An Introduction to the Mechanics of Incompressible Fluids
- Open Access
- 2022
- Open Access
- Book
- Author
- Michel O. Deville
- Publisher
- Springer International Publishing
About this book
This open access book allows the reader to grasp the main bulk of fluid flow problems at a brisk pace. Starting with the basic concepts of conservation laws developed using continuum mechanics, the incompressibility of a fluid is explained and modeled, leading to the famous Navier-Stokes equation that governs the dynamics of fluids. Some exact solutions for transient and steady-state cases in Cartesian and axisymmetric coordinates are proposed. A particular set of examples is associated with creeping or Stokes flows, where viscosity is the dominant physical phenomenon. Irrotational flows are treated by introducing complex variables. The use of the conformal mapping and the Joukowski transformation allows the treatment of the flow around an airfoil. The boundary layer theory corrects the earlier approach with the Prandtl equations, their solution for the case of a flat plate, and the von Karman integral equation. The instability of fluid flows is studied for parallel flows using the Orr-Sommerfeld equation. The stability of a circular Couette flow is also described. The book ends with the modeling of turbulence by the Reynolds-averaged Navier-Stokes equations and large-eddy simulations. Each chapter includes useful practice problems and their solutions.
The book is useful for engineers, physicists, and scientists interested in the fascinating field of fluid mechanics.
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Table of Contents
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Chapter 1. Incompressible Newtonian Fluid Mechanics
- Open Access
Download PDF-versionThe chapter explores the properties and modeling of incompressible Newtonian viscous fluids, focusing on their behavior and the Navier–Stokes equations. It discusses the Reynolds number, flow patterns such as the circular Couette flow and flow around a cylinder, and provides insights into the complex dynamics of fluid flow. The chapter is designed to be a valuable resource for specialists in fluid mechanics, offering both theoretical foundations and practical applications.AI Generated
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AbstractThis chapter, presents the basic concepts of fluid mechanics such as velocity, acceleration, material derivative and the governing equations obtained from the conservation laws of mass, momentum, angular momentum and energy. The introduction of the constitutive relation for viscous incompressible Newtonian fluid leads to the Navier–Stokes equations. Boundary and initial conditions are discussed. Special attention is devoted to the meaning and differences between incompressible and compressible fluids. The Boussinesq equations are described. The chapter ends with considerations on the control volume method, a very efficient tool to solve fluid problems. -
Chapter 2. Dimensional Analysis
- Open Access
Download PDF-versionDimensional analysis is a fundamental tool in fluid mechanics, enabling the simplification of complex problems by identifying relevant dimensionless numbers. By applying the Vaschy–Buckingham theorem, one can organize variables into dimensionless groups, which helps in understanding and predicting fluid flow phenomena. The chapter highlights the importance of dimensional invariance and the use of dimensionless forms of equations, such as the Navier–Stokes equations, to distinguish dominant phenomena and simplify mathematical models. Practical applications, including dynamic similarity and self-similarity, are discussed to illustrate the power of dimensional analysis in both experimental and computational contexts.AI Generated
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AbstractDimensional analysis based on the principle of dimensional invariance allows the introduction of dimensionless numbers, like the famous Reynolds number, via the application of the Vaschy–Buckingham theorem. Dynamic similarity and self-similarity are described. The analysis of the dimensionless compressible Navier–Stokes equations shows how the incompressible equations are recovered when the Mach number goes to zero. The nature of pressure in the compressible and incompressible cases is broadly discussed. -
Chapter 3. Exact Solutions of the Navier–Stokes Equations
- Open Access
Download PDF-versionThis chapter delves into the exact solutions of the Navier–Stokes equations, focusing on viscous incompressible fluid flows. It begins with steady state plane flows, such as Couette and Poiseuille flows, and extends to more complex geometries like cylindrical flows and helical flows. The chapter also explores unsteady plane and axisymmetric problems, including periodic flows and various pipe flows. Notably, it introduces the concept of self-similar solutions and discusses the penetration depth of wall motion in semi-infinite spaces. Additionally, it covers the dynamics of flows on oscillating planes and the transient behavior of flows in circular pipes. The chapter concludes with a section on plane periodic solutions and provides exercises for further exploration.AI Generated
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AbstractThis chapter covers extensively various exact solutions of the Navier–Stokes equations for steady-state and transient cases. Of particular interest are the pulsating flows in a channel and in a circular pipe as these solutions are relevant for blood flow analysis. -
Chapter 4. Vorticity and Vortex Kinematics
- Open Access
Download PDF-versionThe chapter delves into the concept of vorticity, a crucial aspect of fluid dynamics, focusing on its role in viscous flows and its generation through various mechanisms such as baroclinic effects and boundary layers. It introduces the vorticity equation and explores its implications for both incompressible and compressible fluids. The text also discusses the vorticity number, a dimensionless invariant used to measure the strength of rotational flows, and provides practical examples like the Taylor–Green vortex and Kovasznay flow to illustrate complex vortex dynamics. Additionally, it covers the production of vorticity on solid walls and its significance in understanding wall shear stress. The chapter concludes with exercises that allow readers to apply the theoretical concepts to practical problems, making it a valuable resource for specialists in fluid mechanics.AI Generated
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AbstractThe concepts of vorticity and circulation are introduced. The three-dimensional governing equation for the vorticity is obtained. The generation of vorticity on the solid walls is analyzed in detail. For irrotational flows, the modeling rests upon Bernoulli equation. The flow behind a grid solved by Kovasznay provides a benchmark solution for numerical simulation. The chapter ends with the Taylor–Green vortex described by Clebsch potentials. -
Chapter 5. Stokes Flow
- Open Access
Download PDF-versionThe chapter delves into Stokes flow, a special case of fluid dynamics where viscous forces dominate over inertial forces. It begins by introducing the Stokes equation and its relevance in technologies like glass melting and biological systems such as the swimming of fish and propulsion of microorganisms. The text then explores the mathematical treatment of Stokes flow, including the derivation of harmonic pressure and vorticity functions. It provides detailed solutions for plane flows and flows in corners, highlighting the biharmonic nature of the stream function. Furthermore, the chapter discusses the occurrence of Moffatt eddies in corner flows, offering insights into the behavior of fluid in confined geometries. The text also covers Stokes eigenmodes and their applications in numerical methods for solving the Navier-Stokes equations. Additionally, it presents solutions for flow around a sphere and a cylinder, comparing the analytical results and discussing the limitations of the Stokes approximation. The chapter concludes with a three-dimensional solution for Stokes flow and suggests exercises for further exploration.AI Generated
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AbstractCreeping flows where the viscous effects are dominant, are considered. The Moffatt corner eddies are described. The flow around a sphere is detailed and leads to the Stokes formula. Stokes eigenmodes are analyzed and a three-dimensional Stokes solution is given. The flow around a circular cylinder leads to the Stokes’ paradox. -
Chapter 6. Plane Irrotational Flows of Perfect Fluid
- Open Access
Download PDF-versionThe chapter delves into the theory of steady-state two-dimensional irrotational flows of inviscid incompressible fluids, utilizing complex variables to introduce the velocity potential and stream function. It discusses the fundamental relationships between these functions and their applications in various flow patterns, such as parallel flows, vortices, and sources. The chapter also explores the use of conformal transformations to map complex flows onto simpler domains, highlighting the method's power in solving intricate fluid dynamics problems. Furthermore, it presents practical applications, such as flow around a circular cylinder and the Joukowski transformation, showcasing the versatility of complex potentials in fluid mechanics. The chapter concludes with exercises that challenge the reader to apply the learned concepts to real-world problems.AI Generated
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AbstractThis chapter treats the theory of irrotational flows of perfect fluids by the use of complex variables. The theory is based on a complex velocity and the related concepts like circulation, flow rate, complex potential. Several simple examples are given. More elaborated is the flow around a circular cylinder with and without circulation . Using conformal mapping and especially the Joukowski transformation, it is possible to consider an aerodynamics application, namely the flow around an airfoil. Blasius theorem allows for the computation of the forces and moment generated by the flow around an immersed body. It is applied to the case of the cylinder and Joukowski profile. -
Chapter 7. Boundary Layer
- Open Access
Download PDF-versionThe chapter delves into the theory of boundary layers, essential for understanding high Reynolds number flows in fluid dynamics. It begins by addressing the limitations of the perfect fluid theory and introduces Ludwig Prandtl's boundary layer theory as a necessary correction. The chapter then explores the mathematical formulation of the boundary layer equations, including dimensional analysis and the derivation of Prandtl's equations. Practical applications are discussed, such as the development of the boundary layer on a flat plate and the calculation of boundary layer thicknesses, friction coefficients, and drag coefficients. The chapter also introduces the von Kármán integral equation and the von Kármán-Pohlhausen method for approximating velocity profiles. Throughout, the chapter combines theoretical rigor with practical examples, making it an invaluable resource for specialists in fluid dynamics.AI Generated
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AbstractThe Prandtl’s equations for laminar boundary layer are obtained via dimensional analysis. The case of the flat plate is treated as a suitable example for the development of the boundary layer on a simple geometry. Various thicknesses are introduced. The integration of Prandtl’s equation across the boundary layer produces the von Kármán integral equation which allows the elaboration of the approximate von Kármán-Pohlhausen method where the velocity profile is given as a polynomial. The use of a third degree polynomial for the flat plate demonstrates the feasibility of the approach. -
Chapter 8. Instability
- Open Access
Download PDF-versionThe chapter discusses the transition between laminar and turbulent flow, driven by instability mechanisms that vary depending on the geometrical configuration. It focuses on two types of transitions: spectral transition, exemplified by the circular Couette flow, and catastrophic transition, such as in developed Poiseuille flow. The Orr-Sommerfeld equation is central to the analysis, offering insights into the stability of flows through eigenvalue problems. The chapter also explores the stability of the circular Couette flow, including Rayleigh's criterion and the marginal stability curve, and touches on non-linear axisymmetric Taylor vortices. Additionally, it includes exercises that challenge readers to apply the principles discussed to other flow scenarios, such as the Rayleigh-Bénard instability.AI Generated
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AbstractThe first instability mechanism is applied to the plane parallel channel flow. We establish the well known Orr-Sommerfeld equation which is solved by the Chebyshev Tau method. The associated Fortran programme is given in the appendix. Then the stability of the circular Couette flow between two concentric cylinders is undertaken. The inviscid approach yields the Rayleigh stability criterion. The incorporation of the viscous and pressure terms generates through a linearization process a set of differential equations again solved by high-order discretization methods through a generalized eigenvalue problem. The chapter ends with the case of the non-linear axisymmetric Taylor vortices. -
Chapter 9. Turbulence
- Open Access
Download PDF-versionThis chapter offers an in-depth exploration of turbulence, a complex phenomenon prevalent in various fluid flow scenarios. It begins with a historical overview, tracing the development of turbulence modeling from the 19th century to contemporary times, emphasizing the role of computational methods. The chapter delves into the general characteristics of turbulent flows, such as unsteadiness, diffusivity, and dissipative nature, and explains the challenges in defining turbulence. It introduces the Navier-Stokes equations and the Reynolds decomposition, essential for understanding turbulent flow dynamics. The chapter also discusses the Reynolds-Averaged Navier-Stokes (RANS) equations, turbulent kinetic energy, and the dynamic equation of the Reynolds tensor. Furthermore, it explores the structures and scales of homogeneous turbulence, including the energy cascade and Kolmogorov scales. The chapter concludes with a discussion on linear turbulence models and practical applications, such as the logarithmic velocity profile in turbulent boundary layers. This comprehensive analysis makes the chapter a valuable resource for researchers and professionals in fluid dynamics.AI Generated
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AbstractThe Reynolds decomposition and statistical averaging of velocity and pressure generate the Reynolds averaged Navier–Stokes (RANS) equations. The closure problem is solved by the introduction of a turbulence constitutive equation. Several linear turbulence models are presented in the RANS framework: \(K-\varepsilon , K-\omega \). The solution of the RANS equations for the turbulent channel flow is elaborated giving the celebrated logarithmic profile. Non-linear models are built on the anisotropy tensor and the incorporation of the concept of integrity bases. The chapter ends with the theory of large eddy simulations with a few up-to-date models: dynamic model, approximate deconvolution method. -
Chapter 10. Solutions of Exercises
- Open Access
Download PDF-versionThe chapter provides detailed solutions to exercises in fluid dynamics, covering topics such as contact force analysis, natural convection between parallel planes, and the dynamics of bubbles. It explores the application of integral equations and the Navier-Stokes equations to solve problems like the force exerted by a plate on a fluid and the velocity profile in natural convection. Additionally, it delves into the dynamics of spherical bubbles, offering a comprehensive analysis of the forces and pressure distributions involved. The chapter also presents solutions for various flow problems, including plane Couette and Poiseuille flows, and spherical Couette flow, highlighting the use of symmetry and boundary conditions to derive velocity profiles and pressure distributions. The solutions are presented in a step-by-step manner, making the chapter an invaluable resource for students and professionals seeking to deepen their understanding of fluid dynamics.AI Generated
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AbstractChapter ten collects the detailed solutions of the exercises proposed at the end of each chapter.
- Title
- An Introduction to the Mechanics of Incompressible Fluids
- Author
-
Michel O. Deville
- Copyright Year
- 2022
- Publisher
- Springer International Publishing
- Electronic ISBN
- 978-3-031-04683-4
- Print ISBN
- 978-3-031-04682-7
- DOI
- https://doi.org/10.1007/978-3-031-04683-4
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