13 rans eqns
DESCRIPTION
Rans EqnsTRANSCRIPT
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RANS equations Equations for k and rij Eddy viscosity
The RANS Equations
Maurizio Quadrio
DIA, Politecnico di Milano
2012
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RANS equations Equations for k and rij Eddy viscosity
OUTLINE
1 RANS EQUATIONS
2 EQUATIONS FOR k AND rij
3 EDDY VISCOSITY
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RANS equations Equations for k and rij Eddy viscosity
REYNOLDS DECOMPOSITION
Let u(xxx , t) be definite and continuous in Rd ,d = 1,2,3. Letu(xxx) exist in the time-mean sense:
u(xxx) = limT
1T
T0u(xxx , t)dt
REYNOLDS DECOMPOSITION
u(xxx , t) = u(xxx) +u(xxx , t)
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RANS equations Equations for k and rij Eddy viscosity
THE FLUCTUATING FIELD
u(xxx , t) is the fluctuating part u(xxx) is independent upon time By definition u = u and u = 0 Coupling between mean and fluctuating field (closure
problem: more unknowns than equations)
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RANS equations Equations for k and rij Eddy viscosity
DERIVATION FOR CONTINUITY EQ.NS + REYNOLDSS DECOMPOSITION + TIME AVERAGE
1
uuu = 02
(uuu+uuu)= 03
uuu = uuu
4
uuu = 05
uuu = 0
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RANS equations Equations for k and rij Eddy viscosity
MOMENTUM EQUATION
uuu uuu = limT
T0uuu uuudt
= uuu [
limT
T0uuudt
]= uuu
[limT
T0uuudt
]= uuu uuu= 0
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RANS equations Equations for k and rij Eddy viscosity
THE FINAL FORM
uuu uuu = uuu2 = uuu2 = uuu2uuu uuu = uuu2 = uuu2
STEADY (RANS)
(uuu uuu) + (uuuuuu
)=1
p+2uuu
UNSTEADY (URANS)
uuu t
+ (uuu uuu) + (uuuuuu
)=1
p+2uuu
Closure is required!7 / 46
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RANS equations Equations for k and rij Eddy viscosity
THE REYNOLDS STRESSES
uiuj (apparent) Reynolds stresses Momentum diffusion due to turbulent motions (analogy
with viscous stresses) uiuj is a symmetric tensor k = 12uiui uiuj = 23kij +aij aij only determines momentum trasport
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RANS equations Equations for k and rij Eddy viscosity
URANSDO THE UNSTEADY RANS MAKE SENSE?
Two separated time scales T1 and T2 T1 must exist in theflowReynoldss decomposition can be redefined as:
uuu(xxx , t) =1T
T/2T/2
uuu(xxx , t + )d T T1
PROBLEMS
Turbulent flows do not admit scale separation Scale T is not clearly defined
Mathematical problem Operative problem
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RANS equations Equations for k and rij Eddy viscosity
PROBLEMS WITH RANS1) EFFECTS OF TIME AVERAGING
Aim: to remove unessential information But: details are sometimes important (e.g. combustion) Mapping physics statistics not unique!
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RANS equations Equations for k and rij Eddy viscosity
PROBLEMS WITH RANS2) REYNOLDSS DECOMPOSITION
Considering mean values only is limiting Considering mean values only is simple Advantages and disadvantages must be balanced Existence of coherent structures emphasizes limitations
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RANS equations Equations for k and rij Eddy viscosity
PROBLEMS WITH RANS3) SMALL-SCALE INTERACTIONS ARE MISSING
Modelling statistics (uiuj ) rather than physics (u
i , uj ) hides
small-scale interactions
Scalar dissipation: RANS can only predict mean fluxes Combustion: RANS miss peak values
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RANS equations Equations for k and rij Eddy viscosity
PROBLEMS WITH RANS4) RANS SOLUTIONS AND AVERAGED-NS SOLUTIONS
RANS SOLUTIONS ARE EQUAL TO (TIME-AVERAGED)EXPERIMENTAL DATA?
SSSuuu0?=SSS(t)uuu0
Do averaging- and solution-operators commute?
Theorem: if and only if model for uuuuuu is exact! Difference between averaged solution and RANS solution
is square root of uuuuuu error (modelling error)
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RANS equations Equations for k and rij Eddy viscosity
OUTLINE
1 RANS EQUATIONS
2 EQUATIONS FOR k AND rij
3 EDDY VISCOSITY
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RANS equations Equations for k and rij Eddy viscosity
EQUATION FOR EE MEAN-FLOW KINETIC ENERGY
E 12uiui
DEDt
+ TTT =P
T i ujuiuj +ui p/2ujsij
P uiujuixj
2sijsij
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RANS equations Equations for k and rij Eddy viscosity
EQUATION FOR kk TURBULENT KINETIC ENERGY
k 12uiui
DkDt
+ TTT =P
T i 12uiujuj +u
ip/2ujsij
P uiujuixj
2sijsij
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RANS equations Equations for k and rij Eddy viscosity
EQUATION FOR rijrij REYNOLDS STRESS
rij uiujEquation for rij is obtained by:
subtracting eq. for ui from that for ui to get an eq. for ui ; multiplying eq. for ui by that for uj to obtain eq. for uiuj ; time-averaging eq. for uiuj .
( t
+ukxk
)(uiuj
)=
xkuiujuk +Pij + ij ij +2uiuj
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RANS equations Equations for k and rij Eddy viscosity
REYNOLDS STRESSES EQUATION
Triple correlation (closure problem):uiujuk
Production tensor:
Pij uiukujxkujuk
uixk
Velocity-pressure gradient tensor:
ij 1
(uip
xj+uj
p
xi
) Dissipation tensor:
ij uixk
ujxk
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RANS equations Equations for k and rij Eddy viscosity
TURBULENT KINETIC ENERGY AND REYNOLDSSTRESSES
Trace of uiuj eq. is 2 times the k eq.
Pii = 2P
ii =2
xi
uip
ii = 2
Difference between dissipation and pseudo-dissipation:
2sijsij uixk
uixk
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RANS equations Equations for k and rij Eddy viscosity
OUTLINE
1 RANS EQUATIONS
2 EQUATIONS FOR k AND rij
3 EDDY VISCOSITY
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RANS equations Equations for k and rij Eddy viscosity
CLASSIFICATION OF TURBULENCE MODELS
MODELS BASED ONEDDY-VISCOSITY
rij is given through aneddy-viscosity
more developed less recent
MODELS FOR THEREYNOLDS-STRESS TENSOR
a model for rij is givendirectly
less developed more recent
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RANS equations Equations for k and rij Eddy viscosity
THE GRADIENT-DIFFUSION HYPOTHESISAN EXAMPLE FOR THE PASSIVE SCALAR
(uuu ) + uuu = 2
Scalar flux vector uuu : direction and magnitude of the(turbulent) transport of
Hypothesis: this vector is aligned to the mean scalargradient
Turbulent diffusivity t(xxx) (positive scalar quantity):
uuu =t(xxx)
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RANS equations Equations for k and rij Eddy viscosity
THE TURBULENT DIFFUSIVITY
Define the effective diffusivity e(xxx):
e(xxx) = + t(xxx)
The mean scalar equation is closed as:
(uuu ) = (e(xxx))
Problem: t(xxx) must be known
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RANS equations Equations for k and rij Eddy viscosity
THE TURBULENT-VISCOSITY HYPOTHESIS
(uuu uuu) + (uuuuuu
)=1
p+2uuu
Reynolds stresses tensor uuuuuu: effect of turbulentfluctuations on the mean motion
Hypothesis: this tensor is aligned to the meanrate-of-strain tensor
Eddy viscosity t(xxx) (positive scalar quantity):
aij uiuj +23kij = t(xxx)
(uixj
+ujxi
)
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RANS equations Equations for k and rij Eddy viscosity
THE EDDY VISCOSITY
Define the effective viscosity e(xxx):
e(xxx) = +t(xxx)
The mean momentum equation is closed as:
(uuu uuu) =p+ (e(xxx)uuu)
Problem: t(xxx) must be known.
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RANS equations Equations for k and rij Eddy viscosity
APPRAISAL OF THE BOUSINNESQS HYPOTHESIS
1 Intrinsic hypothesis: aij depends upon mean velocitygradients only
2 Specific hypothesis:
aij =2tsijmodelled after laminar flows
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RANS equations Equations for k and rij Eddy viscosity
1) THE INTRINSIC HYPOTHESISTURBULENCE DOES HAVE MEMORY!
Experiment: sudden axisymmetric contraction after a grid(Uberoi 1956, Sk/ = 2.1)
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RANS equations Equations for k and rij Eddy viscosity
EVOLUTION OF ANISOTROPYNORMALIZED ANISOTROPIES bij = aij/2k
Open symbols: larger strain Sk/ = 55
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RANS equations Equations for k and rij Eddy viscosity
INTRINSIC HYPOTHESIS IS INCORRECT
Contraction: normalized anisotropies bij = aij/2k are notconstant
For large strain bij depend on the total amount of meanstrain
Straight section: anisotropy is not zero and decreases onthe turbulence timescale k/
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RANS equations Equations for k and rij Eddy viscosity
THE MOLECULAR ANALOGY
VISCOUS FLUIDKinetic theory of gases:
velocity scale: meanmolecular speed c
length scale: mean freepath
kinematic viscosity 12c
TURBULENT FLOWEmpirical argument:
velocity scale: turbulencevelocity scale u
length scale: turbulencelength scale `
turbulent viscosity t u`
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RANS equations Equations for k and rij Eddy viscosity
EXAMPLESIMPLE SHEAR FLOW WITH S = u1/x2 U/L
VISCOUS FLUID
Fluid = ensemble ofmolecules
Molecular timescalem = /c
Shear timescale S =S 1mS
cS =
LUc
= KnM 1
TURBULENT FLOW
Turbulent flow = ensembleof eddies
Turbulent timescalet = k/
Shear timescale S =S 1tS S k
= O(1)
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RANS equations Equations for k and rij Eddy viscosity
IS THE INTRINSIC HYPOTHESIS WORTH SOMETHING?TURBULENT VISCOSITY IS OFTEN USEFUL
In simple shear flow the mean velocity gradients change slowly:
Local mean velocity gradients characterize the history ofmean distortion
Reynolds stress balance is dominated by local processes WhenP/ 1 the turbulent-viscosity hypothesis is correct
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RANS equations Equations for k and rij Eddy viscosity
2) THE SPECIFIC HYPOTHESISTURBULENCE IS NOT A NEWTONIAN FLUID
INCONSISTENT!
aij =2tsij
In turbulent shear flows sii = 0 but normal Reynoldsstresses are not!
aij is not aligned with sij
NOT INVARIANT
aij =2tsijTensorial relation is not general
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RANS equations Equations for k and rij Eddy viscosity
SPECIFIC HYPOTHESIS
INVARIANT BUT NOT OBJECTIVE
aij = t ,ijklskl
Not rotation-invariant
OBJECTIVE RELATION
aij = 1t ij +2t sij +
3t sikskj
Unable to reproduce simple experimental situations...A costitutive eq. for turbulence does not need to be objective(Coriolis, etc)
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RANS equations Equations for k and rij Eddy viscosity
SPECIFIC HYPOTHESIS: WHY A LINEAR LAW?
NEWTONIAN FLUID
Straining small comparedto molecular scales:
S /c 1
Small departure fromequilibrium
Linear dependence ofstress tensor on velocitygradient tensor
TURBULENT FLOW
Straining large comparedto turbulence scales:
S k/ > 1
Large departure fromequilibrium
More general dependencyof Reynolds stresses onvelocity gradient tensor
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RANS equations Equations for k and rij Eddy viscosity
BEYOND A LINEAR LAW
Non-linear laws: e.g. memory effect
aij = A23kij
0
M()[uixj
(t ) + ujxi
(t )]d
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RANS equations Equations for k and rij Eddy viscosity
LIMITATIONS OF EDDY VISCOSITY CONCEPT
Eddy viscosity not fully adequate for: Flows with abrupt change of shear rate Flows over curved surfaces Flows in ducts with secondary motions and/or separations Rotating or stratified flows Three-dimensional flows Many others...
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RANS equations Equations for k and rij Eddy viscosity
RANS RESULTS HAVE LIMITED RELIABILITY...FREITAS, SELETED BENCHMARKS FROM COMMERCIAL CFD CODES, J.FLUIDS ENG. 1995
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RANS equations Equations for k and rij Eddy viscosity
... BUT CAN BE USED NONETHELESS!TOROROSSO (1)
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RANS equations Equations for k and rij Eddy viscosity
... BUT CAN BE USED NONETHELESS!TOROROSSO (2)
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RANS equations Equations for k and rij Eddy viscosity
... BUT CAN BE USED NONETHELESS!TOROROSSO (3)
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RANS equations Equations for k and rij Eddy viscosity
... BUT CAN BE USED NONETHELESS!TOROROSSO (3)
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RANS equations Equations for k and rij Eddy viscosity
... BUT CAN BE USED NONETHELESS!UPPER RESPIRATORY AIRWAYS (1)
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RANS equations Equations for k and rij Eddy viscosity
... BUT CAN BE USED NONETHELESS!UPPER RESPIRATORY AIRWAYS (2)
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RANS equations Equations for k and rij Eddy viscosity
... BUT CAN BE USED NONETHELESS!UPPER RESPIRATORY AIRWAYS (3)
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RANS equations Equations for k and rij Eddy viscosity
... BUT CAN BE USED NONETHELESS!UPPER RESPIRATORY AIRWAYS (4)
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RANS equationsEquations for k and rijEddy viscosity