The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity , pressure , density , and temperature , as functions of space and time. Before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability , both of which can also be applied to gases.

## Incompressible Flow - Ronald L. Panton - Google книги

The foundational axioms of fluid dynamics are the conservation laws , specifically, conservation of mass , conservation of linear momentum , and conservation of energy also known as First Law of Thermodynamics. 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 assumes that fluids are continuous, rather than discrete. Consequently, it is assumed that properties such as density, pressure, temperature, and flow velocity are well-defined at infinitesimally small points in space and vary continuously from one point to another.

The fact that the fluid is made up of discrete molecules is ignored. For fluids that are sufficiently dense to be a continuum, do not contain ionized species, and have flow velocities small in relation to the speed of light, the momentum equations for Newtonian fluids are the Navier—Stokes equations —which is a non-linear set of differential equations that describes the flow of a fluid whose stress depends linearly on flow velocity gradients and pressure. The unsimplified equations do not have a general closed-form solution , so they are primarily of use in Computational Fluid Dynamics.

The equations can be simplified in a number of ways, all of which make them easier to solve. Some of the simplifications allow some simple fluid dynamics problems to be solved in closed form. In addition to the mass, momentum, and energy conservation equations, a thermodynamic equation of state that gives the pressure as a function of other thermodynamic variables is required to completely describe the problem. An example of this would be the perfect gas equation of state :.

Three conservation laws are used to solve fluid dynamics problems, and may be written in integral or differential form. The conservation laws may be applied to a region of the flow called a control volume.

## Fluid Mechanics

A control volume is a discrete volume in space through which fluid is assumed to flow. The integral formulations of the conservation laws are used to describe the change of mass, momentum, or energy within the control volume. Differential formulations of the conservation laws apply Stokes' theorem to yield an expression which may be interpreted as the integral form of the law applied to an infinitesimally small volume at a point within the flow. All fluids are compressible to some extent; that is, changes in pressure or temperature cause 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.

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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 flow is evaluated. As a rough guide, compressible effects can be ignored at Mach numbers below approximately 0.

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. All fluids are viscous, meaning that they exert some resistance to deformation: neighbouring parcels of fluid moving at different velocities exert viscous forces on each other. Isaac Newton showed that for many familiar fluids such as water and air , the stress due to these viscous forces is linearly related to the strain rate. Such fluids are called Newtonian fluids.

The coefficient of proportionality is called the fluid's viscosity; for Newtonian fluids, it is a fluid property that is independent of the strain rate.

Non-Newtonian fluids have a more complicated, non-linear stress-strain behaviour. The sub-discipline of rheology describes the stress-strain behaviours of such fluids, which include emulsions and slurries , some viscoelastic materials such as blood and some polymers , and sticky liquids such as latex , honey and lubricants.

The dynamic of fluid parcels is described with the help of Newton's second law. An accelerating parcel of fluid is subject to inertial effects. The Reynolds number is a dimensionless quantity which characterises the magnitude of inertial effects compared to the magnitude of viscous effects. In such cases, inertial forces are sometimes neglected; this flow regime is called Stokes or creeping flow. In high Reynolds number flows, the flow is often modeled as an inviscid flow , an approximation in which viscosity is completely neglected.

## Incompressible Flow

Eliminating viscosity allows the Navier—Stokes equations to be simplified into the Euler equations. The integration of the Euler equations along a streamline in an inviscid flow yields Bernoulli's equation. When, in addition to being inviscid, the flow is irrotational everywhere, Bernoulli's equation can completely describe the flow everywhere. Such flows are called potential flows , because the velocity field may be expressed as the gradient of a potential energy expression.

This idea can work fairly well when the Reynolds number is high. However, problems such as those involving solid boundaries may require that the viscosity be included. Viscosity cannot be neglected near solid boundaries because the no-slip condition generates a thin region of large strain rate, the boundary layer , in which viscosity effects dominate and which thus generates vorticity.

Therefore, to calculate net forces on bodies such as wings , viscous flow equations must be used: inviscid flow theory fails to predict drag forces , a limitation known as the d'Alembert's paradox. A commonly used [ citation needed ] model, especially in computational fluid dynamics , is to use two flow models: the Euler equations away from the body, and boundary layer equations in a region close to the body.

The two solutions can then be matched with each other, using the method of matched asymptotic expansions. A flow that is not a function of time is called steady flow. Steady-state flow refers to the condition where the fluid properties at a point in the system do not change over time.

Time dependent flow is known as unsteady also called transient [6]. 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 a background flow, the flow is unsteady. Turbulent flows are unsteady by definition. A turbulent flow can, however, be statistically stationary. According to Pope: [7]. 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.

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Steady flows are often more tractable than otherwise similar unsteady flows. The governing equations of a steady problem have one dimension fewer time than the governing equations of the same problem without taking advantage of the steadiness of the flow field. Turbulence is flow characterized by recirculation, eddies , and apparent randomness.

Flow in which turbulence is not exhibited is called laminar. The presence of eddies or recirculation alone does not necessarily indicate turbulent flow—these phenomena may be present in laminar flow as well. Mathematically, turbulent flow is often represented via a Reynolds decomposition , in which the flow is broken down into the sum of an average component and a perturbation component. It is believed that turbulent flows can be described well through the use of the Navier—Stokes equations. Direct numerical simulation DNS , based on the Navier—Stokes equations, makes it possible to simulate turbulent flows at moderate Reynolds numbers.

Restrictions depend on the power of the computer used and the efficiency of the solution algorithm. The results of DNS have been found to agree well with experimental data for some flows. Most flows of interest have Reynolds numbers much too high for DNS to be a viable option, [9] given the state of computational power for the next few decades.

Transport aircraft wings such as on an Airbus A or Boeing have Reynolds numbers of 40 million based on the wing chord dimension. Solving these real-life flow problems requires turbulence models for the foreseeable future. Reynolds-averaged Navier—Stokes equations RANS combined with turbulence modelling provides a model of the effects of the turbulent flow. Such a modelling mainly provides the additional momentum transfer by the Reynolds stresses , although the turbulence also enhances the heat and mass transfer.

Revised to reflect students' ready access to mathematical computer programs that have advanced features and are easy to use, Incompressible Flow, Fourth Edition includes: Several more exact solutions of the Navier-Stokes equations Classic-style Fortran programs for the Hiemenz flow, the Psi-Omega method for entrance flow, and the laminar boundary layer program, all revised into MATLAB A new discussion of the global vorticity boundary restriction A revised vorticity dynamics chapter with new examples, including the ring line vortex and the Fraenkel-Norbury vortex solutions A discussion of the different behaviors that occur in subsonic and supersonic steady flows Additional emphasis on composite asymptotic expansions Incompressible Flow, Fourth Edition is the ideal coursebook for classes in fluid dynamics offered in mechanical, aerospace, and chemical engineering programs.

Preface to the First Edition. Vector Calculus and Index Notation. Kinematics of Local Fluid Motion. Newtonian Fluids and the NavierStokes Equations. Some Incompressible Flow Patterns. Characteristics of HighReynoldsNumber Flows. Kinematic Decomposition of Flow Fields. Ideal Flows in a Plane. ThreeDimensional Ideal Flows. Boundary Layers. Lubrication Approximation. Dimensional Analysis. Compressible Flow. Streamfunctions and the Velocity Potential. Vorticity Dynamics. Flows at Moderate Reynolds Numbers.