Fluid dynamics

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Fluid dynamics is the subdiscipline of fluid mechanics that studies fluids in motion. Fluids are specifically liquids and gases. The solution of a fluid dynamic problem typically involves calculating for various properties of the fluid, such as velocity, pressure, density, and temperature, as functions of space and time. The discipline has a number of subdisciplines, including aerodynamics (the study of gases) and hydrodynamics (the study of liquids). Fluid dynamics has a wide range of applications. For example, it is used in calculating forces and moments on aircraft, the mass flow rate of petroleum through pipelines, and in prediction of weather patterns, and even in traffic engineering, where traffic is treated as a continuous fluid. Fluid dynamics offers a mathematical structure that underlies these practical discipines which often also embrace empirical and semi-empirical laws, derived from flow measurement, to solve practical problems.

1 Equations of fluid dynamics
2 References
3 See also

Equations of fluid dynamics

The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of momentum (also known as Newton's first law), and conservation of energy. These are based on classical mechanics and are modified in quantum mechanics and general relativity.

For fluids which are sufficiently dense to be a contiunuum, 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 the simplifications make the equations 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, an 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, $ρ$ is density, $R u$ is the gas constant $M$ is the molecular weight and $T$ is temperature.

Compressible vs incompressible flow

A fluid problem is called compressible if the pressure variation in the flowfield are large enough to effect substantial changes in the density of the fluid. Flows of liquids with pressure variations much smaller than those required to cause phase change(cavitation), or flows of gases involving speeds much lower than the isentropic sound speed are termed incompressible.

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 0.1. For liquids, whether the incompressible assumption is valid depends on the fluid properties (specifically the critical pressure and termperature of the fluid) and the flow conditions (how close to the critical pressure the actual flow pressure becomes). Acoustic problems require allowing compressibility, since sound waves can only be found from the fluid equations which include compressible effects.

The incompressible Navier-Stokes equations can be used to solve incompressible problems. They are simplifications of the Navier-Stokes equations in which the density has been assumed to be constant.

Viscous vs inviscid flow

Viscous problems are those in which fluid friction have significant effects on the solution. Problems for which friction can be neglected without contributing significant error (as defined by the person solving the problem) are called inviscid.

The Reynolds number can be used to evaluate whether viscous or inviscid equations are appropriate to the problem. High Reynolds numbers indicate that the inertial forces are more significant than the viscous forces. However, even in high Reynolds number regimes certain problems require that viscosity be included. In particular, problems calculating net forces on bodies (such as wings) 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 as well as inviscid, Bernoulli's equation can be used throughout the field.

Steady vs unsteady flow

Another simplification of fluid dynamics equations is to set all changes of fluid properties with time to zero. This is called steady flow, and is applicable to a large class of problems, such as lift and drag on a wing or flow through a pipe. Both the Navier-Stokes equations and the Euler equations become simpler when their steady forms are used.

Whether a problem is steady or unsteady depends on the frame of reference. For instance, the flow around a ship in a uniform channel is steady from the point of view of the passengers on the ship, but unsteady to an observer on the shore. Fluid dynamicists often transform problems to frames of reference in which the flow is steady in order to simplify the problem.

If a problem is incompressible, irrotational, inviscid, and steady, it can be solved using potential flow, governed by Laplace's equation. Problems in this class have elegant solutions which are linear combinations of well-studied elementary flows.

Laminar vs turbulent flow

Turbulence is flow dominated by recirculation, eddies, and apparent randomness. Flow in which turbulence is not exhibited is called laminar. 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). 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(Pope-2000), given the state of computational power for the next few decades. Any flight vehicle large enough to carry a human(L>3m), moving faster than 72km/hour(20m/sec) is well beyond the limit of DNS simulation(Re = 4 Million). Transport aircraft wings(such as A300 or 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 forseeable 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 also. Large Eddy Simulation also holds promise as a simulation methodology, especially in the guise of discrete 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 other materials, such as milk and blood, and also some plastic solids, have more complicated non-Newtonian stress-strain behaviours. These materials include sticky liquids such as latex, honey, and even water (due to its constituent hydrogen bonds) which are studied in the sub-discipline of rheology.

Other approximations

There are a large number of other possible approximations to fluid dynamic problems. Stokes flow is flow at very low Reynolds numbers, such that inertial forces can be neglected compared to viscous forces. The Boussinesq approximation neglects variations in density except to calculate buoyancy forces and is appropriate for free convection problems.

References

Pope, Stephen B. 2000 "Turbulent Flows" (Cambridge University Press)

Fluid dynamics

From Exampleproblems

Contents

Equations of fluid dynamics

Compressible vs incompressible flow

Viscous vs inviscid flow

Steady vs unsteady flow

Laminar vs turbulent flow

Newtonian vs non-Newtonian fluids

Other approximations

References

See also

Fields of study

Mathematical equations and objects

Types of fluid flow

Fluid properties

Fluid numbers

Fluid phenomena

Applications

See also

Views

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