Modelling diabatic atmospheric boundary layer using a RANS-CFD code with a k-ε turbulence closure F. VENDEL

Transcription

1 Modelling diabatic atospheric boundary layer using a RANS-CFD code with a k-ε turbulence closure F. VENDEL Florian Vendel 1, Guillevic Laaison 1, Lionel Soulhac 1, Ludovic Donnat 2, Olivier Duclaux 2, and Cécile Puel 2 1) Laboratoire de Mécanique des Fluides et d Acoustique, Université de Lyon CNRS, Ecole Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon I, 36 avenue Guy de Collongue, Ecully, France 2) TOTAL, Centre de Recherche de Solaize, France

2 Introduction Context Modelling of pollutant dispersion over industrial areas iplies the description of the stratified surface layer flow and its interaction with buildings or coplex obstacles Better coputers perforances ake it possible today to siulate this flow using CFD odels and RANS equations (Fluent, Phoenics, StarCD ) But, generally standard paraeterization ipleented in these coercial odels are not really adapted so as to represent the atosphere So the question is : how to paraeterize the atospheric processes, particularly theral stratification in a RANS-CFD code? 2

3 Introduction CFD odelling of the SBL in the literature Application of CFD odels in neutral stability conditions Richards P.J. and Hoxey R.P., 1993 Blocken B. and al., 27 Hargreaves D.M. and Wright N.G., 27 Application in stable or unstable stability conditions (less studied) Duynkerke P.G., 1988 : odification of the k-ε odel constants to atch the physical characteristic of atospheric surface layer in neutral and stable conditions Huser A. and al., 1997 : inlet turbulence profiles does not aintain with distance (turbulence increase in stable stratification) Pontiggia M. and al., 29 : add a source ter in turbulent dissipation rate e equation 3

4 Introduction Main points to odel To odel a surface boundary layer with RANS-CFD codes, we focus on two ain points : Top boundary conditions Inlet boundary conditions Equations solved : Moentu equations Energy equation Turbulence closure Outflow boundary conditions Ground boundary conditions 4

5 Introduction Soe questions unsolved Equations solved : Which set of equations odels properly the flow and the turbulence for a diabatic surface layer? How to treat the inconsistency between the k and e profiles (stable/unstable conditions) and the conservation equations? Boundary conditions : How to describe the pressure profile in order to define appropriate downwind boundary conditions for stable and unstable cases? How to describe the inlet profiles to represent diabatic surface layer? How to ipose a constant flux of oentu and energy with the altitude (surface boundary layer assuption)? 5

6 Suary 1. Reference odel of the surface boundary layer 2. Consistency with k and e equations 3. Paraeterization of a diabatic surface layer in a RANS CFD siulation 6

7 The flow is oriented along the x direction and the ean vertical velocity is equal to zero : 1. Reference odel of the surface boundary layer Surface boundary layer assuptions v w (1) The vertical turbulent fluxes (Reynolds stresses and heat flux) are constant with respect to altitude (Garratt J.R., 1992) : 2 u' w' cste u* H w' ' cste C u The Monin-Obukhov siilarity theory predicts that the diensionless p * * (2) z u* z * u z gradient of velocity and potential teperature only depends on z/l MO (Garratt J.R., 1992) : h z z L C u where and (3) L MO MO gh 7 p 3 *

8 1. Reference odel of the surface boundary layer Surface boundary layer assuptions The turbulence satisfies a local equilibriu within the surface layer (Tennekes, H. and Luley, J. L., 1972) : P B e with u P u' w' shear TKE production z g B w' ' theral TKE production e turbulent dissipation rate / destruction (4) Influence of buoyancy effects in the oentu equation can be taken into account using Boussinesq approxiation (the density is constant except in the buoyancy ter of the oentu equation) : cste (5) ) g ( )g ( with 1 for an ideal gas 8 (6)

9 1. Reference odel of the surface boundary layer Conservation equations When using the precedent assuptions, the Reynolds Averaged Navier-Stokes (RANS) conservation equations for the ass, horizontal oentu and energy are verified and the vertical oentu equation reduces to : P z ( ) g with P abs P P gz (7) Where P is defined as difference between absolute and hydrostatic pressure. Integration of this equation will give the vertical profile of P in a stable boundary layer. P is constant for the neutral case, where ( z) 9

10 In order to odel the turbulence fluxes, we use in this work a k-ε turbulent closure : u' w' K w' ' K h 1. Reference odel of the surface boundary layer k-ε turbulence closure u z z K C Where k is the turbulent kinetic energy, ε the turbulent dissipation rate and K and K h are the turbulent diffusivity of oentu and heat. k and ε are given by two conservation equations (steady surface layer) : k² e with and Kh (8) Prt K K k z k z D P B e Where D is the diffusion ter (9) z K e e C z ep k e ² k e1 Ce 2 1 (1)

11 1. Reference odel of the surface boundary layer k-ε turbulence closure Focus on the turbulent dissipation rate equation : z K e e C z ep k e ² k e1 Ce 2 No ter for the buoyancy effects (Duynkerke P.G., 1988) The use of k-ε odel requires values for the paraeters C µ, σ k, σ ε, C ε1, C ε2 For siulating realistic atospheric values of the TKE in the surface layer ( u v w 5.5 * k 1 2 u Garratt J. R., 1992), we use the odified constant set proposed by Duynkerke P. G., c k e c e1 c e Table 1. Duynkerke constants for the k-ε odel 11

12 2 u' w' cste u* w' ' cste u* * z u u* z z h * z P B e with u P u' w' z g B ' ' w P z u' w' K w' ' K u z z (2) (3) ( ) g h k² K C e K h with and K Pr t (4) (7) (8) Integration of (3) gives classical logarithic velocity and teperature profiles : 1. Reference odel of the surface boundary layer Set of equations for the vertical profiles u u z * ln z z z * z ln z h t Where ψ et ψ h are the integrated universal functions of the Monin-Obukhov theory Integrations of (7) using the precedent relations (11) provides : P z g* ln z h t dz z With equation (2), (3), (4), one can derive the profile of ε: u 3 * e z 1 z Cobining equations (2), (8), (13) gives the profile of k : k z 2 z u * 1 C µ u z Equations (8), (13), (14) provide the profile K : K 12 z * (11) (12) (13) (14) (15)

13 K k P B e z k z D (9) K ² e ep e Ce1 Ce 2 z e z k k (1) P B e (4) 1. Reference odel of the surface boundary layer Conclusion This set of solution has been used by several authors to define the upwind boundary conditions for a RANS-CFD calculation of a diabatic surface layer (Huser A. and al., 1997; Pontiggia M. and al., 29) The ain proble is that the conservation equation for k (9) and the conservation equation for ε (1) have not been used to derive this set of solution In neutral condition k is constant, so D is equal to zero and the equation (9) becoes (4). In order to satisfy (1), the next relation ust be verified : e Ce 2 Ce1 Cµ In stable/unstable conditions we have seen that k depends of z, so D is not null : ² So for a diabatic surface layer these conservations equations have no reason to be satisfied by the two turbulent profiles described before 13

14 P B e u P u' w' z g B ' ' w K k P B e z k z D with (4) (9) 2. Consistency with the k and ε conservation equations Equation of k The consistency between equations (4) and (9) iplies that the diffusion ter D should be equal to. If σ k is a constant, one can show that D cannot be null, except for the neutral case. Freedan F.R. and Jacobson M.Z. (23) suggest that the value of D does not exceed 1-3.(P+B) We propose to evaluate the ratio between D and the TKE k, which can be interpreted as the inverse of a characteristic tie t k for k to vary significantly fro the pseudo equilibriu value. Near the ground : t k k 2LMO k z 1 D u * for (15) L MO 14

15 k z 2 u * 1 C µ (14) 2. Consistency with the k and ε conservation equations Interpretation For exaple, with L MO =5 and u * =.25.s -1, the characteristic tie t k for k to vary significantly fro (14) is about 1 s More generally, one can predict that for studying an atospheric SBL in a short doain (<1 k), an inflow boundary condition based on equation (14) for k will reain alost constant when using k-ε turbulence odel with a constant σ k For larger doains, we suggest to introduce a non-constant paraeterization of σ k, in order to ensure the local equilibriu 15

16 K z e e C z 3 ep k e ² k e1 Ce 2 u * e z 1 z (1) (13) In the assuption of an hoogeneous and steady SL, it is required that the profile of ε will be solution of the conservation equation. But introducing (13) in (1) gives : 2. Consistency with the k and ε conservation equations Equation of ε ' R 1 T i z e Ri L with Ri, ² MO R 1 R R dri d 1 In the neutral case, this equation is satisfied by adjusting the value ² i of the constant σ ε, but in the diabatic case, it is no ore possible to i C e 2 ² C µ R i e, N i ' and e, N Ce 2 Ce1 Cµ ² (17) satisfy equation (17) with a constant value of σ ε In the sae way we estiate the ratio e/t. It can be derived near the ground : t e e LMO k 1 T C C L e 2 µ u * for (18) MO 16

17 2. Consistency with the k and ε conservation equations Interpretation For exaple, with L MO =5 and u * =.25.s -1, the characteristic tie t ε for e to vary significantly fro the equilibriu is about 24 s More generally, one can predict that for studying an atospheric SBL even on a relatively short distance (>1 ), solution (13) for turbulent dissipation rate will not aintain with distance when using a k-ε turbulence odel with a constant σ ε Therefore we suggest introducing a non constant paraeterization of σ ε : e 1 for 1 C C µ e 2 1 z ln( z z 2 ) LMO e, N L MO 17

18 u P z z z u * ln z z * z ln h zt z (11) g* ln z h t dz z u * e z 1 z k z 3 2 u * 1 C µ (12) (13) (14) 3. Paraeterization in a RANS-CFD siulation Settings used in the CFD code Fluent to siulate a diabatic surface layer Inlet Dirichlet condition Equations (11), (13), (14) Top boundary conditions : Shallow nuerical layer (2 ) for preserving the oentu and heat fluxes through the thickness of the doain Voluic source ters for oentu and energy equations Equations solved : Standard RANS equations Incopressible and Boussinesq assuptions Energy conservation was treated considering the potential teperature instead of the siple teperature k-ε turbulence closure Outflow condition : Satisfy the vertical equilibriu of the oentu equation with buoyancy effects Equation (12) Non-constant paraeterization for σ k and σ ε Ground boundary conditions : Wall function based on the rough logarithic law for the velocity (see Blocken B. and al., 27) Sensible heat flux H (positive or negative) 18

19 3. Paraeterization in a RANS-CFD siulation Results The paraeterization based on the different conditions was ipleented and tested with coercial CFD software Fluent 6.3. The siulation doain used is 2D doain of 2 k long Siulation for different stability conditions (stable, neutral and unstable) were perfored in order to evaluate the conservation of the upwind boundary condition along a such doain. We illustrate the results for a stable condition : H = -15 W. -2, u * =.4.s -1 and L MO = 392 We can observe that the vertical inlet profiles reain perfectly preserved along the 2 k of the doain 19

20 3. Paraeterization in a RANS-CFD siulation Results A siulation without any specific treatent of the atospheric theral stratification effect was perfored. We copare the results with our paraeterization With theral stratification paraeterization Without theral stratification paraeterization Figure 1. Vertical profiles of pressure, velocity, Reynolds stress, k and ε for different position in the siulation doain. a) Black profiles correspond to our ethodology. b) Red profiles correspond to a RANS / k-ε siulation without theral stratification paraeterization. 2

21 Conclusions In this work, we have proposed an analysis of the application of a RANS-CFD approach with a k-ε closure to the siulation of a diabatic atospheric surface layer We have discussed the consistency of the upwind turbulence profiles with conservation equations for k and ε We have proposed an approach to odify the outlet pressure condition and to include a top flux condition so as to satisfy the ain physical patterns of the surface layer The results illustrate the ability of our approach to aintain the inlet profiles and the probles encountered if no paraeterization is used for the stratification effects 21

22 Liitations and future works Liitations : Approach liited to the surface boundary layer This approach needs a correction of the k-ε constants Ideal solution : paraeterization of the «constants» depending on the distance with obstacles Today : need to choose between Duynkerke and standard paraeterizations according to the iportance of building effects vs. stratification effects Future works : Instable case Make the analysis for ore coplex turbulence odels (Reynolds stress odel) 22

23 Applications See Poster 6 or H Develop a new odelling approach, based on the use of precise and detailed CFD calculations, which are stored in a database and then coupled with a real tie lagrangian particle dispersion odel Precise CFD calculations are ade thanks to the presented ethodology and take into account the diabatic surface layer to create the database before the operational use During the operational use of our odel, a wind field is interpolated fro the data base and coupled with a lagrangian dispersion odel, so as to provide short coputational tie and study dispersion on a coplex industrial areas Lagrangian dispersion on the refinery of Feyzin 23

24 Thanks for your attention Lagrangian dispersion on the refinery of Feyzin 24