ENVIRONMENTAL AND INDUSTRIAL CFD SIMULATIONS Turbulence models in the environmental flow Zbyn ě k Ja ň our Institute of Thermomechanics AS CR, Dolej š kova 5 Prague 8, , Czech Republic,

2 Overview Introduction, Equations, Turbulence, Atmospheric Boundary Layer, Closure Problem, Models, Boundary Conditions, Applications Conclusion

3 Introduction The most fluid on the world belongs to the atmosphere and the ocean, Geophysical fluid dynamics

4 Introduction

5 Equations Inertial coordinate system Reference coordinate system

6 Equations Inertial coordinate system: Continuity equation The equation of motion The energy equation

7 Equations Reference coordinate system: R – perpendicular distance from the rotation axis, The last term on the r.h.s. can be included into the gravitation force

8 Equations Reference coordinate system: Continuity equation The equation of motion Coriolis force: or f~10 -4

9 Turbulence Is the atmosphere turbulent? According to Tennekes, Lumley: A First Course in Turbulence the turbulence flow has following characters: Irregular-Y, Diffusive-Y, Large Re10 9- Y, 3D vorticity fluctuations-Y, Dissipativeneeds energy supply- Y/N, Continuum-Y, Turbulent flows are flows- Y

10 Turbulence Wake behind a jet – turbulent / nonturbulent ? The answer: It is not flow ; it is a picture of the former turbulent wake

11 Turbulence Energy sources: –Atmospheric Boundary Layer (ABL) –Free atmosphere: Clouds, Clear-Air Turbulence (CAT)

12 Turbulence Characteristic scale: –VelocityU, –Length in horizontal directionL, –Length in vertical directionH, –-pressure  P,

13 Turbulence

14 Turbulence

15 Turbulence Turbulent flow -L~10 2  –Atmospheric Boundary Layer (ABL) –Free atmosphere: Clouds, Clear-Air Turbulence (CAT)

16 Turbulence The ABL: Layer of air directly above the Earth surface in which effects of the surface (friction, heating and cooling) are felt on time scales less than a day, and in which significant fluxes of momentum, heat or matter are carried by turbulent motions on scale of the order of the depth of the boundary layer or less

17 Turbulence

18 Turbulence Cloud Cumulus-type cloud associated with thunderstorm:

19 Turbulence CAT Shear turbulence without visible manifestations. It occurs outside of clouds, In only about 20% of the free atmosphere below 12 km, is even less common above 12 km and occurs in only about 2% near 17 km, It generally occurs in stable conditions, It has not cased severe structure damage of aircraft.

20 Turbulence Atmospheric turbulence differs from most laboratory turbulence in: –Heat convection coexists with mechanical turbulence, –The rotation of the earth becomes important for many problems

21 Atmospheric Boundary Layer (ABL) The ABL is the region in which the large- scale flow of the free atmosphere adjusts to the boundary condition imposed by the earth´s surface

22 ABL Small-scale maximum- turbulent peak Large-scale maximum- synoptic peak Spectral gap around 1 cycle/hour

23 ABL Fluctuations with frequency smaller than 0.1 cycle/km belongs to the mean value Fluctuations with frequency large than 0.1 cycle/km belongs to the turbulent fluctuations + Reynolds conditions

24 Equations

25 Closure problem New dependent variables: Closure problem, etc. New dependent variables

26 Model taxonomy Ensemble-averaged equations –Integral models, –First-order closure models, –Second-order closure models, –Reynolds-stress models, Volume-averaged equations –Large Eddy Simulation (LES) Full simulation –Direct Numerical Simulation (DNS)

27 Integral models Reynolds equations are integrated over at least one coordinate direction and the number of independent variables decreases

28 Integral models Mixed Layer

29 Integral models Where is:

30 Integral models Equations for velocity and temperature jumps

31 Integral models Equations for heat and momentum fluxes at the inversion base 9 equations for 10 dependent variables

32 Integral models Models for z i : w e entrainment velocity - - R b – Richardson number

33 First-order closure models K-models based on the hypothesis of Boussinesq(1877), who suggested that turbulent shearing stress in analogy to viscous stress can be related to the mean strain Where t is eddy viscosity – new dependent variable

34 Eddy viscosity t = constant - Ekman spiral(1905)

35 Eddy viscosity Notice: ABL thickness  1km  t = 10

36 Eddy viscosity Prandt´s model - Blackadar (1962) generalized by Estoqe, Bhumralk (1969) and Yu (1977). l-mixing length z 0 – roughness length

37 Eddy viscosity Richardson number

38 Two equations models

39 Large Eddy Simulation The first large-eddy simulations were performed by Deardorff (1972; 1973;1974), and were later investigated by e.g.,: Schemm and Lipps (1976), Sommeria (1976), Moeng (1984), Wyngaard and Brost (1984), Schmidt and Schumann (1989), Mason (1989). Much of the previous work LES has been focused on simulations of the convective boundary layers (Nieuwstadt et al., 1992). The cloudy boundary layers were simulated by e.g., Sommeria 1976; Deardorff 1980; Moeng 1986; Moeng et al. 1996; Lewellen and Lewellen 1996, Cuijpers and Duynkerke (1993).

40 Boundary Conditions

41 Boundary Conditions The equations of motion has to be supplemented with initial and boundary conditions – in many papers the conditions are not introduced

42 Boundary Conditions In limited-area atmospheric models the surface -  S is the only physical boundary of the solution domain. All other boundaries are purely computational

43 Boundary Conditions on the surface

44 Boundary Conditions on the surface Two methods: Boundary conditions on the surface + modification of the equations of motion for small turbulence Reynold number+ increasing number of grid points near the wall, Wall function

45 Boundary Conditions-wall function for 30 < z 1 u * /v < 100

46 Roughness length- experience

47 Roughness length-models Petersen: z 0 = D f H, where D  0.5, f = A f / A T

48 Boundary Conditions on the top of the ABL

49 Outlet Boundary Conditions

50 Inlet Boundary Conditions Dirichlet condition determined from: In-situ measurement – a very few data sets, Universal profiles: –Ekman spiral, –Power law, –…. -mostly for horizontally homogeneous surface

51 Boussinesq approximation limited-area 

52 Boussinesq approximation Large scale flow Small scale fluctuation Turbulent fluctuation

53 Boussinesq approximation Hydrostatic approximation : G eostro ph ic approximation : Large scale flow:

54 Boussinesq approximation Shallow water approximation: (incompressible) Continuity equation Anelastic approximation Small scale fluctuation:

55 Boussinesq approximation Reynolds equations: Notices: Small scale fluctuation of the pressure and potential temperature, Buoyant force instead gravitational force, Incompressible case

56 Boussinesq approximation F´=0 for  i

57 Application

58 Application

59 Application Dispersion from linen source inside the street canyon- FLUENT

60 Application experiment k-  model RNG k-  model Dispersion from linen source inside the street canyon- FLUENT

61 Application laser sheet- DANTEC, The recordings from the video camera for values of the Reynolds number of Re  U 0 H/  (2.3 x 10 4 ; 2.3 x 10 5 ), Smoke generator

62 Application External velocity U g =1.5m/s, liquid is drawn from the cavern into the external stream,

63 Application External velocity U g =4.0m/s

64 Application

65 Application UABL is similar to the flow over a rough surface, with a large roughness length z 0 and a defined surface heat flux Q G ; The horizontally homogeneous atmospheric boundary layer horizontal length scale - L   A simple model of the UABL

66 Application

67 Application radiosounding launched in Barcelona, indifferent stratification influence of topography is more important across Internal-Sub-Layer artificial mean profile determined from the data sets seems to be more suitable for comparison;

68 Application radiosounding launched in Évora, indifferent stratification

69 Application sodar measurement in Prague, without stratification assessment

70 Application

71 Application Algebraic turbulence models

72 Application Plume from point sources in south east Giant Mountains Algebraic turbulence models

73 4. Conclusions Eddy viscosity models appears: – Quite satisfactory in neutral or stable ABL; –Fail in convective situations; Reynolds stress models are more suitable, Boundary Conditions are complicated and important task