Boussinesq Viscosity Model Research Articles

Adjoint Complement to the Universal Momentum Law of the Wall

The paper is devoted to an adjoint complement to the universal Law of the Wall (LoW) for fluid dynamic momentum boundary layers. The latter typically follows from a strongly simplified, unidirectional shear flow under a constant stress assumption. We first derive the adjoint companion of the simplified momentum equation, while distinguishing between two strategies. Using mixing-length arguments, we demonstrate that the frozen turbulence strategy and a LoW-consistent (differentiated) approach provide virtually the same adjoint momentum equations, that differ only in a single scalar coefficient controlling the inclination in the logarithmic region. Moreover, it is seen that an adjoint LoW can be derived which resembles its primal counterpart in many aspects. The strategy is also compatible with wall-function assumptions for prominent RANS-type two-equation turbulence models, which ground on the mixing-length hypothesis. As a direct consequence of the frequently employed assumption that all primal flow properties algebraically scale with the friction velocity, it is demonstrated that a simple algebraic expression provides a consistent closure of the adjoint momentum equation in the logarithmic layer. This algebraic adjoint closure might also serve as an approximation for more general adjoint flow optimization studies using standard one- or two-equation Boussinesq-viscosity models for the primal flow. Results obtained from the suggested algebraic closure are verified against the primal/adjoint LoW formulations for both, low- and high-Re settings. Applications included in this paper refer to two- and three-dimensional shape optimizations of internal and external engineering flows. Related results indicate that the proposed adjoint algebraic turbulence closure accelerates the optimization process and provides improved optima at no computational surplus in comparison to the frozen turbulence approach.

Prediction of the spreading mechanism of 3D turbulent wall jets with explicit Reynolds–stress closures

The paper investigates the predictive performance of different explicit Reynolds–stress closure models when applied to the simulation of 3D wall jets. The flow is of particular interest for its remarkably large ratio of lateral to normal spreading. Experiments report that the lateral rate of spread exceeds the wall-normal rate of spread between five and nine times. This phenomenon is often vigorously misrepresented by RANS simulations. There exists some body of evidence suggesting that the large lateral spreading is due to significant amounts of turbulence-driven axial vorticity (J. Fluid Mech. 435 (2001) 305). The origin of the axial vorticity can be traced back to the anisotropy of turbulent normal stresses perpendicular to the jet axis. The present paper assess the ability of explicit stress–strain relationships to mimic the normal-stress anisotropy in 3D wall jets. It is shown that linear Boussinesq-viscosity models inevitably fail to render the spreading mechanism. Moreover, the paper argues that a physically sound modelling of 3D wall jets requires an explicit closure to include at least one quartic term.

Comparison of LES and RANS in bluff-body flows

The turbulent flow around bluff-bodies features a variety of complex phenomena, e.g. streamline curvature, separation and the formation of large unsteady vortical structures. In particular, an accurate representation of the interaction between mean transient motion and residual turbulence poses a challenge to numerical simulation procedures. In virtually all commercial simulation packages, the representation of turbulence relies on Reynolds-averaged Navier–Stokes (RANS) equations in conjunction with Boussinesq-viscosity models (BVM). Various studies have demonstrated the inability of established RANS methodologies to render the fundamental physics when applied to transient flows. In contrast to this, the computationally more demanding large eddy simulation (LES) is known to be a viable approach to simulate unsteady turbulent flows. The present study aims to assess the predictive prospects of advanced recent RANS practices, i.e. explicit algebraic stress models (EASM), in unsteady bluff-body flows. Results are reported in comparison to conventional BVM, LES and measurements. Examples included refer to three different cylinder flows, which indicate that the predictive accuracy obtained from an EASM is in close proximity to LES results, whereas the computational surplus remains moderate in comparison with a linear BVM.

Non-linear eddy-viscosity modelling of separated flows

The paper reviews some recent developments in the area of non-linear eddy-viscosity modelling and investigates the performance of several model variants when these are applied to flows with complex strain. The formulation of this type of model has been motivated by the desire to combine the advantageous numerical properties and economy of linear Boussinesq-viscosity models with the superior predictive performance of second-moment closure which is mathematically complex and numerically challenging. The rationale, fundamental principles and inherent properties of non-linear stress-strain relationships are considered first in general terms, with particular emphasis being placed on the mechanisms by which normal-stress anisotropy and sensitivity to curvature strain are represented. This is followed by a review of several particular model forms and the principal elements of their derivation. The models are then applied to four two-dimensional flows featuring, inter alia, strong curvature, irrotational straining, separation from curved surfaces and transition. The computational results presented demonstrate that the models possess predictive characteristics which are qualitatively similar to those of second-moment closure. However, quantitative agreement between predictions and reality varies considerably among the models, and this reflects the strong reliance of the models on the quality of the closure approximations adopted and their sensitivity to the details of the calibration process.

Computational modelling of separated flow around a streamlined body at high incidence

AbstractA computational study has been undertaken of 3D vortical separation from the curved surface of a prolate spheroid at high angle of attack (10° and 30°). Attention focuses on the predictive capabilities of a new variant of non-linear, low Re eddy-viscosity model and full second-moment closure, the latter coupled to a low Re k-ε Boussinesq-viscosity model which is applied to the semi-viscous near-wall region. The study demonstrates that both anisotropy-resolving formulations return very similar predictive performance which is in several respects superior to that achieved with the k-ε model based on the linear stress-strain relations. At the higher incidence angle, transition is free, and this is the source of considerable uncertainty in respect of the sensitivity of the predicted leeward flow to the location of transition. While none of the models is fundamentally capable of capturing natural transition, the ability of the non-linear model to suppress turbulence generation by irrotational straining at the windward impingement region, thereby preventing (bypass) transition by diffusion of turbulence from the freestream to the laminar boundary layer, is exploited to demonstrate that the sensitivity of the separated leeward flow to the location of transition is weak.

Computation of recirculating swirling flow with the GLM Reynolds stress closure

A Reynolds stress closure based on the generalized Langevin model (GLM), developed by Haworth and Pope[3,4], is applied to the flow calculation with swirl-induced recirculation. The purpose of the work is to assess the performance of this model under the complex flow conditions caused by the presence of strong swirl which gives rise to both unconventional recirculation in the vicinity of the symmetry axis and strong anisotropy in the turbulence field. Comparison of the computational results are made both with the experimental data of Roback and Johnson[11] and the computational results obtained with the typical isotropization of production model (IPM) and thek-e type Boussinesq viscosity model.

APPLICATION OF A HIGHER-ORDER TURBULENCE CLOSURE MODEL TO PLANE JET

Abstract This paper presents performance tests of a higher-order turbulence closure model for the predictions of turbulent shear flows. Computations of the momentum and temperature fields in the flow domain being considered entail the solutions of the Reynolds-averaged transport equations containing the second-order turbulence fluctuating products. Computations of the Reynolds stresses are performed with three closure models and the k-c model (one of the Boussinesq viscosity models). Computations of the second-order temperature velocity products are achieved by closing the pressure-heat flux term. These predictions are compared with three sets of experimental data. Several advantages have been observed in using the higher-order transport equations for evaluating Reynolds stresses; considerable improvement is achieved when higher-order turbulence closures are employed. This is because the Boussinesq viscosity model assumes that turbulence is isotropic and thus overpredicts the transverse normal component o...

An Investigation on Turbulent Heat Transfer of an Axisymmetric Jet Impinging on a Flat Plate

A study is reported on the heat transfer characteristics of a flat plate when an axisymmetric air jet impinges normally on to the plate. The two-dimensional Navier-Stokes equations are solved by using the kε Boussinesq Viscosity Model (BVM). Refined wall function relations are used for the evaluation of the k-equation. The turbulence Prandtl number is proposed as a function of the local ratio of turbulent energy production to energy dissipation rate. Predictions by the present model show generally good agreement with the experimental data.