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, fluidpermeable structures prescribed by a macroscopic design approach where all solid boundaries are parameterized and modeled explicitly. Local, gradientbased 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 (
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 surfacepenetrating 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 [
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 [
Porous media saturated with a single fluid have been the subject of many theoretical and experimental studies since the works of Darcy [
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 [
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 (
The Monte Carlo sweep described above was carried out for a structure having an impermeable core and an aspect ratio of 5 : 1 (Figure
(a) The system schematic illustrates the principal cross section of a spherical body with radius
Monte Carlo sampling of the 8dimensional functional space of permeability distributions (described by the
Re = 20
Re = 100
Figure
Finally, for larger Reynolds numbers, the size of this manifold increases rapidly, implying that the efficiency of wake and drag mitigation using permeable media is substantially better for larger bodies or at higher speeds. As Re equals 100, wake reduction factor relative to a sphere with equivalent drag already exceeds a factor of two up from a factor of 25% at
While simple homogenization theories that yield the Brinkman equation are considered reliable at small Reynolds numbers, their usefulness in the turbulent regime remains controversial [
Since we are interested in evaluating a potentially very large class of complicated geometric structures, we develop a geometry parametrization scheme along the lines of the levelset method popular in twophase flow modeling.
In this section, the Brinkman term is used again, but this time only to model the impermeable, solid phase of the fluidsolid composite in a way that is compatible with a levelset parameterization of the solid boundaries. Such compatibility enables subsequent use of efficient optimization techniques. Performing optimization on a levelset parameterized binary composite has been known as topology optimization in the design of solid composites since the works of Bendsoe and Kikuchi [
Specifically, our parametrization of the inverse permeability is
Below, we consider both spherical (as in Section
To implement this approach, we use a commercial software package, COMSOL Multiphysics version 4.4 [
Unitless coefficients used in the incompressible SpalartAllmaras turbulence model, whose descriptions are available in the COMSOL CFD User’s Manual [
Variable  Value 


0.1355 

0.622 

7.1 

2/3 

0.3 

2 

0.41 

2 
With this relationship, (
This explicit form of the volume force can be suitably specified in CFD solvers that do not explicitly support porous effective media. With this approach, we were able to achieve consistent convergence while using the SpalartAllmaras turbulence model [
To reduce computational time, we consider two reducedcomplexity scenarios. In the cylindrical or 2D case, the PVCcoated object is a cylinder, and the PVC shapes are obtained by extrusion of a 2D cross section out of its plane. In other words, the levelset function in (
We selected the physical size of the impermeable cores and the freestream flow parameters based on the limitations of the water tunnel used in the experiments described below. The choices for these parameters were based on ensuring that the finitechannel (wall proximity) and shallowwater effects of the tunnel were minimal, while maximizing the size of the PVC region to keep the minimum feature size of the structure as large as possible, so that fused deposition modeling (FDM) prototyping would be seamless. Our parameter selections also ensured operation within the specifications of the tunnel and avoided simulation parameters that could lead to computational nonconvergence. These constraints led us to use a freestream velocity of 0.25 meters per second (m/s), a spherical (impervious) object radius of 27.5 millimeters (mm), and a cylindrical object with an 8 mm radius. The outer radii for the PVCs were twice the impervious core radii, with a spherical obstruction envelope radius of 55 mm and a cylindrical obstruction envelope radius of 16 mm. These parameters correspond to Reynolds numbers of 8000 (for the cylinders) and 27500 (for the spheres). Although different Reynolds numbers were tested (ranging from creep flow to turbulent flow), we settled on these Reynolds numbers for the experiment in order to meet the constraints of the resources available to us, such as the test chamber size, water tunnel velocity, and 3D printer supplier capabilities. These moderately high Re numbers well beyond the stable laminar flow regimes for cylinders and spheres necessitated the use of lowRe turbulence models in the forward solver.
The fluid (water) was approximated to have a density of 1000 kilograms per cubic meter (
For the initial guesses, we use a class of levelset functions constructed from periodic (such as trigonometric) functions as follows:
Threedimensional CAD models of onehalf of the PVCcoated cylinder (a) and the whole PVCcoated sphere with a mounting strut (b).
To achieve the results illustrated in Figure
Other wake metrics are possible, such as the minimum local pressure, domainwide pressure deviation norm, or the volume of the region where pressure deviation is negative; these metrics are more relevant to the cavitation effects; however, they were not used in the optimization of structures reported here.
In performing the optimization of the PVC, we found that the downstream hemisphere (or semiannulus) is more important in the wake formation at nonnegligible Reynolds numbers, whereas in the Stokes limit the equatorial plane is essentially a symmetry plane of the flow, and the up and downstream domains are equally important. This justifies the choices of initial guesses, which, after optimization, led to the shapes in Figure
Four structures were examined over the course of the experiment, each offering less than a 10% channel area obstruction to minimize the wall effects on the nearfield flow. All four structures, designed to be contiguous rigid bodies, were fabricated from ABS, a thermoplastic commonly used in fused deposition modeling 3D printers. The first two bodies were control samples: an impervious sphere and a cylinder spanning the height of the test section. The sphere was offset to be centered in the channel by a threaded rod. Both this threaded rod and a machine screw at the bottom of the cylinder interfaced with a mounting fixture in the bottom of the channel. The second two geometries were 3Dprinted permeable volumetric composites, one of the cylindrical type and one of the spherical type as described in Section
The prototype samples were generated from twodimensional plots of their cross sections. In order to realize these composites in three dimensions, the cross section plots were first vectorized and converted to the DXF format and then imported into 3D CAD software such as Solidworks. Vectorization was accomplished using the trace bitmap function in the Inkscape freeware and by removing extraneous elements of the twodimensional plot output manually. Once imported into threedimensional Computer Aided Design (CAD), the cross sections were either extruded (for cylindrical geometries) or revolved around the symmetry axis (for spherical samples). Finally, connection elements were added manually in the CAD environment to ensure the structural integrity of the samples. Although these viscoelastic spacers alter the flow, they are designed to be as distant from the Particle Image Velocimetry (PIV) observation plane as possible. For example, the spacer cylinder connection point is a full quarter of the sample height away from the plane examined by the PIV method, and the spherical sample’s standoffs were not inplane with the PIV sheet at all. These completed rigid solids were then threedimensionally printed in Stratasys uPrint SE Plus with an ABS thermoplastic material.
Measurements of the flow velocity profiles were performed in the water tunnel housed at the Naval Undersea Warfare Center in Newport, Rhode Island. The water tunnel (Figure
Photograph of the experimental setup, showing a water PIV tunnel with a sample, the optical camera, and the scanning laser emitter.
The prototype samples were mounted in the same manner as the control samples. In the cylindrical cases, in order to avoid oscillation at the samples top end due to vortex shedding, a viscoelastic spacer was placed between the top of the samples and the ceiling of the channel. Furthermore, because the chamber is 305 mm in height and the threedimensional printer was only capable of generating shapes that could fit inside of a 154 mm sided cube, two PVCcoated cylinders were assembled endtoend to span this height.
Twodimensional flow field behavior can be extracted from such a system via Particle Image Velocimetry (PIV). The PIV method involves projecting a laser sheet into a transparent flow field populated by reflective seeds, which are particles in the flow that will reflect the laser light so that it can be detected by a camera oriented perpendicular to the laser sheet. In Figure
Schematic of our PIV data acquisition method.
When gathering data, the camera will capture two images in quick succession, and the PIV software can identify how certain seed elements in each image have moved. By calibrating the system to recognize how many pixels in the image constitute a known measurement and knowing the time between capturing the two images, local velocities may be identified. For the purposes of this experiment, thirtyfive image pairs were averaged to illustrate the effective mean flow field.
This experiment was conducted with Litron Nano L20015 class 4 lasers and a Litron LR1550 generator. The laser was redirected and focused on the samples with an LAVision 1108453 mirror stabilizer arm, 1108405 lens, a 1101342 PIV camera with a Nikon AF 50 mm lens attached, and a cylindrical beam splitter. The beam splitter was mounted on top of a Slik AMT Tripod, and the data was processed using DaVis 7.2 PIV data acquisition software, which sampled the recording with a rectangular grid size of ninetythree points along a side. In order to appropriately compare, the computational data was interpolated to these same grid points.
Experimental and computational results for select figures of merit for both control and prototype cases of the sphere and cylinder are summarized in Table
Results summary and comparison. Note that the physical location of the PVC cylinder cap caused masking errors in the experimental data, resulting in anomalous or unnecessarily absent readings near the geometry.
Uncloaked sphere  Cloaked sphere  Cloak effect  Uncloaked cylinder  Cloaked cylinder  Cloak effect  


0.18  0.27  +52%  0.027  0.072  +160% 

0.22  0.24  +8%  0.048  0.054  +12% 
DWW difference  17%  16%  42%  34%  

1.9  1.3  −31%  1.4  1.3  −10% 

1.8  1.5  −16%  1.6  1.2  −25% 
MLW difference  5%  14%  8%  10% 
Figure
(a) Experimental local wake near the uncloaked sphere with Re = 27500. The range where
Exp
Comp
Figure
(a) Experimental local wake near the cloaked sphere with Re = 27500. The range where
Exp
Comp
Figure
(a) Experimental local wake near the uncloaked cylinder with Re = 8000. The range where
Exp
Comp
Figure
(a) Experimental local wake near the cloaked cylinder with Re = 8000. The range where
Exp
Comp
The optimization algorithms converged to certain local minima in the wake metrics depending on the prescribed initial conditions of the inverse permeability coefficient. Of all of the initial configurations tried in simulations (one or two quadrants, staggered or even obstructions, varying sizes, and spatial frequencies), three staggered downstream obstructions offered the smallest local minimum in both the spherical case and the cylindrical case. Although the DWW of the PVC cases did not drop below that of the exposed cores (control samples), which is particularly evident in Figure
With the agreement of the velocity and local wake results between simulation and experimentation, other characteristics of the computational results can be examined. The greatest of these are the pressure fields, illustrated in Figures
The pressure fields between the uncloaked sphere (a) and the cloaked sphere (b) illustrate a localization of the pressure drop in the axisymmetric cross section from the control case to the PVC case. Note the location of the PVC boundaries in (b) and how the pressure immediately downstream region has a notably reduced magnitude of negative pressure.
Uncloaked
Cloaked
The pressure fields between the core cylinder (a) and the PVC cylinder (b) illustrate a dramatic reduction in the magnitude of the negative pressure immediately downstream the core. The outline of the PVC boundaries is illustrated in (b).
Uncloaked
Cloaked
The results of this investigation illustrate that passive structures in the nearfield of a given obstruction in constant flow can be configured to control the resulting wake. The nearfield zone can be defined as the region where the flow past the structure deviates substantially from its asymptotic behavior. In the lowRe limit, it can be defined as simply twice as large as the structure itself, as suggested by the competition of the farfield (1/
The results show that the maximum value of the local downstream disturbances can be substantially reduced by employing permeable volumetric composites whose geometry is optimized to meet that specific goal. This is partially due to the reduction or elimination of the large vortex in the flow immediately downstream of the large impervious core, which is a consequence of the nearfield array of obstructions of optimized sizes and shapes and positioned at strategic locations. In terms of hydrodynamic “cloaking,” this is a valuable metric as it reduces the possibility of cavitation in the downstream flow. Figure
The experiment allowed us to quantify the differences between our CFD models and the actual timeaveraged velocity profiles. At 0.27 and 0.24, respectively, for the DWW of the spherical case, a 16% modelmeasurement discrepancy is observed. Similarly, for the cylindrical case, DWW values of 0.0721 and 0.054 yield a 34% difference. Potential sources of these differences lie in the model setup, experimental methods, and the data processing. From the computational side, in the case of lowRe RANS models such as SpalartAllmaras, one may require additional tuning when a Brinkman term is added as an extra volumetric force. The virtual noslip boundaries, created by regions of low permeability, generate additional viscous boundary layers that are not properly accounted for in the RANS code.
In the experimental setup, further discrepancy is introduced by having to use additional structural elements whose sole purpose is to maintain the structural rigidity of the sample and to immobilize it relatively to the flow. Both of these sources of errors could be, in principle, quantified by performing full threedimensional CFD modeling on the CAD geometries of the actual experimental samples, with all connectors and mounting fixtures included, a procedure we propose for future experiments. The experimental results were subject to many other variables, including the effects of the tunnel walls, fabrication accuracy issues, inconsistent seed illumination, and human error from visual calibration of the view sizes.
One final source of error is associated with the nature of the cross section illustrated by the computational models versus the projection viewed by the camera in the experiment. Parts of the outofplane physical samples masked the behavior of the flow in some regions very near to the geometric core in the PIV plane of study, making it impossible to gather flow data in those regions. As such, in these regions of question, the computational results will have a known value of twodimensional fluid velocity whereas the experimental results provide no reliable value. This obstacle makes the DWW measure less reliable from the measurement perspective than the MLW metric, as certain regions of the flow field that were measured and accounted for in the area integral in the computational case were null in the experimental case. This may explain why the MLW discrepancy is consistently smaller than the DWW discrepancy.
In conclusion, we have shown the possibility of wake control and reduction using permeable volumetric composites (PVCs), a generalization of the hydrodynamic metamaterial concept. Our experimental tests demonstrate reasonable agreement with CFD models, with the understood sources of discrepancy. Notably, our demonstrations go beyond the laminar flow regime, reaching Reynolds numbers in the Navyrelevant range of
Figure
(a) Flow field around a sphere with a symmetric boundary condition at the far side of the domain. Horizontal and vertical axes are in meters, and the color scale is in meters per second. (b) Flow field around a sphere with a wall boundary condition at the far side of the domain. Horizontal and vertical axes are in meters, and the color scale is in meters per second. (c) Pressure around a sphere with a modified domain size. Horizontal and vertical axes are in meters, and the color scale is in meters per second. (d) Tight view of the flow field around a sphere with a symmetric boundary condition at the far side of the domain. Horizontal and vertical axes are in meters, and the color scale is in meters per second. (e) Tight view of the flow field around a sphere with a wall boundary condition at the far side of the domain. Horizontal and vertical axes are in meters, and the color scale is in meters per second. (f) Tight view of the flow field around a sphere with a modified domain size. Horizontal and vertical axes are in meters, and the color scale is in meters per second. (g) Tight view of the pressure around a sphere with a symmetric boundary condition at the far side of the domain. Horizontal and vertical axes are in meters, and the color scale is in pascals. (h) Tight view of the pressure around a sphere with a wall boundary condition at the far side of the domain. Horizontal and vertical axes are in meters, and the color scale is in pascals. (i) Tight view of the pressure around a sphere with a modified domain size. Horizontal and vertical axes are in meters, and the color scale is in pascals.
Figures
(a) View of the mesh used in the study. The cloak domain was mapped with 15 elements in the larger, inner annular thickness, 8 elements in the smaller, thinner annulus, and 100 elements along the halfcircumference of the sphere cross section. This creates a total of 2300 elements. (b) View of a coarser mesh for comparison. The cloak domain was mapped with 12 elements in the larger, inner annular thickness, 5 elements in the smaller, thinner annulus, and 70 elements along the halfcircumference of the sphere cross section. This creates a total of 1330 elements, cutting 570 from the model.
(a) View of the pressure around a sphere from the study mesh. Horizontal and vertical axes are in meters, and the color scale is in pascals. (b) View of the pressure around a sphere from the coarser mesh for comparison. Horizontal and vertical axes are in meters, and the color scale is in pascals.
By reducing the number of elements in the cloak domain by nearly half, the results do not perceptibly change.
Figures
Velocity fields around a sphere at
Comparison of SpalartAllmaras to SST (a) and
The authors verify that no conflict of interests is present with regard to this investigation and their work.
This work was funded by the Office of Naval Research (ONR) through the Naval Undersea Research Program (formerly a University Laboratory Initiative), award no. N000141310743. Dean R. Culver acknowledges financial support of the Naval Research Enterprise Internship Program (NREIP), Department of the Navy. The authors are thankful to Maria Medeiros (ONR 333), Charles Henoch, James Hrubes, William Wilkinson, and the rest of the hydrodynamics group at the Naval Undersea Warfare Center Division Newport for their technical support of the experiment and to the personnel of the Center for Metamaterials and Integrated Plasmonics (Duke University) for assisting with the additive manufacturing process.