1. Introduction to the Turbulence Models
Seyhan Uygur Onbaşıoğlu

2. Not in order but interacting
Physical domain
Computational domain
Boundary Conditions
Turbulence Model
Numerical Method (discretization etc.)
Grid type and layout

3. History of the dependent variableφ

4. If you deal with CFD you should know about turbulence!
Flow around/in cars, aeroplanes, buildings.
Boundary layers and wakes around/after cars, aeroplanes, buildings.
Flow and combustion in engines, gasturbines, combustors.
Air movements in rooms, enclosures…

5. Mean and fluctuation
If the mean flow is steady we divide turbulence into mean and fluctuation parts.

6. Irregularity, randomness, dissipation
Turbulent flow is irregular, random and chaotic.
Random but deterministic.
There is a spectrum of scales from the largest to the smallest.

14. Scales
Largest scale is the order of the flow geometry, boundary layer thickness, jet width
Smallest scale is the smallest eddy scale where the viscous stresses are dissipated into internal energy.

15. Eddies…
An eddy is a turbulent motion localized within a region of size l.
Sizes of the eddy differ from the flow length L, to the smallest eddy size.
For each eddy size
there is a velocity and time scale

16. From Big to the Smaller Eddies…
Each eddy has Reynolds number. Large eddies are unstable and break up into smaller eddies and so on…

17. The Energy Cascade

18. The Energy Cascade

19. Energy cascade… This energy cascade continues until the
Reynolds number is sufficiently small that
energy is dissipated by viscous effects:
the eddy motion is stable,
and molecular viscosity is responsible for dissipation.

20. Vorticity
= 0
“vortex stretching” will stop.
Ul/ν=1

21. Turbulence
is random
contains different scales 3D

22. Scales and the dissipation
Since the KE is destroyed by the viscous forces, the larger viscosity means larger scales.
The amount of energy is to be dissipated is ε.
More dissipation means the larger velocity gradient.

23. Scales
The dimensions on LHS =RHS
Similarly,
Called Kolmogorov scale.

24. Energy Spectrum
I Range for large, energy containing eddies, II the inertial subrange, III Range for small, isotropic scales

26. Energy Spectrum

27. Energy Spectrum

28. Kolmogorov Spectrum Law or -5/3 Law
Explains the difficulty of LES and DNS

29. The Kinetic Energy of Turbulence

30. A statistical measure is an average of some kind:
over the symmetry coordinates, if any are available (e.g, a time average for stationary flows);
over multiple realizations (e.g, an ensemble);
or over the phase space of solutions if the dynamics are homogeneous.
In a statistically homogenous turbulent flow, measurable statistical quantities such as the mean velocity or the turbulent kinetic energy are the same at every point in the flow.

31. Probability density functions and moments
A complete description of a turbulent variable v at a given location and instant in time is given by the probability density function (PDF), P(v), where P(v)dv is the probability of the variable v taking a value
between v and v+ dv, and

36. Navier-Stokes Denklemleri

37. Structure of the Transport Equation

38. Reynolds Averaging and RANS

39. Reynolds averaging (1)

40. Reynolds Averaging (2)

41. Phase averaging

42. Some Rules

43. Obtaining the turbulent transport
N-S for fluctuations

44. Obtaining the turbulent transport
Energy for temperature fluctuations

45. Obtaining the turbulent transport
averaging
fluctuations

46. Obtaining the turbulent transport

47. Transport equation for the turbulent stress

48. Transport equation for the turbulent stress…

49. The terms of transport equation for the turbulent stress
The first 2 terms are turbulent diffusion of Reynolds stresses
The 3 rd term is molecular d,ffus,on of Reynolds stresses
4 th and 5 th are production of the Reynolds stresses

50. The turbulent stress(es)

51. The terms of transport equation for the turbulent stress
6 th term is the dissipation
and the 7 th term is the pressure strain term (making the flow isotropic.)

52. The turbulent kinetic energy

53. The dissipation of the turbulent kinetic energy
We need a transport equation!

54. Obtaining the transport equation for the dissipation of the turbulent kinetic energy

55. The transport equation for the dissipation of the kinetic energy

56. The transport equation for the dissipation of the kinetic energy…

57. The terms of the dissipation equation
The first two terms are the turbulent diffusion of the dissipation
The third one is the molecular diffusion of the dissipation
The 4 th and the 5 th terms are the production of the dissipation
The last two terms are the destruction of the dissipation.

58. Equation for turbulent heat flux

59. Closure Problem

60. Closure Problem…

61. Method for closing

62. Method for closing

63. Without solving RS transport

64. Without solving RS transport…

65. Without solving RS transport…

66. Without solving RS transport…

67. Without solving RS transport…

68. Without solving RS transport…

69. Without solving RS transport…

70. Exact and Modelled Equations

71. For non-isotropic diffusivity

72. Modelled RS equation

73. Modelled RS equation…

74. Modelled k equation

75. Modelled dissipation equation

76. Free shear flows

77. Flow

78. Round jets