is the governing equations. In fact, the term “fluid dynamical equations” is much too general and indeed ambivalent. In practice there exist numerous models of such equations. They reflect a variety of stipulations on the nature of the flow, such as compressibility, viscosity, or elasticity. They also involve various effects such as heat conduction or chemical reactions. A large portion of these models do not fall, mathematically speaking, under the category of “hyperbolic conservation laws,” which is the subject matter of this monograph. We refer the reader to the book by Landau and Lifshitz [75] for a general survey of fluid dynamical models.

In this monograph we are concerned with time-dependent, inviscid, compressible flow, which is studied primarily in the “quasi-one-dimensional” geometric setting. This leads to a system of partial differential equations expressing the conservation of mass, momentum, and energy. There are various approaches to the numerical resolution of this system, such as the classical method of characteristics or the “artificial viscosity” scheme.

In this chapter we consider the system of equations governing compressible reacting flow. The fluid is a homogeneous mixture of two species. The evolution of the flow under the mechanical conservation laws of mass, momentum and energy is coupled to the (continuous or abrupt) conversion of the “unburnt” species to the “burnt” one. We take the simplest model of such a reaction, namely, an irreversible exothermic process. The equation of state of the fluid depends on its chemical composition. The resulting (augmented) system is still nonlinear hyperbolic (in the sense of Chapter 4) and is amenable to the GRP methodology. The basic hypotheses are presented in Section 9.1, leading to the derivation of the characteristic relations and jump conditions. In Section 9.2 we describe the classical Chapman—Jouguet model of deflagrations and detonations, and the Zeldovich—von Neumann—Döring (Z—N—D) solution is presented in Section 9.3. In Section 9.4 we study the generalized Riemann problem for the system of reacting flow. The treatment here is close to that of the basic GRP case (Section 5.1), but there are significant differences because of the reaction equation. In Section 9.5 we outline briefly the resulting GRP numerical scheme and study a physical problem of ozone decomposition.

Here the fluid dynamical theory and GRP schemes of Chapters 4 and 5 are applied to one-dimensional test cases. The problems are aimed primarily at demonstrating the capabilities of the scheme, but they are also revealing of nontrivial fluid dynamical phenomena that arise even at the relatively simple one-dimensional settings considered here. In Section 6.1 we treat a shock tube problem, using several scheme options to solve it. An interesting class of fluid dynamical problems is that of wave interactions, to which Section 6.2 is devoted. We selected four different cases in this class, shock—shock, shock—contact, shock—rarefaction, and rarefaction—contact interactions. In each case the GRP solution is compared to either an exact one or to a solution of a Riemann problem that approximates the exact one in some “asymptotic” sense. In the remainder of the chapter we employ the quasi-one-dimensional (“duct flow”) scheme, solving three different problems, comparing each numerical solution to the corresponding exact one. Section 6.3 treats a spherically converging flow of cold gas, and Section 6.4 is devoted to the flow induced by an expanding sphere. Finally, in Section 6.5 we present a detailed treatment of the steady flow in a converging—diverging nozzle, obtained numerically as a large-time solution by the GRP scheme.

The notation for the fluid dynamical variables here is identical to that of Chapters 4 and 5.

This chapter introduces the GRP method in the context of the scalar conservation law ut + f(u)x = 0. We start in Section 3.1 with the classical first-order (conservative) “Godunov Scheme,” which leads naturally to its second-order GRP extension. Section 3.2 contains a number of numerical (one-dimensional) examples, for linear and nonlinear equations, illustrating the improved resolution obtained by the GRP method. In Section 3.3 we extend the GRP methodology to the two-dimensional scalar conservation law ut + f(u)x + g(u)y = 0. Analytical and numerical results are compared for simple as well as complex wave interactions.

From Godunov to the GRP Method

In this section we discuss the GRP method, aimed at a high-resolution numerical approximation of the solution to a conservation law of the form (2.1). We always assume that f(u) is strictly convex:f″(u) ≥ μ > 0. It is shown that this method is a natural analytic extension to the Godunov (upwind) scheme. This latter scheme has been extensively studied in Section 2.2, in the context of the linear convection equation. We start here by studying this scheme in the nonlinear case.

As in Section 2.2, we take a uniform spatial grid xj = jΔx, − ∞ < j < ∞, and uniformly spaced time levels tn+1 = tn+k, t0 = 0.

This monograph deals with the generalized Riemann problem (GRP) of mathematical fluid dynamics and its application to computational fluid dynamics. It shows how the solution to this problem serves as a basic tool in the construction of a robust numerical scheme that can be successfully implemented in a wide variety of fluid dynamical topics. The flows covered by this exposition may be quite different in nature, yet they share some common features; they all belong to the class of compressible, inviscid, time-dependent flows. Fluid dynamical phenomena of this type often contain a number of smooth flow regions separated by singularities such as shock fronts, detonation waves, interfaces, and centered rarefaction waves. One must then address various computational issues related to this class of fluid dynamical problems, notably the “capturing” of discontinuities such as shock fronts, detonation waves, or interfaces; resolution of centered rarefaction waves where flow gradients are unbounded; and evaluation of flow variables in irregular computational cells at the intersection of a moving boundary surface with an underlying mesh.

From the mathematical point of view, the various systems of equations governing compressible, inviscid, time-dependent flow phenomena may all be characterized as systems of “(nonlinear) hyperbolic conservation laws.”

Hyperbolic conservation laws (in one space variable) are systems of time-dependent partial differential equations.