湍流入门：从基础到统计分析

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"A first course in turbulence" 本书"湍流初步"是针对流体动力学中的一个核心主题——湍流而设计的一本入门教材，旨在帮助初学者从基础流体动力学到专业文献中对湍流的理解进行平滑过渡。通常，湍流这个主题因其非线性和统计上的不确定性，对初学者来说颇具挑战性。本书的独特之处在于，它不仅适用于具有不同技术背景和兴趣的学生、工程师和科学家，而且在讲解过程中避免了高级数学发展的难度和实验细节的复杂性。作者H. Tennekes和J.L. Lumley采用了一种中庸的路径，即不深入复杂的数学推导，也不陷入实验细节和经验数据的海洋，而是强调物理理解，增强对那些目前无法严格解决的问题的直觉洞察。书中广泛使用了量纲分析，这是一种处理数学上难以捉摸问题的有效工具。量纲推理、标度论证和相似性规则在一开始就引入，并贯穿全书。书中讨论了雷诺应力和气体动理论，将混合长度理论置于适当的视角，并对其与量纲分析的剪切流进行了详尽的比较。随后，对涡旋动力学进行了深入探讨，包括涡旋拉伸和涡旋预算。此外，书中还专门用两章来介绍自由边界剪切流和良好约束的湍流剪切流，涵盖了如尾流、射流、剪切层、热羽流、大气边界层、管道和通道流动以及压力梯度中的边界层等实例。在分析了湍流的空间结构后，书中提供了一个紧凑但全面的统计方法介绍，为读者理解湍流的随机和谱结构打下基础。统计方法的应用涵盖了湍流输运（包括扩散和混合）和湍流谱的研究。通过这本书，学生不仅可以了解到湍流的基本概念，还能掌握分析和理解复杂湍流现象的方法，从而为他们进一步研究这一领域的专业文献打下坚实的基础。

1.2

Methods

analysis

role similar to that of molecular viscosity in laminar flows. This approach

based on

superficial resemblance between the way molecular motions trans-

fer momentum and heat and the way in which turbulent velocity fluctuations

transfer these quantities. Phenomenological concepts like "eddy viscosity"

(to replace molecular viscosity) and "mixing length" (in analogy with the

mean

free

path in the kinetic theory of gases) were developed by Taylor,

Prandtl, and others. These coqcepts are studied in detail in Chapter

Molecular viscosity

property of fluids; turbulence

characteristic of

flows. Therefore, the use of an eddy viscosity to represent the effects of

turbulence on

flow

liable to be misleading. However, current research

seems

to indicate that, in simple flows, we may, for analytical reasons, speak

turbulent fluid rather than of

turbulent flow. Turbulent "fluids,"

however, are non-Newtonian: they exhibit viscoelasticity and suffer memory

effects. In favorable circumstances, the memory

fading in time,

that one

may be able to develop

semilocal theory relating the mean stress to the

mean rate of strain.

Phenomenological theories of turbulence make crucial assumptions

fairly early

stage

in the analysis. In recent years,

group of theoreticians

(Kraichnan, Edwards, Orszag, Meecham, and others) have developed very

formal and sophisticated statistical theories of turbulence, in the hope of

finding

formalism that does not need ad hoc assumptions

(see

Orszag,

1971).

far, however, rather arbitrary postulates

are

needed in

these

theories, too. The mathematical complexity of this work

overwhelming

that

discussion of it has to be

left

out of this book.

Dimensional analysis One of the most powerful tools in the study of turbu-

lent

flows

dimensional analysis. In many circumstances

possible to

argue that some aspect of the structure of turbulence depends only on

few

independent variables or parameters. If such

situation prevails, dimensional

methods often dictate the relation between the dependent and independent

variables, which results in

solution that

known except for

numerical

coefficient. The outstanding example of this is the form of the spectrum of

turbulent kinetic energy in what

called the "inertial subrange."

Asymptotic invariance Another frequently used approach

to exploit some

the

asymptotic properties of turbulent flows. Turbulent flows are char-

acterized by very high Reynolds numbers;

seems reasonable to require that

Introduction

any proposed descriptions of turbulence should behave properly in

the

limit

the Reynolds number approaches infinity. This

often

very powerful con-

straint, which makes fairly specific results possible. The development of the

theory of turbulent boundary layers (Chapter

case

in point. The limit

process involved in an asymptotic approach

related to vanishingly small

effects of the molecular viscosity. Turbulent flows tend to be almost indepen-

dent of the viscosity (with the exception of the very smallest

scales

of mo-

tion); the asymptotic behavior leads to such concepts

"Reynolds-number

similarity" (asymptotic invariance).

Local invariance Associated with, but distinct from, asymptotic invariance

the

concept of "self-preservation" or local invariance. In simple flow geom-

etries,

the characteristics of the turbulent motion

some point in time and

space appear to be controlled mainly by the immediate environment. The

time and length

scales

of the flow may vary slowly downstream, but,

the

turbulence time

scales

are small enough to permit adjustment to the gradually

changing environment,

often possible to assume that the turbulence

dynamically similar everywhere

nondimensionalized with local length and

time

scales.

For example, the turbulence intensity

wake

order

aU/ay,

where

the local width of the wake and

W/ay

the average

mean-velocity gradient across the wake.

Because turbulence consists of fairly large fluctuations governed by non-

linear equations, one may expect

behavior like that exhibited by simple

nonlinear systems with limit cycles. Such behavior should be largely indepen-

dent of initial conditions; the characteristics of the limit cycle should depend

only on the dynamics of the system and the constraints imposed on

it.

In the

same way, one expects that the structure of turbulence in

given class

shear flows might be in some

state

of dynamical equilibrium in which local

inputs

energy should approximately balance local losses.

the energy

transfer mechanisms in turbulence

are

sufficiently rapid,

tha't effects of

past events do not dominate the dynamics, one may expect that this limit-

cycle type of equilibrium

governed mainly

local parameters such

scale

lengths and times. Simple dimensional methods and similarity arguments can

be very useful in this kind of situation. Because one may want to look for

local scaling laws (both in the spatial and the spectral domain), the problem

finding appropriate length and time

scales

becomes an important one.

Indeed, scaling laws are

the heart of turbulence research.

1.3

The

origin

turbulence

1.3

The origin of turbulence

In flows which are originally laminar, turbulence arises from instabilities

large Reynolds numbers. Laminar pipe flow becomes turbulent

Reynolds

number (based on mean velocity and diameter) in the neighborhood of

2,000

unless great care

taken to avoid creating small disturbances that might

trigger transition from laminar to turbulent flow. Boundary layers in zero

pressure gradient become unstable

Reynolds number

U6"lv

600

approximately

(6"

the displacement thickness,

the free-stream velo-

city, and

the kinematic viscosity). Free shear flows, such

the flow in

mixing layer, become unstable

very low Reynolds numbers because of an

inviscid instability mechanism that does not operate in boundary-layer and

pipe flow. Early stages of transition can easily be seen in the smoke rising

from

cigarette.

On the other hand, turbulence cannot maintain

itself

but depends on

its

environment to obtain energy. A common source of energy for turbulent

velocity fluctuations

shear in the mean flow; other sources, such

buoy-

ancy, exist too. Turbulent flows are generally shear flows.

turbulence

arrives in an environment where there

no shear or other maintenance mech-

anism,

decays: the Reynolds number decreases and the flow tends to

become laminar again. The classic example

turbulence produced by

grid

in uniform flow in

wind tunnel.

Another way to make

turbulent flow laminar or to prevent

laminar

flow from becoming turbulent

to provide for

mechanism that consumes

turbulent kinetic energy. This situation

prevails in turbulent flows with

imposed magnetic fields

low magnetic Reynolds numbers and in atmos-

pheric flows with

stable density stratification, to

cite

two

examples.

Mathematically, the details of transition from laminar to turbulent flow

are rather poorly understood. Much of the theory of instabilities in laminar

flows

linearized theory, valid for very small disturbances;

cannot deal

with the large fluctuation levels in turbulent flow. On the other hand, almost

all

of the theory of turbulent flow

asymptotic theory, fairly accurate

very high Reynolds numbers but inaccurate and incomplete for Reynolds

numbers

which the turbulence cannot maintain

itself.

A noteworthy excep-

tion

the theory of the

late

stage

decay of wind-tunnel turbulence

(Batchelor,

1953).

Experiments have shown that transition

commonly initiated by

pri-

Introduction

mary instability mechanism, which in simple cases

two

dimensional. The

primary instability produces secondary motions, which are generally three

dimensional and become unstable themselves.

sequence of this nature gen

erates intense localized three-dimensional disturbances (turbulent "spots"),

which arise

random positions

random times. These spots grow rapidly

and merge with each other when they become large and numerous to form

field

developed turbulent flow. In other

cases,

turbulence originates from

an instability that causes vortices which subsequently become unstable. Many

wake flows become turbulent in this way.

1.4

Diffusivity

turbulence

The outstanding characteristic of turbulent motion

its

ability to transport

or mix momentum, kinetic energy, and contaminants such

heat, particles,

and moisture. The rates of transfer and mixing are several orders of magni-

tude greater than the rates due to molecular diffusion: the heat transfer and

combustion rates of turbulent combustion in an incinerator are orders of

magnitude larger than the corresponding rates in the laminar flame of

candle.

Diffusion in

problem with

imposed length scale Contrasting laminar and

turbulent diffusion

rates

useful exercise not only for getting acquainted

with turbulence but also for recognizing the multifaceted role of the Rey-

nolds number. Suppose one has

room (with

characteristic linear dimension

in which

heating element (radiator)

installed.

there

no air motion

in the room, heat has to be distributed by molecular diffusion. This process

governed by the diffusion equation

the temperature;

the thermal

diffusivity, assumed to be constant):

(1.4.1)

are not looking for

specific solution of (1.4.1) with

given

set

boundary conditions. Instead, we want to discover the gross consequences of

(1.4.1)

with

the

simple tools of dimensional analysis. Dimensionally,

(1.4.1)

may be interpreted

-rp'

Trn

(1.4.2)

1.4

Diffusivity

turbulence

where

characteristic temperature difference. From

(1.4.21,

we obtain

T,--,

(1.4.3)

which relates the time scale

T,,,

of the molecular diffusion to the independent

parameters

and

If the characteristic linear dimension

(the

length scale)

the room

m, the time

scale

of this diffusion process

of the order

lo6

sec

(more than

100

h). In this estimate the value of

for air

room

temperature and pressure has been used

0.20

cm2/sec).

conclude that

molecular diffusion

rather ineffective in distributing heat through

room.

On the other hand, even fairly weak motions, such

those generated by

small density differences (buoyancy), can disperse heat through the room

quickly. Suppose that the turbulent motion of

the

air in the room may also

be characterized by the length

scale

(that

is,

motions are present of

scales

L).

This

fair assumption, since large-scale motions are most effective in

distributing heat and since the largest possible

scales

of motion can be no

larger than the

size

of the room.

also need

characteristic velocity

(this

may be thought of

an rms amplitude of the velocity fluctuations in the

room). For flow with

length scale

and

velocity

scale

the characteristic

time

Tt--.

(1.4.4)

Apparently,

can be determined only if

can be estimated. Suppose the

radiator heats the air in

its

vicinity by

degrees Kelvin. This causes

buoyant acceleration

g Ae/O,

which

of order

0.3

m/secz if

10°K.

This acceleration probably occurs only near the surface of the radiator. If

has

height

0.1

m, the kinetic energy of the air above the radiator

ghA6/6,

which

of order 0.03 (m/sec)’ per unit mass. This corresponds to

velocity of

cm/sec. Much of the kinetic energy, however,

lost because

the stable vertical temperature gradient in the room (the air near the ceiling

tends to be hotter than the air near the floor). A characteristic velocity

order 5cm/sec may be

reasonable average throughout the room. With

5 cm/sec and

becomes

100

sec,

or about

min.

course, we

still

have to rely on molecular diffusion to even out small-scale irregularities

in the temperature distribution. However, the turbulence generates eddies

small

about

cm (this estimate can

obtained with simple equations

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