A DNS/URANS approach for simulating rough-wall turbulent flows
Introduction
The effect of roughness in wall-bounded flows is usually represented by the effective roughness height . The value of corresponds to the roughness height of the sand-grain experiments of Nikuradse (1950) which yields the same drag as that of the surface of interest. The drag due to the roughness is usually measured in terms of the roughness function , which is the shift in velocity profiles within the logarithmic region, between the smooth and rough cases (Jiménez, 2004). Alternatively, can also be obtained by considering the difference between the velocities at the centreline of the channel (or at the top of the boundary layer).
Obtaining for arbitrary surfaces is a cumbersome task both experimentally and numerically. Apart from typical sources of experimental uncertainty, the turn-around for laboratory measurements of is heavily dependent on wind- or water-tunnel availability and requires manufacturing tiled versions of each surface of interest. On the other hand, numerical approaches to computing are either limited in terms of , in the context of direct numerical simulations (DNS) or the robustness of the turbulence model within the roughness layer, in the context of large eddy simulations (LES) and Reynolds averaged Navier–Stokes models (RANS). Devising novel numerical strategies to tackle the roughness problem is a preferable approach due to growing computational power and the flexibility in manipulating or simplifying the problem setup.
In the context of LES, a rough-wall can either be treated by a wall-model or by fully resolving the turbulence in its vicinity (Pope, 2000). Naturally, wall-models require some degree of knowledge of the surface properties in order to estimate a value of , e.g. the roughness function should be known a priori. In wall-resolved LES the computational costs scale similarly as in a DNS with increasing with the added difficulty of choosing models and resolutions capable of handling the inherent anisotropy of the near wall turbulence (Baggett et al., 1997, Cabot and Moin, 2000). Naturally, RANS treatment of near wall turbulence suffers from similar (if not more severe) issues as LES.
Interest in increasing accuracy of flow solutions for industrially relevant Reynolds numbers, not limited to wall turbulence, has pushed for the development of hybrid models in which RANS and LES are used simultaneously in different parts of the flow [see the review by (Fröhlich and von Terzi, 2008)]. In wall-bounded turbulence hybrid models usually employ RANS in the vicinity of the wall and “switch” to LES somewhere in the log-layer. As noted by Fröhlich and von Terzi (2008) and Hamba (2003), a mismatch in the velocity profile develops at the interface between the models, independent of blending function or specific LES/RANS models employed. Hamba (2003) bypassed this issue by feeding DNS information to the turbulence models, which is only practical when solving flows for which DNS are available (in which case carrying out turbulence-modelled simulations becomes redundant).
It is thus not surprising that Flack (2018) argues DNS should be a necessary component of numerical strategies aimed at determining for realistic surfaces at industry-relevant friction Reynolds numbers . Because a DNS requires resolving all scales of motion, from the viscous length scale to the largest eddies which scale with the channel half-height (or boundary layer height) , grid sizes and computation time can quickly become too large to efficiently study a broad variety of surfaces.
In-line with the observations above, Chung et al. (2015) devised a DNS-based approach to computing the roughness function without the typical cost associated with a DNS. Their strategy relies in solving the flow in a channel which is just wide enough to sustain turbulence while being long enough to allow the flow to fully develop over the roughness elements. The reduced width of the domain allows for cheaper simulations while allowing for to be computed within the log-layer, since the turbulence becomes unrealistic in the outer layer (Jiménez et al., 2001). The method provides encouraging results [see (MacDonald et al., 2017)] when using body forces to mimic roughness effects but shows some discrepancies with full-scale DNS when applied to pyramid-like roughness elements.
It should be noted that the approach of Chung et al. (2015) relies on Townsend’s outer-layer similarity hypothesis. Experiments on boundary layers by Krogstad (1992) show that the presence of roughness elements can lead to mean velocity profiles which cannot be obtained from their rescaled smooth-wall counterparts. This suggests that the flow characteristics in the vicinity of the roughness elements may depend upon details of those elements and may not be represented by a single parameter, suggesting that values of obtained in channel flows under zero-pressure gradient may not be applicable to more complex flows. It should be noted that, in principle, this is not an issue for the turbulence-model approaches mentioned above (which, as discussed already, suffer from their own complications) but could prevent the method of Chung et al. (2015) from being applicable to more general problems, since that method relies on Townsend’s outer similarity hypothesis.
Drawing inspiration from the methods discussed above, in the present contribution we propose a new method for rough surface simulations in which can be obtained from simulations using DNS-like resolution but at a fraction of the cost of a typical DNS. The computational savings are obtained by solving the flow in small domains. In order to prevent the flow from laminarising (or becoming incipiently turbulent), we blend the DNS with a turbulence model. Here unsteady RANS models are used to predict the core of the flow. Notice that no assumptions are made with respect to the structure of the surface or the flow properties (such as outer layer similarity).
We begin by outlining the method itself, which we call a stress-blended method (SBM). We then assess the performance of the SBM and the effects of some of the tunable parameters (such as interface shape and location) on smooth-wall simulations. The method is further validated using a parametric forcing approach to mimic roughness effects. Finally we test SBM on a scanned grit-blasted surface for which DNS is available. We conclude by outlining the advantages and limitations of the SBM along with further developments and applications.
Section snippets
The stress-blended method
Acknowledging the need to incorporate DNS in calculating the flow over rough surfaces (Flack, 2018), we develop a method which incorporates DNS accuracy at a cheaper (computational) cost that such a DNS would require, by combining the DNS with a turbulence model in a relatively small domain. As will be seen below, the blending (of the DNS with the turbulence model) occurs at the level of the Reynolds stresses and thus we refer to the present approach as the stress-blended method (SBM).
In order
Numerical method
All simulations were carried out using second-order accurate finite-differences on a staggered grid for the spatial discretisation and a second-order accurate Adams–Bashforth method for the time integration. The grid is always stretched in the wall-normal direction using a hyperbolic tangent. For the smooth wall and parametric forcing cases the grid spacing varied between at the wall and at the centreline, whereas for the surface scan cases the resolution was kept constant within the
Results
To validate the new approach, we first compare the accuracy of SBM with regards to DNS and RANS in smooth wall turbulence. Simulations were carried out as shown in Table 1 with domain sizes substantially smaller than those required for a full scale DNS. These simulations were of both smooth wall turbulence as well as using the parametric forcing approach described above. Finally as simulation of the flow over a grit-blasted surface was carried out, spanning and keeping the
Conclusion
A novel hybrid method for tackling wall-bounded turbulent flows, called stress-blended method (SBM), has been presented. The method relies on exchanging the Reynolds stresses between the DNS and RANS regions over an interface.
Preliminary tests show that the shape and size of the interface have little effect on the flow, for interface locations within the log-layer. The mismatch between the DNS and RANS stresses over the interface requires special considerations with regards to the overall
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors acknowledge the support of EPSRC through the Grant No. EP/P009638/1 and the computational resources allocated in ARCHER HPC through the UKTC funded by the EPSRC Grant No. EP/R029326/1. Statistics obtained with SBM are openly available from the University of Southampton repository (https://doi.org/10.5258/SOTON/D1411).
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