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. Flow in which turbulence is not exhibited is called laminar. Turbulence is flow dominated by recirculation, eddies, and apparent randomness. Problems in this class have elegant solutions which are linear combinations of well-studied elementary flows. If a problem is incompressible, irrotational, inviscid, and steady, it can be solved using potential flow, governed by Laplace's equation. Fluid dynamicists often transform problems to frames of reference in which the flow is steady in order to simplify the problem. 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. Whether a problem is steady or unsteady depends on the frame of reference. Both the Navier-Stokes equations and the Euler equations become simpler when their steady forms are used. 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. When the flow is everywhere irrotational as well as inviscid, Bernoulli's equation can be used throughout the field.Īnother simplification of fluid dynamics equations is to set all changes of fluid properties with time to zero. The Euler equations can be integrated along a streamline to get Bernoulli's equation. 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 standard equations of inviscid flow are the Euler equations. As illustrated by d'Alembert's paradox, a body in an inviscid fluid will experience no force. In particular, problems calculating net forces on bodies (such as wings) should use viscous equations. However, even in high Reynolds number regimes certain problems require that viscosity be included. High Reynolds numbers indicate that the inertial forces are more significant than the viscous forces. The Reynolds number can be used to evaluate whether viscous or inviscid equations are appropriate to the problem. Problems for which friction can be neglected without contributing significant error (as defined by the person solving the problem) are called inviscid. Viscous problems are those in which fluid friction have significant effects on the solution. They are simplifications of the Navier-Stokes equations in which the density has been assumed to be constant. The incompressible Navier-Stokes equations can be used to solve incompressible problems. Acoustic problems require allowing compressibility, since sound waves can only be found from the fluid equations which include compressible effects. Nearly all problems involving liquids are in this regime and modeled as incompressible. As a rough guide, compressible effects can be ignored at Mach numbers below approximately 0.3. In order to determine whether to use compressible or incompressible fluid dynamics, the Mach number of the problem is evaluated. If the density changes have negligible effects on the solution, they are ignored and the problem is called incompressible. Some of them allow appropriate fluid dynamics problems to be solved in closed form.Ī fluid problem is called compressible if changes in the density of the fluid have significant effects on the solution. All of the simplifications make the equations easier to solve. The equations can be simplified in a number of ways. 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 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. These are based on classical mechanics and are modified in quantum mechanics and general relativity. 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. 1.1 Compressible vs incompressible flow.
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