Remark: foils with „black background“could be skipped, they are aimed to the more advanced courses
Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010
Turbulent flows,…
Computer Fluid Dynamics E181107CFD6
2181106
Some figures and equations are copied from Wikipedia
TurbulenceCFD6
( ) ( ( ) ) M
uu u p e u S
tρ µ∂ + ⋅∇ = −∇ + ∇⋅ ∇ +
∂
( ( )) ( ( ) ) M
uuu p e u S
tρ µ∂ + ∇ = −∇ + ∇ ⋅ ∇ +
∂
i
Source of nonlinearities of NS equations are inertial and viscous terms
for incompressible fluids the equivalent conservative form
( ( ) ) ( ( ) ) M
uuu u u p e u S
tρ µ∂ + ∇ − ∇ = −∇ + ∇ ⋅ ∇ +
∂
i i
The convection term is a quadratic function of velocities – source of
nonlinearities and turbulent phenomena
Instabilities Rayleigh Benard convectionCFD6
Horizontal liquid layer heated from below is in still (viscous forces attenuate small disturbances) until the buoyancy exceeds a critical level RaL. Then the more or less regular cells with circulating fluid are formed.
>1100L
Tu
Tb
Even if the stability limit was exceeded the flow pattern is steady and remains laminar. At this stage the nonlinear convective term is not so important. Only if RaL>109 (approximately) eddies start to be chaotic, velocity fluctuates and the flow is turbulent.
Instabilities Karman vortex streetCFD6
Repeating pattern of swirling vortices caused by the unsteady separation of flow of a fluid over bluff bodies. A vortex street will only be observed above a limiting Re value of about 90.
L
Even if the stability limit was exceeded the flow pattern is steady and remains laminar.
TurbulenceCFD6
2
( )Re
uu uD
u
ρ ρµ µ∇= =
∇
i
Relative magnitude of inertial and viscous terms is Reynolds number
Increasing Re increases nonlinearity of NS equations. This nonlinearity leads to sensitivity of NS solution to flow disturbances.
Laminar flow : Re<Recrit
Turbulent flow : Re>Recrit
Distarbunce is attenuated
Distarbunce is amplified
Turbulence - fluctuationsCFD6
Turbulence can be defined as a deterministic chaos. Velocity and pressure fields are NON-STATIONARY (du/dt is nonzero) even if flowrate and boundary conditions are constant. Trajectory of individual particles are extremely sensitive to initial conditions (even the particles that are very close at some moment diverge apart during time evolution). .
Velocities, pressures, temperatures… are still solutions of NS and energy equations, however they are nonstationary and form chaotically oscillating vortices (eddies). Time and spatial profiles of transported properties are characterized by fluctuations
'u u u= +Actual value at given time and space
Mean value
fluctuation
Turbulence - fluctuationsCFD6
Statistics of turbulent fluctuations
Mean values (remark: mean values of fluctuations are zero)
rms (root mean square)
Kinetic energy of turbulence
Intensity of turbulence
Reynolds stresses
, , ,u p Tρ
2'u
2 2 21( ' ' ' )
2k u v w= + +
2( )/
3I k u=
2 2 2, , , , ,' , ' , ' , ' ', ' 't xx t yy t zz t xy t xzu v w u v u wτ ρ τ ρ τ ρ τ ρ τ ρ= = = = =
Turbulent eddies - scalesCFD6
E(κ)
κ=2πf/u1/L 1/η
Large energetic eddies (size L) break to smaller eddies. This transformation
is not affected by viscosity
Inertial subrange (inertial effects dominate and spectral energy depends
only upon wavenumber and ε)
Smallest eddies (size is called Kolmogorovscale) disappear, because kinetic energy
is converted to heat by friction
Spectral energy of turbulent eddies
wavenumber (1/size of eddy)
Kinetic energy of turbulent fluctuations is the sum of energies of turbulent eddies of different sizes
/ /
0
1( )
2 i ik u u E dκ κ∞
= = ∫
Typical values of frequency f~10 kHz, Kolmogorov scale η~0.01 up to 0.1 mm
Kolmogorov scale η decreases with the increasing Re
Turbulent eddies - scalesCFD6
Kolmogorov scales (the smallest turbulent eddies) follow from dimensional analysis, assuming that averything depends only upon the kinematic viscosity ν and upon the rate of energy supply ε in the energetic cascade (only for small isotropic eddies, of couse)
3
4νηε
= ντε
= 4u νε=
Length scale Time scale velocity scale
These expressions follow from dimension of viscosity ν [m2/s] and the rate of energy dissipation ε [m2/s3]
2 2 2 2
3 3[ ] [ ][ ]
1 2 2
0 3
m m mm
s s s
α β
α β α β
α β α β
η ν ε
α βα β
+
+
=
= =
= += +
3 / 4
1/ 4
αβ
== −
Example tutorialCFD6
Derive time scale of the smallest turbulent eddies
ντε
=
Time scale
2 2 2 2
3 3[ ] [ ][ ]
0 2 2
1 3
m m ms
s s s
α β
α β α β
α β α β
τ ν ε
α βα β
+
+
=
= =
= +− = +
1/ 2
1/ 2
αβ
== −
Turbulent eddies - scalesCFD6
Wikipedia (abbreviated)
Turbulent flow is composed by "eddies" of different sizes. The sizes define a characteristiclength scale for the eddies, which are also characterized by velocity scales and time scales(turnover time) dependent on the length scale. The large eddies are unstable and break uporiginating smaller eddies, and the kinetic energy of the initial large eddy is divided into thesmaller eddies that stemmed from it. The energy is passed down from the large scales of themotion to smaller scales until reaching a sufficiently small length scale such that the viscosityof the fluid can effectively dissipate the kinetic energy into internal energy.In his original theory of 1941, Kolmogorov postulated that for very high Reynolds number, thesmall scale turbulent motions are statistically isotropic. In general, the large scales of a flow are not isotropic, since they are determined by the particular geometrical features of theboundaries (the size characterizing the large scales will be denoted as L).
Kolmogorov introduced a hypothesis: for very high Reynolds numbers the statistics of smallscales are universally and uniquely determined by the kinematic viscosity (ν) and the rate ofenergy dissipation (ε). With only these two parameters, the unique length that can be formedby dimensional analysis is
DNS Direct Numerical SimulationCFD6
ηηηη L
In order to resolve all details of turbulent structures it is necessary to use mesh with grid size less than the size of the smallest (Kolmgorov) eddies. N-grid points in one direction should be
3 3 3/4 3 9/43 9/4 9/4
3 9/4 9/4 3/4( ) Re
LN
L L L u LuN
L τ
ηε
η ν ν ν
>
> = = = =
3u
Lε ≃
ReuL
τ ν=Re
uL
ν=
Velocity scale u in previous expression is related to magnitude of turbulent fluctuations (rms of u’, or √k). The Reτ related to the velocity fluctuation is called turbulent Reynolds number.
(based upon dimensional ground)
114 k2100 M4650230000
63 k150 M145061600
47 k40 M80030800
32 k6.7 M38012300
No.oftime steps
No.of grid points in DNS
Table concerns DNS modelling of channel flow experiments (rewritten from Wilcox: Turbulence modelling, chapter 8).
9/4
3/4_
Re
Re
DNS
time steps
N
N
τ
τ
∼
∼
Remark: Re~106 or 107 at flow around a car or flow around wings
DNS Direct Numerical SimulationCFD6
AbstractThe three-dimensional compressible Navier-Stokes equations are approximated by a fifth
order upwind compact and a sixth order symmetrical compact difference relations combined with threestage
Ronge-Kutta method. The computed results are presented for convective Mach number Mc =
0.8 and Re = 200 with initial data which have equal and opposite oblique waves. From the computed
results we can see the variation of coherent structures with time integration and full process of instability,
formation of A -vortices, double horseshoe vortices and mushroom structures. The large structures
break into small and smaller vortex structures. Finally, the movement of small structure becomes dominant,
and flow field turns into turbulence. It is noted that production of small vortex structures is combined
with turning of symmetrical structures to unsymmetrical ones. It is shown in the present computation
that the flow field turns into turbulence directly from initial instability and there is not vortex pairing in
process of transition. It means that for large convective Mach number the transition mechanism for
compressible mixing layer differs from that in incompressible mixing layer.
Direct numerical simulation of transition and turbulence in compressible mixing layer
FU Dexun , MA Yanwen ZHANG Linbo
Vol 43 No.4, SCIENCE IN CHINA (Series A), April 2000
DNS Direct Numerical SimulationCFD6
DNS AND LES OF TURBULENT BACKWARD-FACING STEP
FLOW USING 2ND- AND 4TH-ORDER DISCRETIZATION
ADNAN MERI AND HANS WENGLE
Abstract. Results are presented from a Direct Numerical Simulation (DNS)
and Large-Eddy Simulations (LES) of turbulent flow over a backward-facing
step with a fully developed channel flow utilized
as a time-dependent inflow condition. Numerical solutions using a
fourth-order compact (Hermitian) scheme, which was formulated directly
for a non-equidistant and staggered grid in [1] are compared with numerical
solutions using the classical second-order central scheme. The results
from LES (using the dynamic subgrid scale model) are evaluated against a
corresponding DNS reference data set (fourth-order solution).
velocity
y
Hydrodynamic instability due to prevailing inertial forces (convection term in NS equations) is the cause of turbulence.
Inviscid instabilities
characterised by existence of inflection point of velocity profile
- jets
- wakes
- boundary layers wit adverse pressure gradient ∆p>0
Viscous instabilities
Linear eigenvalues analysis (Orr-Sommerfeld equations)
- channels, simple shear flows (pipes)
- boundary layers with ∆p>0
Transition Laminar -TurbulentCFD6
Poiseuille flow ~ 5700Couette flow – stable?
velocity
y Inflection – source of instability
(max.gradient)
There is no inflection of velocity profile in a pipe, however turbulent regime
exists if Re>2100
Transition Laminar -TurbulentCFD6
How to indentify whether the flow is laminar or turbulent ?
ExperimentallyVisualization, hot wire anemometers, LDA (Laser Doppler Anemometry).
Numerical experimentsStart numerical simulation selected to unsteady laminar flow. As soon as the solution converges to steady solution for sufficiently fine grid the flow regime is probably laminar
Recrit
According to value of Reynolds number using literature data of critical Reynolds number
Transition Laminar -TurbulentCFD6
Stability analysis
Velocity disturbance
Mean (undisturbed)
flow
Production (extracting energy from the mean
flow to fluctuations)
Linear stability analysisMomentum equation for disturbance
linear stability theory can predict when many flows become unstable, it can sayvery little about transition to turbulence since this progress is highly non-linear
Transition Laminar -TurbulentCFD6
1000Planar channel
400Cross flow
300Couetteflow
100Baffled channels
5-10Jets
RecritGeometry
500000Plate
8000Cavity
7000Suspension in pipe
5000Coiled pipe
2000Circ.pipe
RecritGeometry
D
D
D/2
D
D
D
D
D
Turbulent structures evolutionCFD6
Journal of Fluids and Structures 18 (2003) 305–324Force coefficients and Strouhal numbers offour cylinders in cross flowK. Lama, J.Y. Lib, R.M.C. Soa
Re=200
Re=800
Re<4
Re<40
Re<200 2D von Karman vortex street
Fully developed turbulent flowsCFD6
Free flows (self preserving flows)
Jets Mixing layers Wakes
h yu
umax
umax
umin
uyb
max
( )u y
fu h
= min
max min
( )u u y
gu u b
− =−
max
max
constu
xconst
ux
=
=
Circular jet
Planar jet
x
x
Jet thickness ~x, mixing length ~x see Goertler, Abramovic Teorieja turbulentnych struj, Moskva 1984:
Example - tutorialCFD6
Entrainment in jets (increase of volumetric flowrate)
h yu
umax
max
( )u y
fu h
=
1
max max max
0 0
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )h
planar
yV x u x f dy h x u x f d h x u x F c x
hη η= = = =∫ ∫ɺ
12 2
max max max
0 0
( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) ( )h
circ
yV x u x f ydy h x u x f d h x u x F cx
hπ η η η= = = =∫ ∫ɺ
Planar jet
Circular jet
~x ~1/√x
~1/x
Fully developed turbulent flowsCFD6
Flow at walls (boundary layers)y
Laminar sublayer
Buffer layer
Log law * wuτρ
=
*
*
u yuu y
u
ρµ
+ += =
Friction velocity
u+=lnEy+/κ30<y+<500Log law
u+=-3+5lny+5<y+<30buffer
u+=y+y+<5laminarINNER layer
(independent of bulk velocities)
OUTER layer (law of wake) max
*
1ln( )
u u yA
u κ δ− = +
u
Fully developed turbulent flowsCFD6
Flow at walls (turbulent stresses)τt
*yuy
ρµ
+ =
2 2( )
[1 exp( )]26
t m
m
ul
y
yl y
τ ρ
κ+
∂=∂
= − −
2 2' ' 24 [1 exp( )]26t w
yu vτ ρ κ τ
+
= = − −
Prandtl’smodel
vanDriest model of mixing length lm5 30
Example tutorialCFD6
Calculate thickness of laminar sublayer at flow of water in pipe (D=2 cm) at flowrate 1 l/s.
*
5yu ρ
µ=
3
3
4 4 10 1000Re 63662
0.02 10
V
D
ρπ µ π
−
−
⋅ ⋅= = =⋅
ɺ
2 22
2 4 2 44
1 0.3162 2 25.2[ ]
8 Rew
V Vu Pa
D Dτ λρ λρ ρ
π π= = = =
ɺ ɺWall shear stress from Blasius
Turbulent region, well within validity of Blasius correlation
Dimensionless thickness of laminar
sublayer
*
28
0.6320.159[ / ]
Rew V
u m sD
τρ π
= = =ɺ
2 2 88 8 85 Re 0.005 0.02 63662 2 10 636620.31
10000.632 0.001 0.632 0.632
Dy m
V
µ π π π µρ
−⋅ ⋅= = = =ɺ
Friction velocity
TIME AVERAGING of turbulent fluctuationsCFD6
RANS (Reynolds Averaging of Navier Stokes eqs.)
1( ) '
( )
''
' '
t t
t
t t
tt t
t
t dtt
t dt
dt
ρ
ρ
ρρ
+∆
+∆
+∆
Φ = Φ Φ = Φ + Φ∆
ΦΦ = Φ = Φ + Φ
ΦΦ = Φ +
∫
∫
∫
ɶ ɶ
ɶ
Time average
Favre’s average
Proof: ( ')( ') ' 'ρ ρ ρ ρ+ Φ + Φ = Φ + Φ
Remark: Favre’s average differs only for compressible substances and at high Mach number flows. Ma>1
TIME AVERAGING of turbulent fluctuationsCFD6
Trivial facts
Averaged value of fluctuation is zero
Average value of gradient of fluctuations is zero
Average value of product is not the product of aver aged values
' 0Φ ='
0x
∂Φ =∂
' 'uv u v u v= +
( ) ( ) ( ' ')uv u v u v∇ = ∇ + ∇ i i i
TIME AVERAGING of NS equationsCFD6
0 0 0u u u∇ = ∇ = → ∇ = i i i
( ) ( ) ( )
( ) ( )) ) ( ' '(
u uu p ut
u ut
u uu p u
ρ ρ µ
ρ ρ µ ρ
∂ + ∇ = −∇ + ∇ ∇∂∂ + ∇ = −∇ + − ∇∇ ∇∂
i
i i
i i
Continuity equation
Navier Stokes equations
Reynolds stresses
' 't u uτ ρ= −
TIME AVERAGING of transport equationsCFD6
( ) ( ) ( )
( ) ( ') ( ')) ( u
ut
ut
ρ
ρ ρ
ρ
ρ
∂ Φ + ∇ Φ = ∇ Γ∇Φ∂∂ Φ + ∇ Φ = ∇ Γ ΦΦ∂
− ∇∇
i i
ii i
Turbulent fluxes
' 'tq uρ= − Φ
Boussinesq hypothesisCFD6
Turbulent fluxes and turbulent stresses are defined by the same constitutive equations as in laminar flows, just only replacing diffusion coefficients and viscosity by turbulent transport coefficients.
Boussinesq hypothesisCFD6
Rate of deformationbased upon gradient
of averaged velocities
' ' tuρΦ = −Γ ∇Φ
' 'i ti
ux
ρ ∂ΦΦ = −Γ∂
' ' ( ( ) )Ttu u u uρ µ= − ∇ + ∇
1( ( ) )
2TE u u= ∇ + ∇
2t tEτ µ=
' ' ( )j ii j t
i j
u uu u
x xρ µ
∂ ∂= − +∂ ∂
Analogy to Fourierlaw
Analogy to Newton’slaw
q Tλ= − ∇
( ( ) )Tu uτ µ= ∇ + ∇
TRANSPORT coefficients-analogyCFD6
It is assumed that the rate of turbulent transport based upon migration of turbulent eddies is the same for momentum, mass and energy, therefore all transport coefficients should be almost the same
tt
t
µ σ=Γ
Prandtl number for turbulent heat transfer
Schmidt number for turbulent mass transfer
σt=0.9 at walls
σt=0.5 for jets
according to Rodi
Turbulent viscosity modelsCFD6
Turbulent viscosity is not a material parameter. It depends upon the actual velocity field and fluctuations at current point x,y,z. There exist different RANS models forturbulent viscosity prediction
Algebraic models (not reflecting transport of eddies)
1 equation models (transport equation for turbulent viscosity)
2 equations models (viscosity derived from transport equations of other characteristics of turbulent eddies)
Nonlineasr eddy viscosity models (v2-f)
RSM Reynolds Stress Modelling (transport equations forcomponents of reynolds stresses)
Algebraic modelsCFD6
Prandtl’s model of mixing length. Turbulent viscosity is derived fromanalogy with gases, based upon transport of momentum by molecules (kinetictheory of gases). Turbulent eddy (driven by main flow) represents a molecule, andmean path between collisions of molecules is substituted by mixing length.
Excellent model for jets, wakes, boundary layer flows. Disadvantage: fails in recirculating flows (or in flows where transport of eddies is very important).
2t m
ul
yµ ρ ∂=
∂Mixing length
h
h
y
u(y) 0.075ml h=0.09ml h=
0.07ml h=
[1 exp( )]26m
yl yκ
+
= − −
Circular jet
Planar jet
Mixing layers
Currently used algebraic models are Baldwin Lomax, and Cebecci Smith
2 equations modelsCFD6
Two equation models calculate turbulent viscosity from the pairs of turbulent characteristics k-ε, or k-ω, or k-l (Rotta 1986)
k (kinetic energy of turbulent fluctuations) [m2/s2]
ε (dissipation of kinetic energy) [m2/s3]
ω (specific dissipation energy) [1/s]
2
t
t
k
kCµ
µ ρω
µ ρε
=
=
Wilcox (1998) k-omega model, Kolmogorov (1942)
Jones,Launder (1972), Launder Spalding (1974)k-epsilon model
3 /u lε ∼
Cµ = 0.09 (Fluent-default)
Tutorial 2 equations modelsCFD6
Derive relationship for turbulent viscosity
2
t
kCµµ ρ
ε= Launder Spalding k-epsilon model
1
1
2
αβγ
== −
=
kα β γµ ρ ε=2 2 2 2 3
2 3 3 2 2 3[ . ] [ ][ ][ ]
Ns kg kg m m kg mPa s
m ms m s s s
α β γ α γ β α
α β γ γ β
+ −
+
= = = =
1
1 2 2 3
1 2 3
αγ β
γ β
=− = + −
= +
k – transport equationCFD6
All two equation models (and also Prandtl’s one equationmodel) are based upon transport equation for the kinetic energy of turbulence 2 2 21 1
' ' ( ' ' ' )2 2i ik u u u v w= = + +
' ' ( ) ' ' ( )u
u u uu u p u ut
ρ ρ µ∂ + ∇ = − ∇ + ∇ ∇∂
i i i i i i
1' ( ' ')
2
u ku u u
t t t
ρ ρρ∂ ∂ ∂= =∂ ∂ ∂
i i
( ) ( ' ' 2 ' ' ' ' ') 2 ' : ' ' ' :2
kku p u u e u u u e e u u E
t
ρ ρρ µ µ ρ∂ + ∇ = ∇ − + − − −∂
i i i i
Transport equation for k is derived in a similar way like the transport equation of mechanical energy by multiplying NS equation by vector of velocities (unlike mechanical energy only by the vector of velocity fluctuations)
Transport of pressure viscous stress reynolds stress rate of dissipation turbul.production
(p’) (2µe’) (ρu’u’)
k – transport equationCFD6
Rate of dissipation of kinetic energy of velocity fluctuation
' ' tu k kµ ∇∼
( ) ( ) 2 : 2 ' : 'tt
k
kku k E E e e
t
µρ ρ µ µσ
∂ + ∇ = ∇ ∇ + −∂
i i
Dispersion terms (viscous and reynolds stresses) and production term can be expressed using turbulent viscosity (Boussinesq)
' ' ' ' '2 ku u u k u kρ ρ= = Γ ∇ i
Example: approximation of Reynolds stress transport term
-ρε
2 2': ' ' 'ij ije e e e
µ µερ ρ
= =
' '1' ( )
2j i
iji j
u ue
x x
∂ ∂= +∂ ∂
Laminar and not turbulent viscosity!
σk = 1 Fluent
εεεε – transport equationCFD6
1 2( ) ( ) ( 2 : )ttCu E E C
kt ε εε
µρε ρε ε µ ρεσ
ε∂ + ∇ = ∇ ∇ + −∂
i i
Transport equation for dissipation ε looks like the k-transport. Just substitute ε for k. Production and dissipation terms are modified by universal constants C1ε and C2ε
This term follows from dimensional analysis
Production term (generator of turbulence) Dissipation
term
! 1[ ] [ ]k s
ε ≡
C1ε = 1.44 C2ε = 1.92 σε= 1.3 Fluent (default)
k-εεεε modifications (RNG, realizable)CFD6
1 221( ) (( ) ) ( 2 : )ttu C E E C
t kf fε ε
ε
µρε ερε ε µ εσ
µ ρ∂ + ∇ = ∇ + ∇ + −∂
i i
Corrections of dispersion, production and dissipation terms withthe aim to extent applicability of k-ε for low Reynolds numberflows
( ) (( ) ) 2 :tt
k
kku k E E
t
µρ ρ µµ ρεσ
∂ +∇ =∇ + ∇ + −∂
i i
Functions of turbulent Reynolds 2
Re tt
k
Cµ
νεν ν
= =
Effective viscosity (RNG)CFD6
2
0 (laminar)
(high Re)
ef
ef t
k
C kk µ
µ µ
ρµ µ µ
µε
→ →
→ ∞ → =
Blending formula for effective viscosity
2[1 ]ef
C kµρµ µ
µ ε= +
εεεε - estimateCFD6
3 23
PPo n d
dε ρ= =
Dissipated energy in a mixing tank
3 51
PPo
n dρ= ≈
Power number
d
n
2 2 4 2F d u d nρ ρ∼ ∼
5 2tM Fd d nρ= ∼
5 3tP M d nω ρ= ∼
k-εεεε boundary conditionsCFD6
*3
w
u
yε
κ=
k and ε must be specified at inlets. Estimate of kinetic energy k is based either upon measurement (anemometers) or experience (from estimated intensity of turbulence I). Dissipation is estimated from correlations for power consumption estimates.
Values of k and ε at wall (must be also defined as boundary conditions) can be approximated by wall functions
*2
w
uk
Cµ
= Friction velocity
* wuτρ
=
Distance of the nearest boundary
node from wall
This is implemented in majority of CFD programs
Reynolds stresses (k -εεεε)CFD6
2' ' 2 ( ) 2 2
3i i i
i i t ii ti i i
u u uu u k k k
x x xρ ρ µ ρ δ µ ρ∂ ∂ ∂− = − = + − = −
∂ ∂ ∂
Constitutive equation for Reynolds stresses must be modified with respect to isotropic pressure
2' ' ( )
3ji
tij i j t ijj i
uuu u k
x xτ ρ µ ρ δ
∂∂= − = + −∂ ∂
This term is zero for incompressible liquidsRemark: Turbulent stresses determine
kinetic energy of turbulent fluctuations 0u∇ =i
Assesment of k -εεεε modelsCFD6
Problems and erroneous results can be expected
Unconfined flows (wakes, jets). k-ε model overestimatesdissipation, therefore jets are overdumped.
Pressure transport term (u’p’) is neglected (errors in flows characterised by high pressure gradients)
Curved boundaries or swirling flows
Fully developed flows in noncircular ducts should be characterized by secondary flows due to anisotropy of normal Reynolds stresses (these features cannot be predicted by linear viscosity models)
Problems in buoyancy driven flows
Discrepancies?CFD6
Special case: Steady unidirectional flow, constant density, homogeneous turbulence
1 2( ) ( ) ( 2 : )ttu C E E C
t kε εε
µρε ερε ε µ ρεσ
∂ + ∇ = ∇ ∇ + −∂
i i
( ) ( ) 2 :tt
k
kku k E E
t
µρ ρ µ ρεσ
∂ +∇ =∇ ∇ + −∂
i i
2( )t u
y
µερ
∂=∂
2 2
1
( )t Cu
y Cε
ε
µερ
∂=∂
These terms are identically zero in homogeneous turbulence
Further reading C1ε = 1.44 C2ε = 1.92 σk = 1 σε= 1.3
Reynolds Stress Models (RSM)CFD6
' 'ij i jR u u=
( )ij tij ij ij ij ij
DRP R
Dt
ν εσ
= + ∇ ∇ − + Π + Ωi
production
diffusion
Dissipation
Pressure transport
Centrifugal forces
2
3 i jε δ
6 transport equations
1 dissipation
1 2( ) ( ) ( 2 : )ttu C E E C
t kε εε
µρε ερε ε µ ρεσ
∂ + ∇ = ∇ ∇ + −∂
i i
Large Eddy Simulation (LES)CFD6
Instead of time averaging of NS equations, spatial fluctuation filtering of NS equations (only small eddies are removed, motion of large eddies is calculated).
Filter is realized by convolution integral
( , ) ( , ) ( , ) ( , ) ( , )u x t G x u t d G u x t dξ ξ ξ ξ ξ ξΩ Ω
= − ∆ = ∆ −∫∫∫ ∫∫∫Convolution, Green’s
function G
LES G-function exampleCFD6
3( , ) 1/ for | | / 2
( , ) 0 for | | / 2
G x x
G x x
ξ ξξ ξ
− ∆ = ∆ − < ∆− ∆ = − ≥ ∆
∆
/2
/2
( , ) /x
x
u t dξ ξ+∆
−∆
∆∫
x
u
LES filtering - propertiesCFD6
u u≠Basic difference in comparison with time averaging
( ) ( , )u u
G u x t dx x x
ξ ξ ξ∂ ∂ ∂= = −∂ ∂ ∂ ∫∫∫
LES NS equationsCFD6
0i
i
u
x
∂ =∂
2
( ( ))i ii j
j i k k
u upu u
t x x x xρ µ∂ ∂∂ ∂+ = − +
∂ ∂ ∂ ∂ ∂
Continuity equation without changes
( ')( ') ' ' ' '
' ' ' '
i j i i j j i j i j i j i j
i j i j i j i j i j i j
u u u u u u u u u u u u u u
u u u u u u u u u u u u
= + + = + + + =
= + − + + +
Lij Leonard stresses
Cij cross stresses
Rij SGS (sub grid stresses)
LES NS equationsCFD6
( ( )) ( )i ii j ij
j i j j
u upu u
t x x x xρ µ τ∂ ∂∂ ∂ ∂+ = − + +
∂ ∂ ∂ ∂ ∂
Leonard stresses are usually neglected because they are of the second order –almost the same as discretisation error. Cross and subgrid stresses are usually modeled together using turbulent viscosity approach.
It looks like Reynolds stresses
LES Smagorinski SGSCFD6
2
1( )
2
ij T ij
jiij
j i
E
uuE
x x
τ µ=
∂∂= +∂ ∂
Smagorinski model of subgrid stresses is almost the same as the Prandtl’s model of mixing length, only instead of the mixing length is substituted a filter size.
2( ) :T sC E Eµ ρ= ∆
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