Contribution towards a Reynolds-stress closure for low-Reynolds-number turbulence

1976 ◽  
Vol 74 (4) ◽  
pp. 593-610 ◽  
Author(s):  
K. Hanjalić ◽  
B. E. Launder
Keyword(s):  
Reynolds Number ◽  
Shear Stress ◽  
Reynolds Stress ◽  
Dissipation Rate ◽  
Numerical Solutions ◽  
Low Reynolds Number ◽  
Transport Processes ◽  
Turbulent Shear ◽  
Turbulent Shear Stress ◽  
Low Reynolds

The problem of closing the Reynolds-stress and dissipation-rate equations at low Reynolds numbers is considered, specific forms being suggested for the direct effects of viscosity on the various transport processes. By noting that the correlation coefficient$\overline{uv^2}/\overline{u^2}\overline{v^2} $is nearly constant over a considerable portion of the low-Reynolds-number region adjacent to a wall the closure is simplified to one requiring the solution of approximated transport equations for only the turbulent shear stress, the turbulent kinetic energy and the energy dissipation rate. Numerical solutions are presented for turbulent channel flow and sink flows at low Reynolds number as well as a case of a severely accelerated boundary layer in which the turbulent shear stress becomes negligible compared with the viscous stresses. Agreement with experiment is generally encouraging.

scholarly journals Turbulent Flow Computations in Rotating Cavities Using Low-Reynolds-Number Models

1996 ◽  
Author(s):  
Hector Iacovides ◽  
Kostas S. Nikas ◽  
Marcel A. F. Te Braak
Keyword(s):  
Reynolds Number ◽  
Dissipation Rate ◽  
Correction Term ◽  
Low Reynolds Number ◽  
Moment Closure ◽  
Rotating Disc ◽  
Moment Coefficient ◽  
Second Moment Closure ◽  
Low Reynolds ◽  
Radial Outflow

This study is concerned with the use of low-Reynolds-number models of turbulence transport in the computation of flows through rotating cavities. The models tested are the Launder and Sharma low-Re k-ε (L-S) and a low-Re differential second-moment closure (DSM), first used by Iacovides and Toumpanakis, both with and without the Yap correction term to the dissipation rate equation. The cases examined include rotor-stator systems without throughflow, rotor-stator systems with radial outflow, contra-rotating disc systems without throughflow and also with radial outflow, co rotating discs with radial outflow and also rotor-stator systems with radial inflow. Earlier studies have shown that, when no throughflow or when radial outflow is involved, the L-S tends to over-estimate the size of the regions over which the boundary layers remain laminar, while the zonal k-ε/l-eqn model is unable to predict partially laminarized flows. A modification to the ε equation proposed here, which in regions of low turbulence reduces the dissipation rate when the fluid is in solid body rotation, provides a simple empirical way to significantly improve the L-S predictions of partially laminarized flows through rotating cavities, to acceptable levels. The DSM model used, in some cases led to some further predictive improvements and, for rotor-stator systems without throughflow, to a significant improvement in the predicted value of the moment coefficient. The Yap length scale correction term, while in most cases it has either a beneficial or a neutral effect on the flow predictions, in cases involving radial inflow it leads to poorer predictions. Models that do not rely on wall distance thus appear more likely to have a wider range of applicability.

Analytical and numerical studies of the structure of steady separated flows

1966 ◽  
Vol 24 (1) ◽  
pp. 113-151 ◽  
Author(s):  
Odus R. Burggraf
Keyword(s):  
Reynolds Number ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Low Reynolds Number ◽  
Reynolds Numbers ◽  
Total Power ◽  
Large Reynolds Number ◽  
Boundary Velocity ◽  
Displacement Thickness ◽  
Low Reynolds

The viscous structure of a separated eddy is investigated for two cases of simplified geometry. In § 1, an analytical solution, based on a linearized model, is obtained for an eddy bounded by a circular streamline. This solution reveals the flow development from a completely viscous eddy at low Reynolds number to an inviscid rotational core at high Reynolds number, in the manner envisaged by Batchelor. Quantitatively, the solution shows that a significant inviscid core exists for a Reynolds number greater than 100. At low Reynolds number the vortex centre shifts in the direction of the boundary velocity until the inviscid core develops; at large Reynolds number, the inviscid vortex core is symmetric about the centre of the circle, except for the effect of the boundary-layer displacement-thickness. Special results are obtained for velocity profiles, skin-friction distribution, and total power dissipation in the eddy. In addition, results of the method of inner and outer expansions are compared with the complete solution, indicating that expansions of this type give valid results for separated eddies at Reynolds numbers greater than about 25 to 50. The validity of the linear analysis as a description of separated eddies is confirmed to a surprising degree by numerical solutions of the full Navier–Stokes equations for an eddy in a square cavity driven by a moving boundary at the top. These solutions were carried out by a relaxation procedure on a high-speed digital computer, and are described in § 2. Results are presented for Reynolds numbers from 0 to 400 in the form of contour plots of stream function, vorticity, and total pressure. At the higher values of Reynolds number, an inviscid core develops, but secondary eddies are present in the bottom corners of the square at all Reynolds numbers. Solutions of the energy equation were obtained also, and isotherms and wall heat-flux distributions are presented graphically.

A New Low-Reynolds-Number k-ε Model for Turbulent Flow Over Smooth and Rough Surfaces

1996 ◽  
Vol 118 (2) ◽  
pp. 255-259 ◽  
Author(s):  
Hanzhong Zhang ◽  
Mohammad Faghri ◽  
Frank M. White
Keyword(s):  
Reynolds Number ◽  
Turbulent Flow ◽  
Dissipation Rate ◽  
Rough Surfaces ◽  
Low Reynolds Number ◽  
Friction Factors ◽  
Log Law ◽  
Low Reynolds ◽  
Flow Experiments ◽  
Grain Roughness

A new low-Reynolds-number k-ε model is proposed to simulate turbulent flow over smooth and rough surfaces by including the equivalent sand-grain roughness height into the model functions. The simulation of various flow experiments shows that the model can predict the log-law velocity profile and other properties such as friction factors, turbulent kinetic energy and dissipation rate for both smooth and rough surfaces.

scholarly journals Computation of Turbulent Heat Transfer on the Walls of a 180 Degree Turn Channel With a Low Reynolds Number Reynolds Stress Model

2002 ◽  
Author(s):  
D. L. Rigby ◽  
A. A. Ameri ◽  
E. Steinthorsson
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Reynolds Stress ◽  
Low Reynolds Number ◽  
Reynolds Stress Model ◽  
Turbulent Heat Transfer ◽  
Turbulent Heat ◽  
Stress Model ◽  
Generalized Gradient ◽  
Low Reynolds

The Low Reynolds number version of the Stress-ω model and the two equation k-ω model of Wilcox were used for the calculation of turbulent heat transfer in a 180 degree turn simulating an internal coolant passage. The Stress-ω model was chosen for its robustness. The turbulent thermal fluxes were calculated by modifying and using the Generalized Gradient Diffusion Hypothesis. The results showed that using this Reynolds Stress model allowed better prediction of heat transfer compared to the k-ω two equation model. This improvement however required a finer grid and commensurately more CPU time.

An Expression of Reynolds Stresses in Turbulent Lubrication Theory

1992 ◽  
Vol 114 (1) ◽  
pp. 57-60 ◽  
Author(s):  
A. K. Tieu ◽  
P. B. Kosasih
Keyword(s):  
Reynolds Number ◽  
Shear Stress ◽  
Turbulent Flow ◽  
Low Reynolds Number ◽  
Stress Gradient ◽  
Lubrication Theory ◽  
Mixing Length ◽  
Reynolds Stresses ◽  
Shear Stress Gradient ◽  
Low Reynolds

This paper proposes an alternative model of Reynolds stresses for turbulent lubrication theory. The approach relies on Prandtl’s mixing length theory which is based on a modified Van Driest mixing formula [1]. However, unlike the previous theories [2, 3] the proposed equation is capable of accounting for the effect of shear stress gradient on the mixing length. Thus it is well suited to turbulent flow analysis in bearings where the presence of shear stress gradient due to the effect of pressure gradient should be considered. A series of velocity measurements in thin channels in the low Reynolds number turbulent flow range are analysed using the theory. The data analysis shows a strong effect of shear stress gradient on the viscous sublayer in the low Reynolds number regime. As a result, a new model of mixing length applicable to the turbulent lubrication analysis in thin film at low or high Reynolds numbers or under low or high shear stress gradient is presented.

A Reynolds-Stress Model for Near-Wall and Low-Reynolds-Number Regions

1988 ◽  
Vol 110 (1) ◽  
pp. 38-44 ◽  
Author(s):  
Nobuyuki Shima
Keyword(s):  
Reynolds Number ◽  
Reynolds Stress ◽  
Present Model ◽  
Transport Model ◽  
Low Reynolds Number ◽  
Reynolds Stress Model ◽  
Turbulence Energy ◽  
Stress Model ◽  
Low Reynolds ◽  
Near Wall

The Reynolds stress model for high Reynolds numbers proposed by Launder et al. is extended to near-wall and low-Reynolds-number regions. In the development of the model, particular attention is given to the high anisotropy of turbulent stresses in the immediate vicinity of a wall and to the behavior of the exact stress equation at the wall. A transport model for the turbulence energy dissipation rate is also developed by taking into account its compatibility with the stress model at the wall. The model and the low-Reynolds-number model of Hanjali’c and Launder are applied to fully-developed pipe flow. Comparison of the numerical results with Laufer’s data shows that the present model gives significantly improved predictions. In particular, the present model is shown to reproduce the sharp peak in the distribution of the streamwise turbulence intensity in the immediate vicinity of the wall.

scholarly journals Rotation of a low-Reynolds-number watermill: theory and simulations

2018 ◽  
Vol 849 ◽  
pp. 57-75 ◽  
Author(s):  
Lailai Zhu ◽  
Howard A. Stone
Keyword(s):  
Reynolds Number ◽  
Numerical Solutions ◽  
Numerical Study ◽  
Rotational Symmetry ◽  
Low Reynolds Number ◽  
Rotational Velocity ◽  
Microfluidic Channel ◽  
Hydrodynamic Interactions ◽  
Small Scale ◽  
Low Reynolds

Recent experiments have demonstrated that small-scale rotary devices installed in a microfluidic channel can be driven passively by the underlying flow alone without resorting to conventionally applied magnetic or electric fields. In this work, we conduct a theoretical and numerical study on such a flow-driven ‘watermill’ at low Reynolds number, focusing on its hydrodynamic features. We model the watermill by a collection of equally spaced rigid rods. Based on the classical resistive force (RF) theory and direct numerical simulations, we compute the watermill’s instantaneous rotational velocity as a function of its rod number $N$, position and orientation. When $N\geqslant 4$, the RF theory predicts that the watermill’s rotational velocity is independent of $N$ and its orientation, implying the full rotational symmetry (of infinite order), even though the geometrical configuration exhibits a lower-fold rotational symmetry; the numerical solutions including hydrodynamic interactions show a weak dependence on $N$ and the orientation. In addition, we adopt a dynamical system approach to identify the equilibrium positions of the watermill and analyse their stability. We further compare the theoretically and numerically derived rotational velocities, which agree with each other in general, while considerable discrepancy arises in certain configurations owing to the hydrodynamic interactions neglected by the RF theory. We confirm this conclusion by employing the RF-based asymptotic framework incorporating hydrodynamic interactions for a simpler watermill consisting of two or three rods and we show that accounting for hydrodynamic interactions can significantly enhance the accuracy of the theoretical predictions.

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