(Redirected from Fluid flow)
'Fluid dynamics' is the sub-discipline of
fluid mechanics dealing with
fluids (
liquids and
gases) in motion. It has several subdisciplines itself, including
aerodynamics (the study of gases in motion) and
hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating
forces and
moments on
aircraft, determining the
mass flow rate of
petroleum through pipelines, predicting
weather patterns, understanding
nebulae in
interstellar space and reportedly modelling fission weapon detonation. Some of its principles are even used in
traffic engineering, where traffic is treated as a continuous fluid.
Fluid dynamics offers a systematic structure that underlies these practical disciplines and that embraces empirical and semi-empirical laws, derived from
flow measurement, used to solve practical problems. The solution of a fluid dynamics problem typically involves calculation of various properties of the fluid, such as
velocity,
pressure,
density, and
temperature, as functions of space and time.
Equations of fluid dynamics and aerodynamics
The foundational axioms of fluid dynamics are the
conservation laws, specifically,
conservation of mass,
conservation of momentum (also known as
Newton's second law and third law), and
conservation of energy. 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 infinitely 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 are
non-linear differential equations that describe the flow of a fluid whose stress depends linearly on velocity and on pressure. The unsimplified equations do not have a general
closed-form solution, so they are only of use in
computational fluid dynamics or when they can be simplified. 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
is
pressure,
is
density,
is the
gas constant,
is the
molecular mass and
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 modelled 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, i.e.
:
where
is the
convective derivative. 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 problem 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).
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 solution.
The
Reynolds number can be used to evaluate whether viscous or inviscid equations are appropriate to the problem.
Stokes flow is flow at very low Reynolds numbers, 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 at all, compared to inertial terms.
This idea can work fairly well when the Reynolds number is high, even if certain problems, such as those involving boundaries, may require that viscosity be included. Viscosity often cannot be neglected near boundaries because the
no-slip condition can generate a region of large strain rate (a
Boundary layer) which enhances the effect of even a small amount of
viscosity, generating
vorticity.
Therefore, to calculate net forces on bodies (such as wings) we should use viscous equations. As illustrated by
d'Alembert's paradox, a body in an inviscid fluid will experience no 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 far from the body and the
boundary layer equations, which incorporate viscosity, 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 field.
Steady vs unsteady flow
When all time derivatives of a flow field vanish, the flow is considered steady. 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 with respect to the background flow, the flow is unsteady.
Turbulent flows are unsteady by definition. A turbulent flow can, however, be statistically stationary. According to Pope, Turbulent Flows, Cambridge university press, page 75:
The random field U(x,t) 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 less (time) 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 vs turbulent flow
Turbulence is flow dominated 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 does not necessarily indicate turbulent flow--these phenomena may be present in laminar flow as well. Mathematically, turbulent flow is often represented via
Reynolds decomposition, in which the flow is broken down into the sum of a steady component and a perturbation component.
It is believed that turbulent flows obey the
Navier-Stokes equations.
Direct numerical simulation (DNS), based on the incompressible Navier-Stokes equations, makes it possible to simulate turbulent flows with moderate Reynolds numbers (restrictions depend on the power of computer and efficiency of solution algorithm). The results of DNS agree with the experimental data.
Most flows of interest have Reynolds numbers too high for DNS to be a viable option (see: Pope), 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 Navier-Stokes equations combined with
turbulence modeling provides a model of the effects of the turbulent flow, mainly the additional momentum transfer provided by the
Reynolds stresses, although the turbulence also enhances the heat and mass transfer.
Large eddy simulation also holds promise as a simulation methodology, especially in the guise of
detached eddy simulation (DES), which is a combination of
turbulence modeling and large eddy simulation.
Newtonian vs non-Newtonian fluids
Sir
Isaac Newton showed how
stress and the rate of change 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.
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' 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 use 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.
References
★ Batchelor, G.K. (1967) "An Introduction to Fluid Dynamics" (Cambridge University Press).
★ Shinbrot, Marvin (1973) "Lectures on Fluid Mechanics" (Gordon and Breach).
★ Landau, L.D. and Lifshitz, E.M. (1987) "Fluid Mechanics" (Pergamon Press).
★ Acheson, D.J. (1990) "Elementary Fluid Dynamics" (Clarendon Press).
★ Pope, S.B. (2000) "Turbulent Flows" (Cambridge University Press)
See also
Other fundamental engineering topics
★
Analysis of resistive circuits
★
Dynamics
★
★
Thermodynamics
★
Engineering economics
★
Heat transfer
★
Materials science
★
★
Strength of materials
★
Statics
External links
★
Fluid Mechanics @ Chemical Engineering Information Exchange
★
Geophysical and Astrophysical Fluid Dynamics
★
SOLSI - An example of Enterprise realising Fluid Dynamics studies
★
Plasma Pong - A video game that uses real-time Fluid Dynamics to drive the environment
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