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In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of – and . It has several subdisciplines, including (the study of air and other gases in motion) and (the study of water and other liquids in motion). Fluid dynamics has a wide range of applications, including calculating and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding in interstellar space, understanding large scale geophysical flows involving oceans/atmosphere and modelling fission weapon detonation.
Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time.
Before the twentieth century, " hydrodynamics" was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, both of which can also be applied to gases.
In addition to the above, fluids are assumed to obey the continuum assumption. At small scale, all fluids are composed of molecules that collide with one another and solid objects. However, the continuum assumption assumes that fluids are continuous, rather than discrete. Consequently, it is assumed that properties such as density, pressure, temperature, and flow velocity are well-defined at small points in space and vary continuously from one point to another. The fact that the fluid is made up of discrete molecules is ignored.
For fluids that are sufficiently dense to be a continuum, do not contain ionized species, and have flow velocities that are small in relation to the speed of light, the momentum equations for are the Navier–Stokes equations—which is a non-linear set of differential equations that describes the flow of a fluid whose stress depends linearly on flow velocity gradients and pressure. The unsimplified equations do not have a general closed-form solution, so they are primarily of use in computational fluid dynamics. The equations can be simplified in several ways, all of which make them easier to solve. Some of the simplifications allow some simple fluid dynamics problems to be solved in closed form.
In addition to the mass, momentum, and energy conservation equations, a thermodynamics equation of state that gives the pressure as a function of other thermodynamic variables is required to completely describe the problem. An example of this would be the perfect gas equation of state:
where is pressure, is density, and is the absolute temperature, while is the gas constant and is molar mass for a particular gas. A constitutive relation may also be useful.
{D t} = \mathbf{F} - \frac{\nabla p}{\rho}
Mathematically, incompressibility is expressed by saying that the density of a fluid parcel does not change as it moves in the flow field, that is, where is the material derivative, which is the sum of time derivative and convective derivatives. This additional constraint simplifies the governing equations, especially in the case when the fluid has a uniform density.
For flow of gases, to determine whether to use compressible or incompressible fluid dynamics, the Mach number of the flow is evaluated. As a rough guide, compressible effects can be ignored at Mach numbers below approximately 0.3. For liquids, whether the incompressible assumption is valid depends on the fluid properties (specifically the critical pressure and temperature of the fluid) and the flow conditions (how close to the critical pressure the actual flow pressure becomes). acoustics problems always require allowing compressibility, since sound waves are compression waves involving changes in pressure and density of the medium through which they propagate.
Non-Newtonian fluids have a more complicated, non-linear stress-strain behaviour. The sub-discipline of rheology describes the stress-strain behaviours of such fluids, which include and slurries, some viscoelasticity materials such as blood and some , and sticky liquids such as latex, honey and lubricants.
The Reynolds number is a dimensionless quantity which characterises the magnitude of inertial effects compared to the magnitude of viscous effects. A low Reynolds number () indicates that viscous forces are very strong compared to inertial forces. In such cases, inertial forces are sometimes neglected; this flow regime is called Stokes flow.
In contrast, high Reynolds numbers () indicate that the inertial effects have more effect on the velocity field than the viscous (friction) effects. In high Reynolds number flows, the flow is often modeled as an inviscid flow, an approximation in which viscosity is completely neglected. Eliminating viscosity allows the Navier–Stokes equations to be simplified into the Euler equations. The integration of the Euler equations along a streamline in an inviscid flow yields Bernoulli's equation. When, in addition to being inviscid, the flow is Lamellar field everywhere, Bernoulli's equation can completely describe the flow everywhere. Such flows are called , because the velocity field may be expressed as the gradient of a potential energy expression.
This idea can work fairly well when the Reynolds number is high. However, problems such as those involving solid boundaries may require that the viscosity be included. Viscosity cannot be neglected near solid boundaries because the no-slip condition generates a thin region of large strain rate, the boundary layer, in which viscosity effects dominate and which thus generates vorticity. Therefore, to calculate net forces on bodies (such as wings), viscous flow equations must be used: inviscid flow theory fails to predict drag forces, a limitation known as the d'Alembert's paradox.
A commonly used model, especially in computational fluid dynamics, is to use two flow models: the Euler equations away from the body, and boundary layer equations in a region close to the body. The two solutions can then be matched with each other, using the method of matched asymptotic expansions.
Turbulence flows are unsteady by definition. A turbulent flow can, however, be statistically stationary. The random velocity field is statistically stationary if all statistics are invariant under a shift in time. This roughly means that all statistical properties are constant in time. Often, the mean field is the object of interest, and this is constant too in a statistically stationary flow.
Steady flows are often more tractable than otherwise similar unsteady flows. The governing equations of a steady problem have one dimension fewer (time) than the governing equations of the same problem without taking advantage of the steadiness of the flow field.
It is believed that turbulent flows can be described well through the use of the Navier–Stokes equations. Direct numerical simulation (DNS), based on the Navier–Stokes equations, makes it possible to simulate turbulent flows at moderate Reynolds numbers. Restrictions depend on the power of the computer used and the efficiency of the solution algorithm. The results of DNS have been found to agree well with experimental data for some flows.See, for example, Schlatter et al, Phys. Fluids 21, 051702 (2009);
Most flows of interest have Reynolds numbers much too high for DNS to be a viable option, given the state of computational power for the next few decades. Any flight vehicle large enough to carry a human ( > 3 m), moving faster than is well beyond the limit of DNS simulation ( = 4 million). Transport aircraft wings (such as on an Airbus A300 or Boeing 747) have Reynolds numbers of 40 million (based on the wing chord dimension). Solving these real-life flow problems requires turbulence models for the foreseeable future. Reynolds-averaged Navier–Stokes equations (RANS) combined with turbulence modelling provides a model of the effects of the turbulent flow. Such a modelling mainly provides the additional momentum transfer by the Reynolds stresses, although the turbulence also enhances the heat transfer and mass transfer. Another promising methodology is large eddy simulation (LES), especially in the form of detached eddy simulation (DES) — a combination of LES and RANS turbulence modelling.
Some of the terminology that is necessary in the study of fluid dynamics is not found in other similar areas of study. In particular, some of the terminology used in fluid dynamics is not used in fluid statics.
A point in a fluid flow where the flow has come to rest (that is to say, speed is equal to zero adjacent to some solid body immersed in the fluid flow) is of special significance. It is of such importance that it is given a special name—a stagnation point. The static pressure at the stagnation point is of special significance and is given its own name—stagnation pressure. In incompressible flows, the stagnation pressure at a stagnation point is equal to the total pressure throughout the flow field.
To avoid potential ambiguity when referring to the properties of the fluid associated with the state of the fluid rather than its motion, the prefix "static" is commonly used (such as static temperature and static enthalpy). Where there is no prefix, the fluid property is the static condition (so "density" and "static density" mean the same thing). The static conditions are independent of the frame of reference.
Because the total flow conditions are defined by bringing the fluid to rest, there is no need to distinguish between total entropy and static entropy as they are always equal by definition. As such, entropy is most commonly referred to as simply "entropy".
Other approximations
There are a large number of other possible approximations to fluid dynamic problems. Some of the more commonly used are listed below.
Multidisciplinary types
Flows according to Mach regimes
While many flows (such as flow of water through a pipe) occur at low (subsonic flows), many flows of practical interest in aerodynamics or in Turbomachinery occur at high fractions of (Transonic) or in excess of it (Supersonic speed or even Hypersonic speed). New phenomena occur at these regimes such as instabilities in transonic flow, shock waves for supersonic flow, or non-equilibrium chemical behaviour due to ionization in hypersonic flows. In practice, each of those flow regimes is treated separately.
Reactive versus non-reactive flows
Reactive flows are flows that are chemically reactive, which finds its applications in many areas, including combustion (IC engine), propulsion devices (rockets, jet engines, and so on), detonations, fire and safety hazards, and astrophysics. In addition to conservation of mass, momentum and energy, conservation of individual species (for example, mass fraction of methane in methane combustion) need to be derived, where the production/depletion rate of any species are obtained by simultaneously solving the equations of chemical kinetics.
Magnetohydrodynamics
Magnetohydrodynamics is the multidisciplinary study of the flow of electrically conducting fluids in Electromagnetism fields. Examples of such fluids include plasmas, liquid metals, and Saline water. The fluid flow equations are solved simultaneously with Maxwell's equations of electromagnetism.
Relativistic fluid dynamics
Relativistic fluid dynamics studies the macroscopic and microscopic fluid motion at large velocities comparable to the velocity of light. (1987). 9780080339337, Pergamon. ISBN 9780080339337 This branch of fluid dynamics accounts for the relativistic effects both from the special theory of relativity and the general theory of relativity. The governing equations are derived in Riemannian geometry for Minkowski spacetime.
Fluctuating hydrodynamics
This branch of fluid dynamics augments the standard hydrodynamic equations with stochastic fluxes that model
thermal fluctuations.
As formulated by Lev Landau and Evgeny Lifshitz,
a white noise contribution obtained from the fluctuation-dissipation theorem of statistical mechanics
is added to the viscous stress tensor and heat flux.
Terminology
The concept of pressure is central to the study of both fluid statics and fluid dynamics. A pressure can be identified for every point in a body of fluid, regardless of whether the fluid is in motion or not. Pressure can be measured using an aneroid, Bourdon tube, mercury column, or various other methods.
Characteristic numbers
Terminology in incompressible fluid dynamics
The concepts of total pressure and dynamic pressure arise from Bernoulli's equation and are significant in the study of all fluid flows. (These two pressures are not pressures in the usual sense—they cannot be measured using an aneroid, Bourdon tube or mercury column.) To avoid potential ambiguity when referring to pressure in fluid dynamics, many authors use the term static pressure to distinguish it from total pressure and dynamic pressure. Static pressure is identical to pressure and can be identified for every point in a fluid flow field.
Terminology in compressible fluid dynamics
In a compressible fluid, it is convenient to define the total conditions (also called stagnation conditions) for all thermodynamic state properties (such as total temperature, total enthalpy, total speed of sound). These total flow conditions are a function of the fluid velocity and have different values in frames of reference with different motion.
Applications
See also
Further reading
H. Chanson (2025). 9780415492713, CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages. ISBN 9780415492713
External links
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