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Analysis and Qualitative Effects of Large Breasts on Aerodynamic Performance and Wake of a “Miss Kobayashi’s Dragon Maid” Character

January 2018

DOI:10.13140/RG.2.2.30181.50404/1

Authors:

A computational fluid dynamics methodology is used to study the salient flow features around the breasts of a human figure and to describe the aerodynamic differences imparted by their geometric presence. Two models are proposed for examination: a 3-dimensional reference based on a character design with a significantly buxom figure and a modification of this design where the breast size is reduced significantly. The two models are tested at speeds ranging from 1 to 30 m⋅s^-1 using Reynolds-averaged Navier Stokes (RANS). Drag, lift, and skin friction forces, along with turbulence kinetic energy (TKE), are investigated and compared between the different models. The present results are expected to provide useful information on the validity of the statement, "Flat is Justice" in terms of an aerodynamic standpoint. In addition to this, the results can offer worthwhile data investigating the anthropometrical presence of large breasts on sport aerodynamics.

Comparison of wake velocity magnitudes at 5 z-locations centered along x = 0 and ranging from −1 ≤ í µí¼í µí¼ ≤ 2 meters. Lucoa positioned and to scale with x-axis. Feet z = 0.1m, Legs z = 0.5m, Hips z = 0.9m, Chest z = 1.18m, Head z = 1.55m.

…

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Analysis and Qualitative Effects of Large Breasts on Aerodynamic Performance and

Wak e of a “Miss Kobayashi’s Dragon Maid” Character

N. Rabino

ARTICLE INFO

Keywords:

Computational fluid dynamics, ANSYS, drag

coefficient, human aerodynamics, SST k-ω

model, anime, Quetzalcoatl, titties, thicc

AMS Subject Classifications:

00A72, 76-05, 76G25

ABSTRACT

A computational fluid dynamics methodology is used to study the salient flow features around

the breasts of a human figure and to describe the aerodynamic differences imparted by their

geometric presence. Two models are proposed for examination: a 3-dimensional reference based

on a character design with a significantly buxom figure and a modification of this design where

the breast size is reduced significantly. The two models are tested at speeds ranging from 1 to 30

m⋅s^-1 using Reynolds-averaged Navier Stokes (RANS). Drag, lift, and skin friction forces,

along with turbulence kinetic energy (TKE), are investigated and compared between the differ-

ent models. The present results are expected to provide useful information on the validity of the

statement, “Flat is Justice” in terms of an aerodynamic standpoint. In addition to this, the results

can offer worthwhile data investigating the anthropometrical presence of large breasts on sport

aerodynamics.

1. Introduction

The aerodynamics of the human form has been an area of valuable

research in various aspects of sports and competition. Air resistance

(hereinafter referred to as “drag”) is a concerning factor in many time-

based trials, and enhancing potential efficiency can be done through the

elucidation of the flow around the human figure. Studies concerning the

drag of the human body using wind tunnels can be found dating back to

the 1920s [1]. A small sampling of subsequent studies exploring the

effect of drag covers areas such as running [2], cycling [3], skiing [4],

and skating [5], all of which reinforces the relevance of aerodynamic

investigation on the human shape in regards to performance.

In many of such studies, the authors seek to investigate the effect of

positioning in relation to drag [6], and some utilize numerous subjects

of differing anthropometric proportions to describe a generalized result

on such positioning [7, 8]. Hitherto, none within the author’s investiga-

tions has described the effect of specific physiological features on aero-

dynamic performance in great detail. Stemming from certain internet

communities and pertinent to the current era comes the succinct state-

ment, “Flat is Justice”, which consequentially begets interesting debate

that can reverberate and diffuse throughout media. Essentially, the

statement describes the appreciation of flat-chested women [9], which

posits a peculiar aspect that has yet to be fully explored in human aero-

dynamics; namely, the effect of breasts in regards to drag and overall

aerodynamic performance.

This work is intended to contribute to the understanding of how

large breasts can affect the dynamics of the human wake through the

use of computational fluid dynamics (CFD) simulation tools. This pre-

liminary work focuses solely on comparing the relevant effects of large

breasts of a selected human design to that of the same design but with,

euphemistically, “lesser tracts of land”. The following sections will

present an overall understanding on the human wake in relation to sim-

plified geometry along with engineering applications, introduce the

chosen human geometry and models, relevant boundary conditions, the

governing equations, and the numerical methods used to solve the equa-

tions. An in-depth review on the computational uncertainty is described,

following with extensive results and discussion, conclusions, and rec-

ommendations for future work.

1.1 Background on the Human Wake

The human body can best be described as a bluff-body in respect to

the flow around it. Literature on the behavior of the wakes behind bluff

bodies indicates that the flow will be unsteady due to the turbulent tran-

sition and separation of the boundary layer [10]. A simplification analo-

gous to the human shape can be represented by a grouping of uniform

circular cylinders [11] and therefore existing studies on this type of

geometry can provide general insight into the wake region. Sumner et

al. [12] described the wake and development of vortex structures of

cylinders with aspect ratios (i.e. height to diameter) of 3, 5, and 9, and

determined that a transition in vortex shedding occurs at /=3. An

investigation by Okamoto and Sunabashiri [13] also supports this find-

ing, adding that cylinders with an aspect ratio of 3 experience a recircu-

lation region that extends four diameters downstream. Assuming the

human form takes on a roughly large cylindrical shape near this aspect

ratio, it is to be expected that the recirculation region will behave simi-

larly and extend approximately four body widths downstream.

A readily apparent deviation in geometry compared to studies done

on singular cylinders is the presence of the gap between the legs. An

extensive and comprehensive review done by Zhou and Alam [14] on

the various arrangements of two cylinders indicate the wake structure

falls into a multitude of regimes. In a side-by-side configuration, being

similar to the two legs of a human, it is deduced that there are three

primary regimes where the wake experiences proximity interference.

When closely spaced together, the first regime shows that the cylinders

act similarly to that of a single bluff body with a width corresponding to

the two cylinders. When the gap width is larger than 20% of the diame-

ter, each cylinder has individual wakes that strongly affect one another

and is associated with the second regime. At gap widths exceeding ap-

proximately 100~120% of the diameter, each cylinder acts as an inde-

pendent body with the vortex streets being loosely influenced by one

another. Seeing that human legs are not strictly cylinders with a fixed

diameter but more akin to inverted tapered cylinders, the wakes behind

the legs will likely behave in a similar fashion observed in both the first

and second regime. With the ankles and calves being narrower and hav-

ing a larger gap between them, the second regime is applicable. A tran-

sition into the first regime can be expected associated with the bulkiness

of the thighs and reduction in gap width.

Engineering literature can also provide additional details on the

flow characteristics around the body. Many of such studies are motivat-

ed by exposure control and contaminant transport [15, 16], thermal

issues [17, 18], and comfort prediction [19], rather than overall drag

effects. Inherently, many of the tested flow characteristics are evaluated

in a quiescent environment or at air velocities that are of a lower order

compared to those found in sport-related studies. Nonetheless, these

studies provide useful insight on the natural turbulence caused by the

human form and the expected anatomical location of flow separation.

Inthavong et al. [20] utilized a high speed camera to record the wake

generation of a 1/5th scaled realistic human manikin that was accelerat-

ed to a velocity of ~1 m⋅s-1. From their results, it was found that the

shoulder undergoes flow separation and produces vortices in a regular

pattern. The hands produce a well-defined yet unstable vortex sheet that

curls towards the centerline of the body. The head acts similarly to clas-

sical sphere/cylinder cases with the addition of a trailing wake forming

from behind the neck. The neck was found to remove the expected

counter free shear layer that is present in cylinder studies and thus elim-

inates the formation of an oscillating vortex sheet. In all, it can be said

that the observed human wake is a highly complex and richly diverse

system that is easily influenced by the inherent geometry used; it is

expected that from this study, an overall summary can be presented on

how and to what degree the previously described flow structures are

affected by the presence of large breasts.

2. Methodology

2.1 Design Proposal and Model Scaling

The use of realistic human models affords greater realization of the

pertinent flow characteristics as they are considerably different than

those of generalized models. Ya n et al. [21] concluded that an excessive

degree of simplification in using a manikin can affect the ability to

achieve accurate results, and thus precludes the use of a simplified

model for this study. However, the acquisition of a 3-D scanned human

model with a significant bust indubitably proved difficult. The use of a

highly unconventional approach was used to ameliorate this issue.

The animated adaptation of Miss Kobayashi’s Dragon Maid, being

a recently popular show [22] and spawning a sizable subculture on the

internet [23], proved suitable in terms of providing potential models.

The dragon characters (themselves being based off of mythically and

culturally prominent dragons) assume a human form to interact with

other humans in this well-received [24] slice-of-life urban fantasy. A

majority of the human forms of the female dragon characters possessed

significant busts. However, Quetzalcoatl (referred to canonically as

“Lucoa” and will be named as such throughout the rest of this paper)

substantiated herself as the adaptation’s gag character by her significant

size [25, 26], thus making her the perfect candidate in providing a suita-

ble model. Being clearly the largest amongst her fellow dragons as es-

tablished in Figure 1, Lucoa provides the best contrast between a large

bust and having none at all. To provide the most direct comparison in

regards to the effect of large breasts on the wake, a dramatic reduction

in bust size as reflected in Figure 1(b) was proposed for use in this

study.

In order to obtain accurate results from the setups described later in

this paper, it is important to have the subject in question reflect real

world scales properly. Lucoa’s height while in human form is not given

explicitly in any related media within the scope of the author’s research.

Thus, Lucoa’s height must be estimated in relation to objects of which a

reasonable measurement can readily be found. Conveniently, there is a

scene found within Episode 6, Season 1 of the animated adaptation

wherein Lucoa steps through a doorframe. Assuming the door is of a

typical size1 used for external entrances, in addition to Lucoa being

scaled properly in the scene, we can estimate her height using a vanish-

ing point technique.

Using the door as depicted in Figure 2 to judge Lucoa’s height, it

was determined that she stands approximately 177 cm measured to the

top of her hat, with her horns boosting her overall figure to a height of

182 cm. These numbers can be considered reasonable based on canoni-

cal descriptions of Lucoa’s towering stature [27] compared to the aver-

age height of 158 cm for a Japanese woman [28].

2.2 3-Dimensional Models and Geometry Analysis

Since Lucoa is a fictional character that is commonly portrayed in

a 2-dimensional2 world, determining her form drag between the two

proposed designs as described in Figure 1 requires that we add another

dimension to her model. Conveniently enough, an available 3-D model

of Lucoa [29] was used that would make the simulation possible. This

MikuMikuDance 3-D model (henceforth referred to as the “Normal”

model) was then imported into the 3-D modeling program Blender,

scaled to the determined height as described in the previous section,

then exported into an STL file. This STL file was then repaired using

the built-in repair feature present in Microsoft 3D Builder due to the

unclean geometry inherent with the model. To achieve the modified

design (henceforth referred to as the “Flat” model), the original Miku-

MikuDance model was modified using the built-in tools in Blender to

dramatically reduce Lucoa’s breast size. The export and repair process

remained the same as for the original model.

As shown in Figure 3, all positions between the two models remain

the same and left unperturbed to leave the reduction in breast size as the

sole geometric difference to be investigated. Although the typical or-

thostatic (standing) orientation of a human has the upper limbs in a

1 A typical metric external door’s size is 926 mm wide by 2040 mm tall.

2 Referring to the media she is portrayed in, such as printed materials and televi-

sion.

(a) Original reference design.

Courtesy: Kyoto Animation.

(b) The modified design proposed

for comparison.

Figure 1. Comparison of different designs for Lucoa.

Figure 2. A perspective measurement of Lucoa in reference to a door frame

using the Vanishing Point Tool in Adobe Photoshop.

relaxed position [30], the arms are left posed at a 45° adduction angle

from the torso, as this is the default ‘A’ pose when importing the model.

This arm position also has an advantage in this study as it potentially

enables a more thorough analysis on the effect of breasts on the wake

region, whereas a neutral standing posture would have the arms inter-

fere with the downstream effect of the breasts. The hair is left modeled

as solid to reduce simulation complexity and setup. 3 While humans

naturally lean forward against the direction of the wind to maintain

equilibrium [31], this factor is not considered in this study as this lean-

ing would change the frontal area exposed to the fluid flow and thus

complicate comparisons against static reference models.

Dimensionally, the bounds of the two models are similar, with the

height and arm span being 1.82 and 1.387 meters respectively. The

Normal model has a depth of 0.525 meters whereas the Flat model is

only 0.414 meters. The frontal projected area, , of both models is

0.584 m2. The volumetric difference between the two is 9.19 L, indicat-

ing that each breast on the Normal model has an enormous volume of

approximately 4.6 L. The under-bust circumference of the Normal mod-

el is approximately 64 cm and the bust measures 115 cm. The Flat mod-

el has the same under-bust measurement whereas the bust measures 68

cm. Attempting to match the dimensions and bust volume of the Normal

model to existing cup sizing scales is difficult as these measurements

are exceptionally large and exceed volumes measured in other studies

[32]. Using the JIS L 4006:1998 [33] scale and extrapolating4 cup siz-

ing from the largest listed size (I-cup), the Normal model can be de-

scribed as being 10 cups larger; an estimated “S65”. The Flat model is a

stark contrast to this, where it matches a petite “AA65” size.

The dramatic difference in bust size between the models serves to

provide the most significant change in outcomes; it is assumed that due

to the absurd bust size, any size smaller than the Normal model would

have an outcome that would fall in a range between both models.

2.3 Evaluated Metrics and Implementation

Four metrics under investigation for this study include drag and lift

forces (including their associated coefficients), skin friction coefficient,

and finally, turbulence kinetic energy. To evaluate the drag coefficient,

CD, and drag force, FD, the following equations are used,

CD=2FD



2

(1)

FD=(−cos+sin)



(2)

3 Hair physics is beyond the scope of the author, and thus this study, due to the

inordinate amount of computing resources and time needed to setup and simulate

hair strands in a physically accurate fashion.

4 In [33], each cup size is binned with every 2.5 cm deviation from the under-

bust measurement starting from 7.5 cm.

where  is the fluid density,  is the free stream velocity,  is the

frontal projected area, and  is the pressure at the surface .  is the

local wall shear stress being defined as,



=0

(3)

with  as the dynamic viscosity,  the flow velocity along the boundary,

and  being the height above the boundary. The value of CD is not con-

stant and is dependent on Reynolds number, which is defined as,

Re=



(4)

where  is an arbitrary characteristic length. In this study,  is equal to

the height of the models.

The lift coefficient is comparable to the drag coefficient, being that

the force is evaluated in a direction that is perpendicular to the mean

flow direction, e.g. vertically upwards. Thus,

CL=2FL



2

(5)

FL=(−sin+cos)



(6)

Instead of the frontal projected area, , a reference surface area, , is

used. For consistent comparison however,  and  are left defined as

being equivalent, thus =. This result does not affect the calculated

forces but rather only the coefficient, and as such, the lift coefficient is

dependent on the frontal area.

The skin friction coefficient, Cf, is evaluated in a similar manner to

the drag coefficient since the force attributed to skin friction is a com-

ponent of the profile drag, FD. Therefore,

Cf=2



2(7)

Analyzing the skin friction coefficient allows insight into areas where

the boundary layer thickness changes; as turbulent flow increases, the

thickness of the boundary layer increases, and consequently areas where

Cf transitions to larger values or experiences spikes are indicative of

where flow separation is prevalent [34, 35].

Turbulence kinetic energy (TKE) signifies of the loss of kinetic en-

ergy from the mean flow and represents the energy present with eddies

in turbulent flow; it is a direct measure of the intensity of turbulence. In

a general form quantifying the mean of turbulence normal stresses, TKE

is defined as,

=1

2()2

+()2

+()2



(8)

The exact value of TKE is calculated based on the closure of the Reyn-

olds-averaged Navier-Stokes equations, which is further discussed in

Section 3.3.

The numerical simulations in this present work, along with the au-

tomatic evaluation of the equations described in this section, were car-

ried out using ANSYS FLUENT R17. The 3-D models defined in Sec-

tion 2.2 were imported into FLUENT and followed the methodology as

described in the following section.

3. Computational Fluid Dynamics (CFD) Setup

and Analysis

3.1 Boundary Conditions

The use of boundary conditions based on real-world environments

enhances the overall applicability of the results stemming from the sim-

ulations. It was therefore important to determine the most appropriate

and accurate environment in which to simulate the models with. It was

(a) Reference (Normal) model.

(b) Modified (Flat) model.

Figure 3. 3-D representations of Lucoa to be used in CFD simulations, detailing

(clockwise) top, side, and front views.

found that the overall location used in the animated adaptation of Miss

Kobayashi’s Dragon Maid was based on the city of Koshigaya [36],

situated in the Saitama Prefecture of Japan. A logical time of year to

assume a person being outside without excess clothing would be some-

time in the summer. Using the month of August, it was found that

weather conditions in Koshigaya and nearby surrounding regions fea-

ture averages [37] of 22.6°C for temperature, 73% for relative humidity,

and 1005.9 hPa for local atmospheric pressure. Thus, the air density was

calculated to be =1.1581 kgm−3 and the dynamic viscosity to be

=1.8684710−5 kgm−1s−1.

Since the human body can vary based upon the clothing worn, sur-

face roughness and the effects of fabrics are parameters that are ignored

in this study. Although multiple studies have shown fabrics have a no-

ticeable effect on the overall drag of a human body [7, 38], the walls in

this computational work can be regarded as smooth. In all simulations,

the models and ground of the domain are modeled as non-moving walls

with no-slip conditions. The clothing that is part of the models is treated

in the same manner.

Tab le 1. Summary of boundary conditions in the present study.

Wind speed



ms−1

1.0, 2.5, 5.0, 7.5,

10.0, 15.0, 20.0,

25.0, 30.0

Reynolds

Number

−

1.281 ⋅ 105 ~

3.384 ⋅ 10

Domain bounds

−

−1.441.44

−2.345.65

02.32

Turbulent

intensity

inlet

−

outlet,backflow

−

Turbulent

viscosity ratio



inlet

−



outlet,backflow

−

Outlet gauge

pressure



Inlet velocities range from 1 m⋅s-1 to 30 m⋅s-1 in the positive y-

direction (since in this respect, the positive z-direction refers to the “up”

orientation; refer to Figure 4 for clarification), highlighting typical wind

speeds encountered on a day-to-day basis such as walking [39] all the

way up to standing in a violent storm [40]. At the inlet, turbulence is

specified using both turbulence intensity, , and turbulent viscosity

ratio, /. Turbulence intensity is defined as the ratio of the root-mean-

square of velocity fluctuations, , to the mean flow velocity, , and

the turbulent viscosity ratio being directly proportional to the turbulent

Reynolds number (2/). These values are summarized in Table

The boundary condition at the outlet is treated as a pressure outlet

where a static gauge pressure is specified. In this case, turbulence is

specified similarly as the inlet but regarded in terms of “backflow”,

should the flow reverse direction at the boundary during iterative calcu-

lations. The remaining borders of the “virtual wind tunnel” are modeled

as symmetric to simulate zero-shear slip walls. In FLUENT, this bound-

ary condition assumes a zero flux for all quantities, which imposes a

zero normal gradient across the defined boundary and thus enforces a

parallel flow.

In FLUENT, the flow is initialized with a velocity field equal to the

specified velocity for the run, e.g., a run specified at 1.0 m⋅s-1 would

have the entire field initialized with that value, and so on. Turbulence

parameters at the boundaries are also initialized based on turbulence

values as specified in Table 1. The blockage ratio was determined to be

8.7%, which would necessitate the usage of a correction factor to data;

however, a blockage ratio of up to 10% in regards to bluff bodies has

shown to provide reasonably similar outcomes compared to testing

using lower blockage ratios [41] and therefore a correction factor was

not used.

3.2 Grid Generation

The computational domain was discretized with an unstructured

grid as shown in Figure 5. To reduce numerical diffusion and to more

accurately resolve the viscous boundary layer, the surface grids on the

models and ground were extruded using prismatic elements that are

sized appropriately to the aspect ratio of their associated surface cell.

These prisms are grown to 5 layers and follows recommendations put

forth by Lanfrit [42]. Two prismatic bodies of influence (BOI) of in-

creasing refinement are used to improve the resolution of the grid in

both the wake region and the surrounding area around the models to

sufficiently capture turbulence and flow separation. This is done to en-

sure that computational processing is focused on more important re-

gions in the flow regime while keeping the far field sufficiently coarse

enough as to not dramatically hamper computational time. The overall

grid is limited to a maximum spacing of 0.1 m and a minimum of 2 mm.

The smaller, finer BOI is sized by the bounds −0.8940.894,

−0.12.76, 01.87 and the larger, coarser BOI defined by

−0.9440.944, −14.26, 01.92, all in meters.

A conversion algorithm in FLUENT was used to convert the pre-

liminary tetrahedral and prismatic grid into a polyhedral one. Polyhedra

exhibit advantages over tetrahedra, namely, they approximate gradients

better than tetrahedra due to the fact they are bounded by many neigh-

bors. Additionally, polyhedra have more lax geometric criteria due to

their insensitivity to stretching, making grid pre- and post-processing

easier; this is well suited to the highly complex geometry of the models

used. It has been observed that polyhedral grids provide the same level

of accuracy as tetrahedral ones, but of significantly lower element

count, thereby hastening simulations [43]. Furthermore, polyhedral

grids have shown to improve convergence while having notably greater

accuracy under unsteady simulations [44]. This is further supported by

similar external aerodynamic studies run under FLUENT, where

speedups between 2 to 3 times towards a converged solution have been

observed [45].

3.3 Turbulence Model and Computational Approach

The flow around the models is modeled with Reynolds-averaged

Navier-Stokes (RANS) equations in incompressible form. Written in

Cartesian tensor form and having flow variables of the form =+

(with  and being the mean and fluctuating components respectively)

Figure 4. Boundary conditions of the computational domain, with the inlet being

represented in blue, outlet in red, walls in white, and symmetry in yellow.

Figure 5. Side view of the full grid domain along the median plane.

being substituted into the instantaneous continuity and momentum

equations of the exact Navier-Stokes equations, we obtain [46]



+

()=0

(9)



()+ 





()

=−



+











+



−2





+



−





(10)

The closure of the set of equations are done by means of Menter’s

two-equation blended k-ε / k-ω shear stress transport (SST) model [47]

which computes eddy viscosity with a linear stress-strain closure. Ment-

er’s model combines and smoothly blends the individual strengths of

the k-ε and k-ω models, where the k-ε model sufficiently predicts turbu-

lence in both the free stream and wake, and the k-ω model more accu-

rately predicts boundary layer separation in adverse pressure gradients

near no-slip surfaces.

The spatial convective terms in Equations (9) and (10) are discre-

tized using a second-order upwind scheme (except for pressure being

solved using PRESTO! — PREssure STaggering Option, which is simi-

lar to staggered-grid schemes used on structured grids [48]) with the

diffusive terms discretized using a weighted least squares cell-based

construction technique. The inadequacies of the least squares approach

are well-known; however, it provides accuracy that is comparable to

nodal schemes [49] and due to the use of a polyhedral mesh, the least

squares approach provides an adequate balance between computational

expense and accuracy. The solver utilizes a pressure-based coupled

approach to the equations, which is found to have superior convergence

and reduced computational time than segregated algorithms [50]. The

computational fluid dynamics code used in ANSYS FLUENT has been

validated and used in many widespread applications, e.g., reverse flow

in converging channels [51], moment and lift predictions on full-scale

3-D models of airplanes [45], bluff-body drag prediction [52], heat

transfer in Couette flows [53], and complete direct injection internal

combustion engine simulation [54].

Simulating the flow around a human body was found to be naturally

unsteady as shown by Edge et al. [55], and the ideal approach would be

solving the flow equations in a time-dependent fashion then finding the

computed mean quantities manually. However, simulating the flows in a

time-accurate manner and finding the computed means would be pro-

hibitively time-consuming for this study, especially since there are a

total of 18 different runs that would need to be completed. In addition to

this, obtaining time-averaged results from transient simulations for all

of the required runs would require computational resources beyond the

capabilities of the author’s reach.5 Thus, the most realistic and only

feasible approach was to attempt a steady-state calculation, then deter-

mine iteration-averaged quantities once a quasi-steady-state in the flow

field was developed. This method is not without major drawbacks; no-

tably, obtaining exact quantities can only be achieved by finding the

limit of an infinite sample and it is therefore expected that the comput-

ed-mean will contain sampling errors. Moreover, this approach would

more than likely imply that small shedding features are not resolved and

thus their effects on the overall flow are ignored; this suggests that there

will be accuracy implications in terms of solutions to the problems un-

der examination.

In order to give the simulations a semblance of a fighting chance, a

pseudo-transient time-stepping method was applied to the flow equa-

tions being solved. This implicit under-relaxation serves as a predictor-

corrector method with the objective of fast convergence and accurate

temporal integration that achieves an approximate steady state [56].

Furthermore, the use of pseudo-transient under-relaxation factors

(URFs) was found to accelerate convergence in a more robust manner

[57]and the rate of convergence has been shown to reassuringly arrive

at a solution earlier than without the use of these URFs [58]. Hence, by

taking advantage of the natural structure of the problem by evolving the

5 The astute reader should note that for all of the simulations, a laptop using

2011-era hardware (Intel i7-2820QM processor, 16 GB RAM) was used.

flow equations in time, we are able to achieve a solution that reasonably

integrates transient effects in a way that can be realized through the use

of a steady-state calculation. In FLUENT, the pseudo time step was left

to be calculated automatically following the recipe found in Section

21.6.1 of the Theory Guide [46]. This is done to hasten setup and retain

consistency across all 18 simulations.

3.4 Convergence Discussion

Initial calculations were run to preview the exact behavior of the it-

erative calculation process; indeed, as previously demonstrated by Edge

et al., the inherent unsteadiness of the flow resulted in difficulties ob-

taining reduced residuals, with the scaled continuity residuals hovering

below but near 1⋅10-2, and other residuals remaining below 1⋅10-4. A

technique stemming from experience was devised to assist in reducing

the residuals while maintaining spatial accuracy. For the first 50 itera-

tions, a blending method between first and second order spatial discreti-

zation schemes (with a bias for the first order scheme), lower URFs, and

an “aggressive” pseudo-time step, were used to rapidly approach a qua-

si-stable solution. The following 100 iterations switched the spatial

discretization scheme to a higher order method along with raised URFs

to increase accuracy. The remaining iterations were then run using a

“conservative” pseudo-time step scaled by 0.5 to further converge and

resolve flow details that are more apparent on a smaller timescale. This

technique enabled the residuals to drop by nearly an order of magnitude,

and thus it was used for all other simulations.

As discussed in Section 3.3, taking mean quantities of a finite sam-

ple will incur sampling errors of which are difficult to quantify com-

pared to a mean derived from an infinite sample. However, a potential

approach to help determine the exact nature of these errors would be

taking an integrated surface quantity, such as drag coefficient, and see-

ing how this quantity changes as the iterations progress. Such a measure

serves as an observed global variable in that the measured quantity is

dependent on how accurate the grid can resolve the flow. Figure 6

shows the iteration history of drag coefficient between the two models

taken with an inlet velocity of 10 m⋅s-1 and run using a grid with medi-

um refinement (refer to Section 3.5 for refinement details). The figure

shows that the solution reaches quasi steady-state values after 150 itera-

tions, with an oscillatory period occurring roughly every 140~150 itera-

tions. A 100 iteration moving average calculated by FLUENT was ap-

plied to the values to assist in determining if overall convergence was

reached. Looking at the tendency of the averages, it is reasonable to

conclude that the solution is indeed quasi-stable. With the average vary-

ing very little compared to actual iteration values, and for the sake of

minimizing computational expense in this study, this observed tendency

incurs a reasonable assurance that the simulations are adequately con-

verged. Finally, the net flux imbalance was calculated to be less than

1% of the smallest flux through the domain, further supporting conver-

gence [59].

Figure 6. Plot of drag coefficient versus number of iterations on a medium mesh

with an inlet velocity of 10 m⋅s

-1

3.5 Numerical Uncertainty and Grid Selection

Moving forward, to preserve accuracy and to assure low computa-

tional costs, the effect of grid resolution is studied by comparing results

on drag coefficient using different grids of different refinement. The

importance of grid convergence and examination of discretization errors

in CFD simulations have been demonstrated across numerous organiza-

tions [60, 61] and journals [62], and the necessity of quantifying these

errors have led to well-established methods that attempt to describe

effects of resolution to that of the extrapolated solution [63]. Although it

has been repeated in this paper that quantifying exact errors is essential-

ly a fruitless adventure, the additional time spent on understanding the

potential discretization error nonetheless is useful in helping select a

grid that balances the required resolution to minimal processing power.

In this grid refinement study, each model used three different grids

that are refined methodologically by decreasing the cell spacing within

the BOIs. Between both models, the grid generation methods remained

the same and follows the procedures as described in Section 3.2. Repre-

senting fine (grid 1), medium (grid 2), and coarse (grid 3) respectively,

the Normal model utilized grids containing 2.82⋅106, 2.32⋅106, and

1.53⋅106 elements. The Flat model used 2.67⋅106, 2.16⋅106, and

1.41⋅106, elements. The difference in overall element counts are at-

tributed to the reduced surface area of the Flat model compared to the

Normal one. For sake of completeness and following the method as

recommended by The American Society of Mechanical Engineers [64],

the subsequent equations are presented to determine the discretization

error between the described meshes. Letting  denote the representative

grid size,

 = 1

(



)



=1 1/3

(11)

where  is the volume of each cell. Let 1<2<3 and let  desig-

nate the refinement ratio between successive grids, such that 21 =

2/1 and 32 =3/2. Calculating the apparent order of accuracy, ,

involves solving the following expression using a fixed-point iteration

and using the first term as the initial guess:

 = 1

ln(21)ln|

/

|+ln21

−

32



−

(12)

where =sign(32/21), 32 =3−2, 21 =2−1, and  repre-

senting the quantity being investigated. Finding the extrapolated, as-

ymptotic value involves calculating



21 =(21

1−2)/(21

−1)

(13)

with 

32 being similar. Then, the following error estimates can be

found:



21 =1−2

1

(14)





21 =

21 −1





(15)

GCI

=1.25

21



−1

(16)

with 

21 describing the approximate relative error, 

21 describing the

extrapolated relative error, and GCI21 being the grid convergence index

(GCI). 

32, 

32, and GCI32 are calculated similarly.

The GCI is an indicator with a 95% confidence interval of how far

the finer of the two compared grids is to the asymptotic value  and

predicts how further refinement affects the solution [65]. The evaluated

quantities are taken after 400 iterations, which in Figure 6 and discussed

previously, has shown to provide a reasonable amount of iterations to

determine a sufficiently averaged quantity. The results from the previ-

ous equations are summarized in Table 2.

Tab le 2. Grid summary and calculations of discretization error between the two

models.

 = drag coefficient at

10 ms−1

Lucoa, Normal

Lucoa, Flat

1,2,3

2.82106, 2.32106,

1.53106

2.67106, 2.16106,

1.41106

Average wall clock

time per 100 iterations

(minutes)

72, 38, 19

67, 34, 17

Average memory usage

per compute thread

(MB)

1674, 985, 648

1546, 916, 542

1,2,3

0.025541, 0.027221,

0.031327

0.026007, 0.027912,

0.032206

1,2,3

0.9352, 0.9418, 1.0015

0.9501, 0.9581, 1.0178

21,32

1.06577, 1.15083

1.07325, 1.15384

21,32

0.00659, 0.09700

0.00798, 0.05974



12.8180

11.4372



21,

0.9299, 0.9299

0.9437, 0.9437



21,

0.70%, 6.34%

0.84%, 6.24%



21,

0.56%, 1.27%

0.68%, 1.53%

GCI21,GCI32

0.70%, 1.57%

0.84%, 1.88%

All grids were noted to converge in a monotonic manner, indicating

the average flow field certainly benefits from grid refinement. As shown

by the lower values of GCI21 to that of GCI32, it would be most benefi-

cial to run all the simulations using the fine grid. On the other hand, as

accuracy increases, so do memory and compute requirements. The fin-

est grid managed to exceed the available physical memory on the ma-

chine that the simulations were run on. In addition, calculations were

found to require nearly double the amount of time to arrive at a solution

compared to the medium grid. The tradeoff between the reduction in

GCI and resource consumption was too great considering the limited

resources to begin with and the potential amount of time required for all

18 simulations. Therefore, as shown from this grid study, the medium

grid serves as the most practical balance between accuracy and resource

expenditure; for all simulations, the medium grid was selected.

4. Results and Discussion

The immediate discrepancies between Lucoa’s model and other

human models used in similar studies (other than the fact that she has an

enormous chest) are the presence of horns, a baseball cap, raised arms,

and solidly modeled hair. Each affects the wake in their respective man-

ner, with these individual effects being described in the following sub-

sections. It is important to recognize that although these differences

between Lucoa and other human models are present, they nonetheless

do not affect the direct comparison between the Normal and Flat mod-

els, which was a primary objective of this work.

The following subsection details the variances of drag and lift as air

velocity changes and compares the outcomes to previous studies on the

human form. The subsequent subsections delve into the differences in

wake structure through the analysis of streamlines, velocities, TKE, and

skin friction. While no sound comments can be made on the concise

form of instantaneous flow structures, quasi steady-state general fea-

tures present in the flow are described.

4.1 Velocity-Based Trends

The convergence method presented in Section 3.4 was used for all

velocities listed in Table 1 between the two models. The drag and lift

were calculated using the same 100 iteration moving average in

FLUENT and the values at the end of 400 iterations were used. Figures

7-9 plot the results between the two models. In all of the figures, a cubic

spline was used to interpolate the data. The figures labeled with “(a)”

correspond to drag-related values while figures labeled with “(b)” cor-

respond to lift-related values.

Figure 7 focuses on the coefficients, and it is readily apparent that

at all velocities, the Flat model incurs higher coefficients than the Nor-

mal model. In the region near 5 m⋅s-1, the drag coefficients between the

two models were closest to one another. Judging from the errors de-

scribed in Section 3.5, it is safe to presume that these differences are

within the error bounds and therefore the difference between the two

models near this velocity is negligible. In 7(a), it is interesting to ob-

serve that the drag coefficients decrease following a power relation

corresponding to velocity. This relation most likely arises from the pe-

culiarity of the solidly modeled hair; downwash due to the hair diverts

air behind the torso then directs the air downwards, which causes less

air to flow directly behind the body. Presumably, this downwash closes

off the size of the primary recirculation region (PRR) behind the body.

As the air velocity increases, the effect of the downwash becomes more

prominent and therefore a general decrease in the drag coefficient is

observed. The large breasts on the Normal model may also play a role

in enhancing this downwash effect, as the air is gradually diverted

around the bust and is able to maintain momentum before being redi-

rected by the hair. The stronger resulting downward jet of air thus closes

off the recirculation region more effectively. The degree of which this

downwash affects drag is difficult to determine from this study, but it is

reasonable to deduce that due to the drag reduction of the Normal model

compared to the Flat model, the effects are markedly noticeable. The

geometrical presence of the breasts may also have an effect in reducing

drag as they smoothly redirect a portion of the air around the torso,

(a) Drag coefficient vs. air velocity.

(b) Lift coefficient vs. air velocity.

Figure 7. Plots comparing coefficient quantities between the two models.

(a) Drag force vs. air velocity.

(b) Lift force vs. air velocity.

Figure 8. Plots comparing the forces experienced on the two models at various air velocities.

(a) Relative drag difference of the Flat model to the Normal model.

(b) Relative lift difference of the Flat model to the Normal model.

Figure 9. Plots indicating the relative force differences of the Flat model compared to the Normal model.

whereas the Flat model simply forces majority of the air to stagnate at

the chest, which incurs higher drag.

The difference in lift is notably more dramatic than drag. In Figure

7(b), the lift coefficient for both models gradually increases from 1 to

10 m⋅s-1 then levels off as the velocity changes. This increase can also

be contributed to the downwash from the hair, since as the velocity

increases, there is a higher flow rate and thus more fluid mass pushing

upwards against the models. The brim of the baseball cap on both mod-

els may additionally contribute to lift as it lies orthogonal to the mean

flow direction and causes a buildup in pressure in front of the face. The

Normal model provides a lower lift coefficient possibly due to the

breasts providing downforce; since the Normal model has a significant

and primarily upward sloping surface in front of the torso that the Flat

model lacks, the downward force from the air flowing atop the bust

most likely provides this reduction in lift. Granted, due to the geometry

being modeled as completely solid, this effect does not account for

breast deflection caused by these forces. Nonetheless, the presence of

Lucoa’s large breasts evidently provides a reduction in lift.

Figure 8 provides a comparison of the actual forces experienced on

the models. As seen in both 8(a) and 8(b), the forces increase following

a roughly squared relation, which is what would be expected looking at

the velocity relationship in Equations (1) and (5). As elaborated in the

previous paragraphs, the difference in drag is less dramatic than that of

lift, and this is reflected by the more difficult to discern plots in 8(a)

than in 8(b). This observation is further expanded upon in Figure 9,

which compares the relative force difference the Flat model encounters

compared to the Normal model. Looking at 9(a), the Flat model experi-

ences a nearly 4% increase in drag at 1 m⋅s-1, however, this value de-

creases immediately and falls to below 1% at 5 m⋅s-1. From 5 to 10

m⋅s-1, the difference between the two widens again, levels off from 10

to 20 m⋅s-1, then impetuously increases at 25 m⋅s-1 and remains marked-

ly greater at the highest tested velocity of 30 m⋅s-1. The reason as to

why the drag difference reaches its minima at 5 m⋅s-1 cannot be directly

inferred from this work. Throughout the tested velocity range, the Flat

model incurs an average drag increase of 2%, which, in the context of

day-to-day life at low air velocities, is practically insignificant. Howev-

er, in regards to performance, the large breasts on the Normal model can

prove to be advantageous, assuming the overall position remains similar

to the erect postures of the models. The degree of which this advantage

can be quantified depends heavily on the type and duration of competi-

tion to be examined, which is not discussed further in detail. The rela-

tive lift force difference averages 24%, with the values ranging from

32% at 1 m⋅s-1, gradually decreasing to 21% near 15 m⋅s-1, and then

slightly increasing to 23% at the highest tested velocity. From this re-

sult, the presumed effect of Lucoa’s breasts reducing lift is more pro-

nounced at lower air velocities, while still remaining effective as the

velocity increases.

Tab le 3. Comparisons of drag coefficient with other researchers’ results.

Researchers

Year

Reference

Posture

This study

Standing

(Normal model)

0.89 ~ 1.05

Standing

(Flat model)

0.91 ~ 1.09

Hill

1927

[1]

Standing

0.98

Schmitt

1954

[7]

Standing

(clothed)

Approx. 1.36

Standing

(nude)

Approx. 1.20

Shanebrook and

Jaszczak

1976

[66]

Running (cylin-

drical model)

1.2

Penwarden et al.

1977

[67]

Standing

(clothed)

1.18

Davies

1980

[2]

Running

0.87

Brownlie et al.

1987

[68]

Standing

0.96 ~ 0.98

Gómez et al.

2013

[69]

Running

1.20

Inoue et al.

2016

[70]

Running (without

ground effects)

1.17

Table 3 summarizes the encountered drag that both models experi-

enced throughout the tested velocity range, together with other re-

searchers’ experiments. As a result, the present study reveals that Lu-

coa’s form tends to correspond favorably to experiments done with

standing models. It is seen that Lucoa’s results overlap with those of

Hill [1] and Brownlie et al. [68]. Schmitt’s [7] results were an atypical

case as the drag coefficients in that study were based on the entire sur-

face area of the body divided by the product of volume and subject

height, instead of the customary frontal projected area used elsewhere.

This resulted in coefficients ranging from 10 to 13, necessitating a re-

calculation in order to more directly compare to the other results in

Table 3. It was demonstrated through Schmitt’s results that the use of

nude subjects (akin to having smooth walls as mentioned in Section 3.1)

result in lower drag. Although the values for nude subjects were still

greater than those found in the current s t u d y, t his finding provides addi-

tional support as to why Lucoa’s overall form tends to encounter lower

coefficients in contrast to other studies. With the objective of obtaining

CD values for a wide range of people in everyday situations, work done

by Penwarden et al. [67] confirms that clothing certainly increases drag.

Furthermore, the authors in that study comment that extrapolating their

data to simulate bare-skinned models would result in values more akin

to that of Hill’s data, thereby supporting the notion that Lucoa’s results

are indeed realistic.

Comparisons to results based around running are also provided, as

this form of competition is a natural progression from simply standing.

Understandably, a runner will experience higher drag due to their ever-

changing position and the strict anti-symmetry in orientation. Shane-

brook & Jaszczak [66] investigated drag through the use of a general-

ized cylindrical human model, analogous to the assumptions described

in Section 1.1. As such, their results parallel those of other investiga-

tions done on runners and show that a cylindrical model can provide

convincing data. Davies’ [2] work discovered that the effects of Reyn-

olds number remained constant below velocities of approximately 18

m⋅s-1. Conversely, this effect was not seen in this study, which most

likely harkens back to the drawback of the modeled hair. Gómez et al.

[69] developed a parametric model of drag that considered both veloci-

ty, , and acceleration, 2, based on the record-setting performance of

Usain Bolt. They revealed that 92% of energy used in running is ab-

sorbed by drag, which bolsters both the significance of the drag reduc-

tion and the implicational anthropomorphic advantage that Lucoa has.

Inoue et al. [70] examined the effects of a moving belt system to simu-

late ground effects of a runner, in addition to providing data on how leg

position (e.g., forward or behind) changes drag. From their st u d y, it was

observed that having the legs placed either forward and behind veritably

increases drag due to the posture. Consequently, it is expected that a

prospective study incorporating the effects of running based on the pre-

sent results (as described in herein and Table 3) would feature compara-

ble CD values. While the current stationary results may not be directly

applicable to the moving case, they nevertheless offer a suitable starting

point on which to base further research.

4.2 Salient Flow Analysis

Discussion pertaining to the variances in flow structures is exam-

ined under an air velocity of 10 m⋅s-1. In addition to being previously

examined in Sections 3.4 and 3.5, this velocity is readily achievable

under human locomotion [69], making the selection straightforward.

While the results are derived from a quasi-steady-state approach, gen-

eral flow structures can be reasonably interpreted.

4.2.1 Streamlines and Flow Velocities

As reflected in Figures 10-13, the fluid dynamics of the wake are

shown with the use of streamlines colored according to velocity magni-

tude. The figures labeled as “(a)” and “(b)” correspond to the Normal

and Flat models respectively. The overall wake structures revealed in

Figure 10 fall into two distinct regions based on height, with the PRR

found above the hips and extending to the top of the head, and a second,

more complex regime behind the legs. The works described in Section

1.1 correctly predicted these structures. Specifically, the PRR for both

(a) Wake structures associated with the Normal model.

(b) Wake structures associated with the Flat model.

Figure 10. Streamlines emanating at =0 and spanning a line from 02.3 meters showing primary flow structures. Pressure experienced on the models is also

shown.

(a) Set of vertically aligned streamlines on the Normal model.

(b) Set of vertically aligned streamlines on the Flat model.

Figure 11 . Detailed view of flow structures around the chest of the models using the same streamlines from Figure 10.

(a) Set of horizontal streamlines on the Normal model flowing atop the bust.

(b) Set of horizontal streamlines on the Flat model at the same -location.

Figure 12. Detailed view of flow structures around the chest using streamlines emanating at =1.18 meters and spanning a line from −0.50.5 meters.

(a) Set of horizontal streamlines on the Normal model flowing beneath the bust.

(b) Set of horizontal streamlines on the Flat model at the same -location.

Figure 13. Detailed view of flow structures around the chest using streamlines emanating at =1.14 meters and spanning a line from −0.50.5 meters.

models was found to extend roughly 1.4 meters downstream, or approx-

imately 4 times the width of the torso, which agrees strongly with Oka-

moto’s and Sunabashiri’s findings. The type of vortex shedding in this

recirculation region cannot be determined, but it is expected to nonethe-

less be asymmetric, with the large-scale structures being advected

downstream based on the findings of Edge et al. The wake behavior

behind the legs were also discovered to obey the findings summarized

by Zhou and Alam, with the region from the ankles to the knees having

wake structures associated with each leg, and the region from the knees

to the hips behaving as a unified bluff-body. The figure also shows a

distinct vertical “jet sheet” stemming from the gap between the legs,

with an average magnitude approximately 35% greater than the free

stream velocity. A vortex pair originating from the top of the legs is also

evident. This vortex pair can be seen being caused by the downwash of

the hair; the momentum of the downward flow interacts with the air

flowing past the top of the thighs, which results in these prominent

structures. Additional momentum provided by the jet of air between the

legs further enhances the strength of these vortices. Furthermore, the

effects of the downwash are evident in the way the vortex pair assumes

a downward angle as the flow advects; this is indicative of the signifi-

cant amount of air being captured by the geometry of the hair.

From what can be drawn from comparing Figure 10(a) and 10(b),

the Flat model has a weaker, less organized vortex stemming from the

legs. This observation may further explain why the Normal model has

lower lift than the Flat model. Aider et al. [71] described the effect of

vortex pairs affecting lift and drag, and shown that inflow caused by

streamwise counter-rotating vortices result in a net downward force.

Thus, the weaker vortex pair on the Flat model contributes less towards

reducing lift than the Normal model. Following this, it is reasonable to

suggest that the behavior of the drag curve in Figure 7(a) can be at-

tributed to the formation of these vortices; the energy being redirected

into the formation of these structures diverts a portion of the mean flow

away from the PRR, leading to drag reduction as the velocity increases.

Figure 11 provides a close-up look at the same streamlines in Figure

10 around the torso of both models. As can be seen in 11(a) and correct-

ly determined in Section 4.1, the breasts provide a gradual interface for

the air to move around the torso compared to the relatively abrupt ob-

struction the Flat model imposes as reflected in 11(b). This is further

exemplified in Figure 12, which shows a set of horizontal streamlines

positioned at z=1.18m. In 12(a), the air flows smoothly atop the breasts

at a velocity approximately 40% to 60% of the free stream velocity,

whereas the Flat model in 12(b) simply forces a relatively larger portion

of the incoming air to stall at the torso. It is interesting to note that a

small recirculation region develops directly atop the breasts, which most

likely is dependent on the angle between them and the chest. From what

can be seen in 12(a), this small recirculation region prevents a small

portion of air from surmounting the shoulders and instead redirects the

air downwards and off to the sides of the torso. Figure 13 provides an

upwards facing view of the models along with horizontal streamlines

emanating from z=1.14m. In 13(a), air flowing beneath the bust gains a

velocity that is approximately 20% higher than the mean free stream

velocity before being diverted perpendicular to the body and interacting

with the incoming air stream and the rest of the torso. By the action of

the breasts, the air is diverted in such a way that it is able to maintain

momentum while it moves around the rest of the chest. This is in con-

trast to 13(b), where a portion of the air is forced to stall in front of the

chest before being redirected around the torso, much like as was seen in

other figures involving streamlines with the Flat model.

Figure 14 compares the velocity magnitude of the wake behind the

two models at five different z-locations. At these locations, data is sam-

pled along a line centered at the midline (x=0) and spans downstream

from −12 meters. The z-locations are taken at heights of z =

0.1m, z = 0.5m, z = 0.9m, z = 1.18m, and z = 1.55m, all of which corre-

spond to the feet, legs, hips, chest, and head, respectively. From this

comparison, the differences in wake velocities are slight, with the only

significant difference occurring at the height of the legs. At this height,

it can be seen that the Flat model has a wake velocity that is consistently

higher than that of the Normal model. This observation can be attributed

by the lower energy present in the vortices generated by the Flat model;

these weaker formations are affected by the mean flow to a greater de-

gree and thus the velocity magnitude in this region is higher than if the

vortices were to have greater strength, as seen with the Normal model.

4.2.2 Turbulence Kinetic Energy

Figure 15 shows a slice of the domain through the median plane of

the models, indicating the TKE present in the flow. At first glance, con-

tours for both models present the same general structures as described

earlier, such as the PRR behind the torso and the smaller structures as-

sociated with the legs. Due to the automatic scaling of the colors, the

differences present in the PRR of the Flat model compared to the Nor-

mal model are slightly exaggerated. However, the differences are still

distinguishable in that the PRR for the Flat model has a measurably

higher level of TKE than that of the Normal model. Indeed, 15(b) indi-

cates that the two apparent bands of significant TKE associated with the

PRR are generally more intense, even after factoring the differences

between the color scales. Common to both models is a region behind

the hips where the relatively highest TKE was measured. This lower

band within the PRR was found to be more compact for the Normal

model, whereas the Flat model had a marginally less intense and more

widespread band. This effect may be attributed to the way the down-

wash from the hair interacts with the PRR.

Directly below the PRR is the TKE present in the vortex pairs orig-

inating from the legs. In 15(a), the Normal model has a notably higher

level of TKE associated with these vortices, coupled with the observa-

tion that this energy is maintained as the flow advects downstream. In

contrast, 15(b) shows that the TKE in the vortices is lower and that the

energy is dissipated sooner. This remark further supports the interpreta-

tions hitherto; that the formation of stronger vortices the Normal model

generates plays a role in reducing both drag and lift by diverting energy

away from the PRR.

Smaller regions with notable TKE present between both models are

those associated directly behind the legs, at the ankles, and directly

above the brim of the baseball cap. Between the two, the TKE at the

ankles remains unperturbed by the differences in the wake structures

above it. This region is related to the individual wakes associated with

each foot, behaving much like the two cylinder arrangement as de-

scribed earlier. A downward “jet” of TKE originating from the thighs

and curling parallel to the mean flow direction at the height of the knees

shows some variability between the two models. This region is linked to

the “jet” of air that is formed between the legs interacting with the

downwash from the hair. With the Normal model in 15(a), this “jet” of

turbulence remains both vertical and closer to the legs. In 15(b), this

region takes on a slightly more horizontal transition and extends farther

into the wake. A small recirculation region, much like that associated

with the breasts on the Normal model, is found above the brim of the

baseball cap, which is due to the blunt transition the shape of the head

imposes to the incoming flow.

Figure 14. Comparison of wake velocity magnitudes at 5 z-locations centered

along x = 0 and ranging from

−12

meters.

Lucoa positioned and to scale

with x-axis.

Feet z = 0.1m, Legs z = 0.5m, Hips z = 0.9m, Chest z = 1.18m, Head z = 1.55m.

Figure 16 shows the TKE around the torso of both models along the

coronal plane. As obvious as the difference between both models, the

presence of a turbulent region behind the breasts and below the axilla

comes as no surprise. It is apparent that this region is influenced by the

small recirculation region that forms above the breasts spilling air

downwards and behind the torso, which can be seen in Figure 12(a).

Furthermore, air from beneath the breasts also contributes to the size of

this region, as in Figure 13(a) it can be understood that this layer of

accelerated air provides additional shear against the flow from above

and increases the turbulence behind the breasts. Since the Flat model

lacks such features, Figure 16(b) presents an understandably unremark-

able result, with only a small sliver of turbulence along the side of the

torso. Common to both models however, is turbulence being generated

by the hair itself, stemming from the hair being left modeled as solid.

The true extent of how much turbulence is generated from the solid hair

and its effects on the PRR is unfortunately intractable from this current

study due to its highly complex geometry. Though, it is expected that

simulations dealing with physically accurate hair may undoubtedly

increase turbulence, and hence, drag, based on casual observations done

on how hairstyles can affect aerodynamic performance [72].

The same approach used to obtain information as seen in Figure 14

was used in Figure 17, with TKE being measured instead of velocity. As

indicated from this figure, it is seen that the vortices originating from

the legs indubitably diverts energy away from the PRR and supports the

findings stemming from Figure 15. This is reflected with stronger over-

all turbulence present at z-locations corresponding with the hips, chest,

and head for the Flat model, along with markedly lower TKE at the

level of the legs in contrast to the Normal model. Slight differences are

noted, with the Flat model having a higher peak TKE at the legs imme-

diately behind the body than the Normal model. Then, from 1 to 2 me-

ters downstream, the Normal model generates a greater amount of TKE.

This observation may ultimately be attributed to the difference in shape

the leg vortices assume between the two models.

4.2.3 Skin Friction Coefficient

An evaluation of the effects of the macroscopic geometry between

the two models on skin friction is shown with Figure 18. This figure

represents a scatter plot of Cf measured at every vertex on the models,

with regions of relatively high Cf labeled according to which location

on the model they are found on. Two directions are evaluated, with the

x-axis on 18(a) representing the streamwise position, and the x-axis on

18(b) representing the z-location. A silhouette of the models present in

the background of the plots are positioned and scaled to their respective

x-axis to further provide context with the labeling. Additionally, the data

was averaged and plotted along with the scatter plot to reveal discrep-

ancies that may be more difficult to discern.

Expectedly, both models feature the same general scatter plots, with

nearly negligible differences. However, common to both models and the

plots are the presence of numerous spikes associated with the hair. In-

asmuch as they are an artifact of the hair being modeled solidly, these

spikes are aptly denoted as “hair anomalisms” within the plots. As such,

(a) TKE associated with the Normal model.

(b) TKE associated with the Flat model.

Figure 15. TKE present in the flow behind the models along the median plane.

(a) TKE present behind the breasts of the Normal model.

(b) TKE present around the torso of the Flat model.

Figure 16. TKE present around the torso of both models along the coronal plane.

Figure 17. Comparison of TKE at 5 z-locations centered along x = 0 and ranging

from

−12

meters. Lucoa positioned and to scale with x-axis.

Feet z = 0.1m, Legs z = 0.5m, Hips z = 0.9m, Chest z = 1.18m, Head z = 1.55m.

the hair is undeniably a major contributor to skin friction on the models.

This is especially apparent in 18(a), with the source of skin friction

being contributed solely by the hair from the rear of the model. Another

exemplification of the hair’s effect on Cf can be seen in 18(b), with a

grouping of peaks obtaining a value of 0.06 corresponding to the hair

tips and bangs around the face. Additional sources of significant Cf can

be easily observed in the same figure, such as the ankles, around the

thighs, the fingertips, and especially Lucoa’s horns. These dramatic

changes to skin friction indicate where on the models flow separation is

more likely to occur, which in turn, relates to where turbulence can

potentially be generated.

The breasts, compared to the relatively noisy scatter generated by

the rest of the body, provides a smooth transition in regards to skin fric-

tion over its surface. A small spike found at the downstream location of

y = -0.77m in 18(a) corresponds to the small recirculation region above

the breasts seen in Figure 15(a). In a sense, it can deduced that the

breasts act to reduce Cf, which can be seen as a small dip in the

smoothed data centered around z = 1.2m in 18(b). However, even with

such a diverse of a plot, the average Cf across the whole body can be

seen as roughly 0.01 from both figures, which indicates how little skin

friction actually affects overall drag with bluff bodies.

5. Summary and Recommendations

This paper has offered a unique compendium of data providing in-

sight into the effects of a specific physiological feature on the aerody-

namic performance of a human. As such, the results have indicated that

large breasts can be notably aerodynamic through the reduction of drag

and lift. The Flat model incurred a 4% maximum drag increase com-

pared to the Normal model, with an average of approximately 2% span-

ning velocities from 1 to 30 m⋅s-1. The Flat model also experienced

more lift, with a maximum difference being 32% and averaging 22%.

As illustrated, the mechanism behind the drag and lift behaviors ob-

served between both models was elucidated through the analysis of

streamlines around the body and the structures associated with TKE; the

Normal model provides advantageously lower drag and lift by the gen-

eration of stronger vortices from the legs, which in turn originates from

the action of the breasts redirecting the flow around the torso. From

what has been presented in this preliminary work, it is safe to conclude

that the phrase “Flat is Justice” is deficient aerodynamically.

A major shortcoming intrinsic to this study was the decision to

leave the hair present in the models as immovable and solid. As noted

earlier, a significant portion of the air flowing around the body was

captured by the geometry of the hair, and this affected the wake struc-

tures behind the model. This effect therefore blunts the overall applica-

bility of the results found in this study to actual human models. It is

important to note, however, that large breasts do give a consistently

notable aerodynamic advantage, as reflected in the overall lower forces

experienced by the Normal model. Additionally, even though the wake

structures generated by the hair resulted in a departure from the ex-

pected drag and lift behaviors, comparisons done with other experi-

menters’ results show that the outcomes from this study were strongly

promising.

For future studies, several recommendations are provided. In the

near-term, a re-evaluation of the current work without the hindrance of

modeled hair should be done. Work done using a time-dependent com-

putational approach should also be completed to further gauge the ef-

fects and inaccuracies of using the pseudo-transient method in relation

to drag and lift. A keynote proposition would be the use of a wind tun-

nel experiment to acquire validation data. Ideally, this experiment

would use live subjects of varying breast sizes in order to provide addi-

tional data in which to compare to the data herein. Additional research

to evaluate the degree of which the information provided from this cur-

rent work applies to real world situations should also be completed.

Acknowledgments

The author would like to send a massive thanks to the user

“icemega5” found on both Twitter and BowlRoll.net for the Quetzalco-

atl model. Without their work, this study would not have been possible.

The author is grateful for his fellow colleagues for dedicating their time

in helping proofread and provide guidance on this paper. Additionally,

the author would like to make a shout out to users that frequent the

r/anime_irl and r/animemes communities for their inspirational and

fervent dissoluteness. Lastly, the author expresses gratitude towards

both Cool-Kyou Shinsha and Kyoto Animation for their work on Miss

Kobayashi’s Dragon Maid.

The author received no funding for this research. The results of the

present study do not constitute endorsement of any potential entity

whether expressed or implied. Quetzalcoatl is also not the author’s

“waifu”, although he respects her character as THE

GODDESS OF THICC.

Appendix A: Supplementary Material

Additional contour plots in higher resolution and quality from the other

velocities not examined in Section 4.2 can be found by following this

link: https://imgur.com/a/bz31B.

A video further visualizing the flow structures in 3-D can be seen here:

https://youtu.be/1414xHh6tw0.

The entire CFD study (project files and generated data) can be found by

following:

https://mega.nz/#!lz5hzA5Q!JpG_t0IbflLuI24sC1pkV-

C7nKW1OxqZSMzBeHlMt_A.

(a) Skin friction of both models sampled along the streamwise direction.

(b) Skin friction of both models sampled along the z-direction.

Figure 18. Scatter plots of skin friction comparing both models. Sources of regionally significant Cf are labeled. An additional plot denoting a smoothed average of the

scatter data is present to provide clarity in comparison. Silhouettes of the models are positioned and scaled according to their respective axis in the plots.

Appendix B: Additional Figures

Figure 19. Velocity contours and vectors of the Normal model along the median

plane.

Figure 20. Velocity contours and vectors of the Flat model along the median

plane.

Figure 21. Pressure coefficient of the flow regime around the Normal model

along the median plane.

Figure 22. Pressure coefficient of the flow regime around the Flat model along

the median plane.

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Wind-tunnel experiments are the most effective approaches both to elucidate the flows around runners in track-and-field athletics, and to evaluate their air resistances. In the present study, we develop a moving-belt system, and show its basic performance, such as the distributions of time-mean flow velocity and turbulent intensity above the moving-belt using a hot-wire anemometer. As a result, we have confirmed the effective improvement of the velocity distributions in terms of flow uniformity and turbulence reduction, especially near the moving-belt surface. Following, using this the developed moving-belt system, we investigate the air resistance on a runner in solo running and in duet running. For solo running, we reveal an increase in air resistance of more than 10% in comparison with the conventional result with no moving-belt system. For duet running, we reveal the optimum duet-running formation where a following runner behind a pacemaker experiences small air resistances or strong drafts.

Computational fluid dynamics study of human-induced wake and particle dispersion in indoor environment

Article

Full-text available

Aug 2016

The impact of human-induced wake flow and particle re-dispersion from floors in an indoor environment was investigated by performing computational fluid dynamics simulations with dynamic mesh of a moving manikin model in a confined room. The manikin motion was achieved by a dynamic layering mesh method to update new grids with each time step. Particle transport from the floors and its re-dispersion was tracked by a Lagrangian approach. A series of numerical simulations of three walking speeds were performed to compare the flow disturbance induced by the walking motion. The significant airflow patterns included: an upward-directed flow in front of the body combined with a high velocity downward-directed flow at the rear of the body; a stagnant region behind the gap between the legs and counter-rotating vortices in the wake region. The airflow momentum induced by the moving body disturbed PM2.5 particles that were initially at rest on the floor to lift and become re-suspended due to its interaction with the trailing wake. The residual flow disturbances after the manikin stopped moving continued to induce the particle to spread and deposit over time. The spatial and temporal characteristics of the particle dispersion and concentration showed that higher walking speed was conducive to reducing human's exposure to contaminants in breathing region.

Wake of two interacting circular cylinders: A review

Article

Dec 2016
INT J HEAT FLUID FL

Evaluation of manikin simplification methods for CFD simulations in occupied indoor environments

Article

Jun 2016
ENERG BUILDINGS

While simplified computational thermal manikins (CTMs) are widely employed in CFD modes of occupied indoor spaces in order to save the computational cost, a criterion of simplification is still absent and the effects of CTM simplification are yet not clear. In this study, six CTMs including a 3D scanned CTM and five simplified CTMs generated from various simplification approaches were employed to analyse the impact of CTM simplification on the prediction of airflow field and contaminant transport. Comparison of the predicted airflow field against the published data in the literature demonstrated that CTM simplification has a strong effect on the thermal airflow field prediction in the vicinity of manikin surfaces. For densely occupied indoor spaces such as a train cabin, the error induced by CTM simplification could be enlarged and further cause significant global error to the prediction of contaminant transport. This is especially true when contaminants are released from the CTMs. This study demonstrated that the mesh decimating algorithm is promising to simply CTMs that is not only able to reduce considerable computational cost but capable of maintain an acceptable predictive error.

Bluff Body Aerodynamics

Chapter

Apr 2013

Richard Flay

The aerodynamic characteristics of bluff bodies differ substantially from streamlined bodies, and an understanding of bluff body aerodynamics is essential to make progress in understanding wind engineering. Streamlined bodies like aircraft wings have a rounded nose, a thin profile, and a sharp trailing edge. Their wakes are small and for small angles of attack, the lift force developed is considerably greater than the drag force. On the other hand, bluff bodies have a large separated wake, with unsteady flow, and the drag force is comparable with the lift force. It is necessary to understand the size and nature of these forces to ensure that engineered designs are fit for purpose under wind action.

Accelerating CFD Solutions

Article

Jan 2011

M. Keating

Fundamentals of Boundary-Layer Theory

Chapter

Jan 2000

CFD Aerodynamic Analysis of Ahmed Body

Article

Dec 2014

Abstract— The CFD (Computational Fluid Dynamics) analysis is used to find the parameter in the automobile industry. This analysis is used to understand the fuel consumption, stability of the vehicle and passenger comfort. The air flow over ground vehicle is analysed and coefficient of drag is calculated using CFD (Ansys Fluent). For this calculation, Ahmed body (simplified car body) as ground vehicle is considered which is commonly used as test case in industry. The Ahmed body is made up of a round front part, a movable slant plane placed in the rear of the body to study the separation phenomena at 250, 350 angles, and a rectangular box, which connects the front and rear slant plane. The most significant feature of the body is the 250, 350 angle of rear slant. Air is used as a working fluid. The inlet velocity of fluid is 40 m/s. k-ε turbulent model used as a standard model. Two separate cases have been solved for two different front radius of Ahmed body R80, R120 and ground clearance at 20mm, 40mm and results are comparing. The results are present in the form of drag coefficient value and flow field which include velocity contour and velocity vector fields. the validation is carried out by simulation around the Ahmed body with the rear slant angle of 250,350 .the actual wind tunnel experimental data are compared with the results. Keywords— Aerodynamics, Ahmed body, k-ε model, CFD, drag coefficient.

Application of grid convergence index in FE computation

Article

Mar 2013

Leslaw Kwasniewski

This paper presents an application of the grid convergence index (GCI) concept based on the Richardson extrapolation to a selected simple problem of a cantilever beam loaded with vertical forces at the tip end. The GCI method, popular in computational fluid dynamics, has been recently recommended for finite element (FE) applications in solid and structural mechanics. Based on the results obtained usually for three meshes, the GCI method enables one to determine, in an objective manner, the order of convergence to estimate the asymptotic solution and the bounds for discretization error. The example shows that the characteristics of the convergence depend on the selection of the quantity of interest, which can be local or a global functional such as the deflection considered here. The results differ for different FE formulations, and the difference is bigger when the nonlinearities (e.g., due to plastic response) are taken into account

Analysis and Qualitative Effects of Large Breasts on Aerodynamic Performance and Wake of a “Miss Kobayashi’s Dragon Maid” Character

Abstract and Figures

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