rudolf Žitný, Ústav procesní a zpracovatelské techniky Čvut fs 2010
DESCRIPTION
Computer Fluid Dynamics E181107. 2181106. CFD6. Turbulent flows,…. Remark : fo i l s with „ black background “ could be skipped, they are aimed to the more advanced courses. Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010. - PowerPoint PPT PresentationTRANSCRIPT
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 E181107CFD62181106
Some figures and equations are copied from Wikipedia
TurbulenceCFD6
( ) ( ( ) ) Mu u u p e u St
( ( )) ( ( ) ) Mu uu p e u St
Source of nonlinearities of NS equations are inertial and viscous terms
for incompressible fluids the equivalent conservative form
( ( ) ) ( ( ) ) Mu uu u u p e u St
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 uDu
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 T
2'u
2 2 21 ( ' ' ' )2
k u v w
2( ) /3
I k u
2 2 2, , , , ,' , ' , ' , ' ', ' 't xx t yy t zz t xy t xzu v w u v u w
Turbulent eddies - scalesCFD6
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 )
Smallest eddies (size is called Kolmogorov scale) disappear, because kinetic energy is
converted to heat by friction
E()
=2f/u1/L 1/
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 d
Typical values of frequency f~10 kHz, Kolmogorov scale ~0.01 up to 0.1 mm
Kolmogorov scale decreases with the increasing Re
2/3
5/3( )E C
Viscous subrange - scalesCFD6
Kolmogorov scales (the smallest turbulent eddies) follow from dimensional analysis, assuming that averything depends only upon the kinematic viscosity and upon the rate of energy supply in the energetic cascade (only for small isotropic eddies, of course)
34
4u
Length scale Time scale velocity scale
These expressions follow from dimension of viscosity [m2/s] and the rate of energy dissipation [m2/s3]
2 2 2 2
3 3[ ] [ ][ ]
1 2 20 3
m m mms s s
3 / 41/ 4
Viscous scale - tutorialCFD6
Derive time scale of the smallest turbulent eddies
Time scale
2 2 2 2
3 3[ ] [ ][ ]
0 2 21 3
m m mss s s
1/ 21/ 2
Inertial subrange - scalesCFD6
Spectral energy E decreases in the turbulent cascade when turbulent eddies are decomposed to smaller and smaller eddies. Energy transformation is not affected by viscosity in the inertial subrange
2 2
/ /
[1/ ]0 [ / ] spectralwave numberkinetic energy energy is
2 /of fluctuations independent of size of eddieviscosity in the
inertial range
1 ( ) 2 i i
mm s
k u u E d
Dimension analysis in the inertial subrangen
m
CE )(
2 2
2 3[ ] [ ] [ . ]m m n
n m
k Ed
m m ms s m
35
343
32
n
m
2/3
5/3( )E C
And this is the famous Kolmogorov’s law of 5/3
Turbulent eddies - scalesCFD6
Wikipedia (abbreviated)
Turbulent flow is composed by "eddies" of different sizes. The sizes define a characteristic length 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 up originating smaller eddies, and the kinetic energy of the initial large eddy is divided into the smaller eddies that stemmed from it. The energy is passed down from the large scales of the motion to smaller scales until reaching a sufficiently small length scale such that the viscosity of 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, the small 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 the boundaries (the size characterizing the large scales will be denoted as L).
Kolmogorov introduced a hypothesis: for very high Reynolds numbers the statistics of small scales are universally and uniquely determined by the kinematic viscosity () and the rate of energy dissipation (). With only these two parameters, the unique length that can be formed by dimensional analysis is
Turbulent eddies - scalesCFD6
Why and when it is important to know the turbulent scales:
Rate of chemical reactions depends upon the rate of micromixing determined by the frequency and the size of turbulent eddies
Breakup of droplets depends upon the energy and size of turbulent eddies (too large eddies only move but do not break a droplet)
Coalescence of bubbles depends upon collision frequency, collision speed, and these parameters are correlated with the energy dissipation rate and the length microscale l corresponding to radius of bubbles
3/22 )( lCv
Ronnie Andersson and Bengt Andersson. Modeling the Breakup of Fluid Particles in Turbulent Flows. AIChE Journal June 2006 Vol. 52, No. 6
H. Venneker, Jos J. Derksen and Harrie E. A. Van den Akker. Population balance modeling of aerated stirred vessels based on CFD . AIChE J. Volume 48, Issue 4, April 2002, 673–685
DNS Direct Numerical SimulationCFD6
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 (Kolmogorov) 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 LuNL
3uL
Re uL Re uL
Velocity scale u in previous expression is related to magnitude of turbulent fluctuations (rms of u’, or k). The Re related to the velocity fluctuation is called turbulent Reynolds number.
(based upon dimensional ground)
No.of grid points in DNS
No.of time steps
12300 380 6.7 M 32 k
30800 800 40 M 47 k
61600 1450 150 M 63 k
230000 4650 2100 M 114 k
Table concerns DNS modelling of channel flow experiments (rewritten from Wilcox: Turbulence modelling, chapter 8).
9/4
3/4_
Re
ReDNS
time steps
N
N
Remark: Re~106 or 107 at flow around a car or flow around wings
DNS Direct Numerical SimulationCFD6
Abstract The 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
Runge-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 STEPFLOW USING 2ND- AND 4TH-ORDER DISCRETIZATIONADNAN 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-facingstep with a fully developed channel flow utilizedas a time-dependent inflow condition. Numerical solutions using afourth-order compact (Hermitian) scheme, which was formulated directlyfor a non-equidistant and staggered grid in [1] are compared with numericalsolutions using the classical second-order central scheme. The resultsfrom LES (using the dynamic subgrid scale model) are evaluated against acorresponding DNS reference data set (fourth-order solution).
Transition Laminar-TurbulentCFD6
Bacon
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 ?
Experimentally
Visualization, 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 say very little about transition to turbulence since this progress is highly non-linear
Transition Laminar-TurbulentCFD6
Geometry Recrit
Jets 5-10
Baffled channels
100
Couette flow
300
Cross flow 400
Planar channel
1000
Geometry Recrit
Circ.pipe 2000
Coiled pipe 5000
Suspension in pipe
7000
Cavity 8000
Plate 500000
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 of four 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 yfu h
min
max min
( )u u ygu u b
max
max
constux
constux
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 yfu h
1
max max max0 0
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )h
planaryV x u x f dy h x u x f d h x u x F c xh
12 2
max max max0 0
( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) ( )h
circyV x u x f ydy h x u x f d h x u x F cxh
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 * wu
*
* u yuu yu
Friction velocity
laminar y+<5 u+=y+
buffer 5<y+<30 u+=-3+5lny+
Log law 30<y+<500 u+=lnEy+/
INNER layer (independent of bulk velocities)
OUTER layer (law of wake) max
*
1 ln( )u u y Au
u
Fully developed turbulent flowsCFD6
Flow at walls (turbulent stresses)t
*yuy
2 2( )
[1 exp( )]26
t m
m
uly
yl y
2 2' ' 24 [1 exp( )]26t wyu v
Prandtl’s model
vanDriest model of mixing length lm
5 30
Example tutorialCFD6
Calculate thickness of laminar sublayer at flow of water in pipe (D=2 cm) at flowrate 1 l/s.
*
5 yu
3
3
4 4 10 1000Re 636620.02 10
VD
2 22
2 4 2 44
1 0.3162 2 25.2[ ]8 Rew
V Vu PaD D
Wall shear stress from Blasius
Turbulent region, well within validity of Blasius correlation
Dimensionless thickness of laminar
sublayer
*28
0.632 0.159[ / ]Re
w Vu m sD
2 2 88 8 85 Re 0.005 0.02 63662 2 10 63662 0.3110000.632 0.001 0.632 0.632
Dy mV
Friction velocity
TIME AVERAGING of turbulent fluctuationsCFD6
Benton
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
Time average
Favre’s average
Proof: ( ')( ') ' '
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 averaged values
' 0 ' 0
x
' 'uv u v u v
( ) ( ) ( ' ')uv u v u v
TIME AVERAGING of NS equationsCFD6
0 0 0u u u
( ) ( ) ( )
( ) ( )) ) ( ' '(
u uu p ut
u ut
u uu p u
Continuity equation
Navier Stokes equations
Reynolds stresses
' 't u u
TIME AVERAGING of transport equationsCFD6
( ) ( ) ( )
( ) ( ') ( ')) ( u
ut
ut
Turbulent fluxes
' 'tq u
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 deformation based upon gradient
of averaged velocities
' ' tu
' 'i ti
ux
' ' ( ( ) )Ttu u u u
1 ( ( ) )2
TE u u 2t tE
' ' ( )j ii j t
i j
u uu u
x x
Analogy to Fourier law
Analogy to Newton’s law
q T
( ( ) )Tu u
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
Prandtl number for turbulent heat transfer
Schmidt number for turbulent mass transfer
t=0.9 at walls
t=0.5 for jets
according to Rodi
Turbulent viscosity modelsCFD6
Botero
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 for turbulent 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)
Nonlinear eddy viscosity models (v2-f)
RSM Reynolds Stress Modelling (transport equations for components of reynolds stresses)
Algebraic modelsCFD6
Prandtl’s model of mixing length. Turbulent viscosity is derived from analogy with gases, based upon transport of momentum by molecules (kinetic theory of gases). Turbulent eddy (driven by main flow) represents a molecule, and mean 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
uly
Mixing length
h
h
y
u(y) 0.075ml h0.09ml h
0.07ml h
[1 exp( )]26myl y
Circular jet
Planar jet
Mixing layers
Currently used algebraic models are Baldwin Lomax, and Cebecci Smith
ndeformatio
of rate
2 1][s
mtt
2 equations modelsCFD6
Two equation models calculate turbulent viscosity from the pairs of turbulent characteristics k-, or k-, or k-l (Rotta 1986)
k (kinetic energy of turbulent fluctuations) [m2/s2]
(dissipation of kinetic energy) [m2/s3]
(specific dissipation energy) [1/s]
2
t
t
k
kC
Wilcox (1998) k-omega model, Kolmogorov (1942)
Jones,Launder (1972), Launder Spalding (1974) k-epsilon model
3 /u l
C = 0.09 (Fluent-default)
Tutorial 2 equations modelsCFD6
Derive relationship for turbulent viscosity
2
tkC
Launder Spalding k-epsilon model
11
2
k 2 2 2 2 3
2 3 3 2 2 3[ . ] [ ][ ][ ]Ns kg kg m m kg mPa sm ms m s s s
1
1 2 2 31 2 3
k – transport equationCFD6
All two equation models (and also Prandtl’s one equation model) are based upon transport equation for the kinetic energy of turbulence 2 2 21 1' ' ( ' ' ' )
2 2i ik u u u v w
' ' ( ) ' ' ( )uu u uu u p u ut
1' ( ' ')2
u ku u ut t t
( ) ( ' ' 2 ' ' ' ' ') 2 ' : ' ' ' :2
k ku p u u e u u u e e u u Et
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’) (2e’) (u’u’)
k – transport equationCFD6
Rate of dissipation of kinetic energy of velocity fluctuation
' ' ~ tu k k
( ) ( ) 2 : 2 ' : 'tt
k
k ku k E E e et
Dispersion terms (viscous and reynolds stresses) and production term can be expressed using turbulent viscosity (Boussinesq)
' ' ' ' '2 ku u u k u k
Example: approximation of Reynolds stress transport term
-
2 2' : ' ' 'ij ije e e e
' '1' ( )2
j iij
i j
u uex x
Laminar and not turbulent viscosity!
k = 1 Fluent
– transport equationCFD6
1 2( ) ( ) ( 2 : )ttCu E E C
kt
Transport equation for dissipation looks like the k-transport. Just substitute for k. Production and dissipation terms are modified by universal constants C1 and C2
This term follows from dimensional analysis
Production term (generator of turbulence) Dissipation
term
! 1[ ] [ ]k s
C1 = 1.44 C2 = 1.92 = 1.3 Fluent (default)
k- modifications (RNG, realizable)CFD6
1 221( ) (( ) ) ( 2 : )ttu C E E C
t kf f
Corrections of dispersion, production and dissipation terms with the aim to extent applicability of k- for low Reynolds number flows
( ) (( ) ) 2 :tt
k
k ku k E Et
Functions of turbulent Reynolds 2
Re tt
kC
Effective viscosity (RNG)CFD6
2
0 (laminar)
(high Re)
ef
ef t
k
C kk
Blending formula for effective viscosity
2[1 ]ef
C k
- estimateCFD6
Dissipated energy in a mixing tank
3 5 ~ 1PPon d
Power number
d
n
2 2 4 2~ ~F d u d n 5 2~tM Fd d n
5 3~tP M d n
Check units of dissipation [W/kg =m2/s3]
k- boundary conditionsCFD6
*3
wuy
k and 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 at wall (must be also defined as boundary conditions) can be approximated by wall functions
*2
wukC
Friction velocity
* wu
Distance of the nearest boundary
node from wall
This is implemented in majority of CFD programs
Reynolds stresses (k-)CFD6
2' ' 2 ( ) 2 23
i i ii i t ii t
i i i
u u uu u k k k
x x x
Constitutive equation for Reynolds stresses must be modified with respect to isotropic pressure
2' ' ( )3
jitij i j t ij
j i
uuu u kx x
This term is zero for incompressible liquidsRemark: Turbulent stresses determine
kinetic energy of turbulent fluctuations 0u
Assesment of k- modelsCFD6
Problems and erroneous results can be expected
Unconfined flows (wakes, jets). k- model overestimates dissipation, 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 k
( ) ( ) 2 :tt
k
k ku k E Et
2( )t uy
2 2
1
( )t Cuy C
These terms are identically zero in homogeneous turbulence
Further reading C1 = 1.44 C2 = 1.92 k = 1 = 1.3
Reynolds Stress Models (RSM)CFD6
' 'ij i jR u u
( )ij tij ij ij ij ij
DRP R
Dt
production
diffusion
DissipationPressure transport
Centrifugal forces
23 ij
6 transport equations
1 dissipation
1 2( ) ( ) ( 2 : )ttu C E E C
t k
Large Eddy Simulation (LES)CFD6
Demuth
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 d
Convolution, Green’s
function G
LES G-function exampleCFD6
3( , ) 1/ for | | / 2( , ) 0 for | | / 2
G x xG x x
Δ
/2
/2
( , ) /x
x
u t d
x
u
LES filtering - propertiesCFD6
u uBasic difference in comparison with time averaging
( ) ( , )u u G u x t dx x x
LES NS equationsCFD6
0i
i
ux
2
( ( ))i ii j
j i k k
u upu ut x x x x
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 ut x x x x
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
123
1 ( )2
ij T ij kk ij
jiij
j i
E
uuE
x x
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( ) 2 :T sC E E
Smagorinski’s constant Cs=0.1 in Fluent
Remark: in the vicinity of wall the Cs mixing length is limited by y (karman constant times the distance from wall)
LES RNG SGSCFD6
21/3(1 max(0, ))s eff
eff C
2( ) 2 :s rngC E E
Crng=0.157 in Fluent
For small s the effective viscosity equals the laminar viscosity , while eff=s holds for a high level of turbulence.
Crng=100 in Fluent
LES boundary conditionsCFD6
Inlet: Simulated noise of velocities (normal distribution superposed to specified intensity of turbulence I)
Wall: For a fine mesh resolving laminar sublayer a linear (laminar) velocity profile is assumed, otherwise velocity is calculated from the buffer layer model
1 ln u yu Eu
Von Karman constant 0.41Friction velocity
Fluent turbulent modelsCFD6
Spalart Almaras (viscosity transport 1 PDE)
k- dissipation of k (standard+RNG+realizable)
k- specific dissipation (standard+SST shear stress transport)
k-kl- transition model laminarturbulent
v2-f boundary layer detachment (low Re)
RSM
DES (Detached Eddy Simulation, Spalart Almaras+realizable+SST), like LES at wall
LES (Large Eddy Simulation)
Fluent example flow in a pipeCFD6
Material: water-liquid (fluid)
Property Units Method Value(s) --------------------------------------------------------------- Density kg/m3 constant 998.20001 Cp (Specific Heat) j/kg-k constant 4182 Thermal Conductivit w/m-k constant 0.6 Viscosity kg/m-s constant 0.001003 Molecular Weight kg/kgmol constant 18.0152
Tube geometry: L=0.5m, D=0.04m.
Velocity at inlet (uniform) u=1m/s
Mesh 50 x 20, compression 2 (last/first)
6
1 0.04 1000Re 400000.001
1 0.04 1.1Re 237818.5 10
water
air
WALL
AXISVELOCITY
INLET
PRESSURE OUTLET
4
4
0.316 0.5 1000 1400.042 40000
0.316 0.5 1.1 0.3110.042 2378
water
air
p Pa
p Pa
Pap arla 185.0min
Fluent example flow in a pipeCFD6
Voda
DisplayVectorsDef.ModelVisc. Def.Solv.Axisym. Def.Boundaryu=1
Fluent example flow in a pipeCFD6
AirLaminar Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 0.61464286
Spalart Almaras Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 0.62425083
k-epsilon Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 1.2050774
RNG Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 1.2050774
Realizable Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 1.1221648
k-omega Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 0.78146631
WaterSpalart Almaraz Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 141.57076
RNG Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 245.41853
k-epsilon Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 274.16226
k-omega Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 258.04153
4
4
0.316 0.5 1000 1400.042 40000
0.316 0.5 1.1 0.3110.042 2378
water
air
p Pa
p Pa
Pap arla 185.0min
ax
max
140 500 6400.311 0.55 0.866
waterm
air
p Pap Pa
ResultsSurface average (pressure at
inlet)
Fluent example flow in a pipeCFD6
053.0
8.2204.0140
4
*
w
w
u
PaLDp
Buffer layer 5<y+<30
Wall functions must be used as boundary conditions at wall
26001.0
100040/02.0053.0
y
this value was calculated for finer mesh with 40 cells in radial direction
Fluent example flow in a pipeCFD6/ Journal File for GAMBIT 2.4.6, Database 2.4.4, ntx86 SP2007051421/ Identifier "pipe2dn"/ File opened for write Mon Nov 28 08:28:27 2011.identifier name "pipe2dn" new nosavepreviousface create width 1 height 0.02 xyplane rectanglewindow matrix 1 entries 1 0 0 0 0 1 0 0 0 0 1 0 -0.2624999880791 \ 0.2624999880791 -0.1590357869864 0.1590357869864 -1 1face move "face.1" offset 0.5 0.01 0undo begingroupedge delete "edge.1" keepsettings onlymeshedge mesh "edge.1" firstlast ratio1 1.981 intervals 100undo endgroupundo begingroupedge delete "edge.3" keepsettings onlymeshedge modify "edge.3" backwardedge mesh "edge.3" firstlast ratio1 2 intervals 100undo endgroupundo begingroupedge delete "edge.4" keepsettings onlymeshedge modify "edge.4" backwardedge picklink "edge.4"edge mesh "edge.4" lastfirst ratio1 2 intervals 40undo endgroupundo begingroupedge delete "edge.2" keepsettings onlymeshedge mesh "edge.2" firstlast ratio1 0.5 intervals 40undo endgroupface mesh "face.1" mapphysics create btype "WALL" edge "edge.3"physics create btype "AXIS" edge "edge.1"physics create btype "VELOCITY_INLET" edge "edge.4"physics create btype "PRESSURE_OUTLET" edge "edge.2"solver select "FLUENT/UNS"export fluent5 "pipe2dn.msh" nozvalsave/ File closed at Sat Nov 26 13:56:19 2011, 1.39 cpu second(s), 3331408 maximum memory.save name "pipe2dn.dbs"export fluent5 "pipe2dn.msh" nozval/ File closed at Mon Nov 28 08:31:30 2011, 0.50 cpu second(s), 4639176 maximum memory.
the simplest way how to modify geometry is the journal file modification (L=1m, mesh 100x40)
2
1 2 1 2( )2up p L LD
By comparing results for tubes of different lengths (L1,L2) it is possible to eliminate the entrance effect
Fluent example flow in a pipeCFD6
Papp mm 1405.01 Water L=0.5mSpalart Almaraz Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 141.57076
RNG Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 245.41853
k-epsilon Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 274.16226
k-omega Mass-Weighted Average Static Pressure (pascal)-------------------------------- -------------------- velocity_inlet.3 258.04153
Water L=1mSpalart Almaraz Mass-Weighted Average Static Pressure (pascal)-------------------------------- --------------------
velocity_inlet.3 282.RNG Mass-Weighted Average Static Pressure (pascal)-------------------------------- --------------------
velocity_inlet.3 410.
k-epsilon Mass-Weighted Average Static Pressure (pascal)-------------------------------- --------------------
velocity_inlet.3 445
k-omega Mass-Weighted Average Static Pressure (pascal)-------------------------------- --------------------
velocity_inlet.3 481
141Pa error 1%
165Pa error 18%
171Pa error 22%
223Pa error 59%
RSM failed, negative pressure drop predicted -230.0078 Pa !