USING ISOTROPIC SYNTHETIC FLUCTUATIONS AS INLET BOUNDARY CONDITIONS FOR UNSTEADY SIMULATIONS. Abstract

Transcription

1 Advaces ad Applicatios i Fluid Mechaics Volume 1, Issue 1, 2007, Pages 1-35 This paper is available olie at Pushpa Publishig House USING ISOTROPIC SYNTHETIC FLUCTUATIONS AS INLET BOUNDARY CONDITIONS FOR UNSTEADY SIMULATIONS Divisio of Fluid Dyamics Departmet of Applied Mechaics Chalmers Uiversity of Techology SE Göteborg, Swede Abstract The paper proposes a method to prescribe sythesized turbulet ilet boudary coditios. Whe makig LES, DES or hybrid LES-RANS a precursor chael DNS is ofte used. The disadvatage of this method is that it is difficult to re-scale the DNS fluctuatios to higher Reyolds umbers. I the preset work sythesized isotropic turbulet fluctuatios are geerated at the ilet plae with a prescribed turbulet legth scale ad eergy spectrum. A large umber of idepedet realizatios are geerated. A correlatio i time betwee these realizatio is itroduced via a asymmetric, o-trucated time filter. I this way the turbulet time scale of the sythesized isotropic turbulet fluctuatios is prescribed. The method is first validated for DNS at Re τ = 500. It is the employed i hybrid LES-RANS of chael flow at Re τ = 2000 o a coarse mesh. The sesitivity to differet prescribed ilet legth scales, time scales ad amplitudes of the fluctuatios is ivestigated Mathematics Subject Classificatio: 76F65. Keywords ad phrases: hybrid LES-RANS, forcig fluctuatios, trigger resolved turbulece, embedded LES. Commuicated by Shahrdad G. Sajjadi Received August 22, 2006

2 2 The ilet boudary coditios have much i commo with forcig fluctuatios i the iterface regio i hybrid LES-RANS. I both cases the object is to trigger the equatios to resolve turbulece. The method of geeratig ilet boudary coditios is also relevat i embedded LES, where LES is used o a mesh embedded i a global steady or usteady RANS computatio. 1. Itroductio Ilet boudary coditios are importat whe makig Large Eddy Simulatio. I high Reyolds umber flow the grid is mostly too coarse to resolve ay large portio of the turbulet spectrum. This is especially so ear the ilet, where few cells are commoly located i order to reduce the umber of cells; the majority of the cells are allocated to resolve boudary layers, wakes ad recirculatig regios. The grids are eve coarser i hybrid LES-RANS ad DES ad the grid spacig may be larger tha the turbulet itegral legth scale. The object of the ilet boudary coditios is the ot to supply turbulece with the correct time ad legth scales, but to supply scales relevat to the grid. I other words, the ilet turbulece should have itegral legth ad time scales related to the grid size, x, y, z, ad the computatioal time step t, i.e., scales that the Navier-Stokes equatios o the give grid uderstad. The object of the ilet boudary coditios is to trigger the equatios to resolve turbulece. I hybrid LES-RANS, URANS (Usteady Reyolds-Averaged Navier- Stokes) is used ear the walls ad LES is used at a certai distace away from the wall [9, 13, 24-26, 28], see Fig. 1. The locatio at which the switch is made from URANS to LES is called the iterface. Istead of resolvig the ear-wall turbulet structures, they are modelled with the RANS turbulece model. A drawback of hybrid LES-RANS is that, at the iterface where the turbulece model equatios switch from URANS to LES, the LES gets poor iterface coditios from the URANS. The usteadiess that is covected from the URANS regio to the LES regio cotais o proper turbulet scales. To supply the LES regio with proper turbulet scales it was suggested that turbulet fluctuatios (forcig) should be added as mometum sources i the URANS regio, i the iterface regio or i the iterface [2, 7, 10, 11, 18, 22, 24] of the

3 USING ISOTROPIC SYNTHETIC FLUCTUATIONS 3 URANS regio ad the LES regio. This forcig at the iterface is similar to prescribig fluctuatig velocity fluctuatios (forcig) at the ilet. The object is i both cases the same: to efficietly as possible force the mometum equatios ito turbulet mode. Figure 1. The LES ad URANS regios. The iterface is located at = y y ml ad the umber of cells i the URANS regio i the wallormal directio is j ml. The curretly most popular method of geeratig turbulet ilet boudary coditios is to make a precursor DNS of chael flow or boudary layer flow. This is a accurate method provided that the Reyolds umber of the DNS is relevat. If the Reyolds umber of the DNS is too low, the it is ot clear how to re-scale the DNS fluctuatios. While it is easy to re-scale the amplitude of the fluctuatios, it is difficult to re-scale the turbulet legth ad time scales. Isotropic sythesized fluctuatios based o the method of Kraicha [17] have ofte bee used to geerate turbulet fluctuatios, see e.g. [1, 3, 15, 21]. I this method a eergy spectrum is prescribed that yields the amplitude of the fluctuatios as a fuctio of wave umber. No-isotropic fluctuatios were geerated i [2, 5, 6, 19, 20, 23] ad the fluctuatios were scaled so that the time-averaged sythesized fluctuatios match a prescribed Reyolds stress tesor. A disadvatage of this kid of scalig

4 4 is that the prescribed spectrum, ad hece the two-poit correlatio, are modified if the Reyolds stress tesor is o-homogeeous. To achieve correlatio i time, Fourier series were applied i time i the same way as i space i most of the works cited above. I [21] a method was also ivestigated where a three-dimesioal box with geerated fluctuatios was covected across the ilet plae; i this way fluctuatio correlatios i the streamwise directios were trasformed ito correlatios i time. I the work by Billso et al. correlatio i time is defied by a asymmetric ifiite time filter. The method offers a coveiet way to prescribe turbulet legth ad time scales idepedetly. This method is adopted i the preset work. A method based partly o sythesized fluctuatios was recetly preseted ad it is called the vortex method [14]. It is based o a superpositio of coheret eddies where each eddy is described by a shape fuctio that is localized i space. The eddies are geerated radomly i the iflow plae ad the covected through it. The method is able to reproduce first ad secod-order statistics as well as two-poit correlatios. Chael flow is a commo test case for testig hybrid LES-RANS. Sice this flow is etirely depedet o the ear-wall turbulece, it is a challegig test case. It is also a useful test case for evaluatig ilet boudary coditios. I the preset study we use hybrid LES-RANS, ad the computatioal grid is purposely very coarse. A mea velocity profile is prescribed, that is take from the law of the wall. Turbulet istataeous fluctuatios are superimposed o the mea profile. These fluctuatios are take from sythesized, isotropic turbulece prescribig the turbulet itegral legth scale. A large umber of idepedet sythesized velocity fields are created that are idepedet of each other, which meas that the time correlatio is zero. The turbulet time scale is prescribed usig a suitable liear combiatio of the ruig time average (from zero to time t) of the ilet fluctuatios ad the fluctuatig field at time t. The ifluece of differet itegral legths, time scales ad amplitudes of the turbulet ilet boudary coditios is ivestigated. The paper is orgaized as follows. The equatios ad the stadard

5 USING ISOTROPIC SYNTHETIC FLUCTUATIONS 5 hybrid LES-RANS model are first preseted. The method for geeratig sythetic fluctuatios is the described ad a brief presetatio is give of the umerical method ad the hybrid LES-RANS model. I the results sectio, we start with DNS simulatio cotiue with hybrid LES-RANS, ad ed with some results of usig hybrid LES-RANS with forcig at the iterface. The paper eds with the coclusios. 2. Equatios The Navier-Stokes equatios with a added turbulet/sgs viscosity read ui 1 p u ( u ) = βδ1 ( ν ν ) i iuj i T, (1) t x j ρ xi x j x j ui = 0, (2) xi where ν T = νt ( νt deotes the turbulet RANS viscosity) for y yml (see Fig. 1), ad for y > yml we use ν T = ν sgs. The switch betwee URANS ad LES is achieved i the same way at the upper matchig plae (at 2δ yml ). Coefficiet β = 1 is used for chael flow simulatios with periodic streamwise boudary coditios; whe iletoutlet boudary coditios are used, β = 0. The desity is set to oe i all simulatios. 3. Sythesized Turbulece A turbulet velocity field ca be simulated usig radom Fourier modes. This was proposed i [17] ad later developed further i [1, 3, 15, 21]. The velocity field is give by N radom Fourier modes as where u ˆ, ψ ad N u ( x ) = 2 uˆ cos( κ x ψ ) σ, (3) i j = 1 j j σ i are the amplitude, phase ad directio of Fourier mode, respectively. The otatio used here follows that i [4, 6, 10] ad more iformatio is give i these papers. The sythesized turbulece at oe time step is geerated as follows. i

6 6 Figure 2. The probability of a radomly selected directio of a wave i wave-space is the same for all da i o the shell of a sphere. Figure 3. The wave-umber vector, κ i, ad the velocity uit vector, σ i, are orthogoal (i physical space) for each wave umber. The uit vector, σ i, is defied such that σ κ i i = 0 (superscript deotes Fourier mode ). Furthermore, of σ i i the σ 3 is parallel to κ i (i.e., σ 3 = ξ3 ). The directio ξ 1 ξ 2 plae is radomly chose through α.

7 USING ISOTROPIC SYNTHETIC FLUCTUATIONS 7 1. For each mode, create radom agles ϕ, α ad θ (see Figs. 2 ad 3) ad radom phase ψ. The probability distributios are give i Table 1. Table 1. Probability distributios of the radom variables p ( ϕ ) = 1 ( 2π) 0 ϕ 2π p ( ψ ) = 1 ( 2π) 0 ψ 2π p ( θ ) = 1 2 si( θ) 0 θ π p ( α ) = 1 ( 2π) 0 α 2π 2. Defie the highest wave umber based o mesh resolutio κ max = 2π ( 2 ), where is the grid spacig. The fluctuatios are geerated o a grid with equidistat spacig, η = y N, z = z max Nk, where η deotes the wall-ormal directio ad N j ad deote the umber of cells i the y ad z directios, respectively. The fluctuatios are set to zero at the wall ad are the iterpolated to the ilet plae of the CFD grid (the y z plae). 3. Defie the smallest wave umber from κ 1 = κ e p, where κ e = α 9π ( 55Lt ), α = Factor p should be larger tha oe to make the largest scales larger tha those correspodig to κ e. I the preset work p = Divide the waveumber space, κ max κ 1, ito N modes, equally large, of size κ. max j N k 5. Compute the radomized compoets of κ j accordig to Fig Cotiuity requires that the uit vector, σ i, ad orthogoal. σ 3 is arbitrarily chose to be parallel with κ j are κ i (see Fig. 3), ad α ad the requiremet of orthogoality give the remaiig two compoets.

8 8 7. A modified vo Kármá spectrum is chose, see Eq. 4 ad Fig. 4. The amplitude û of each mode i Eq. 3 is the obtaied from u ˆ j 1 2 = ( E( κ ) κ). Figure 4. Modified vo Kármá spectrum. 8. Havig ˆ u, κ, computed. j u E( κ) = α κ σ i ad ψ, allows the expressio i Eq. 3 to be 2 rms e ( κ κe ) 2 [ 1 ( κ κ ) ] 4 2 [ 2( κ κ η ) ] e 17 6 e κ = ( κi κi ), κη = ε ν. (4) I this way ilet fluctuatig velocity fields ( u, v, w ) are created at the ilet y z plae. A fluctuatig velocity field is geerated each time step as described above. They are idepedet of each other, however, ad their time correlatio will thus be zero. This is uphysical. To create correlatio i time, ew fluctuatig velocity fields, U, V, W, are computed based o a asymmetric time filter, as i [4, 6] 1 ( U ) m = a ( U ) m b( u ) m 1 ( V ) m = a ( V ) m b( v ) m

9 USING ISOTROPIC SYNTHETIC FLUCTUATIONS 9 m m 1 m ( W ) = a ( W ) b( w ), (5) where m deotes the time step umber ad a = exp( t T ). This asymmetric time filter resembles the spacial digital filter preseted i [16]. The secod coefficiet is take as b = ( 1 a 2 ) 0. 5 which esures that 2 2 i U i = u ( deotes averagig). The time correlatio of U i will be equal to exp( t T ), ad thus Eq. 5 is a coveiet way to prescribe the turbulet time scale of the fluctuatios. The ilet boudary coditios are prescribed as u( 0, y, z, t) = U ( y) u ( y, z t) i i, v( 0, y, z, t) = V ( y) v ( y, z t) i i, w( 0, y, z, t) = W ( y) w ( y, z, t), (6) i i m where u = ( U ), v = ( V ) m ad w = ( W ) m (see Eq. 5). For the i i hybrid LES-RANS simulatios at Re τ = 2000, the mea velocities are set as V i = Wi = 0 ad [27] y y 5 U i = l( y ) 5 < y < 30 (7) 1 l( y ) B y 30, κ where κ = 0. 4 ad B = 5.2. For the DNS at Re τ = 500, the mea i streamwise velocity, U i, is take from a DNS of fully developed flow (streamwise periodic boudary coditios). 4. The Numerical Method A icompressible, fiite volume code is used [9]. For space discretizatio, cetral differecig is used for all terms. The Crak- Nicholso scheme is used for time discretizatio of all equatios. The umerical procedure is based o a implicit, fractioal step techique with a multigrid pressure Poisso solver [12] ad a o-staggered grid arragemet.

10 10 5. Hybrid LES RANS A oe-equatio model is employed i both the ier URANS regio ad outer LES regio ad reads k t T x j k ( ujkt ) = ( T ) x ν ν j x T j P kt C ε 3 2 kt Pk T = τ s τ = 2ν s. (8) ij ij, ij T ij I the ier regio ( y y ml ) k T correspods to the RANS turbulet kietic eergy k; i the outer regio ( y > y ml ) it correspods to the subgrid-scale kietic turbulet eergy k sgs. No special treatmet is used i the equatios at the matchig plae except that the form of the turbulet viscosity ad the turbulet legth scale are differet i the two regios, see Table 2. k = 0 at the walls ad k T x = 0 at the ilet. T Table 2. Turbulet viscosities ad turbulet legth scales i the URANS ad LES regios. ad δ V deote the distace to the earest wall ad the computatioal cell volume, respectively URANS regio LES regio 1 2 l 2.5[ 1 exp( 0.2k ν)] ( ) ν.5k [ 1 exp( 0.014k ν)] T = = δv k sgs C ε Results 6.1. Direct Numerical Simulatios The mesh has i the streamwise (x), wall-ormal (y) ad spawise (z) directios, respectively. The turbulet viscosity, ν T, i Eq. 1 is set to zero. The size of the computatioal domai is x max = 8π, y max = 2δ = 2 (geometric stretchig of 12%) ad z max = 0.5π. This gives

11 a x ad USING ISOTROPIC SYNTHETIC FLUCTUATIONS 11 z of approximately 50 ad 12, respectively, ad y 0. 3 for the ear-wall ode. The time step was set to 3 tu τ δ = which gives a maximal CFL of approximately oe. The Reyolds umber is Re τ = u τ δ ν = 500. Neuma boudary outlet boudary coditios were iitially used but they were foud to disturb the solutio upstream of the outlet. Istead covective outlet boudary coditios were prescribed as ui u U i 0 = 0, (9) t x z t 0 max. 0 0 boudary coditios with differet time scales, legth scales ad amplitudes are ivestigated, see Table 3. max where U ( y, t) = u( x, y, z, t ) dzdt Fluctuatig ilet velocity Table 3. Presetatio of test cases for the DNS simulatios. The baselie turbulet legth ad time scales are set to L 1 = = Lt ad T 1 = T = 0.22 (see Sectio 3), respectively. The baselie amplitude of the ilet fluctuatios is u i, rms = vi, rms = wi, rms = 1.5. Costats a ad b from Eq. 5 are also give. L T u i, rms a b L 1 T L 1 T L 1 T L 1 T L 1 T L 1 2 T L 1 4 T L 1 T

12 12 Figure 5. Mea velocities ad resolved shear stresses. DNS ilet fluctuatig velocities. : x δ = 1; --- : x δ = 10; - - : x δ = 24. Figure 6. Mea velocities ad resolved shear stresses. Sythesized isotropic ilet fluctuatig velocities. u i, rms = vi, rms = wi, rms = 1.5. Ilet time scale T 1 12 ad ilet legth scale L 1. : x δ = 1; --- : x δ = 10; - - : x δ = 24.

13 USING ISOTROPIC SYNTHETIC FLUCTUATIONS 13 2 T i, rms i i. 0 Figure 7. Autocorrelatio B() τ = 1 ( Tu ) u ( t) u ( t τ) dt : sythetic fluctuatios with T 1 12; --- : DNS fluctuatios at y 130; : DNS fluctuatios at y 380; : exp( 12τ) T 1. 2 max Figure 8. Two-poit correlatio B ( ζ) = 1 ( z max wi, rms ) wi ( z) ( z ζ) dz. x = 0. : sythetic fluctuatios with L ; --- : sythetic w i fluctuatios with L 1 4; : fully developed chael DNS at y 20; o : fully developed chael DNS at y 60; : fully developed chael DNS at y z 0

14 14 The velocity profile ad the resolved shear stresses are show i Fig. 5 (DNS ilet fluctuatios) ad Fig. 6 (sythesized fluctuatios usig ilet time scale T 1 12 ad ilet legth scale L 1). The profiles are show for x = δ, x = 10δ ad x = 24δ. The velocity profiles agree very well with DNS data for both cases. The resolved shear stresses are slightly better with DNS fluctuatios tha with sythesized fluctuatios. Still, the sythetic isotropic fluctuatios do a remarkable job cosiderig that the prescribed ilet shear stress is zero; they maage to trigger the equatios to yield a reasoable shear stress profile eve at x = δ. The legth ad time scales used i Fig. 6 give autocorrelatio ad two-poit correlatios that agree well with fully developed DNS at y = 130 ad 60, respectively, see Figs. 7 ad 8. The two-poit correlatios ad the autocorrelatios of the isotropic fluctuatios are costat with respect to y, sice the y ad z coordiates are treated as homogeeous directios whe the sythetic fluctuatios are geerated. The true autocorrelatios ad two-poit correlatios decrease as the wall is approached. The twopoit correlatios evaluated from DNS at differet y locatios are icluded i Fig. 8. This illustrates the fact that the prescribed turbulet time ad legth scales of the sythetic homogeeous isotropic fluctuatios agree with the physical correlatio oly i some average sese. Figure 9. Frictio velocity. Ivestigatio of sesitivity to differet ilet time scales. u i, rms = vi, rms = wi, rms = 1.5. Ilet legth scale L 1. : T 12; --- : T 4; - - : T ; : DNS.

15 USING ISOTROPIC SYNTHETIC FLUCTUATIONS 15 Figure 10. Frictio velocity. Ivestigatio of sesitivity to differet amplitudes of ilet sythetic isotropic fluctuatios. Ilet time scale T 1 12 ad ilet legth scale L 1. : u i, rms = vi, rms = wi, rms = 0.75; --- : ui rms vi, rms = wi,, = rms = 1.5; u i, rms = vi, rms wi, rms - - : = = 2.25; : sythetic fluctuatios re-scaled to fit RMS profiles of DNS. The ifluece of differet time scales is show i Fig. 9. DNS ilet fluctuatios ad sythetic ilet fluctuatios with T 1 4 ad T 1 12 give a frictio velocity withi 1% of the target value, u τ = 1, at approximately x = 5δ. The larger the time scale, the greater the over-predictio of the frictio velocity ear the ilet. The amplitude of the sythetic fluctuatios used i the predictio preseted so far was set to 1.5. How does a chage i amplitude affect the results? Differet amplitudes have bee used i Fig. 10. It ca be see that the sesitivity to the ilet amplitude is much stroger tha the sesitivity to the ilet time scale. Whe a small amplitude of the fluctuatios is used ( u i, rms = 0.75) it is just about that the ilet fluctuatios maage to trigger the equatios to resolve turbulece. It takes 10δ util the frictio velocity is withi 4% of its target value. A large amplitude gives a overshoot i the frictio velocity ear the ilet but, cotrary to whe a small amplitude is employed, o udershoot i the frictio velocity appears.

16 16 Figure 11. Frictio velocity. Ivestigatio of sesitivity to differet ilet turbulet legth scales. Ilet time scale T : legth scale L 1; --- : legth scale L 1 2; - - : legth scale 1.5L 1; : legth scale L 1 4. Figure 11 ivestigates the ifluece of differet turbulet legth scales of the ilet fluctuatios. As ca be see, the effect of chagig the legth scale is larger tha that of chagig the time scale. The larger the legth scale of the ilet fluctuatios, the more efficiet they are i triggerig the equatios. The simulatios diverged whe larger legth scale tha 1.5L 1 was tested. Resolved shear stresses ad wall-ormal RMS fluctuatios at a statio close to the ilet, are show i Figs. 12 (differet time scales) ad 13 (differet amplitudes). Here it is cofirmed what was see i the frictio velocities: large time scales ad large amplitudes are efficiet i triggerig resolved turbulece, ad the equatios are more sesitive to chages i amplitude tha to chages i time scale. Recall that at the ilet, ui, rms = vi, rms = wi, rms = cost across the chael except for y δ < 1 N j = 1 80 (lower wall) ad y δ > 2 1 N j (upper wall), see Sectio 3, where the istataeous fluctuatios are obtaied by liear iterpolatio (zero value of the fluctuatios at the wall).

17 USING ISOTROPIC SYNTHETIC FLUCTUATIONS 17 Figure 12. Shear stresses ad wall-ormal stresses. x = 0.35δ. Ivestigatio of sesitivity to differet ilet time scales. u i, rms = vi, rms w = 1.5. : T 12; --- : T 4; - - : = i, rms T. Figure 13. Shear stresses ad wall-ormal stresses. x = 0.35δ. Ivestigatio of sesitivity to differet amplitude of ilet sythetic isotropic fluctuatios. v w u i, rms = vi, rms = wi, rms 0.75; --- : ui, rms : = = 1.5; - - : ui, rms = vi, rms = wi, = i, rms = i, rms rms = 2.25.

18 18 Figure 14. Mea velocities ad resolved shear stresses. Sythesized isotropic ilet fluctuatig velocities. Sythetic fluctuatios re-scaled to fit RMS profiles of DNS. Ilet time scale T 1 ad ilet legth scale L 1. o : ilet profiles; : x δ = 1; --- : x δ = 10; - - : x δ = 24. I the results preseted so far, costat RMS amplitudes of the isotropic fluctuatios i the y directio have bee used for 1 N j < y δ < 2 1 N j. A alterative could be to re-scale the fluctuatios so that the RMS profiles agree with DNS data of fully developed chael flow. The turbulet shear stress, ( u v ) i, would still be zero, sice the fluctuatios are isotropic. It should be oted that it is ot cosistet to rescale the fluctuatios to yield ohomogeeous ormal stress profiles cosiderig that the fluctuatios were geerated usig Fourier series based o the assumptio of homogeeous turbulece. Furthermore, whe rescalig fluctuatios, the two-poit correlatios i the y directio are modified ad will be differet from those show i Fig. 8. Figure 14 gives velocity ad shear stress profiles obtaied with re-scaled sythetic ilet fluctuatios. As ca be see, these fluctuatios give much worse profiles tha does costat RMS amplitude. The velocity profile at x = δ is overpredicted ad the resolved shear stress is much too low. The frictio velocity show i Fig. 10 also shows that whe re-scaled ilet fluctuatios are used it takes a log distace before the frictio velocity gets close to the target value.

19 USING ISOTROPIC SYNTHETIC FLUCTUATIONS 19 It ca be cocluded from the DNS results preseted above that ilet boudary coditios usig sythetic, homogeeous, isotropic fluctuatios give results almost as accurate as ilet coditios take from a pre-cursor DNS. Furthermore, it was foud that ilet fluctuatios with a large time scale, large legth scale ad large amplitude are efficiet i triggerig the mometum equatios to resolve turbulece. Havig gathered some experiece o how to prescribe ilet boudary coditios i DNS, we ow tur our attetio to the mai object of the preset study: ilet boudary coditios for hybrid LES-RANS Hybrid LES-RANS A ode mesh (x, streamwise; y, wall-ormal; z, spawise) was used. The size of the computatioal domai is x max = 8π, ymax = 2δ = 2 (geometric stretchig of 17%) ad z max = 2π. This gives a x ad z of approximately 785 ad 393, respectively ad y < 1 ear the walls, expressed i ier scalig. I outer scalig δ x 2.5 ad δ z 5. This mesh is very coarse ad the resolutio is realistic for idustrial applicatios. The locatio of the matchig plae is at y = (lower wall) which correspods to y = 150 ad 16 cells ( = j ml ) i the URANS regio at each wall, see Fig. 1. The time step was 3 set to tu τ δ = which gives a maximal CFL of approximately 0.6. The Reyolds umber is Re τ = u τ δ ν = Neuma boudary coditios are prescribed at the outlet. Differet time ad legth scales of the sythesized ilet turbulet fluctuatios are evaluated, see Table 4. The base-lie turbulet scales T 1 ad L 1 are used i Fig. 15. The velocity profile is well predicted ad resolved turbulece is created. Further dowstream, the velocity i the ceter regio icreases slightly. The resolved turbulece is still too small close to the ilet ( x = δ). At this locatio the wall shear stress is overpredicted (see Fig. 16) because of the strog ilet fluctuatios. The magitudes of the resolved shear stresses icrease further dowstream ad the frictio velocity goes toward oe. This sceario is similar to what was see i the DNS simulatios Oe differece, however, is that the

20 20 magitude of the resolved shear stress is actually too large at x = 10δ. The reaso why the mometum equatios are much more affected is because of the high Reyolds umber ad the very coarse grid resolutio. With the prescribed ilet mea velocity profile, hybrid LES-RANS does ot give a turbulet shear stress (sum of modelled ad resolved) that is able to satisfy the time-averaged streamwise mometum equatio. The result is that the flow is decelerated i the wall regios. The terms i the averaged streamwise mometum equatios are preseted i Fig. 17 (the wall ormal advective term, v u y, is ot icluded because it is egligible). At x = δ, the advectio term, which i fully developed coditios should be zero, has the same magitude as the turbulet diffusio. The forcig ilet fluctuatios create a situatio i which the averaged mometum equatio is far from its fully developed flow coditio. The advectio term still plays a importat role at x = 10δ, especially i the ceter regio. At the ed of the domai ( x = 24δ), the advectio term is rather small but still o-zero. Hece the flow is ot fully developed. This ca also be see i Fig. 16, where the predicted frictio velocity has ot reached a costat value at the ed of the domai. Table 4. Presetatio of test cases for hybrid LES-RANS simulatios. Re τ = The baselie turbulet legth scales ad turbulet time scale are set to L 1 = = Lt ad T 1 = T = (see Sectio 3), respectively. The baselie amplitude of the ilet fluctuatios is u v = w 1.5. The costats a ad b from Eq. 5 are i, rms = i, rms i, rms = also give. L T u i, rms a b L 1 T L 1 T L T L 1 T L 1 T L 1 T

21 USING ISOTROPIC SYNTHETIC FLUCTUATIONS 21 Figure 15. Mea velocities ad resolved shear stresses. Hybrid LES- RANS. Ilet time scale T 1 ad ilet legth scale L 1. : x δ = 1; --- : x δ = 10; - - : x δ = 24. Figure 16. Frictio velocity. Hybrid LES-RANS. : T 1 ad L 1 ; --- : T 1 ad 2L 1 ; : T 1 ad L = 0 ; : T = 0 ad L 1; : T 1 4 ad L 1.

22 22 Figure 17. Terms i the u equatio. Hybrid LES-RANS. Ilet time scale T 1 ad ilet legth scale L 1. : u u x; --- : u v y; - - : y ( ν νt ) u y ; : p x [ ].

23 USING ISOTROPIC SYNTHETIC FLUCTUATIONS 23 The frictio velocity (solid lie) i Fig. 16 exhibits typical odd-eve oscillatios. This is due to the coarse mesh ad cetral differecig. These oscillatios could easily be damped out usig a wiggle detector [8] but, sice they do ot cause ay problems, o dampig was used. Oscillatios are preset i all cases, but all lies/markers except the solid lie are plotted for every secod x ode to ehace visibility. Whe the turbulece legth scale of the ilet turbulece is icreased from L 1 to 2L 1, the predicted velocity profiles ad resolved shear stresses are ot to ay great extet affected, see Fig. 18. It ca be see that the fluctuatios with legth scale 2L 1 are somewhat more efficiet i creatig resolved turbulece ear the ilet tha fluctuatios with legth scale L 1. The result is a larger frictio velocity, a eve larger resolved shear stress at x = 10δ tha i Fig. 15, ad the advectio term ear the ilet is still larger. Figure 18. Mea velocities ad resolved shear stresses. Hybrid LES- RANS. Ilet time scale T 1 ad ilet legth scale 2L 1. : x δ = 1; --- : x δ = 10; - - : x δ = 24. Next we test removig all turbulet spacial structures by settig the turbulet legth scale to zero. Figure 19b shows that, compared to Fig. 15b, the geerated resolved shear stresses are very small; at the outlet

24 24 the resolved shear stress has reached a value of oly 0.2. Cosequetly the velocity i the ceter regio i Fig. 19a is much too large due to the small resolved shear stress. The frictio velocity is also much too small, see Fig. 16. Figure 19. Mea velocities ad resolved shear stresses. Hybrid LES- RANS. Ilet time scale T 1 ad ilet legth scale L = 0. : x δ = 1; --- : x δ = 10; - - : x δ = max Figure 20. Two-poit correlatio B ( ζ) = 1 ( zmaxwrms ) w ( z) w ( z ζ) dz. Hybrid LES-RANS. y 60. Time scale T. : L ; --- : L = 0; - - : 2L1. Markers illustrates the grid spacig. 1 z 0 1

25 USING ISOTROPIC SYNTHETIC FLUCTUATIONS 25 Figure 20 shows the two-poit correlatios of the resolved turbulece at the ilet ad at x = 22. 9δ (both at y 60) for the ilet turbulece scales L = 0, L 1 ad 2L 1. At the ilet (Fig. 20a) the two-poit correlatio is of course zero for L = 0. With L 1 the correlatio is approximately 0.2 at ζ = 0. 2δ = (the circles idicate the spawise cell spacig), ad the two-poit correlatio with 2L 1 is slightly more tha twice as large at ζ = z. However, far dowstream (Fig. 20b), the differece i spawise correlatios betwee all three cases is ot large. Thus, eve for the case with o spawise ilet correlatio ( L = 0), some spawise structures have bee geerated, although the velocity profile ad resolved stress profile are very poorly predicted (Fig. 19). The fact that at x = 22. 9δ some resolved turbulece has bee created makes it likely that the velocity profiles ad resolved stresses will be the same i all three cases for a very log chael. As metioed i Sectio 1, the purpose of the ilet turbulet fluctuatios is to trigger the equatios ito resolvig turbulece. This triggerig must be doe at a scale that the equatios uderstad. There is o poit i usig ilet turbulet fluctuatios whose legth scale is smaller tha the grid scale. The turbulet legth ad time scales of the ilet fluctuatios should ot be as correct as possible, but they should be related to the grid ad the discretizatio scheme. Figure 21. Mea velocities ad resolved shear stresses. Hybrid LES- RANS. Ilet time scale T = 0 ad ilet legth scale L 1. : x δ = 1; --- : x δ = 10; - - : x δ = 24.

26 26 Figure 21 presets velocity ad shear stress profiles where the turbulet time scale of the ilet fluctuatios are set to zero ( a = 0, b = 1 i Eq. 5, see Table 4). As ca be see, the equatios have eve greater problems copig with zero time scale tha with zero legth scale. The magitude of the resolved shear stresses remais smaller tha 0.15 throughout the chael. Figure 16 shows the predicted frictio velocity for all test cases. As has bee observed i the velocity profiles, the poorest results are obtaied for the cases for which the time scale, T, or legth scale, L, are set to zero. With a ilet turbulet legth scale of L 1 the predicted frictio velocity improves for the larger time scale. Whe the ilet legth scale is icreased to 2L 1 it is foud that these fluctuatios are too efficiet i triggerig the mometum equatios. Figure 22. Autocorrelatio B (). τ Hybrid LES-RANS. : ilet, time scale T 1; : ilet, time scale T 1 4; --- : x = 22.9δ, ilet time scale T 1 4; - - : x = 22.9δ, ilet time scale T 1. The autocorrelatios are preseted i Fig. 22. The autocorrelatio at the ilet obviously icreases whe the prescribed time scale is icreased from T 1 4 to T 1 via a ad b, see Table 4. I the URANS regio (Fig. 22a), the small ilet time scale ( T 1 4) is icreased far dowstream because of the large turbulet viscosity (which yields large modelled dissipatio) which dampes small time scales more tha large time scales. I the case with the large ilet time scale ( T 1 ) the predicted time

27 USING ISOTROPIC SYNTHETIC FLUCTUATIONS 27 scale does ot chage much far dowstream, because the prescribed time scale at the ilet fits the URANS regio. I the LES regio (Fig. 22b), the small ilet time scale is kept far dowstream, whereas the large ilet time scale decreases far dowstream. Agai, this idicates that the small ilet time scale is close to the atural time scale of the LES regio but that the large time scale is too large ad i the latter case the LES regio switches to its ow time scale. Figure 23. Mea velocities ad resolved shear stresses. Hybrid LES- RANS. Ilet time scale T 1 ad ilet legth scale L 1. u i, rms = vi, rms w : x δ = 1; --- : x δ = 10; - - : x δ = 24. = i, rms = Figure 24. Mea velocities ad resolved shear stresses. Hybrid LES- RANS. Ilet time scale T 1 ad ilet legth scale L 1. u i, rms = vi, rms w : x δ = 1; --- : x δ = 10; - - : x δ = 24. = i, rms =

28 28 Figure 25. Frictio velocity. Hybrid LES-RANS. Ivestigatio of the ifluece of differet amplitude of the ilet fluctuatios. Ilet time scale T 1 ad ilet legth scale L 1. : i, rms = i, rms = i, rms = 1. 5 (baselie); --- : u i, rms = vi, rms = wi, rms = 0.75; : u i, rms = vi, rms w = i, rms = We ivestigate ext how a chage i amplitude of the ilet fluctuatios affects the flow. The baselie amplitude is u i, rms = vi, rms w 1.5. It is reduced by 50% i Fig. 23 ad it is icreased by = i, rms = 50% i Fig. 24 compared to the baselie case i Fig. 15. As expected, a large amplitude is more efficiet i triggerig the equatios tha a small amplitude, ad it is see that the ilet fluctuatios with the smallest amplitude (Fig. 23) geerate much too small resolved shear stress at x = δ. With the largest amplitude (Fig. 24), the equatios are strogly disturbed by the ilet fluctuatios, which lead to a large over-predictio i the shear stress at x = 10δ. The three cases are compared i Fig. 25 where it ca be see that the smallest amplitude costitutes a poor ilet boudary coditio. The largest amplitude gives a large over-predictio i frictio velocity, but the flow has recovered far dowstream from the strog disturbaces geerated by the ilet fluctuatios ad the flow at this locatio is actually well predicted. u v w

29 USING ISOTROPIC SYNTHETIC FLUCTUATIONS Hybrid LES-RANS with forcig A careful look at Fig. 16 reveals that the frictio velocity decreases with x for all cases. The resolved turbulece which was triggered at the ilet by the isotropic fluctuatios is slowly beig dampeed as x icreases. Assume that we are usig hybrid LES-RANS to simulate the flow i a diffuser or the flow over hill. Fluctuatig ilet boudary coditios are prescribed. However from the decreasig frictio velocities show i Fig. 16, we should be cocered about what happes if the throat of the diffuser or the hill is located far from the ilet. If that is the case, the flow will be icorrect whe it fially reaches the diffuser/hill because, as see i Fig. 16, the frictio velocity will be too small (or, more correctly, too low a ratio of the frictio ad ceterlie velocity). It is well kow that stadard LES-RANS gives too low a frictio velocity for a ifiitely log chael [2, 11, 22, 24, 26]. Oe way to overcome this deficiecy i hybrid LES-RANS is to itroduce forcig [2, 10, 11, 22, 24] sometimes called back scatter or erichmet i the ear-wall (URANS) regio or at the iterface. Figure 26. Added forcig fluctuatios, u f, v f, w f, i a cotrol volume ( j 1) i the LES regio adjacet to the iterface. ml The preset work adopts the approach preseted i [10]. The forcig is itroduced as sources i the three mometum equatios at the iterface (see Figs. 1 ad 26) SU = γρu f v f A = γρu f v f V y

30 30 SV = γρv f v f A = γρv f v f V y SW = γρw f v f A = γρw f v f V y, (10) where V ad y deote the cell volume ad grid spacig i the y directio, respectively, ad γ = k T ( x, yml, z) kf, kf = ( uf, rms vf, rms wf, rms ) (11) 2 u f, v f, w f deote sythetic isotropic fluctuatios geerated i the same way as the ilet fluctuatios. See [10] for more details. Figure 27. Mea velocities ad shear stresses. Hybrid LES-RANS. Periodic streamwise boudary coditios. Solid lies: forcig with isotropic fluctuatios; dashed lies: o forcig. I Fig. 27a the velocity profiles are preseted usig forcig ad o forcig. As ca be see, the agreemet with the law of the wall usig hybrid LES-RANS with forcig is excellet. Without forcig, agreemet is less good. The resolved ad modelled shear stresses are show i Fig. 27b for the two hybrid models. The resolved stress usig forcig is larger tha without forcig. The reaso is that the forcig fluctuatios trigger the mometum equatios to resolve turbulece. Sice the total shear stress must follow 1 y, the modelled shear stress with forcig is smaller tha without forcig.

31 USING ISOTROPIC SYNTHETIC FLUCTUATIONS 31 Figure 28. Frictio velocity. Hybrid LES-RANS. T 1 ad L 1. : forcig; --- : o forcig. I Fig. 28 hybrid LES-RANS with ad without forcig is used for a log chael ( cells, x max = 32πδ). Without forcig, the frictio velocity is still decreasig at the outlet whereas with forcig the frictio velocity decreases dow to u τ Hece, with forcig, the ilet ca be located very far dowstream without ay large adverse effect. Cosiderig the accurate velocity profile predicted usig forcig for a ifiitely log chael (i.e., with streamwise periodic boudary coditios), see Fig. 27a, it may seem somewhat uexpected that the frictio velocity predicted usig forcig (Fig. 28) does ot approach the target value u τ = 1. The reaso is that i the buffer regio the prescribed ilet mea velocity profile (see Eq. 6) does ot etirely agree with the mea profile show i Fig. 27a. 7. Coclusios I the preset paper isotropic sythetic fluctuatios have bee used to prescribe realistic turbulet ilet boudary coditios. For DNS simulatios this approach was compared with ilet coditios take from a pre-cursor DNS. The two methods were foud to be comparable (the pre-cursor DNS slightly better) ad both methods were foud to give a wall frictio velocity, u τ, withi 1% of the target value at less tha five

32 32 half-chael widths dowstream of the ilet. Quite reasoably resolved shear stress profiles are obtaied with sythesized fluctuatios as early as oe half-chael width dowstream of the ilet. The ifluece of differet turbulet ilet legth scales, time scales ad amplitudes of the sythetic fluctuatios have bee ivestigated. It was foud that ilet fluctuatios with a large time scale, large legth scale ad large amplitude are efficiet i triggerig the mometum equatios to resolve turbulece. A large legth scale may lead to too violet forcig of the equatios ad lead to over-predicted resolved stresses, at least i coectio with DNS. It has bee show that it is ecessary to prescribe both a realistic legth scale ad time scale of the ilet fluctuatios. If zero legth or time scale is used, the the resultig resolved fluctuatios are much too small. It is also importat that the amplitude of the fluctuatios is sufficietly large. A value of u i, rms u τ = is too small, whereas u i, rms uτ = 1.5 is sufficiet. I all simulatios carried out i the preset work except oe, the RMS of the sythesized fluctuatios was costat across the ilet (except for y < η, lower wall, ad y > 2δ η, upper wall). I oe simulatio the fluctuatios were re-scaled so that the ormal stresses of the sythesized fluctuatios were equal to those of a DNS at a lower Reyolds umber. This re-scalig modifies the two-poit correlatios i the wall-ormal directio ad hece also the prescribed turbulece legth scale. It was foud that this approach gave cosiderably poorer results tha whe o re-scalig was used. Stadard hybrid LES-RANS yields too small wall shear stress (or, equivaletly, too large ceterlie velocity) i ifiitely log chaels. Oe way to reduce this deficiecy is to itroduce forcig at the iterface or i the iterface regio. It is argued that fluctuatig ilet boudary coditios should be viewed as forcig coditios with the object of triggerig the equatios to resolve turbulece. Hece, it is ot importat that the legth scale ad time scale are physically correct. They should be chose o the basis of their beig efficiet i triggerig the equatios, ad they should be related to the grid, the time step, the discretizatio

33 USING ISOTROPIC SYNTHETIC FLUCTUATIONS 33 scheme, the turbulece model ad the locatio of the ilet. If the ilet is located, for example, far upstream of a slated surface alog which separatio is expected to take place, it may occur that the resolved turbulece triggered at the ilet is dissipated before the flow reaches the slated surface. I this case additioal forcig is eeded at the iterface betwee the LES regio ad the URANS regio to keep the resolved turbulece alive. The flow i a log chael was computed to exemplify this problem. Ackowledgemet This work was fiaced by the DESider project (Detached Eddy Simulatio for Idustrial Aerodyamics) which is a collaboratio betwee Aleia, ANSYS-AEA, Chalmers Uiversity, CNRS-Lille, Dassault, DLR, EADS Military Aircraft, EUROCOPTER Germay, EDF, FOI-FFA, IMFT, Imperial College Lodo, NLR, NTS, NUMECA, ONERA, TU Berli, ad UMIST. The project is fuded by the Europea Commuity represeted by the CEC, Research Directorate-Geeral, i the 6th Framework Programme, uder Cotract No. AST3-CT Refereces [1] C. Bailly ad D. Juvé, A stochastic approach to compute subsoic oise usig liearized Euler s equatios, AIAA paper , [2] P. Batte, U. Goldberg ad S. Chakravarthy, Iterfacig statistical turbulece closures with large-eddy simulatio, AIAA Joural 42(3) (2004), [3] W. Bechara, C. Bailly ad P. Lafo, Stochastic approach to oise modelig for free turbulet flows, AIAA Joural 32(3) (1994), [4] M. Billso, L.-E. Eriksso ad L. Davidso, Jet oise predictio usig stochastic turbulece modelig, AIAA paper , 9th AIAA/CEAS Aeroacoustics Coferece, [5] M. Billso, L.-E. Eriksso ad L. Davidso, Modelig of sythetic aisotropic turbulece ad its soud emissio, The 10th AIAA/CEAS Aeroacoustics Coferece, AIAA , Machester, Uited Kigdom, [6] M. Billso, Computatioal techiques for turbulece geerated oise, Ph.D. Thesis, Dept. of Thermo ad Fluid Dyamics, Chalmers Uiversity of Techology, Göteborg, Swede, 2004.

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