Fluid dynamics
Equations of fluid dynamics
The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum (also known as Newton’s Second Law of Motion), and conservation of energy (also known as First Law of Thermodynamics). These are based on classical mechanics and are modified in quantum mechanics and general relativity. They are expressed using the Reynolds Transport Theorem.
In addition to the above, fluids are assumed to obey the continuum assumption. Fluids are composed of molecules that collide with one another and solid objects. However, the continuum assumption considers fluids to be continuous, rather than discrete. Consequently, properties such as density, pressure, temperature, and velocity are taken to be well-defined at infinitesimally small points, and are assumed to vary continuously from one point to another. The fact that the fluid is made up of discrete molecules is ignored.
For fluids which are sufficiently dense to be a continuum, do not contain ionized species, and have velocities small in relation to the speed of light, the momentum equations for Newtonian fluids 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 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 a number of ways, all of which make them easier to solve. Some of them allow appropriate fluid dynamics problems to be solved in closed form.
In addition to the mass, momentum, and energy conservation equations, a thermodynamical equation of state giving the pressure as a function of other thermodynamic variables for the fluid is required to completely specify the problem. An example of this would be the perfect gas equation of state:
where p is pressure, is density, Ru is the gas constant, M is the molar mass and T is temperature.
Compressible vs incompressible flow
All fluids are compressible to some extent, that is changes in pressure or temperature will result in 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 modeled as an incompressible flow. Otherwise the more general compressible flow equations must be used.
Mathematically, incompressibility is expressed by saying that the density of a fluid parcel does not change as it moves in the flow field, i.e.,
where D / Dt is the substantial derivative, which is the sum of local 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 to be 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). Acoustic problems always require allowing compressibility, since sound waves are compression waves involving changes in pressure and density of the medium through which they propagate.
Viscous vs inviscid flow
Viscous problems are those in which fluid friction has significant effects on the fluid motion.
The Reynolds number, which is a ratio between inertial and viscous forces, can be used to evaluate whether viscous or inviscid equations are appropriate to the problem.
Stokes flow is flow at very low Reynolds numbers, Re 1, such that inertial forces can be neglected compared to viscous forces.
On the contrary, high Reynolds numbers indicate that the inertial forces are more significant than the viscous (friction) forces. Therefore, we may assume the flow to be an inviscid flow, an approximation in which we neglect viscosity completely, compared to inertial terms.
This idea can work fairly well when the Reynolds number is high. However, certain problems such as those involving solid boundaries, may require that the viscosity be included. Viscosity often cannot be neglected near solid boundaries because the no-slip condition can generate a thin region of large strain rate (known as Boundary layer) which enhances the effect of even a small amount of viscosity, and thus generating vorticity. Therefore, to calculate net forces on bodies (such as wings) we should use viscous flow equations. As illustrated by d’Alembert’s paradox, a body in an inviscid fluid will experience no drag force. The standard equations of inviscid flow are the Euler equations. Another often used model, especially in computational fluid dynamics, is to use the Euler equations away from the body and the boundary layer equations, which incorporates viscosity, in a region close to the body.
The Euler equations can be integrated along a streamline to get Bernoulli’s equation. When the flow is everywhere irrotational and inviscid, Bernoulli’s equation can be used throughout the flow field. Such flows are called potential flows.
Steady vs unsteady flow
Hydrodynamics simulation of the Rayleighaylor instability
When all the time derivatives of a flow field vanish, the flow is considered to be a steady flow. Otherwise, it is called unsteady. 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 than the governing equations of the same problem without taking advantage of the steadiness of the flow field.
Although strictly unsteady flows, time-periodic problems can often be solved by the same techniques as steady flows. For this reason, they can be considered to be somewhere between steady and unsteady.
Laminar versus turbulent flow
Turbulence is flow characterized by recirculation, eddies, and apparent randomness. Flow in which turbulence is not exhibited is called laminar. It should be noted, however, that the presence of eddies or recirculation alone does not necessarily indicate turbulent flowhese phenomena may be present in laminar flow as well. Mathematically, turbulent flow is often represented via a Reynolds decomposition, in which the flow is broken down into the sum of an average component and a perturbation component.
It is believed that turbulent flows can be described well through the use of the Naviertokes equations. Direct numerical simulation (DNS), based on the Naviertokes 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 agree with the experimental data.
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 (L > 3 m), moving faster than 72 km/h (20 m/s) is well beyond the limit of DNS simulation (Re = 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). In order to solve these real-life flow problems, turbulence models will be a necessity for the foreseeable future. Reynolds-averaged Naviertokes equations (RANS) combined with turbulence modeling provides a model of the effects of the turbulent flow. Such a modeling mainly provides the additional momentum transfer by the Reynolds stresses, although the turbulence also enhances the heat and mass transfer. Another promising methodology is large eddy simulation (LES), especially in the guise of detached eddy simulation (DES)hich is a combination of RANS turbulence modeling and large eddy simulation.
Newtonian vs non-Newtonian fluids
Sir Isaac Newton showed how stress and the rate of strain are very close to linearly related for many familiar fluids, such as water and air. These Newtonian fluids are modeled by a coefficient called viscosity, which depends on the specific fluid.
However, some of the other materials, such as emulsions and slurries and some visco-elastic materials (eg. blood, some polymers), have more complicated non-Newtonian stress-strain behaviours. These materials include sticky liquids such as latex, honey, and lubricants which are studied in the sub-discipline of rheology.
Subsonic vs transonic, supersonic and hypersonic flows
While many terrestrial flows (e.g. flow of water through a pipe) occur at low mach numbers, many flows of practical interest (e.g. in aerodynamics) occur at high fractions of the Mach Number M=1 or in excess of it (supersonic flows). New phenomena occur at these Mach number regimes (e.g. shock waves for supersonic flow, transonic instability in a regime of flows with M nearly equal to 1, non-equilibrium chemical behavior due to ionization in hypersonic flows) and it is necessary to treat each of these flow regimes separately.
Non-relativistic vs relativistic flows
Classical fluid dynamics is derived based on Newtonian mechanics, which is adequate for most applications. However, at speeds comparable to the speed of light, c, Newtonian mechanics is inaccurate and a relativistic framework has to be used instead.
Magnetohydrodynamics
Main article: Magnetohydrodynamics
Magnetohydrodynamics is the multi-disciplinary study of the flow of 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.
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.
Lubrication theory and 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.
The Boussinesq equations are applicable to surface waves on thicker layers of fluid and with steeper surface slopes.
Darcy’s law is used for flow in porous media, and works with variables averaged over several pore-widths.
In rotating systems, the quasi-geostrophic approximation assumes an almost perfect balance between pressure gradients and the Coriolis force. It is useful in the study of atmospheric dynamics.
Terminology in fluid dynamics
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.
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 sensehey 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.
In Aerodynamics, L.J. Clancy writes: To distinguish it from the total and dynamic pressures, the actual pressure of the fluid, which is associated not with its motion but with its state, is often referred to as the static pressure, but where the term pressure alone is used it refers to this static pressure.
A point in a fluid flow where the flow has come to rest (i.e. 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 stagnation point. The static pressure at the stagnation point is of special significance and is given its own nametagnation 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, such as air, the temperature and density are essential when determining the state of the fluid. In addition to the concept of total pressure (also known as stagnation pressure), the concepts of total (or stagnation) temperature and total (or stagnation) density are also essential in any study of compressible fluid flows. To avoid potential ambiguity when referring to temperature and density, many authors use the terms static temperature and static density. Static temperature is identical to temperature; and static density is identical to density; and both can be identified for every point in a fluid flow field.
The temperature and density at a stagnation point are called stagnation temperature and stagnation density.
A similar approach is also taken with the thermodynamic properties of compressible fluids. Many authors use the terms total (or stagnation) enthalpy and total (or stagnation) entropy. The terms static enthalpy and static entropy appear to be less common, but where they are used they mean nothing more than enthalpy and entropy respectively, and the prefix “static” is being used to avoid ambiguity with their ‘total’ or ‘stagnation’ counterparts.
See also
Fields of study
Acoustic theory
Aerodynamics
Aeroelasticity
Aeronautics
Computational fluid dynamics
Flow measurement
Hemodynamics
Hydraulics
Hydrology
Hydrostatics
Electrohydrodynamics
Magnetohydrodynamics
Rheology
Quantum hydrodynamics
Mathematical equations and concepts
Airy wave theory
Bernoulli’s equation
Reynolds transport theorem
Benjaminonaahony equation
Boussinesq approximation (buoyancy)
Boussinesq approximation (water waves)
Conservation laws
Euler equations (fluid dynamics)
Darcy’s law
Dynamic pressure
Fluid statics
Helmholtz’s theorems
Kirchhoff equations
Manning equation
Mild-slope equation
Morison equation
Naviertokes equations
Oseen flow
Pascal’s law
Poiseuille’s law
Potential flow
Pressure
Static pressure
Pressure head
Relativistic Euler equations
Reynolds decomposition
Stokes flow
Stokes stream function
Stream function
Streamlines, streaklines and pathlines
Types of fluid flow
Cavitation
Compressible flow
Couette flow
Free molecular flow
Incompressible flow
Inviscid flow
Isothermal flow
Laminar flow
Open channel flow
Secondary flow
Superfluidity
Supersonic
Transient flow
Transonic
Turbulent flow
Two-phase flow
Fluid properties
Density
List of hydrodynamic instabilities
Newtonian fluid
Non-Newtonian fluid
Surface tension
Viscosity
Vapor pressure
Compressibility
Fluid phenomena
Boundary layer
Coanda effect
Convection cell
Convergence/Bifurcation
Drag (force)
Hydrodynamic stability
Lift (force)
Ocean surface waves
Rossby wave
Shock wave
Soliton
Stokes drift
Turbulence
Venturi effect
Vortex
Vorticity
Water hammer
Wave drag
Applications
Acoustics
Aerodynamics
Cryosphere science
Fluid power
Hydraulic machinery
Meteorology
Naval architecture
Oceanography
Plasma physics
Pneumatics
Pump
Slosh dynamics
Miscellaneous
Important publications in fluid dynamics
Isosurface
Keuleganarpenter number
Rotating tank
Sound barrier
Beta plane
Immersed boundary method
Bridge scour
References
Acheson, D. J. (1990). Elementary Fluid Dynamics. Clarendon Press. ISBN 0198596790.
Batchelor, G. K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press. ISBN 0521663962.
Chanson, H. (2009). Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows. CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages. ISBN 978-0-415-49271-3.
Clancy, L. J. (1975). Aerodynamics. London: Pitman Publishing Limited. ISBN 0273011200.
Lamb, Horace (1994). Hydrodynamics (6th ed.). Cambridge University Press. ISBN 0521458684. Originally published in 1879, the 6th extended edition appeared first in 1932.
Landau, L. D.; Lifschitz, E. M. (1987). Fluid Mechanics. Course of Theoretical Physics (2nd ed.). Pergamon Press. ISBN 0750627670.
Milne-Thompson, L. M. (1968). Theoretical Hydrodynamics (5th ed.). Macmillan. Originally published in 1938.
Pope, Stephen B. (2000). Turbulent Flows. Cambridge University Press. ISBN 0521598869.
Shinbrot, M. (1973). Lectures on Fluid Mechanics. Gordon and Breach. ISBN 0677017103.
Notes
^ Eckert, Michael (2006), The Dawn of Fluid Dynamics: A Discipline Between Science and Technology, Wiley, p. ix, ISBN 3527405135
^ Shengtai Li, Hui Li “Parallel AMR Code for Compressible MHD or HD Equations” (Los Alamos National Laboratory)
^ See Pope (2000), page 344.
^ Clancy, L.J. Aerodynamics, page 21
External links
Wikimedia Commons has media related to: Fluid dynamics
Wikimedia Commons has media related to: Fluid mechanics
eFluids, containing several galleries of fluid motion
National Committee for Fluid Mechanics Films (NCFMF), containing films on several subjects in fluid dynamics (in realmedia format)
Fluid Mechanics @ Chemical Engineering Information Exchange
Geophysical and Astrophysical Fluid Dynamics
List of Fluid Dynamics books
Fluid Mechanics, A short course for physicists
COMSOL Multiphysics – Fluid Dynamics page
Oofelie – Flow Simulation
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Dimensionless numbers in fluid dynamics
Archimedes Atwood Bagnold Bejan Biot Bond Brinkman Capillary Cauchy Damkhler Dean Deborah Eckert Ekman Etvs Euler Froude Galilei Graetz Grashof rtler Hagen Keuleganarpenter Knudsen Laplace Lewis Mach Marangoni Morton Nusselt Ohnesorge Pclet Prandtl (magnetic turbulent) Rayleigh Reynolds (magnetic) Richardson Roshko Rossby Rouse Ruark Schmidt Sherwood Stanton Stokes Strouhal Suratman Taylor Ursell Weber Weissenberg Womersley
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