Can adaptive grid refinement produce grid-independent solutions for incompressible flows?

This paper studies if adaptive grid reﬁnement combined with ﬁnite-volume simulation of the incompressible RANS equations can be used to obtain grid-independent solutions of realistic ﬂow problems. It is shown that grid adaptation based on metric tensors can generate series of meshes for grid convergence studies in a straightforward way. For a two-dimensional airfoil and the ﬂow around a tanker ship, the grid convergence of the observed forces is suﬃciently smooth for numerical uncertainty estimation. Grid reﬁnement captures the details of the local ﬂow in the wake, which is shown to be grid converged on reasonably-sized meshes. Thus, grid convergence studies using automatic reﬁnement are suitable for high-Reynolds incompressible ﬂows.

The results of such simulations depend on the physical models being 9 used, such as the turbulence model in the Reynolds-averaged Navier-Stokes 10 (RANS) equations. Often, such models are applied in situations which are far 11 more complex than the ones for which they were developed and which may 12 be outside their range of validity. Research of physical modelling, specifically 13 for today's realistic simulations, is therefore of prime importance.
14 To accurately assess the precision of a physical model, we need to know 15 a numerical solution in which the numerical errors are small with respect to 16 the modelling errors: a solution that is close to grid convergence. In simple 17 cases, it is possible for an experienced user to generate meshes which provide accurately that the numerical errors become much smaller than the physical  To obtain anisotropic grid refinement, we use metric tensors as refinement 113 criteria. This technique was introduced for the generation of anisotropic 114 tetrahedral cells [5], it has later been used successfully for the adaptive re-115 finement of such meshes [11,12]. The technique also provides a practical and 116 flexible framework for the refinement of hexahedral meshes. 117 In our procedure, the refinement of the cells is decided as follows. First, 118 the 3 × 3 criterion tensors C i in each cell i are computed (in some way) from 119 the flow solution. In a hexahedral cell, let the cell sizes d i,j (j =1 , 2, 3) be 120 the vectors between the opposing face centres in the three cell directions. 121 The goal of the grid refinement is then to create a grid which is uniform 122 6 under the transformation C, which implies that: where T r is a constant. In the refinement procedure, this is obtained in the 124 following way. Each time the procedure is called, the criterion C i and the cell 125 sizes d i,j are computed on the current grid. A cell i is refined in the direction are weighted in the way in which they appear in the flux.

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The Hessian criterion based on the pressure p (PH criterion) is computed 148 as: is the density and V = √ u 2 + v 2 + w 2 . Thus, the criterion is chosen as: The maximum of two tensors is computed using the approximative procedure 170 defined by [3]. Further testing has to determine if this criterion is adequate, 171 or if diffusive and turbulence terms must also be added. if one of the cells in a column needs to be refined, all the cells are refined.

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Thus, the column / layer structure of the boundary layer grid is preserved.            Thus, in these regions the grids are geometrically similar. If this is enough 336 for convergence studies, will be seen below.
with an unknown order p is least-squares fitted through the force coefficients 343 obtained on the five grids considered. This fit is used to compute the es-   This is different for C l : the power-law fit has no solution for T r =0.0884, 408 the coefficients vary for each added point, and C l0 is always lower than C l , 409 even though C l on the finest grids increases with further refinement. The fit 410 on the finest five grids looks more reasonable, however. It is plausible that 411 the data for C l contains a higher-order error term as well as high-frequency 412 noise, so that fine grids are needed to reach the asymptotic range.

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The noise in the data may be created because subsequent grids are not      For low-Reynolds FCH computations, getting less than 1% uncertainty 501 for the forces with our procedure requires more than 100M cells. However, 502 these grids resolve the wake with very fine cells, which may not be necessary 503 for obtaining only the forces; also, 1% uncertainty is often too strict (section    30 represents a more realistic study than the 10-grid Nakayama case, whose 540 coarsest grids would be insufficient to capture the details of the more com-541 plex KVLCC2 flow, while the finest grids would be too expensive in three 542 dimensions.

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The refined meshes in the propeller plane (the aftmost cut plane on the 544 hull in figure 8) are shown in figure 9 for the four finest thresholds. These Thus, the solution on the finest mesh is sufficiently precise to assess modelling 593 errors accurately.

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These results agree with the findings of the Nakayama case (section 5.5).

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Convergence of the local flow in the wake is indeed obtained at 10M cells.  The original grid which is not refined everywhere for coarse meshes, does not deteriorate the convergence. The mesh is refined there where this is 639 crucial for the flow, which is enough to ensure grid convergence of the forces.

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The lift of an airfoil converges quickly with grid adaptation, because the