High-order methods for large-eddy simulation in complex geometries

High-order methods for large-eddy simulation in complex geometries Christine Baur, Patrick Bontoux, Michael Kornhaas, Mathieu Minguez, Richard Pasquetti, Michael Sch¨afer, Eric Serre, and Eric S´everac

Abstract Developing high-order methods for large-eddy simulation (LES) is of interest to avoid mixing between subgrid scale modeling contributions and approximation errors of the numerical method. Two different approaches are investigated. The first one focuses on the so-called Spectral Vanishing Viscosity LES (SVV-LES) approach, which allows to extend the well known capabilities of spectral methods from laminar to turbulent flows, while the second one rather investigates the possibility of extending a second order finite volume code to higher order approximations. For the SVV-LES approach, a volume penalization like technique is used to address complex geometries.

1 Introduction Spectral methods are known to be well suited for laminar or transitional flows in simple geometries (Cartesian, cylindrical, spherical...). Our goal is here to describe some routes allowing to address more complex flows, especially turbulent flows in complex geometries. In the frame of spectral / spectral element methods the spectral vanishing viscosity (SVV) technique is especially interesting, because allowing to dissipate the energy which accumulates in the high frequency range of the spectral approximation without losing the so-called spectral accuracy, i.e. the exponential decrease of the approximation error with the discretization parameter as soon as Patrick Bontoux · Mathieu Minguez · Eric Serre · Eric S´everac Lab. M2P2, Universit´e Paul C´ezanne / Aix-Marseille III, IMT, La Jet´ee´ , Tech. Chteau-Gombert, 38 rue Fr´ed´eric Joliot-Curie 13451 Marseille, France e-mail: [email protected] Mathieu Minguez · Richard Pasquetti UMR CNRS 6621, Universit´e de Nice-Sophia Antipolis, Parc Valrose, 06108 Nice, France e-mail: [email protected] Christine Baur · Michael Kornhaas · Michael Sch¨afer Technische Universit¨at Darmstadt, e-mail: [email protected]

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the solution is smooth. Initially developed for 1D hyperbolic problems [28, 56], typically the Burgers equation, the SVV technique has been extended to the 3D incompressible Navier-Stokes (NS) equations, both in Cartesian and cylindrical geometries, with very satisfactory results for turbulent rotor-stator flows. To handle complex geometries, one may think to use a domain decomposition technique. Such an approach was previously developed for 2D geometries resulting from gathering rectangles, see [9], and so it was natural to extend it to more complex situations. This method, similar to the well known spectral element method, but in the frame of collocation approximations, was unfortunately not successful because yielding to very ill posed problems [48]. On the contrary, the so-called pseudo-penalization method [40], i.e. a volume penalization method with implicit formulation of the penalization term, has allowed to address complex geometries, especially the square cylinder wake and the flow over a simplified car model (the “Ahmed body” [1]). However, for very high Reynolds number flows the boundary layers cannot be resolved with such an embedding technique and a near wall correction appeared necessary to obtain satisfactory results. At the same time, a high order (4 th order) finite volume method is currently developed, the LES capability being presently based on a classical subgrid scale (SGS) model, i.e. the dynamic Smagorinsky model. The paper provides a synthesis of works carried out these last four years and so partially relies on papers already published, especially [19, 23, 31, 32, 34, 35, 39, 43]. It is organized as follows : In Section 2 we present the SVV-LES modeling and give results obtained for a rotor-stator flow, i.e. a complex flow in a simple geometry. In Section 3, we describe the pseudo-penalization technique together with the near wall correction and give results obtained for the square cylinder and for the Ahmed body. In Section 4 we describe the high order finite volume method presently developed and give some preliminary results for efficiency investigations and benchmark cases.

2 SVV-LES method - application to a rotor-stator flow In the frame of collocation methods, our SVV-LES methodology, see e.g. [37], yields to consider the following stabilized dimensionless NS equations : Dt u = −∇p + ν∆SVV u ∇·u = 0

(1) (2)

with usual notations (t : time, u : velocity, p : pressure, Dt : material derivative, ν : inverse of the Reynolds number) and where ∆SVV if a SVV modified Laplacian, which depends on the discretization parameter, say N, and such that ∆SVV → ∆ if N → +∞. Similar developments can be carried out if a weak approximation of the NS equations is considered, see [37, 59]. The ∆SVV operator is hereafter defined and the SVV-LES approach is then applied to the computation of a turbulent rotor-stator flow.

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2.1 SVV stabilization Basically, the SVV method relies on the introduction of a discretization dependent artificial diffusive term only active in the high frequency range of the numerical approximation. For 1D partial differential equations, with N for the discretization parameter and uN (x) for the numerical solution, such an additional term reads: VN ≡ εN ∂x (QN (∂x uN ))

(3)

where εN is a O(1/N) coefficient and QN is the spectral viscosity operator such that, following [28], in the 1D non-periodic case and with {Lk }k≥0 for the set of the Legendre polynomials : ∀φ ,

φ=

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√ where Qˆ k = 0 if k ≤ mN and 1 ≥ Qˆ k > 0 if N ≥ k > mN , with e.g. mN = N and Qˆ k = exp(−(k − N)2 /(k − mN )2 ). In the periodic case trigonometric polynomials are substituted to the Legendre polynomials. In the frame of collocation Chebyshev methods we use a similar definition, but with the Chebyshev polynomials instead of the Legendre’s. The main point is of course that a hierarchical basis must be used. Note that theoretical papers use a step variation rather than the smoother exponential dependence of Qˆ k , which is commonly used in the applications. Extending the definition (3) to a multidimensional space is not so natural, as it appears when looking at the different forms proposed in the literature, see e.g. [20, 21, 22]. For us, we advocate the following definition: VN ≡ ∇ · (εN QN (∇uN )) ,

εN QN ≡ diag{εNi QiNi }

(4)

with QiNi the 1D viscosity operator acting in direction i. Thus, we actually use a diagonal matrix form of the operator εN QN , just as one introduces a non-scalar diffusivity when anisotropic media are considered. Coming back to the NS equation, the SVV stabilization can be simply implemented by combining the diffusion and (vectorial) SVV term to obtain,

∆SVV = ∑ ∂i ∂˜i , i

εN ∂˜i ≡ (1 + i QiNi )∂i . ν

(5)

The determination of the ”right values” of the SVV parameters, εN and mN , is discussed in [36]. In this paper we suggest, on the basis of numerical experiments, to simply minimize the magnitude of the SVV stabilization term. A similar formulation may be established for the cylindrical coordinate system used for rotor-stator flows, the main complication coming from the fact that the vector Laplacian operator is no-longer diagonal. We refer to [39, 50] for a complete description. The algorithm was first introduced in [39].

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The main features of the Cartesian and cylindrical codes that have been developed, using this SVV-LES methodology, are mentioned hereafter. More details may be found in [6] for the Cartesian code and in [44, 46] for the cylindrical code. • The time-scheme is second order accurate: Second order backward differences are used for the approximation of the time derivative of the velocity. • The SVV modified diffusion terms are treated implicitly. • The convective terms are treated explicitly. A second order Adams-Bashforth extrapolation is used in the cylindrical code whereas an “Operator Integration Factor” Semi-Lagrangian method is used in the Cartesian code. • Projection methods are used to get a divergence-free velocity field. It relies on the so-called unique grid PN − PN−2 approximation in the Cartesian code whereas one uses a preliminary solve for the pressure in the cylindrical code. • Fourier-Galerkin approximations are used in the homogeneous direction, i.e. in the spanwise and azimuthal directions for the Cartesian and cylindrical codes, respectively. Chebyshev collocation approximations are used in the nonhomogeneous directions. • Specific to the Cartesian code: (i) A domain decomposition technique is used in the streamwise direction and (ii) the bluff body is modeled by using a pseudopenalization technique.

2.2 Application to a rotor-stator flow We investigate a rotor-stator flow within a cavity made of two discs enclosing an annular domain of radial extent ∆ R = R1 − R0 , where R0 and R1 are the internal and the external radii, respectively. Two stationary cylinders of height H, termed the shaft and the shroud, bound the domain, see Fig. 1. Two parameters define the shape of the system : These may be taken as the curvature parameter Rm = (R0 + R1 )/∆ R and the aspect ratio L = ∆ R/H. Here, Rm = L = 5. The reference time and velocity are Ω −1 and Ω R1 , respectively, and the Reynolds number is defined as Re = Ω R21 /ν , where ν is the kinematic viscosity. The normalized dimensionless coordinates in any meridian plane are (r = rdim /∆ R − Rm , z = 2zdim /H − 1) ∈ [−1, 1]2 (rdim and zdim are the corresponding “dimensioned” variables). No-slip boundary conditions are applied to all walls : ur = uθ = uz = 0, except on the rotating disk (z = 1) where uθ = (Rm + r)/(Rm + 1). The junction of the stationary cylinders with the rotor is regularized employing a boundary layer function, uθ = exp(−(z − 1)/µ ), where the value of the shape parameter µ = 6. 10−3 provides a reasonable representation of experimental conditions (there is a thin gap between the edge of the rotating disc and the stationary sidewall), while retaining spectral accuracy. Rotor-stator flows are very challenging for numerical modeling, particularly in turbulent regimes, see [47, 49, 50] and references herein. A characteristic feature of such flows is indeed the coexistence of adjacent coupled flow regions involving

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Fig. 1 Schematic of the rotor-stator cavity.

laminar, transitional and turbulent regions which are completely different in terms of the flow characteristics. Moreover, the turbulence is strongly inhomogeneous and anisotropic because of confinement, flow curvature and rotational effects. A high Reynolds number flow, Re = 106 , is computed to show the capability of the SVV technique to investigate the turbulence features of rotor-stator flows. The computational grid has been chosen relatively coarse for this Reynolds number with 81 × 160 × 81 points in (r, θ , z) directions, respectively, and a time step ∆ t = 10−4 was used. As a comparison, this grid is twice coarser than the one used by Andersson and Lygren [2] in their LES of ”case D” study of a ”wide gap” cavity. It permits however to correctly solve the boundary layers : Values of z+ = hzVτ /ν around 0.65 and less than 0.6 are found for the Ekman and B¨odewadt layers, respectively. Concerning the SVV parameters we use εN = 2/N and mN = N/2 in all directions, with here N = (80, 80, 80). At Re = 106 , both layers are turbulent as shown by the isolines of the mean turbulent kinetic energy in Fig. 2. The figure provides qualitative evidence that the turbulence is mainly concentrated near the two discs. The maximum of turbulent kinetic energy is located at the rotor-shroud junction where the flow, accelerated by the rotor, impinges the stationary outer cylinder. The flow consists of rather tangled

Fig. 2 Isolines of the instantaneous fields of Q-criterion for θ = π /4 (top) and θ -averaged turbulent kinetic energy (bottom), Re = 106 .

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Fig. 3 Axial profiles at mid radius of the mean radial velocity (top left), mean tangential velocity (top right), square root of the radial (bottom left) and azimuthal (bottom right) normal Reynolds Stresses in the Ekman (solid line) and in the B¨odewadt (dashed line) layers, Re = 106 . Mean velocities are normalized by the local velocity of the rotor. Square root of Reynolds stresses are normalized by the local friction velocity, Vτ , on the corresponding disc.

co-rotating vortices which originate near the shroud and move radially inward on the stator and outward on the rotor, following the main flow direction. The growth of these vortices strongly affects not only the structure of the two disc boundary layers but also the geostrophic core. This is revealed by plotting the isolines of the Q-criterion in the meridian plane (see Fig. 2). In the vicinity of the inner and outer cylinders there is now a strong mixing between both boundary layers, involving a large number of vortices of different scales. This underlines the important effects that the inner and outer cylinders have on the characteristics of the turbulence. The axial profiles at mid radius of the mean radial and tangential component of the velocity show a more sharpened profiles and a larger core region than at Re = 7. 104 (see Fig. 3). The core rotates at Kuθ /(Ω rdim ) = 0.3919 at mid-radius, which 2 /ν = 694444. This value of corresponds to a local Reynolds number Reloc = Ω rdim K is in close agreement both with the numerical work of [2] (K = 0.40 at Reloc = 640000 in ”case D”) and experimental results of Itoh et al. [17] (K = 0.41). The

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small underestimation of K is mainly due to differences in the problem modeling (a cylindrical cavity for [17], an homogeneous cavity in radial direction for [2], and a shrouded rotor-stator annular cavity for the SVV-LES). Indeed, the stationary inner cylinder greatly stabilizes the flow and the shroud tends to slow down the flow by dissipating a lot of energy in the impinging jet and in the Stewardson layer, that follows the Ekman layer at the upper part of the shroud. As no reference profile is available at this Reynolds number, we have compared two local parameters introduced by Daily and Nece [8] and based on the wall friction velocity: The skin-friction drag coefficient Cθ = 2(Vτ /rdim Ω )2 that is derived from the torque exerted by the fluid on the two discs and the local stress Reynolds number Reτ = rdimVτ /ν . SVV-LES calculations give values at mid-radius equal to Cθ = 1.6 10−3 in the Ekman layer and Cθ = 8.74 10−4 in the B¨odewadt layer. Such values are slightly smaller than those found in [2], where Cθ = 2.05 10−3 and Cθ = 1.15 10−3, respectively. The relative dissipation rate between both layers due to the skin-friction, Cθ (rotor)/Cθ (stator), is equal to 1.83 for the SVV-LES. This value is in close agreement with the value 1.79 of [2] for the ”case D” and with the value 2 measured experimentally in [17]. The local stress Reynolds numbers at mid-radius are found equal to Reτ = 525 at the rotor and Reτ = 288 at the stator. These values, which show that the level of turbulence at mid-radius is larger in the Ekman layer than in the B¨odewadt layer, are very close to those obtained by Andersson and Lygren [2], Reτ = 525 and Reτ = 284, respectively.

3 Pseudo-penalization method and bluff body wake flows To address bluff body wake flows with the SVV-LES spectral (Cartesian) solver one uses a volume penalization like method. It is described hereafter together with the corrections that have been implemented to improve the results in near wall region. The computation of the square cylinder wake allows to validate this Near Wall (NW) treatment. The approach is then applied to a challenging benchmark, the Ahmed body [1, 26], slant angle 25o. Till now RANS approaches fail to describe this flow and no LES results are completely satisfactory, see e.g. [7, 14, 29] and [10, 11, 15, 16, 24], respectively.

3.1 Modeling of the bluff body A volume penalization method is used to model the obstacle, but no penalization term is however explicitly introduced in the momentum equation. As suggested in [40], this is implicitly done through the time discretized NS equations.

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With χ for the characteristic function of the obstacle and assuming that the linear diffusive term is treated implicitly and the non-linear convective term explicitly, the pseudo-penalization method consists of solving at each time-step:

ν∆ un+1 −

α n+1 u − ∇pn+1 = (1 − χ¯ )fn+1 τ ∇ · un+1 = 0

(6) (7)

where n is the time index, τ (≡ ∆ t) the time-step and α a scheme dependent coefficient (α = 3/2 for a second-order backward finite difference scheme). The pair (un , pn ) is the numerical approximation of (u, p) at time tn and fn+1 is an easily identifiable source term, which also depends on the time scheme. Finally, χ¯ is a regularized characteristic function, in practice obtained from local averages of the function χ . Clearly, inside the obstacle (un+1 , pn+1 ) solves the steady Stokes equations with a O(1/τ ) penalization term. Then it appears that inside the obstacle the velocity approximately vanishes, i.e. is essentially O(τ ) ≪ 1. As soon as the Reynolds number is really high, as it is e.g. for the Ahmed body flow at Re = 768000, the boundary layers cannot be resolved properly by the mesh. This has motivated a lot of researches during the three last decades on Near Wall Modeling (NWM), see e.g. [42]. The problem is especially difficult when the flow shows large detachments. Several approaches have been suggested, based on the boundary layer equations or requiring to resolve joined equations, see [58], patching techniques, DES (Detached Eddy Simulation) methods, see [30]... Such approaches have essentially been developed for finite volume approximations and it is not straightforward to implement them in a spectral solver and when using a penalization type method. The results presented in this paper have been obtained with a cruder approach but which has allowed significant improvements : • The characteristic function of the obstacle is not smoothed, • the control parameters of the SVV technique, εN and mN , are relaxed in NW region. Point 2 may be formulated by completing the momentum equation (1) with the body force term NW − ∆SVV )u fNW = χNW ν (∆SVV (8) with χNW a second characteristic function used to localize the NW adjustment and NW is defined like ∆ where the operator ∆SVV SVV but makes use of a smaller value of εN and / or a greater value of mN . In practice, we have only increased the value of mN within the NW region.

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3.2 Validation for the square cylinder The geometry corresponds to the one proposed by the ERFOCTAC test case [57]. A square cylinder of unitary side is placed at the origin of the coordinate system, within the computational domain (−4.5, 15) × (−7, 7) × (−2, 2). The entire domain is decomposed in the x-streamwise direction in eight sub-domains with the interfaces located at x = {−4.5, −2, −0.55, 0.55, 2, 4.5, 7.5, 10.5, 15}. In each sub-domain we use polynomials of degree Nx = 30, Ny = 350 and Nz = 16, so that one has about 2, 7.106 mesh points. Using a mapping in the y-vertical direction, the first grid point is located at a distance of ∆ y = 0.015 that does not allow a fine description of the boundary layer, as already mentioned in the conclusions of the ERCOFTAC test case [57], for ∆ y = 0.01. At the inlet an uniform velocity profile Ue is imposed, whereas the outlet is treated with a convective condition at velocity Ue . The characteristic Reynolds number of the flow, based on Ue and D, is equal to Re=21400. According to stability p SVV parameters have been chosen as p the √ tests, (mNx , mNy , mNz ) = (3 Nx , 4 Ny , 2 Nz /2) and εN = 1/N in NW regions whereas √ in the complete domain (mN , εN ) = ( N, 1/N). The simulations have been performed on the NEC-SX8 of IDRIS, and required 5 GB of memory and 73 CPU hours to get convergence. Instantaneous isosurfaces of the pressure, as obtained without and with near wall treatment are provided in Fig. 4. Clearly, the filtering of the characteristic function induces a shift of the detachment line. Statistics above the cylinder are presented in Fig. 5, for the mean streamwise velocity and for the streamwise Reynolds stress. Results are given for the standard SVV-LES, without filtering of the characteristic function and with the near wall relaxation of the SVV parameters. Comparisons are provided with the experimental results of [27]

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Fig. 4 Instantaneous isosurfaces of the pressure without (left) and with (right) near wall treatment

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3.3 Simulations of the Ahmed body flow Computations were carried out in the domain (−7.25, 7.25) × (0, 3.47) × (−2.35, 2.35), using the height of the car model as reference length and the rear part being located at x = 0. The computational domain is partitioned in 8 subdomains. In each one the discretization is 41 × 191 × 340, obtained with Chebyshev polynomials of degrees Nx = 40 and Ny = 190 and trigonometric polynomials of degree Nz = 170. Since Gauss-Lobatto-Chebyshev points naturally accumulate at the end-points, subdomains interfaces have been located at the front and rear part of the Ahmed body. In the vertical direction, non-linear mappings are used to accumulate grid-points on the upper part of the bluff body. The mesh makes use of about 21 millions of grid-points. Starting from the fluid at rest, a computation was first carried out on a rough mesh (discretization is roughly divided by 2 in each direction) till a turbulent flow is well established, say at the dimensionless time t = 100. Then, the solution was interpolated on a fine mesh and the simulation was continued till t = 160. Statistics were

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only computed during the last 40 time units of the simulation, in order to avoid any pollution coming from the coarse mesh computation. The dimensionless time-step was equal to τ = 2.10−3 and the CPU time to 9 s for one time-step, i.e. approximately 9.510−8s per iteration and degree of freedom on the NEC SX8 computer of the IDRIS center. The SVV stabilization is governed by the threshold frequency mN and the SVV amplitude εN . Independently √ of the spatial (x, y, z) direction, outside the boundary layer we have used mN = N and εN√= 1/N. p Within √ the boundary layer, we have used anisotropic values, i.e. mN = {2 Nx , 5 Ny , 4 Nz } and again εN = 1/N. Results obtained at the reference Reynolds number Re = 768000 and comparisons with results obtained at Re = 8322, as in the experiments of [13, 51], are provided. Same meshes and time-steps have been used in the two cases and the values of the SVV control parameters are the same, both inside and outside the boundary layers. However, for computational cost reasons we only used 12 time units to get the statistical results at the lower Reynolds number. As described hereafter, the topology of the two flows are in fact close, showing that the flow is not very sensitive to the Reynolds number. More details may be found in [31, 34]. In Fig. 6 we compare isolines of the mean streamfunction in the vertical median plane z = 0. One clearly observes three recirculation zones, on the upper part of the body, over the slant and behind the obstacle. Moreover, recirculation bubbles similar to the one in the upper part occur at the lateral walls. Despite similar, these recirculation zones appear larger, i.e. longer and thicker, at the lower Reynolds number. One should mention that experimentally such recirculations may or may not be observed, depending on the laminar or turbulent feature of the upstream flow. Behind the obstacle the topology is slightly different : The recirculation zone shows two contra-rotating recirculation bubbles and one observes that the lower one is less pronounced at Re = 8322, which probably results from a thicker boundary layer developing under the body. In Fig. 7 are shown velocity fields in the plane x = 1.34 for the two values of the Reynolds number. Clearly, the cone like trailing vortices which escape from the

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edge of the slant are similar. However, at the low Re the vortices appear (i) slightly more distant and (ii) are located lower. We compare in Fig. 8 mean streamwise velocity and turbulent kinetic energy profiles. Under the body one observes that the flow is nearly laminar for the low Re simulation : a parabolic like profile is obtained and the turbulent kinetic energy is close to 0. This is not very surprising, since the local Reynolds number based on the distance from the body to the ground equals 1445. Over the slant, one recovers the conclusion inferred by the mean streamlines that the reattachment is delayed. For the higher Re detailed comparisons with the experiments of Lienhart et al. [26] are provided in [33], but to be self contained we present in Fig. 9 such comparisons, for the mean streamwise velocity and for the turbulent kinetic energy. We also point out the influence of the NW correction. Even if the SVV-LES profiles compare rather well with the experimental data, one observes at the beginning of the slant a deficit of streamwise velocity associated to an overestimation of the tur-

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bulent kinetic. The phenomenon is in fact confined in the median region, say for −0.5 < z < 0.5 and develops from the front part of the body, downstream of the recirculation bubble localized on the upper part, see Fig. 6. Of course, the results obtained here at the lower value of the Reynolds number are certainly more reliable, since the SVV stabilization plays here a less important role. This may be visualized by computing the dissipation rate of the turbulent kinetic energy, see [31, 34, 38] for details.

4 High-order finite volume method and applications To increase the geometrical flexibility of the concerned Chebyshev collocation pseudo–spectral method, a combination of a mapping technique together with a multi-block decomposition algorithm has been investigated [48]. The mapping and the ’generalized’ domain decomposition technique increase the condition numbers of the discrete systems which turned out to be a crucial issue with respect to an efficient solution [43]. Due to this fact that the extension of the spectral methods to more complex geometries leads not only to ill-conditioned and full system matrices but also to high memory requirement. An efficient computation of an accurately resolved turbulent flow seems not to be possible with such an approach. A compromise between the mentioned drawbacks of the spectral methods and the required high accuracy for the computation of turbulent flows with LES is seen in the finitevolume method in conjunction with compact finite difference methods.

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4.1 Numerical method and LES modeling The basis is the finite volume solver FASTEST [52]. It solves the filtered NavierStokes equations on boundary fitted, block structured hexahedral grids. The convective and diffusive fluxes are approximated with a second-order central difference scheme. In order to minimize discretization errors a fourth order compact scheme is developed for flux approximations (section 4.2). Subgrid stresses are computed using the Smagorinsky model with the dynamic approach of Germano [12]. The implicit second order Crank-Nicolson scheme is applied for time discretization. Pressure velocity coupling is realized with a SIMPLE type algorithm which is embedded in a geometric multigrid scheme with standard restriction and prolongation [4]. The resulting linear systems of equations are solved with an ILU method.

4.2 Fourth-order compact finite volume scheme A method to achieve a higher order spatial accuracy is the compact finite difference method (e.g. [41]). Compared to conventional finite difference methods the finite compact difference schemes reflect the high-frequency, short-wave parts of the solution in a better way and have a better fine-scale resolution [25]. For the discretization of the averaged convective and diffusive cell face fluxes a fourth-order compact finite difference scheme suggested by [41] is employed. To show the procedure of the interpolation the discretization of an east cell face of a control volume for an orthogonal 2D grid (Fig. 10) is taken exemplarily. Unknowns of convective fluxes are interpolated as 1 3 1 uw + ue + uee = (uP + uE ). 4 4 4

(9)

Unknowns of diffusive fluxes are interpolated as 1 ∂ uw 6 1 ∂ uee ∂ ue + + = (uP − uE ) , 10 ∂ x ∂ x 10 ∂ x 5∆ x

Fig. 10 Notation of a control volume.

(10)

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15

where ∆ x is the grid spacing in x direction. The approximation of the unknowns in the convective fluxes and the first derivative in the diffusive fluxes at a node does not only depend on the function values at nodes in the cell center in the neighborhood but also on function values and the partial derivatives on cell faces. Thus the global dependence of spectral methods is imitated. With the conventional discretization the following discretized momentum equation is obtained [45]: aP uP − aW uW − aE uE − aN uN − aS uS = SP

(11)

With the compact scheme we obtain the following set of equations which has to be solved simultaneously for the convective and diffusive fluxes Fi and Di as well as the nodal values uP :

∑

i=e,w,n,s

λFi Fi +

∑

λDi Di

= SP

(12)

i=e,w,n,s

1 1 Fw + Fe + Fee = 34 (uP + uE ) 4 4 1 1 Dw + De + Dee = 5∆6 x (uP − uE ) 10 10

(13) (14)

4.3 Efficiency investigations of the finite volume solver For the efficiency investigations we chose the well known test case “Periodic flow over a 2D hill” [18] which is a periodic segment of a channel constricted by “2D hills” at the lower wall. The Reynolds number Rh = 11600 is based on the hill height h. The computational domain is periodic in streamwise as well as in spanwise directions what avoids uncertainties due to unknown boundary conditions. Although the simple geometry, the flow shows different features like separation on curved surfaces, strong recirculation and reattachment. The test case geometry is shown in Fig. 11. The computational domain is of the size 9h × 3.03h × 4.5h. The flow is driven by a pressure gradient that is adjusted to the desired Reynolds Number of Rh = 11600. On the top and bottom walls no slip boundary conditions are applied. Two grid resolutions were considered both with a fully resolved wall boundary layer and y+ values at the wall below unity. A reference solution was computed utilizing a highly resolved LES on a 11.8 × 106 control volume (CV) grid. The below mentioned parameter studies are carried out on a coarser 1.47 × 106 CV grid. In many cases, especially when structured or block structured grids are used for wall bounded flows, the highest Courant (CFL) numbers occur only in few control volumes whereas for the most part of the computational domain they are much below unity. Also the locations with the highest Courant number vary with time. Fig. 12 shows the control volumes with CFL > 0.3 for a single time step with a maxi-

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mum Courant number of CFLmax ≈ 1. With this background the use of implicit time disretization schemes and their ability to handle time step sizes with corresponding CFL > 1 seems to be a promising way to increase the efficiency of the simulations since with the same number of time steps a greater time interval can be covered. If the quality of the results is not affected, the computational time - especially when turbulence statistics are of interest - can be reduced significantly. To gain information how greater time step sizes affect the quality of the computational results and the efficiency of the simulations, computations with corresponding maximum CFL numbers of approximately 1, 2, and 5 were carried out. Averaging was performed for 40 flow-through times. Fig. 13 shows the profiles of the mean streamwise velocity component /ub and the mean streamwise fluctuations /u2b for the three considered time step sizes as well as the reference solution obtained on the fine 11.8 × 106 CV grid at the positions x/h = 0.05 and x/h = 2. At both considered locations the obtained velocity profiles are in very good agreement with the reference solution (LES - fine grid). There are no significant differences between the results for the different time step sizes. Also the fluctuations in streamwise direction are in good agreement to each other for all three time step sizes and the reference solution. The deviation to the reference solution at the position x/h = 2 is in the expected range since only the resolved part is plotted and this part is greater on the finer grid. It can be stated that the time step size - at least in the considered range - has no negative influence on the quality of the computational results. But it has great influence on the efficiency of the computations. In table 1 the computational times for a given time interval of 0.05s (equivalent to 1000 time steps with CFL ≈ 1) are summarized. For the time step size with CFL ≈ 2 almost half of the computational time could be saved whereas for the greatest considered time step size (CFL ≈ 5) the convergence behavior became worse and therefore it was less efficient as the simulation with CFL ≈ 2. Further numerical investigations like the influence of the convergence criterion on quality and efficiency and more detailed analysis of the flow features haven been published in [23].

Fig. 11 Test case geometry with mean streamlines and analysis locations.

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Fig. 12 Spatial distribution of the CFL number for CFLmax ≈ 1. CFL > 0.3.

Fig. 13 Influence of the time step size. Profiles of the mean velocity component < u > /ub and the mean streamwise fluctuations /u2b at x/h = 0, 05 and x/h = 2. Table 1 Computational time for a given time interval of 0.05s (2 IBM Power 5 Processors with 1.9 GHz).

∆ t (CFL) Computational time

5 × 10−5 s (≈ 1) 10−4 s (≈ 2) 2.5 × 10−4 s (≈ 5) 76.4 × 103 s

38, 3 × 103 s

46, 8 × 103 s

4.4 Investigation of efficiency improvements by an algebraic multigrid method The finite volume method leads to a large sparse system of equations. To solve this equation system efficiently multigrid methods are used. In this section a geometric and an algebraic multigrid method are investigated. For the simulations performed in this study the SAMG package by Fraunhofer SCAI [55] was used as algebraic multigrid (AMG) solver. While the coarser grid levels of the geometric multigrid method are dependent on the mesh and the geometry the coarser grid levels of the AMG solver are constructed based just on the system matrix. A schematic comparison between the two multigrid solvers is shown in Fig. 14. For the efficiency investigations a laminar and a turbulent test series are considered. The geometry corresponds to a channel flow through

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Fig. 14 Comparison of geometric and algebraic multigrid method.

a labyrinth (see Fig. 15-17). The idea is achieving a more and more complex geometry and investigating the computational effort for the different test cases with the two methods. The number of control volumes is almost the same for each test case (around 575 × 103 CV). In the laminar case the inlet velocity of the fluid is u = 0.45 m/s in x-direction with a density ρ = 1 kg/m3 and kinematic viscosity µ = 10−3 kg/ms yielding the Reynolds number Re = 77. The resulting linear sets of equations are solved with a full multigrid strategy for the geometric multigrid method and V-cycles for the algebraic method. Results of the calculations are shown in Fig. 15-17. In Fig. 18 the computing times for the different test cases are shown. It can be seen that the more complex the geometry the more advantageous the AMG solver becomes. For the turbulent test series the inlet velocity is u = 16 m/s and thus the Reynolds number is Re = 2743. For the LES simulation a dynamic Smagorinsky model is chosen with time step ∆ t = 5 · 10−5s. The results are shown in Fig. 19 and 20. The geometric and algebraic multigrid methods are compared with single grid calculations. Due to the fact that in the residuals of the geometric multigrid calcula-

Fig. 15 Test case LAB1, velocity profile at y = 0.1 m.

Fig. 16 Test case LAB2, velocity profile at y = 0.1 m.

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Fig. 17 Test case LAB3, velocity profile at y = 0.1 m.

Fig. 18 Computing time for laminar flow test series.

tion oscillations appeared [53] a method where the calculations start as a geometric multigrid method and switch to the single grid method [53] are also considered (GMGSG). The results are summarized in Fig. 21 (computational time corresponds to 20 time steps). Even the geometric multigrid method without switching is faster than the AMG solver. The best results are obtained with the GMGSG approach. Thus for laminar test cases the AMG solver appears to be a promising method, but for the considered turbulent test cases it seems to be not.

Fig. 19 Contour plot of velocity norm.

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Fig. 20 Isosurface of velocity at 25 m/s.

Fig. 21 Computing time for LAB3.

4.5 Benchmark computations for flow in a 3D diffuser An incompressible flow through a 3D diffuser (ERCOFTAC Test Case 13.2: Flow in a 3D diffuser - Diffuser 1) with deflected upper wall and one deflected side wall is studied by means of LES at a bulk Reynolds number Rh = 10000 based on the height h of the inlet duct. Experimental data are available from the work of Cherrye et al. [5]. The flow shows an adverse pressure gradient caused by the duct expansion. The focus of the study is on the size and shape of the three dimensional flow separation pattern. The adverse pressure gradient and the high sensitivity of the flow to small changes of the geometry makes it a challenging test case for turbulence modeling. The test case geometry is depicted in Fig. 22. According to the experimental setup of Cherrye et al. [5] a fully developed turbulent channel flow is used as inlet boundary condition. This inlet data is generated by a simultaneously running periodic simulation of a channel with the same cross section as the diffuser inlet. To allow the flow through the diffuser to influence the flow field in the development channel a part (5 channel heights) of the development channel is modeled in front of the diffuser. Furthermore the convergent part behind the diffuser (compare Fig. 22) is modeled to avoid back flow at the outlet plane. The

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21

Fig. 22 Geometry of the diffuser (diffuser 1) taken from [5].

Fig. 23 Computational domain. Only each second grid line is shown.

computational domain including the development channel and the convergent part is depicted in Fig. 23. Two cases are considered. One with dynamic Smagorinsky model and the second one without subgrid model. The grid size was approximately 4 million CV including the development channel and the convergent part behind the diffuser. The wall boundary layers are fully resolved with y+ values below unity at all walls. The time step size is chosen in a way that the corresponding Courant number is around one. This leads to a time step size of 1.1 × 10−4 s. Averaging is performed for approximately 80000 time steps. The simulations were carried out on 16 IBM Power 5 CPUs with a load balancing efficiency of 95%. This leads to a computational times per time step of approximately 14s. Fig. 24 shows contours of the mean streamwise velocity component < u > in comparison to experimental data taken from [5] at the cross sections x/h = 2, 5, 8, 12 and 15. The computations show a good agreement with experimental data for all considered locations. The overall flow features are well captured. Also the size and shape of the recirculation bubble is well reproduced (< u >= 0 thicker line).

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Fig. 24 Contours of mean streamwise velocity < u > on cross sections x/h = 2, 5, 8, 12 and 15 in comparison to experimental data.

Fig. 25 shows mean streamwise velocity profiles on the slice z/h = 1 in comparison to experimental data from Cherrye et al. [5]. Both simulations show a relatively good agreement with the experiment. But at streamwise position x/h = 2 the velocity close to the upper wall differs from experimental values. Since this position is close to the separation point, the deviation of the computational data leads to misprediction of the recirculation at streamwise positions x/h = 6 and x/h = 8. At the positions further downstream the agreement with the experiment is better. There are no significant differences between the two simulations with and without subgrid model. In Fig. 26 mean streamwise velocity profiles on the slice z/h = 2.625 are shown. At this spanwise position recirculation is present at the greatest part of the domain (compare Fig. 24 and 22). Again at the position x/h = 2 the velocity at the upper wall differs from experimental data what again leads to misprediction of the recirculation at positions further downstream. At this slice the two simulations with and without model differ from each other. The simulation without subgrid model shows a better agreement with experiments than the one with dynamic Smagorinsky model, especially recirculation at streamwise positions x/h = 6 is predicted much better. The differences to the experiments are most probably caused by a to coarse grid resolution in streamwise direction in combination with too big aspect rations of the control volumes (compare Fig. 23). The influence of the grid resolution and the subgrid model are investigated in detail in present studies. Detailed simulation data of the LES with dynamic Smagorinsky model, with profiles of all velocity components and Reynolds stresses, at various streamwise and spanwise positions can be found in [19].

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Fig. 25 Mean streamwise velocity profiles < u > /ubulk on the slice z/h = 1 in comparison to experimental data.

Fig. 26 Mean streamwise velocity profiles < u > /ubulk on the slice z/h = 2.625 in comparison to experimental data.

Acknowledgements The SVV-LES calculations were carried out on the NEC SX8 vector computer of the CNRS computational center IDRIS (project 074055 and 084055). Computational resources of the mesocentre SIGAMM of the OCA were also used. The authors gratefully acknowledge support from the DFG and the CNRS trough the DFG-CNRS program ”LES of complex flows”.

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Index

3D diffuser, 20 Ahmed body, 2, 7, 10 B¨odewadt layer, 5 Bluff body wake, 7 Compact finite volume scheme, 14 Courant number, 15

Multigrid algebraic, 17 geometric, 17 Pseudo-penalization method, 7 Rotor-stator flow, 2, 4

FASTEST, 14

Smagorinsky model, 21 Spectral methods, 1 Spectral vanishing viscosity (SVV), 1–3 Square cylinder wake, 7, 9

High-order methods, 1

Volume penalization method, 2

Ekman layer, 5

27