Fluid dynamics

See also: Transport phenomena

See also: applied to a control volume, is a statement that any change in momentum of the fluid within that control volume will be due to the net flow of momentum into the volume and the action of external forces acting on the fluid within the volume.

${\displaystyle {\frac {\partial }{\partial t}}\iiint _{\scriptstyle V}\rho \mathbf {u} \,dV=-\,{}}$ ${\displaystyle _{\scriptstyle S}}$ ${\displaystyle (\rho \mathbf {u} \cdot d\mathbf {S} )\mathbf {u} -{}}$ ${\displaystyle {\scriptstyle S}}$ ${\displaystyle {}\,p\,d\mathbf {S} }$ ${\displaystyle \displaystyle {}+\iiint _{\scriptstyle V}\rho \mathbf {f} _{\text{body}}\,dV+\mathbf {F} _{\text{surf}}}$

In the above integral formulation of this equation, the term on the left is the net change of momentum within the volume. The first term on the right is the net rate at which momentum is convected into the volume. The second term on the right is the force due to pressure on the volume's surfaces. The first two terms on the right are negated since momentum entering the system is accounted as positive, and the normal is opposite the direction of the velocity u and pressure forces. The third term on the right is the net acceleration of the mass within the volume due to any body forces (here represented by f_body). Surface forces, such as viscous forces, are represented by F_surf, the net force due to shear forces acting on the volume surface. The momentum balance can also be written for a moving control volume.^[3]

The following is the differential form of the momentum conservation equation. Here, the volume is reduced to an infinitesimally small point, and both surface and body forces are accounted for in one total force, F. For example, F may be expanded into an expression for the frictional and gravitational forces acting at a point in a flow.

${\displaystyle \ {\frac {D\mathbf {u} }{Dt}}=\mathbf {F} -{\frac {\nabla p}{\rho }}}$

In aerodynamics, air is assumed to be a Newtonian fluid, which posits a linear relationship between the shear stress (due to internal friction forces) and the rate of strain of the fluid. The equation above is a vector equation in a three-dimensional flow, but it can be expressed as three scalar equations in three coordinate directions. The conservation of momentum equations for the compressible, viscous flow case is called the Navier–Stokes equations.^[2]

Conservation of energy

Although energy can be converted from one form to another, the total energy in a closed system remains constant.

${\displaystyle \ \rho {\frac {Dh}{Dt}}={\frac {Dp}{Dt}}+\nabla \cdot \left(k\nabla T\right)+\Phi }$

Above, h is the specific

thermal conductivity of the fluid, T is temperature, and Φ is the viscous dissipation function. The viscous dissipation function governs the rate at which the mechanical energy of the flow is converted to heat. The second law of thermodynamics requires that the dissipation term is always positive: viscosity cannot create energy within the control volume.^[4] The expression on the left side is a material derivative

.

Classifications

Compressible versus incompressible flow

All fluids are compressible to an extent; that is, changes in pressure or temperature cause changes in density. However, in many situations the changes in pressure and temperature are sufficiently small that the changes in density are negligible. In this case the flow can be modelled as an incompressible flow. Otherwise the more general compressible flow equations must be used.

Mathematically, incompressibility is expressed by saying that the density ρ of a fluid parcel does not change as it moves in the flow field, that is,

${\displaystyle {\frac {\mathrm {D} \rho }{\mathrm {D} t}}=0\,,}$

where D/Dt is the

. 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

are compression waves involving changes in pressure and density of the medium through which they propagate.

Newtonian versus non-Newtonian fluids

All fluids, except

. The coefficient of proportionality is called the fluid's viscosity; for Newtonian fluids, it is a fluid property that is independent of the strain rate.

Inviscid versus viscous versus Stokes flow

The dynamic of fluid parcels is described with the help of

Newton's second law

. An accelerating parcel of fluid is subject to inertial effects.

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 (Re ≪ 1) 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 or creeping flow.

In contrast, high Reynolds numbers (Re ≫ 1) 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

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^[6] 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.

Steady versus unsteady flow

A flow that is not a function of time is called steady flow. Steady-state flow refers to the condition where the fluid properties at a point in the system do not change over time. Time dependent flow is known as unsteady (also called transient^[8]). Whether a particular flow is steady or unsteady, can depend on the chosen frame of reference. For instance, laminar flow over a sphere is steady in the frame of reference that is stationary with respect to the sphere. In a frame of reference that is stationary with respect to a background flow, the flow is unsteady.

Turbulent flows are unsteady by definition. A turbulent flow can, however, be statistically stationary. The random velocity field U(x, t) is statistically stationary if all statistics are invariant under a shift in time.^[9]: 75 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.

Laminar versus turbulent flow

Turbulence is flow characterized by recirculation,

component and a perturbation component.

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.^[10]

Most flows of interest have Reynolds numbers much too high for DNS to be a viable option,

(DES) — a combination of LES and RANS turbulence modelling.

Other approximations

There are a large number of other possible approximations to fluid dynamic problems. Some of the more commonly used are listed below.

• The Boussinesq approximation neglects variations in density except to calculate buoyancy forces. It is often used in free convection problems where density changes are small.
• Hele–Shaw flow exploits the large aspect ratio
  of the domain to show that certain terms in the equations are small and so can be neglected.
• Slender-body theory is a methodology used in Stokes flow problems to estimate the force on, or flow field around, a long slender object in a viscous fluid.
• The
  shallow-water equations can be used to describe a layer of relatively inviscid fluid with a free surface, in which surface gradients
  are small.
• Darcy's law is used for flow in porous media, and works with variables averaged over several pore-widths.
• In rotating systems, the
  atmospheric dynamics
  .

Multidisciplinary types

Flows according to Mach regimes

Main article: Mach number

While many flows (such as flow of water through a pipe) occur at low Mach numbers (subsonic flows), many flows of practical interest in aerodynamics or in turbomachines occur at high fractions of M = 1 (transonic flows) or in excess of it (supersonic or even hypersonic flows). 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

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

Main article: Magnetohydrodynamics

electrically conducting fluids in electromagnetic fields. Examples of such fluids include plasmas, liquid metals, and salt 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

Minkowski spacetime

.

Fluctuating hydrodynamics

This branch of fluid dynamics augments the standard hydrodynamic equations with stochastic fluxes that model thermal fluctuations.[12] As formulated by Landau and Lifshitz,^[13] 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.

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

.

Characteristic numbers

This section is an excerpt from Dimensionless numbers in fluid mechanics.[edit]

Dimensionless numbers (or characteristic numbers) have an important role in analyzing the behavior of fluids and their flow as well as in other transport phenomena.^[14] They include the Reynolds and the Mach numbers, which describe as ratios the relative magnitude of fluid and physical system characteristics, such as density, viscosity, speed of sound, and flow speed

.

To compare a real situation (e.g. an

ISO 80000-11

.

Terminology in incompressible fluid dynamics

The concepts of total pressure and

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.

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.

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.

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

isentropically

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".

See also

Main article: Outline of fluid dynamics

• List of publications in fluid dynamics
• List of fluid dynamicists

References

1. ISBN 3-527-40513-5
   .
2. ^
   ISBN 978-0-07-125408-3
   .
3. ^ Nangia, Nishant; Johansen, Hans; Patankar, Neelesh A.; Bhalla, Amneet Pal S. (2017). "A moving control volume approach to computing hydrodynamic forces and torques on immersed bodies". Journal of Computational Physics. 347: 437–462.
   S2CID 37560541
   .
4. ISBN 0-07-069710-8
   .
5. PMID 29271417
   .
6. ISSN 0044-2267
   .
7. ^ Shengtai Li, Hui Li "Parallel AMR Code for Compressible MHD or HD Equations" (Los Alamos National Laboratory) [1] Archived 2016-03-03 at the Wayback Machine
8. ^ "Transient state or unsteady state? -- CFD Online Discussion Forums". www.cfd-online.com.
9. ^
   ISBN 0-521-59886-9
   .
10. doi:10.1063/1.3139294
11. ISBN 0-08-033933-6
    .
12. ^ Ortiz de Zarate, Jose M.; Sengers, Jan V. (2006). Hydrodynamic Fluctuations in Fluids and Fluid Mixtures. Amsterdam: Elsevier.
13. ^ Landau, Lev Davidovich; Lifshitz, Evgenii Mikhailovich (1959). Fluid Mechanics. London: Pergamon.
14. ^ "ISO 80000-1:2009". International Organization for Standardization. Retrieved 2019-09-15.

Further reading

• Acheson, D. J. (1990). Elementary Fluid Dynamics. Clarendon Press.
  ISBN 0-19-859679-0
  .
• ISBN 0-521-66396-2
  .
• ISBN 978-0-415-49271-3
  .
• ISBN 0-273-01120-0
  .
• ISBN 0-521-45868-4
  . Originally published in 1879, the 6th extended edition appeared first in 1932.
• Milne-Thompson, L. M. (1968). Theoretical Hydrodynamics (5th ed.). Macmillan. Originally published in 1938.
• Shinbrot, M. (1973). Lectures on Fluid Mechanics. Gordon and Breach.
  ISBN 0-677-01710-3
  .
• Nazarenko, Sergey (2014), Fluid Dynamics via Examples and Solutions, CRC Press (Taylor & Francis group),
  ISBN 978-1-43-988882-7
• Encyclopedia: Fluid dynamics Scholarpedia

External links

Wikimedia Commons has media related to Fluid dynamics.

• National Committee for Fluid Mechanics Films (NCFMF), containing films on several subjects in fluid dynamics (in RealMedia format)
• Gallery of fluid motion, "a visual record of the aesthetic and science of contemporary fluid mechanics," from the American Physical Society
• List of Fluid Dynamics books