COMPARISON OF TURBULENCE MODELS FOR SIMULATING FLOW IN WATERJETS

Transcription

1 COMPARISON OF TURBULENCE MODELS FOR SIMULATING FLOW IN WATERJETS X. Luo, B. Epps, C. Chryssostomidis and G. Karniadakis MITSG -08 Sea Grant College Program Massachusetts Institute of Technology Cambridge, Massachusetts 0239 NOAA Grant No. NA06OAR47009 Project No M/PM- Reprinted from Proceedings of the th International Conference on Fast Sea Transportation, Honolulu, HI, 20

2 th International Conference on Fast Sea Transportation FAST 20, Honolulu, Hawaii, USA, September 20 Comparison of Turbulence Models for Simulating Flow in Waterjets Xian Luo, Brenden Epps, Chryssostomos Chryssostomidis and George Em Karniadakis,2 Design Laboratory, MIT Sea Grant, MIT 2 Division of Applied Mathematics, Brown University ABSTRACT We have developed a fast numerical algorithm for simulating flows in complex moving domains. The new hybrid smoothed profile/spectral element method exhibits high-order accuracy as well as great computational efficiency as it removes the tyranny of mesh generation. Here we extend this work by incorporating a variational multiscale large eddy formulation for modeling the subgrid turbulent scales. We present verification and validation studies of the combined method and compare different turbulence modeling methodologies. Subsequently, we apply it to study laminar, transitional and turbulent flows in an axial-flow waterjet propulsion system (ONR AxWJ-). The robustness and efficiency of our methods enable parametric studies of many cases, which may aid greatly in the early stages of design and evaluation of such propulsion systems. KEY WORDS: waterjet, CFD, high-order methods, turbulence modeling. INTRODUCTION Computational fluid dynamics (CFD) tools can now be used effectively for the design and optimization of waterjets for ship propulsion. While standard CFD approaches may be inefficient in simulating accurately the complex viscous interactions inside a watejet system, recent advances in immersive boundary methods have allowed for much faster simulations without the need for frequent re-meshing, hence eliminating a large computational bottleneck in analysis. For example in [] we developed a smooth profile method (SPM) to represent all the moving parts of a waterjet system using fixed Eulerian grids (see figure ), which in conjunction with the high-order accuracy of spectral element discretizations resulted in accurate and efficient simulations. However, without the use of a turbulence model we were limited to a relatively low Reynolds number regime. For some numerical simulations of waterjets, potential flows are assumed with limited viscous corrections, e.g. based on a two-dimensional integral boundary layer analysis [2]. There have been some RANS solvers applied to waterjet simulations, but numerical simulations of the interaction between rotor and stator in a fully unsteady manner are too complicated and computationally expensive. So many assumptions have been made, e.g. the rotor and stator problem is decoupled and the flow is rotationally cyclic so that one can model a single blade passage only [3]. Recently, in [4] we presented the first 3D simulation results by combining SPM with an unsteady RANS (URANS) approach using the Spalart- Allmaras (SA) turbulence model to account for the subgrid stresses [5, 6]. In the present work we continue on developing higher fidelity turbulence models for the SPM/spectral element method and focus, in particular, on formulating largeeddy simulations models for waterjet systems. Figure. Waterjet AxWJ-: Geometry description for the waterjet we consider in this study; SPM models the rotor and stator subdomains. In classical LES, large- and small-scale motions are separated by applying a spatial filtering operation to the Navier- Stokes equations before discretization. Hence, there are two levels of approximation, filtering and truncation, and they both contribute to the overall modelling error. Filtering may be explicitly carried out or only implicitly assumed. The result is a set of equations for the large-scale motion. The residual motion, i.e., motions on scales that are smaller than the filter width, appear in these equations as a residual stress term. This term is a priori unknown, and although the intention is not to describe the residual motion in itself, it must be modeled explicitly to incorporate the effect of the residual motion on the resolved scales. There are several technical issues in filter-based LES that have to be addressed. For instance, filtering and spatial differentiation do not, in general, commute on bounded domains or for non-uniform grids, so careful analysis is needed to obtain the correct form of the equations. It is also not obvious how to prescribe correct boundary conditions for the filtered velocity at solid walls. Another issue with the filter-based LES models is that the residual stress model may adversely affect the resolved part of the energy spectrum. These questions have been the subject of a considerable amount of research in the last two decades. For example, Carati et al. [7] have analyzed the error contributions from filtering and truncation and found that the filtering error can be expressed in terms of the resolved velocity field and

3 that the purpose of the subgrid model is to account for the effect of the unresolved motion, i.e. the truncation errors. Here we consider a different approach to LES, the variational multiscale (VMS) LES method originally proposed by Hughes et al. [8]. This formulation is based on the variational (or weak) form of the Navier Stokes equations in unfiltered form and it fits naturally within the spectral element framework. Hence, there is no filtering error and it is evident that the role of the model term is to compensate for the truncation errors, fully in line with the analysis in [7]. The VMS-LES approach involves two steps: (i) a scale partitioning of the solution space, and (ii) variational projection of the Navier- Stokes equations onto the different scale ranges. Since the scale partitioning operators do not act on the governing equations, the same way as a filter does, commutation between scale partitioning and differentiation is not an issue. In the next section we present an overview of the SPM spectral element method and in section 3 we present a validation test for flow past a NACA002 airfoil. In section 4 we formulate the VMS-LES approach and in section 5 we present comparisons of results based on URANS and LES. We conclude in section 6 with a brief summary. 2. SMOOTHED PROFILE METHOD (SPM) SPM represents the moving bodies by smoothed profiles, or the so-called indicator functions, which equal unity inside the moving domains, zero in the fluid domain, and vary smoothly between one and zero in the solid-fluid interfacial domain. In [9] we proposed a general form, which is effective for any domain shape such as rotors, i.e., φ i (x,t) = [ tanh( d ] i(x,t) )+, () 2 ξ i where index i refers to the i th moving body (e.g., a single blade of rotor or stator). Also, ξ i is the interface thickness parameter and d i (x,t) is the signed distance to the i th moving body with positive value outside and negative inside. For simple geometries (cylinders, ellipsoids, etc.) d i (x,t) can be obtained analytically. However, for general complex shapes, such as impellers which can be represented by many surface point coordinates, spline interpolations are used to calculate d i (x,t) and thus φ i (x,t). A smoothly spreading indicator function is achieved by summing up the indicator functions of all the N p non-overlapping moving bodies: φ(x,t) = N p i= φ i(x,t). Based on this indicator function, the velocity field of the moving bodies, u p (x,t), is constructed from the rigid-body motions of each moving domain: φ(x,t)u p (x,t) = N p i= {V i (t)+ω i (t) [x R i (t)]}φ i (x,t), (2) where R i, V i = dr i dt and ω i are spatial positions, translational velocity and angular velocity of the i th moving body, respectively. The total velocity field is then defined by a smooth combination of both the velocity field of moving bodies u p and the fluid velocity field u f : u(x,t) = φ(x,t)u p (x,t)+( φ(x,t))u f (x,t). (3) We see that inside the moving domains (φ = ), we have u = u p, i.e., the total velocity equals the velocity of the moving body. At the interfaces (0 < φ < ), the total velocity changes smoothly from the body velocity u p to the fluid velocity u f. SPM imposes the no-penetration constraint on the surfaces of the simulated moving bodies. It can be shown (ref. [9]) that imposing the incompressibility condition of the total velocity u = 0 ensures the no-penetration surface condition ( φ) (u p u f ) = 0, and vice versa. SPM solves for the total velocity, u, in the entire domain D, including inside the moving domains, using the incompressible Navier-Stokes equations with an extra force density term, i.e., u t +(u )u = ρ p+ν 2 u+f s in D (4a) u = 0 in D, (4b) where ρ is the density of the fluid, p is the pressure field, ν is the kinematic viscosity of the fluid, and the fluid solvent is assumed to be Newtonian with constant viscosity for simplicity. Here f s is the body force density term representing the interactions between the moving bodies and the fluid. SPM assigns R t f sdt = φ(u p u) to denote the momentum change (per unit mass) due to the presence of the moving bodies. Thus, at each time step the flow is corrected by a momentum impulse to ensure that the total velocity matches that of the rigid domains within the moving domain, hence enforcing the rigidity constraint. To numerically solve equations (4), we developed a highorder temporal discretization [9] instead of the original fullyexplicit scheme [0]. We introduced a semi-implicit treatment, using a stiffly-stable high-order splitting (velocitycorrection) scheme []. In particular, the viscous term is treated implicitly and the order of the time integration scheme is up to third. This choice enhances the stability and also increases the temporal accuracy of the original SPM implementation. The hydrodynamic force F h and torque Q h on the moving bodies exerted by the surrounding fluid are derived from the momentum conservation. Specifically, the momentum change in the moving domains equals the time integral of the hydrodynamic force and the external force, and hence: Z n F hi = t n Q hi = Z t D D ρφ n+ i (u u n p)dx (5a) r n+ i [ρφ n+ i (u u n p)]dx (5b) where the indices n,n + refer to the solutions at different time steps, u is the intermediate velocity field in the splitting scheme, and r n+ i is the distance vector from the rotational reference point on the i th moving body to any spatial point x. For spatial discretization, we apply the spectral element method (see []). This hybrid method benefits from both finite element and spectral methods: on one hand, for domains with complex geometry, we can increase the number of subdomains/elements (h-refinement) with the error in the numerical solution decaying algebraically. On the other hand, with fixed elemental size we can increase the interpolation order

4 within the elements (p-refinement) to achieve an exponentially decaying error, provided the solutions are sufficiently smooth throughout the domain. Hence the spectral element method exhibits dual path to convergence, and we can verify convergence without re-meshing. Furthermore, the use of smooth profiles in SPM preserves the high-order numerical accuracy of the spectral element method. For complex geometry domains with moving subdomains as the waterjet system, the spectral element method allows us to accurately represent arbitrary fixed rigid boundaries of the flow domain while SPM allows us to represent efficiently the moving/complex subdomains, e.g. impellers and stators. Table. NACA002 airfoil: average drag coefficient C D and lift coefficient C L for various attack angles α and Reynolds numbers Re. Re α C D C L SPM DNS experiment [2] SPM e 7 DNS e 8 experiment [3] SPM DNS experiment [3] SPM DNS experiment [3] SPM DNS VALIDATION OF SPM As the rotor and stator blades in waterjets have crosssections that are shaped like airfoils, we consider unsteady incompressible flows around a stationary NACA002 airfoil in order to validate our numerical methods. Our numerical tests suggest that good resolution near the wedge-shaped trailing edge is required and also that a non-uniform interface thickness over the airfoil should be used. To this end, we use an average of ξ = 0.00 with 2ξ for the leading edge and 0.25ξ for the trailing edge. Fig. 2 shows SPM results of instantaneous vorticity contour for the case of Re = ρu c/µ = 20,000 at various angles of attack α. Qualitative agreement is shown when compared to results of DNS with body-conforming meshes. Also, we calculate the average drag coefficient C D = Drag /2ρU bc and 2 lift coefficient C L = Lift /2ρU 2bc, where Drag = F x cos(α) + F y sin(α),lift = F x sin(α)+f y cos(α). In table we compare our SPM simulation results with DNS results and experimental data. Our simulations show that C D is smaller and C L is larger for increasing Re in the given Re range, which is confirmed by both experimental measurements [3] and DNS. In general, there is good agreement among all three sets. Figure 2. Flow past airfoil NACA002 at Re=20,000: instantaneous vorticity contours and streamlines for angle of attack α = 0 by SPM (upper left), α = 0 by SPM (upper right), α = 20 by SPM (lower left) and α = 0 by DNS (lower right). 4. TURBULENCE MODELING We performed simulations of the waterjet using three different approaches: () Unsteady Reynolds Average Equations or URANS based on the Spalart-Allmaras (SA) turbulence model to account for the subgrid stresses. SA is a transport equation for a viscosity-like variable ν, which is used to calculate the turbulent eddy viscosity ν t in the URANS equation. Implementation details and verification studies can be found in [4]. (2) Spectral vanishing viscosity (SVV) for the stabilization of spectral element methods applied to the solution of the incompressible Navier-Stokes equations. SVV was introduced in [4], and it adds a small amount of dissipation in order to control high-wavenumber oscillations associated with discontinuities arising at the domain boundaries or due to under-resolution. (3) Variational Multiscale LES or VMS- LES that we will present in some detail below. 4.. Variational Multiscale-Large Eddy Simulations (VMS-LES) We implement the VMS-LES methodology in conjunction with the modal version of the (Galerkin) spectral element method, which is a natural setting; an implementation using the previous generation of nodal spectral elements was presented in [5]. In the VMS-LES the governing equations are projected onto an a priori scale partitioning of the solution space. We write the solution space ϒ as a disjoint sum: ϒ = ϒ ϒ ϒ. Here ϒ and ϒ comprise the large and small resolved scales, respectively, whereas ϒ refers to the unresolved scales. Decomposing the test function U ϒ and trial functions W ϒ in these spaces gives: U = U +Ũ +Û, W =W + W +Ŵ. We employ a Galerkin formulation and construct the variational formulation by choosing test and trial functions in the same function space ϒ. We can develop a set

5 of scale-projected equations from the projections into different scale ranges, as follows: W,N(U) = W,F, W,N(U) = W,F, Ŵ,N(U) = Ŵ,F, (6) where N(U) is the operator W,N(U) = w, u t w, p + s w,2ν s u + w,uu, and W,F = w,f with f as the body force. Here s u = 2 ( u i x j + u j x i ). The following assumptions were made in VMS-LES [6]: () The separation in wavenumber space between large and unresolved scales is sufficiently large so that there is negligible direct dynamic influence from the unresolved scales on the large scales; (2) the dynamic impact of the unresolved scales on the small scales is on the average dissipative in nature. With these assumptions, we add the large- and smallscale projections of the Navier-Stokes equations and the following equation is derived [6, 8]: W,N(Ũ) + s w,2ν T s ũ = W,F. (7) Here all the original terms coupling the large and small resolved scales are retained, whereas the small-scale projection equation has been supplemented with a dissipative term on the left-hand-side that accounts for the interactions between the small and the unresolved scales. We are primarily concerned with the complete resolved solution Ũ = U +Ũ, not with the large and small scales per se. Since the product of a symmetric and an anti-symmetric tensor is zero, the model term in the form given in (7) reduces to: w,2ν T s ũ = w x j,ν T ũ i x j + w x j,ν T ũ j x i. (8) of the (non-commuting) scale extraction and differentiation operator; (2) the use of the symmetric gradient; (3) the form of the small-scale extraction operator; and (4) the difference between ε N SVV = O(/N) and ν T. We note that ν T contains a /N 2 -factor (through the term 2 ) and a velocity gradient, which makes it variable throughout the domain. However, it should be noted that apart from the sequence of the operators, the SVV method could have been formulated within the VMS-LES formalism. The principle of basing the scaledependent operators on the local Jacobi modal representation on each element is also common to both methods Verification of VMS-LES To illustrate the stabilizing effect of VMS-LES, we consider the two-dimensional double shear layer problem examined in the context of under-resolution [7]. The coupling between the velocity components, introduced by the second term in RHS, are handled by including the cross terms in the explicit part of the time splitting, leaving the Helmholtz solvers for the velocity components uncoupled. The first term in RHS, which is symmetric, can be treated either implicitly (added into the viscous substep) or explicitly. Implicit treatment is computationally more expensive, as the Helmholtz matrix needs to be calculated at each time step; however, it allows larger time steps. For the eddy viscosity ν T (x,t) we use the Smagorinsky model: ν T = (C S ) 2 s ũ, (9) where the scale extraction operators are avoided completely by using this full resolved velocity, which is computationally a more attractive alternative than the original ν T = (C S ) 2 s ũ. In contrast, SVV-LES uses a filtering operator Qˆ ε that selects only the modes with high wave numbers with a smooth spectral representation from zero at low wave numbers to one at high wave numbers. If we view Qˆ ε as a small-scale extraction operator (albeit different from the one used in VMS), we can write the SVV weak formulation as W,N(Ũ) +ε N SVV w, u = W,F in the notation of this paper. There are four main differences between SVV and the present VMS-LES method Eq. (7): () the reversed sequence Figure 3. SPM simulations of double shear layer flow: vorticity contour at time T=2: (upper left) - Low-resolution DNS, P=6; (upper right) - High-resolution DNS, P=0; (lower) - VMS, P=6. This problem consists of solving the two-dimensional incompressible Navier-Stokes equations on a periodic box of length two (i.e., [,] [,] [,]) with initial conditions given: u(x,y) = tanh(ε(y + 0.5)) for y 0, u(x,y) = tanh(ε( y + 0.5)) for 0 < y and v(x,y) = δcos(πx). Here ε = 40.0 and δ = 0.05, and the kinematic viscosity is taken as ν = 0 4. Given these initial conditions, the perturbed shear layers rolls up into two vortices with trailing arms. For the purposes of our numerical experiment, we use an evenly-spaced quadrilateral mesh with 024 elements of sixth order. Using second-order time integration with a time step Dt = , we integrate the solution to final time T = 2 upon which we can compare different results. Fig. 3 shows that VMS solutions agree with the resolved solution (DNS with P=0). VMS s removal of the high-frequency information produces a stable solution, even though the highfrequencies are under-resolved for polynomial order P=6.

6 5. WATERJET SIMULATIONS Next we apply URANS, SVV-LES and VMS-LES to simulate transitional and turbulent flows in an axial-flow waterjet propulsion system (AxWJ-), see Fig.. While the simple boundaries (shaft, hub and casing) are treated in the standard way (i.e., applying directly Dirichlet boundary conditions), the rotor and stator blades are modeled with the smoothed profiles to avoid body-conforming meshes. We set the rotor to have rotational speed ω and translational velocity V = 0 in (2), which means the reference frame moves at the waterjet advance velocity V a. We calculate the Reynolds number Re = UD ν = nd2 2ν and the advance ratio J = V a nd, with the diameter of inlet D = 2 inch and n = ω/2π. The nondimensionalization factors for length and velocity are: meter ( inch) and nd m/s. Figure 4. URANS simulations of waterjet AxWJ- at Re = 23, 623, J = : Contours of instantaneous streamwise velocity (left) and viscosity ratio (right). We use a mesh with 209,66 tetrahedral elements and polynomial order of 3 for the spectral element discretization. The boundary conditions include an upstream prescribed velocity inlet (x = 6 inch), which is non-uniform near the walls to account for the boundary layer. In URANS the SA viscosity is given as ν = 3ν at the inlet as suggested in [8, 9]. The downstream outlet (x = 8) is set to have a fixed pressure and zero Neumann condition for velocities. For initial conditions, we use zero velocity field and unit SA viscosity ν = ν if not otherwise indicated. We first presented URANS results of instantaneous flow fields for velocity and SA viscosity inside the waterjet at various Re and J in [4]. For larger Reynolds number, e.g. for the case Re = 23,623,J = , we add tripping at x = 3 on the surface of the hub and casing near the inflow region. Fig. 4 shows that the velocity and viscosity ratio are much larger near the impeller than with Re = 57,906. This points to a faster transition from laminar to turbulent flow with the flow in the impeller region being fully turbulent. The robustness and efficiency of our numerical approach enable fast parametric studies for many cases with various advance ratio J and Reynolds number Re. We calculate the timeaveraged thrust coefficient K T, torque coefficient K Q = torque ρn 2 D 5 and efficiency η = K T K Q J 2π. Table 2 lists some typical numerical results compared to the experimental data for an ducted propeller [20] with a similar pitch ratio P/D =.0. We see that similar to the ducted propeller results, the waterjet propulsion system has decreasing thrust and torque with increasing advance ratio at a fixed Re. Also, the optimum performance (i.e. maximum efficiency) happens at an intermediate J for each Re; this optimum J increases with increasing Re. Furthermore, the table shows that the performance is sensitive to the tip clearance of the rotor, i.e. smaller tip gap leads to larger thrust and also better efficiency. Our numerical results show the right trend towards the design point Re = 4.4e6,J = with a tip gap of 0.03 inch. Table 2. Waterjet AxWJ-: average thrust coefficient K T and torque coefficient K Q for various angular velocity ω, Reynolds number Re. tip gap =0.225 inch=0.09d. Re J K T K Q η ducted propeller P/D= tip gap tip gap tip gap Now we compare numerical solutions for different turbulence modeling. Figure 5 shows 3D SPM simulation results of instantaneous streamwise velocity iso-surfaces after 0.5 rotation with no turbulence modeling, URANS and LES. Figure 6 shows 2D contours of instantaneous streamwise velocity after 2.5 rotations with different turbulence modeling; at this time instance DNS without modeling failed to achieve stable solution. We see that the turbulence models successfully stabilize the simulation by modeling the subgrid stresses. The plots also indicate that LES resolves smaller scales and finer flow structures compared to URANS. However, the overall hydrodynamic features are similar for all turbulence models, such as the reversal flow upstream of the rotor and the strong flow inside and after the stator. Figure 7 shows a time history of the streamwise velocity at a fixed point just downstream of the stator and off centerline. We see that the DNS simulation is not stable beyond

7 ical solutions. With the two different turbulence modeling, URANS and VMS-LES give similar force results at the beginning. However, with URANS, KT is decaying and also it has smaller fluctuations, because of the over-dissipation of the method. On the other hand, VMS-LES sustains the fluctuations, because it resolves more of the turbulence motions compared to URANS and it decays (in the mean) at a much small rate before it reaches an asymptotic value. Specifically, the difference in the average value of KT (beyond 5 rotations) is around 3%. Figure 5. SPM simulations of flow in a waterjet at Re=57,906 and J=0.667: instantaneous streamwise velocity iso-surfaces after 0.5 rotation by DNS no turbulence modeling (upper-left) and URANS-SA (upper-right); SVV-LES (lower-left) and VMS-LES(lower-right). about 0.4 rotations of the stator whereas all other three simulations are stable asymptotically. In particular, it is interesting to observe the small fluctuations of the URANS results whereas the SVV-LES and VMS-LES sustain higher fluctuations as expected. Similarly, Figure 8 shows the simulated thrust coefficient KT over time and it confirms that without turbulence modeling we are not able to obtain stable numer- Figure 7. SPM simulations of flow in waterjet at Re=57,906 and J=0.667: streamwise velocity of a history point over simulation time with or without turbulence models. Re=57906, J=0.667 no model URANS VMS avg= KT.4.2 avg= rotation Figure 8. SPM simulations of flow in waterjet at Re=57,906 and J=0.667: thrust coefficient KT as function of time with and without turbulence models. Figure 6. SPM simulations of flow in a waterjet at Re=57,906 and J=0.667: instantaneous streamwise velocity contours after 2.5 rotations by URANS (upper); SVV (lowerleft) and VMS (lower-right). Note that DNS without any turbulence modeling could not obtain stable solution at this time instance. 6. SUMMARY We have developed and validated an efficient numerical method based on smooth profiles and spectral element discretization for simulating turbulent flows with complex moving subdomains. This approach avoids the tyranny of meshgeneration and is substantially faster (up to 000 times) than

8 arbitrary Lagrangian-Eulerian (ALE) codes. In particular, in this paper we formulated a new LES model based on variational concepts (VMS-LES) in the context of hierarchical spectral element expansions. We compared the new VMS- LES code against URANS and other spectrally-filtered simulations, which in conjunction with spectral type accuracy capture the laminar-turbulence transition. We applied these turbulence modeling methods to study flows in waterjet propulsion system (ONR AxWJ-), and demonstrated that the combination of simple meshing and fast solvers allows for extensive parametric study in early-design phase of waterjets. The main flow features of the waterjet system were captured using both URANS and VMS-LES but accurate and sustainable turbulent motions could only be captured with VMS-LES. Future work will address further validation of the VMS-LES approach in the context of new emerging designs of waterjets for propulsion systems. ACKNOWLEDGEMENT This work is supported by the Office of Naval Research N , Sea Basing:T-Craft Dynamic Analysis, and MIT Sea Grant College Program, NA06OAR REFERENCES [] X. Luo, C. Chryssostomidis, G. E. Karniadakis, Fast 3D flow simulations of a waterjet propulsion system, in: Proceedings of Grand Challenges in Modeling Simulation, Summer Simulation Multiconference, July 2009, Istanbul, Turkey. [2] H. Sun, S. Kinnas, Performance prediction of cavitating water-jet propulsors using a viscous/inviscid interactive method, in: Transactions of the 2008 Annual Meeting of the Society of Naval Architects and Marine Engineers, Houston, Texas, [3] S. Schroeder, S.-E. Kim, H. Jasak, Toward predicting performance of an axial flow waterjet including the effects of cavitation and thrust breakdown, in: First International Symposium on Marine Propulsors, [4] X. Luo, C. Chryssostomidis, G. E. Karniadakis, Spectral element/smoothed profile method for turbulent flow simulations of waterjet propulsion systems, in: Proceedings of Grand Challenges in Modeling Simulation, Summer Simulation Multiconference, July 200, Ottawa, Canada. [5] P. R. Spalart, S. R. Allmaras, A one-equation turbulence model for aerodynamic flows, AIAA Paper (992) [6] P. R. Spalart, S. R. Allmaras, A one-equation turbulence model for aerodynamic flows, La Recherche Aerospatiale No. (994) 5 2. [8] T. Hughes, L. Mazzei, K. Jansen, Large eddy simulation and the variational multiscale method, Comput. Visual. Sci. 3 (2000) [9] X. Luo, M. R. Maxey, G. E. Karniadakis, Smoothed profile method for particulate flows: Error analysis and simulations, Journal of Computational Physics 228 (2009) [0] Y. Nakayama, K. Kim, R. Yamamoto, Hydrodynamic effects in colloidal dispersions studied by a new efficient direct simulation, in: Flow Dynamics, Vol. 832 of American Institute of Physics Conference Series, 2006, pp [] G. E. Karniadakis, S. J. Sherwin, Spectral/hp Element Methods for CFD, Oxford University Press, New York (second edition), [2] S. Sunada, T. Yasuda, K. Yasuda, K. Kawachi, Comparison of wing characteristics at an ultralow Reynolds number, Journal of Aircraft 39 (2) (2002) [3] Y. Zhou, M. M. Alam, H. Yang, H. Guo, D. Wood, Fluid forces on a very low Reynolds number airfoil and their prediction, International Journal of Heat and Fluid Flow In Press, Corrected Proof (200). [4] G.-S. Karamanos, G. E. Karniadakis, A spectral vanishing viscosity method for large-eddy simulations, J. Comput. Phys. 62 (2000) 22. [5] C. E. Wasberg, T. Gjesdal, B. A. P. Reif, O. Andreassen, Variational multiscale turbulence modelling in a high order spectral element method, J. Comput. Phys. 228 (2009) [6] S. Collis, Monitoring unresolved scales in multiscale turbulence modeling, Phys. Fluids 3 (200) [7] D. L. Brown, Performance of under-resolved twodimensional incompressible flow simulations, J. Comp. Phys. 22 (995) [8] C. L. Rumsey, Apparent transition behavior of widelyused turbulence models, International Journal of Heat and Fluid Flow 28 (2007) [9] P. Spalart, C. L. Rumsey, Effective inflow conditions for turbulence models in aerodynamic calculations, AIAA Journal 45 (0) (2007) [20] J. S. Carlton, Marine propellers and propulsion, London, U.K., Butterworth-Heinemann Ltd., second edition, [7] D. Carati, G. Winckelmans, H. Jeanmart, On the modelling of the subgrid-scale and filtered-scale stress tensors in large-eddy simulation, J. Fluid Mech. 44 (200) 9 38.