School of Mechanical Aerospace and Civil Engineering Turbulen Flows As noed above, using he mehods described in earlier lecures, he Navier-Sokes equaions can be discreized and solved numerically on complex domains. Conens: Compuaional Modelling of Turbulen Flows T. J. Craf George Begg Building, C43 In principle, by refining he grid, and using high-order convecion schemes, numerical errors can generally be reduced o negligible levels. For laminar flows, i is usually feasible o obain grid-independen soluions in he above manner. However, many flows of engineering ineres are urbulen, and hese inroduce a number of furher complicaions. Inroducion o CFD Numerical soluion of equaions Reading: Finie difference mehods J. Ferziger, M. Peric, Compuaional Mehods for Fluid Finie volume mehods Dynamics Pressure-velociy coupling H.K. Verseeg, W. Malalasekara, An Inroducion o Compuaional Fluid Dynamics: The Finie Volume Mehod Solving ses of linear equaions nseady problems S.V. Paankar, Numerical Hea Transfer and Fluid Flow Turbulence and oher physical modellingnoes: hp://cfd.mace.mancheser.ac.uk/mcfd Body-fied coordinae sysems - People - T. Craf - Online Teaching Maerial Turbulen flows are sill described by he Navier-Sokes equaions. However, we need o consider some of he differences in behaviour one finds in laminar and urbulen flows. - p. 3 Overview Turbulen Eddies and Scales In earlier lecures we have examined schemes for discreizing, and subsequenly solving, generic fluid flow equaions. We have noed ha he same mehods can be applied o oher ranspor equaions when considering emperaure or mass fracion ranspor, for example. Prof. Osborne Reynolds demonsraed he ransiion from laminar o urbulen pipe flow a Mancheser in 883. When examining finie volume schemes we saw how he full Navier-Sokes equaions could be discreized, and how a pressure-velociy coupling scheme could be applied o obain he pressure field. In his lecure we consider how we migh use he mehods already me o model urbulen flows which are ofen of ineres in engineering problems. One quesion o be addressed is wheher we could simply apply he mehods already me, or wheher furher modelling or approximaion needs o be inroduced. As migh be expeced, we will examine a number of alernaive reamens, wih he aim of oulining when, and in wha circumsances, hey migh be used. He idenified he parameer D/ν as a dimensionless group ha could be used o characerize wheher he flow would be laminar or urbulen. Flow visualizaion of a urbulen boundary layer shows a range of eddy sizes, mixing he fluid. - p. 2 - p. 4
In a finie volume scheme (or any oher numerical mehod), he grid mus be fine enough o resolve all he deails of he flow. In a urbulen flow, where here is a large range of urbulen eddy sizes, he compuaional mesh would have o be significanly finer han he smalles of hese eddies. In a flow round an airfoil secion, for insance, he oal domain size migh be of he order of several meres. However, a a high Reynolds number, he smalles eddies may be less han a millimere in size. Since he urbulen eddies are hree-dimensional, a direc soluion of he Navier-Sokes equaions for he above flow would require a full 3-dimensional, ime-dependen, soluion on a grid wih a leas several housand nodes in each of he hree coordinae direcions. These requiremens are well beyond he capabiliies of any curren compuers (and will, mos likely, say so for a long ime). Hence, a number of differen approaches for compuing urbulen flows have been developed. Large Eddy Simulaion (LES) This sraegy also employs a 3-D, ime-dependen, soluion of he Navier-Sokes equaions, bu wih a relaively coarse grid and ime sep, so he fine-scale eddies are no resolved. A model hus has o be inroduced o accoun for he effec of he small-scale eddies. In mos cases his consiss of effecively increasing he viscosiy by an amoun ermed he "sub-grid-scale viscosiy" (which may vary in space and ime). This sub-grid-scale model accouns for he effecs of he small, energy-dissipaing eddies, which are oo small for he compuaional grid o resolve. Mos of he sandard finie volume soluion echniques examined earlier can usually be adoped for LES, alhough a few adapaions are someimes necessary. LES schemes are now beginning o be used a research level for a variey of complex engineering flow. The compuaional ime is sill unaccepably long, however, for mos rouine indusrial applicaions. - p. 5 - p. 7 Direc Numerical Simulaion (DNS) For very low Reynolds number (and simple geomerical) flows, i is possible o simply solve he Navier-Sokes equaions numerically, fully resolving all he eddy sizes and imescales. DNS can provide highly deailed flow daa, bu requires supercompuing class machines. Reynolds Averaging In mos engineering siuaions i is he average velociy ad pressure ha are of ineres, and he fine deails of all he urbulen eddies are no required. As originally proposed by Reynolds, he velociy and pressure fields can be spli ino a mean (or average) and a flucuaing par: e i = i + u i ep = P + p () () = + u() u Since he range of urbulen eddy sizes grows rapidly as he Reynolds number increases, compuaional requiremens soon become prohibiive, as noed above. High order numerical schemes are also normally required. The underlying second-order accuracy of mos finie volume schemes is ofen no sufficien, and specral mehods are frequenly employed. These generally resric one o using simple geomeries. Whils useful for fundamenal flow sudies, DNS is no herefore a ool for rouine engineering calculaions. - p. 6 0 0 In many cases he flow field may be seady on average, in which case he decomposiion of variables can be easily done by defining 0 Z T i = lim e i d (2) T T 0 - p. 8
In flows where here is a large-scale unseadiness a slighly differen definiion is needed. For example, in periodic flows a phase-averaging may be inroduced: NX i () = lim e i ( + NT) (3) N N In ohers, an ensemble-averaging may be used: NX i () = lim e i () (4) N N In such cases he mean quaniies are sill funcions of ime, bu hey only represen he larges unseady moions, and none of he fine-scale urbulence is included in hem. Since only he mean flow is resolved in he calculaions, symmeries (and/or 2-dimensionaliy or seadiness) can be aken advanage of o reduce compuaional requiremens. This approach is herefore he mos widely used in engineering applicaions. T - p. 9 Eddy Viscosiy Models The Reynolds averaged Navier-Sokes equaions can be wrien as i + i j = P + ρ x i ν «i u i u j From hese, i appears ha he u i u j erms simply ac as addiional sresses on op of he viscous sresses ν i /. A simple way o model he Reynolds sresses is hus by inroducing a urbulen or eddy viscosiy, ν, and modelling u i u j in erms of ν and he mean velociy gradiens: i u i u j = ν + «j + 2/3kδ ij (8) x i where δ ij is a special ensor, known as he Kronecker dela, which is defined as δ ij = ( if i = j 0 if i j The quaniy k is he urbulen kineic energy, defined as k = /2(u 2 + u2 2 + u2 3 ) (0) (7) (9) - p. The Reynolds Averaged Navier-Sokes Equaions Having decomposed velociies, pressure, ec. ino mean and flucuaing pars, he nex sep is o subsiue hese ino he Navier-Sokes equaions and average he resuling expressions in order o arrive a a se of equaions governing he behaviour of he mean flow field. Deails of his derivaion are no given here, bu he resul is a se of equaions raher similar o he original Navier-Sokes sysem: Coninuiy: Momenum: i = 0 x i (5) i + ( i j ) = P + ν «i u i u j ρ x i (6) However, he Reynolds sresses, u i u j, are no known (and canno be calculaed, since we are no now resolving he urbulen eddies). Models mus herefore be inroduced o approximae he Reynolds sresses. The subjec of urbulence modelling as a whole is beyond his course. Here we can only noe a few aspecs of some of he more common approaches found in many CFD codes. Wih his model for u i u j, he momenum equaion can simply be wrien as i + i j = P + ρ x i wih he effecive viscosiy ν eff = ν + ν. i ν eff + ««j x i Since he above momenum equaion is essenially he same form as ha already considered, he same numerical procedures already examined can be employed o obain a se of discreized equaions for he mean velociy a each node. The main difference is ha in he diffusive fluxes he viscosiy ha has o be compued a he cell faces is now he effecive viscosiy, ν eff. If he above eddy-viscosiy approach is adoped, he remaining modelling problem is how o prescribe he eddy-viscosiy, ν. Noe ha whils ν is a fluid propery, ν is no i is simply a funcion of he local flow behaviour. () A is simples level, he modelling of he urbulen sresses is done by relaing u i u j o he mean velociy gradiens and some prescribed lenghscale, derived from he geomery of he flow. More complex models solve addiional ranspor equaions for quaniies such as he urbulence energy. - p. 0 - p. 2
Zero-Equaion Models In he simple mixing-lengh, or one-equaion model, a mixing-lengh l m is prescribed algebraically, and ν is obained from or, in a more general formulaion: ν = l 2 m ν = lm 2 y (2) s i + j x i l m «2 (3) Two-Equaion Models One problem wih he above models is ha he urbulence lenghscale has o be prescribed, which is ofen difficul o do in a reliable manner for complex flows and geomeries. In a wo-equaion model, a second variable (ofen ε) is solved for: Dk D = P k ε +» (ν + ν /σ k ) k Dε D = c ε ε and he urbulen viscosiy is modelled as k P k c ε2 ε 2 k +» (ν + ν /σ ε ) ε (6) (7) ν = c µk 2 /ε (8) The lenghscale l m is aken o vary linearly wih disance from he wall, bu wih a damping erm o accoun for viscous effecs in he near-wall viscous sublayer. y The model consans c ε, c ε2, c µ, and σ ε are uned for simple equilibrium shear flows. A fairly sandard k-ε model akes c ε =.44, c ε2 =.92, c µ = 0.09, and σ ε =.3. In he near-wall region, where viscous effecs are imporan, a damping funcion is inroduced ino c µ, and addiional source erms are someimes included in he ε equaion. Two-equaion models hus avoid he need o prescribe a lenghscale, and are he mos widely used class of model in ypical indusrial CFD simulaions. - p. 3 - p. 5 One-Equaion Models In a one-equaion model, a ranspor equaion is solved for he urbulen kineic energy k: Dk D = P k ε +» (ν + ν /σ k ) k where P k = u i u j i / and he dissipaion rae ε is modelled as ε = k 3/2 /l ε. The urbulen viscosiy is hen aken as (4) ν = c µk /2 l µ (5) More Advanced Models There are a number of more advanced modelling schemes. For example, full sress ranspor models solve modelled ranspor equaions for each individual sress componen: Du i u j D = P ij + φ ij ε ij + diff ij (9) For a 3-dimensional flow his requires 6 ranspor equaions o be solved for he u i u j ensor (and usually a furher one for he dissipaion rae, ε). Such schemes are herefore more compuaionally expensive han wo-equaion models, bu are poenially more reliable. Boh lenghscales l µ and l ε are ypically prescribed as increasing linearly wih disance from he wall alhough hey may have differen viscous damping erms associaed wih hem o accoun for very near-wall viscous effecs. l y Noe ha he addiional ranspor equaions for urbulence quaniies are of he same generic form as hose already considered when discreizing oher equaions. Broadly he same mehods already covered can hus be applied o discreize and solve hem. In he ypical segregaed finie volume solver srucure examined in earlier lecures, one hus simply inroduces he addiional sep(s) wihin he main ieraion loop of discreizing each urbulence variable ranspor equaion and solving he resuling se of equaions o updae each variable. - p. 4 - p. 6
Near-Wall Treamens Close o a wall viscous effecs become imporan, and damping funcions have o be inroduced ino some of he model coefficiens and funcions. There are also usually large gradiens of velociy, urbulence energy (and ε), and a very fine grid mus herefore be employed in order o resolve hese. In he fully urbulen region of a simple zero pressure gradien boundary layer, he velociy profile is known o saisfy he log-law: + = κ log(ey+ ) (2) where he non-dimensional velociy, +, and wall disance, y +, are defined as + = /(τ w/ρ) /2 and y + = y(τ w/ρ) /2 /ν (22) The high aspec raio cells can also affec he numerical soluion sabiliy. In large 3-D calculaions he need o use very fine near-wall grids can make compuaions exremely expensive. If his is assumed o hold a node P, we obain p (τ w/ρ) /2 = κ log(ey p(τ w /ρ) /2 /ν) (23) In an ieraive soluion procedure, equaion (23) can be solved o obain an esimae for he wall shear sress, τ w, which is hen used in updaing he velociy, as described above. Such an approach removes he need o fully resolve he near-wall viscous layer of he flow. Wall funcions are a mehod of reaing he near-wall region of he flow in an approximae manner, wihou fully resolving he hin viscosiy-affeced flow region. - p. 7 However, i does assume ha all near-wall flows saisfy he log-law, which is no rue! There are more advanced wall funcion approaches, which are more widely applicable, bu hese will no be covered here. - p. 9 Wall Funcions The idea of a wall funcion approach is o place he firs compuaional node ouside he viscous sublayer, and make suiable assumpions abou how he near-wall velociy profile behaves, in order o obain he wall shear sress. y n As seen earlier, he momenum equaion in he near-wall cell can be discreized ino an equaion of he form a p p = a e e + a w w + a n n + a s s + S u (20) where a s arises from he shear sress a he souh face of he cell (ie. he wall shear sress). x P y v However, since we are no resolving he near-wall sublayer, esimaing he velociy gradien from a linear variaion of, as we do a oher posiions, will no give an accurae approximaion of he gradien, and hence he shear sress, a he wall. If, insead, we could obain he wall shear sress from an assumed velociy profile, we could se a s o zero and hen add he wall shear sress conribuion, τ w( x), direcly ino he source erm. - p. 8