Turbulence in Coastal and Civil Engineering

The following sections are included:

• Preface
• Contents
• List of symbols and abbreviations

The following sections are included:

• Introduction
• Flow over a flat plate
• Flow in wave boundary layers
• Flow in wave boundary layers in solitary motion
• Steady flow in pipes
• Oscillatory flow in pipes
• Flow around a circular cylinder in a steady current
• Flow induced by breaking waves
• References
• Appendix to Chapter 1: Transition to turbulence over a flat plate
  • Description of transition process
    • Problem statement
    • Emmons’ (1951) observations
    • Detailed description of a single turbulent spot
    • Critical Reynolds number for transition
    • Transition and turbulent spots over rough boundaries
  • Hydrodynamic stability theory for transition
    • The Orr-Sommerfeld equation
    • The eigenvalue problem
    • Experimental verification of the hydrodynamic stability theory
  • Direct numerical simulation (DNS) of transition
  • References

In this chapter, we will derive the basic equations for turbulent flows. These are the continuity equation, the equations of motion, and the energy equation. In a turbulent flow, any hydrodynamic quantity (such as any component of the velocity, or the pressure, or the temperature, etc.) fluctuates with time. Fig. 2.1 schematically illustrates a typical time series of any quantity (the i-component of the velocity in the example in the figure). Traditionally, this kind of behaviour is handled by decomposing the quantity in consideration into two components, namely a mean value and a fluctuating value around the mean. The next section will focus on the method of averaging to achieve mean values. Subsequently, the fundamental equations of hydrodynamics (the continuity equation and the equations of motion) will, by time averaging, be “transformed” to those for turbulent flows, leading to three sets of equations, namely the continuity equation, the Reynolds equations, and the energy equations, the fundamental equations for turbulent flows.

In this chapter, we will study steady, turbulent boundary layers over a wall, or in a channel, or in a pipe. Examples of turbulent boundary layers over a wall may be: flow over a flat plate, flow over the seabed, flow in wide rivers, flow over the earth surface (the atmospheric boundary layer), etc…

As has been seen in the previous chapters, hydrodynamic quantities such as the velocity components in a turbulent flow may appear to be random variables. Although more recent research has shown that the near-wall turbulent-boundary-layer process is quasi-repetitive in nature (Chapter 3, bursting process), unless special pattern recognition techniques are used to detect these repetitive, quasi-cyclic events, the turbulence signals even near a wall appear to be largely random. Therefore, it is imperative to use statistical and other analysis techniques to study turbulence…

In shallow waters, the orbital motion of water particles under a progressive wave degenerates into a bottom-parallel, straight line, oscillatory motion at the seabed (expressed as U₀ = U_0m sin(ωt) in Fig. 5.1 for a small amplitude linear (sinusoidal) wave, where ω is the angular frequency, U₀ the streamwise velocity, and t the time). For each half cycle of this oscillatory motion, a “new” time-dependent boundary layer develops over the seabed. This cyclic boundary layer is called the wave boundary layer…

The preceding chapter has focused primarily on wave boundary layers which develop beneath spatially homogeneous oscillatory (typically, sinusoidal) free-stream flows. This corresponds precisely to the situation in conventional oscillating tunnel facilities, e.g. Fig. 5.21, and is a sufficiently accurate description of real wave boundary layers for many purposes. However, in reality, the flow beneath water waves will generally not be spatially homogeneous. This is due either to their progressive nature alone (i.e. variations over their wave length), or for other more deliberate reasons e.g. due to sloping beds or changing bed roughness. Moreover, while waves may, to leading-order, be described as sinusoidally shaped (see the Appendix to Chapter 5), in reality their shape will often deviate significantly from this idealized form due to nonlinear (finite amplitude) effects. This will also affect their corresponding near-bed free stream flows, and hence the boundary layers that develop beneath…

A detailed understanding of the flow and sediment transport within the surf and swash zones is of major engineering importance. Here the surf zone is defined as the area extending in the cross shore direction from incipient wave breaking to the limit of runup, while the swash zone is the wet-dry portion of the beach ranging between the limit of drawdown and runup. The flow and turbulence induced by breaking waves make these regions of large importance within coastal sediment transport and morphology. In particular, surf and swash zone processes govern the exchange of sand between the emerged and submerged portions of the beach, and are therefore largely responsible for determining shoreline configurations, including e.g., beach accretion or recession…

Diffusion/dispersion is another important property of turbulent flows. Some examples are: (1) Waste products released into a recipient environment, such as a stream or the ocean, undergo rapid diffusion; (2) Smoke (or exhaust gases) released into the atmosphere diffuses rapidly when there is strong wind; (3) Likewise, dredged material (sand, silt, etc.), when dumped into the sea, is subjected to large diffusion…

The present chapter describes the mathematical modelling of turbulence, also colloquially known simply as “turbulence modelling”. Turbulence modelling in various forms has already been presented and/or employed in several of the previous chapters, hence this chapter is intended to supplement much of the preceding book. In what follows the closure problem of turbulence is first covered, establishing the necessity of resorting to turbulence modelling in the first place. The remainder of the chapter is then dedicated to describing some selected models for establishing turbulence closure in practice. In particular one closure from each of the three main model types are presented. These correspond to the mixing length model (an algebraic model), the k-ω model (a turbulence energy equation model), and a large eddy simulation (LES) model (a direct simulation model). Simple examples employing each of the specific models described are also presented. The chapter ends with a brief discussion of how computational costs associated with various methods (direct numerical simulation - DNS, LES, and turbulence energy equation models) scale with respect to increases in the Reynolds number…

The following sections are included:

• Introduction
• Data sets
• MatRANS k-ω turbulence model in Matlab
• Exercise 1: Analysis of a turbulent boundary layer in an open channel
• Exercise 2: A simple numerical model of dispersion in a turbulent boundary layer flow
• Exercise 3: Statistical, correlation, and spectral analysis of turbulent air jet flow
• Exercise 4: Turbulence modelling of oscillatory wave boundary layer flows
• References

The following sections are included:

• Author index
• Subject index