Computational Fluid Dynamics

Finite-difference methods are numerical methods that find solutions to differential equations using approximate spatial and temporal derivatives that are based on discrete values at spatial grid points and discrete time levels. As the grid spacing and time step are made small, the error due to finite differencing becomes small with correct implementation. In this chapter, we present the fundamentals of the finite-difference discretization using the advection-diffusion equation, which is a simple model for the Navier–Stokes equations.

For compressible and incompressible flows, there is a difference in how the numerical solution techniques are formulated, based on whether or not the mass conservation equation includes a time-derivative term. Fluid motion is described by the conser4 vation equations for mass, momentum, and energy. For incompressible flow, the 5 conservation equation for kinetic energy can be derived from the momentum conser6 vation equation. Hence, we only need to be concerned with the mass and momentum 7 conservation equations. Furthermore, if the temperature field is not a variable of 8 interest, we do not need to consider the internal energy in the formulation. We do 9 note that the treatment of momentum conservation should be consistent with the 10 conservation of kinetic energy in a discrete manner, as it influences the achievement 11 of reliable solution and numerical stability.

For simulating flows around bodies, it is possible to use the Cartesian grid formulation from Chap. 3. However, the Cartesian grid representation of a body with complex surface geometry can be a challenge. The body can be approximated in a staircase (pixelated, rasterized) manner as shown in Fig. 4.1 (top) but one can imagine that the numerical solution can be affected greatly from such staircase representation, especially for high-Reynolds number flows in which the boundary layer can be very thin. If we need to accurately represent the boundary geometry, the location of the boundary and its orientation with respect to the cells need to be captured with very fine mesh size if a Cartesian grid is used. This approach however inefficiently refines the grid for the sole purpose of representing the smooth body surface accurately on a grid that is not aligned with the boundary geometry. In recent years, adaptive mesh refinement [1] and immersed boundary methods [5, 8, 10] (see Chap. 5) have been developed to utilize Cartesian grids for flows over bodies with complex geometries. The use of Cartesian grid methods remain a challenge for accurately simulating turbulent flows that require highly refined grid near the wall boundary.

Analysis of flow over bodies with complex surface geometry can pose a challenge in terms of spatial discretization. Creating a high quality boundary fitted mesh can not only be difficult but also time consuming especially when there are complex flow structures over intricate boundary geometry that need to be resolved in the simulations. This is especially true for bodies encountered in engineering applications such as fluid flow around an automotive engine and undercarriage as well as aircraft landing gears. Furthermore, if we have problems involving fluid-structure interaction, the location of the moving or deforming interface needs to be determined numerically. In such cases, the need to re-mesh the flow field around the body at every time step can introduce an added computational burden. Similar situation arises when one attempts to simulate particle-laden flows in which the interface between different phases needs to be tracked and resolved accurately.

Numerical simulations of turbulent flows can be performed to capture (1) the temporal fluctuations and (2) the time-averaged features in the flow field. For example, let us consider a simulated flow through an asymmetric diffuser. The three-dimensional instantaneous flow field obtained from numerically solving the Navier–Stokes equations [36] is shown in Fig. 6.1. The computation captures flow separation and the existence of the recirculation zone in the diffuser. The streamwise velocity profiles from the computation are compared to experimental measurements in Fig. 6.2.

The constitutive equations used in the Reynolds-averaged Navier–Stokes (RANS) equations are referred to as turbulence models. Although a large number of studies have been performed on the development of turbulence models, there has not been a universal turbulence model that is applicable to all turbulent flows. However, we in general suggest the use of the k-\(\varepsilon \) model for “simple” flows and the Large-Eddy Simulation (LES) for more complex flows (Chap. 8) found in many practical engineering applications. In this book, we refer to “simple flows” as stationary flows that have average streamlines that are relatively straight on the absolute coordinate system with low level of acceleration. Flows that impinge on walls, separate from corners, pass through a curved channel, and is in a rotational field would not be considered simple. In this and the next chapters, discussions on the strengths and limitations of these methods are offered. Since our objective is not to introduce all turbulence models available, we ask readers to refer to [3, 6, 28] for comprehensive reviews on RANS.

Let us briefly reexamine the spatial scales present in turbulent flows before discussing how large-eddy simulation can be formulated. Consider the turbulent energy spectra for various turbulent flows across a wide range of Reynolds numbers. Shown in Fig. 8.1 are the energy spectra for various three-dimensional turbulent flows non-dimensionalized by the Kolmogorov scale [4]. Note that the vortices represented with low wave numbers are dependent on the problem. On the other hand, the small vortices represented by the high wave numbers exhibit a universal behavior (independent of the flow field). This is due to the isotropic nature of turbulence near the Kolmogorov scale. Based upon this observation, we can consider modeling the small-scale vortices that possess universality in their behavior and directly resolving the large scale vortices that are influenced by the setup of the flow field. Note however that it would not be appropriate to simply simulate only the large-scale vortices on a coarse grid with the Navier–Stokes equations, because there are interactions amongst vortices over wide range of scale due to the nonlinearity in turbulent flows. Large-eddy simulation (LES) resolves the large-scale vortices in the turbulent flow field and incorporates the influence of small-scale vortices that are not resolved by the grid through a model. This approach taken by LES has been shown to be successful in simulating a variety of complex turbulent flows.