TUBULENT FLUXES OF HEAT, MOISTURE AND MOMENTUM: BASICS OF TURBULENCE

What is turbulence? Alfred Blackadar: “No definition of turbulence can be given at this time!”. Thus, we can only agree on some attributes of turbulence (Lumley and Panofsky 1964):

1. Turbulence is stochastic by nature: Even if the equations are deterministic, they are nonlinear Results are highly dependent on small differences in initial state No way to observe the initial state sufficiently

2. Turbulence is three-dimensional Although some 2-D cases can be considered (e.g. cyclones), their ensemble behaviour is different from small scale turbulence in a large 3-D environment

3. In a turbulent environment any 2 particles that are free to move tend to become increasingly distant from each other with time

No way to treat the turbulence in a deterministic way

4. Turbulence is rotational by nature• Vorticity is an essential attribute of turbulence

5. Turbulence is dissipative• The energy tends to shift from large-scale well organized eddies

(small wave numbers) towards smaller eddies and, finally to molecular motions

Vortices are stretchedby turbulence, theirdiameter is reduced

Spectral energy

Energyshift

6. Turbulence is a phenomenon of large Reynolds numbersLarge spatial dimensionsSmall viscosity

The space available for motions is LARGEin comparison to the dimensions of eddies

The Reynolds numberTurbulent motions are highly dependent on the sizes of eddies and characteristic scales of the environment where there eddies exist.The Reynolds number is the ratio between two lengths:

Characteristic scale of the dimensions of thespace available for all scales of motion

Length which measures the thickness of the laminar (viscous) sub-layer (a layer where turbulence is not

maintained (due to its thickness), even if it is present in the other parts of the space considered

L

Re =

L is defined by boundary configuration (e.g. depth of the channel, diameter of the pipeline). In the atmosphere gravity is essential, and turbulence is largely controlled by stable lapse rates.

(laminar sublayer thickness) is related to viscosity:

= / U (1)where is kinematic viscosity U is the velocity of the !LARGEST! scale of motions

U

L

In the pipes (pipelines): L usually is taken asa diameter, and mean current velocity stands for U

Recr 1000

The dimensionless ratio between L and is the Reynolds number:

Re = L/ = LU/ (2)

Osborne Reynolds's father was a priest in the Anglican church but he had an academic background having graduated from Cambridge in 1837, being elected to a fellowship at Queens' College, and being headmaster of first Belfast Collegiate School and then Dedham School in Essex. In fact it was a family with a tradition of the Church and three generations of Osborne's father's family had been the rector of Debach-with-Boulge.

Osborne was born in Belfast when his father was Principal of the Collegiate School there but began his schooling at Dedham when his father was headmaster of the school in that Essex town. After that he received private tutoring to complete his secondary education. He did not go straight to university after his secondary education, however, but rather he took an apprenticeship with the engineering firm of Edward Hayes in 1861. Reynolds wrote (actually in his application for the chair in Manchester in 1868) of his father's influence on him while he was growing up:

In my boyhood I had the advantage of the constant guidance of my father, also a lover of mechanics, and a man of no mean attainments in mathematics and its application to physics.

Reynolds, after gaining experience in the engineering firm, studied mathematics at Cambridge, graduating in 1867. As an undergraduate Reynolds had attended some of the same classes as Rayleigh who was one year ahead of him. As his father had before him, Reynolds was elected to a scholarship at Queens' College. He again took up a post with an engineering firm, this time the civil engineers John Lawson of London, spending a year as a practicing civil engineer.

In 1868 Reynolds became the first professor of engineering in Manchester (and the second in England). Kargon writes:-

... a newly created professorship of engineering was advertised at Owens College, Manchester, at Ј500 per annum. Reynolds applied for the position and, despite his youth and inexperience, was awarded the post.

We should note in passing that Owens College would later become the University of Manchester. Reynolds held this post until he retired in 1905. His early work was on magnetism and electricity but he soon concentrated on hydraulics and hydrodynamics. He also worked on electromagnetic properties of the sun and of comets, and considered tidal motions in rivers.

After 1873 Reynolds concentrated mainly on fluid dynamics and it was in this area that his contributions were of world leading importance. We summarise these contributions. He studied the change in a flow along a pipe when it goes from laminar flow to turbulent flow. In 1886 he formulated a theory of lubrication. Three years later he produced an important theoretical model for turbulent flow and it has become the standard mathematical framework used in the study of turbulence. An account of Reynolds' work on hydrodynamic stability published in 1883 and 1895 is looked at in [8]. The 1883 paper is called An experimental investigation of the circumstances which determine whether the motion of water in parallel channels shall be direct or sinuous and of the law of resistance in parallel channels. The 'Reynolds number' (as it is now called) used in modelling fluid flow which is named after him appears in this work.

Reynolds became a Fellow of the Royal Society in 1877 and, 11 years later, won their Royal Medal. In 1884 he was awarded an honorary degree by the University of Glasgow. By the beginning of the 1900s Reynolds health began to fail and he retired in 1905. Not only did he deteriorate physically but also mentally, which was sad to see in so brilliant a man who was hardly 60 years old.

Not only is Reynolds important in terms of his research, but he is also important for the applied mathematics course he set up at Manchester. Anderson writes in [3]:- Reynolds was a scholarly man with high standards. Engineering education was new to English universities at that time, and Reynolds had definite ideas about its proper form. He believed that all engineering students, no matter what their speciality, should have a common background based in mathematics, physics, and particularly the fundamentals of classical mechanics. ... Despite his intense interest in education, he was not a great lecturer. His lectures were difficult to follow, and he frequently wandered among topics with little or no connection.

Lamb, who knew Reynolds well both as a man and as a fellow worker in fluid dynamics, wrote:-

The character of Reynolds was like his writings, strongly individual. He was conscious of the value of his work, but was content to leave it to the mature judgement of the scientific world. For advertisement he had no taste, and undue pretension on the part of others only elicited a tolerant smile. To his pupils he was most generous in the opportunities for valuable work which he put in their way, and in the share of cooperation. Somewhat reserved in serious or personal matters and occasionally combative and tenacious in debate, he was in the ordinary relations of life the most kindly and genial of companions.

Osborne Reynolds

Born: 23 Aug 1842 in Belfast, IrelandDied: 21 Feb 1912 in Watchet, Somerset, England

On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Royal Society, Phil. Trans., 1895.

Reynolds proceeded to measure the critical velocity for onset of eddies using three tubes of different diameter and in each case varying the water temperature. To a first approximation, the Reynolds Numbers based on these critical values of velocity were found to be the same (about 13000) for each of the tubes and for all water temperatures. He then set out to find the critical condition for an eddying flow to change into non-turbulent flow, referring to this as the `inferior limit'. To do this, he allowed water to flow in a disturbed state from the mains through a length of pipe and measured the pressure-drop over a five-foot distance near the outlet.

Reynolds’ three tubes

Pressure measurements

www

vvv

uuu

Mean values Turbulent portion

The Reynolds approach to the equations of a turbulent fluid

Reynolds separated each of the velocity components u, v and w into two parts:

(3)

If we now observe a sequence of velocities of particles at times that are sufficiently separated to be considered as uncorrelated with each other, the mean value of each such sequence is independent of the other samples and the mean values of deviations from the mean (u’, v’, w’) are zero:

(4),...0,)(,0, uuvuvuuuu

Equation (4) represents the so-called ensemble averaging.

Reynolds approach:

Averaging of the equations of motion and the equation of continuity,i.e. to replace in each equation

u, v and w

by u, v and w and u’, v’ and w’

and to get the equations for the changes in

u, v and w

dt

d The rate of change following the instantaneous motion

Time and spatial derivatives at a

fixed point

Lagrangianderivative

Eulerianderivative

(5)

Continuity equation

tx

,

Two forms of the continuity equation

0

0

z

w

y

v

x

u

zw

yv

xu

t

z

w

y

v

x

u

dt

d

(6)

(7)

0

0

z

w

y

v

x

u

zw

yv

xu

t

z

w

y

v

x

u

dt

d

0

k

k

kk x

u

xu

t

the derivative of product

k stands for a dummy index which implies summation over the indices corresponding to the three Cartesian coordinate directions

0

k

k

x

u

t

(8)

uuu

ukuk

- Reynolds postulates

!

The Reynolds procedure (substituting from (3) and averaging the result) gives:

0

k

k

x

u

t

Averaging operators are applied to the derivatives, i.e. it is the derivative what is averaged

0

k

k

x

u

t

Equation of continuity can be applied to the mean motion andmean velocity without change

0

k

k

x

u

t

(10)

(11)

Specific humidity (the mass of water vapor per unit mass of air)Kinetic energy per unit massSpecific entropy (Cpln)

Flux and general conservation equation

Properties whose amounts are identified with a mass of fluid:

q

These properties are assumed to be conservative, i.e. they do not change just because of their motion

qSdt

dq

Equation of the conservation of property q:

(12)

Source strength of the quantity q per unit mass

+Continuity

equation multiplied by q term-by-term

Let’s assume that q is specific humidity. Then (12) can be expanded to

qk

k Sx

qu

t

q

k

kq x

quS

t

q

0

k

k

x

uq

tq

(13)

(14)

The rate of internal

production of q perunit volume at thesame fixed point

The rate of change

of the amount of qper unit volume

measuredat a fixed position

?(15)

The rate of convergence

of a vector whosecomponents are

(ukq)

A is a small area on the surface to one of the 3 axis.

ukA is a volume of a cylinder with a height of

uk(unit time).

uk is the mass transported through a unit area of the

surface in one unit of time.

ukq is the amount of property transported per unit

area and per unit time across a surface to xk.

This is flux of q in xk.directionA

V

xk

uk

k

k

k

kq x

qu

x

quS

t

q

''

The flux due to mean state The eddy flux

Let’s average (15), substituting: q=q+q’, uk=uk+uk’:

(q+q’)(uk+uk’) =

=quk+ q’uk+quk’ +q’uk’ (16)

The averaged conservation equation:

=0

(17)

The closure problem (K-theory)

Joseph Boussinesq(1842-1929)

Ludwig Prandtl1875-1953

Ernst Schmidt1892-1975

We need to estimate the eddy flux ofproperty q at ref. level zref

For q we use Taylor series expansion:

z

qzzzqzq refref

)()()(

Parcel moves down through zref with a vertical velocity -|w’|. It has been last mixed with its environment at distance l1 from zref where it took its mean value. Deviation from the mean at reference level and the corresponding contribution to the flux:

(18)

z

qlwqw

z

qlzqzqq

z

qlwqw

z

qlzqzqq

ref

ref

2222222

1111111

,)()(

,)()(

(19)

Averaging over all parcels:

12~,

TLKz

qKqw

Size of the largest energy containing eddy (mixing Prandtl length)

Kinematic exchange coefficients (eddy diffusivity)(20)

Surface stressWhat is stress?

Stress is the force acting across the boundary surface and proportional to the area of surface, across which it acts.

kk

iiki x

x

uuxu

)0()( Stress is the tensor of rank 2normal stresses tangential stresses

kkkjki

jkjjji

ikijii

Motion of the fluid, whenvelocity is expandedinto Taylor series. ij is the

unit symmetric tensor.

3 components of the force x 3 surfaces = 9 stress components

ki

k

k

ik

i

k

k

iiki x

x

u

x

ux

x

u

x

uuxu

2

1

2

1)0()(

Scalar product: ½ displacement vector●

i

k

k

i

x

u

x

u

To get the rate of pure deformation, we have to subtract out the divergence:

kk

i

i

k

k

iik x

x

u

x

u

x

u 3

2

Stress

kj

j

i

k

k

i

kii

k

ik

i xx

u

x

u

x

u

xx

pg

x

uu

t

u

3

213

total stress

The motion equations

General form of the Navier-stokes equations (without Coriolis force):

ik

i

k

k

i

kii

k

ik

i uux

u

x

u

xx

pg

x

uu

t

u

113

(21)

Averaging: (22)

z

uKuu

x

u

x

uzxik

i

k

k

iki

Using K-theory:

(23)

Eddy diffusivity10-5m2s-1

Molecular viscosity10-5m2s-1

z

ww

y

wv

x

wug

z

p

dt

wd

z

wv

y

vv

x

vuuf

y

p

dt

vd

z

wu

y

vu

x

uuvf

x

p

dt

ud

1

1

1

Verticalmomentum

fluxes

(24)

z

qw

y

qv

x

qu

dt

qd

z

w

y

v

x

u

dt

d

(25)Vertical fluxes of

heat and moisture

momentumflux

sensibleheatflux

latentheat flux

(evaporation)

measurements parameterization