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.

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.

Adjoint-Based Aerodynamic Shape Optimization for Low Reynolds Number Airfoils

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Juanmian Lei ◽  
Jiandong He
Keyword(s):  
Reynolds Number ◽  
Shape Optimization ◽  
Low Reynolds Number ◽  
Boundary Layer Separation ◽  
Small Scale ◽  
Test Case ◽  
Aerodynamic Shape Optimization ◽  
Aerodynamic Shape ◽  
Aerial Vehicles ◽  
Low Reynolds

In the past decades, most of the research studies on airfoil shape design and optimization were focused on high Reynolds number airfoils. However, low Reynolds number airfoils have attracted significant attention nowadays due to their vast applications, ranging from micro-aerial vehicles (MAVs) to small-scale unmanned aerial vehicles. For low Reynolds number airfoils, the unsteady effects caused by boundary layer separation cannot be neglected. In this paper, we present an aerodynamic shape optimization framework for low Reynolds number airfoil that we developed based on the unsteady laminar N–S equation and the adjoint method. Finally, using the developed framework, we performed a test case with NACA0012 airfoil as a baseline configuration and the inverse of lift to drag ratio as the cost function. The optimization was carried out at Re = 10,000 and Ma = 0.2. The results demonstrate the effectiveness of the framework.

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.

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