# Applications of Group-Theoretical Methods in Hydrodynamics

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. 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.

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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.

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Such a modelling mainly provides the additional momentum transfer by the Reynolds stresses , although the turbulence also enhances the heat and mass transfer. Another promising methodology is large eddy simulation LES , especially in the guise of detached eddy simulation DES —which is a combination of RANS turbulence modelling and large eddy simulation.

While many flows e. New phenomena occur at these regimes such as instabilities in transonic flow, shock waves for supersonic flow, or non-equilibrium chemical behaviour due to ionization in hypersonic flows. In practice, each of those flow regimes is treated separately. Reactive flows are flows that are chemically reactive, which finds its applications in many areas such as combustion IC engine , propulsion devices Rockets , jet engines etc.

Magnetohydrodynamics is the multi-disciplinary study of the flow of electrically conducting fluids in electromagnetic fields.

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Examples of such fluids include plasmas , liquid metals, and salt water. The fluid flow equations are solved simultaneously with Maxwell's equations of electromagnetism. Relativistic fluid dynamics studies the macroscopic and microscopic fluid motion at large velocities comparable to the velocity of light.

The governing equations are derived in Riemannian geometry for Minkowski spacetime. There are a large number of other possible approximations to fluid dynamic problems. Some of the more commonly used are listed below. The concept of pressure is central to the study of both fluid statics and fluid dynamics. A pressure can be identified for every point in a body of fluid, regardless of whether the fluid is in motion or not. Pressure can be measured using an aneroid, Bourdon tube, mercury column, or various other methods. Some of the terminology that is necessary in the study of fluid dynamics is not found in other similar areas of study.

In particular, some of the terminology used in fluid dynamics is not used in fluid statics. The concepts of total pressure and dynamic pressure arise from Bernoulli's equation and are significant in the study of all fluid flows. These two pressures are not pressures in the usual sense—they cannot be measured using an aneroid, Bourdon tube or mercury column.

To avoid potential ambiguity when referring to pressure in fluid dynamics, many authors use the term static pressure to distinguish it from total pressure and dynamic pressure. Static pressure is identical to pressure and can be identified for every point in a fluid flow field.

A point in a fluid flow where the flow has come to rest i. It is of such importance that it is given a special name—a stagnation point. The static pressure at the stagnation point is of special significance and is given its own name— stagnation pressure. Ovsyannikov and his school. The authors rank themselves as disciples of the school. The present monograph deals mainly with group-theoretic classification of the equations of hydrodynamics in the presence of planar and rotational symmetry and also with construction of exact solutions and their physical interpretation.

It is worth noting that the concept of exact solution to a differential equation is not defined rigorously; different authors understand it in different ways. The concept of exact solution expands along with the progress of mathematics solu- tions in elementary functions, in quadratures, and in special functions; solutions in the form of convergent series with effectively computable terms; solutions whose searching reduces to integrating ordinary differential equations; etc.

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## Geometrical methods in hydrodynamics

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Applications of Group-Theoretical Methods in Hydrodynamics
Applications of Group-Theoretical Methods in Hydrodynamics
Applications of Group-Theoretical Methods in Hydrodynamics
Applications of Group-Theoretical Methods in Hydrodynamics
Applications of Group-Theoretical Methods in Hydrodynamics
Applications of Group-Theoretical Methods in Hydrodynamics

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