Turbulent Transport in Single-Phase Flow. Peter Bernard, University of Maryland

Transcription

1 Turbulent Transport n Sngle-Phase Flow Peter Bernard, Unversty of Maryland

2 Assume that our goal s to compute mean flow statstcs such as U and One can ether: 1 u where U Pursue DNS (.e. the "honest" approach) of averagng solutons of the NS eqn: U u or pursue RANS (.e. the "dshonest" approach) of solvng the averaged NS eqn: where the Reynolds stress tensor, s modeled.

3 DNS: Hghly accurate but of lmted practcal usefulness. RANS: Inaccurate, unrelable, requres emprcal modelng, but of wdespread use. LES, a thrd approach has conceptual problems - though these are usually gnored. In partcular, the average of the fltered velocty: does not necessarly equal the mean velocty,.e. U U u Moreover, f where u r u u s the resolved part of the velocty fluctuaton, then u r U U Conundrum: f the subgrd energy s large, then cannot be found. If the subgrd energy s small, then LES s a DNS.

4 Our nterest here s n the RANS approach. There are basc optons: Drect models for R j u u R j j??? or model the R j equaton: Drect models are most popular and we consder just ths case.

5 The Reynolds stress R j has a physcal nterpretaton as the flux of the th component of momenum n the jth drecton caused by the fluctuatng velocty feld. For non-dense gases the stress tensor n the Naver-Stokes equaton has a smlar nterpretaton as representng the flux of the th component of momentum n the jth drecton due to molecular moton. In the molecular case: C C c C U and the stress tensor s:

6 Can a smlar model for the Reynolds stress tensor be justfed? U U u

7 There are very strong reasons for wantng such a model to be true. In ths case the mean momentum equaton becomes: Ths approach s: easy to nstall wthn a NS solver relatvely well behaved relatvely nexpensve to solve

8 Consder the valdty of the molecular transport analogy n the context of a turbulent transport n a undrectonal mean flow such as n a channel or boundary layer: U(y) d U (y) 0 dy uv(y) 0 In ths case: ρ c1c du μ dy ρuv μ t d U dy

9 U(y) Molecules transport momentum, unchanged, over the mean free path, before colldng wth other molecules and exchangng momentum. U(y) s lnear over here: c c

10 To analyze the physcal mechansms behnd turbulent transport consder the set of flud elements that arrve at a gven pont a at tme t. t- b b t a (y) U local lnear approxmaton b b Unlke the molecular case: momentum s not preserved on paths untl mxng. the dea of "mxng" s undefned no obvous separaton of scales

11 Use backward partcle paths to evaluate an exact Lagrangan decomposton of the Reynolds shear stress that exposes the underlyng physcs. Thus goes to 0 as ncreases (establshes a mxng tme). transport caused by flud partcles carryng, unchanged, the mean momentum at pont b to pont a. transport assocated wth changes n velocty (acceleratons) along partcle paths.

12 v U b U The correlaton s created by flud partcle movements wthn a spatally varyng mean feld: when v > 0 the dfference n mean velocty along the path s negatve and vce versa. v U b U 0

13 Acceleraton transport orgnates largely n the effect of vortcal structures n acceleratng flud partcles as they move toward the wall (sweeps) or retardng flud partcles as they eject from near the wall. Close to the surface, vscous effects retard fast movng flud partcles leadng to a decrease n Reynolds stress.

14 Decomposton of acceleraton transport nto vscous and pressure effects.

15 Evaluaton of the Lagrangan decomposton n channel flow yelds: transport due to partcle acceleratons transport due to partcle dsplacements

16 The Lagrangan analyss can yeld a quanttatve estmate of the potental errors n a gradent model of the Reynolds stress. (Mxng length - dstance traveled durng the mxng tme) gradent term effect of non-lnearty of the mean velocty over the mxng length

17 An exact decomposton of the turbulent shear stress: uv Τ v d U dy Φ 1 v(u U b ) Correct gradent Non-lnearty Acceleraton contrbuton of mean velocty effects where vl t v(t)v(s) ds Τ v t τ Lagrangan ntegral tme scale

18 Errors n the gradent model: uv Τ v d U dy Φ 1 v(u U b ) Clearly, sgnfcant errors are present. RANS models attempt to compensate for errors by a judcous choce of the eddy vscosty.

19 Dssatsfacton wth lnear transport models has fueled nterest n models that are non-lnear n: S j 1 U x j U x j (rate of stran tensor) W j 1 U x j U x j (rotaton tensor) A typcal example of a non-lnear model (e.g. Algebrac RS Models): Sometmes non-lnear models are derved by smplfcaton of RSE models.

20 Assumng some legtmacy for lnear RS models - what s t? For molecular transport: μ ρ 1 αλc UL suggestng that the eddy vscosty depends on the product of velocty (U) and length (L) scales. RANS models vary dependng on the choce for U and L. The - closure assumes U L 3/ ε (eddy turnover tme) Thus: t C μ ε

21 equaton modelng Producton From equaton x σ t x s a turbulent Prandtl number

22 equaton Producton Transport/Dffuson Dsspaton

23 Modelng of the equaton s done n stages by consderng ts propertes n smplfed settngs: 1. sotropc decay.. homogeneous shear flow. 3. constant stress layer near sold walls.

24 The exact equatons governng the decay of Isotropc Turbulence: vortex stretchng dsspaton After defnng: (skewness) (palenstrophy) Reynolds number These may be smplfed to:

25 In the case of self-smlarty, e.g.: f(r,t) u 0 u r ~ f r/λ t S and G are constant and the system of equatons s closed and solvable. Two equlbra exst: Low R T : vortex stretchng neglgble: d dt 7-5 t -5/ Hgh R T : vortex stretchng and dsspaton equlbrate d dt - t -1 In tradtonal modelng vortex stretchng s elmnated creatng an opportunty to match decay rate wth experments: dε dt -C ε ε t -1/ C ε 1

26 Homogeneous shear flow S d U dy s constant everywhere exact equatons d U Assume: P uv C μ S dy ε ς ω enstrophy

27 Modeled Equatons for Homogeneous Shear Flow Wthout vortex stretchng: blow up. C ε1 chosen to match experments Wth vortex stretchng: prod = dss equlbrum

28 equaton modelng homogeneous shear flow model - calbrated to gve correct growth - C ε ε Isotropc turbulence model - calbrated to gve a decay rate consstent wth data

29 Closure (hgh Re form) C ε C C μ ε1 ε 1.9

30 Calbraton of the Closure In the "constant stress layer" Assume: uv and the model: Then: Moreover, f then: Substtutng these results nto the equaton gves:

31 Near-wall modelng Boundary condtons: (0) 0 ε(0) (0) y (0) y Among the problems wth the hgh Re modelng near a boundary: t C μ ε Τ v Introduce a wall functon to force the equvalence: t C μ f μ ε In effect, f μ v

32 Other problems that have to be fxed near a wall: 0 at wall so dsspaton blows up At the wall surface: yet no explct model for has been assumed n hgh Re model.

33 Low Re model for the equaton near walls. here (e.g.) ε ~ ε υ (0) y wall functons t C μ f μ ε

34 What to expect from the popular RANS models: 1. The predctons of RANS models n ther standard form, can be both acceptable or unacceptable dependng on the desred accuracy, navety of the user and other factors.. It s very common to make ad hoc changes to the values of constants and even to add addtonal modelng expressons n order to mprove accuracy, or to force the soluton to acqure desred physcal attrbutes. The dea s that some aspect of physcs s lackng n the orgnal model that needs to be compensated for. 3. Changng the propertes of models can brng the solutons closer to one set of data and further from another set of data. 4. Sometmes model alteratons - wth no bass n physcs - are made as a last resort to force better results: e.g. "clppng" 5. RANS solutons sometmes are regarded as successful f only one part of the soluton s captured - the part that s of nterest.

35 6. Addng addtonal physcs to RANS calculatons can be especally dffcult - two layers of naccuracy: the underlyng turbulence and the new physcal model. Dfferent models of the physcs (e.g. partcle dsperson, chemstry, combuston) can react dfferently to the same underlyng RANS modelng. 7. A numercal calculaton wth a RANS scheme may converge for one set of nput parameters and not converge for a smlar case of the same flow. 8. The qualty of one partcular RANS model may appear to be better than t s because f performs better than other models. 9. Very often computatonal speed s consdered more mportant than accuracy. 10. In some flows, complants about steady RANS solutons have led to the use of URANS (Unsteady RANS) n whch features such as vortex sheddng are consdered to be part of the mean (albet transent) feld.

36 11. Many research studes have compared LES predctons to RANS predctons. Sometmes RANS s as good as LES, sometmes LES s better, sometmes the added accuracy of LES s not justfed by the cost. 1. RANS s ncreasngly beng used to model the wall regon of LES snce the local DNS resoluton that one would hope for s often not feasble.