Turbulent flow is a fluid motion with particle trajectories varying randomly in time, in which irregular fluctuations of velocity, pressure and other parameters arise. Since turbulence is a property of the flow rather than a physical characteristic of the liquid, an energy source for maintaining turbulence is required in each case, where such flow is realized. Turbulence may be generated by the work either of shear stresses (friction) in the main (mean) flow, i.e., in the presence of mean velocity gradients (a shear flow), or of mass (buoyant, magnetic) forces. With a shear flow, the energy is supplied to the pulsatory motion from the mean motion through large-scale vortexes, whose sizes are comparable with characteristic dimensions of the flow (for the flow behind a grid, this dimension is a characteristic mesh size of the grid; for a boundary layer flow, this dimension is a boundary layer thickness; for tubes, it is a radius; and for jets and a free shear layer, it is a transverse dimension of this layer). In near-wall flows (i.e., boundary layer, as well as tube and channel flows), turbulence generates in the region of the greatest near-wall velocity gradients throughout the flow extent. However, the turbulent flow develops only on the upset of stability of a laminar flow existing at Reynolds numbers below a certain critical value Re_{c}, which is Re_{c} =
ūD/v = 2.3 × 10^{3} for the tube flow. A developed turbulent flow is established in a tube, away from the inlet, when Re > 10^{4}, and in a boundary layer when Re_{x} = u_{∞}x/ν > 10^{6}. Though the velocity fluctuations in the tube constitute as little as a few percent of the average flow velocity, they are indispensable to the development of the entire flow.

The velocity profile for turbulent flow is fuller than for the laminar flow (Figure 1), whereas a relationship between the average and axial velocities
ū/u_{0}
depends on the Re number, being about 0.8 at Re = 10^{4} and increasing as Re rises. With the laminar flow, the ratio is constant and equal to 0.5. A general specific feature of the near-wall turbulent flows is the presence, on the wall, of a thin viscous sublayer, wherein molecular viscosity forces are dominant and the velocity distribution is linear (δ_{w} in Figure 1b).

When describing turbulent flows unaffected by solid surfaces, the term "free turbulence" is used. Figure 2 exemplifies *free turbulent flows*. Velocity gradients emerge in all these cases. Specifically, in a jet outflow from an opening, vortex bands formed on its edge diffuse under the effect of molecular viscosity with distance from the opening, so that the thickness of the mixing layer with considerable mean velocity gradients attains a large value, the flow acquires instability, and turbulence develops. The flow becomes turbulent behind the point, at which u_{1}x/ν = 7 × 10^{4}. Beyond the point of the motion transition to turbulent flow the rate of increase in the mixing layer thickness rises, so that at a distance equal to several times of the opening width the mixing layers interlock, and a completely developed turbulent jet flow is established. In common with the near-wall flows, in all cases the width "b" of the mixing zone is small relative to its length, and the transverse velocity gradient is large as compared to the gradient along the flow. The turbulent flow is much more capable for transfering momentum, heat and suspended particles, and for propagation of chemical reactions than is laminar flow.

The difference
between the actual value at the point U and the mean value
is referred to as a fluctuating component. The turbulent pulsations are characterized by space and time scales. Instead of the latter, its reciprocal value is often used, *viz.*, frequency. At large Reynolds numbers, pulsations with a wide spectrum of scales are present in the flows. The key role in the turbulent flows is played by large-scale pulsations with sizes comparable to the dimensions of the region wherein the motion occurs. The corresponding frequencies are of the order of U/L. Small-scale pulsations contain only a small part of the entire kinetic energy of the fluid.

From the mechanics standpoint, turbulent flow is a nonlinear mechanical system with an extremely large number of degrees of freedom. Various methods are employed to model and describe the turbulent flows: statistical, spectral, diffusional, direct numerical modeling and semiempirical theories.

The *statistical theory of turbulence* is based on representing the flow as an infinitely changing assembly of vortexes. The vortexes and vortex tubes stretch in a definite direction by the action of deformations produced by the main flow, and in random directions when they interact. In consequence of the vortex stretching in all directions, turbulence is an essentially three-dimensional process. The kinetic energy of the main flow is transferred to ever smaller vortexes and eventually transforms to the internal energy of the thermal motion under the effect of viscosity forces. The random velocity field, set up by elementary vortexes in the cascade process of energy transfer from larger to smaller vortexes, cannot be described by explicit mathematical relations. In each observation of the phenomenon, one of the multiplicity of potential results will be reproduced, i.e., instantaneous velocities of the turbulent flow form a random vector field.

Random quantities are four-dimensional functions of the space-time point A(x,y,z,t). A complete statistical description of the fields of hydrodynamic characteristics requires the specification of all multidimensional probability distributions for these characteristics on all kinds of sets. The probability P of emergence of the prescribed point at a certain space and time point is determined by the three-dimensional probability density:

where V_{p} is the velocity vector; r is the radius vector; and U_{1}, U_{2}, U_{3} and x_{1}, x_{2}, x_{3} are the velocity components and the coordinates. The turbulent flow field is assumed known if the 3n-dimensional probability density f^{3n} is specified. However, it is actually unfeasible to determine f^{3n}. 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

where k = k_{i} + k_{j} + ... k_{p}. 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 . The moments formed from random variables referring to several different random fields, e.g., of velocity and pressure, are called joint moments. 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:

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

The correlation factor for the Reynolds stresses is

where σ^{2} =
. An integral length scale is determined with the aid of the space correlation function by the relation

Λ is a space measure of interrelationships or a length of the correlation between the velocity fluctuations at two points of the flow field. 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). The microscales τ and λ are the measures of fast variations in small vortexes

Some central moments also have a clear physical meaning. Thus, the quantities

are the dispersion, asymmetry and excess of the quantity u_{i}, respectively, whereas a square root of the dispersion is the root-mean-square or standard deviation of this quantity. The asymmetry and excess coefficients characterize a deviation of the probability density distribution from the normal (Gaussian) distribution law, for which S = 0 and E – 3 = 0. Asymmetry is negative, if the distribution is stretched to the left of the mean value, and positive, if it is stretched to the right. The values of the excess E > 3 testify to a plane-vertex shape of the distribution curve, and E > 3 to an acute-vertex shape. 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 distributions, for example, of velocity in time at each space point constitute a complex nonperiodic function f(t), which may be represented for T → ∞ as the Fourier integral

where ω is the frequency. The existence of the Fourier integral necessitates a finite value of the integral . Thus, the formulas

define the reciprocal Fourier transforms. The function F(ω) referred to as a complex continuous spectrum of the function f(t), N is a continuous function of the circular frequency ω. The correlation function of two nonperiodic signals f_{i}(t) and f_{j}(t) is described by the formula

Applying the Fourier transform yields

where

and

the frequency spectra of the processes may be determined through measuring the correlation function or, alternatively, the frequency spectra of the pulsations may be measured directly by various spectrometers. The inverse Fourier transform

at t = 0 results in the relation

which shows that the power of turbulent pulsations, equal numerically to their dispersion σ^{2}, is the sum of the powers of individual harmonic components of the pulsations.

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:

After the application of the Fourier transforms and the reciprocal transformations

where FT is the Fourier transform, is the Fourier-function transform and τ is the time interval, the output signal acquires the form

here, H(ω) = FT[H(t)] is the transfer function of the system. 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:

the fluid is considered as a continuous medium;

physical properties of the fluid are taken to be such that all necessary derivatives of the functions, characterizing its state, are available;

a mass of the collection of fluid particles remains unchanged with time;

two contacting fluid subregions are affected by identical but opposing force fields applied to their mutual boundary;

a total force acts on the fluid subregion in the direction of the resultant force;

local values of the stress and deformation rate tensors are interrelated linearly;

boundary conditions must satisfy the immobility condition on a solid-fluid interface.

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.*, the effects of compressibility, buoyancy forces, chemical reactions, phase interaction in multiphase flows, etc. When solved by the finite-difference method, Eq. (24) are represented in difference form for a finite number of nodal points. The boundary conditions are not simple to assign. A uniform turbulence may be modeled using periodic boundary conditions imposed on the solution to Eqs. (24), specifically

where n is the vector with integral components. For these boundary conditions, the numerical solution of the equations is sought in the form of the truncated Fourier series

where the wave vectors R have integral components, if the period of this function is equal to 2π, in accordance with condition, Eq. (25). Using Eq. (26), Eq. (25) may be represented as

where σ_{αγ} is the Kronecker symbol, whereas by the repeated Greek symbols the summation from 1 to 3 is made. In deriving Eq. (7), the pressure is eliminated by means the Poisson equation resulting from the medium incompressibility:

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. The criteria are a rapid convergence to the exact solution for V,N,P → ∞ and the availability of effective methods to solve the system of ordinary differential equations for the functions a_{m,n,p} (t). Initial conditions are selected randomly. Use is made of spectral representation of Eq. (26) with the coefficients of the form

where rγ(K) are the statistically independent random variables with the Gaussian distribution and with the dispersion proportional to a specified, nonrandom energy spectrum E(K). Substituting Eq. (29) into Eq. (26) yields a random initial velocity field characterized by the Gaussian distribution and by the energy spectrum E(K). 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. O. Reynolds proposed that the Navier-Stokes equation

defines any liquid flow. Here, F_{ν} is the volume force. All actual parameters are resolved into the time-averaged
and fluctuating components, in this case

where T is the averaging period which is rather large as against the period of turbulent pulsations but small relative to the time interval, characteristic of the mean turbulent motion. The equations of motion resulting from the averaging (η = const and ρ = const)

are called the Reynolds equations of mean turbulent motion. The terms
form the Reynolds stress tensor and define additional "turbulent stresses" in the transfer of momentum
by the pulsatory motion
. The turbulent stresses τ_{T} are responsible for a rise of the total drag in turbulent flow as compared to laminar flow.

The expressions
represent turbulent normal stresses with the same indexes and turbulent tangential stresses with different indexes. The quantities
i≠ k are referred to as turbulent viscosity or a coefficient of turbulent momentum transport in the direction of x_{1}. 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 developed turbulent flow, η^{T} >> η.

In a similar manner, the averaged equations for turbulent transfer of the scalar substance r (heat, substance) can be obtained

Here, f is the molecular transfer coefficient,
and F_{γ} is a source term for heat or substance release (or absorption). The main problem in solving the Reynolds equations and the averaged equation of scalar substance transfer is deriving the relations for
,
and
, i.e., closing the equations for averaged values. 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:

where ε_{τ} is the turbulent viscosity coefficient, ε_{q} is the turbulent thermal diffusivity coefficient and Pr_{t} = ε_{τ}/ε_{q} is the turbulent Prandtl number. A great number of simple algebraic relations are proposed for defining ε_{τ}, ε_{q}, and Pr_{t} (see Turbulence Modeling).

The equation describing the interaction between the processes of generation, transfer and dispersal of temperature nonuniformities in the turbulent flow has the form

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. The models utilizing the equations for averaged turbulence characteristics may be divided conventionally into three groups:

the methods using the Reynolds stress fields for calculating the entire Reynolds stress tensor ,

the methods of closure by the mean turbulent kinetic energy determined from the expression ε ≡ 1/2 and

the methods employing to calculate heat or mass transfer the equations for along with the equations for turbulent heat fluxes and turbulent mass transport.

The Reynolds stress and kinetic energy equations can be obtained immediately from the Navier-Stokes equations. The balance equation for the turbulent energy ε = 1/2 (u_{k}^{2}) is of the form

Here, the first term defines the local time variation of the turbulent energy, the second defines the convective transfer, the third defines the work of turbulent stresses (the turbulence generation), the fourth defines the potential and kinetic energy transfer by the velocity fluctuations (the turbulent diffusion), the fifth defines the viscous diffusion and the sixth defines the energy of turbulent pulsations.

As an example, we present the Reynolds stresses model suggested by Hanjalic and Launder: the equation for the Reynolds stress τ_{t}/ρ =

the transfer equation for the turbulent kinetic energy e:

and the transfer equation for the rate of kinetic energy dissipation ε:

The above equations are written for a plane flow of an incompressible liquid. Unknown terms of the equations are approximated using the idea of Prandtl and Kolmogorov. In recent years various modifications of such a model have found wide application in calculating near-wall flows and heat and mass transfer along plane and curvilinear walls, under the conditions with external turbulence, negative or positive pressure gradient, buoyancy forces as well as with flows in circular tubes and channels.

Results of the statistical analysis of turbulence and, to a greater extent, visual observations of flows revealed that the flow is not merely random. Various types of organized collective motions were detected in turbulent shear layers (in a boundary layer, jets, wakes and mixing layers). These motions involve quasi-periodic formations (coherent structures) which originate randomly in space and time, move, change and afterwards collapse. Thus, the fundamental feature of the turbulent flow is an intricate combination of randomness and regularity which are difficult to describe analytically. Experiment has gained a decisive importance in studying the structure of turbulent flows, (see also article on Turbulence).

The first careful visual investigations of the near-wall region were carried out by S. Kline and his colleagues in the 60-s. They managed to observe, in the viscous sublayer, liquid jets moving with different velocities and to trace the character of their development. Based on numerous experiments performed over the last 20 years, the flow in the viscous sublayer may be represented schematically as follows. Liquid portions having a velocity higher than a local average velocity arrive periodically at the wall from the most remote regions. In this case, paired vortexes with the axes directed along the flow originate on the surface of the solid wall. The vortex origination is random in space and time. While moving, the vortexes recede from the wall. At a certain distance from the wall, a retarded strip is "overtaken" by the liquid having an appreciably higher velocity. A layer of intense shear emerges, and here the flow loses stability. The retarded jet starts to pulsate, thereafter it "explodes", and the fluid escapes from the wall, strongly disturbing the overlying layer. A fresh fluid portion from more remote regions arrives at the wall, in the place of the fluid portion ejected from the viscous sublayer. Subsequently, the process of viscous layer "renewal" recurs. It is established that the average distance between the retarded strips, reduced to dimensionless form using the viscous sublayer parameters friction velocity V* = (τ_{w}/ρ)^{1/2} and ν, is
= ΔzV*/ν
100. The vortex extent along the flow is by an order of magnitude greater than, and the vortex dimension normal to the surface is of the order of, the viscous sublayer thickness. The mean dimensionless lifetime of *coherent structures* is t^{+} = tV^{2}*/ν
100. Presently it has been ascertained that the turbulent energy generation is particularly intense at the instants of violent fluid ejections from the wall. Studying the interaction between the inner and outer flow zones is now one of the most vital problems of near-wall turbulence.

The most important types of anisotropic free turbulent flows are turbulent wakes behind bodies over which the fluid flows (or which move through the fluid), turbulent jets and turbulent mixing zones emerging at the boundary between different-velocity flows (Figure 2). The self-similar solutions to all the above-listed cases appear as

where b(x) is the half-width of the jet, wake or mixing zone, and is the velocity on the jet axis or the velocity deviation from the velocity of the undisturbed flow in a wake behind the body. The coordinate system for the mixing zone is chosen such that the equality = is fulfilled. Exponents in the self-similar laws for a selection of flows are tabulated in Table 1.

For all the above-enumerated flows with a fairly large Reynolds number, the velocity and friction stress profiles at a rather long distance x are presented in the form

where F_{1}(r) and F_{1}(r) are the universal functions for each flow type, r is the transverse coordinate (the distance from the axis OX for three-dimensional flows and from the plane z = 0 for plane flows) and v is the transverse velocity. It is established experimentally that the self-similarity conditions for turbulent characteristics are attained at distances x much longer than for the average velocity profile. Thus, for a circular jet issuing into the submerged space when X > 8D (D is the opening diameter), the velocity profile has already become self-similar, whereas self-similarity of the Reynolds stresses requires x > 500D.

The universal laws of near-wall turbulence for a plane-parallel flow are determined from the Reynolds equation

τ(y) = ρνdu_{x}/dy —
= τ_{w} = const, where τ_{w} is the wall stress.

Various semiempirical models are commonly used to define the velocity profile (to close the Reynolds equation). It was assumed in the first classical Prandtl-Karman models that in the turbulent core, wherein the molecular viscosity does not affect the flow,

and the turbulent tangential stress is of the form

where l is the mixing length. The only characteristic dimension in the region of developed turbulence is a distance from the wall, i.e., l = ky, and the velocity profile conforms to the logarithmic law

where V* = is the dynamic shear velocity (friction velocity), and A and k are the universal constants.

A dimensional analysis makes it possible to obtain the general form of the relation for the mean velocity profile

that expresses the universal law of near-wall turbulence holding not only for the mean velocity but also for other moments of hydrodynamic fields. Here, the values of the function j in *"the wall law"* differ from one another. In the conditions when the mean velocity gradient is independent of the viscosity (the region of developed turbulence), the logarithmic law of velocity distribution follows from "the wall law".

The wall law is valid for the without pressure gradient flow over a plane surface and for the developed flow in tubes and channels. For a more detailed information see the entries Boundary Layer and Tubes, Single-Phase Flow In.

#### REFERENCES

Hinze, J. O. (1975) *Turbulence*, McGraw-Hill, New York.

Frost, U. and Moulden, T. (1977) *Turbulence, Principles and Application.*

Cebeci and Bradshow (1987) *Convective Heat Transfer.* Abramovich, G. N. (1963) *The Theory of Turbulent Jets*, M.I.T., Cambridge, MA.

Bradshow, P. (1971) *An Introduction to Turbulence and its Measurement*, Oxford, Pergamon Press.

Kline, S. J., Reynolds, W. S., Shraub, F. A., and Runstadler, P. W. (1967) *J. Fluid Mech.*, 30.

Townsend, A. A. (1976) *The Structure of Turbulent Shear Flow*, 2nd edn., Cambridge, Univ. Press.

Bradshaw, P., Cebeci, T., and Whitelaw, J. H. (1981) *Engineering Calculation Methods for Turbulent Flow*, Academic Press, New York.