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This chapter presents different ways of formulating the incompressible Navier-Stokes equations based on primitive variables, that is, velocity and pressure, as well as velocity-vorticity algorithms. It considers both coupled, splitting, and least-squares formulations for primitive variables. Both the Uzawa coupled algorithm and a new substructured solver are discussed. The discussion on primitive variables time-splitting includes recent theoretical advances in the pressure-correction and velocity correction schemes as well as the rotational formulation of the pressure boundary condition. The final section is devoted to nonlinear terms; it includes a discussion of spatial and temporal discretization with focus on the semi-Lagrangian method for the incompressible Navier-Stokes equations.

This chapter discusses compressible Euler and Navier-Stokes equations as well as general hyperbolic conservation laws. The principle issue is how effectively to use the high-order expansions of the spectral/hp method whilst honouring the inherent monotonicity and conservation properties of the analytic system. Different ways of dealing with these fundamental issues for both the Euler and the Navier-Stokes equations are considered. A new section for the shallow water equations is also included and the section on the discontinuous Galerkin method has been rewritten. Finally, the last section discusses modeling of plasma flows, i.e., the so-called magneto-hydrodynamic (MHD) equations.

This chapter discusses numerical simulations of the incompressible Navier-Stokes equations. Exact Navier-Stokes solutions are presented that are used as benchmarks to validate new codes and evaluate the accuracy of a particular discretization. Some aspects of direct numerical simulation (DNS) and large-eddy simulation (LES) are discussed. The issue of stabilization at high Reynolds number is then presented using the concepts of dynamic subgrid modelling, over-integration, and spectral vanishing viscosity. A new parallel paradigm based on multi-level parallelism is introduced that can help realize adaptive refinement more easily. The final section includes a heuristic refinement method for Navier-Stokes equations.

This chapter presents reduced models of the compressible and incompressible Navier-Stokes equations which are used in the various discretization concepts discussed in the rest of the book. The convergence philosophy of spectral and finite element methods, the combination of which provides a dual path of convergence, is also introduced. Topics covered include the basic equations of fluid dynamics and numerical discretizations.

This chapter focuses on the modelling of one-fluid magnetohydrodynamics problems. The crucial point under consideration is the coupling between hydrodynamics phenomena and electromagnetic phenomena. From a mathematical viewpoint, the coupling induces a nonlinearity, additional to the nonlinearities already present in the hydrodynamics. A series of difficult, thus interesting, problems follow. With a reasonable amount of theoretical efforts, these problems can be dealt with. For instance, it can be shown that a system coupling the time-dependent incompressible Navier-Stokes equations with a simplified form of the Maxwell equations (the so-called low-frequency approximation) is well-posed when the electromagnetic equation is taken time-dependent, in parabolic form. In contrast, the same model is likely to be ill-posed when the electromagnetic equation is taken time-independent, in elliptic form.

This chapter presents how MHD equations can be derived from the general conservation equations for fluid mechanics coupled with the Maxwell equations modelling the electromagnetic phenomena. A hierarchy of models is considered, from the most general one (a full time-dependent system consisting in the incompressible Navier-Stokes equations with a Lorentz body force calculated from the Maxwell equations and the Ohm's law) to the most simplified one. Depending on the physical context, one model or the other is appropriate. The most sophisticated model raises unsolved questions of existence and uniqueness, mainly related with the hyperbolic nature of the Maxwell equations, but some simpler models can be fully analysed.

Continuous media are here seen from the traditional viewpoint of balance equations; the chapter introduces the concepts of mesoscopic particles, fields and fluxes. The conservation laws of the first two chapters are revisited in this framework. The Cauchy stress tensor is then considered and estimated from thermodynamical considerations for a viscous fluid, with emphasis on incompressible (or weakly compressible) flows, leading to the Navier–Stokes and Euler equations of motion. also talks about similarity, about the particular role of pressure for incompressible flows and describe surface tension. Lastly, the Navier–Stokes equations are revisited from the Boltzmann equation introduced in Chapter 2, and the author gives a personal viewpoint of variational principles for continuous fluid mechanics.

This chapter describes the governing equations considered throughout the book. The equations of fluid dynamics are recalled, together with the k-epsilon turbulence model, which is used later on for high Reynolds number flows when the topology of the answer is not known. The fundamental equations of fluid dynamics are recalled; this is because applied OSD for fluids requires a good understanding of the state equation: Euler and Navier–Stokes equations in this case, with and without turbulence models together with the inviscid and/or incompressible limits. The chapter recalls wall-laws also used for OSD as low complexity models. By wall-laws domain decomposition with a reduced dimension model near the wall is understood. In other words, there is no universal wall-laws and when using a wall-function, it needs to be compatible with the model used far from the wall. Large eddy simulation is giving a new life to the wall-functions especially to simulate high-Reynolds external flows.

Chapter 2 discusses the experimental observations of the boundary-layer separation in subsonic and supersonic flows that lead to a formulation of the concept of viscous-inviscid interaction. It then turns to the so-called ‘self-induced separation’ of the boundary layer in supersonic flows. This theory is formulated based on the asymptotic analysis of the Navier–Stokes equations at large values of the Reynolds number. As a part of the flow analysis, this chapter also introduces the ‘triple-deck model’. It then shows how this model may be used to describe the classical problem of the boundary-layer separation in an incompressible fluid flow past a circular cylinder.

This chapter reviews the Navier-Stokes equations for a gaseous system in vibrational and/or chemical non-equilibrium, and proposes various forms. The equations are written in a dimensionless form so that dimensionless numbers are defined, characterizing the relative influence of the physical processes involved in the equations. Typical particular flows are presented such as one and two-dimensional flows, isentropic, and dissipative flows. Problems related to gas-solid and gas-liquid interfaces, flow stability, entropy and global balances, dimensional analysis, and magnetohydrodynamics are also briefly discussed.

This chapter expands the distribution function of a pure gas in a series of a small parameter representing the ratio of the collisional time to a reference flow time; this is the so-called Chapman-Enskog method. Zero order and first order solutions are discussed, corresponding to isolated and non-isolated media respectively. The zero order solutions are related to the Euler equations and the first order to the Navier-Stokes equations. Transport terms such as viscosity and conductivity coefficients, included in these equations are determined for pure gases in equilibrium at zero order (WNE solution) and approximate expressions are also proposed. More details are given in the appendices on collisional integrals, orthogonal bases used in the expansions, equation systems, collisional models, and relaxation equations.

This chapter starts with a discussion of the continuum hypothesis and the conditions under which it can be used. The origin of the internal forces acting in moving fluids is then identified and the notion of the surface stress and the stress tensor are introduced, as is the rate-of-strain tensor. The constitutive equation, which relates the surface stress to the deformational motion of a fluid, is then derived. Newton’s Second Law and the law of conservation of energy are applied to deduce the equations governing fluid motion. These are known as the Navier–Stokes equations. They are derived for both incompressible fluid flows and compressible flows of a perfect gas. The chapter concludes with a demonstration of how the Navier–Stokes equations can be expressed in curvilinear coordinates.

Chapter 7 presents with V. Dolejší presents discontinuous Galerkin methods (DCGMs): violated boundary conditions and continuity of the piecewise polynomials are compensated by additional penalty terms in the discrete weak form. This chapter restricts them to linear and quasilinear equations and systems of order 2. hp-variants of DCGMs and numerical experience with the steady compressible Navier-Stokes equations are added. The proof of stability, nearly identical to Chapter 4 based upon the anticrime transformation, is omitted. The relations between the strong and weak forms yield consistency.

Ions are used in normal liquid helium to investigate the hydrodynamics in presence of electrostriction. The Navier-Stokes equation must be modified in order to account for the spatial variations of density and viscosity around the ion because of the local pressure increase due to electrostriction. The solutions of the modified hydrodynamic equations are compared to experiments in normal liquid ⁴He. The issue matters if the liquid actually freezes around the ion. The analogies with the transport of the O₂ ^- ion in dense Neon gas are described.

Chapter 1 demonstrates, for the simple mechanical example of a bent rod, the change in character from linear to nonlinear regimes. This is followed by several examples for different types of nonlinear elliptic differential equations in mathematics, science and in engineering, e.g. the Monge‐Ampère, the reaction‐diffusion, the von Kármán and the Navièr‐Stokes equations. These problems require appropriate analytical results for (space‐) discretization methods. The necessary tools from functional analysis and calculus in Banach spaces are summarized.

George Gabriel Stokes made fundamental mathematical contributions to fluid dynamics that had profound practical consequences. The basic equations formulated by him play a central role in numerical weather prediction, in the simulation of blood flow in the body and in countless other important applications. In this chapter the primary focus is on the two most important areas of Stokes’s work on fluid dynamics, the derivation of the Navier–Stokes equations and the theory of finite amplitude oscillatory water waves.

The scientific legacy of George Gabriel Stokes is considered. Certain aspects of Stokes’s research work are reviewed and related to more recent fields of research. These include the Navier–Stokes equations and other approaches to rational continuum mechanics, the issue of existence of solutions, the boundary no-slip condition; Stokes flow and the issue of pendulum drag; the Hele-Shaw cell, viscous fingering, wavelength selection in pattern formation; moving contact lines; the highest water wave, rogue waves, the NLS equation; Stokes lines, exponential asymptotics, dendrite growth, slow manifods, and diffraction.

At T=2.17 K and P=0, liquid helium undergoes the superfluid transition. Dose this transition belong to the same universality class of the usual liquid-vapor transition? This question has been addressed in experiments in which the ion mobility has been studied. It appears that the mobility is not singular at the transition though its slope appears to be infinite. The explanation for this result is given by solving the Navier-Stokes equations for a granular fluid composed by interwoven islands of normal fluid and superfluid. Investigations of the ion mobility at the melting transition have put into evidence the existence of an electrostriction-induced, superfluid transition in the liquid surrounding the positive ions.