Viscous fluid mechanics

Author: Zhang Zixiong et al.
Publisher:
Publish Date: 1999-08-01
Features: The book systematically presents the fundamental theories of incompressible viscous fluid mechanics, primarily exemplified by water. The entire book is divided into twelve chapters. The first five chapters cover the basic theories and equations of viscous fluid mechanics. Chapters 6 to 8 discuss the fundamental theories and equations of turbulence. Chapters 9 to 12 describe various typical turbulent flows: jets, wakes, turbulent flow in circular pipes, turbulent flat plate boundary layers, and open channel turbulent flow. In the appendix, "Knowledge of Field Theory and Basic Tensor Operations" is provided. This book can serve as a textbook or reference for graduate students in engineering fields such as water resources, hydropower, civil engineering, environmental science, oceanography, port engineering, coastal engineering, shipbuilding, mechanical engineering, and other disciplines that focus on fluids, particularly liquids. It is also useful for hydraulic teachers to enhance their theoretical understanding and gain a deeper insight into the fundamental aspects of modern fluid mechanics. Additionally, it can be used as a reference for researchers, educators, and engineers engaged in scientific research, teaching, and engineering work in related fields.
Excerpt: The further development of theoretical fluid mechanics began in 1821 when Navier (Claude-Louis-Marie-Henri Navier, 1785–1836) and others considered incorporating intermolecular forces into the Euler equations. In 1845, Stokes (George Gabriel Stokes, 1819–1903) represented these intermolecular forces using the viscosity coefficient μ and formally completed the Navier-Stokes equations, ultimately establishing the fundamental equations of viscous fluid mechanics and laying the foundation for modern viscous fluid mechanics. However, due to the nonlinearity of the equations, solving them posed significant mathematical challenges. As a result, until the late 19th century, theoretical and experimental fluid mechanics developed independently. In the early 20th century, German engineer Ludwig Prandtl (1875–1953) made outstanding contributions to fluid mechanics, particularly viscous fluid mechanics, by proposing the Boundary Layer Theory. Prandtl suggested that under high Reynolds numbers, the effect of viscosity is primarily confined to a very thin layer of flow near the solid wall of objects or other flow boundaries, known as the boundary layer. The flow outside the boundary layer can be treated as ideal flow. This concept overcame the major mathematical difficulties in solving viscous flow problems and fundamentally addressed significant issues such as flow resistance and energy loss. The introduction of the boundary layer theory unified theory and experiment, gradually integrating the two branches of fluid mechanics—ideal fluid mechanics and hydraulics—thereby advancing fluid mechanics in an epoch-making way. Among various engineering fields, aeronautical engineering was the first to apply boundary layer theory and achieve significant technical breakthroughs. Subsequently, shipbuilding, chemical engineering, and mechanical engineering also benefited from this theory. In recent years, boundary layer theory has begun to be applied to solve flow problems in hydraulic, hydropower, environmental, and civil engineering. The rapid development of modern high-performance computers has accelerated the growth of computational fluid dynamics (CFD), making it an essential tool for solving viscous flow problems. Advances in high technology have also led to remarkable improvements in experimental techniques and measurement instruments in fluid mechanics research. Lasers, ultrasonic technology, electronic technology, and image acquisition and processing techniques have become widely used. These developments have provided powerful means for humans to further observe and explore flow phenomena, especially for refined, mechanistic studies. The integration of theoretical, computational, and experimental methods is fostering new breakthroughs in fluid mechanics. Major global challenges today, such as water scarcity, environmental protection, disaster prevention and mitigation, and marine development, are all closely related to fluid mechanics. It is believed that in addressing these critical issues, viscous fluid mechanics will also experience rapid new developments.
1-1-2 Examples of Viscous Flow
To illustrate the differences between viscous flow and ideal flow and to fully recognize the complexity of viscous flow, the flow around a two-dimensional cylinder is first studied. The flow of uniform flow over a two-dimensional cylinder with radius \( r_0 \) is a classic problem in fluid mechanics. For incompressible ideal fluids, the exact solution for cylinder flow is the superposition of uniform flow and a dipole. The velocity distribution is expressed in right-handed cylindrical coordinates \((r, \theta, z)\), where \( U_\infty \) is the velocity of the undisturbed inflow at infinity, \( u_r \) is the radial velocity, and \( u_\theta \) is the circumferential velocity, with counterclockwise direction being positive. The streamlines of this flow are shown in Figure 1-1.