Large Eddy Simulation

1. In The Name of God Large Eddy Simulation By Reza Barati Under Guidance of Prof. G. Heidarinejad Large Eddy Simulation University of Tarbiat Modares April, 28 2012 E-mail: r88barati@gmail.com

2. Outline • Introduction • Simulation of the turbulence • Direct Numerical Simulation (DNS) • Reynolds Averaged Navier-Stokes (RANS) • Large Eddy Simulation (LES) • Application examples • Summary and conclusions

3. Introduction • Most flows encountered in natural world and industrial processes are turbulent. Turbulent wake past a leaking oil tanker Turbulent wake behind a bluff body Turbulent jet Large Eddy Simulation 3

4. Introduction (2) • An understanding of turbulence and the ability to predict turbulence for any given application is invaluable for the engineer. For example: • Turbulence increases drag due to increased frictional forces. • A few characteristic of the turbulence are: • Inherently unsteady, three dimensional and aperiodic swirling motions (fluctuations) resulting in enhancement of mixing, heat transfer and shear. • Instantaneous fluctuations are random (unpredictable) both in space and in time, but statistical averaging of turbulence fluctuations results in accountable transport mechanisms. • Wide range of length scales (vortices or eddies) exist in all turbulent flows (from very small to very large). Large Eddy Simulation 4

5. Introduction (3) • Difficulty in turbulence modeling is trying to accurately capture the contributions of all scales in the spectrum. Large Eddy Simulation 5

6. Largest Eddies Large Eddies Smallest Eddies Mean Flow Small Eddies Molecules ... . Introduction (4) • Small eddies receive the kinetic energy from slightly larger eddies. The slightly larger eddies receive their energy from even larger eddies and so on. The largest eddies extract their energy from the mean flow. This process of transferred energy from the largest turbulent scales (eddies) to the smallest is called cascade process. Internal Energy Kinetic Energy Dissipation Turbulence Energy Transfer Cascade Process Large Eddy Simulation 6

7. Simulation of the Turbulence • No analytical solution possible to solve the Navier-Stokes equations. • For most engineering applications it is unnecessary to resolve the details of the turbulent fluctuations. • Need to simplify by mean flowequations. • Intrinsically we are either averaging or filtering out particular scales of turbulence. • Since the averaged or filtered scales are not resolved, their effect must be modelled. • A turbulence model is a computational procedure to close the system of mean flow equations. • For a turbulence model to be useful it: • Must have wide applicability, • Be accurate, • Simple, • And economical to run. Large Eddy Simulation 7

8. Simulation of the Turbulence (2) • Different approaches to make turbulence computationally tractable: • DNS: Direct Numerical Simulation • RANS Model: Reynolds Averaged Navier-Stokes Model • LES Model: Large Eddy Simulation Model • LES • unsteady • DNS • unsteady • RANS • Steady • unsteady Large Eddy Simulation 8

9. Simulation of the Turbulence (3) • Direct numerical simulation (DNS): Currently DNS is the most exact approach to modeling turbulence since no averaging is done or approximations are made. The basic requirement for such a numerical simulation to succeed is for the numerical mesh spacing to be finer than the smallest scales of motion in the solution. • Most straight-forward approach: • No Modelling at all • Resolve all scales of turbulent flow explicitly. • A DNS is equivalent to a lab experiment. • Advantage: • (In principle) a very accurate turbulence representation. • DNS is highly informative regarding the physics of fluid flow. • DNS can be used to understand the mechanisms of turbulent production and dissipation. Large Eddy Simulation 9

10. Simulation of the Turbulence (4) • Direct numerical simulation (DNS): • Limitation: • limited to simple geometries. • Requires supercomputers; Computer cost (memory, CPU, hardware,…) increases wit the Reynolds number. (the total number of nodes is proportional to ). Therefore, Reynolds number has to be relatively low. Large Eddy Simulation 10

11. Simulation of the Turbulence (5) • Direct numerical simulation (DNS): • Limitation (continued): • Although DNS is the simplest form numerical point of view, the equations also need special treatment in that finite difference techniques (and the other standard techniques) cannot be used. • Technique: • DNS uses the technique that is called spectral methods, in that we express the velocity field as a Fourier series (in spectral space) and the procedure is then to calculate the coefficients of the Fourier series. Large Eddy Simulation 11

12. Simulation of the Turbulence (6) • Direct numerical simulation (DNS): • Consequences: • All the effort in DNS is directed towards the resolution of small scales that called the Kolmogoroff scale. • 99% of the energy is contained outside the dissipation range (the smallest scales). • For practical engineering purposes, DNS is not only too costly, but also the details of the simulation are usually not required. • Given the current processing speed and memory of the largest computers, only very modest Reynolds number flows with simple geometries are possible. • DNS is currently applied to simple flows such as channel flows and free shear flows. Large Eddy Simulation 12

13. Simulation of the Turbulence (7) • Reynolds averaged Navier-Stokes simulation (RANS): RANS turbulence models make no attempt to resolve any scales of turbulence at all. The equations of motion are averaged over all scales of turbulence and application of averaging to the equations of motion results in additional unknowns, called the Reynolds stresses and are grouped into the Reynolds stress tensor. • Closure models are: • Zero-equation turbulence models • No transport equation used. • Advantages: simplest of Models satisfying the requirements. • Disadvantages: need to calibration for different flows. Large Eddy Simulation 13

14. Simulation of the Turbulence (8) • Reynolds averaged Navier-Stokes simulation (RANS): • Closure models are (continued) : • One-equation turbulence models • Transport equation modeled for turbulent kinetic energy k. • Advantages: additional assumptions can be avoided. • Disadvantages: the length scale is still a algebraic quantity, and computationally more difficult. • Two-equation models • More complete by modeling transport equation for turbulent kinetic energy k and eddy dissipation. • Advantages: overcomes the short comings of zero and one equation model. • Disadvantages: Not appropriate to use in a viscous sublayer, and still need to make assumptions. Large Eddy Simulation 14

15. Simulation of the Turbulence (9) • Reynolds averaged Navier-Stokes simulation (RANS): • Closure models are (continued) : • Second-order closure • Reynolds Stress Model (RSM): The RSM turbulence model solves transport equations for each individual stress. • Does not use Boussinesq approximation as first-order closure models • Advantages: non-isotropic turbulence effects makes this suitable for highly swirling flows. • Disadvantages: 7-equation (3D) model. Large Eddy Simulation 15

16. Simulation of the Turbulence (10) • Reynolds averaged Navier-Stokes simulation (RANS): • Advantage: • Computationally inexpensive, fast. • The only dynamic behaviour that will affect the equation will be that of the mean flow. Hence for flows which are statistically steady the Reynolds averaged equations of motion can be solved at steady state. • Limitation : • Turbulent fluctuations not explicitly captured. • Parameterizations are very sensitive to large-eddy structure that depends on environmental conditions such as geometry and stratification. Parameterizations are not valid for a wide range of different flows. Large Eddy Simulation 16

17. Simulation of the Turbulence (11) • Reynolds averaged Navier-Stokes simulation (RANS): • Consequence: • Not suitable for detailed turbulence studies. RANS is only suitable for the design of processes the performance of which depends mainly on the mean flow characteristics and is not strongly affected by the turbulence. • The predictions of RANS models in their standard form, can be both acceptable or unacceptable depending on the desired accuracy, naivety of the user and other factors. Large Eddy Simulation 17

18. Simulation of the Turbulence (12) • Large Eddy Simulation (LES): Based on space-filtered equations. Time dependent calculations are performed. Large eddies are explicitly calculated. For small eddies, their effect on the flow pattern is taken into account with a “subgrid model” of which many styles are available. • Seeks to combine advantages and avoid disadvantages of DNS and RANS by treating large scales and small scales separately, based on Kolmogorov's similarity theory of turbulence. In other words, in LES, the large scales are directly represented while the small scales are modelled using standard modelling techniques (RSM, …). Large Eddy Simulation 18

19. Simulation of the Turbulence (13) • Large Eddy Simulation (LES): • The filter would act as an automation technique that tells the equations what to fully resolve and what to model. • Filtering is also characterized by what is called a filter width which define the smallest size of the eddy to be resolved. All eddies with scales less than are modelled. • Intuitive image: Grid of filter width D “fishes” from flow the large, energy rich eddy elements, while the small eddies “escape” through the grid mesh. Large eddies, coarse structure Small eddies, fine structure Large Eddy Simulation 19

20. Simulation of the Turbulence (14) • Large Eddy Simulation (LES): • Filtering – Decomposition: The idea is to decompose the velocity field into a filtered field and a residual velocity field called the residual stress or subgrid scale SGS component. Instantaneous component S Large Eddy Simulation 20

21. Simulation of the Turbulence (15) • Large Eddy Simulation (LES): • The large or resolved scale field is a local average of the complete field. • Where G(x-) is the filter kernel. Large Eddy Simulation 21

22. Simulation of the Turbulence (16) • Large Eddy Simulation (LES): • Two Classical Filters for Large-Eddy Simulation: • Box or top-hat filter Large Eddy Simulation 22

23. Simulation of the Turbulence (17) • Large Eddy Simulation (LES): • Two Classical Filters for Large-Eddy Simulation (continued): • Gaussian filter Large Eddy Simulation 23

24. Simulation of the Turbulence (18) • Large Eddy Simulation (LES): • Effects of filter width on results: Large Eddy Simulation 24

25. Simulation of the Turbulence (19) • Large Eddy Simulation (LES): • Filtered Navier-Stokes Equations: filtering the original Navier-Stokes equations gives filtered Navier-Stokes equations that are the governing equations in LES. • Continuity equation: Large Eddy Simulation 25

26. Simulation of the Turbulence (20) • Large Eddy Simulation (LES): • Filtered Navier-Stokes Equations (continued): • Momentum equation: Large Eddy Simulation 26

27. Simulation of the Turbulence (21) • Large Eddy Simulation (LES): • Filtered Navier-Stokes Equations (continued): • Momentum equation (continued) : Large Eddy Simulation 27

28. Simulation of the Turbulence (22) • Large Eddy Simulation (LES): • Filtered Navier-Stokes Equations (continued): • Momentum equation (continued) : Large Eddy Simulation 28

29. Simulation of the Turbulence (23) • Large Eddy Simulation (LES): • Filtered Navier-Stokes Equations (continued): • Momentum equation (continued) : Large Eddy Simulation 29

30. Simulation of the Turbulence (24) • Large Eddy Simulation (LES): • Filtered Navier-Stokes Equations (continued): • Momentum equation (continued) : Large Eddy Simulation 30

31. Simulation of the Turbulence (25) • Large Eddy Simulation (LES): • Filtered Navier-Stokes Equations (continued): • Momentum equation (continued) : Leonard stress Reynolds stress Needs modeling Sub-grid scale (SGS) stress Cross stress Large Eddy Simulation 31

32. Simulation of the Turbulence (26) • Large Eddy Simulation (LES): • Filtered Navier-Stokes Equations (continued): • Momentum equation (continued) : • The Leonard stress term represents the interaction of two resolved eddies to produce small scale turbulence. • The Reynolds stress term represents the interaction of two subgrid scale eddies. • The cross stress tensor represents the transfer of energy between large, resolved eddies and small, modeled eddies. Large Eddy Simulation 32

33. Simulation of the Turbulence (27) • Large Eddy Simulation (LES): • Sub-Grid Scale (SGS) models: The SGS model has to parameterize the effect of the SGS motions (small-scale turbulence) on the large eddies (resolved-scale turbulence). • Take idea from RANS modeling, introduce eddy viscosity model. • Smagorinsky model (1963) • Algebraic Dynamic Model (Germano, 1991) • Localized Dynamic Model (Kim and Menon, 1993) • Wall Adapting Local Eddy-viscosity model (Nocoud and Ducros, 1999) Large Eddy Simulation 33

34. Simulation of the Turbulence (28) • Large Eddy Simulation (LES): • Sub-Grid Scale (SGS) models (continued): • Smagorinsky model: This model is based on the notion of subgrid scale local equilibrium between the energy provided from large scales to small scales and the dissipation of energy by the small scales, and is equivalent to Prandtl` s mixing length theorem. Large Eddy Simulation 34

35. Simulation of the Turbulence (29) • Large Eddy Simulation (LES): • Sub-Grid Scale (SGS) models (continued): • 2D • 3D Dimensions of the computational Large Eddy Simulation 35

36. Simulation of the Turbulence (30) • Large Eddy Simulation (LES): • Sub-Grid Scale (SGS) models (continued): • Smagorinsky model (continued): • The major shortcoming is that there is no C0universally applicable to different types of flow. • Difficulty with transitional flows. • An ad hoc damping is needed in near-wall region. 36 Large Eddy Simulation

37. Simulation of the Turbulence (31) • Large Eddy Simulation (LES): • Sub-Grid Scale (SGS) models (continued): • Wall function • Damping function Large Eddy Simulation 37

38. Simulation of the Turbulence (32) • Large Eddy Simulation (LES): • Advantages: • Large eddies can be resolved directly. • Grid Size: Because large eddies are considered, the grid size become larger. • Only small eddies needs to be modelled. Large Eddy Simulation 38

39. Simulation of the Turbulence (33) • Large Eddy Simulation (LES): • Advantages (continued): • Calibration: Large eddies are more problem-dependent. They are dictated by the geometries and boundary conditions of the flow involved. Small eddies are less dependent on the geometry, tend to be more isotropic, and are consequently more universal. The chance of finding a universal turbulence model is much higher for small eddies. Less dependent on geometry, more universal. • Highly turbulent flows can be simulated. Large Eddy Simulation 39

40. Simulation of the Turbulence (34) • Large Eddy Simulation (LES): • Limitations: • Unsteady simulation. • Turn around time is weeks/months (RANS is hours or days). • Cannot afford grid independence testing. • High computational cost in boundary layers. Large Eddy Simulation 40

41. Simulation of the Turbulence (35) • Large Eddy Simulation (LES): • Consequences: • LES is midway between DNS and RANS in terms of rigor and computational requirement. • The filter is a function of grid size. Eddies smaller than the grid size are removed and modeled by a subgrid scale (SGS) model. Larger eddies are directly solved numerically by the filtered transient NS equation. • LES is computationally veryexpensive. LES can produce an overwhelming quantity of detailed information about a flow structure. Generally, in engineering flows, such levels on instantaneous information is not required. • RANS is increasingly being used to model the wall region of LES. • Many research studies have compared LES predictions to RANS predictions. Sometimes RANS is as good as LES, sometimes LES is better, sometimes the added accuracy of LES is not justified by the cost. Large Eddy Simulation 41

42. Application Examples • LES has been most successful for high-end applications where the RANS models fail to meet the needs. For example: • Combustion • Mixing • External Aerodynamics (flows around bluff bodies) Large Eddy Simulation 42

43. Application Examples (2) • Large Eddy Simulation (LES) implemented in: • Boundary layer (Schumann 1990) • Turbulent mixing layer (Vremen et al., 1997) • Nuclear applications (Gotzbach and Worner 1999) • Open channel flow (Stoesser et al., 2005) • Sediment transport (Zedler and Street, 2006) • Pipe flow (Hradisky, 2011) MOVIES Large Eddy Simulation 43

44. Application Examples (3) • LES of a realistic car model exposed to Crosswind Large Eddy Simulation 44

45. Application Examples (4) Large Eddy Simulation 45

46. Application Examples (5) • Bluff-body jet 20 m/s bluff body 62 m/s R = 1.8 mm Large Eddy Simulation 46

47. Application Examples (6) • Bluff body jet axial velocities Large Eddy Simulation 47

48. Summary and conclusions Turbulence modeling comes in varying degrees of complexity. Determining the right choice of turbulence model depends on the detail of results expected. • DNS: Direct Numerical Simulation Solves the unsteady Navier-Stokes equations directly. Powerful research tool. • RANS Model: (Reynolds Averaged Navier-Stokes Model) Motivation: Engineers are normally interested in knowing just a few quantitative properties of a turbulent flow. Method: Using Reynolds-Averaged form of the Navier-Stokes equations with appropriate turbulence models. • LES: Large Eddy Simulation Motivation: The large scale motions are generally much more energetic than the small scales, and they are the most effective transporters of the conserved properties. Method: Large Scales -> Solve, Small Scales ->Model Large Eddy Simulation 48

49. Summary and conclusions (2) The Kolmogorov universal power law spectrum Large Eddy Simulation 49

50. Summary and conclusions (3) Turbulence Simulation Exact Equations Approximate Equations Unsteady, Spatially Filter DNS • RANS LES “hybrid” methods combine RANS and LES (e.g., Detached-Eddy Simulation (DES)) Do not use Boussinesq Use Boussinesq assumption Reynolds Stress Model (7 new PDE’s) Zero Equation Models (e. g. mixing length model) 1 Equation Models (e. g. Spalart-Almaras) 2 Equation Models (e. g. K-e& K-w) Large Eddy Simulation 50