Journal of Fluids

Volume 2016 (2016), Article ID 3587974, 15 pages

http://dx.doi.org/10.1155/2016/3587974

## A Volumetric Approach to Wake Reduction: Design, Optimization, and Experimental Verification

^{1}Duke University, P.O. Box 90300, Hudson Hall, Durham, NC 27705, USA^{2}Duke University, P.O. Box 90291, Hudson Hall, Durham, NC 27708, USA^{3}Office of Naval Research, Naval Undersea Warfare Center, Newport, RI 02841, USA

Received 22 October 2015; Revised 23 February 2016; Accepted 20 March 2016

Academic Editor: Jose M. Montanero

Copyright © 2016 Dean R. Culver et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Wake reduction is a crucial link in the chain leading to undetectable watercraft. Here, we explore a volumetric approach to controlling the wake in a stationary flow past cylindrical and spherical objects. In this approach, these objects are coupled with rigid, fluid-permeable structures prescribed by a macroscopic design approach where all solid boundaries are parameterized and modeled explicitly. Local, gradient-based optimization is employed which permits topological changes in the manifold describing the composite solid component(s) while still allowing the use of adjoint optimization methods. This formalism works below small Reynolds number (Re) turbulent flow (–10,000) when simulated using small Reynolds-averaged Navier-Stokes (RANS) models. The output of this topology optimization yields geometries that can be fabricated immediately using fused deposition modeling (FDM). Our prototypes have been verified in an experimental water tunnel facility, where the use of Particle Image Velocimetry (PIV) described the velocity profile. Comparisons with our computational models show excellent agreement for the spherical shapes and reasonable match for cylindrical shapes, with well-understood sources of error. Two important figures of merit are considered: domain-wide wake (DWW) and maximum local wake (MLW), metrics of the flow field disturbance whose definitions are described.

#### 1. Introduction

Hydrodynamic drag and the closely related phenomenon of wake are two of the most important characteristics of a vessel. While drag is directly responsible for the bulk of the energy expenditure of any propulsion system, wake properties describe the disturbance left behind in the flow field and, therefore, particularly in the cases of surface-penetrating vehicles, how easy it is to detect the vessel itself. Much of the prior research in vessel hydrodynamics has been focused on the control and reduction of drag. The methods for achieving this are numerous, including active and passive structural and passive chemical approaches. Existing work in hydrodynamic flow control via structures includes tangential surface motion control on a cylinder [1] and active and passive structures in sea creatures [2]. Chemistry and fluid heterogeneity offer another avenue into drag reduction, such as studies considering hydrophobic surfaces [3], surfactant injection [4], bubble injection [5], and supercavitation [6]. Attention to the wake in hydrodynamics was given mostly to the extent that it is useful to control the drag coefficient. In aerodynamics, wake control has found substantial relevance in aircraft maneuverability [7], efficiency of morphing wings [8], and noise mitigation [9]. Another active area of research is passive and active turbulence control; efforts were made to leverage the critical Reynolds number and postpone the onset of turbulence or flow separation through, for example, synthetic jets [10] and oscillating walls [11]. Active control of fully turbulent flows in order to lower the drag is also currently investigated; see, for example, [12] for further references on that topic.

While these approaches to drag reduction are somewhat successful, they focus almost exclusively on the boundary, attempting to control or modify the flow through the modifications of the boundary shape, or active interactions with the boundary layer, as in the case of synthetic jets and artificial cilia [13]. The possibility of controlling the flow with a multilayered, three-dimensional, volumetric structure has not received much attention until recently, due to the difficulty of modeling, designing, and fabricating any complex, three-dimensional structures. This is especially true for active scenarios. Urzhumov and Smith [14, 15] have investigated the possibilities in volumetric flow control using a simplified, two-scale description based on the local permeability of a porous medium. Motivated by recent advances in transformation electromagnetics [16] and acoustics [17], they considered both passive and active scenarios for the control of the Stokes flow around a spherical object and found that complete cancellation of both the drag term and the domain-wide wake requires an active system. These studies [14, 15] did not address the question as to whether or not any wake reduction can be achieved in hydrodynamics by passive means and also left out any consideration of flow nonlinearity and turbulence. With ever-improving computational hardware and numerical methods, it has recently become possible to perform complete physical modeling of a highly complex fluid-solid composite without the need for local homogenization. Moreover, with solution times in the range of seconds, it is feasible to perform optimization of the shapes and topologies of the solid phase, provided that smart choices in the shape parameterization and optimization algorithms are made. While global optimization of fully three-dimensional fluid-solid composites is still beyond reach for a single-CPU computer, two-dimensional and axisymmetric stationary flows through passive structures are now amenable to exhaustive investigations. The task remains feasible even with inclusion of developed-turbulence models based on RANS equations, such as, for example, the Spalart-Allmaras model [18] or a low-Re - model, both available in a commercial Computational Fluid Dynamics (CFD) simulator, COMSOL Multiphysics. This paper reports our initial effort in that direction, supplemented by experimental results on a rapid prototype sample measured in a particle velocimetry tunnel.

#### 2. Materials and Methods

##### 2.1. Wake and Drag Manipulation with Homogenizable Porous Media

Porous media saturated with a single fluid have been the subject of many theoretical and experimental studies since the works of Darcy [20] and Brinkman [21]. In the creeping flow limit, homogenization of porous media has been rigorously achieved by applying asymptotic theories to the Navier-Stokes model [22]. However, Brinkman-Stokes equation [23] is generally believed to be applicable even at nonvanishing Reynolds numbers (), with or without the nonlinear Forchheimer correction [24]. And yet, homogenization of turbulent porous flows remains a highly complicated subject [24, 25]. The Brinkman equation readswhere is the true (microscopic) fluid viscosity. Here, we assume that the porosity is close to 100%, implying that a small solid filling fraction in the porous media is considered. The last term on the right-hand side of (1) is the Brinkman term corresponding to the linear (Darcy) filtration law; it depends on the effective medium parameter , which is the permeability.

Permeability and porosity are the constitutive parameters of composite media that can be controlled by engineering the microstructure of the medium. Recently, composite media with engineered effective medium properties have become known as metamaterials. The main distinction between naturally occurring composite media and metamaterials is the precisely controlled and often periodic microstructure of the latter.

The effect of radial permeability distributions on the flow past spherical structures has been a subject of several studies [14, 26, 27]. Recently, the effects of these distributions on the turbulent wake of spherically symmetric structures [28] and the transition Reynolds number of cylindrical structures [15] were considered. However, the effect of more spatially complex permeability distributions remains unknown, due to both the computational cost of full three-dimensional solutions and the much larger functional space of permeability maps to be explored. Performing such a study is our starting point for evaluating the potential of passive permeable structures for wake reduction.

Here, we consider an impermeable spherical body enclosed in a permeable, spherical shell (the envelope). Ideally, we wish to know the effect of every physically possible permeability distribution on the main properties of the stationary flow, such as wake and drag. In order to sample the infinitely dimensional functional space of all possible permeability distributions, we consider the following parametrization, which introduces a finite, selectable number of parameters. One set () of these parameters describes the radial variation of permeability, and two other sets ( and ) describe the angular variation. Specifically,create an 8-dimensional parametric space for any in the layer, that is, forwhere is the number of layers. In the above, we keep only the first four coefficients on the Fourier series in the angle. The generalization to an arbitrary number of angular degrees of freedom is intuitive. In our numerical calculations, we further restrict ourselves to four radial layers. We then sample this eight-dimensional parameter space by creating a uniform grid within its boundaries and solving the Brinkman-Stokes equation (1) for each spatial distribution. The specific (exponential) form in (2) was chosen to ensure that the permeability distributions remain positive everywhere.

The Monte Carlo sweep described above was carried out for a structure having an impermeable core and an aspect ratio of 5 : 1 (Figure 1). The manifold describing the relationship between the wake and the drag of nonspherically symmetric structures indicates that passive permeable media with controlled permeability distributions can offer significant trade-offs between the drag and the wake of the structure. The manifold describing the full range of achievable wake and drag combinations is illustrated in Figure 2.