1. Computer Fluid Dynamics E181107
CFD6
Turbulent flows,…
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
Some figures and equations are copied from Wikipedia

2. Turbulence
CFD6
Source of nonlinearities of NS equations are inertial and viscous terms
for incompressible fluids the equivalent conservative form
The convection term is a quadratic function of velocities – source of nonlinearities and turbulent phenomena

3. Instabilities Rayleigh Benard convection
CFD6
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.
>1100 L 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.

4. Instabilities Karman vortex street
CFD6
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.

5. Turbulence
CFD6
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.
Distarbunce is attenuated
Distarbunce is amplified
Laminar flow: Re<Recrit
Turbulent flow: Re>Recrit

6. Actual value at given time and space
Turbulence - fluctuations
CFD6
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
Actual value at given time and space
Mean value
fluctuation

7. Turbulence - fluctuations
CFD6
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

8. Turbulent eddies - scales
CFD6
Kinetic energy of turbulent fluctuations is the sum of energies of turbulent eddies of different sizes
Large energetic eddies (size L) break to smaller eddies. This transformation is not affected by viscosity
E()
=2f/u
1/L /
Spectral energy of turbulent eddies
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
wavenumber (1/size of eddy)
Typical values of frequency f~10 kHz, Kolmogorov scale ~0.01 up to 0.1 mm
Kolmogorov scale  decreases with the increasing Re

9. Viscous subrange - scales
CFD6
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)
Length scale
Time scale
velocity scale
These expressions follow from dimension of viscosity  [m2/s] and the rate of energy dissipation  [m2/s3]

10. Viscous scale - tutorial
CFD6
Derive time scale of the smallest turbulent eddies
Time scale

11. And this is the famous Kolmogorov’s law of 5/3
Inertial subrange - scales
CFD6
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
Dimension analysis in the inertial subrange
And this is the famous Kolmogorov’s law of 5/3

12. Turbulent eddies - scales
CFD6
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

13. Turbulent eddies - scales
CFD6
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
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

14. DNS Direct Numerical Simulation
CFD6  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 
(based upon dimensional ground)
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.
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).  
Remark: Re~106 or 107 at flow around a car or flow around wings

15. DNS Direct Numerical Simulation
CFD6
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
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.

16. DNS Direct Numerical Simulation
CFD6
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).

17. Transition Laminar-Turbulent
CFD6
Bacon

18. Inflection – source of instability (max.gradient)
Transition Laminar-Turbulent
CFD6
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
velocity y
Inflection – source of instability (max.gradient)
velocity y
Poiseuille flow ~ 5700
Couette flow – stable?
There is no inflection of velocity profile in a pipe, however turbulent regime exists if Re>2100

19. Transition Laminar-Turbulent
CFD6
How to indentify whether the flow is laminar or turbulent ?
Experimentally
Visualization, hot wire anemometers, LDA (Laser Doppler Anemometry).
Numerical experiments
Start 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

20. Transition Laminar-Turbulent
CFD6
Stability analysis
Linear stability analysis
Momentum equation for disturbance
Velocity disturbance
Production (extracting energy from the mean flow to fluctuations)
Mean (undisturbed) flow
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

21. Transition Laminar-Turbulent
CFD6
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 D D D
D/2 D

22. Turbulent structures evolution
CFD6
Journal of Fluids and Structures 18 (2003) 305–324
Force coefficients and Strouhal numbers of four cylinders in cross flow
K. Lama, J.Y. Lib, R.M.C. Soa
Re=200
Re<4
Re<40
Re=800
Re<200 2D von Karman vortex street

23. Fully developed turbulent flows
CFD6
Free flows (self preserving flows)
Jets Mixing layers Wakes h y u
umax
umin u y b
umax
Jet thickness ~x, mixing length ~x see Goertler, Abramovic Teorieja turbulentnych struj, Moskva 1984:
Circular jet
Planar jet x x

24. Example - tutorial
CFD6
Entrainment in jets (increase of volumetric flowrate) h y u
umax
Planar jet ~x
~1/x
Circular jet
~1/x

25. Fully developed turbulent flows
CFD6
Flow at walls (boundary layers) y
Friction velocity
Log law
Buffer layer
Laminar sublayer u
INNER layer (independent of bulk velocities)
laminar
y+<5
u+=y+
buffer
5<y+<30
u+=-3+5lny+
Log law
30<y+<500
u+=lnEy+/
OUTER layer (law of wake)

26. vanDriest model of mixing length lm
Fully developed turbulent flows
CFD6
Flow at walls (turbulent stresses) t
Prandtl’s model
vanDriest model of mixing length lm

27. Example tutorial
CFD6
Calculate thickness of laminar sublayer at flow of water in pipe (D=2 cm) at flowrate 1 l/s.
Turbulent region, well within validity of Blasius correlation
Wall shear stress from Blasius
Friction velocity
Dimensionless thickness of laminar sublayer

28. TIME AVERAGING of turbulent fluctuations
CFD6
Benton

29. TIME AVERAGING of turbulent fluctuations
CFD6
RANS (Reynolds Averaging of Navier Stokes eqs.)
Time average
Favre’s average
Proof:
Remark: Favre’s average differs only for compressible substances and at high Mach number flows. Ma>1

30. TIME AVERAGING of turbulent fluctuations
CFD6
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

31. TIME AVERAGING of NS equations
CFD6
Continuity equation
Navier Stokes equations
Reynolds stresses

32. TIME AVERAGING of transport equations
CFD6
Turbulent fluxes

33. Boussinesq hypothesis
CFD6
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.

34. Rate of deformation based upon gradient of averaged velocities
Boussinesq hypothesis
CFD6
Analogy to Fourier law
Analogy to Newton’s law
Rate of deformation based upon gradient of averaged velocities

35. TRANSPORT coefficients-analogy
CFD6
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
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

36. Turbulent viscosity models
CFD6
Botero

37. Turbulent viscosity models
CFD6
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)

38. Algebraic models
CFD6
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). h
u(y)
Circular jet
Planar jet
Mixing length h
Mixing layers y
Currently used algebraic models are Baldwin Lomax, and Cebecci Smith

39. 2 equations models
CFD6
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] 
Wilcox (1998) k-omega model, Kolmogorov (1942)
Jones,Launder (1972), Launder Spalding (1974) k-epsilon model
C = 0.09 (Fluent-default)

40. Tutorial 2 equations models
CFD6
Derive relationship for turbulent viscosity
Launder Spalding k-epsilon model

41. k – transport equation
CFD6
All two equation models (and also Prandtl’s one equation model) are based upon transport equation for the kinetic energy of turbulence
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’) (2e’) (u’u’)

42. Example: approximation of Reynolds stress transport term
k – transport equation
CFD6
Dispersion terms (viscous and reynolds stresses) and production term can be expressed using turbulent viscosity (Boussinesq)
Example: approximation of Reynolds stress transport term
k = 1 Fluent
-
Rate of dissipation of kinetic energy of velocity fluctuation
Laminar and not turbulent viscosity!

43.  – transport equation
CFD6
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
Production term (generator of turbulence)
Dissipation term
This term follows from dimensional analysis
C1 = C2 = = 1.3 Fluent (default)

44. k- modifications (RNG, realizable)
CFD6
Corrections of dispersion, production and dissipation terms with the aim to extent applicability of k- for low Reynolds number flows
Functions of turbulent Reynolds

45. Effective viscosity (RNG)
CFD6
Blending formula for effective viscosity

46.  - estimate Dissipated energy in a mixing tank CFD6 n Power number dd
Check units of dissipation  [W/kg =m2/s3]

47. Distance of the nearest boundary node from wall
k- boundary conditions
CFD6
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
This is implemented in majority of CFD programs
Friction velocity
Distance of the nearest boundary node from wall

48. This term is zero for incompressible liquids
Reynolds stresses (k-)
CFD6
Constitutive equation for Reynolds stresses must be modified with respect to isotropic pressure
This term is zero for incompressible liquids
Remark: Turbulent stresses determine kinetic energy of turbulent fluctuations

49. Assesment of k- models
CFD6
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

50. Discrepancies?
CFD6
Special case: Steady unidirectional flow, constant density, homogeneous turbulence
These terms are identically zero in homogeneous turbulence
Further reading
C1 = C2 = k = 1 = 1.3

51. Reynolds Stress Models (RSM)
CFD6
Centrifugal forces
6 transport equations
production
Dissipation
Pressure transport
diffusion
1 dissipation

52. Large Eddy Simulation (LES)
CFD6
Demuth

53. Convolution, Green’s function G
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
Convolution, Green’s function G

54. LES G-function example
CFD6 u Δ x

55. LES filtering - properties
CFD6
Basic difference in comparison with time averaging

56. LES NS equations CFD6 Continuity equation without changes
Lij Leonard stresses
Cij cross stresses
Rij SGS (sub grid stresses)

57. It looks like Reynolds stresses
LES NS equations
CFD6
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

58. Smagorinski’s constant Cs=0.1 in Fluent
LES Smagorinski SGS
CFD6
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.
Remark: in the vicinity of wall the Cs mixing length is limited by y (karman constant times the distance from wall)
Smagorinski’s constant Cs=0.1 in Fluent

59. LES RNG SGS CFD6 Crng=0.157 in Fluent Crng=100 in Fluent
For small s the effective viscosity equals the laminar viscosity , while eff=s holds for a high level of turbulence.

60. LES boundary conditions
CFD6
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
Von Karman constant 0.41
Friction velocity

61. Fluent turbulent models
CFD6
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 laminarturbulent
v2-f boundary layer detachment (low Re)
RSM
DES (Detached Eddy Simulation, Spalart Almaras+realizable+SST), like LES at wall
LES (Large Eddy Simulation)

62. Fluent example flow in a pipe
CFD6
Material: water-liquid (fluid)
Property Units Method Value(s)
Density kg/m constant
Cp (Specific Heat) j/kg-k constant
Thermal Conductivit w/m-k constant 0.6
Viscosity kg/m-s constant
Molecular Weight kg/kgmol constant
WALL
VELOCITY INLET
AXIS
Tube geometry: L=0.5m, D=0.04m.
Velocity at inlet (uniform) u=1m/s
Mesh 50 x 20, compression 2 (last/first)
PRESSURE OUTLET

63. Def.ModelVisc. Def.Solv.Axisym. Def.Boundaryu=1
Fluent example flow in a pipe
CFD6
Voda
Def.ModelVisc. Def.Solv.Axisym. Def.Boundaryu=1
DisplayVectors

64. ResultsSurface average (pressure at inlet)
Fluent example flow in a pipe
CFD6
Air
Laminar Mass-Weighted Average
Static Pressure (pascal)
velocity_inlet
Spalart Almaras Mass-Weighted Average
velocity_inlet
k-epsilon Mass-Weighted Average
velocity_inlet
RNG Mass-Weighted Average
Realizable Mass-Weighted Average
velocity_inlet
k-omega Mass-Weighted Average
velocity_inlet
ResultsSurface average (pressure at inlet)
Water
Spalart Almaraz Mass-Weighted Average
Static Pressure (pascal)
velocity_inlet
RNG Mass-Weighted Average
velocity_inlet
k-epsilon Mass-Weighted Average
velocity_inlet
k-omega Mass-Weighted Average
velocity_inlet

65. Fluent example flow in a pipe
CFD6
this value was calculated for finer mesh with 40 cells in radial direction
Buffer layer 5<y+<30
Wall functions must be used as boundary conditions at wall

66. Fluent example flow in a pipe
CFD6
/ Journal File for GAMBIT 2.4.6, Database 2.4.4, ntx86 SP
/ Identifier "pipe2dn"
/ File opened for write Mon Nov 28 08:28:
identifier name "pipe2dn" new nosaveprevious
face create width 1 height 0.02 xyplane rectangle
window matrix 1 entries \
face move "face.1" offset
undo begingroup
edge delete "edge.1" keepsettings onlymesh
edge mesh "edge.1" firstlast ratio intervals 100
undo endgroup
edge delete "edge.3" keepsettings onlymesh
edge modify "edge.3" backward
edge mesh "edge.3" firstlast ratio1 2 intervals 100
edge delete "edge.4" keepsettings onlymesh
edge modify "edge.4" backward
edge picklink "edge.4"
edge mesh "edge.4" lastfirst ratio1 2 intervals 40
edge delete "edge.2" keepsettings onlymesh
edge mesh "edge.2" firstlast ratio1 0.5 intervals 40
face mesh "face.1" map
physics 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" nozval
save
/ File closed at Sat Nov 26 13:56: , 1.39 cpu second(s), maximum memory.
save name "pipe2dn.dbs"
/ File closed at Mon Nov 28 08:31: , 0.50 cpu second(s), maximum memory.
the simplest way how to modify geometry is the journal file modification (L=1m, mesh 100x40)
By comparing results for tubes of different lengths (L1,L2) it is possible to eliminate the entrance effect

67. Fluent example flow in a pipe
CFD6
Water L=1m
Spalart Almaraz Mass-Weighted Average
Static Pressure (pascal)
velocity_inlet
RNG Mass-Weighted Average
velocity_inlet
k-epsilon Mass-Weighted Average
velocity_inlet
k-omega Mass-Weighted Average
velocity_inlet
Water L=0.5m
Spalart Almaraz Mass-Weighted Average
Static Pressure (pascal)
velocity_inlet
RNG Mass-Weighted Average
velocity_inlet
k-epsilon Mass-Weighted Average
velocity_inlet
k-omega Mass-Weighted Average
velocity_inlet
141Pa error 1%
165Pa error 18%
171Pa error 22%
223Pa error 59%
RSM failed, negative pressure drop predicted Pa !