Hydrodynamics

• Type of physical science: Hydrodynamics, Fluid dynamics, Liquids, Classical physics
• Field of study: Fluids

Hydrodynamics is the branch of physics that describes the movement of idealized fluids. The mathematical techniques used for describing ideal fluid motion have been used successfully to describe not only the dynamics of many real fluids but also various forms of transport and dynamics in other fields.

Overview

Hydrodynamics is the study of the motion of an ideal fluid flowing on or within a confining boundary. To simplify the description of movement and its analysis, the existence of an ideal fluid—an idealized material that has zero viscosity and that is incompressible—is assumed. In many cases, the dynamics of such a material are similar to the observed motions of water and other fluids.

Early researchers sometimes used analogies comparing fluid motion to solid objects. An ideal fluid flows in response to gravity from a region of high gravitational potential to a region of lower potential. This is similar to the motion of a solid object dropped from a height. In fact, it was observed that the velocity of a rock dropped from a specific height was the same as the velocity of water escaping through a hole in the bottom of an open container holding water to the same height as the rock was dropped; this observation was codified in Torricelli’s law. An ideal fluid flowing from an opening could theoretically rise to the original water height if energy losses are neglected. The fact that real water, for example, does not reach that height is a consequence of the viscosity of water.

Another observation involved water flowing in a horizontal pipe with a varying diameter. From the principle of conservation of mass, it is known that the volumetric rate (cubic feet per second) at which water flows through any cross section of the pipe, regardless of diameter, is the same. The volumetric rate of flow is given by multiplying the cross-sectional area of the pipe by the average fluid velocity over the cross-section. This product is constant for any cross section. The velocity of flow is then indirectly proportional to the cross-sectional area of the pipe. In a horizontal pipe, fluid flows faster through a small diameter than through a larger one. Leonardo da Vinci had earlier studied similar flow dynamics in streams, though the formal continuity equation was developed later.

A third finding that had a significant effect on the development of hydrodynamics was that when small vertical open tubes were attached to a horizontal pipe that was conveying water, the height at which the water stood in the tubes was observed to be related to the diameter of the pipe. There was also a slight decrease in the height of water in the tubes over a length of pipe having a uniform diameter. It was proposed that this effect was caused by the water’s internal friction, or viscosity. For regions of the pipe with a smaller diameter, the water stood lower in the tube. The height the water stood in the tube was interpreted as proportional to the pressure in the pipe at the point where the tube connected to the pipe. The pressure at the bottom of the tube equaled the height of the fluid in the tube multiplied by the weight density of the fluid.

By experiment, for water flowing in a horizontal pipe, vertical tubes attached to a smaller-diameter region of pipe showed lower pressure than in a larger-diameter region under steady flow conditions. This was interpreted to mean that water flowing relatively faster through a smaller-diameter section of pipe has less pressure associated with it than when it is flowing slower through a larger-diameter region of pipe. This led Daniel Bernoulli to propose that the pressure of a fluid increases as its velocity decreases and, decreases as its velocity increases. This understanding was extended to form Bernoulli’s theorem, which states that the sum of the kinetic energy, potential energy, and pressure energy for a given volume element of fluid is constant for all locations along the element’s path. This relationship is used extensively in hydrodynamic analysis; with modifications for viscosity, the relationship can be used to understand the movement of any real fluid.

The objective of most hydrodynamics analysis is to determine the fluid “flow field,” given certain boundary conditions, sources of fluid, sinks, and other specifics. Such a field can be visualized by thinking of the fluid as being composed of numerous small volume elements. The motion of each small volume element is represented by a small arrow, the length of which is proportional to the flow velocity and the direction of which is the same as the fluid’s motion. The velocity can be determined by measuring the distance traveled by a small volume element in a specific time interval. The assemblage of all these motion arrows at a specific point in time gives an indication of the flow field. The average length of the arrows in a small region gives the average velocity of flow, and the average direction of flow is the average direction of all the arrows in the region. In steady hydrodynamics analyses, the flow field is assumed to be “stationary,” or unchanging.

When an ideal fluid is moving uniformly, the motions of individual small volume-elements define lines through space that are called “streamlines.” A three-dimensional collection of streamlines can be selected that defines a bundle similar to the surface of a pipe. Such a collection of streamlines is known as a “stream-tube.” The tube can have a varying diameter along its length. There are no volume elements that move into or out of the side of a stream-tube; they enter and exit only at the ends of the tube. For steady incompressible flow, the rate of flow of fluid through a stream-tube is constant. The cross-section of the stream-tube is an area. The volume-per-unit-time rate of flow of fluid through the stream-tube equals the average velocity of the fluid crossing the cross-sectional area of the tube times the area of the tube. Constant flow rate within the tube requires that the average flow velocity at any cross-section of the tube times the cross-sectional area be a constant. This means that when the cross-sectional area of the tube gets larger, the average flow velocity must become lower. Under Bernoulli conditions, the fluid pressure increases as the cross section gets larger.

The two-dimensional equivalents of stream-tubes are filaments. In performing a graphical analysis, the potential lines are first constructed; these are perpendicular to any fluid boundaries. The streamlines are then constructed perpendicular to the potential lines. The fluid is assumed to flow from a higher potential to a lower one. The collection of streamlines and perpendicular potential lines defines a two-dimensional “flow net.”

In cases in which a fluid supply or loss is caused by a drain or some other point of exit, the flow net can be adjusted to show these situations. For a source—such as the end of a water hose—supplying fluid onto a smooth area, the idealized streamlines are radial out from the source location. The potential lines are concentric circles about the source. In the case of a sink, such as by a drain, the streamlines are directed radially inward to the sink location. The potential lines are, again, concentric circles. For impermeable boundaries, the streamlines are parallel to the boundary, with the potential lines perpendicular to the boundary.

While the two-dimensional flow net is often used to predict the flow of water on a two-dimensional surface, the equivalent three-dimensional flow net is useful for three-dimensional analyses. Numerous graphical and analytical techniques have been developed that utilize flow nets to analyze ideal fluid-flow problems.

Treating water as an ideal fluid has several logical consequences. One of these is the assumption that any rotation present in the fluid at the start of observation will continue and will not die out or increase. For example, if an ideal fluid is imagined to be circulating in a coffee cup, according to the ideal-fluid model, the circulation would not stop over time. The analysis of real fluids, however, must account for the effects of viscosity, which would cause such rotational motion to die out if it were not renewed by an outside force.

A major difference between ideal water and natural water, therefore, is that the ideal liquid is considered to have no internal viscosity and to experience no friction between itself and the edges of a pipe or a canal. For ideal water, the velocity of the fluid is considered to be uniform across the cross-section of a pipe or a stream. For real water, however, there is friction that causes the water-flow velocity near the bounding structure to be very low (zero at the surface) and to increase to a maximum near the center of the cross section of the flow.

The inclusion of the effects of viscosity causes the real-world equations used to describe fluid motion to be considerably more complicated than those that describe the motion of an ideal fluid. Solutions to real-world equations are often achieved using numerical integration techniques on computers. Because of this complexity, and because ideal-fluid analysis often yields good approximations of fluid movement, ideal fluids are often assumed in order to obtain initial predictions for fluid movement in a given situation.

A common starting point in many fluid analyses is to suppose that the fluid being studied obeys Newton’s law of viscosity. For water and air in typical environmental conditions, this is often not too far wrong, however there can be considerable deviation from Newton’s law of viscosity in describing the dynamics of other fluids. These fluids are called “non-Newtonian” fluids.

The mathematical and conceptual techniques used to quantify fluid flow developed shortly after Isaac Newton proposed his laws and after the introduction and early development of calculus. Calculus was well suited to describe some of the processes and effects observed in flowing water. Similarities between the flow of water, of heat, and of electromagnetic radiation caused the development of many analytical tools in these fields to be intertwined.

Applications

Fluid motion is responsible for most of the transport and mixing that takes place in the environment, in industrial processes, in living organisms, and in vehicles. Knowledge of fluid dynamics is responsible for the increased efficiency of energy recovery from fuels used in powering aircraft, ships, and automobiles. Fluid mechanics has been instrumental in the design of the shape of these structures and vehicles to reduce drag and help them move more effectively through the air or water. Fluid dynamics describes the transport of pollutants in the air, surface, and subsurface, as well as the transport of liquids on the surface and subsurface of the Earth.

The Bernoulli equation, with a small correction for viscous losses, is used to design the extensive pipe networks employed in industrial facilities, water distribution in cities, and in irrigation. Successful analysis and prediction of the flow of water in open channels and on fields has as its basis a sound foundation in fluid dynamics. The siting and operation of power plants and other large facilities require considerable flow analysis. Determination of the potential effects of a facility on regional air and water quality is required before construction can begin. These “environmental impact assessments” also examine groundwater and its location, motion, and sensitivity to pollution from surface spills and discharges from a facility. Analyses of the mixing of discharges from a facility in streams and lakes, as well as emissions into the airshed, require an understanding of fluid mechanics. After the fact, fluid analyses are critical in designing effective remediation and cleanup operations to stop further pollution.

The concepts and processes of hydrodynamics are also used in describing atmospheric processes and in attempting to predict the weather. The ocean plays a major role in these processes, with a slight change in temperature sometimes having far-reaching ramifications, as in the case of the Pacific Ocean phenomenon known as “El Niño.” The ideas of fluid mechanics are used with those of heat transport in explaining the coupled ocean-atmosphere fluid system that controls the weather. The fluid mechanics of these processes share many common features and are often referred to as “geophysical fluid dynamics.” The general ideas and concepts of fluid dynamics are also used to explain phenomena in outer space. Fluid mechanics is used in describing and analyzing the stars, the giant gaseous planets, and in other astrophysical problems. Back on Earth, fluid mechanics is used to analyze the shape and drift of the continents, volcanic activity, and the generation of the Earth’s magnetic field.

Improvements in the combustion of fossil fuels have reduced fuel consumption and pollutant generation. Fluid-mechanics research has contributed to this effort with large economic benefit. The technique of imparting a swirl to the air in jet-engine combustors, for example, improved the engines’ fuel economy substantially. This technology migrated into the design of new high-efficiency home oil burners. Many other innovative engine designs have also been based on an understanding of fluid mechanics.

Small-scale physical models using water have often been used to predict how flow occurs in other areas of physics, such as electromagnetics. Wave movement in water is often as an analogy used to represent the movement of light waves in optics. The effects of a smooth wave moving through double slits are readily apparent. This technique provides a convenient means to observe interference and diffraction by waves in one medium when analogous effects occur in another.

Fluid mechanics is also used in biology for many important applications. These range from a better understanding of normal biological processes to the development of therapeutic medical procedures. Among these are heart and heart-valve function and the design and evaluation of artificial replacements for these organs. Other medical areas in which an understanding of hydrodynamics is important include the measurement of cardiovascular flow and the measurement and modification of the characteristics of pulmonary flow. Additional work occurs in the microscopic realm at the level of cells, micropores, and microorganisms. In many of these instances, the fluids studied are non-Newtonian.

Context

While water-conveyance and water-storage structures have been used for centuries, it was not until the sixteenth century that mathematical techniques were developed to describe fluid flow quantitatively and to assist in the design of such structures. Leonardo da Vinci conducted studies in hydraulics and fluid motion in the fifteenth and early sixteenth century, while Evangelista Torricelli in the seventeenth century proposed the law governing the velocity of a fluid exiting from a tank drain.

Daniel Bernoulli introduced the term “hydrodynamics” in 1738 to cover the sciences of hydrostatics and hydraulics. He laid the foundation for the theorem that bears his name, while he and his father Johann shared in the development of the concept of fluid pressure. Jean d’Alembert contributed to potential flow theory; later developments in complex variables were formalized by others. Leonhard Euler integrated much of the previous work in hydrodynamics, formulated the equation of motion of an ideal fluid, and developed its mathematical theory.

In the nineteenth century, Sir George Stokes developed the equations of motion of a viscous fluid; he is sometimes regarded as having founded the modern theory of hydrodynamics. Stokes was followed by William Rankine, who developed the theory of sources and sinks, and Hermann von Helmholtz, who introduced the mathematical analysis of streamlines, founded the theories of vortex motion and discontinuous motion, and made fundamental contributions to hydrodynamics. In 1879, Horace Lamb published Hydrodynamics, which has remained a major work of reference on the subject.

At the beginning of the twentieth century, various developments caused the unification of the fields of hydrodynamics and hydraulics into the science of fluid mechanics. Osborne Reynolds helped this process by promoting the use of models and the theoretical description of the flow of viscous fluids. He also studied and described the abrupt transition from laminar to turbulent flow. Ludwig Prandtl provided a means of linking hydraulics and hydrodynamics through the use of boundary-layer concepts. Theodore von Kármán furthered the unification of hydrodynamics and hydraulics by his work on vortex sheets, drag, and momentum.

The field of hydrodynamics has developed slowly. This has resulted not from a lack of research funding or an absence of military or commercial interest but rather from the extraordinary difficulty of the subject itself. Progress in the field has been difficult and seems likely to remain so. Many unsolved problems remain, and the ability to predict many flows is limited. Research shows that “active fluids,” such as bacteria-driven flows, can generate forces and motion through internal energy, extending classical fluid theory.

Throughout the development of hydrodynamics, many of the concepts, mathematical techniques, and forms of analysis have been used in other fields of physics, science, and engineering. Thus, despite its inherent difficulty, hydrodynamics continues to be a rich and fruitful topic of investigation. Hydrodynamics increasingly uses machine learning methods, such as physics-informed neural networks, to model complex fluid flows. Fluid mechanics is widely applied in climate modeling, biomedical flows (such as blood and microfluidics), and renewable energy systems like wind and ocean flows.

Principal terms

BERNOULLI EQUATION: A conservation-of-energy equation, used extensively in hydrodynamics analyses, that states that along the path of a small volume-element of ideal fluid, the sum of the kinetic, potential, and pressure energy terms is a constant

CONSERVATION OF ENERGY: In hydrodynamics, the principle that for an ideal fluid, the total energy associated with a small volume-element of fluid remains constant throughout its motion

CONSERVATION OF MASS: In hydrodynamics, the principle that the mass of ideal fluid entering a small body element equals the mass of fluid leaving the element

IDEAL FLUID: A fluid assumed to have zero viscosity and to be incompressible

POTENTIAL LINES: Lines of constant velocity potential in a flow field; a fluid element moves from a high potential line to a lower one

PRESSURE: The force per unit area exerted by a fluid on a container; total pressure is the sum of a static pressure, and a dynamic pressure, caused by motion of the fluid

SINK: The location where fluid is removed from a fluid system

SOURCE: The location where fluid is applied to a fluid system

STREAMLINE: A line tangent to the velocity of a fluid element at every point; streamlines are perpendicular to equipotential lines in potential flow

VISCOSITY: The internal resistance to motion of a fluid caused by intermolecular forces

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