Book Condition: Used: Good. In recent years, turbulence has become a very lively area of scientific research and application, attracting many newcomers who need a basic introduction to the subject. Turbulent Flows ably meets this need, developing both physical insight and the mathematical framework needed to express the theory. The authors present basic theory and illustrate it with examples of simple turbulent flows and classical models of jets, wakes, and boundary layers.
A deeper understanding of turbulence dynamics is provided by their treatment of spectral analysis and its applications.
Assessment of turbulent flow effects on the vessel wall using four‐dimensional flow MRI
Turbulent Flows offers a solid grounding in the subject of turbulence, developing both physical insight and the mathematical framework needed to express the theory. It begins with a review of the physical nature of turbulence, statistical tools, and space and time scales of turbulence. Basic theory is presented next, illustrated by examples of simple turbulent flows and classical models of jets, wakes, and boundary layers. The remainder of the book is devoted to spectral analysis and its applications and the numerical simulation of turbulent flows.
This well-balanced text will interest graduate students in engineering, applied mathematics, and the physical sciences. It will also be a useful reference for practicing engineers and scientists. Visit Seller's Storefront. Please contact me if you are not satisfied with your order in any manner.
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In most cases, the random field can be described adequately by statistical moments of various orders, which may be obtained, for example, from experimental data: the n-point moment of r-th order has the form. The angular brackets denote a statistical mean defined in terms of the probability density. The statistical moments of nth order are characterized by properties of the nth order tensor. Two-point moments are of the form:. When studying turbulent flow, consideration is given to two-point moments of not higher than of the third order. The moments formed from fluctuating quantities are named central moments:.
One-point central moments of the velocity components determine the Reynolds stress.
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The moments formed from random variables referring to several different random fields, e. A two-point joint moment of the pressure and velocity fields has the form. Space-time moments are the mean value of the product of random variables relating to different points and different time instants:. Time moments are defined as the mean values of the product of hydrodynamic velocities pertaining to the same point but to different time instants:.
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The two-point moments of the second order are called correlation functions. The correlation factor is defined as a dimensionless correlation function of the form. An integral length scale is determined with the aid of the space correlation function by the relation. Retentivity of the turbulent pulsations at a certain point is characterized by the integral time scale:. The scale T E may be regarded as a measure of duration of the coupling between the turbuient pulsations U i t.
Asymmetry is negative, if the distribution is stretched to the left of the mean value, and positive, if it is stretched to the right. In the first case this indicates the predominance of small pulsations relative to the normal distribution law, and in the second case the predominance of large pulsations.
Another efficient means of describing turbulence is a spectral analysis method. The spectral and statistical theories are interrelated mathematically through the Fourier transform. The spectral analysis makes it possible to describe the kinetic energy exchange by vortexes of different sizes or by pulsations of different frequencies. In analyzing turbulence, use is made of frequency spectra and of spectra in the space of wave numbers.
Relations are derived using the principles of the harmonic analysis. The existence of the Fourier integral necessitates a finite value of the integral.
MECH 656 Fundamentals of Turbulent Flow (4 credits)
Thus, the formulas. The correlation function of two nonperiodic signals f i t and f j t is described by the formula. The inverse Fourier transform. The Monte-Carlo turbulence modeling method is based on the construction of an artificial stochastic model of the process with preset statistical properties of turbulence. These properties represent a limited set of statistical parameters determined experimentally or theoretically. The Monte-Carlo turbulence modeling method utilizes principles of the theory of control systems. The basic idea resides in the study, synthesis and development of the system with such transfer function that, on excitation of the system by specified random noise-type disturbances, a random process possessing physical properties of the modeled phenomenon is realized at its exit.
In modeling the system operation, a random signal I t of Gaussian noise type is fed to the entrance to the control system with the impulse transfer function constructed so that the output signal y t has the required statistical characteristics. The relation between the above two characteristics may be written in the form of the convolution-type integral:. The model construction consists in determining the transfer function of the system, or in computing the convolution-type integral, or in employing the Fourier series with random coefficients to represent random signals.
The random coefficients may be prescribed in such a way that the statistical moments of the signal have preset values. With the help of quick Fourier transform methods, the random input signal and, subsequently, the Fourier spectrum are formed. Afterwards, the Fourier spectrum is multiplied by the transfer function and, using the inverse transformation, the sought output signal is obtained.
The numerical modeling of the turbulent flow is based on solving the system of Navier-Stokes equations. It is assumed that the system of Navier-Stokes equations describes the turbulent flow with regard to the following premises:. In a direct investigation of turbulence, consideration is generally given to incompressible fluid flows for which. Equations 24 account for the principal nonlinear mechanisms of the turbulent flow evolution, however, other additional effects may also be essential in more complicated problems, viz. When solved by the finite-difference method, Eq.
The boundary conditions are not simple to assign. A uniform turbulence may be modeled using periodic boundary conditions imposed on the solution to Eqs. For these boundary conditions, the numerical solution of the equations is sought in the form of the truncated Fourier series. Using Eq. In deriving Eq.
This method for solving the system of ordinary differential equations is named spectral. In a more general case, the spectral methods are described through representing the velocity as the truncated series in smooth functions:. A correct choice of the basic functions predetermines the efficency of the method. Initial conditions are selected randomly. Use is made of spectral representation of Eq.
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Substituting Eq. In the direct numerical modeling, large-scale characteristics of the turbulent flow are assumed to be independent of the Reynolds number, if the boundary and initial conditions do not depend on it. This assertion allows modeling only of small-scale formations. The direct numerical modeling of turbulence requires much more tedious and laborious computations than do the solutions using semiempirical theories of transfer. The direct numerical solution is useful for accumulating data via a "numerical experiment" and varying and improving semiempirical theories and for constructing formulas to approximate the results of direct computations.
The solution of quite a number of typical problems allows refinement of various aspects of the turbulent flow mechanism. Another approach to the numerical modeling of turbulent flow consists in developing simplified models based on physical considerations.
Such models are most widely employed for describing complex flows encountered in the engineering practice. Further information on these models is given in the article on Turbulence Modeling. In contradistinction to the above theories, the transfer models use averaged characteristics of the turbulent flow.
Reynolds proposed that the Navier-Stokes equation. All actual parameters are resolved into the time-averaged and fluctuating components, in this case. The terms form the Reynolds stress tensor and define additional "turbulent stresses" in the transfer of momentum by the pulsatory motion. The expressions represent turbulent normal stresses with the same indexes and turbulent tangential stresses with different indexes. The turbulent transport coefficients are not physical properties of the flowing medium but are rather dependent on the molecular viscosity, Reynolds number and coordinates.
In a similar manner, the averaged equations for turbulent transfer of the scalar substance r heat, substance can be obtained.
MECH Fundamentals of Turbulent Flow (4 credits) | eCalendar - McGill University
The main problem in solving the Reynolds equations and the averaged equation of scalar substance transfer is deriving the relations for , and , i. By closure methods, the models may be divided into the models utilizing the mean velocity field and the models employing the field of mean turbulence characteristics. The first group methods of Prandtl, Kantian, van Driest and Cebeci were constructed based on the analogy between turbulence and molecular chaos.
They involve such notions as mixing length as well as turbulent viscosity, thermal conductivity and diffusion coefficients. They presume a linear relationship between the tensor of turbulent stresses and the tensor of average deformation rates the Boussinesq hypothesis , as well as between the turbulent heat or passive admixture flux and the average temperature admixture concentration gradient. The Boussinesq hypothesis has the form:. The equation describing the interaction between the processes of generation, transfer and dispersal of temperature nonuniformities in the turbulent flow has the form.
Introduction to turbulence/Wall bounded turbulent flows
Another approach resides in that, for the turbulent viscosity which is a scalar quantity, a transfer equation is written analogous to the equations of scalar quantity transport in the turbulent flow. Fairly complicated problems met with in the engineering practice are solved with the aid of the models using the turbulent viscosity.
Thus, for example, the calculations are performed for a three-dimensional boundary layer on an aircraft fuselage.