New Relations between Macroparameters in Shock Wave
Alexander I. Erofeev, Oscar G. Friedlander
Central Aerohydrodynamic Institute (TsAGI)
1, Zhukovsky str., 140180 Zhukovsky, Russia
Abstract. The profiles of macroparameters in a strong shock wave in one-component monatomic gas are investigated. It
is found out that the heat flux is changed approximately as the second power polynomial, and the velocity gradient is
changed as the third power polynomial of the gas velocity in the shock wave. Three independent coefficients of these
polynomials are the functions of Mach number of the undisturbed flow. These dependencies allow us to describe the
velocity and temperature profiles in strong shock waves in elementary functions. With these relations we can describe the
stress and heat flux through the velocity and temperature gradients. The relations generalize the Newton-Fourier linear
laws which are true for the infinitely weak shock wave.
The present paper continues the analysis started in [1 - 6]. The dissipative quantities and the velocity gradient in
the shock wave at Mach number M > 1.5 are investigated. The parameters dependent on the kind of the molecule
interaction potential are found. Papers [7,8] have considered the Couette flow and have shown that the normal
solution of the kinetic equation exists at any local Knudsen number. In this case it is possible to express the stress as
the function of the velocity gradient only (for more details see [9]). The question is how general these results are.
This paper presents the attempt to understand if it is possible to describe the flow in the shock wave by macroscopic
equations obtained by closing the conservation equations by non-linear dependencies of the stress and heat flux from
the velocity and temperature gradients.
THE PROBLEM STATEMENT AND METHOD OF SOLUTION
Let us consider the one-component monatomic gas. The results given in [10, 11] have shown that the DSMC
method allows us to use VHS model of the molecule interaction instead of the real power potentials to determine the
values of the macroparameters. The algorithm of majorant collision frequency is used. The shock wave is generated
by the moving piston. The dimension of the design area and the cell, time step and the number of simulating
molecules were chosen to reduce the statistic error of the hydrodynamic parameters calculation to 0.1 %. Besides the
hydrodynamic parameters the stress, heat flux and the velocity gradient were investigated. The quantitative data
were obtained at 1.5 < M < 50 for three values of k power for the potential of intermolecular interaction
U(r) = cr~k : k=4, 10 and oo. The data obtained for k = 4 refine the results of [12]. Density, temperature, pressure
are referred to their values in an undisturbed flow - p^ T^^p^. The velocity is referred to c^ = (2RT00)1
2
. Stress
pxx and heat flux qx are referred, respectively, to p^ and p^c^ . Coordinate x is referred to mean free path
A^ = (le/SjrXjUoo / p00)(2/7tRT00f'5(l-2/3k)(l-l/k)
. In dimensionless variables the undisturbed velocity in a
shock wave coordinate system is u\ = S , where S is the velocity ratio. The conservation equations of mass,
momentum and energy can be reduced to the form:
qx-(3/ 2)upxx + 4S(Ul - u)(u -u2) = 0
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
154
(1)
Here, the velocity behind shock wave U2 is defined by the Rankine-Hugoniot relation. From conservation equations
it follows that if we find out in any way the dependence of pxx from u, the dependencies qx (u)9 T(u) are
simultaneously established, too.
2. NUMERICAL RESULTS.
2.1. Stress, heat flux, temperature.
The analysis of the numerical results allowed to establish the following simple approximate dependence of pxx
from u
upxx =
~2 = (40 / 33)5(1 + 0.5S~
- «)(« - «2 ) ,
(2)
Here the relation for C(S) is given for the VHS (pseudomaxwellian) molecules k =4. Fig.la shows dependence (2)
presented by a symbol for the shock waves at M = 3 and M =11. The line presents the DSMC results.
40
U.ZrU
Upxx -
0.16-
,+'
/ >,-*-*
M=3
x
"-t-
K\
\
ft
;
Sh
i
0.0
20
5O
•K
K
0 04
10
\
X
0.2
b)
30
•f X
J<
5
0.08-
T
a)
+/ ^
0.12-
n nn
M=ll
.+--*
0.4
0.6
0.8
1.0
10
FIGURE 1. Maxwell molecules: a) stress and velocity relation, b) parabolic dependence of temperature on velocity.
For the purpose of convenience normalized velocity V = (HI - u)/(u2 —u{) is used instead of velocity u in Fig. la.
The value of upxx in Fig.la is additionally referred to S 3 . The conservation law and the parabolic dependence of
upxx against velocity establish the parabolic dependence of the heat flux against velocity. Moreover, the heat flux
and product of stress by velocity should also be proportional each other (see also [5, 7]). The calculations have
shown that this relation is not observed only in the area close to U2. The parabolic dependence upxx against
velocity is identical to the proportional dependence of the transverse stress against the gas density in the shock wave
described in [2, 3],
The results analogous to (2) were obtained also for the gas consisting of hard sphere molecules (k = oo) and in the
case when k = 10. When k = oo the main difference is that coefficient 40/33 is replaced by coefficient 40/32 = 1.25 in
relation (2). Thus, both functional dependence (2) and its parameter C(S) weakly depend on the molecule interaction
potential. This feature allows to think that the dissipative processes in the shock wave are approximately symmetric,
especially when we are speaking about the heat flux. Also, we can say that point XQ where
u =uav = ( t / 1 + i / 2 ) / 2 is the center of the shock wave (this point was taken as zero coordinate in [12]).
From relation (2) and the momentum conservation law it follows that there exists the parabolic dependence of
temperature against the flow velocity. Fig.l shows dependence T(u) for maxwellian molecules at M = 11. The
solid line shows the DSMC results, the dashed line - the values obtained by the momentum conservation equation
and approximation (2). It should be pointed out that the parabolic dependence of the temperature on velocity
establishes the relation of their profiles in the shock wave.
155
2.2. The velocity gradient and the hydrodynamic profiles.
To derive the relation between stress and its Newton approximation we should analyze the velocity gradient. It
turned out that the velocity gradient profile can be approximated by the third power polynomial of velocity:
where V is the normalized velocity. For Maxwellian molecules coefficients a, b can be well approximated by
functions
\-l/3
4
5
1
(M-l) 4
'
fe =
T( 1 "T7)—————————T4
3
M 15.29 +(M-l)
Approximation (3) of the velocity gradient by the third power polynomial as well as relation (2) turned out to be
applicable not only for the gas of Maxwellian molecules but also for the gas consisting of molecules interacting with
another potential. But coefficients a, b are strongly depend on the molecule potential.
0.25
1.00
0.75
0.50
0.25
0.00
0.00
0.25
0.50
0.75
1.00
0
25
50
75
100
125
FIGURE 2. Maxwellian molecules, M =11: a) velocity gradient vs. velocity, b) profiles of hydrodynamic parameters.
Figure 2a presents the dependence of the velocity gradient referred to V(l - V) against velocity V for Maxwellian
molecules and M = 11. The solid line shows the DSMC results, the dashed line - approximation (3). Approximation
(3) allows us to define the dependence of the coordinate against velocity in elementary functions. And relation (2)
makes it possible to derive the temperature profile by the velocity profile. Fig. 2b shows the normalized profiles of
hydrodynamic parameters of the gas of Maxwellian molecules. The solid lines show the DSMC results, and the
dashed lines present the results obtained by approximations (1 - 3). It is obvious that the results are in well
agreement. It means that the deviations from approximations (2), (3) have weak effect on the profiles of the
parameters in a shock wave. Figures 3a, 3b show the analogous data for the hard sphere molecules.
1.01-V
b)
0.80.60.40.2-
1.2
0.00.0
0.2
0.4
0.6
0.8
U
8
10
12
FIGURE 3. Hard sphere molecules: a) velocity gradient vs. velocity, b) profiles of hydrodynamic parameters, M=50.
156
The case when M = 50 shows the greatest deviations from approximation (3). But it has practically no influence on
the approximation of the profiles of density, velocity and temperature (see Fig. 3). It should be mentioned that
approximation (3) as well as any other macroscopic theory does not allow to describe the far asymptotic of
macroparameters in a shock wave (see [13, 14]).
2.3. Generalized Newton and Fourier relations.
Newton relation for stress pxx =-(4/3)fi(du/dx) and Fourier relation for the heat flux qx =-k(dTldx) are
true only for the infinitely weak shock wave (M —> 1). The results obtained in paragraphs 2.1 and 2.2 allow to close
the conservation equations by the relations generalizing the Newton and Fourier relations for an arbitrary value of M
number. The equations constructed in this way will enable to define the macroparameters in the shock waves
approximately by solving the continuum equations and not Boltzmann equation. For the purpose of convenience let
us employ not the direct results of DSMC calculations, but their approximations (2), (3). The dependencies
pxx,qx p^x,qx ,p,T on coordinate x and M obtained from the kinetic solution define as implicit functions the
needed
relations.
First,
let
us
analyze
the
features
of
pXx/P= Fp(Pxx/P>M),
dependencies
qx I pc = Fq(qx I pc,M) , where c = -
b)
-30
-1.2
-0.8
-0.4
FIGURE 4. Maxwellian molecules: a) actual stress vs. Newton stress, b) actual heat flux vs. Fourier heat flux.
These functions are shown in Figures 4a, 4b. The values of stress pxx (Fig. 4a) close to bisector, correspond to
subsonic areas of flow, and values close to ordinate correspond to supersonic velocities. The data shown in Fig.4a
make it obvious that in subsonic flow stress is close to Newton stress at any M number. Maybe, it means that it
corresponds to the normal Gilbert solution. But strong difference of stress from Newton values in supersonic flow
can mean that their normal solution (if it exists) is not of Gilbert type.
The detailed analysis of the dependencies for the heat flux (dot line in Fig.4b shows the Fourier approximation)
was carried out. The analysis has shown that the heat flux values are close to the values of Fourier approximation
only in the weak shock waves. In strong shock waves their difference is significant in any point (see also [1]). Fig.
5a presents the profiles of the heat flux for the strong shock wave.
The heat fluxes corresponding to Fourier relation or Burnett approximation were calculated by the velocity and
temperature profiles obtained by DSMC method. The results presented in Fig.Sa show that neither Fourier relation
nor Burnett approximation can describe properly the heat transfer in the shock wave. It should be pointed out also
that the dependencies for stress and heat flux as functions of Newton stress and Fourier heat flux shown in Figures
4a, 4b (at a fixed M number) are two-valued and are not local. So, the solution of the problems by these relations
will require additional analysis. The dependencies of stress and heat flux against local values of hydrodynamic
parameters and their gradients have much better properties:
lp,qx
PjP =
157
(4)
O.OE+0
b)
0.75
-l.OE+4
1 - DSMC
2-N-S
3 - Burnett
200
400
600
FIGURE 5. Maxwellian molecules: a) difference approximations of heat flux profiles for M=50, b) generalization of Newton
relation for stress
The first relation for Maxwellian molecules is presented in Fig.Sb. It proves single-valued dependence of the relative
stress on its arguments.
ACKNOWLEDGMENTS
The research was supported by Russian Foundation for Basic Research (Grant 99-01-00154) and Program of
State Support for the Leading Scientific Groups (Grant 00-15-96069).
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