Download Macro Traffic Flow Model Based on the Hydromechanics Theory WANG Fu

Physical and Numerical Simulation of Geotechnical Engineering
9th Issue, Dec. 2012
Macro Traffic Flow Model Based on the Hydromechanics
Theory
WANG Fu 1, 2, LI Jie 2
1. Department of Civil Engineering, Wuhan Polytechnic University, Wuhan 430023, P.R.China
2. School of Civil Engineering & Mechanics, Huazhong University of Science &Technology, Wuhan 430074, P.R.China
[email protected]
ABSTRACT: Traffic flow theory, an interdisciplinary subject, applies mathematical and physical
method to describe the traffic characteristics. It is a developing theory without comprehensive theory
has been formatted in this field so far. When analyzing the similar characteristics between traffic and
liquid flow, the macro traffic flow model based on the hydromechanics theory is established by
adopting the hydromechanics method, and in the case of permanent traffic flow, the macro traffic flow
model is solved and the definition of traffic press is given. Traffic flow takes on a flow-resistance
condition on occasions when the density is high. As a result, according to the viscosity theory of
liquid, the viscous macro traffic flow model is set up.
KEYWORDS: Macro traffic flow, Hydromechanics, Traffic press, Viscosity theory
INTRODUCTION
Molecules, known as units of liquid medium in
hydromechanics, are not closely ranged one by one, but
hold interspaces between each other. From the view of
microcosmic, liquid is not a matter of consistency
distribution. However, on engineering application,
hydromechanics merely discusses the law of motion,
dynamic property of liquid and the interactions between
liquid and solid walls, while movement of microcosmic
molecules are out of concern. As for the liquid’s macro
movement, there are two assumptions. Firstly, the
minimum liquid infinitesimal is one with infinitesimal
volume (called as liquid particle), thus interspaces between
molecules units can be omitted. Secondly, without taking
the motion of every single molecule into account, we see
liquid as a consistent medium with endless liquid
infinitesimals that distribute continuously. So this is the
consistent assumption of liquid [1][2].
Traffic flow is generated by vehicles, and distances exist
between vehicles, from the microcosmic point of view,
although traffic flow is not consistently distributed, we can
define vehicles together with their distances as vehicle field.
Regarding that traffic flow is filled with vehicle field. We
can assume it as a flow of continuity. Besides, the vehicle
field can be contracted to occupy the entire roadway.
Consequently, it’s what we called the continuity assumption
of traffic flow. After comparing the similar features
between liquid flow and traffic flow, subsequently,
hydromechanics theory is employed to study the qualities
of traffic flow [3].
1
ANALOGY
BETWEEN
THE
CHARACTERISTICS OF TRAFFIC FLOW AND
LIQUID FLOW
Traffic flow is a special kind of liquid flow, there being a
correspondence relation between its parameters, illustrated
in table 1.
Table 1 Analogy between the characteristics of traffic flow and liquid flow
Physical characteristic
Continuous body
Discrete element
Variable
Hydromechanics system
One-way incompressible liquid
Molecule
Mass m
Speed u
Pressure p
Traffic flow system
One-lane compressible vehicle flow
Vehicle
Vehicle amount m
Speed u
Traffic pressure p
mu
mu
Momentum
2 BUILDING TRAFFIC FLOW MODEL
In hydromechanics, the movement differential equation [4]
is:
 (ku )
 (u )( ku )  k f    p
t
or (kui )  (ku j ui )  kf  pij
i
t
x j
xi
To a perfect, one-dimensional liquid which doesn’t move
© ST. PLUM-BLOSSOM PRESS PTY LTD
permanently, movement differential equation or Euler
equation is: [5][6]
z 1 p u
u
g   
u
0
x k x t
x
Where,  is operator notation, k is density, u is speed, f
is mass force, p is second- factorial stress tensor, g is
gravitation acceleration, z is site altitude. The above
expressions are frequently used in electromagnetic
hydromechanics
and
chemical
hydromechanics
respectively.
Macro Traffic Flow Model Based on the Hydromechanics Theory
DOI: 10.5503/J.PNSGE. 2012.09.003
according to Newton’s second law of motion,
1 p
(k x)  a  p,
a
k x
whenΔ x→0
This article is devoted to introducing the above theories
to application of traffic flow, and establishing macro traffic
flow model.
Regarding traffic flow as the movement under certain
restraint fictitious force, when the composite force
approaches zero, traffic flow represents a uniform motion.
On circumstances that the composite force generates a
increment-Δp, traffic flow is accelerated. In figure 1, take
one vehicle field in a single lane Δx for example, the
amount of vehicle is kΔx, the length along the lane is Δx,
defining acceleration speed of the vehicle field as a, and it’s
corresponding increment of composite force as –Δp,
du u
u
p p , and
a

u

dt t
x
x x
u
u 1 p
u

0
t
x k x
(1)
(2)
The above formula is movement differential equation or
Euler equation under non-cohesive condition.
- p
k x
x
Figure 1 Stress analysis to vehicle field of traffic flow
with vehicle amount being m, to generate acceleration a.
gas pressure remain the same whether gas flow is
permanent or not, likewise, traffic pressure based on
permanent traffic flow is also available to calculate traffic
pressure of impermanent traffic flow.
When traffic flow reaches a given density, it presents
somewhat cohesive features for mutual interferences. The
origin of viscosity is from hydromechanics. The viscosity
of liquid is due to tangential resistance among liquid
micelles when relative slip occurs. Viscosity forms the
internal friction of liquid, which is direct ratio with touch
area and inverse ratio with velocity gradient, that is
u
Z  A
y
3 TRAFFIC PRESSURE AND VISCOSITY
RESISTANCE
To permanent traffic flow, k  0 , u  0 . As a result,
t
t
Euler equation is given by
du 1 dp

0
dx k dx
That is kudu  dp  0
u
(3)
Based on the relationship between the three parameters
of traffic flow, we can obtain q=ku, in combination with
formula (3), and solving the equation
(4)
p  p1  q(u1  u)
Where, under known state 1,
p1
is acting force and
u1
is
Where: u is velocity gradient,
speed, analyzing the dimension of p: the generally used
basic dimensions of length, mass and time are [L], [M] and
[T] respectively in hydromechanics. Likening traffic flow
with liquid flow, the number of vehicles is linked to mass.
so, the basic dimension in traffic flow are length [L],
amount of vehicle [M] and time [T], and dimensions of
other physical variable can be derived from certain physical
equation, basing on the basic dimensions, for instance, due
to q=ku, we can deduce that the dimension of flow volume
is [ MT-1 ], besides, from the definition of speed, the
dimension of speed is [ LT-1 ], in addition, based on formula
(3), the dimension of p is [ LMT-2 ], known as the dimension
of pressure, with pcu  km/h 2 as its unit, therefore p is
defined as traffic pressure.
The physical meanings of traffic pressure: on the
premises of permanent traffic flow, contrasting formula (1)
to Newton’s second law of motion F=ma, it’s obvious that
traffic pressure is the force which enables the vehicle field,
y

is the kinematic
viscosity of liquid.
Internal friction on unit area is tangential stress:
u
y
In fixed reference frame, when the density is low, traffic
flow wave transmits forward so that it wouldn’t influence
the upper stream, while the density is high, traffic flow
wave transmits backward, and force upper stream to change
speed. Comparing with viscosity of liquid, in the article, we
define the interference from the lower stream as viscosity
of traffic flow. Supposing viscous resistance is
 
0

 w   u1  u q
 2

15
k
x
whenk  k1 , u  u1 ;
whenk  k1, u  u1 .
Physical and Numerical Simulation of Geotechnical Engineering
9th Issue, Dec. 2012
After introducing the definition of viscous resistance and
reviewing the stress analysis of vehicle field, we can get the
movement differential equation describing viscous traffic
flow:
traffic flow.
REFERENCES
u
u 1 p
u

 w  0
t
x k x
[1].
This is movement differential equation of viscous traffic
flow.
[2].
4 CONCLUSIONS
[4].
[3].
(1) Based on the similar characteristics of traffic flow
and liquid flow, we build the macro traffic flow model
based on hydromechanics.
(2) Solving the macro traffic model on permanent
condition and analyzing the physical feature of the solution,
we got the definition of traffic pressure.
(3) When the density is high, traffic flow represents
viscous, comparing with the viscosity of liquid, we
established the movement differential equation of viscous
[5].
[6].
16
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