Applicability of the Vortex Methods for
Aerodynamics of Heavy Vehicles
Kyoji Kamemoto and Akira OJima
Department of Mechanical Engineering, Yokohama National University
79-5 Tokiwadai, Hodogaya-ku, Yokohama, Kanagawa, 240-8501, Japan
T: +81-45-339-3881 F: +81-45-331-6593 E-mail: kame@post.me.ynu.ac.jp
College Master Hands Inc.
2-1-31 Midorigaoka, Zama, Kanagawa, 228-0021, Japan
T: +81-46-228-9519 F: +81-46-228-9519 E-mail: ojima@cmhands.com
Abstract.
This paper describes recent works of practical applications of vortex element
methods to study of aerodynamics of heavy vehicles, carried by the authors’
group, explaining the mathematical basis of the method based on the BiotSavart law. It is pointed as one of the most attractive features of the vortex
method that the numerical simulation using the method is considered to be a
new and simple technique of large eddy simulation, because they consist of
simple algorithm based on physics of flow and it provides a completely gridfree Lagrangian calculation. As typical results of aerodynamics of heavy vehicles, unsteady flows around a heavy vehicle model such as a tractor-trailer with
different gap lengths and unsteady aerodynamic characteristics of a tractortrailer with meandering motion are explained.
1 Introduction
The aerodynamic force and noise on road vehicles are the result of complex
interactions between the flow separation and the dynamic behavior of the vortical wake. In order to design suitable shapes of vehicles, it is necessary to predict a physical mechanism of a flow separation and an interaction with vortical
wake. Recently, computational fluid dynamics (CFD) is becoming an indispensable tool for vehicle design because of the advances in numerical methods
and the remarkable progress in the computer performance. A variety of numerical methods have been applied to simulate the flows around a vehicle.
However, the numerical simulation of automotive flows still is not so easy
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from the view of engineering applications. The flow around a vehicle is an essentially unsteady flow originated from the large scale separations of the
boundary layer. The applicability of the conventional turbulence models of
time-mean type seems questionable, as far as unsteady separated flows are concerned. And the large eddy simulation (LES) of Eulerian type inevitably meets
crucial difficulties in its application to flows of higher Reynolds numbers, because the scheme essentially needs fine grids to obtain reasonable resolution of
turbulence structures.
On the other hand, the vortex methods have been developed and applied
for analysis of complex, unsteady and vortical flows in relation to problems in
a wide range of industries, because they consist of simple algorithm based on
physics of flow. Therefore, the vortex methods may be the means to provide
one of the most suitable techniques for the prediction of unsteady aerodynamic characteristics of heavy vehicles.
Leonard (1980) summarized the basic algorithm and examples of its applications. Sarpkaya (1989) presented a comprehensive review of various vortex
methods based on Lagrangian or mixed Lagrangian-Eulerian schemes, the
Biot-Savart law or the vortex in cell methods. Kamemoto (1995) summarized
the mathematical basis of the Biot-Savart law methods. Various studies related
to the simulation of three dimensional unsteady flows around a bluff body
with vortex methods have been reported. Gharakhani et al. (1996) applied a
three-dimensional vortex-boundary method to the simulation of the flow
around tractor-trailer. Bernard et al. (1999) applied a vortex tube and sheet
method to the simulation of higher Reynolds number flows around a prolate
spheroid. Ojima and Kamemoto (2000) simulated unsteady vortical wakes
behind a sphere and a prolate spheroid by using an advanced vortex method.
Cottet and Poncet (2002) calculated the unsteady vortex features shedding
from a circular cylinder by using a vortex-in-cell method. Ploumhans et al.
(2002) applied a vortex method with parallel tree codes to the simulation of
unsteady flows past a sphere.
As well as many finite difference methods, it is a crucial point in vortex
methods that the number of vortex elements should be increased when higher
resolution of turbulence structures is required, and then the computational
time increase rapidly. In order to reduce the operation count of evaluating the
velocity at each particle through a Biot-Savart law, fast N-body solvers that reduce the operation count from O(N2) to O (N log N) are proposed (Greengard et al. 1987).
This paper describes the governing equation and the numerical method of
the Biot-Savart law vortex methods developed and examined up to this time by
the group of the present authors. As application examples, the numerical
simulation of unsteady flows around heavy vehicles; a simplified heavy vehicle
model such as a tractor-trailer with a gap length and a tractor-trailer with meandering motion, are explained.
Applicability of the Vortex Methods for Aerodynamics of Heavy Vehicles
505
2 Algorithms of Vortex Methods
2.1 Mathematical Basis
The governing equations of viscous and incompressible flow are described by
the vorticity transport equation and the pressure Poisson equation which can
be derived by taking the rotation and the divergence of Navier-Stokes equations, respectively.
Where u is a velocity vector and a vorticity w is defined as follows.
As explained by Wu and Thompson (1973), the Biot-Savart law can be derived from the definition equation of vorticity as follows.
Here, subscript “0” denotes variable, differentiation and integration at a location r0, and n0 denotes the normal unit vector at a point on a boundary surface S. And G is the fundamental solution of the scalar Laplace equation with
the delta function d ( r - r0 ) in the right hand side, which is written for a
three-dimensional field as follows.
Here, R = r – r0, R =|R|=| r – r0 |. In Eq. (2.4), the inner product n0 ⋅ u0
and the outer product n0¥u0 stand for normal velocity component and tangential velocity vector on the boundary surface, and they correspond to the
source distribution on the surface and the vortex distribution that has the rotating axis in parallel to the surface, respectively. Therefore, it is mathematically understood that velocity fields of viscous and incompressible flow are
obtained from the field integration concerning vorticity distributions in the
flow field and the surface integration concerning source and vortex distributions around the boundary surface. In this study, a boundary surface is represented by the panel method. The source and vortex corresponding to the second and third terms of right hand side of Eq. (2.4) are distributed on the
boundary surface. The strengths of source and vortex are obtained by using
the following two conditions; zero normal component of relative velocity to
the boundary surface u ⋅ n = 0 and the relation of the conservation of the vortex strength, respectively.
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The pressure in the field is obtained from the integration equation (Uhlman 1992), instead of the finite difference calculation of the Eq. (2.2) as follows.
Here, b =1 in the flow field and b =1/2 on the boundary S. G is the fundamental solution given by Eq.(5). H is the Bernoulli function defined as
follows.
The values of H on the boundary surface are calculated from Eq. (2.6) by
using the panel method. After the pressure distribution around the boundary
surface is calculated from Eqs. (2.6, 2.7), integration of the pressure acting on
the body surface yields the force acting on the body.
2.2 Introduction of Nascent Vortex Elements
The simulation of the flow past a body must involve a vorticity creation
around its surface accompanied by the processes of the viscous diffusion and
the convection. In the advanced method developed by the group of the present
authors, a thin vorticity layer is considered along the solid surface, and a vortex
element is introduced into the surrounding flow field considering diffusion
and convection of vorticity from the thin vorticity layer as shown in Fig.1. If a
linear distribution of velocity in the thin vorticity layer is assumed, the normal
velocity Vn on a panel can be expressed using the relation of continuity of flow
and non-slip condition on the solid surface for the element of the vorticity
layer
Here, usi=ui ⋅ nsi and DSi=h ⋅Dli. Where, DSp , ui and nsi denote the panel
area, the velocity vector and the normal unit vector on a side sectional plane of
an element of the vorticity layer, respectively. On the other hand, the vorticity
of the thin layer diffuses through the panel into the outer flow field with a
diffusion velocity. In order to consider this vorticity diffusion, the diffusion
velocity is employed in the same manner as the Vorticity Layer Spreading
Method (Kamemoto 1994), which is expressed as the following equation.
Here, n is kinematic viscosity of the fluid. If Vn+Vd becomes positive, a
nascent vortex element is introduced into the flow field, where the thickness
and vorticity of the element are given from the relation of the strength of vorticity conservation as follows.
Applicability of the Vortex Methods for Aerodynamics of Heavy Vehicles
507
Here, w is the vorticity originally involved in the element of the vorticity
layer, V and Vvor are the volume of the vorticity element and the nascent vortex
element. Every vortex element is introduced at the distance of 0.5hvor from the
panel as a vortex plate. Every vortex plate element, which moves beyond a
boundary at the distance of four times h from the solid surface, is replaced with
a vortex blob of the core-spreading model (Nakanishi and Kamemoto 1992).
In this scheme, by the assumption of a linear distribution of velocity in the
thin vorticity layer, shearing stress on the wall surface is evaluated from the
following equation.
Fig. 1.
Introduction of a nascent vortex element
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Every nascent vortex element which is far from the solid surface can be replaced with an equivalent discrete vortex element. The discrete vortex element
is modeled by a vortex blob which is a spherical model with a radially symmetric vorticity distribution. The i-th vortex blob is defined by the position ri=(rx,
ry, rz), its vorticity wi=(wx, wy, w z) and its core radius e i. The vorticity distribution around the vortex blob is represented by the following equations.
p(x ) is smoothing function (Winckelmans and Leonard 1988).
The motion of the discrete vortex element is represented by Lagrangian
form of a simple differential equation ∂r/∂t = u. Then, the trajectory of the
discrete vortex element over a time step is approximately computed from the
second-order Adams-Bashforth method. On the other hand, the evolution of
vorticity is calculated by Eq. (2.1) with the three-dimensional core spreading
method, in which the core radius increases with time. In this study, stretch
and diffusion terms in Eq. (2.1) are separately considered. The change of core
radius due to the stretch is calculated from the following equations.
Here, e and l are the core radius and the length of the vortex blob model,
respectively. The viscous term in Eq. (2.1) is expressed by the core spreading
method. The core spreading method is based on Navier-Stokes equation for
the viscous diffusion of an isolated two-dimensional vortex filament in a rest
fluid, and the rate of core spread is represented as follows.
c 2n
Ê de ˆ
=
, (c=2.242)
Á ˜
Ë dt ¯ diffusion 2e t
(2.19)
Taking two factors into account, each value of a new element is obtained
from the following equations.
ÈÊ de ˆ
˘
Ê de ˆ
e t +Dt = e t + ÍÁ ˜
+Á ˜
˙ ⋅ Dt
ÎÍË dt ¯ stretch Ë dt ¯ diffusion ˚˙
lt + Dt = lt +
dl
⋅ Dt
dt
(2.20)
(2.21)
Applicability of the Vortex Methods for Aerodynamics of Heavy Vehicles
509
Here, Eq. (2.22) is based on the conservation of circulation.
It should be noted here that in order to keep higher accuracy in expression
of a local vorticity distribution, a couple of additional schemes of redistribution of vortex blobs are introduced in our advanced vortex method.
When the vortex core of a blob becomes larger than the minimum width of local flow section, the vortex blob is discretized into a couple of smaller blobs.
On the other hand, if the rate of three-dimensional elongation becomes large
to some extent, the vortex blob is discretized into multiple blobs to approximate the elongated vorticity distribution much more properly.
3 Calculation Results and Discussions
The vortex method explained above is applied to the simulation of unsteady
flows around a heavy vehicle such as a simplified tractor-trailer model. The
effect of the gap length between the tractor and the trailer on drag forces was
investigated numerically. And, the simulation of the flow around the tractortrailer during its meandering was performed.
As the tractor-trailer model, the Ground Transportation System model
(SNL model) used to the experimental investigation by the group of Sandia
National Laboratories (Gutierrez et al. 1996) was employed in this study.
Here, it should be noted that wheels were not considered and the side corner
of the tractor was beveled in this calculation. In order to investigate the effect
of the gap length between the tractor and the trailer on drag forces, unsteady
flows around tractor-trailer with two gap length cases (G/S1/2=0.25 and 0.65)
were calculated. Here, S is a frontal area of the tractor. Reynolds number was
set as Re=U0S1/2/n =3.0¥106.
Figure 2 shows the flow patterns represented by discrete vortex elements
and pressure distributions on the body surface. Here, the maximum number
of vortex element is 190,850. For each gap length, two kinds of vortex structures are formed on the side wall of the trailer. One of them is the vortex
structure formed from the front pillar of the tractor, and another is formed
from lower part of the gap and is rolled up from the under-hood of the trailer.
For large gap length (G/S1/2=0.65), the wake formed behind the tractor flows
into the gap between the tractor and the trailer. Figure 3 shows the instantaneous velocity vectors and vorticity distributions within the gap at mid height.
The symmetric vortex structure is formed within the gap at short gap length
(G/S1/2=0.25). However, for large gap length (G/ S1/2=0.65), the vortex flow
within the gap becomes strongly asymmetric flow. Figure 4 provides the drag
force coefficient versus time, CD=Fx/(0.5rU02S), computed by using the present method. For both gap lengths, the drag coefficient acting on the tractor is
almost constant. However, the drag coefficient acting on the trailer at the large
gap length (G/S1/2=0.65) is larger than that at the short gap length
(G/S1/2=0.25). Figure 5 shows the comparison of drag coefficient acting on
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tractor and trailer with the experimental results. In Fig.5, “2-00 Cab Exp.”
shown by broken lines indicates that the side corner radius is twice the length
of that of SNL model. It can be observed that the calculated results are closer
to the experimental results of 2-00 Cab indicated by broken lines. It is considered that these results were caused by the beveled shape of the side corners of
the tractor used in this calculation.
Furthermore, the aerodynamic features of a model of tractor-trailer with
meandering motion were investigated. Tractor and trailer are connected by a
pin joint at the position of x/S1/2=0.382 away from the front of the trailer.
The tractor turns to the left after turning 90 degrees to the right in the turning
radius R/ S1/2=6.4. As a typical flow field, Fig.6 shows instantaneous flow
patterns represented by discrete vortex elements. Here, the maximum number
of vortex element is 331,321. The complex and unsteady vortex structure is
formed behind the tractor-trailer according to the meandering motion. Figure
7 shows the time history of drag force and side force coefficients acting on the
tractor and the trailer during meandering motion. In Fig.7, it is shown that
the drag acting on the meandering trailer becomes negative intermittently. It
is revealed from the calculation that considerable fluctuations of aerodynamic
forces inevitably act on both the tractor and the trailer as a result of unsteady
interaction of the flow separated from the tractor with the trailer. Although
the detail of the flow characteristics are left as study in the future, the present
method can be useful for the investigation of the unsteady and complex vortical flow around the tractor-trailer and for the effect of the deformation of the
gap on the characteristics of the drag acting on the tractor-trailer.
4 Conclusions
In this paper, recent works concerning the development of the methods were
overviewed, and the mathematical background and numerical treatment of a
vortex method developed by the group of the present authors were explained.
And it became clear that the vortex methods have so much interesting features
that they consist of simple algorithm based on physics of flow and provide
easy-to-handle and completely grid-free Lagrangian calculation of unsteady
and vortical flows without use of any RANS type turbulence models.
From the results of recent works of application, it has been confirmed that
the vortex method is available and useful for research and development of
aerodynamics of heavy vehicles. Finally, the present authors would like to
state that the advanced vortex method is one of the most capable methods to
contribute to the new generation of computational fluid dynamics (CFD) and
it yields a promising way to a grid-free Lagrangian large eddy simulation of unsteady and complex flows of higher Reynolds numbers.
Applicability of the Vortex Methods for Aerodynamics of Heavy Vehicles
511
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(a) G/S1/2=0.25
(b) G/S1/2=0.65
Fig. 2. Instantaneous flow patterns represented by discrete vortex elements
Trailer front
Tractor back
Trailer front
Tractor back
(a) G/S1/2=0.25
Fig. 3. Vorticity and velocity distributions within the gap
(b) G/S1/2=0.65
Applicability of the Vortex Methods for Aerodynamics of Heavy Vehicles
(a) G/S1/2=0.25
(b) G/S1/2=0.65
Fig. 4. Time histories of fluid forces acting on the tractor-trailer
Fig. 5. Comparison of experimental results
Fig. 6. Instantaneous flow pattern around a tractor-trailer with meandering
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K. Kamemoto and A. OJima
(a) Drag coefficient
Fig. 7. Force coefficients versus time
(b) Side force coefficient