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Fluid dynamics offers a mathematical structure, which underlies these practical disciplines, 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 calculating for various properties of the fluid, such as Velocity , Pressure , Density , and Temperature , as functions of space and time. EQUATIONS OF FLUID DYNAMICS The foundational axioms of fluid dynamics are the Conservation Law s, 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 . 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 Fluid s 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, 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 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 0.3. 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. 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. Remark that solutions of this problem are reversible in time, i.e. still make sense reversing the motion. 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 fairly work when Reynolds Number is high, even if certain problems require that viscosity be included. In particular, problems involving Boundaries (walls, etc.). Indeed, we can not neglect viscosity near to a wall at rest, because it locally stops the fluid too, giving a slower motion in its neighborhood, which is affected by Viscosity and 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 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 (source: 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 72 km/hour (20 m/sec) 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 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 Fluid s 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, 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. Stokes Flow is flow at very low Reynolds numbers, such that inertial forces can be neglected compared to viscous forces. Remark that solutions to this problem are reversible in time, i.e. still make sense reversing the motion. On the contrary, the Inviscid Flow is an approximation in which we neglect Viscosity at all, compared to inertial terms. This idea, which leads to solve Hyperbolic equations, can fairly work far from Boundaries (walls, etc.) when Reynolds Number is high. The Boussinesq Approximation neglects variations in density except to calculate Buoyancy forces and is appropriate for free Convection problems. REFERENCES Batchelor, G.K. 1967 "An Introduction to Fluid Dynamics" (Cambridge University Press). Landau, L.D. and Lifshitz, E.M. 1987 "Fluid Mechanics" (Pergamon Press). SEE ALSO Fields of study
Mathematical equations and objects
Types of fluid flow
Fluid properties
Dimensionless fluid parameters
Fluid phenomena The following observed fluid phenomena can be characterised and explained using fluid mechanics:
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