Double-averaged Navier Stokes Equations Research Articles

Integral methods for friction decomposition and their extensions to rough-wall flows

A primary objective of integral methods, such as the momentum integral method, is to discern the physical processes contributing to skin friction. These methods encompass the momentum, kinetic energy and angular momentum integrals. This paper reformulates existing integrals based on the double-averaged Navier–Stokes equations, and extends their application to flows over rough walls. Our derivation yields distinct decompositions for the bottom-wall viscous friction coefficient, denoted as $C_S$ , and the roughness element drag coefficient $C_R$ . The decompositions comprise three terms: a viscous term, a turbulent term and a roughness (dispersive) term – regardless of the flow configuration, be it channel or boundary layer. Notably, when these integrals are evaluated for laminar flow scenarios, only the viscous term remains significant. In addition, we elucidate the spatial distributions of the terms within these decompositions. To demonstrate the practicality of our formulations, we apply them to analyse data from direct numerical simulations of turbulent half-channel flows. These flows feature aligned and staggered cubical roughness at various packing densities. Our analyses, based on kinetic-energy-oriented decompositions, reveal that when the surface coverage density $\lambda _p$ is small, the dominant terms within the decompositions are the viscous and turbulent terms. With increasing $\lambda _p$ , the viscous dissipation term decreases, while the turbulent production term increases and then decreases. These variations arise from a subdued near-wall cycle and the development of a shear layer at the height of the cubes.

Application of a Distributed Element Roughness Model to Additively Manufactured Internal Cooling Channels

Abstract Design for cooling effectiveness in turbine blades relies on accurate models for dynamic losses and heat transfer of internal cooling passages. Metal additive manufacturing (AM) has expanded the design space for these configurations, but can give rise to large-scale roughness features. The range of roughness length scales in these systems makes morphology resolved computational fluid dynamics (CFD) impractical. However, volumetric roughness models can be leveraged, as they have computational costs orders of magnitude lower. In this work, a discrete element roughness model (DERM), based on the double-averaged Navier–Stokes equations, is presented and applied to additively manufactured rough channels, representative of gas turbine blade cooling passages. Unique to this formulation of DERM is a generalized sheltering-based treatment of drag, a two-layer model for spatially averaged Reynolds stresses, and explicit treatment of dispersion. Six different AM rough surface channel configurations are studied, with roughness trough to peak sizes ranging from 15% to 60% nominal channel passage half-width, and the roughness Reynolds number ranges from Rek = 60 to 300. DERM predictions for spatially and temporally averaged mean flow quantities are compared to previously reported direct numerical simulation results. Good agreement in the mean velocity profiles, stress balances, and drag partitions are observed. While DERM models are typically calibrated to specific deterministic roughness morphologies at comparatively small roughness Reynolds numbers, the present more generalized DERM formulation has wider applicability. Here, it is demonstrated that the model can accommodate random roughness of large scale, typical of AM.

An experimental investigation of turbulent free-surface flows over a steep permeable bed

Steep streams involve shallow, supercritical turbulent flows over a permeable bed made up of coarse particles. They usually exhibit higher flow resistance and stronger mass and momentum exchanges between the stream and subsurface flow than low-gradient streams. Describing their flow dynamics using generalised Manning–Strickler equations has led to empirical relationships with weak predictive power (errors between predictions and data of over one order of magnitude). We studied shallow turbulent flows by employing a mesoscopic approach based on the double-averaged Navier–Stokes equations. More specifically, we were concerned with the possibility of modelling the turbulent and dispersive shear stress equations using simple algebraic equations. To that end, we studied shallow, supercritical turbulent flows over a sloping bed made up of randomly packed spherical particles. Using visualisation techniques based on particle velocimetry imaging and refractive index matched scanning, we were able to reconstruct the velocity field throughout the bed and stream, far from the sidewalls, and estimate the contributions of the dispersive and turbulent shear stresses to the total shear stress. The dispersive shear stress represented less than 20 % of the turbulent shear stress, but because it was concentrated within a thin layer (called the roughness layer) where it outweighed the turbulent shear stress, it had a significant influence on the mean velocity profile. We proposed an algebraic closure equation for dispersive shear stress, based on the mixing-length model used for turbulent shear stress, and we found that it captured closely the mean-velocity and turbulence-intensity profiles of shallow flows over horizontal or sloping permeable beds. Our data suggest that flow dynamics was affected largely by turbulence damping, drag forces and dispersion within the roughness layer, which may explain why steep streams differ from low-gradient streams.

Analysis of the different sources of stress acting in fully rough turbulent flows over geometrical roughness elements

The discrete element roughness method is considered in this article for the prediction of turbulent flows over rough walls. This approach is derived by ensemble- and volume-averaging the Navier–Stokes equations, providing double-averaged Navier–Stokes equations, and yielding three unknown terms in the momentum equation: the Reynolds stress and dispersive stress tensors and the average drag force acting on the roughness elements. This work aims at analyzing these different terms, quantifying their respective contributions to the near-wall momentum budget, and providing guidance for their modeling. For this purpose, relevant data of turbulent flows are required. Roughness-resolved Reynolds-averaged Navier–Stokes simulations of transitionally and fully rough channel flows over academic roughness configurations are performed at high friction Reynolds numbers ranging from 3500 to 8000. Comparisons with existing and new velocity measurements performed in rough-wall turbulent boundary layers provide support to the simulation results, a particular emphasis being given on the validity of the numerical results in the wake of the roughness elements. These numerical results highlight the influence of roughness elements geometry and density on the roughness drag coefficient and the dispersive stress. It is particularly suggested that the standard roughness drag closure model should be revisited for double-averaged flows. Furthermore, the dispersive stress is shown to mainly originate from the wake of the roughness elements, an observation that could be leveraged for its modeling. However, since this stress contributes only marginally to the global stress and to the skin friction coefficient, such a modeling may not be critical at first order.

Spatially-averaged turbulent flow characteristics over ribbed surface in presence of unidirectional wave over steady current

The paper presents an experimental study on the characterization of the effect of surface-wave of different frequencies on spatially-averaged turbulent statistics over a rough bed. The model roughness comprised of periodically positioned, transverse square ribs that extended across the whole width of the channel. The three-dimensional turbulent velocity fluctuations were mapped by an acoustic Doppler velocimeter (ADV) in the flow domain at high resolution. These data were analyzed to examine time-space averaged turbulence parameters of the double averaged Navier Stokes equations and momentum transfer under the influence of the rough boundary. Further, Shannon entropy at different vertical position were also evaluated to develop insight into the energy transfer mechanism. Results show a reduction in the apparent roughness height and an increase of the intensity of turbulence under surface wave influence. Furthermore, the superposition of waves led to the increase in turbulence production and showed maximum values at the roughness layer as compared to buffer layer in current-only flow over plane bed.

Characteristics of mean flow and turbulence statistics of depth-limited flows with submerged vegetation in a rectangular open-channel

ABSTRACTThis paper reports a numerical investigation of the characteristics of mean flow and turbulence statistics for a depth-limited flow with submerged vegetation in a rectangular channel. To achieve this, the double-averaged Navier–Stokes equations were solved using the Reynolds stress closure model. A zone with increased velocity was observed at the corner between sidewall and free surface, thus forming two velocity maxima, one at the centre of the channel and one at the corner. This is consistent with previous experimental observations. The velocity maximum at the corner can be attributed to the enlarged bottom vortex caused by submerged vegetation, which transports high-momentum fluids from the centre to the corner of the channel. The velocity maximum at the corner was found to occur in the flow with submerged vegetation regardless of roughness density and relative submergence. The velocity maximum at the corner affects the mean flow, but has a negligible effect on turbulence statistics.

Modeling flows over gravel beds by a drag force method and a modified S–A turbulence closure

A double-averaged Navier–Stokes equations (DANS) model has been developed for depth-limited open channel flows over gravels. Three test cases are used to validate the model: an open-channel flow over a densely packed gravel bed with small-scale uniform roughness (D/d50∼13, d50=median diameter of roughness elements, D=water depth), open-channel flows over large-scale sparsely distributed roughness elements (D/Δ∼2.3–8.7, Δ=roughness height) and steep slope gravel-bed river flows with D/d50∼7–25. Various methods of treatment of the gravel-induced resistance effect have been investigated. The results show that the wall function approach (WFA) is successful in simulating flows over small gravels but is not appropriate for large gravels since the vertical profile of the longitudinal velocity does not follow the logarithmic–linear relationship. The drag force method (DFM) performs better but the non-logarithmic velocity distribution generated by sparsely distributed gravels cannot be simulated accurately. Noting that the turbulence length scale within the gravel layer is governed by the gravel size, the DANS model incorporating the DFM and a modified Spalart–Allmaras (S–A) turbulence closure is proposed. The turbulence length scale parameter in the S–A model is modified to address the change in the turbulence structure within the gravel layer. The computed velocity profiles agree well with the corresponding measured profiles in all cases. Particularly, the model reproduces the S-shape velocity profile for sparsely distributed large size roughness elements. The modeling methodology is robust and can be easily integrated into the existing numerical models.

Double-Averaged Open-Channel Flows with Small Relative Submergence

We investigate the turbulent structure of shallow open channel flows where the flow depth is too small (compared with the roughness height) to form a logarithmic layer but large enough to develop an outer layer where the flow is not directly influenced by the roughness elements. Since the log layer is not present, the displacement height d, which defines the position of the zero plane, and the shear velocity u* cannot be found by fitting the velocity data to the log law. However, these parameters are still very important because they are used for scaling flow statistics for the outer and roughness layers. In this paper we propose an alternative procedure for evaluating d in laboratory conditions, where d is found from additional experiments with the fully developed log layer. We also point out the appropriate procedure for evaluating the shear velocity u* for flows with low submergence. These procedures are applied to our own laboratory flume experiments with uniform sphere roughness, where velocities were measured using Particle Image Velocimetry. Results were interpreted within the framework of the double-averaged Navier–Stokes equations and include mean velocities, turbulence intensities, Reynolds stresses, and form-induced normal and shear stresses. The data collapse well and show that in flows without a developed log layer the structure of turbulence in the outer layer remains similar to that of flows with a log layer. This means that even though the roughness layer in the experiments reported herein was sufficiently high to prevent the development of the log layer, influence of the bed roughness did not spread further up into the outer layer. Furthermore, the results show that flow statistics do not depend on relative submergence except for the form-induced stresses which increase when relative submergence decreases.

Subelement Form-Drag Parameterization in Rough-Bed Flows

Spatial averaging of the Reynolds-averaged Navier–Stokes equations gives the double-averaged Navier–Stokes equations, for which boundary drag appears naturally and explicitly in momentum conservation equations. Increasing use of the double-averaged equations, e.g., for relating flows to three-dimensional bed roughness, for evaluation of profiles of flow stresses and velocities in ecologically significant regions below roughness tops, and for modeling purposes, requires parameterization of boundary drag at subelement scales. Based on seven flows over repeated square-rib roughness and ten flows over repeated fixed simulated sand waves, with measured velocities and bed pressures, expressions for form-drag coefficient CD =f (elevation below roughness top, relative roughness submergence, roughness steepness) are obtained for each of the two-dimensional roughness types. Using these equations, form drag variation with elevation below roughness tops can be calculated using either the double average of the square ...

Quadrant analysis of persistent spatial velocity perturbations over square-bar roughness

We explore a new application of the quadrant method in the context of the double-averaged Navier–Stokes equations for studying open channel flow near rough beds. Quadrant analysis is applied to spatial disturbances of time-averaged velocity components, using the experimental data from flow over two-dimensional regular transverse square-bar roughness. The spatial velocity disturbances change periodically performing a full cycle over a single roughness element, so that the quadrant diagrams are regular closed orbits. A colour code is used to produce a quadrant map of the flow cross-section, which reveals contributions from each quadrant to the time-averaged momentum transfer.