Handout: Large Eddy mulaton I 058:68 Turbulent flows G. Constantnescu Introducton to ubgrd-cale (G) Models G tresses should depend on: Local large-scale feld or Past hstory of local flud (va PDE) Not all models have these propertes Importance of model n a calculaton depends on energy n subgrd-scales Low Reynolds number: E G /E 10-50%; results relatvely nsenstve to model, however results can be very senstve to the numercs f artfcal dsspaton s present (e.g., convectve terms are dscretzed usng upwnd schemes) Hgh Reynolds number: E G /E 1; model more mportant Requrements that a good G/F model must fulfll: - represent nteracton wth small scales - the most mportant feature of a G model s to provde adequate dsspaton (by ths we mean transport of energy from the resolved grd scales to the unresolved grd scales; the rate of dsspaton ε n ths context s the flux of energy through the nertal subrange) - the dsspaton rate must depend on the large scales of the flow rather than beng mposed arbtrarly by the model. The G model must depend on the large-scale statstcs and must be suffcently flexble to adust to changes n these statstcs. - especally n energy conservng codes (deal for LE) the only way for t.k.e. to leave the resolved modes s by the dsspaton provded by the G model. - prmary goal of an G model s to obtan correct statstcs of the energycontanng scales of moton 1
All the above observaton suggest the use of an eddy vscosty type G model Take dea from RAN modelng, ntroduce eddy vscosty ν T : u u τ ν = T + = ν T x x mplest model has constant eddy vscosty: Really DN of lower Re number flow pectrum not controlled Not successful n practce Better model due to magornsky magornsky model Dmensonally eddy vscosty s l t 1 Obvous choce: ν T = Cql - Turbulence length scale easy to defne (unlke n RAN): largest sze of unresolved scales s approxmately l = - Velocty scale not obvous (smallest resolved scales, ther sze s of the order of the varaton of velocty over one grd element) u - use q = l = l y - Better choce for 3D flows: = ( ) 1/ Observaton: In RAN pretty much the nverse story, velocty scale s obvous (k 1/ ) whle the turbulent length scale s tough to defne and estmate - Combne prevous expressons to obtan ν T = C = C
The above model s due to magornsky (1963) - Desgned for global weather modelng - Used for the frst tme n 196-63 at NCAR. - Orgnal grd contaned 0*0*0 ponts. - Nowadays smulatons usng couple of mllon ponts are current! Central problem: Need to specfy value for parameter C More physcal argument to derve magornsky model: Let s look at the energy transfer from large scales - Due to nonlnear term n N equatons To obtan energy equaton, take scalar product of N equatons wth u uu t + u x uu p = u x u + ν u x x econd term descrbes the energy transfer between scales (recall prevous dscusson n wavenumber space), dmensonally vel_scale 3 /length_scale Energy transfer = dsspaton = ε Q / L 3 - L=ntegral scale, Q=large scale velocty If largest unresolved eddes are nvscd (close to realty f Re very large): 3 ε q / Based on dmensonal analyss: ν T q By equatng the two expressons for ε: q / Q = ( / L) ν T 1/ 3 q Q( / L) Further assume: 1/3 Q L( ) = L Then: 1/ = Q 4/3 1/ 3 L 3
ν T = C 4/3 L /3 Ths expresson s somewhat dfferent than the classcal magornsky model (two length scales are present) but, supposng one can estmate L, ths form of the model s a more accurate expresson for the G eddy vscosty. Ths expresson s also obtaned analytcally usng more advanced theores of turbulence (EQDVM model). Integral scale L dffcult to compute, but we can assume: L/ =ct ; n realty L/ =f(re, etc.) o we recover usual form of model (the value of the constant changed) ν T C In realty C=C(L/ )=C(Re, etc.) May explan why varaton of C needed to obtan accurate predcton of turbulent flows (ths s gong to be addressed later va dynamcally calculatng the model coeffcent C) magornsky Model can be derved n several ways: Heurstcally (two versons gven above) Inertal range arguments (Llly) Turbulence Theores (RNG) Constant predcted by all methods (based on theory, decay sotropc turbulence) C = C 0. Theores of turbulence suggest a spectral eddy vscosty (Chollet and Leseur) Means a dfferent eddy vscosty actng on each wave number k Analogy wth ordnary vscosty Energy removed from wave number k s ε ( k) = νk E( k) Ths suggest a spectral eddy vscosty of the form: ν ( k) = T ( k) / k E( k) as ε ( ) ~ T ( k) T > k > 4
T > (k)=net energy transfer to small scales It can be shown based on DN calculatons (sotropc turbulence) and a prory analyss that: pectral eddy vscosty constant at small k Increases rapdly at large k (small scales whch n a numercal smulaton on a fnte grd correspond to the cutoff wave number k c ) - Reason: man nteracton s between smallest resolved scales and largest unresolved scales. Eddy vscosty s largest between k c / and k c. Fndngs can be used to construct models Parameter estmaton (based on Llly s theory) Assume hgh Reynolds Number Cutoff les n nertal subrange (no prod, no dss, E=E(ε,k), k s knetc energy) Energy spectrum correspondng to the velocty s: /3 E( k) = CKε k 5/3 C K =Kolmogorov constant 1.4 -. In knetc energy equaton for resolved scales: ε = = C R ν T Estmate the square of the stran rate: 3 = kc 3 /3 4/ 3 k E( k) dk CKε kc 0 Use k c = π /, ε R = ε and prevous two expressons to obtan C 1 = π 3C K 3/ 4 Wth C 1.6 C = 0. 165 K = Other methods gve almost same value 5
Classcal magornsky model Performance Predcts many flows reasonable well But there are problems: - Optmum parameter value vares wth flow type *Isotropc turbulence C 0. *hear flows (e.g. channel) C 0. 065 *Factor 10 dfference n eddy vscosty!! - Length scale uncertan wth ansotropc flter *Two possbltes are: 1/3 1 3) ( ; 1/ 1 + + 3 ) ( - Needs modfcaton to account for: *Rotaton, stratfcaton C =F(R,.) R=Rchardson number *Near-wall regon C =F(y + ); vscosty comes nto play resultng n the need for further reducton of the model coeffcent. Van Drest dampng s usually used but the results are not very good. Ways to mprove the model: - dynamcally calculate the model coeffcent. Ths s the dynamc magornsky model - ntroduce transport equatons for relevant quanttes. In partcular, solvng an equaton for the subgrd knetc energy allows a much better estmaton of the velocty scale for the G fluctuatons. These are the one-equaton G models. - Both types of models are gong to be dscussed n detals later. 6
A totally dfferent approach (not based on eddy vscosty G models): Use small scales of LE tself, the smallest resolved scales are not very dfferent from the largest unresolved scales. Use that nformaton for model constructon. Ths s the man dea behnd the scale smlarty model (Bardna et al.) cale smlarty model Frst model based on the small resolved scales: We already made the pont that the most mportant nteractons nvolve nteractons between the Largest subgrd scales and mallest resolved scales Need to defne these scales Defne velocty felds: Unresolved scales ( D ) : u = u u (by defnton) Largest subgrd scale part defned by flterng: u = u u mallest resolved scales ( D > ) ;defned by second flter on the resolved feld (of larger wdth, generally) u u Last two expressons are dentcal!!! o we assume that these scales have smlar structure (near grd cutoff). In other words, we smply assume that the G stresses for the full velocty feld are the same as the ones correspondng to the resolved feld u τ = u u u u ~ u u u u 7
These deas are clearer by lookng at the followng analyss of the relaton among the dfferent velocty felds Full Feld u Resolved Feld u Flter Flter Agan Unresolved Feld = u u u Largest Transfer mallest (border between resolved/unresolved) u u u = u u = u u Another relevant dscusson s related to what s called a pror testng of LE models usng a precalculated DN database (no actual LE smulaton s needed). Resolved Feld u A pror testng Full DN Feld u Flter Unresolved Feld = u u u Evaluaton Model G stresses Exact G stresses ν T (for magornsky) uu uu (from DN) u u u u (scale smlarty) 8
Fnally, compare results by correlaton or scatter plot. Ths way one can draw conclusons about the performance of a partcular LE model. Other way to derve the scale smlarty model Later, we are gong to derve the scale smlarty model as a partcular case of a more general class of models based on reconstructon of the total velocty feld from the resolved one (deflterng). The man dea n these models s: Use defnton of resolved velocty: u ( x) = G( x, x ) u ( x dx ) Apply Taylor seres to u (x) : du ( ) ( ) ( u x u x + x x) +... dx Keep only frst term nto defnton of (x ) to obtan same result: u τ = u u u u Performance of scale-smlarty model: Improves energy spectra (compared to magornsky) Can account for the transfer of energy from mall resolved scales large resolved scales (backscatter accounted n a physcal way) Correlates well wth exact stress (a pror analyss) Not dsspatve (does not dsspate energy automatcally as magornsky model wth constant coeffcent does, e.g., n lamnar regon of a flow the eddy vscosty and turbulence dsspaton predcted by magornsky model wll be dfferent from zero and postve, whch s obvously wrong) Inadequate as stand-alone G model (not very robust numercally as t does not ntroduce enough dsspaton n some cases, needs to be combned wth a purely dsspatve model, e.g., magornsky lke; ths s the man dea behnd mxed models to be dscussed later) Bass for other models 9