turbulence - theory and modeling mvk140/mmv010f lecture 2 ...€¦ · turbulence - theory and...

Turbulence - Theory and Modeling MVK140/MMV010F Lecture 2 :

Governing Equations and Statistical Tools

Holger Grosshans

Division of Fluid Mechanics

25.10.11

Lund University / Division of Fluid Mechanics / 25.10.11

Content

• Tensor notation

• Governing Equations

• Statistical Tools


Tensor notation


Tensor notation

• Gradient of a scalar

• Divergence of a vector

• Divergence of a matrix


Governing Equations


Governing Equations

Compressible Flows :

• Mass conservation

• Momentum conservation

• Energy conservation

• Equation of state

Unknowns :

• Density

• Velocities (3)

• Pressure

• Energy


Governing Equations

Compressible Flows :



• Energy conservation

• Equation of state

Unknowns :

• Density

• Velocities (3)

• Pressure

• Energy

Incompressible Flows :



Unknowns :

• Velocities (3)

• Pressure


• Mass is constant :

• Use Reynolds Transport Theorem :

• Use differential form :

• Rate of mass outflow in x-direction :

x

y

z dx

dz

dy

udydz dydzudu

0inut

CV

mmdt

dxdydzt

dt

CV

dxx

uud

Mass conservation

𝑫𝑴

𝑫𝒕= 𝟎


• Mass outflow over all control surfaces :

• Divide by volume :

• Vector notation :

• Einstein notation :

0

wdxdyvdxdzudydz

dxdydzz

wwdxdzdy

y

vvdydzdx

x

uudxdydz

t

0

0

0

i

i

x

u

t

Vt

z

w

y

v

x

u

t

Mass conservation


0

0

i

i

x

u

V

0

0

i

i

x

u

V

0

t

konstant

Mass conservation

• Stationary (steady) flow :

• Incompressible flow :


• When is incompressibility a reasonable assumption ?

x

u

xu

x

u

x

u

x

u

xu

V

dVd

VV

dadp 2

Speed of sound

VdVdp

11 2

2

2

22 Ma

a

V

V

dp

a

dp

Mach number

Usual limit: 3.0Ma

Compressibility criterion


• Newtons 2nd law : ma = F

• Use Reynolds Transport Theorem :

• Use differential form :

• Rate of mass outflow in x-direction :

x

y

z dx

dz

dy

dydzVu

dydzVudVu

FVmVmd

t

Vinut

CV

dxdydz

t

Vd

t

V

CV

Momentum conservation


• Use same strategy as for mass conservation :

• Split the derivatives :

• Material / substantial derivative :

Fdxdydz

z

Vw

y

Vv

x

Vu

t

V

Fdxdydz

z

Vw

y

Vv

x

Vu

t

V

z

w

y

v

x

u

tV

Mass conservation (=0)

FdxdydzDt

VD



• Acceleration in different frames of reference :

Lagrangian :

Eulerian :

• Eulerian frame :

material/substantial

derivative

t

VatzyxVV

,,, 000

t

VatzyxVV

,,,

z

Vw

y

Vv

x

Vu

t

V

Dt

VD

VVt

V

Dt

VD

Local acceleration


j

ij

ii

x

uu

t

u

Dt

Du

Convective acceleration


• Forces acting on the element = volume + surface forces

• Volume forces : gravity

• Surface forces :

dxdydzgFd g zyx gggg ,,

yyyx

yz

xy

xx

zz

zy

zxxz

The stress tensor:

zzyzxz

zyyyxy

zxyxxx

ij

p

p

p




we use again the infinitesimal

control volume

x

y

z dx

dz

dy

dydzdxdx

xxxx

dydzxx

dxdzyx

dxdzdydy

yx

yx

dxdydzzyx

dF zxyxxxxs

,




dxdydzzyx

dF zxyxxxxs

,

dxdydzzyx

dFzyyyxy

ys

,

dxdydzzyx

dF zzyzxzzs

,

x:

y:

z:




Divide by the volume and use ijijij p

Kronecker’s delta

otherwise 0

if 1 jiijzyxx

p

d

dFzxyxxxxs

,

zyxy

p

d

dF zyyyxyys

,

zyxz

p

d

dFzzyzxzzs

,

dxdydzd



• Surface forces +

volume forces :

pd

Fd s

pgDt

VD

zyxx

pg

z

uw

y

uv

x

uu

t

u zxyxxxx

zyxy

pg

z

vw

y

vv

x

vu

t

v zyyyxy

y

zyxz

pg

z

ww

y

wv

x

wu

t

w zzyzxzz

gd

Fd g

convective acceleration Local acceleration gravity Pressure force Viscous force

j

ij

i

i

j

ij

i

xx

pg

x

uu

t

u



• Motion and deformation of a fluid element :

Translation:

Rotation:

Shear:

Volume

dilatation:



dy

dx

tdyy

vdy

tdxx

udx

tdyy

u

tdxx

v

d

d


• Shear deformation :



rate of strain

for small angles

dt

d

dt

dxy

2

1

dtx

udx

dxx

v

dt

d

1

dty

vdy

dyy

u

dt

d

1

dy

dx

tdyy

vdy

tdxx

udx

tdyy

u

tdxx

v

d

d




• For a Newtonian fluid the stresses are linearly

dependent on the rate of deformation

y

u

dt

dx

v

dt

d

dt

0

xyxy 2

ibleincompress if

0

3

22

ijijij V

z

w

y

w

z

v

x

w

z

u

y

w

z

v

y

v

x

v

y

u

x

w

z

u

x

v

y

u

x

u

ij

2

2

2

Dynamic

viscosity

dy

dx

tdyy

vdy

tdxx

udx

tdyy

u

tdxx

v

d

d


Rate of strain:

i

j

j

iij

x

u

x

uS

2

1

ijij S 2


Momentum conservation:

pgDt

VD

2

2

2

2

2

2

z

u

y

u

x

u

x

pg

z

uw

y

uv

x

uu

t

ux

2

2

2

2

2

2

z

v

y

v

x

v

y

pg

z

vw

y

vv

x

vu

t

vy

2

2

2

2

2

2

z

w

y

w

x

w

z

pg

z

ww

y

wv

x

wu

t

wz

Can for incompressible flow of a Newtonian fluid be written as:

VpgDt

VD 2

The Navier-

Stokes

equations

j

ij

i

i

j

ij

i

xx

pg

x

uu

t

u

jj

i

i

i

j

ij

i

xx

u

x

pg

x

uu

t

u

2



• Energy conservation for a control volume:

• For an infinitesimal element :

vWQdxdydz

z

w

y

v

x

u

t

e

dAnVp

eeddt

dWWQ

CSCV

ssvs

Note that 0sW for infinitesimal CV

pe

Energy conservation


dxdydz

z

w

y

v

x

u

t

p

z

w

y

v

x

u

te

zw

yv

xu

t

eWQ v

Mass conservation (=0)

dxdydzVppVDt

DeWQ v

Energy conservation

• Applying product rule :

• After some manipulation :


• Heat flux :

• Radiation is neglected

• Conductive heat transfer :

(Fourier’s law)

• Sum over surfaces :

• Use of Fouriers law :

Tkq

dy

dx

x

Tkqx

dx

x

qq xx

dxdydzqdxdydzz

q

y

q

x

qQ zyx

dxdydzTkQ

Energy conservation


• Viscous work :

dy

dx

xw dxx

ww xx

dxdydzVdxdydzz

w

y

w

x

wW zyxv

xzxyxxx wvuw

Energy conservation


• Sum of all terms :

• Rewrite the viscous term :

VTkVppVDt

De

VVV T

Viscous dissipation,

always positiv

222222

2x

w

z

u

z

v

y

w

y

u

x

v

z

w

y

v

x

uVT

For incompressibel and Newtonian:

Energy conservation


• Mass :

• Momentum :

• Energy :

VTkVppVDt

De

pgDt

VD

0

V

t

Conservation equations


• Mass :

• Momentum :

• Energy :


0

i

i

x

u

t

j

ij

i

i

j

ij

i

xx

pg

x

uu

t

u

iji

jjjj

j

j

j

j

j uxx

Tk

xx

up

x

pu

x

eu

t

e


For incompressible flow, Newtonian fluid with

constant density, viscosity and conductivity:

• Mass :

• Momentum :

• Energy :

VTkDt

DTc Tp 2

0 V

VpgDt

VD 2


Note that energy equation is now decoupled from

equations for mass and momentum

0

i

i

x

u

jj

i

i

i

j

ij

i

xx

u

x

pg

x

uu

t

u

2

j

iij

jjj

jpx

u

xx

Tk

x

Tu

t

Tc

2


• Definition of rotation and vorticity :

• For a 2-D flow :

VVrot 2

0,,vuV

y

u

x

vV ,0,0

zyx,,

Vorticity


dy

dx

tdyy

vdy

tdxx

udx

tdyy

u

tdxx

v

d

d

Vorticity


• Rotation of a fluid element :

angular velocity

for small angles

dt

d

dt

dz

2

1

dtx

udx

dxdtx

v

d

1

dty

vdy

dydty

u

d

1

dy

dx

tdyy

vdy

tdxx

udx

tdyy

u

tdxx

v

d

d

Vorticity


• Rotation of a fluid element :

y

u

dt

dx

v

dt

d

dt

0

dy

dx

tdyy

vdy

tdxx

udx

tdyy

u

tdxx

v

d

d

y

u

x

vz

2

1Angular velocity

z

v

y

wx

2

1

x

w

z

uy

2

1

Note that for 2D-flow: 0 yx

VVrot 2

1

2

1

Vorticity: 2

Irrotational if 0

Vorticity

Rate of rotation:

i

j

j

iij

x

u

x

u

2

1

j

kijkix

u


• Take the curl of the Navier-Stokes equations :

j

ij

jj

i

j

ij

i

x

u

xxxu

t

2

Vorticity


Statistical description of turbulent flow


Turbulent jet flow


Definition of

randomness

Example: A fluid mechanics experiment with well defined

boundary contitions.

Resul

t An event A: 6.0 CA

Three possibilities:

A is certain

A is impossible

A is random if neither impossible nor

certain, then C is a random variable


0

i

i

x

u

jj

i

i

i

j

ij

i

xx

u

x

pg

x

uu

t

u

2

Deterministic Random


Lorenz equations

xyzdt

dz

xzyxdt

dy

xydt

dx

10

3

8

Two cases: 23 28

Initial values:

1.0)0(

1.0)0(

1.0)0(

z

y

x


Lorenz equations

Two cases: 23 28


Lorenz equations

231.0)0( x 1000001.0)0( x

Difference


Lorenz equations

281.0)0( x 1000001.0)0( x

Difference


Lorenz equations

Observations

: 23 28

What can we learn from this exercise regarding flows?


There are always perturbations

originating from boundary

conditions, initial conditions etc.

present in a flow.

Turbulent flows are acutely sensitive

to perturbations

Turbulence is only meaningful to

describe in a statistical sense


Random variables

Tools for characterising random variables

Cumulative Distribution Function (cdf)

Probability

that bVU U is a random

variable,

V is the sample

space bb VFVUPP

abab VVVFVF for

0F

1F

impossible

certain


Random variables


Probability Density Function (pdf)

dV

VdFVf

0Vf 1

Vf

Properties:

Probability

that ba VUV

dVVfVFVF

b

a

V

V

ab


Random variables


Moments:

dVVVfU

1st

moment,

mean

Mean of U

fluctuation

2nd moment,

variance

UUu

dVVfUVuU

222var

Standard deviation = r.m.s. 2u

3rd moment,

skewness

dVVfUVu

333

4th moment,

flatness

(curtosis)

dVVfUVu

444


Random processes

Let the random variable, U be a function of time, it

is the called a random process.

At each time instant

V

tVFtVf

,;

NNNNN VtUVtUVtUPtVtVtVF ,...,,,;...;,;, 22112211

Contains no information about the

coupling in time. Hence, several different

temporal behaviours can have the same

one-time pdf

Joint N-time cdf

Normally, impossible


Turbulent flow

Turbulent flow

Gaussian process


Statistically stationary: All statistics are

invariant under a shift in time

Multi-time statistics:

Autocovariance stutusR

Autocorrelation

function

2tu

stutus

10 1s

Integral time scale dss

0


Multi-time statistics:

Autocovariance

dsesRE si

1

Frequency

spectrum

deEsR si

1


Autocorrelation function


Frequency spectra


Random fields

U is now a time dependent random vector field

One-point, one-time joint pdf

321

3 ,,,;

VVV

tFtf

xVxV

Contains no information about the

coupling in time or space.

Joint N-point, N-time pdf

Normally, impossible

)()()()2()2()2()1()1()1( ,,,...,,,,,, NNN tttf xVxVxV


Statistically stationary: All statistics are

invariant under a shift in time

Statistically homogeneous: All statistics are

independent of position

Homogeneous turbulence: The statistics of the

velocity fluctuations are independent of

position

Isotropic turbulence: The statistics of the

velocity fluctuations are independent of

coordinate system rotations and reflections,

i.e. direction independent


tututR jiij ,,,, rxxxr Two-point

correlation

Integral length

scale

dstrR

tRtL

0

11

11

11 ,,,,0

1, xe

xx 1

Wave number spectra in

homogeneous turbulence

Velocity spectrum

tensor

rrκrκ dtRet ij

iij ,,

κκrrκ dtetR ij

iij ,,

Two-point

correlation

Wave number vector: κ

Wave length: κ

2

Energy spectrum

function

κκκ dκttκE ii ,,

iiii uutRκdtκE2

1,0

2

1,

0

Turbulent kinetic

energy


Averaging

Time

average: du

Ttu

Tt

t

1

Ensemble

average:

Mean: dVtVVftU ;

Questions:

•What is the difference between mean and average?

•Why would we need different types of averages?

tuN

tu

N

n

n

1

)(1


Averaging

Variance: duuT

u

Tt

t

22 1

N

n

n uuN

u

1

2)(2 1

Standard

deviation or root-

mean-square

(rms):

rmsuu 2

uu

u


Reynolds equations

Notation that will be used

from now on in the lecture

notes:

Average

: Fluctuation:

Instantaneou

s:

u

u

'u

Reynolds decomposition

'uuu

u

'u

turbulence - theory and modeling mvk140/mmv010f lecture 2 ...€¦ · turbulence - theory and...

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