Introduction

Computer simulation of turbulent flows is becoming increasingly attractive due to the greater physical realism relative to conventional modelling, at a cost that is reducing with continuing advances in computer hardware and algorithms. The first turbulence simulations appeared over 30 years ago, since when we have seen increases in computer performance of over four orders of magnitude such that many of the canonical turbulent flows first studied by laboratory experiments can now be reliably simulated by computer. Examples include turbulent channels (Kim et al., 1987), turbulent boundary layers (Spalart, 1988), mixing layers (Rogers & Moser, 1994), subsonic and supersonic jets (Freund, 2001, Freund et al., 2000) and backward-facing steps (Le et al., 1997). Where simulations can be reliably made, they provide more data than are available from laboratory experiments, even with modern non-intrusive flow diagnostics. In these situations they provide insight into the basic fluid mechanics. This can be at a very simple flow visualisation level, where a conceptual picture of what is happening in a flow can be quickly obtained from computer animations of key features, or at more advanced levels where the simulations provide statistical data to assist Reynolds-averaged model development. Indeed, several important recent turbulence models have come out of groups who do both simulations and modelling, examples being the Spalart & Allmaras (1994) model and Durbin's K-ε-ν2 (Durbin, 1995), and it is rare to come across a turbulence modelling paper that has not used simulation data as a reference.

Introduction

The aim of this chapter is to provide, in plain English, a guide to the capabilities and shortcomings of turbulence models for reproducing satisfactorily engineering flows where buoyant or stratification effects are important. While these two descriptors are often used interchangeably in the literature, in the present chapter buoyant is used to denote a situation where the effect of gravity is to cause a force field whose primary effect is on the mean flow, while stratified implies that the principal effects on the flow arise from gravitational action on the turbulent fluctuating velocities. The distinction is neither pedantic nor unimportant; for, a stratified flow will ordinarily require a more rigorous modelling of gravitational effects than a buoyant flow. Put another way, gravitational effects on horizontal flows are more troublesome than on vertical flows. The account may hopefully also be useful where the flows of interest are affected by other types of force field, perhaps particularly flows affected by Coriolis forces or swirl.

The chapter gives especial attention to two-equation models of turbulence as this is currently the main level of commercial CFD. Linear two-equation eddyviscosity models are considered first, beginning with the situation where buoyant/stratification effects are absent. This is important to enable the reader to assess whether, for the flows of interest, a linear eddy viscosity model would be suitable even in a uniform density situation. Thereafter, the treatment of buoyant flows with linear two-equation models is considered.

Abstract

The problems concerning the understanding and prediction of strongly-distorted turbulent boundary layers are reviewed. Some of the views expressed emanate from the Isaac Newton Institute (INI) programme on turbulence in 1999; others are an experimentalist's view of current modelling techniques and the requirements for future work. The purpose of this chapter is, first, to pinpoint research that has already been carried out in this area and, second, to highlight gaps in our knowledge where there is a need for specific experiments to enable the subsequent development of existing turbulence models. Attention is not restricted to Reynolds-averaged Navier–Stokes (RANS) solvers but also considers the problems regarding the modelling for large-eddy simulation (LES). The subject of this chapter is vast, and therefore ample use is made of existing reviews by authors who are experts in specific subject areas. These include ‘extra rates of strain’, changes in boundary conditions, as well as even more complex phenomena such as shock/boundary-layer interaction and boundary layers with a variety of embedded vortices. While we have many of the numerical tools required for design and prediction, it is clear that physical understanding of the dominant mechanisms is lacking and therefore our ability to predict them is also. As far as our existing knowledge is concerned, emphasis is placed on an empirical approach for reasons of pragmatism. Some ‘application challenges’ presented during the course of the INI programme on turbulence are addressed.

Introduction

Computational modelling is assuming a greater and greater role in the study of multi-phase flows. Although it is not yet feasible to predict complex multi-phase flow fields over the full range of velocities and flow patterns, computational methods are helpful for a variety of reasons which include:

1. They enable insights to be obtained on the nature and relative importance of phenomena and are a natural aid to experimental measurement. Indeed, it is often possible to compute quantities which cannot be readily measured.

2. When coupled with experimental observations and empirical relationships, computational methods can give predictions which are reaching the stage of being useful in the design and operation of systems involving multi-phase flows, particularly for dispersed flow situations. This fact is reflected in the growing number of commercial computer codes which are available for application in this field.

In this chapter, we will deal first with the application of single-phase prediction methods in the interpretation of two-phase flows. Here, a brief description is given of the available turbulence models and examples cited of the application of this approach (flows in coiled tubes, horizontal annular flow and waves in annular flow).

An important class of two-phase flows is that where one of the phases is dispersed in the other, for example dispersions of bubbles in a liquid (bubble flow), dispersions of solid particles in a gas or liquid (gas–solids or liquid–solids dispersed flows) and dispersions of droplets of one liquid in another liquid (liquid–liquid dispersed flow).

Introduction

The theory of liquid sloshing dynamics in partially filled containers is based on developing the fluid field equations, estimating the fluid free-surface motion, and the resulting hydrodynamic forces and moments. Explicit solutions are possible only for a few special cases such as upright cylindrical and rectangular containers. The boundary value problem is usually solved for modal analysis and for the dynamic response characteristics to external excitations. The modal analysis of a liquid free-surface motion in a partially filled container estimates the natural frequencies and the corresponding mode shapes. The knowledge of the natural frequencies is essential in the design process of liquid tanks and in implementing active control systems in space vehicles. The natural frequencies of the free liquid surface appear in the combined boundary condition (kinematic and dynamic) rather than in the fluid continuity (Laplace's) equation.

For an open surface, which does not completely enclose the field, the boundary conditions usually specify the value of the field at every point on the boundary surface or the normal gradient to the container surface, or both. The boundary conditions may be classified into three classes (Morse and Fesbach, 1953):

1. the Dirichlet boundary conditions, which fix the value of the field on the surface;

2. the Neumann boundary conditions, which fix the value of the normal gradient on the surface; and

3. the Cauchy conditions, which fix both value of the field and normal gradient on the surface.

Each class is appropriate for different types of equations and different boundary surfaces.

Introduction

The previous chapter treated the free oscillations of liquid free surface in different container geometries. The fluid field equations were developed and the natural frequencies were determined from the free-surface boundary conditions. The knowledge of liquid-free-surface natural frequencies is important in the design of liquid containers subjected to different types of excitations. In the design process, it is important to keep the liquid natural frequencies away from all normal and nonlinear resonance conditions. The excitation can be impulsive, sinusoidal, periodic, or random. Its orientation with respect to the tank can be lateral, parametric, pitching/ yaw, roll, or a combination. Under forced excitation, it is important to determine the hydrodynamic loads acting on the container, and their phase with respect to the excitation. The hydrodynamic forces are estimated by integrating the pressure distribution over the wetted area. One should also determine the free-surface wave height, which affects the location of the center of mass. In the neighborhood of resonance, the free surface experiences different types of nonlinear behavior and this will be considered in Chapters 4, 6, and 7.

Under impulsive excitation, liquid sloshing dynamics was studied by Morris (1938), Werner and Sundquist (1949), Jacobsen (1949), Jacobsen and Ayre (1951), and Hoskins and Jacobsen (1957). Bauer (1965a) considered the response of liquid propellant to a single pulse excitation. Transient and steady state response to sinusoidal excitation of a liquid free surface was studied by Sogabe and Shibata (1974a, b).

Introduction

The linear theory of liquid sloshing is adequate for determining the natural frequencies and wave height of the free surface. Under translational excitation, the linear theory is also useful for predicting the liquid hydrodynamic pressure, forces, and moments as long as the free surface maintains a planar shape with a nodal diameter that remains perpendicular to the line of excitation. However, it does not take into account the important vertical displacement of the center of gravity of the liquid for large amplitudes of free-surface motion. It also fails to predict complex surface phenomena observed experimentally near resonance. These phenomena include nonplanar unstable motion of the free surface associated with rotation of the nodal diameter (rotary sloshing) and chaotic sloshing. These phenomena can be uncovered using the theory of weakly nonlinear oscillations for quantitative analysis and the modern theory of nonlinear dynamics for stability analysis. The main sources of nonlinearity in the fluid field equations appear in the free-surface boundary conditions.

The early work of nonlinear lateral sloshing is based on asymptotic expansion techniques. For example, Penney and Price (1952) carried out a successive approximation approach where the potential function is expressed as a Fourier series in space with coefficients that are functions of time. These coefficients are approximated by Fourier time series using the method of perturbation. The resulting solution is expressed in terms of a double Fourier series in space and time. This method was applied by Skalak and Yarymovich (1962) and Dodge, et al. (1965).

Introduction

Parametric sloshing refers to the motion of the liquid free surface due to an excitation perpendicular to the plane of the undisturbed free surface. Generally, parametric oscillation occurs in dynamical\ systems as a result of time-dependent variation of such parameters as inertia, damping, or stiffness. This variation may be due to the influence of externally applied forces or acceleration fields referred to as parametric excitations. For the case of parametric sloshing, the effective gravitational field becomes time dependent. Although parametric oscillation is regarded to be of a secondary interest, it can have catastrophic effects on mechanical systems near critical regions of parametric instability. For example, sloshing waves are generated when the liquid container is vertically excited at a frequency close to twice the natural frequency of the free surface. This type is referred to as Faraday waves since he (1831) was the first to observe and report them. Faraday waves are distinct from other waves with crests normal to a moving boundary (wave-maker) as cross-waves reported by Miles (1985a, 1988b, 1990a), Miles and Becker (1988), and Guthar and Wu (1991), see also Section 4.4. Hocking (1976, 1977, 1987b) observed surface waves produced by a vertically oscillating plate.

In 1831, Faraday observed the fluid inside a glass container oscillates at one-half of the vertical excitation frequency. Another similar series of experiments conducted by Mathiessen (1868, 1970) showed that the fluid oscillations are synchronous.

Introduction

Regular gravitational potential has a stabilizing effect in that it brings the liquid volume toward the bottom of its container. When the body force diminishes, the liquid volume assumes any location inside its container in an unpredictable manner. The problems of liquid sloshing dynamics under microgravity are different from those encountered under a regular gravitational field. These problems include liquid reorientation and the difficulty of moving and handling, since the body force is almost negligible. Under microgravity, surface tension forces become predominant. The Bond number, given by the ratio of the gravitational to capillary forces, plays a major role in the free-liquid-surface characteristics. For very small values of the Bond number, capillary forces predominate and the free surface will not be flat, but will rise around the vertical walls of a container.

The problems of liquid behavior at a low and zero gravity include the mechanics and thermodynamics of capillary systems, heat transfer in cryogenic tanks and mechanisms of energy transport, capillary hydrostatics, low-gravity sloshing, and fluid handling. The early work on free-liquid-surface behavior under a low gravity field considered different problems of the surface–vapor interface (Petrash and Otto, 1962, 1964, Shuleikin, 1962, Petrash, et al. 1962, 1963, Petrash and Nelson, 1963, Neu and Good, 1963, Paynter, 1964a, b, Zenkevich, 1965, Abramson, 1966a, Seebold et al., 1967 and Hastings and Rutherford, 1968).

Introduction

An impulsive acceleration to a liquid can result in impact hydrodynamic pressure of the free surface on the tank walls. It can also occur during maneuvering or docking of spacecraft in an essentially low gravity field. Methods for estimating liquid impact and its hydrodynamic pressure are not well developed and are only identified by experimental studies (Pinson, 1963, Dalzell and Garza, 1964, and Stephens, 1965). It was found that when hydraulic jumps or traveling waves are present, extremely high impact pressures can occur on the tank walls. A hydraulic jump may occur in a liquid container undergoing oscillatory motion if the liquid height is relatively shallow and the excitation frequency is close to the natural frequency of the free surface.

Typical pressure–time history records under sloshing impact condition were reported by Cox, et al. (1980). Feng and Robertson (1971, 1972) studied the liquid propellant dynamics during docking. Rumyantsev (1969a) analyzed the collision of a body containing a viscous liquid with a rigid barrier. Hamlin, et al. (1986) indicated that the hydraulic jump could create localized high impact pressures on the container walls, which has a direct effect on the container dynamics and may result in structural damage. Verhagen and WijnGaarden (1965) reported that close to resonance a hydraulic jump was observed, which travels periodically back and forth between the container walls. This hydraulic jump is a nonlinear phenomenon, analogous to the shock wave appearing in one-dimensional gas flow under similar resonance conditions (Chester, 1964, 1968).

Introduction

The problem of fluid dynamic behavior in spinning tanks is encountered in the study of stability and control of rockets, space vehicles, liquid-cooled gas turbines, centrifuges, and oceans. Some spacecraft vehicles are designed to spin in order to gain gyroscopic stiffness during the transfer from low earth orbit. This spin helps to control liquid propellant location in its container. Another class of problems deals with the dynamics and stability of rotating rigid bodies as applied to the evolution of celestial bodies and astronavigation control. Stabilization is achieved when the spacecraft spins about its axis of minimum moment of inertia. A satellite that spins about its axis of minimum moment of inertia may experience instability in the presence of energy dissipation. This is similar to a spinning top on a rough surface and as a result of friction the top's nutation angle increases as it seeks to conserve angular momentum.

The theory of rotating fluids is well documented in Lamb (1945), Morgan (1951), Howard (1963), and Greenspan (1968). Usually, the fluid motion is characterized by certain types of boundary layers and dissipative behavior in addition to complicated viscous layers controlled processes for spin-up and spin-down (Wilde and Vanyo, 1993). Lighthill (1966) presented a survey on the dynamics of rotating fluid. The results always have to be distinguished by the magnitude of the ratio of spin speed and gravity, that is, by the parameter Ω2R/g, where Ω is the spin speed, R is the tank radius, and g is the gravitational acceleration.