Durst F. Fluid Mechanics: An Introduction to the Theory of Fluid Flows

Подождите немного. Документ загружается.

18.6 Scales of Turbulent Flows 553

and the dissipation of turbulent kinetic energy as



= µ

∂u



∂x

∂u



∂x

(18.117)

the equation for the turbulent kinetic energy can be written in the following

way:

¯ρ

∂k

∂x

= −

∂D

∂x

+ P

− 

, (18.118)

where k =(1/2



) was introduced, i.e. k =1/2(u



+ u



+ u



). The mean

transport of turbulent kinetic energy is kept “in balance” by the diﬀusion,

production and dissipation of turbulent kinetic energy at a point in the ﬂow.

Earlier, it was shown that in the equation for the energy of the mean ﬂow

the viscous dissipation, caused by the mean ﬂow ﬁeld, when compared with

the production term of the turbulent kinetic energy, can be neglected. This

is not the case for the turbulent dissipation in the above equation for the

turbulent kinetic energy, i.e. 

cannot be neglected with respect to P

.This

can, on the other hand, be shown through the following order of magnitude

considerations:



∼

¯ρu

µu







= Re





, (18.119)

where l

is the ratio of a characteristic length scale of the considered

turbulence to a length scale characterizing the mean ﬂow. For the ratio l

the following relationship holds in general:

= Re

−m

, (18.120)

where m =1/2 can be set, i.e. in the equation for the turbulent kinetic energy

the viscous dissipation cannot be neglected. It is an essential part of (18.118),

describing the transport of turbulent kinetic energy.

18.6 Characteristic Scales of Length, Velocity

and Time of Turbulent Flows

In the preceding sections, order of magnitude considerations were made in

which scales of length, velocity and time of turbulent ﬂows were employed.

So, for the characterization of the mean ﬂow ﬁeld, a characteristic mean

velocity

, a characterizing length L

and a time scale t

were introduced.

Here, L

is of the order of the dimensions of the total ﬂow extension, i.e. for

internal ﬂows L

has the linear cross-section dimension of the ﬂow channel or

pipe where the ﬂuid ﬂows. The area-averaged or the time-averaged velocity

554 18 Turbulent Flows

of the ﬂow ﬁeld can be introduced as the characteristic mean velocity. For

the characteristic time, t

, the following ratio holds:

. (18.121)

If one considers the turbulent velocity ﬂuctuations that occur superimposed

on the mean ﬂow ﬁeld, it is easy to see that the integral time-scale of the tur-

bulence, introduced in Sect. 18.3, always has to be of the order of magnitude

of the characteristic time scale of the mean ﬂow ﬁeld, i.e. the largest vortices

that the turbulent part of a ﬂow ﬁeld possesses have time scales which cor-

respond to those of the mean ﬂow ﬁeld. Generally, the following relationship

holds:

≤ t

. (18.122)

Following a suggestion of Kolmogorov, so-called micro-scales can be

introduced to characterize the turbulent ﬂow ﬁeld:

= Kolmogorov’s length scale,

= Kolmogorov’s velocity scale,

=(l

) = Kolmogorov’s time scale.

The length, velocity and time scales introduced by Kolmogorov are deter-

mined in such a way that they characterize that part of the spectrum of the

turbulent velocity ﬂuctuations in which the energy production of the turbu-

lent vortices is equal to the dissipation. Thus, assuming isotropic turbulence,

one can introduce:

 ≈ P ≈

for the relationship for production, (18.123)

 ≈ ν

for the relationship for dissipation. (18.124)

From these results, we can deduce:

= 

or (18.125)

1=

; l







. (18.126)

From the equality of the terms for production and dissipation, it follows that:

and thus u

=(ν)

. (18.127)

For the Kolmogorov time scale, the following expression results:







. (18.128)

18.6 Scales of Turbulent Flows 555

The Reynolds number resulting on the basis of the above-introduced micro-

length scale and micro-velocity scale is:

=1. (18.129)

The characteristic turbulent eddy quantities, determined by Kolmogorov’s

scales of turbulence, are those that represent the viscous eﬀects which damp

the turbulent velocity ﬂuctuations. These smallest eddies are assumed to

convert the kinetic energy of turbulence into heat. Because of these charac-

teristic properties, the following deﬁnitions are available in the literature for

the smallest scales of turbulence:

Kolmogrov’s scales = micro-scales = viscous eddy scales:







; u

=(ν)

; τ







. (18.130)

From these scales, characterizing the smallest vortices of a turbulent ﬂow,

the Taylor micro-scale has to be distinguished, which is deﬁned as follows:

. (18.131)

The extensions of the diﬀerent scales can perhaps best be illustrated in

schematic form; see Fig. 18.12, which shows that the Taylor micro-scale de-

ﬁnes an eddy size which is located between the smallest viscous eddies and

the large eddies having quantities of the dimension of the geometric extension

of the mean ﬂow. Taking this into account, it can be shown that the following

hold:

= ν

;





= Re

−

, (18.132)

where Re =(U

)/ν:

Considering that the following holds:

= l

(18.133)

then inserting this relationship into the equation for

and taking into

account

= 1, a further important relationship follows from the above

derivations:





; l

= L

. (18.134)

The diﬀerent length scales of turbulence have proven to be very useful in

formulations of turbulence models, which are summarized in the subsequent

sections. The presentations to follow were chosen such that they are suitable

for introduction into turbulence modeling. More detailed descriptions can be

556 18 Turbulent Flows

Range of turbulence production

Range of dissipation of turbulence

Large length scales

Small length scales

Fig. 18.12 Length scales of turbulence and contributions to production and

dissipation

taken from the available specialized literature on turbulence modeling, given

in the list of references at the end of this chapter.

The above derivations indicate the diﬀerences in the structure of turbulent

ﬂows at small and high Reynolds numbers. For ﬂows with the same inte-

gral dimensions (see Fig. 18.12), the ﬂow at a large Reynolds number proves

to be “micro-structured,” i.e. the smallest eddies have small dimensions,

whereas for the small Reynolds number the ﬂow appears “macro-structured.”

The Taylor length scale proves always to be larger than the Kolmogorov

micro length, and the diﬀerence between the two becomes larger with in-

creasing Reynolds number. From the above relationships, one can compute

=(Re)

1/4

. For a Reynolds number of approximately Re =10

, l

approximately 10 times larger than l

(Fig. 18.13).

In order to characterize the complex nature of turbulent ﬂows, the

following Reynolds numbers are often employed:

=1,Re

K,c

,Re

, (18.135)

= Re. (18.136)

Moreover, for the relationships of the characteristic length scales, the

following expression holds:

= Re

−

= Re

−

= Re

−

= Re

−1





= Re

−

= Re

−1

= Re

−

. (18.137)

These relationships are often employed when considering turbulent ﬂows,

in order to carry out order of magnitude considerations regarding the

characteristic properties of turbulence.

18.7 Turbulence Models 557

(a) (b)

Fig. 18.13 (a) Micro-structure of the turbulence of small Reynolds numbers and

(b) micro-structure of the turbulence of higher Reynolds numbers

18.7 Turbulence Models

18.7.1 General Considerations

When limiting oneself to considerations of the components of mean velocity

and the moments of turbulent ﬂuctuation quantities, as far as the solutions

of ﬂuid mechanical problems are concerned, there is the possibility of solv-

ing ﬂow problems theoretically: instead of the Navier–Stokes equations, the

Reynolds equations are solved. In order to derive the latter set of partial

diﬀerential equations, the instantaneous velocity

,t) was replaced in

Sect. 18.2 by the sum of the local mean velocity

) and the local ﬂuctu-

ation velocity u



,t), i.e. in Sect. 18.2, it was shown that the following can

be set:

= U

+ u



. (18.138)

In Sect. 18.3, it was shown moreover that the introduction of this relation-

ships into the continuity equation and the Navier–Stokes equations results,

after time-averaging all terms in the equations, in a new system of equations

which has to be solved numerically for the ﬂow problems to be investigated.

The derivations again yield, for ρ = constant, four diﬀerential equations:

558 18 Turbulent Flows

Continuity equation for j =1, 2, 3:

∂U

∂x

=0. (18.139)

Reynolds equation:

¯ρ

∂U

∂x

= −

∂

∂x

∂

∂x



∂

∂x

− ¯ρu





+¯ρg

. (18.140)

The derivations have led, however, to the introduction of new unknowns given

by the following fourth-order tensor:



⎛

⎝

2



2



2

⎞

⎠

, (18.141)

which can be referred to as “Reynolds momentum transport terms,” but are

often also called Reynolds stress terms.

The introduction of additional unknowns into the basic equations of

ﬂuid mechanics lead to a non-closed system of equations and this re-

quires additional information in order to obtain solutions from (18.139) and

(18.140). This required information, often formulated as additional diﬀeren-

tial equations, represents general statements on the interrelation between the

Reynolds momentum transport terms and the mean velocity ﬁeld. The deriva-

tions of these additional diﬀerential equations require additional physical

insights into the turbulent velocity correlations (18.141) and their dependence

on the mean ﬂow ﬁeld. The development of model considerations for these

unknown terms is an important ﬁeld of research in ﬂuid mechanics. Consider-

ations of this kind are carried out in the form of turbulence models by various

research groups. The basics of such models are described in the following sec-

tions only in an introductory manner. Further insight into this important

sub-domain of ﬂuid mechanics shows that the introduction of turbulence

models serves to derive “closing assumptions” for the Reynolds equations.

Succeeding in formulating additional equations, i.e. producing physically ap-

propriate turbulence models, a new system of equations results, based on the

Reynolds equations, which can be employed to solve turbulent ﬂow problems.

The formulation of physically appropriate additional equations, i.e. to

provide physically appropriate information for the correlations



for the

treatment of the Reynolds equations, can partly be realized by hypothetical

assumptions, as was often done in the past when setting up turbulence mod-

els. There is, however, the possibility to obtain the information required for

turbulence models by means of detailed experimental investigations in diﬀer-

ent turbulent ﬂows. For this purpose, local measurements of the instantaneous

velocity of turbulent ﬂows are necessary. Such measurements can be achieved

with the help of hot wire and hot ﬁlm anemometry and also laser Doppler

anemometry. Those measuring techniques provide the necessary resolution

18.7 Turbulence Models 559

with respect to time and space variations of the ﬂow, to carry out all the mea-

surements required for detailed turbulence modeling. The measuring methods

can be considered to be fully developed, so that the required measurements

can be carried out without serious application problems. Such measurements

contribute considerably to deepening the comprehension of the physics of tur-

bulence and make it possible to introduce additional information in the form

of new equations to yield numerical solution procedures for turbulent ﬂows.

For measurements in wall boundary layers, hot wire and hot ﬁlm anemome-

ters have been employed with great success for determining the mean velocity

and the ﬂuctuation quantities u



(see Sect. 18.8). Flows of this kind

can be investigated reliably with hot wire anemometers because of their

characteristic properties. However, in the case of very thin boundary lay-

ers, inherent disturbances may occur which are caused by the introduced

measuring sensors. By special shaping of the employed measuring sensors,

these disturbances and the resulting measurement errors can be kept low.

Most measuring methods, requiring the introduction of measuring sensors

into ﬂows, measure the ﬂow velocity only indirectly, i.e. with most measuring

instruments physical quantities are recorded which are functions of the ﬂow

velocity. Unfortunately, the measuring quantities are often also functions

of the properties of the state of the ﬂow medium. The latter have to be

known and have to be adapted already when calibrating the measuring

method, in order to make the interpretation of the measured data possible

in terms of velocity. When ﬂuctuations of the ﬂuid properties occur during

the attempted velocity measurements, e.g. in two-phase ﬂows, ﬂows with

chemical reactions etc., they have to be known to be able to determine

reliably the required velocity values.

The above-mentioned diﬃculties in the employment of indirect measuring

techniques for ﬂow velocities, such as hot wire and hot ﬁlm anemometry,

have led to the development of the laser Doppler technique, which measures

ﬂow velocities directly. By measuring the time which a particle needs to

ﬂow through an interference pattern with a well-deﬁned fringe distance, the

velocity of light-scattering particles can be determined. Such measurements

can be carried out locally and do not depend on the unknown thermodynamic

properties of the ﬂow ﬂuid. Measurements are possible in one- and two-phase

ﬂows, and also in combustion systems and in the atmosphere. The measuring

technique can moreover be employed in particle-loaded ﬂows, i.e. in media as

they often occur in practice. Its application requires, however, optical access

to the measuring point and transparency of the ﬂow medium. In this respect

the employment of laser Doppler anemometry is limited, but its application

makes the determination of ﬂow velocities possible in a large number of ﬂows

that are not accessible to other measuring methods.

Due to considerable developments in applied mathematics over the last

few decades, new methods to solve numerically systems of partial diﬀeren-

tial equations, such as those describing ﬂuid ﬂows, have appeared. These

560 18 Turbulent Flows

1940

1950

1970

1960 1980

1990

2000

Computer power (Flop/s)

2010

Fig. 18.14 Increase in computing and computer power in the employment of

mathematical methods and of high-speed computers

developments were supported by an increase in computing power, as shown

in Fig. 18.14.

Considering also the increased performance of high-performance computer

systems, as also indicated in Fig. 18.14, it becomes understandable that it is

possible nowadays to obtain direct numerical solutions of turbulent ﬂows at

least at small Reynolds numbers. Such solutions lead to important insights

that can be employed for the development of reﬁned turbulence models. In

the following section, it is shown how the knowledge gained by experimental

and numerical investigations of turbulent ﬂows can be used to formulate

turbulence models for the unknown Reynolds momentum transport terms.

Whereas in the past turbulence model developments had to be carried out

without such knowledge, numerically obtained information on the behavior

of turbulent ﬂows is nowadays available.

18.7.2 General Considerations Concerning Eddy

Viscosity Models

In Sect. 18.5, the basic equations of ﬂuid mechanics were employed, i.e. the

continuity equation, the Navier–Stokes equation and the mechanical en-

ergy equation, in order to introduce into them time-averaged quantities of

18.7 Turbulence Models 561

turbulent property ﬂuctuations. After time averaging of the resulting equa-

tions and taking into account ∂/∂t(

···) = 0, the following relationships could

be derived:

• Continuity equations:

–Meanﬂowﬁeld:

∂

∂x

=0. (18.142)

– Fluctuating ﬂow ﬁeld:

∂u



∂x

=0. (18.143)

• Navier–Stokes equations (for g

=0)

–Meanﬂowﬁeld:

¯ρ

∂U

∂x

= −

∂

∂x

∂

∂x



∂

∂x

− ¯ρu





. (18.144)

– Fluctuating ﬂow ﬁeld:

¯ρ

∂



∂x

= −¯ρu



∂U

∂x

−

∂

∂x







¯ρu



+ p





−µ

∂u



∂x

∂u



∂x

+ µ

∂



∂x

. (18.145)

• Thermal energy equation:

– Mean temperature ﬁeld:

ρc

∂

∂x

∂

∂x



∂

∂x

− ρc





. (18.146)

– Fluctuating temperature ﬁeld:

∂

∂x







= −

∂

∂x







− a

∂

∂x







−



∂

∂x

− a



∂T



∂x



(18.147)

with a = λ/ρc

Regarding the solutions of laminar ﬂow problems, it had been possible, by

employing the continuity and the Navier–Stokes equations, to solve a number

of problems analytically, i.e. the set of diﬀerential equations that was available

constituted a closed system of partial diﬀerential equations for these cases.

When considering the corresponding system of equations for turbulent ﬂows,

e.g. the continuity equation and the momentum equation for the mean ﬂow

By a closed diﬀerential system, one understands a system in which the number of

unknown variables and the number of equations available are equal.

562 18 Turbulent Flows

ﬁeld, it can be seen from the above statements that the system of equations

is not closed. There appear six additional unknown quantities, namely the

correlations of the velocity ﬂuctuations u



and u



, i.e. the elements of the

following tensor:



⎛

⎝

2



2



2

⎞

⎠

. (18.148)

This shows that the derivations of the time-averaged equations, generally

holding for turbulent ﬂows, have led to a closing problem which has to be

solved before solutions of the above equations for turbulent ﬂows can be

sought. The development of suitable closing assumptions is tackled in ﬂow

research by turbulence model developments:

• By turbulence modeling, one understands the development of closing as-

sumptions, which are formulated in the form of additional equations and

which are employed in addition to the averaged continuity and Navier–

Stokes equations, i.e. the Reynolds equations, for the solution of ﬂow

problems.

Such closing assumptions should make use of experimentally or numeri-

cally obtained information, so that soundly based assumptions for the prop-

erties of turbulent ﬂow quantities can be employed as additional equations,

needed for numerical solutions of turbulent ﬂow problems.

In the literature, a large number of diﬀerent turbulence models have been

proposed, developed and employed for ﬂow problem solutions, based on the

Reynolds equation for turbulent ﬂows. They can be classiﬁed as follows. All

the following models have one thing in common, according to a suggestion of

Boussinesq: they set the transport mechanism of turbulent velocity ﬂuctua-

tions equal to the transport of molecules in an isotropic Newtonian ﬂuid, i.e.

it is assumed that the following relationship holds:

¯ρ



= −¯ρν



∂

∂x

∂

∂x



, (18.149)

is deﬁned as the eddy viscosity, which has to be regarded as an unknown

quantity, and it represents a property of the turbulent ﬂow and not the ﬂuid.

It is the task of the above eddy viscosity models, often also called “ﬁrst-order

closure models,” to provide good model assumptions for the eddy viscosity ν

To reach this goal, the considerations below based on characteristic velocity

and length scales need to be considered:

ρu

∼ ρν

; ν

∼ u

, (18.150)

i.e. deﬁning equation of the eddy viscosity:

−





∂

∂x

∂

∂x



. (18.151)