Turbulence and its Modelling

Transcription

1 School of Mechancal Aerospace and Cvl Engneerng 3rd Year Flud Mechancs Introducton In earler lectures we have consdered how flow nstabltes develop, and noted that above some crtcal Reynolds number flows become turbulent. Turbulence and ts Modellng T. J. Craft George Begg Buldng, C41 Contents: Readng: Naver-Stokes equatons F.M. Whte, Flud Mechancs Invscd flows J. Matheu, J. Scott, An Introducton to Turbulent Flow Boundary layers P.A. Lbby, Introducton to Turbulence Transton, Reynolds averagng P. Bernard, J. Wallace, Turbulent Flow: Analyss Measurement & Predcton Mxng-length models of turbulence S.B. Pope, Turbulent Flows Turbulent knetc energy equaton D. Wlcox, Turbulence Modellng for CFD One- and Two-equaton models Notes: Flow management - People - T. Craft - Onlne Teachng Materal We have seen how the Reynolds averagng approach can be used to descrbe turbulent flows. ~ U ~ U(t) = + U u(t) U t 0 0 We saw the Reynolds Averaged Naver-Stokes (RANS) equatons, and the appearance n these of the Reynolds stresses, u u j. Turbulence and ts Modellng 2010/11 2 / 27 t u 0 t Here, we begn to consder how one mght use the Naver-Stokes or the RANS equatons to compute engneerng-related turbulent flows. Ths wll lead to the concluson that we need to ntroduce mathematcal models (approxmatons) for the Reynolds stresses. Before examnng how such models mght be devsed, we wll frst consder some of the physcal features of turbulent flows and the behavour of the Reynolds stresses n certan, farly smple, flows. We can then examne how some of these features are (or ar not) accounted for n some of the more wdely-used, but stll relatvely smple, turbulence models. Turbulence and ts Modellng 2010/11 3 / 27 The Energy Cascade Turbulence s characterzed by a wde range of eddy szes. Flow nstabltes develop across a range of wavenumbers, leadng to large eddes breakng down nto smaller ones, these breakng nto stll smaller ones, etc... The turbulent knetc energy, k = 0.5(u 2 + v 2 + w 2 ), s manly transferred from the mean flow va the larger eddes n the flow. As these eddes break up, knetc energy gets transferred nto smaller eddes and, as these break up, nto even smaller ones. Thus, one gets a cascade of turbulent knetc energy transferred down the scale spectrum, from the largest eddes to the smallest ones. At the smallest scales (whch wll also be examned later) the knetc energy s dsspated nto heat. Turbulence and ts Modellng 2010/11 4 / 27

2 The turbulence energy spectrum shows how much knetc energy s contaned n the dfferent scales of moton. 0.5u u = E(k,t)dk (1) 0 k n the fgure represents the wavenumber of the dfferent scales of moton. Large k corresponds to small eddes, and small k to large eddes. The lowest wavenumbers (largest eddes) are typcally related to the flow geometry. As the Reynolds number ncreases, the sze of the smallest scales present (largest wavenumbers) decreases. Thus, as the Reynolds number ncreases there s a wder range of turbulent eddy szes n the flow. Turbulent Eddy Szes The largest eddes n the flow generally scale wth the flow geometry, eg.: Wth ppe dameter or channel wdth. Wth boundary layer thckness. At the smallest scales the turbulence energy s dsspated to heat by the acton of vscosty. Dmensonal analyss suggests that the sze of these smallest eddes wll be related to (ν 3 /ε) 1/4 where ν s the knematc vscosty of the flud and ε s the dsspaton rate of the turbulent knetc energy. Note that as ν decreases (as the Reynolds number ncreases), the sze of the smallest eddes decreases. Thus, as the Reynolds number ncreases one expects to get a wder range of turbulent eddy szes. The large eddes can be very ansotropc. At the smaller scales the turbulence s generally reckoned to become more sotropc. Turbulence and ts Modellng 2010/11 5 / 27 Turbulence and ts Modellng 2010/11 6 / 27 Computaton of Turbulent Flow The Naver-Stokes equatons do stll descrbe a turbulent flow ncludng all the turbulent eddy detals. In prncple, one could solve these drectly (numercally) and obtan full detals of the turbulent flow feld (averagng the results to obtan mean veloctes, etc., f desred). If we consder a flow wth velocty and length scales U and L, we would need a computatonal doman of sze L. As noted above, the smallest eddy sze s of order (ν 3 /ε) 1/4. If we assume ε can be scaled wth U 3 /L (on dmensonal grounds), then the smallest eddy szes are of order (ν 3 L/U 3 ) 1/4. L U The number of computatonal nodes needed to cover a length L, whlst resolvng the smallest eddes s therefore of order L/(ν 3 L/U 3 ) 1/4 = (UL/ν) 3/4 = Re 3/4. Snce turbulent flows are 3-D, we need ths many ponts n each drecton, so the total number of computatonal nodes scales as Re 9/4. However, a consderaton of the range of scales nvolved shows ths s not actually possble, wth current technology, for most flows of engneerng nterest. The grd requrements thus ncrease rapdly as the Reynolds number of the flows goes up. Turbulence and ts Modellng 2010/11 7 / 27 Turbulence and ts Modellng 2010/11 8 / 27

3 Drect Numercal Smulaton (DNS) For rather low Reynolds numbers the Naver-Stokes equatons can be solved numercally, provdng full resoluton of a turbulent flow. The numercal soluton methods requre more accuracy than the standard fnte volume schemes studed n Modellng & Smulaton or CFD-1, and ths often restrcts ther use to smple geometres. Fully developed turbulent channel flows at Reynolds numbers (based on bulk velocty and channel half-wdth) of around 2500 were performed n the 1980 s. More recent channel flow smulatons at Re have been reported. These were performed on a 2048 processor machne, requred processor hours (the entre machne for around 4 months), and produced around 25TB of raw data. In engneerng applcatons, much hgher Reynolds numbers than ths are often encountered. Large gas ppelnes, for example, can be operatng at Reynolds numbers of several mllon. Turbulence and ts Modellng 2010/11 9 / 27 DNS can gve very useful, detaled, data for fundamental turbulence studes, and can be used to gude turbulence model development. Cross-stream flow & streamwse fluctuatons Vortcty sosurfaces However, t clearly s not a tool for everyday engneerng use. Turbulence and ts Modellng 2010/11 10 / 27 RANS Modellng There are modellng approaches such as Large Eddy Smulaton (LES), where only the large-scale turbulence structures are resolved, and approxmatons ntroduced for the effects of the smaller eddes. These are computatonally cheaper than DNS, but often stll too expensve for routne use n an engneerng envronment. Most turbulent flow smulatons thus employ the RANS equatons, solvng for the mean, or average, flow feld. For ncompressble flow wthout body forces, the governng equatons are then: DU = 1 P + ( ν U ) u Dt ρ x x j x u j (2) j U = 0 (3) x (note the use of tensor summaton conventon). In 3-D we have 3 momentum equatons and contnuty. However, the unknowns are 3 velocty components, pressure, and the 6 Reynolds stresses. To close the system we thus need to ntroduce models (approxmatons) for the Reynolds stresses, uuj. There are many such turbulence models avalable, havng dfferent levels of complexty, and provdng dfferng levels of accuracy n varous flows. To understand whch model to choose for a partcular applcaton t s necessary to have some apprecaton of how they were devsed, and hence some knowledge of ther strengths and/or weaknesses. In order to assess a model s accuracy, we need to know what behavour would be found n a real flud flow. Thus, before consderng how models may be devsed, we examne some features of a few smple turbulent flow stuatons, and the behavour exhbted by the turbulence statstcs. Turbulence and ts Modellng 2010/11 11 / 27 Turbulence and ts Modellng 2010/11 12 / 27

4 Shear Stresses Across a Boundary Layer Wth the usual boundary layer approxmatons the mean U-momentum equaton may be wrtten U 2 x + UV = 1 P ρ x + ( ν U ) uv (4) In a turbulent boundary layer, the turbulent shear stress uv s thus the most mportant component of the Reynolds stress tensor. If we assume that convecton s neglgble, then the U-momentum equaton becomes P x = τ (5) where τ s the total shear stress, gven by the sum of the molecular and turbulent shear stresses: τ = µ U ρuv (6) }{{}}{{} Turbulent Molecular Turbulence and ts Modellng 2010/11 13 / 27 y U o x In a zero pressure gradent boundary layer the total shear stress s thus constant across the layer, τ/ = 0. At the wall, uv = 0 and µ U/ = τw where τw s the wall shear stress. We thus obtan µ U ρuv = Constant = τw (7) Immedately adjacent to the wall the turbulent shear stress s neglgble (snce uv = 0 at the wall), and the molecular stress s the domnant. Outsde the thn vscous sublayer, n the fully turbulent part of the boundary layer, molecular effects are neglgble and the turbulent shear stress s then essentally equal to the total stress. In the fully turbulent near-wall regon of a zero pressure gradent boundary layer the turbulent shear stress s thus equal n magntude to the wall shear stress: uv = τw/ρ Vscous sublayer Turb. shear stress Mol. shear stress Turbulence and ts Modellng 2010/11 14 / 27 τ w/ρ y Wth a favourable pressure gradent we get τ/ < 0 so the total shear stress decreases wth dstance from the wall. Conversely, n an adverse pressure gradent we get τ/ > 0 and the total shear stress ncreases wth dstance from the wall (although as separaton s approached the above approxmatons may not be vald). In practce, the above approxmatons do not hold n the outer boundary layer regons, but are not too far from realty n the near-wall regons. Near-Wall Non-Dmensonalzaton It s often convenent to non-dmensonalze near-wall flow quanttes usng wall parameters, rather than the global velocty/length scales. (τw/ρ) 1/2 s a velocty scale, and ν/(τw/ρ) 1/2 a length scale. The velocty and dstance from the wall are then non-dmensonalzed as U + = U/(τw/ρ) 1/2 and y + = y(τw/ρ) 1/2 /ν In the fully turbulent near-wall regon of a zero pressure gradent boundary layer the velocty profle s found to ft a log-law: U + = 1 κ log(ey+ ) (8) Zero pressure gradent Favourable pressure gradent wth constants κ 0.41 and E 9. Turbulence and ts Modellng 2010/11 15 / 27 Turbulence and ts Modellng 2010/11 16 / 27

5 Plottng the shear stresses aganst y + we can see the vscous sublayer typcally extends to around y (although the exact value s rather flow-dependent). Zero pressure gradent Favourable pressure gradent Turbulent Knetc Energy Transport Equaton In an earler lecture we examned the mean knetc energy transport equaton n a turbulent flow. The turbulent knetc energy s defned by k 0.5u 2 0.5(u 2 + v 2 + w 2 ). To derve ts transport equaton, we can use Dk Dt = D ( ) u 2 Du Dt /2 = u Dt A transport equaton for the fluctuatng velocty, u, can be obtaned by subtractng the Reynolds-averaged momentum equaton from the Naver Stokes equaton: Du Dt = DŨ DU Dt Dt Note that here the mean veloctes are used n the convectve contrbutons to the total dervatve D/Dt. e. we nterpret D Dt t + U j x j (9) Turbulence and ts Modellng 2010/11 17 / 27 Turbulence and ts Modellng 2010/11 18 / 27 The dervaton s left as an exercse, but the result s: u u + U t j = 1 p U u xj ρ j + ν 2 u x xj xj 2 ( ) uu j u u j xj (10) Multplyng ths equaton by the fluctuatng velocty u (note the mpled summaton over the ndex ), and averagng, we can arrve at: k t +U k U j = u x u j ν u u ( u j x j x j x j x j 2 u /2+u p/ρ ν k ) (11) x The second term on the rght hand sde of equaton (11) represents the dsspaton of turbulent knetc energy by vscosty at the smallest scales. It s usually denoted as ε. The last term represents dffuson, as a result of vscous and turbulent mxng. The term u u j U / x j must therefore represent the generaton of turbulent knetc energy. It s often denoted by P k and called the producton, or generaton, rate of k. Turbulence and ts Modellng 2010/11 19 / 27 P k appears n both mean and turbulent knetc energy equatons (wth opposte sgns), and s thus nterpreted as the rate at whch knetc energy s lost from the mean flow and transferred to the turbulent eddes. DK Dt = 1 (PU ρ x j)+ j x j ( ν K ) U x u u j j ( ) U 2 + u U u j ν x j x j }{{} P k Dk Dt = u U u j ν u u ( u x j x j x j x j 2 u /2+u p/ρ ν k ) x } {{ } } {{ } P k ε In most crcumstances P k s postve, representng a transfer of knetc energy from the mean flow to the turbulence. However, there are flow condtons under whch P k can be locally negatve n certan regons. Turbulence and ts Modellng 2010/11 20 / 27

6 Boundary Layer Form of the k Equaton Wth the usual approxmatons for a steady boundary layer (or other thn shear flow) the k equaton reduces to (Uk) + (Vk) = Pk ε + ( u2uu/2 pu2/ρ ν k ) (12) x The generaton rate P k now reduces to P k = uv U Note that, as n the boundary layer U-momentum equaton, t s the shear stress component, uv, that s the most mportant element of the Reynolds stress tensor. We can now explan why the peak levels of turbulent knetc energy generally occur near the wall. y U o x As seen earler, n the fully turbulent regon of the boundary layer we can approxmate the turbulent shear stress as a constant (uv τw/ρ), and the mean velocty fts a logarthmc profle, so dfferentatng gves U/ 1/y. In the fully turbulent regon Pk thus decreases wth dstance from the wall as P k 1/y. At the wall tself, P k = 0, snce the turbulent shear stress vanshes. Hence, Pk must have a maxmum somewhere between the wall and the fully turbulent regon e. wthn the vscous sublayer. In an earler lecture we saw ths peak occurs roughly where the molecular and vscous shear stresses are equal. Turbulence and ts Modellng 2010/11 21 / 27 Turbulence and ts Modellng 2010/11 22 / 27 Plane Channel Flow at Re b 5500 Normal Reynolds Stresses Shear stresses Mean knetc energy budget Turbulent knetc energy Turbulent knetc energy budget In a boundary layer, or other shear flow, the 3 normal stresses (u 2, v 2 and w 2 ) are not equal. In general, the streamwse stress s largest, and the wall-normal the smallest. Transport equatons can be derved for each Reynolds stress component, u u j, smlar to that obtaned for k. The result s of the form Du u j = P Dt j ε j + φ j + d j (13) where the generaton rate P j = u u k U j/ x k u ju k U / x k and ε j, d j and φ j represent dsspaton, dffuson and a redstrbuton term. Turbulence and ts Modellng 2010/11 23 / 27 Turbulence and ts Modellng 2010/11 24 / 27

7 In a smple shear flow the generaton terms are gven by P11 = 2uv U P22 = P33 = 0 As expected, snce k = 0.5(u 2 + v 2 + w 2 ), we see that P11 + P22 + P33 = 2P k. P12 = v 2 U Note, however, that all the turbulent knetc energy generaton actually happens n the u 2 component. Other processes then redstrbute ths knetc energy nto v 2 and w 2. Ths explan why the streamwse normal stress s generally larger than the other two components. Turbulence and ts Modellng 2010/11 25 / 27 Wall-Lmtng Stress Behavour To examne the mmedate near-wall behavour of the stresses, we can express the near-wall veloctes as Taylor seres expansons n powers of y (the wall-normal dstance): u = a1y + b1y 2 + c1y v = a2y + b2y 2 + c2y w = a3y + b3y 2 + c3y where the a s, b s, c s etc. are functons of x, z and t. The above expansons ensure that the veloctes vansh at the wall, but they must also satsfy contnuty ( u/ x + v/+ w/ z = 0). Substtutng the expansons for the veloctes nto the contnuty equaton: y a1/ x + y 2 b1/ x a2 + 2y b2 + 3y 2 c y a3/ z + y 2 b3/ z +... = 0 Consderng the O(1) terms leads to a2 = 0. Turbulence and ts Modellng 2010/11 26 / 27 Hence close to the wall the veloctes behave as u y, w y, but v y 2. The Reynolds stresses thus behave as u 2 y 2, v 2 y 4, w 2 y 2 and uv y 3 for small y. Snce k = 0.5(u 2 + v 2 + w 2 ), then k y 2. Plottng the near-wall stresses on log-log axes shows they do closely match these wall-lmtng forms. Turbulence and ts Modellng 2010/11 27 / 27