A suite of boundary conditions for the simulation of atmospheric ows using a modied RANS/k ε closure

Transcription

1 6th OpenFOAM Workshop Penn State, State College, USA A suite of boundary conditions for the simulation of atmospheric ows using a modied RANS/k ε closure J. Sumner, C. Masson École de technologie supérieure June 14th / 38

2 Introduction Wind farm power performance 2 / 38

3 Introduction Wind farm power performance 3 / 38

4 Introduction Oshore wind farms 4 / 38

5 Objective Derive and implement a general k ε surface layer model valid for both stable and neutral thermal stratication 5 / 38

6 Similarity theory Potential temperature proles Neutral Stable z [m] Θ [ C] Figure: Comparison neutral and stable surface layer 6 / 38

7 Similarity theory Velocity proles Neutral Stable z [m] 40 l m = κz/φ m l m = κz U [m/s] Figure: Comparison neutral and stable surface layer 7 / 38

8 Model development Imposing a new length scale The turbulent length scale is determined by the ε equation ( ) d νt dε ε 2 + Pε Cε2 = 0 dz σ ε dz k How to modify ε equation for desired mixing length and turbulence properties? 8 / 38

9 Model development The Apsley & Castro model Recast the ε equation as [ Pε = When l m << l max, When l m l max, Cε1 + (Cε2 Cε1) l m l max Π k ε Pε Cε1 k ε 2 Pε Cε2 0 k ] Πk ε k 9 / 38

10 Model development The Apsley & Castro model Recast the ε equation as [ Pε = When l m << l max, When l m l max, Cε1 + (Cε2 Cε1) l m l max Π k ε Pε Cε1 k ε 2 Pε Cε2 0 k ] Πk ε k 10 / 38

11 Model development The Apsley & Castro model Recast the ε equation as [ Pε = When l m << l max, When l m l max, Cε1 + (Cε2 Cε1) l m l max Π k ε Pε Cε1 k ε 2 Pε Cε2 0 k ] Πk ε k 11 / 38

12 Similarity theory Analytical proles Considering an arbitrary desired l m : Turbulence properties: U(z) = u z z 0 1 l m dz z 1 Θ(z) = σ θ θ dz z 0 l m k(z) = u2 Cµ ε(z) = u3 l ε 12 / 38

13 Model development A revised Apsley & Castro model Assuming equilibrium and writing correction as F d dz ( ) νt dε + F (Cε2 Cε1) ε2 = 0 σ ε dz k Substituting ν t = u l m, ε = u 3 /l ε and k = u 2 / Cµ and solving for F [ F = 1 dl m dl ε d 2 ( l ε κ 2 + l m dz dz dz 2 2l m dl ε l ε dz ) 2 ] 13 / 38

14 Model development A revised Apsley & Castro model Assuming equilibrium and writing correction as F d dz ( ) νt dε + F (Cε2 Cε1) ε2 = 0 σ ε dz k Substituting ν t = u l m, ε = u 3 /l ε and k = u 2 / Cµ and solving for F [ F = 1 dl m dl ε d 2 ( l ε κ 2 + l m dz dz dz 2 2l m dl ε l ε dz ) 2 ] 14 / 38

15 Model development Specic form Taking the mixing length as κz l m = 1 + κz/l max and relating the length scales with a simplied TKE budget equation 1 = 1 1 l ε l m κl yields F = ( 1 + R f ( )( ) 2 lm l ) 3 m l max l max where l m = u 3 0 /P k and R f G k/π k 15 / 38

16 Model development A caveat Turbulent viscosity For analytical proles be solutions, ν t = u l ε u l m k 2 l m k 2 ν t = Cµ = Cµ ε l ε ε (1 R f ) Note: (1 Rf ) is exactly equal to the ratio of length scales for the simplied TKE budget equation 16 / 38

17 Model development Turbulence transport equations 1D k ε model d dz d dz ( ν t dk ( νt dε σ ε dz dz ) + Π k ε = 0 ) ε 2 + Pε Cε2 = 0 k where Π k = P k + G k ( = ν t P k u w du dz ) 2 du dz G k αgθ w = ν t dθ αg σ θ dz 17 / 38

18 OpenFOAM implementation kepsilonlengthlimited // Shear production Gk_ = nut_*2*magsqr(symm(fvc::grad(u_))); // Link to theta field volscalarfield& theta = const_cast<volscalarfield&> (db().lookupobject<volscalarfield>("theta")); // Buoyancy production Gb_ = (alpha_*nut_/sigmatheta_) * g_ & fvc::grad(theta); // Total production volscalarfield G("RASModel::G", Gk_ + Gb_); 18 / 38

19 OpenFOAM implementation kepsilonlengthlimited // Calculate velocity scale u0_ = Foam::pow(Cmu_,0.25) * Foam::sqrt(k_); // Calculate mixing length lm_ = Foam::pow(u0_,3.0) / max(gk_, epsilonsmall_); // Update epsilon, G, Gk, Gb, and lm at the wall epsilon_.boundaryfield().updatecoeffs(); // Flux Richardson number volscalarfield Rf = min(1.0, -Gb_ / max(gk_, epsilonsmall_)); volscalarfield Rfprime = min(1.0, -Gb_ / max(g, epsilonsmall_)); // Weighting function f_ = Foam::sqr(1.0+Rfprime) * (lm_/lmax_+1.0) * Foam::pow((1.0-lm_/lMax_),3.0); 19 / 38

20 OpenFOAM implementation kepsilonlengthlimited // Dissipation equation tmp<fvscalarmatrix> epseqn ( fvm::ddt(epsilon_) + fvm::div(phi_, epsilon_) - fvm::sp(fvc::div(phi_), epsilon_) - fvm::laplacian(depsiloneff(), epsilon_) == (C1_ + (C2_ - C1_)*f_)*G*epsilon_/k_ - fvm::sp(c2_*epsilon_/k_, epsilon_) );... // Re-calculate viscosity nut_ = Cmu_*sqr(k_)/epsilon_ * max(1.0 - Rf, 1e-10); nut_.correctboundaryconditions(); 20 / 38

21 Boundary conditions Generalized Richards & Hoxey Outlet Inlet 21 / 38

22 Boundary conditions Generalized Richards & Hoxey Outlet Inlet 22 / 38

23 Boundary conditions Generalized Richards & Hoxey τ = ρu 2,sl k z = 0 ε z = u 3,sl κz 2 ν t z = κu,sl (1 + l κz max ) 2 dθ dz = q w σ θ c p ν t Outlet Zero gradients Inlet Equilibrium proles τ = ρu 2,w k z = 0 Wall functions dθ dz = q w c p σ θ ν t 23 / 38

24 Boundary conditions Wall functions Local friction velocity Turbulence properties ε P = Π k,p = u3,w ε P = u,w = Cµ ku,w l ε z = u 3,w 2zP z 0 [ z 0 κu P ln(z P /z0) + κ(z P z0)/l max ( 1 1 ) dz l m κl ( 2zp z0 1 2κ(z P z0) ln Cµ k u 2 Π k,p,w z0 ) + 1 l max 1 κl ] 24 / 38

25 OpenFOAM implementation Boundary conditions Version 1.6 provided derivedfvpatchfields for turbulence models 1 epsilonwallfunction 2 nutroughwallfunction Version 1.7 provided fixedshearstress and inow for U and ε 1 atmboundarylayerinletvelocity 2 atmboundarylayerinletepsilon What is needed: 1 Modied ε and G k wall functions 2 A wall function for proper shear stress for ABL 3 A way to cleanly handle a z0 distribution 4 Ecient IO between libraries 25 / 38

26 Regarding consistency Handling distributed roughness Two options: 1 Treat it like a wall function parameter 2 Treat it as a separate eld Constructor... zzerofield_ = new volscalarfield ( IOobject ( "z0", db().time().timename(), patch().boundarymesh().mesh(), IOobject::MUST_READ, IOobject::AUTO_WRITE ), patch().boundarymesh().mesh() ); z0_ = (*zzerofield_).boundaryfield()[patch().index()]; / 38

27 Regarding consistency Model parameters Boundary conditions often require same information 1 Directives (#) and macro substitutions ($) 2 IOdictionary (RASProperties and surfacelayerproperties) Bundle boundary conditions together with explicit links to dictionaries Constructor... : epsilonwallfunctionfvpatchscalarfield(p, if), z0_(p.size(), 0.0), kappa_(0.40), beta_(0), L_(1e100), lmax_(1e100) { readsurfacelayerdict(); zzerofield_ = new volscalarfield / 38

28 Regarding consistency Model parameters ε gradient Shear stress ν t wall function ν t gradient Θ gradient surfacelayer- Properties RASProperties ε inlet U inlet ε wall function 28 / 38

29 OpenFOAM implementation buoyantsimplefoam1d 1 Drop pressure equation simplefoam... //p.storepreviter(); // Pressure-velocity SIMPLE corrector { #include "UEqn.H" //#include "peqn.h" #include "thetaeqn.h" } // Update boundary conditions U.correctBoundaryConditions(); theta.correctboundaryconditions(); turbulence->correct(); / 38

30 OpenFOAM implementation buoyantsimplefoam1d 2 Add potential temperature equation ( ) d νt dθ = 0 dz σ θ dz thetaeqn.h fvscalarmatrix thetaeqn ( - (1.0/sigmaTheta) * fvm::laplacian(turbulence->nut(), theta) ); thetaeqn.relax(); 30 / 38

31 1D simulations Stable conditions: QW = 20 W/m 2, u,sl = 0.28 m/s, z 0 = 0.3 m, l max = 8 m 10 1 z/lmax 10 0 Proposed model Apsley and Castro Analytical U/u Figure: Comparison of proposed model, Apsley & Castro and analytical solutions 31 / 38

32 1D simulations Stable conditions: QW = 20 W/m 2, u,sl = 0.28 m/s, z 0 = 0.3 m, l max = 8 m z/lmax Proposed - G k Proposed - ε Proposed - G b Analytical - G k Analytical - ε Analytical - G b Normalized TKE budget Figure: Comparison of proposed model, Apsley & Castro and analytical solutions 32 / 38

33 1D simulations Neutral length-limited conditions: QW = 0 W/m 2, u,sl = 0.65 m/s, z 0 = 0.3 m, l max = 36 m 10 1 z/lmax 10 0 Proposed model Apsley and Castro Analytical U/u Figure: Comparison of proposed model, Apsley & Castro and analytical solutions 33 / 38

34 The Cε3 coecient Calibration The proposed ε equation can be recast in the form proposed by Rodi for modeling stratied ows: This yields Cε3 = (C ε2 Cε1) Cε1 [ Pε = Cε1(1 + Cε3R f )Π kε 1 (1 + R f )2 ( lm l max + 1 k )( 1 l m l max ) 3 ] 1 R f 34 / 38

35 The Cε3 coecient Comparison 10 Cε3Cε1/(Cε2 Cε1) Betts & Haroutunian (1983) Rodi (1987) Kitada (1987) Burchard & Baumert (1995) Apsley & Castro (1997) Freedman & Jackson (2003) Alinot & Masson (2005) Sumner & Masson (2011) z/l Figure: Comparison of proposals for C ε3 35 / 38

36 Conclusions Summary A slightly modied k ε model for neutral and stable, length-limited surface layer ow A corresponding suite of boundary conditions A new Cε3 expression has been derived that ensures similarity theory for a stable atmosphere is a solution of the model equations 36 / 38

37 Conclusions Future work Extend to two and three-dimensional ow cases Prepare tutorial and libraries for release to extend project 37 / 38

38 Thank you Questions? 38 / 38