Fluid dynamics

Continuum mechanics

Conservation of mass Conservation of momentum Navier–Stokes equations Tensors Solid mechanics Solids · Stress · Deformation · Finite strain theory · Infinitesimal strain theory · Elasticity · Linear elasticity · Plasticity · Viscoelasticity · Hooke's law · Rheology Fluid mechanics Fluids · Fluid statics Fluid dynamics · Viscosity · Newtonian fluids Non-Newtonian fluids Surface tension Scientists Newton · Stokes · Navier · Cauchy· Hooke

typical aerodynamic teardrop shape, showing the pressure distribution as the thickness of the black line and showing the velocity in the boundary layer as the violet triangles. The green vortex generators prompt the transition to turbulent flow and prevent back-flow also called flow separation from the high pressure region in the back. The surface in front is as smooth as possible or even employ shark like skin, as any turbulence here will reduce the energy of the airflow. The Kammback also prevents back flow from the high pressure region in the back across the spoilers to the convergent part. Putting stuff inside out results in tubes, they also face the problem of flow separation in their divergent parts, so called diffuser (automotive)s. Cutting the shape into halfs results in an aerofoil with the low pressure region on top leading to lift (force)

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:

$p={\frac {\rho R_{u}T}{M}}$

where $p$ is pressure, $\rho$ is density, $R_{u}$ is the gas constant, $M$ is the molecular 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 modelled as an incompressible flow. Otherwise the more general compressible flow equations must be used.

Mathematically, incompressibility is expressed by saying that the density $\rho$ of a fluid parcel does not change as it moves in the flow, i.e.

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

where $\mathrm {D} /\mathrm {D} t$ 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

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• 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

Fields of study Acoustic theory Aerodynamics Aeroelasticity Aeronautics Computational fluid dynamics Flow measurement Hemodynamics Hydraulics Hydrology Hydrostatics Hydrodynamics Electrohydrodynamics Magnetohydrodynamics Rheology Quantum hydrodynamics Mathematical equations and concepts Bernoulli's equation Reynolds transport theorem Boussinesq approximation Conservation laws Euler equations Darcy's law Helmholtz's theorems Kirchhoff equations Manning equation Navier-Stokes equations Poiseuille's law Relativistic Euler equations Reynolds decomposition Stream function Streamlines, streaklines and pathlines

Types of fluid flow Compressible flow Couette flow Free molecular flow Incompressible flow Inviscid flow Laminar flow Open channel flow Potential flow Stokes flow Superfluidity Supersonic Transient flow Transonic Turbulent flow Two-phase flow Fluid properties Density Newtonian fluid Non-Newtonian fluid Surface tension Viscosity Vapor pressure

Fluid phenomena Boundary layer Coanda effect Convection cell Convergence/Bifurcation Drag (force) Lift (force) Rossby wave Shock wave Soliton Turbulence Venturi effect Vortex Vorticity Wave drag Applications Acoustics Aerodynamics Fluid power Gaming Meteorology Naval Architecture Oceanography Plasma physics Pneumatics Pump Miscellaneous Important publications in fluid dynamics Isosurface Rotating tank Sound barrier Beta plane

Other fundamental engineering topics

• Analysis of resistive circuits
• Dynamics
  • Thermodynamics
• Engineering economics
• Heat transfer
• Materials science
• Molecular dynamics
  • 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

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