RESEARCH ARTICLE. Zonal PANS: evaluation of different treatments of the RANS-LES interface

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1 March 3, 26 Journal of Turbulence paper Journal of Turbulence Vol., No., 2, 32 Volume 7, Issue 3, pp , 26 RESEARCH ARTICLE Zonal PANS: evaluation of different treatments of the RANS-LES interface L. Davidson Division of Fluid Dnamics, Department of Applied Mechanics Chalmers Universit of Technolog SE Gothenburg, Sweden lada@chalmers.se (Received Month 2x; final version received Month 2x) The partiall averaged Navier-Stokes (PANS) model, proposed b Girimaji [], can be used to simulate turbulent flows either as RANS, LES or DNS. Its main parameter is f k whose phsical meaning is the ratio of the modeled to the total turbulent kinetic energ. In RANS f k =, in DNS f k = and in LES f k takes values between zero and one. Three different was of prescribing f k are evaluated for decaing grid turbulence and fulldeveloped channel flow: f k =.4, f k = k 3/2 tot /ε [2] and from its definition f k = k/k tot where k tot is the sum of the modeled, k, and resolved, k res, turbulent kinetic energ. It is found that the f k =.4 gives the best results. In [3], a method was proposed to include the effect of the gradient of f k. This approach is used at RANS-LES interface in the present stud. Four different interface models are evaluated in full developed channel flow and embedded LES of channel flow; in both cases, PANS is used as a zonal model with f k = in the URANS region and f k =.4 in the LES region. In full developed channel flow, the RANS-LES interface is parallel to the wall (horizontal) and in embedded LES it is parallel to the inlet (vertical). The importance of the location of the horizontal interface in full developed channel flow is also investigated. It isfound that the location and the choice of the treatment atthe interface ma be critical at low Renolds number or if the interface is placed too close to the wall. The reason is that the modeled turbulent shear stress at the interface is large and hence the relative strength of the resolved turbulence is small. In RANS, the turbulent viscosit and consequentl also the modeled Renolds shear stress is onl weakl dependent on Renolds number. It is found in the present work that that also applies in the URANS region. Kewords: LES; PANS; 2G-RANS; Zonal model; Embedded LES; Hbrid RANS-LES; RANS-LES interface condition Introduction Large Edd Simulation (LES) is ver expensive in wall-bounded flow. To be able to extend LES to high Renolds number man proposals have been made in the literature to combine LES with unstead RANS (URANS) near the walls. The first and most common method is Detached Edd Simulation (DES) [4 6]. Later, other researchers have proposed hbrid LES/RANS [7, 8] and Scale-Adapted Simulations (SAS) [9, ]; for a review, see []. DES and hbrid LES/RANS use the cell size as the SGS length scale. SAS does not use the cell size (except as a limiter) but uses the von Kármán lengthscale instead. The main object of the methods mentioned above is that the reduce the turbulent viscosit in the LES region. There are three options to achieve this. () The turbulent viscosit, ν t, is reduced b modifing its formulation; (2) The dissipation term in the equation for the modeled, turbulent kinetic ISSN: (online onl) c 2 Talor & Francis DOI:.8/ YYYYxxxxxx

2 March 3, 26 Journal of Turbulence paper 2 L. Davidson energ, k, is increased b decreasing the turbulent lengthscale, or (3) the destruction term in the lengthscale equation (ε or ω) is decreased. This increases the dissipation term in the k equation as well as decreases the turbulent viscosit directl since ε (or ω) appears in the denominator of the expression for ν t. DES [2] is based on the first option. X-LES [3] and the one-equation hbrid LES-RANS [8, 4] use option number two: ν t is decreased and the dissipation term in the k equation is increased. The third option is used in the partiall averaged Navier-Stokes (PANS) model [] and the Partiall Integrated Transport Model (PITM) [, 6]. PANS is used as a zonal LES-RANS method in the present work where RANS is used near the walls and LES is emploed further awa from the walls. We will also use PANS in embedded LES. Much work has been presented on embedded LES latel. Menter et al.[7] present embedded LES of channel flow and the hump flow. The use the SST RANS model upstream of the embedded LES interface. Downstream of the interface the use the model of Shur et al. [6] which consists of the Smagorinsk model in the LES region and a mixing-length model in the RANS region. At the interface, the modeled turbulence is converted into resolved turbulence using snthetic turbulence from a vortex method [8]. Shur et al. [9] proposed a new reccling method in a interface zone between RANS and LES. The RANS and LES zones overlap and identical grids are used in the overlap region. Onl the fluctuating components are reccled since the mean components are available in the entire overlap zone. The evaluated the method for flat plate boundar flow and flow over a two-dimensional airfoil. Poletto et al. [2] made embedded LES of the hump flow. The used Delaed Detached Edd Simulation (DDES) in the entire region (both upstream and downstream of the interface). At the LES interface the added snthetic fluctuations from the divergence free snthetic edd method [2]. Gritskevich et al. [22] used the Improved Delaed Detached Edd Simulation (IDDES) to make embedded LES of the hump flow. The first computed the entire flow with 2D RANS. Then the used the RANS solution to prescribe the inlet mean velocit profile at the embedded LES interface. The modeled k was taken from the RANS solution and ω was computed from k and the grid size. Snthetic fluctuations were generated with a snthetic turbulence generator [23]. Shur et al. [24] present embedded LES of the flow over a hump. The used overlapping RANS and IDDES in the embedded interface region. At the interface the added snthetic turbulence. The show ver good agreement with measurement data. Xiao and Jenn [2] presented a novel method for treating the RANS-LES interface in hbrid LES-RANS. The solved the RANS and LES equations in the entire domain. The RANS mesh in the LES region does not need to be the same as the LES mesh and vice versa. Drift forces were added in the overlapping regions. As mentioned above, the PANS model is in the present work used both as a zonal hbrid LES-RANS model. f k is set to one in the RANS regions, and it is set to.4 in the LES region. The interface between the RANS and LES regions is defined along a grid plane, an approach also chosen in ZDES [26]. As an alternative to using a constant f k in the LES region, f k can be computed using the cell size and the integral length scale, k 3/2 tot /ε [2], where k tot is the sum of the resolved and the modeled turbulent kinetic energ, i.e. k tot = k res +k. The reason wh f k is not computed in the present stud is that it has been shown that a constant f k =.4 works well in the LES region [27 3]; it was even shown in [3] that using constant f k =.4 works better than computing f k, not to mention that

3 March 3, 26 Journal of Turbulence paper Journal of Turbulence it is numericall more stable. Section 4 compares three alternatives (f k =.4, computingf k accordingtotheequationabove, andcomputingf k fromitsdefinition k/k tot ) for decaing homogeneous grid turbulenceand full developed channel flow. It should be mentioned that Basara et al. [2] have shown that computing f k works well when using the k ε ζ f model. In [3], a method was proposed to include the effect of the gradient of f k. The gradient appears because the PANS filtering operation does not commute with the the spatial gradient. At RANS-LES interfaces, the spatial gradient of f k is large. The approach of Girimaji and Wallin [3] is further developed in the present stud and it is used at RANS-LES interfaces. The appearance of the gradient of f k at RANS-LES interfaces is similar to the method for treating RANS-LES interfaces in Hamba [3]. He shows that when the filter size varies, an additional term appears because the filtering and the spatial gradients do not commute. The divergence of a flux, q i, for example, will include an additional term q i x i = q i x i x i q i () Chaouat and Schiestel [32] used the approach in [3] to modif their PITM approach. The derived the modified transport equations including commutation terms as in Eq.. The gradients of the filter width,, was replaced with gradient of the cut-off wavenumber, κ c, which appear in the PITM approach. The method proposed b Hamba [3] was recentl further developed and evaluated b Davidson [33] at RANS-LES interfaces using the PDH k ω model [34]. The modified approach of including the effect of the gradient of f k is in the present stud evaluated in full developed channel flow and embedded LES of channel flow; in both cases, PANS is used as a zonal model. In full developed channel flow, the RANS-LES interface is parallel to the wall and in embedded LES it is parallel to the inlet. When the RANS-LES interface is parallel to the wall, the term including the gradient of f k across the interface can be both positive and negative (because v is both positive and negative). This term is included onl when it reduces the modeled turbulence. Hence, the term can be seen as a forcing term which represents backscatter. It is used to stimulate growth of the resolved turbulence in the LES region adjacent to the RANS region. The present method reduces the gra area problem described in [3]. The paper is organized as follows. The next section describes the equations and the interface models. The following section describes the numerical method. Then thereisasection comparingthreedifferentwas oftreatingf k,followed basection which presents and discusses the results. The final section presents the conclusions.

4 March 3, 26 Journal of Turbulence paper 4 L. Davidson The PANS Model The low-renolds number partiall averaged Navier-Stokes (LRN PANS) turbulence model reads [27] Dk Dt = [( ν + ν t x j Dε Dt = x j ν t = C µ f µ k 2 σ ku [( ν + ν t σ εu ) ] k +P k +P ktr ε x j ) ] ε ε +C ε P k x j k C ε2 ε,p k = 2ν t s ij s ij, s ij = 2 ε 2 k ( ūi x j + ū j x i ) (2) C ε2 = C ε + f k f ε (C ε2 f 2 C ε ),σ ku σ k f 2 k f ε,σ εu σ ε f 2 k f ε σ k =.4,σ ε =.4,C ε =.,C ε2 =.9,C µ =.9,f ε = where D/Dt = / t + ū j / x j denotes the material derivative. The damping functions are defined as f 2 = f µ = [ exp ( 3. [ exp ( 4 ) ] 2 { ( Rt.3exp[ 6. ) ] 2 { + R 3/4 t R t = k2 νε, = U ε ν, U ε = (εν) /4 ) 2 ]} [ ( Rt ) ] } 2 exp 2 The term P ktr in Eq. 2 is an additional term which is non-zero in the interface region because Df k /Dt. The function f ε, the ratio of the modeled to the total dissipation, is set to one since the turbulent Renolds number is high. f k is set to in the RANS region and to.4 in the LES region (in Section 4 is ma also be computed in the LES region). It was shown in [3] that the k and ε equations in the LES region are in local equilibrium, i.e. (3) P k ε = (4) 48 ε C ε k P k Cε2 ε 2 = () k 49 (. denotes averaging in all homogeneous directions, i.e. t, z and, in full developed channel flow, also x). It seems that Eqs. 4 and cannot both be satisfied since C ε Cε2 ; indeed, the equations are not satisfied instantaneousl for which the convective terms pla an important role. Equation is satisfied but 2 ε C ε k P k Cε2 ε 2. (6) k 3 4 Equations 4 and are both satisfied because the correlation between ε, k and P k (left side of Eq. ) is stronger than that between ε 2 and k (right side). The

5 March 3, 26 Journal of Turbulence paper Journal of Turbulence 6 correlation between the quantities on the left side is larger than that on the right side because Cε2 > C ε. For more detail, see [3] The Interface Condition The commutation error in PANS was recentl addressed in [3]. In PANS, the equation for the modeled turbulent kinetic energ, k, is derived b multipling the k tot equation (k tot = k res +k) b f k. The convective term in the k equation with constant f k is then obtained as f k Dk tot Dt = D(f kk tot ) Dt = Dk Dt (7) 62 where f k = k k tot. (8) 63 Now, if f k varies in space, we get instead f k Dk tot Dt = D(f kk tot ) Dt Df k k tot Dt = Dk Dt k Df k tot Dt (9) Since f k in the present work is constant in time, Df k /Dt = ū i f k / x i. The second term on the right side of Eq. 9 is the commutation term; it represents (excluding the minus sign) energ transfer from resolved to modeled turbulence. It can be written (on the right side of the k equation) k tot Df k Dt = (k+k res) Df k Dt = kdf k Dt + ū iū i 2 Df k Dt () 68 The commutation term in the k res equation is the same but with opposite sign, i.e. k Df k Dt ū iū i 2 Df k Dt () The question is now which term should be added to the momentum equations to get the commutation term in the k res equation. We start b the second term in Eq.. This term can be represented b the source term Si = 2ū Df k i Dt (2) in the momentum equation. To show that this term corresponds to the commutation term in Eq., consider the momentum equation for the fluctuating velocit, ū i. Multipling S i b ū i = ū i ū i and time-averaging gives the source term in the k res equation as (ū i ū i ) 2ū i Df k = 2ū Dt iū i Df k = Dt 2 ū iū i Df k Dt Equation 3 is equal to the time average of the second term in Eq. as it should. (3)

6 March 3, 26 Journal of Turbulence paper 6 L. Davidson LES, f k =.4 URANS, f k = wall int x Figure. Full developed channel flow. The URANS and the LES regions. x I x tr RANS f k = ū, v, w.4 < fk < LES f k =.4 2δ x L Figure 2. Embedded LES. Vertical thick line shows the interface at x I =.9. f k varies linearl in the gra area (width x tr) from to.4. δ = A term corresponding to the first term in Eq. can be added as a source term in the momentum equation as Si 2 = kū i Df k ū mū m Dt (4) 79 Multipling S 2 i b ū i and time averaging gives kū iū i Df k ū mū m Dt k Df k Dt () 8 where we assume that the correlation between k and ū i is weak. Equation is 8 equal to the time average of the first term in Eq. as it should. 82 In [3], the second term on the right side of Eq. 9 is represented b introducing an 83 additional turbulent viscosit, ν tr, in a diffusion term in the momentum equation 84 as (ν tr s ij ), s ij = ( ūi + ū ) j (6) x j 2 x j x i 8 where ν tr = P k tr s 2. (7) 86 The term, P ktr, is computed as P ktr = k tot Df k Dt (8)

7 March 3, 26 Journal of Turbulence paper Journal of Turbulence 7 P x A s Interface Figure 3. Control volume, P, in the LES region adjacent to the interface. S P ktr is an additional production term in the k equation, see Eq. 2. In [3], P ktr is re-written using Eq. 8 as P ktr = k f k Df k Dt (9) Modeling the Interface The gradient of f k across a RANS-LES interface gives rise to an additional term in the momentum equations and the k equation. In the present work, these terms are included onl when the flow goes from a RANS region to an LES region. The effect of these terms will reduce k and act as a forcing term in the momentum equations. Four different models are investigated Interface Model This is based on the approach suggested in[3]. The additional turbulent viscosit, ν tr, gives an additional production term, P ktr, in the k equation, see Eq. 9. Since we are interested in stimulating resolved turbulence in the LES region adjacent to the RANS region, onl negative values of ν tr are included. A negative ν tr means phsicall transfer of kinetic energ from modeled to resolved. It is found that the magnitude of the positive values of ν tr is actuall larger than the magnitude of the negative ones, which means that ν tr >. The negative values correspond to Df k /Dt < (see Eqs. 7, 8 and 9), i.e. when a fluid particle in a RANS region passes the interface into an adjacent LES region. However, ν tr takes such large (negative) values that ν t +ν tr <. To stabilize the simulations, it was found necessar to introduce a limit ν t +ν tr > in the diffusion term in the momentum equation. No such limit is used in the k equation, and hence P k +P ktr is allowed to go negative Interface Model 2 This model is identical to Model except that Eq. 8 is used instead of Eq. 9. k tot in Eq. 8 is defined as k tot = k + 2 ū iū i r.a (2)

8 March 3, 26 Journal of Turbulence paper 8 L. Davidson where subscript r.a. denotes running average. Is is averaged in all homogeneous directions including time(note that time is not a homogeneous direction in decaing grid turbulence, see Section 4.). Since k tot is mostl larger than k/f k [3], this approach will give a larger magnitude of the (negative) production than Model. It will be seen that this modification is of utmost importance Interface Model 3 The right side of Eq. is added to the k equation, but onl when the flow goes from a RANS region to an LES region (i.e. Df k /Dt < ), i.e. ( ) Dfk P ktr = k tot min Dt, where k tot is computed as in Eq. 2. The production term P ktr < which means that it reduces k as it should. The sum of the Si and Si 2 terms (see Eqs. 2 and 4) in the momentum equations read ( )( Dfk S i = min Dt,.+ k ū mū m r.a. (2) ) ū i (22) 223 Since Df k /Dt <, the source S i has the same sign as ū i ; this means that the 224 source enhances the resolved turbulence as it should. 22 It ma be noted that Df k /Dt in Eq. 2 assumes the correct (i.e. negative) sign 226 irrespectivel of the orientation of the RANS-LES interface. Consider, for example, 227 the RANS-LES interfaces at the lower and upper wall in full-developed channel 228 flow (Fig. ). For these interfaces, the gradient of f k in Eq. 2 reads v f k /. 229 When a fluid particles at the lower interface goes from the RANS region to the 23 LES region, v > and f k / < so that v f k / < as intended. Also for 23 the upper interface we get v f k / < since v < and f k / >. For non- 232 cartesian grids the material derivative, Df k /Dt <, has to be formulated in local 233 grid coordinates. 234 It is however found that the forcing often becomes too strong when S i is added 23 to the momentum equation. Theeffect of adding or neglecting S i in the momentum 236 equations is evaluated in Section. 237 The differences between Model 3 and 2 are that When S i =, no explicit modification is made in the momentum equation in Model 3 (recall that ν < ν tr < is used in the momentum equation in Models and 2). Model 2 (and Model ) ma need regularization in case s in the denominator of ν ttr in Eq. 8. No such regularization, however, is used in the present work. It can be argued that the commutation term in Model 3 is introduced in a more phsical wa compared to Models and 2 where artificiall negative viscosities are used Interface Model 4 This interface model was developed in [3] for horizontal interfaces. The modeled turbulent kinetic energ in the LES region adjacent to the interface is reduced b setting the usual convection and diffusion fluxes of k at the interface to zero. New fluxes are introduced in which the interface condition is set to k int = f k k RANS (f k =.4), where k RANS is the k value in the cell located in the URANS region adjacent to the interface. No modification is made for the convection and diffusion of ε across the interface. The implementation is presented in some detail below.

9 March 3, 26 Journal of Turbulence paper Journal of Turbulence 9 We write the discretized equation in the direction (see Figs. and 3) as [36] a P k P = a N k N +a S k S +S U, a P = a S +a N S P where a S and a N include to the convection and diffusion through the south and north face, respectivel, and S U and S P k P include the production and the dissipation term, respectivel. For a cell in the LES region adjacent to the interface (cell P), a S is set to zero, setting the usual convection and diffusion fluxes to zero. New fluxes, including f k, are incorporated in additional source terms as S U = (C s +D s )f k k S, S P = (C s +D s ) C s = max( v s A s,), D s = µ tota s (23) wherec s and D s denoteconvection (first-order upwind)and diffusion, respectivel, through the south face, and A s is the south area of the cell. As can be seen, the k S is multiplied b f k and hence the new convective flux is a factor f k smaller than the original one. Also the diffusion flux is smaller; it is D s (f k k S k P ) compared with the original flux D s (k S k P ). The treatment of the k equation for a vertical interface (the embedded interface in the channel flow) is exactl the same as for a horizontal interface. The difference is that an interface condition is needed also for the ε equation which reads [28] ε I = C 3/4 µ k 3/2 I l sgs, = V /3, C S =., l sgs = C S (24) where subscript I denotes inlet or interface; V is the volume of the cell. The sensitivit to the value of the Smagorinsk coefficient, C S, was investigated in [3] and was found to be weak Mean flow equations and numerical method The momentum equations with an added turbulent viscosit reads ū i t + ū jū i = δ i x j ρ p + x i x j ( (ν +ν t ) ū ) i +S i (2) x j where the first term on the right side is the driving pressuregradient in the streamwise direction, which is used onl in the full-developed channel flow simulations. The last term on the right side is the additional source term in Model 3, see Eq. 22. The effect of this term is shown in Figs. and 2. An incompressible, finite volume code is used [7]. The numerical procedure is based on an implicit, fractional step technique with a multigrid pressure Poisson solver and a non-staggered grid arrangement. For the momentum equations, second-order central differencing is used for the full developed channel flow. For the embedded channel flow, we use the second-order upwind [37] scheme in the RANS region upstream of the interface and second-order central differencing in the LES region downstream of the interface. The Crank-Nicolson scheme is used in the time domain and the first-order hbrid central/upwind scheme is used in space for solving the k and ε equations.

10 March 3, 26 Journal of Turbulence paper L. Davidson ū 2 xz..4.3 ū 2 xz t t (a) f k =.4. (b) Computed f k, Eq ū 2 xz..4.3 ū 2 xz t (c) Computed f k = k/k tot, Eq. 28; : DNS, t (d) The Dnamic Smagorinsk model. Figure 4. DHIT, decaing turbulence. : 32 3 ; : 64 3 ; : Should f k be constant or should it be computed? Thephsical meaningof f k is theratio of modeled tototal turbulent kinetic energ, see Eq. 8. Hence it is expected that it should be smaller when the grid is refined. One wa to compute f k is as proposed b [2] f k = C /2 µ ( L t ) 2/3, L t = k3/2 tot ε, = ( V)/3 (26) 282 Kenjeres and Hanjalic [38] have made a slightl different proposal which reads f k = L t (27) This expression was found to give far too small f k and hence the formula in Eq. 26 was chosen for evaluation. Since the running average of k tot is computed in Model 2 and 3, the option of computing f k from its definition is also evaluated, i.e. f k = k k tot (28) where k tot is computed using the running average (see Eq. 2); the instantaneous value of k is used. In this section we evaluate the difference between computing f k from Eq. 26 or Eq. 28, or using constant f k =.4 in the LES regions. For comparison, the results using the Dnamic Smagorinsk model [39, 4] are also included in Section 4..

11 March 3, 26 Journal of Turbulence paper Journal of Turbulence.8.8 fk xz.6.4 fk xz t t (a) Computed f k, Eq. 26. (b) Computed f k = k/k tot, Eq. 28. Figure. DHIT, computed f k. : 32 3 ; : 64 3 ; : νt xz/ν νt xz/ν t (a) f k = t (b) Computed f k, Eq. 26. νt xz/ν νt xz/ν t (c) Computed f k = k/k tot, Eq t (d) The Dnamic Smagorinsk model. Figure 6. DHIT, turbulent viscosit. : 32 3 ; : 64 3 ; : Decaing homogeneous isotropic turbulence The first test case is decaing homogeneous isotropic turbulence (DHIT). The computational domain is a cubic box with side length 2π. Three different meshes have been emploed: 32 3, 64 3 and 28 3 cells. Periodic boundar conditions are prescribed at all boundaries. The predictions are compared with experiments [4]. We have made the experimental data nondimensional using the characteristic scales, u ref = (.u rms, ) /2 and L ref = M g /(2π), where u rms, denotes the initial measured RMS fluctuation and M g is the mesh size of the turbulence generating grid in the experiments. The turbulence spectra in the experiments were measured at three downstream locations corresponding to the times tu /M g = 42, 98 and 7. The corresponding non-dimensional times in the simulations are,.87 and 2, respectivel. The initial velocit is obtained b inverse Fourier transformation using the measured spectrum at tu /M g = 42. The initial velocit field is generated b a widel used computer program from the group of Prof. Strelets in St. Petersburg. Initial

12 March 3, 26 Journal of Turbulence paper 2 L. Davidson 2 2 E E κ (a) 32 3 mesh. 4 κ (b) 64 3 mesh. 2 2 E 3 E κ (c) 28 3 mesh. κ (d) 28 3 mesh. DNS (no turbulence model). Figure 7. DHIT, Energ spectra. : f k =.4; : computed f k, Eq. 26; : Dnamic model; : computed f k = k/k tot, Eq. 28; : t =.87; : t = 2; εsgs xz.. εsgs xz t t (a) f k =.4. (b) Computed f k, Eq εsgs xz.. εsgs xz t t (c) Computed f k = k/k tot, Eq. 28. (d) Dnamic model. Figure 8. DHIT, SGS dissipation, ε sgs = τ ij ū i x j (τ ij denotes the SGS stress tensor). : 32 3 ; : 64 3 ; : boundar conditions for the turbulent quantities are set as in [27], i.e. k = f k k res (f k =.4) and ε = Cµ 3/4 k 3/2 /l sgs with the SGS length scale, l sgs, taken from the

13 March 3, 26 Journal of Turbulence paper Journal of Turbulence Smagorinsk model as l sgs = C s (C s =.). Figure 4 presents the deca of resolved turbulence versus time. The predictions are compared with two experimental data at t =.87 and 2. The deca is initiall stronger when f k is computed from Eq. 26 compared to f k =.4 but at the last measurement station (t = 2) the two approaches give similar resolved turbulence. Computing f k from Eq. 28 gives a more consistent behaviour compared to Eq. 26: the deca rate increases smoothl when the grid is refined. The deca with DNS (no model) with the 28 3 mesh is also included in Fig. 4c; DNS gives, as expected, a slightl slower deca than when computing f k from Eq. 28. The predicted deca b the Dnamic model is in slightl better agreement with experiments than all PANS models. Figure explains the strong deca with the 32 3 mesh when f k is computed using Eq. 26: the reason is that the computed f k is close to one. As expected, the finer mesh, the smaller f k. For the 28 3 mesh, Eq. 26 gives f k.4. Computing f k from Eq. 28 gives much smaller f k than Eq. 26. Figure 6 presents the turbulent viscosities. Considering the f k in Fig., it is consistent that f k from Eq. 26 gives much larger turbulent viscosit than when using Eq. 28 (with f k =.4 between the two). f k from Eq. 28 gives even a smaller viscosit than the Dnamic model on all three meshes. Furthermore, when using Eq. 26 the turbulent viscosit increases with time, especiall so for the coarse mesh. In RANS, the turbulent kinetic energ and its dissipation deca as k t /4 and ε t 9/4, respectivel, so that ν t t /4, i.e. the turbulent viscosit decreases with time. We expect that the turbulent viscosit should decrease with time also in LES; this is the case for both the Dnamic model, constant f k and using Eq. 28 (Figs. 6a,c,d), but not when computing f k from Eq. 26 (Fig. 6b). Figure 7 shows the energ spectra for the four models on the three meshes and DNS on the fine mesh. On the coarse mesh, computing f k from Eq. 26 gives slightl better agreement with experiments than f k =.4, but on the other two meshes, f k =.4 gives better or much better agreement. Comparing f k =.4 with using Eq. 28, we find that the latter model gives too small a deca at high wavenumbers. The PANS simulations in Fig. 7 can be summarized as follows: f k from Eq. 26 gives too large a deca at high wavenumbers, f k from Eq. 28 too small a deca, and f k =.4 in between the two. This is consistent with the turbulent viscosities in Fig. 6. The spectra using the Dnamic model give better agreement with experiments than all the PANS models. For the fine mesh, the spectrum using Eq. 28 gives good agreement with experiments. The PANS model is supposed to behave as DNS when f k becomes small. f k is indeed small for the finest mesh, see Fig. b. It is therefore interesting to make a comparison between f k from Eq. 28 and DNS, see Fig. 7d. It can be seen that DNS gives reasonable spectra, but the increase at high wavenumbers reveals as expected that the dissipation is insufficient, i.e. the mesh is too coarse for DNS. Nevertheless, comparison between Figs. 7c and 7d shows that the PANS simulations using Eq. 28 and DNS are fairl similar; an even finer mesh would probabl increase the similarit. Using Eq. 26, the similarit with DNS is much smaller (cf. Figs. 7b and 7d) and the reason is that the predicted f k is much larger than when using Eq. 28. However, if the mesh is strongl refined, it is expected that the PANS results using Eq. 26 will be similar to DNS since it is expected that the mesh refinement will make f k go towards zero (the mesh refinements from 32 3 to 28 3 does indeed reduce f k, see in Fig. a). Finall we plot the SGS dissipation, see Fig. 8. This quantit dissipates resolved turbulent kinetic energ. According to the theor, the cut-off is located in the /3 region. Hence, when the grid is refined/coarsened and the location of cutoff moves in wavenumber space, the SGS dissipation should in theor not change.

14 March 3, 26 Journal of Turbulence paper 4 L. Davidson U + U (a) f k =.4. (b) Computed f k, Eq U (c) Computed f k, Eq. 28. Figure 9. Full developed channel flow. Re τ = 8. Model 4 is used at the interface. Thick vertical line show the location of the RANS-LES interface. : 32 3 ; : 64 3 ; : 28 3 ; +: U + = ln( + )/ This means we would like our model to dissipate the same amount of turbulence as we refine/coarsen the mesh. In other words, the SGS model should be gridindependent. As seen in Fig. 8, the constant f k does a fairl good job in keeping ε sgs constant as the grid is coarsened. Computing f k from Eqs. 26 and 28 both give a larger spread in ε sgs when refining the grid; the Dnamics model gives a ε sgs as grid-independent ε sgs as f k =.4. It ma be noted that for the fine mesh, ε sgs using f k from Eq. 28 goes to zero for t > 2.; the reason is the small f k, see Fig. b. Often, we take it for granted that the turbulent viscosit should increase when the grid is coarsened. But in order to keep ε sgs constant when the grid is coarsened, the finite volume method must respond in an appropriate manner and reduce the resolved velocit gradients, ū i / x j so that ε sgs stas constant (for wavenumbers in the /3 region ū i / x j 2/3 and, since ε sgs, ν t must be proportional to 4/3 ). When coarsening the grid, the ke question is neither that the viscosit should increase nor that the resolved velocit gradients should decrease, but that the product of the modeled, turbulent stresses and the velocit gradient, ε sgs = τ ij ū i / x j, should sta constant Full developed channel flow at Re τ = 8 The Renolds number is defined as Re τ = u τ δ/ν where δ denotes half channel height. The streamwise, wall-normal and spanwise directions are denoted b x, and z, respectivel. The size of the domain is x max = 3.2, max = 2 and z max =.6. The mesh has 32 32, or cells in the x z planes. Details of the grids are given in Table. The location of the interface is +. f k is set to one in the RANS regions near the walls, see Fig.. In the LES region f k is either set to.4 or it is computed from Eqs. 26 or 28. Model 4 (see Section 2.2.4) is used at the

15 March 3, 26 Journal of Turbulence paper Journal of Turbulence 8 8 νt /ν 6 4 νt /ν (a) f k =.4. (b) Computed f k, Eq νt /ν (c) Computed f k, Eq. 28. Figure. Full developed channel flow. Re τ = 8. Model 4 is used at the interface. : 32 3 ; : 64 3 ; : fk.6.4 fk (a) Computed f k, Eq (b) Computed f k, Eq. 28. Figure. Full developed channel flow. Computed f k. Re τ = 8. Model 4 is used at the interface. : 32 3 ; : 64 3 ; : k + tot (a) f k =.4. k + tot (b) Computed f k. Figure 2. Full developed channel flow. Re τ = 8. Total, turbulent kinetic energ. Model 4 is used at the interface. : 32 3 ; : 64 3 ; : 28 3 ; : DNS at Re τ = 42 [42].

16 March 3, 26 Journal of Turbulence paper 6 L. Davidson Re τ + x + z + N f Table. Grids. Full developed channel flow. f denotes geometric stretching RANS-LES interfaces. A running average is used when computing k tot (Eq. 2) in the definition of f k. Since the simulations are alwas started from results obtained from earlier simulation (using, for example, another interface model), the running average is alwas started from the first timestep. The same procedure is used when Model 2 and 3 are used in the next section. The sensitivit to running averaging for k tot in evaluated in Section. Figures 9,, and 2 present the velocit profiles, the turbulent viscosities, the computed f k and the total, turbulent kinetic energ. As can be seen, when computing f k from Eq. 26 on the coarse mesh it attains values close to one. The result is a poorl predicted velocit profile, ver large viscosities ( ν t max /ν 4, not shown) in the LES region; furthermore, the resolved turbulence is zero in the entire domain. The predictions on the fine mesh are not ver good either. The velocit profile near the center on the fine mesh of the channel is somewhat overpredicted (Fig. 9b); it seems that the turbulent viscosit gets too small in the LES region, see Fig. b. Computing f k from Eq. 28 gives accurate results for the coarse grid; this is probabl rather fortuitous because the turbulence viscosit is essentiall zero in the LES region for all three meshes. On the medium and the fine mesh, Eq. 28 predicts ver poor velocit profile; the reason is the small f k (Fig. b) and consequentl a ver small turbulent viscosit (Fig. b). The results using Eq. 26 on the coarse grid ma seem somewhat surprising: we get f k (Fig. a) in the entire region, but et the solution does not correspond to a RANS solution as expected. The reason is the interface term (Model 4), which decreases the turbulent viscosit near the RANS-LES interface, see Fig. b. Without interface term, we get a stead RANS solution as expected (not shown). This is also the case when using Eq. 28: without the interface term the flow gives a stead RANS solution as it should (not shown). Contrar to computing f k from Eqs. 26 or 28, using f k =.4 in the LES region gives velocit profiles which are in good agreement with the log-law. It ma be noted that the turbulent viscosit stas constant as the grid is refined; the turbulent viscosit is grid-independent. The is probabl a big advantage when using the model in flow with complex geometries where the cell size ma var strongl. The total, turbulent kinetic energ agrees fairl well with DNS on Re τ = 42 for all three meshes, however it is somewhat overpredicted in the RANS region Results In the previous section is was found that a constant f k =.4 in the LES region is superior compared with computing f k from Eqs. 26 or 28. In the literature, there are several proposals on how to compute f k. The proposals b Basara et al. [2] and Kenjeres and Hanjalic [38] are given in Eqs. 26 and Eq. 27, respectivel. Other formulations are proposed b Girimaji and Abdol-Hamid [44] who compute it as 3( min /L t ) 2/3 where min is the smallest grid cell size and L t = (k +k res ) 3/2 /ε. Foroutan and Yavuzkurt[4] derives an expression from the energ spectrum which

17 March 3, 26 Journal of Turbulence paper Journal of Turbulence 7 3 P + k +P+ ktr U (a) Velocit. +: U + = ln( + )/ (c) Production terms. : the location of the computational cell centers. (νt +νtr)/ν τ 2 ν+ τ+ 2, ū v + + (b) Turbulent viscosit. : the location of the computational cell centers.... (d) Resolved (upper lines) and viscous plus modeled (lower) shear stresses. Figure 3. Full developed channel flow. N k = 32. Re τ = 4. : Model. : Model 2. : Model 3, S i = ; : no interface model. 3 U + 2 (νt +νtr)/ν P + k +P+ ktr + (a) Velocit. +: U + = ln( + )/ (c) Production term. : the location of the computational cell centers. τ 2 ν+ τ+ 2, ū v + + (b) Turbulent viscosit. : the location of the computational cell centers.... (d) Resolved (upper lines) and viscous plus modeled (lower) shear stresses. 427 reads Figure 4. Full developed channel flow. Re τ = 8. For legend, see Fig. 3. [ (Λ/ ) 2/3 ] 9/2 f k =.23+(Λ/ ) 2/3 (29)

18 March 3, 26 Journal of Turbulence paper 8 L. Davidson 3 U S + i 2 P + k +P+ ktr 8 + (a) Velocit. +: U + = ln( + )/ (c) Production term. τ 2 ν+ τ+ 2, ū v (b) Momentum sources S i, see Eq. 22. : S ; : S 2 ; : S 3... (d). : Viscous plus modeled shear stress; : resolved stress. Figure. Full developed channel flow. Re τ = 8. Model 3. S i. : the location of the computational cell centers. Thick vertical line show the location of the RANS-LES interface. 3 U + 2 (νt +νtr)/ν P + k +P+ ktr + (a) Velocit. +: U + = ln( + )/ (c) Production term. : the location of the computational cell centers. τ 2 ν+ τ+ 2, ū v + + (b) Turbulent viscosit. : the location of the computational cell centers.... (d) Resolved (upper lines) and viscous plus modeled (lower) shear stresses. Figure 6. Full developed channel flow. Re τ = 2. : Model. : Model 2. : Model 3, S i = ; : Model 4.

19 March 3, 26 Journal of Turbulence paper Journal of Turbulence 9 (ν +νt)/(uτδ) U + 2 Re τ = 8 Re τ = (a) Velocit. +: U + = ln( + )/ Re τ = 2 Re τ = 8 Re τ = 4 Re τ = 2 + (c) Viscosit (ν +νt)/(uτδ).2 τ 2 + +τν+ 2 Re τ = 8 Re τ = 4 Re τ = Re τ = 2 (b) Viscosit Re τ = 8 Re τ = (d) Viscous plus modeled shear stress Figure 7. Full developed channel flow. Model 4. Re τ = 2 (lower lines), Re τ = 4 (middle lines), and Re τ = 8 (upper lines). Vertical dashed thick lines show the location of the interfaces. : + interface = 2; : + interface = ; : + interface = ; : D RANS using AKN model [43]. τ 2 + τν+ 2, ū v +... Figure 8. Full developed channel flow. Re τ = 2, + interface = 2. Resolved (upper lines) and viscous plus modeled (lower) shear stresses. For legend, see Fig It ma that Eq. 26 or an other formulation could be better calibrated, but that is out of the scope of the present work. In this section we will evaluate the four interface models presented in Section 2.2 in two flows, full developed channel flows (at different Renolds numbers) and embedded LES. In the LES region we set f k = Full Developed Channel Flow Simulations at different Renolds numbers are made, Re τ = 2, Re τ = 4 and Re τ = 8. The size of the domain is x max = 3.2, max = 2 and z max =.6. The mesh has cells in the x z planes. A simulation with twice as large domain in the x z plane (x max = 6.4 and z max = 3.2) with cells was also made for Re τ = 4, and identical results were obtained as for the smaller domain. Details of the grids are given in Table. The location of the interface is

20 March 3, 26 Journal of Turbulence paper 2 L. Davidson Re τ = 4 Figure 3 presents the velocit profiles, the turbulent viscosities, the production terms in the k equation and the shear stresses. The vertical thick dashed lines show the location of the interface. The momentum source in Model 3, S i (see Eq. 22) is set tozero. Model2and3giveavelocit profileingoodagreement withthelog-law. Model does not sufficientl reduce the turbulent viscosit in the LES region and the result is that the turbulent viscosit is much too large but somewhat smaller than no interface model. With Model 2, the magnitude of the additional turbulent viscosit, ν tr, near the interface in Fig. 3b is actuall larger than ν t, which makes their sum, ν t + ν tr, go negative. Recall that in the momentum equations ν t +ν ttr is not allowed to go negative; it is negative onl in P ktr, see Section As a consequence of the large, negative ν ttr, P k +P ktr < at the interface, see Eq. 3c. The production terms in Model 2 and 3 are ver similar, which is expected since ν tr in Model 2 is defined from P ktr, see Eqs. 7 and 9. The shear stresses in Fig. 3d show that without interface condition the flow goes stead. The reason is that the turbulent viscosit in the LES region is too large because there is no commutation term to reduce k. Note that to enhance the readabilit of Fig. 3d, the viscous and modeled shear stresses are plotted with negative sign. Recall that the total shear stress, τ tot, will alwas irrespectivel of model obe the linear law τ tot =, τ tot = τ ν 2 +τ 2 ū v = (ν +ν t ) ū ū v (3) With Model the flow is neither in RANS mode nor in LES mode, but it ends up in being in the middle (modeled and resolved stresses are both rather large), which is a most unfortunate condition for a hbrid LES-RANS model. The stresses predicted b Models 2 and 3 are virtuall identical. The turbulent viscosit is too large with Model (Fig. 3b). If f k is reduced to.3, the turbulent viscosit is reduced (not shown) because the gradient of f k across the interface increases; furthermore, a smaller f k gives a smaller turbulent viscosit in the LES region which promotes growth of resolved turbulence. With f k =.3, Model gives a velocit profile that is almost identical to that of Model 2; the maximum streamwise fluctuation, ū ū max, is % larger with Model. If f k is reduced to.3 also in Model 2 (not shown), the prediction with the two models are even closer to each other ( ū ū max is a couple of percent larger with f k =.3 compared to f k =.4)...2. Re τ = 8 Figure 4 shows the same quantities as Fig. 3, but now for Re τ = 8. Again, Models 2and3give goodvelocit profiles.modelandthe nointerface condition give rather good velocit profiles although the velocit is somewhat too high in the LES region. The reason for the larger velocit levels is seen in Fig. 4b which presents the turbulent viscosities. The no interface model does not manage to reduce ν t in the LES region; also Model is much less efficient in reducing ν t than are Models 2 and 3. This is also clearl seen in the shear stresses (Fig. 4d) which show that the transition from RANS to LES is fastest for Models 2 and 3; the transition is much slower for Model and for no interface condition. The production of k becomes negative for Models 2 and 3 as for Re τ = 4. It can be noted that the magnitude of the negative production at the interface for both Re τ = 4 (Fig. 3c) and Re τ = 8 (Fig. 4c) is slightl larger with

21 March 3, 26 Journal of Turbulence paper Journal of Turbulence Model 2 than Model 3. The reason is probabl that Model 2 reduces the turbulent 488 viscosit in the momentum equations b ν ttr (but its magnitude is not allowed to 489 exceed ν, i.e. ν tr > ν when ν tr < ) which slightl reduces the damping of the 49 resolved fluctuation which in turn increases the fluctuating velocit gradients and 49 the magnitude of P k. It could be noted that if f k is reduced to.3, Model gives 492 virtuall identical velocit profile and resolved stresses as Model Figure presents results where the momentum source S i (Eq. 22) has been 494 activated. It can be seen that the predicted velocit profile is slightl worse than 49 without momentum source term (Fig. 4a). The source terms in the momentum 496 equation are as is P ktr ver large; recall that the driving pressure gradient 497 (the first term on the right side of Eq. 2) is one. It ma at first seem somewhat 498 surprisingthat the source term S i is non-zero since it includes the time average of a 499 resolved fluctuation which shouldbezero, i.e. u i =. Thereason is that S i, at the lower RANS-LESinterface for example, is addedonlwhendf k /Dt = v f k / x 2 <, i.e. when v > (recall that at the lower RANS-LES interface f k / x 2 < ); the 2 average of the samples of ū i when v > is not zero. 3 It ma be noted that the magnitude of the resolved shear stress near the RANS- 4 LES interface exceeds one (see Fig. d) and hence it is larger than the wall shear stress. This seems to violate Eq. 3. However, when S, the right side of Eq. 3 6 includes also the integral of S, i.e. the global force balance reads δx max z max 2τ w x max z max + S dxddz = (3) V where the first term is the driving pressuregradient (the firstterm on the right side of Eq. 2). Since S <, see Fig. b, the wall shear stress, τ w, is reduced when S is added and this explains wh the magnitude of the resolved shear (scaled with τ w ) exceeds one...3. Re τ = 2 Figure 6 presents the same quantities as Fig. 3, but now for Re τ = 2. The results of Model 4 are included here; it was shown in [3] that this interface model gives good results for Re τ = 4 and 8. Model 4 predicts a good velocit profile also for this Renolds number. However, Models, 2 and 3 fail; the all give a stead RANS solution. The reason is that the interface conditions do not manage to decrease the modeled turbulent kinetic energ and hence not the turbulent viscosit on the LES side of the interface. Obviousl it is more difficult for the flow to switch from RANS to LES at Re τ = 2. The same effect is seen if the mesh in the wall-parallel plane is refined to cells (not shown). One difference between the Re τ = 2 and the higher Renolds numbers is that in the former case the resolved stress is much smaller at the interface...4. Failure of Models 3 24 In order to find the reason for this failure of Models 3, the sensitivit to 2 the location of the interface was investigated. Figure 7 presents the velocities, 26 turbulentviscosities andtheshearstressesusingmodel4forre τ = 2, 4 and 27 8 for different locations of the interface. The velocit profiles are well predicted for all cases expect for Re τ = 8 and + 28 interface = 2. Note that the turbulent viscosities are plotted using outer scaling (i.e. the are scaled with friction velocit, u τ, and half channel width, δ) contrar to Figs. 3, 4 and 6 where inner scaling is used (i.e. scaled b ν). The results from a one-dimensional RANS simulation using theaknmodel[43](i.e. PANSwithf k = ) arealsoincluded.whenplottedversus +, the RANS turbulent viscosit is virtuall independent of Renolds number and

22 March 3, 26 Journal of Turbulence paper 22 L. Davidson u v +, v rms,w +2 rms,u rms (a) Renolds stresses. Markers: EARSM. : u v + ; : u +2 rms; : w rms; +2 : u +2 rms Bww(ẑ) ẑ (b) Two-point correlation. Markers denote location of cell centers. Figure 9. Embedded LES. Channel flow. Added snthetic Renolds stresses at the vertical RANS-LES interface. 34 hence all the green dotted lines in Fig. 7c collapse. Actuall this is also the case 3 for the PANS simulations when the interface is defined in outer scaling (i.e. in ): 36 in Fig. 7b, the turbulent viscosit in the URANS region for {Re τ = 2, + interface = 2}, and {Re τ = 4, + 37 interface = } and {Re τ = 8, + 38 interface = } (i.e. =.2) {Re τ = 2, + interface = }, and {Re τ = 4, + 39 interface = } (i.e. 4 =.2) {Re τ = 4, + interface = 2}, and {Re τ = 8, + 4 interface = } (i.e. = 42.62) 43 are ver similar. As a result, this applies also for the viscous plus modeled shear 44 stresses, see Fig. 7d. It is interesting to note that the viscosit in the RANS region 4 in the PANS simulations is much smaller than that in the D RANS simulation. 46 The reason is, as noted in [3], that low modeled k values are transported from the 47 LES region into the RANS region near the lower wall when v < (ε in the LES 48 region in the two simulations are ver similar [3]). 49 Now we need to address the question wh Models -3 fail at Re τ = 2. The interface location was moved awa from the wall to + interface, but the same results were obtained (not shown). Next, the interface location was moved closer to the wall, to + 2 interface 2. Model -3 still fail to predict a good velocit profile 3 (not shown). It was mentioned above that the reason for the poor prediction of the 4 velocit profile at Re τ = 2 could be that the resolved shear stress is too low at the interface, and that the low level of resolved turbulence in the interface region 6 would make the transition from RANS mode to LES mode difficult. However, this 7 theor does not hold when we look at the resolved shear stresses at the interface 8 in Fig. 8 which for Re τ = 2 are as large or larger than those at Re τ = 8 9 for which good velocit profiles are obtained, see Fig. 4. The reason for the poor 6 prediction at Re τ = 2 is that the turbulent viscosit (in outer scaling) and 6 consequentl the modeled shear stress is much larger than at the higher Renolds 62 number, see Figure 7b-d. Because of this, Models -3 do not manage to sufficientl 63 decreaseν t inthelesregionadjacenttotheinterface. Largemodeledshearstresses 64 at the interface (Fig. 7d) are also the reason for the somewhat poorl predicted 6 velocit profiles (Fig. 7a) using Model 4 at Re τ = 4 and 8 with the 66 interface is located at + = 2.