The partially averaged Navier-Stokes (PANS) model, proposed by Girimaji [1], can be used 11 to simulate turbulent flows either as RANS, LES or DNS. Its main parameter is f k whose 12 physical meaning is the ratio of the modeled to the total turbulent kinetic energy. In RANS 13 f k = 1, in DNS f k = 0 and in LES f k takes values between zero and one. 14 Three different ways of prescribing f k are evaluated for decaying grid turbulence and fully-15 developed channel flow: f k = 0.4, f k = k 3/2

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... t /ε [2] and from its definition f k = k/ktot where 16 ktot is the sum of the modeled, k, and resolved, kres, turbulent kinetic energy. It is found 17 that the f k = 0.4 gives the best results. 18 In [3], a method was proposed to include the effect of the gradient of f k . This approach is 19 used at RANS-LES interface in the present study. Four different interface models are evaluated 20 in fully developed channel flow and embedded LES of channel flow; in both cases, PANS is 21 used as a zonal model with f k = 1 in the URANS region and f k = 0.4 in the LES region. In 22 fully developed channel flow, the RANS-LES interface is parallel to the wall (horizontal) and 23 in embedded LES it is parallel to the inlet (vertical). 24 The importance of the location of the horizontal interface in fully developed channel flow is 25 also investigated. It is found that the location -and the choice of the treatment at the interface 26 -may be critical at low Reynolds number or if the interface is placed too close to the wall. 27 The reason is that the modeled turbulent shear stress at the interface is large and hence the 28 relative strength of the resolved turbulence is small. In RANS, the turbulent viscosity -and 29 consequently also the modeled Reynolds shear stress -is only weakly dependent on Reynolds 30 number. It is found in the present work that that also applies in the URANS region. 31 RANS-LES interface condition 33 1. Introduction 34 Large Eddy Simulation (LES) is very expensive in wall-bounded flow. To be able 35 to extend LES to high Reynolds number many proposals have been made in the 36 literature to combine LES with unsteady RANS (URANS) near the walls. The first 37 and most common method is Detached Eddy Simulation (DES) [4-6]. Later, other 38 researchers have proposed hybrid LES/RANS [7, 8] and Scale-Adapted Simulations 39 (SAS) [9, 10]; for a review, see [11]. DES and hybrid LES/RANS use the cell size 40 as the SGS length scale. SAS does not use the cell size (except as a limiter) but 41 uses the von Kármán lengthscale instead. 42 The main object of the methods mentioned above is that they reduce the turbu-43 lent viscosity in the LES region. There are three options to achieve this. 44 (1) The turbulent viscosity, ν t , is reduced by modifying its formulation; 45 (2) The dissipation term in the equation for the modeled, turbulent kinetic 46 Journal of Turbulence paper 2 L. Davidson energy, k, is increased by decreasing the turbulent lengthscale, or 47 (3) the destruction term in the lengthscale equation (ε or ω) is decreased. This 48 increases the dissipation term in the k equation as well as decreases the 49 turbulent viscosity directly since ε (or ω) appears in the denominator of 50 the expression for ν t . 51 DES [12] is based on the first option. X-LES [13] and the one-equation hybrid 52 LES-RANS [8, 14] use option number two: ν t is decreased and the dissipation term 53 in the k equation is increased. The third option is used in the partially averaged 54 Navier-Stokes (PANS) model [1] and the Partially Integrated Transport Model 55 (PITM) [15, 16]. PANS is used as a zonal LES-RANS method in the present work 56 where RANS is used near the walls and LES is employed further away from the 57 walls. We will also use PANS in embedded LES. 58 Much work has been presented on embedded LES lately. Menter et al. [17] present 59 embedded LES of channel flow and the hump flow. They use the SST RANS model 60 upstream of the embedded LES interface. Downstream of the interface they use the 61 model of Shur et al. [6] which consists of the Smagorinsky model in the LES region 62 and a mixing-length model in the RANS region. At the interface, the modeled 63 turbulence is converted into resolved turbulence using synthetic turbulence from a 64 vortex method [18]. 65 Shur et al. [19] proposed a new recycling method in a interface zone between 66 RANS and LES. The RANS and LES zones overlap and identical grids are used in 67 the overlap region. Only the fluctuating components are recycled since the mean 68 components are available in the entire overlap zone. They evaluated the method 69 for flat plate boundary flow and flow over a two-dimensional airfoil. 70 Poletto et al. [20] made embedded LES of the hump flow. They used Delayed 71 Detached Eddy Simulation (DDES) in the entire region (both upstream and down-72 stream of the interface). At the LES interface they added synthetic fluctuations 73 from the divergence free synthetic eddy method [21]. 74 Gritskevich et al. [22] used the Improved Delayed Detached Eddy Simulation 75 (IDDES) to make embedded LES of the hump flow. They first computed the entire 76 flow with 2D RANS. Then they used the RANS solution to prescribe the inlet 77 mean velocity profile at the embedded LES interface. The modeled k was taken 78 from the RANS solution and ω was computed from k and the grid size. Synthetic 79 fluctuations were generated with a synthetic turbulence generator [23]. 80 Shur et al. [24] present embedded LES of the flow over a hump. They used over-81 lapping RANS and IDDES in the embedded interface region. At the interface they 82 added synthetic turbulence. They show very good agreement with measurement 83 data. 84 Xiao and Jenny [25] presented a novel method for treating the RANS-LES inter-85 face in hybrid LES-RANS. They solved the RANS and LES equations in the entire 86 domain. The RANS mesh in the LES region does not need to be the same as the 87 LES mesh and vice versa. Drift forces were added in the overlapping regions. 88 As mentioned above, the PANS model is in the present work used both as a 89 zonal hybrid LES-RANS model. f k is set to one in the RANS regions, and it is 90 set to 0.4 in the LES region. The interface between the RANS and LES regions is 91 defined along a grid plane, an approach also chosen in ZDES [26]. As an alternative 92 to using a constant f k in the LES region, f k can be computed using the cell size 93 and the integral length scale, k 3/2 tot /ε [2], where k tot is the sum of the resolved 94 and the modeled turbulent kinetic energy, i.e. k tot = k res + k. The reason why f k 95 is not computed in the present study is that it has been shown that a constant 96 f k = 0.4 works well in the LES region [27-30]; it was even shown in [30] that 97 using constant f k = 0.4 works better than computing f k , not to mention that 98 May 9, 2019 Journal of Turbulence paper Journal of Turbulence 3 it is numerically more stable. Section 4 compares three alternatives (f k = 0.4, 99 computing f k according to the equation above, and computing f k from its definition 100 k/k tot ) for decaying homogeneous grid turbulence and fully developed channel flow. 101 It should be mentioned that Basara et al. [2] have shown that computing f k works 102 well when using the k − ε − ζ − f model. 103 In [3], a method was proposed to include the effect of the gradient of f k . The 104 gradient appears because the PANS filtering operation does not commute with the 105 the spatial gradient. At RANS-LES interfaces, the spatial gradient of f k is large. 106 The approach of Girimaji and Wallin [3] is further developed in the present study 107 and it is used at RANS-LES interfaces. 108 The appearance of the gradient of f k at RANS-LES interfaces is similar to the 109 method for treating RANS-LES interfaces in Hamba [31]. He shows that when the 110 filter size varies, an additional term appears because the filtering and the spatial 111 gradients do not commute. The divergence of a flux, q i , for example, will include 112 an additional term 113 127 modeled turbulence. Hence, the term can be seen as a forcing term which represents 128 backscatter. It is used to stimulate growth of the resolved turbulence in the LES 129 region adjacent to the RANS region. The present method reduces the gray area 130 problem described in [35]. 131 The paper is organized as follows. The next section describes the equations and 132 the interface models. The following section describes the numerical method. Then 133 there is a section comparing three different ways of treating f k , followed by a section 134 which presents and discusses the results. The final section presents the conclusions. 135 May 9, 2019 Journal of Turbulence paper 4 L. Davidson 2. The PANS Model 136 The low-Reynolds number partially averaged Navier-Stokes (LRN PANS) turbu-137 lence model reads [27] 138