We derive the governing equations for the mean and turbulent kinetic energy and discuss simplifications of the equations for several canonical flows, including channel flow and homogeneous isotropic turbulence. A classical expression for the dissipation rate in isotropic turbulence is provided. In addition, the governing equations for turbulent enstrophy and scalar variance are derived with parallels to the derivation of the turbulent kinetic energy equation. A model for turbulent kinetic energy evolution and dissipation in isotropic turbulence is introduced. Finally, we derive the governing equations for the Reynolds stress tensor components and discuss the roles of the terms in the Reynolds stress budgets in homogeneous shear and channel flows. A crucial link between pressure-strain correlations and the redistribution of turbulent kinetic energy between various velocity components is established. Quantifying how energy is transferred between the mean flow and turbulent fluctuations is crucial to understanding the generation and transport of turbulence and its accompanying Reynolds stresses, and thus properties that phenomenological turbulence models should conform to.

Building on the governing equations and spectral tools introduced in earlier chapters, we analyze the energy cascade, which describes the transfer of turbulent kinetic energy from large to small eddies. This includes an estimate of the energy dissipation rate, as well as the characteristic length and time scales of the smallest-scale motions. Nonlinearity in the Navier-Stokes equations is responsible for triadic interactions between wavenumber triangles that drive energy transfer between scales. Empirical observations suggest that the net transfer of energy occurs from large to small scales. In systems where the large scales are sufficiently separated from the small scales, an inertial subrange emerges in an intermediate range of scales where the dynamics are scale invariant. Kolmogorov’s similarity hypotheses and the ensuing expressions for the inertial-subrange energy spectrum and viscous scales are introduced. The Kolmogorov spectrum for the inertial subrange, which corresponds to a -5/3 power law, is a celebrated result in turbulence theory. We further discuss key characteristic turbulence scales including the Taylor microscale and Batchelor scale.

We discuss properties of numerical methods that are essential for high-fidelity (LES, DNS) simulations of turbulent flows. In choosing a numerical method, one must be cognizant of the broadband nature of the solution spectra and the resolution of turbulent structures. These requirements are substantially different than those in the RANS approach, where the solutions are smooth and agnostic to turbulent structures. We focus on spatial discretization of the governing equations in canonical flows where Fourier analysis is helpful in revealing the effect of discretization on the solution spectra. In high-fidelity numerical simulations of turbulent flows, it is necessary that conservation properties inherent in the governing equations, such as kinetic energy conservation in the inviscid limit, are also satisfied discretely. An important benefit of adhering to conservation principles is the prevention of nonlinear numerical instabilities that may manifest after long-time integration of the governing equations. We end by discussing the appropriate choice of domain size, grid resolution, and boundary conditions in the context of canonical flows with uniform Cartesian mesh spacing.

This chapter explores accounts of parenting, largely drawing on research that has focused on the views or experiences of children themselves rather than the perspectives of adults. In taking this approach, the chapter aims to consider how adultism shapes our understanding of children’s experiences in regards to diverse sexes, genders, and sexualities. A range of research focusing on the experiences of heterosexual children of LGB parents and the children of trans parents is discussed. The chapter also reviews research that reports on the experiences of young LGBTIQ people growing up, with a particular focus on well-being and resiliency. Overall, this chapter highlights the intersections of marginalisation and resistance for children whose lives are shaped by norms related to sex, gender, and sexuality.

This chapter provides a socio-historical account of the pathologisation and de-pathologisation of diversity in sex, gender, and sexuality within and beyond psychology. Focusing on people born with intersex variations, a diversity of genders, and a diversity of sexual orientations (e.g., lesbian, gay, bisexual, queer), this chapter first maps the socio-medicalisation of sex, gender, and sexuality to explore the pathologisation of LGBTIQ people across time. Next, the chapter maps the socio-historical de-pathologisation of sex, gender, and sexual diversity and the development of LGBTIQ psychology as an affirmative field. Different approaches to the treatment of LGBTIQ people in healthcare and the development of professional psychological networks that focus on LGBTIQ psychology are presented.

With a specific focus on violence and abuse, this chapter explores some the challenges that LGBTIQ people often experience, but also the strengths that LGBTIQ people display. The chapter reviews research on intimate partner violence experienced by LGBTIQ people (including identity-related abuse) and the violence perpetrated against animals in these contexts. Situating challenges alongside strengths is an important counter to the often negative messages and stereotypes that circulate about LGBTIQ people, as it encourages a focus on identifying sites of resistance and opportunities for change. The chapter therefore also explores the resiliencies that LGBTIQ people display in the face of adversity, including through relationships with animal companions.

The spectral description of turbulence allows us to decompose velocity and pressure fields in terms of wavenumbers and frequencies, or length and time scales. We discuss the notion of scale decomposition and introduce several properties of the Fourier transform between physical (spatial/temporal) space and scale (spectral) space in various dimensions, including complex conjugate relations for real functions and Parseval’s theorem. The Fourier transform allows us to develop useful relations between correlations and energy spectra, which are used extensively in the statistical theory of turbulence. The one-dimensional and three-dimensional energy spectra are specifically discussed in conjunction with Taylor’s hypothesis to enable spectra computation from single-point time-resolved measurements. The discrete version of the transform, or the discrete Fourier series, is then introduced, as it is typically encountered in numerical simulations and postprocessing of discrete experimental data. Treatment of periodic data is first considered, followed by nonperiodic data with the help of windowing. The procedure for the computation of various discrete spectra is outlined.

This chapter explores prejudice and discrimination and their effects on LGBTIQ people and communities. First, this chapter reviews research on attitudes towards LGBTIQ people, with reference to studies of homophobia, biphobia, and transphobia. With specific reference to hate crimes, it next discusses homophobic, biphobic, and transphobic victimisation. Systematic prejudice (structural prejudice embedded in social and legal institutions) is then discussed in relation to key constructs such as heterosexism, heteronormativity, and cisgenderism. The final section of the chapter focuses on minority stress and the ways in which this and other processes (e.g., internalised homophobia, decompensation) contribute to psychological distress among LGBTIQ people, including those who a multiply marginalised. The impacts of these factors on mental health in LGBTIQ populations are also discussed.