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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
Lund University / Division of Fluid Mechanics / 25.10.11
Tensor notation
Lund University / Division of Fluid Mechanics / 25.10.11
Tensor notation
• Gradient of a scalar
• Divergence of a vector
• Divergence of a matrix
Lund University / Division of Fluid Mechanics / 25.10.11
Governing Equations
Lund University / Division of Fluid Mechanics / 25.10.11
Governing Equations
Compressible Flows :
• Mass conservation
• Momentum conservation
• Energy conservation
• Equation of state
Unknowns :
• Density
• Velocities (3)
• Pressure
• Energy
Lund University / Division of Fluid Mechanics / 25.10.11
Governing Equations
Compressible Flows :
• Mass conservation
• Momentum conservation
• Energy conservation
• Equation of state
Unknowns :
• Density
• Velocities (3)
• Pressure
• Energy
Incompressible Flows :
• Mass conservation
• Momentum conservation
Unknowns :
• Velocities (3)
• Pressure
Lund University / Division of Fluid Mechanics / 25.10.11
• 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
Lund University / Division of Fluid Mechanics / 25.10.11
• 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
Lund University / Division of Fluid Mechanics / 25.10.11
0
0
i
i
x
u
V
0
0
i
i
x
u
V
0
t
konstant
Mass conservation
• Stationary (steady) flow :
• Incompressible flow :
Lund University / Division of Fluid Mechanics / 25.10.11
• 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
Lund University / Division of Fluid Mechanics / 25.10.11
• 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
Lund University / Division of Fluid Mechanics / 25.10.11
• 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
Momentum conservation
Lund University / Division of Fluid Mechanics / 25.10.11
• 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
Momentum conservation
j
ij
ii
x
uu
t
u
Dt
Du
Convective acceleration
Lund University / Division of Fluid Mechanics / 25.10.11
• 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
Momentum conservation
Lund University / Division of Fluid Mechanics / 25.10.11
• Surface forces :
we use again the infinitesimal
control volume
x
y
z dx
dz
dy
dydzdxdx
xxxx
dydzxx
dxdzyx
dxdzdydy
yx
yx
dxdydzzyx
dF zxyxxxxs
,
Momentum conservation
Lund University / Division of Fluid Mechanics / 25.10.11
• Surface forces :
dxdydzzyx
dF zxyxxxxs
,
dxdydzzyx
dFzyyyxy
ys
,
dxdydzzyx
dF zzyzxzzs
,
x:
y:
z:
Momentum conservation
Lund University / Division of Fluid Mechanics / 25.10.11
• Surface forces :
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
Momentum conservation
Lund University / Division of Fluid Mechanics / 25.10.11
• 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
Momentum conservation
Lund University / Division of Fluid Mechanics / 25.10.11
• Motion and deformation of a fluid element :
Translation:
Rotation:
Shear:
Volume
dilatation:
Momentum conservation
Lund University / Division of Fluid Mechanics / 25.10.11
dy
dx
tdyy
vdy
tdxx
udx
tdyy
u
tdxx
v
d
d
Momentum conservation
• Shear deformation :
Lund University / Division of Fluid Mechanics / 25.10.11
• 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
Momentum conservation
Lund University / Division of Fluid Mechanics / 25.10.11
• Shear deformation :
• 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
Momentum conservation
Rate of strain:
i
j
j
iij
x
u
x
uS
2
1
ijij S 2
Lund University / Division of Fluid Mechanics / 25.10.11
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
Momentum conservation
Lund University / Division of Fluid Mechanics / 25.10.11
• 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
Lund University / Division of Fluid Mechanics / 25.10.11
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 :
Lund University / Division of Fluid Mechanics / 25.10.11
• 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
Lund University / Division of Fluid Mechanics / 25.10.11
• Viscous work :
dy
dx
xw dxx
ww xx
dxdydzVdxdydzz
w
y
w
x
wW zyxv
xzxyxxx wvuw
Energy conservation
Lund University / Division of Fluid Mechanics / 25.10.11
• 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
Lund University / Division of Fluid Mechanics / 25.10.11
• Mass :
• Momentum :
• Energy :
VTkVppVDt
De
pgDt
VD
0
V
t
Conservation equations
Lund University / Division of Fluid Mechanics / 25.10.11
• Mass :
• Momentum :
• Energy :
Conservation equations
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
Lund University / Division of Fluid Mechanics / 25.10.11
For incompressible flow, Newtonian fluid with
constant density, viscosity and conductivity:
• Mass :
• Momentum :
• Energy :
VTkDt
DTc Tp 2
0 V
VpgDt
VD 2
Conservation equations
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
Lund University / Division of Fluid Mechanics / 25.10.11
• Definition of rotation and vorticity :
• For a 2-D flow :
VVrot 2
0,,vuV
y
u
x
vV ,0,0
zyx,,
Vorticity
Lund University / Division of Fluid Mechanics / 25.10.11
dy
dx
tdyy
vdy
tdxx
udx
tdyy
u
tdxx
v
d
d
Vorticity
Lund University / Division of Fluid Mechanics / 25.10.11
• 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
Lund University / Division of Fluid Mechanics / 25.10.11
• 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
Lund University / Division of Fluid Mechanics / 25.10.11
• Take the curl of the Navier-Stokes equations :
j
ij
jj
i
j
ij
i
x
u
xxxu
t
2
Vorticity
Lund University / Division of Fluid Mechanics / 25.10.11
Statistical description of turbulent flow
Lund University / Division of Fluid Mechanics / 25.10.11
Turbulent jet flow
Lund University / Division of Fluid Mechanics / 25.10.11
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
Lund University / Division of Fluid Mechanics / 25.10.11
0
i
i
x
u
jj
i
i
i
j
ij
i
xx
u
x
pg
x
uu
t
u
2
Deterministic Random
Lund University / Division of Fluid Mechanics / 25.10.11
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
Lund University / Division of Fluid Mechanics / 25.10.11
Lorenz equations
Two cases: 23 28
Lund University / Division of Fluid Mechanics / 25.10.11
Lorenz equations
231.0)0( x 1000001.0)0( x
Difference
Lund University / Division of Fluid Mechanics / 25.10.11
Lorenz equations
281.0)0( x 1000001.0)0( x
Difference
Lund University / Division of Fluid Mechanics / 25.10.11
Lorenz equations
Observations
: 23 28
What can we learn from this exercise regarding flows?
Lund University / Division of Fluid Mechanics / 25.10.11
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
Lund University / Division of Fluid Mechanics / 25.10.11
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
Lund University / Division of Fluid Mechanics / 25.10.11
Random variables
Tools for characterising random variables
Probability Density Function (pdf)
dV
VdFVf
0Vf 1
Vf
Properties:
Probability
that ba VUV
dVVfVFVF
b
a
V
V
ab
Lund University / Division of Fluid Mechanics / 25.10.11
Random variables
Tools for characterising 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
Lund University / Division of Fluid Mechanics / 25.10.11
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
Lund University / Division of Fluid Mechanics / 25.10.11
Turbulent flow
Turbulent flow
Gaussian process
Lund University / Division of Fluid Mechanics / 25.10.11
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
Lund University / Division of Fluid Mechanics / 25.10.11
Multi-time statistics:
Autocovariance
dsesRE si
1
Frequency
spectrum
deEsR si
1
Lund University / Division of Fluid Mechanics / 25.10.11
Autocorrelation function
Lund University / Division of Fluid Mechanics / 25.10.11
Frequency spectra
Lund University / Division of Fluid Mechanics / 25.10.11
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
Lund University / Division of Fluid Mechanics / 25.10.11
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
Lund University / Division of Fluid Mechanics / 25.10.11
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
Lund University / Division of Fluid Mechanics / 25.10.11
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
Lund University / Division of Fluid Mechanics / 25.10.11
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
Lund University / Division of Fluid Mechanics / 25.10.11
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