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.

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.

An overview of the three modern categories of methods for numerical prediction of turbulent flows is provided: direct numerical simulation (DNS), solution of the Reynolds-averaged Navier-Stokes (RANS) equations, and large-eddy simulation (LES). We describe zero-equation, one-equation, two-equation, and Reynolds stress transport models for the RANS equations. RANS computations require significantly fewer grid points and lower computational cost since the solutions are smooth and turbulent structures are not captured, but there is a need to tune model parameters for different flows to match experimental data. In LES, only the large-scale motions are resolved, whereas unresolved small scales are modeled. We introduce the notion of filtering, subgrid-scale parameterization, as well as the seminal dynamic Smagorinsky subgrid-scale model. Wall-resolved and wall-modeled LES are briefly discussed. With ever increasing computer power, as well as advances in numerical methods and subgrid-scale models, LES is rapidly becoming a viable tool for practical computations. In selecting a method, one should consider quantities to be predicted, accuracy of the predictions, and the computational cost.

Turbulent flow is an important branch of fluid mechanics with wide-ranging occurrences and applications, from the formation of tropical cyclones to the stirring of a cup of coffee. Turbulence results in increased skin friction and heat transfer across surfaces, as well as enhanced mixing. As such, it is of practical significance, and there is a need to establish predictive methods to quantify turbulent flows. Equally important is a physical understanding of turbulent flows to guide strategies to model and control turbulence-driven phenomena. We focus on the study of turbulent flows and draw on theoretical developments, experimental measurements, and results from numerical simulations. Turbulent flows are governed by the Navier-Stokes equations. The solution of these equations for turbulent flows displays chaotic and multiscale behavior. When averaged, the nonlinear terms in the Navier-Stokes equations lead to the so-called closure problem, where additional unknowns are introduced in the mean flow equations. These unknowns are typically modeled using intuition, experience, and dimensional arguments. We present the scaling and dimensional analysis necessary for model development.

Virtually all technologically relevant applications involve interactions of turbulent flows with solid walls, including flows over aircrafts and automobiles. We study these interactions using canonical wall-bounded flows, including fully developed channels, pipes, and flat-plate boundary layers, with a focus on channel flow. A common scaling may be employed in the near-wall region using the friction velocity and viscous length scale to derive the so-called wall units. In this region, which comprises the viscous sublayer, buffer layer, and overlap layer, the law of the wall governs the mean velocity profile, and the constant-stress-layer assumption is often employed. We discuss key features of the mean velocity profile, particularly the log law in the overlap region, which stands as a celebrated result in turbulence theory. Away from the wall, the outer layer scales with the boundary-layer thickness and freestream velocity. We discuss the skin friction and wake laws to describe the mean outer velocity profile and introduce the Clauser chart method. We also examine in more detail the scales and structural features of turbulence near a wall, including streaks and hairpin vortices.

In turbulent free-shear flows, fluid streams interact to generate regions of turbulence that evolve without being limited or confined by solid boundaries. Such interactions create mean shear, which is a source of turbulent kinetic energy that results in enhanced flow mixing. Far downstream, the flow retains little memory of its origins and exhibits self-similar behavior. Its mean velocity profile, turbulence intensities, and Reynolds stresses, when scaled appropriately, become independent of downstream distance as it freely expands into its surroundings. Free-shear flows occur in combustors, vehicle wakes, and jet engine exhaust. We focus our attention on three canonical categories of such flows: jets, wakes, and mixing layers. A detailed similarity analysis of the plane jet is provided alongside summarized results for the plane wake and mixing layer. We introduce examples involving turbines in wind farms and drag on wake-generating bodies. The notion of entrainment, which is central to the expansion of free-shear flows, is discussed. We also examine the scales and structural features of turbulent free-shear flows, including streamwise rib vortices and spanwise rollers.

In many practical applications, one is interested only in the average or expected value of flow quantities, such as aerodynamic forces and heat transfer. Governing equations for these mean flow quantities may be derived by averaging the Navier-Stokes and temperature or scalar transport equations. Reynolds averaging introduces additional unknowns owing to the nonlinearity of the equations, which is known as the closure problem in the turbulence literature. Turbulence models for the unclosed terms in the averaged equations are a way to manage the closure problem, for they close the equations with phenomenological models that relate the unknown terms to the solution variables. It is important that these models do not alter the conservation and invariance properties of the original equations of motion. We take a closer look at the equations of motion to understand these fundamental qualities in more depth. We describe averaging operators for canonical turbulent flows at the core of basic turbulence research and modeling efforts, and discuss homogeneity and stationarity. We also examine the Galilean invariance of the equations of motion and the role of vorticity in turbulence dynamics.