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A scheme for computing steady viscous gas flows
U.S.S.R. Comput. Maths Math. @Pergamon
Pbys. Vol. 18, pp. 192-194
0041-5553/78/0301-0192
$7.50/O
Press Ltd. 1978. Printed in Great Britain.
A SCHEME FOR COMPUTING
STEADY VISCOUS GAS FLOWS*
V.M. BORISOV and A.S. LITVINOVICH Moscow (Received 17 February 1977)
A SCHEME for computing steady viscous gas flows based on the Krook-Bhatnagar-Gross presented. The flow of air in a cylindrical tube of finite length is considered as an example.
model is
Among the diverse approaches to the numerical solution of problems of the dynamics of a viscous gas, the mathematical formulation based on the use of Boltzmann’s integrodiiferential equation [ 1, 2) is of special interest. The fact is that by using asymptotic methods it is possible to pass from Boltzmann’s equation, for example, to the Navier-Stokes system of equations. On the other hand Boltzmann’s equation, for a fixed value of the parametric variables and with a certain method of quasi-linearization, may be regarded as a collection of isolated Cauchy problems, stable in a numberical calculation. This fact may be interpreted as the splitting at the differential level of the initial problem (the solution of the Navier-Stokes equations) into a sequence of simple, well-studied problems. The situation is even more simplified by the use of the Krook-Bhamagar-Gross model, from which the Navier-Stokes equations are also obtained, of course, with Frandtl number equal to unity [2]. In this paper the method of successive approximation [3] is used to calculate the steadystate flows of a viscous gas on the basis of this model. In dimensionless form the transfer equations are written as follows: (1)
div (of) = (fo-f) / Kn E.
is the distribution function of the gas molecules, is the locally Here f@, El@, fok Em) Maxwellian function, based on the local density of the gas n, its velocity u and temperature T. Also w is the unit vector of the direction of the velocity of the molecules, 5 is the modulus of the velocity of the molecules. The dimensionless Knudsen function Kn=p (2’)lJ/nkTRo defines the mode of the gas flow. is the thermal velocity of the molecules at the characteristic temperature To, Here U= (ZRT,,) ‘/a R,, is the characteristic dimension of the domain. The viscosity coefficient p(T) was calculated by Sutherland’s formula, k is Boltzmann’s constant, and R is the specific gas constant. All the physical variables are dedimensionalized with respect to the characteristic dimensions of length, density, temperature, and velocity. A computing mesh formed by a curvilinear coordinate system is introduced in the domain of integration. The computing template is a tetrahedron (Fig. 1). The vertex of the tetrahedron is the
4
ts 1 4
--
---
2
Fig. 1.
* 2%. vj&bisl. Mat. mat. Fiz., 18, 2, 507-508,
1978. 192
J
Short
desired
point
ing formula
where
4, and the base of the tetrahedron is obtained
is formed
of the outward
by the specified
Eq. (1) over the volume
by integrating
Si is the area of the face of the tetrahedron
unit vector
193
communications
normal
situated
points
of the tetrahedron
opposite
to it, L is the characteristic
the vertex
dimension
1, 2, 3. The computand has the form
numbered
i, Ni is the
of the tetrahedron,
equal to
Fig. 2.
the length
of the rectilinear
segment
with the base of the tetrahedron. formula.
The unknown
along the vector
The fixed
macroparameters
w
values o*,
are determined
from
point 4 to the point of its intersection
6, are chosen
by a highly accurate
by summation
quadrature
of the values off
described.
U
0.08
0.06
Fig. 3.
A numerical length
Maxwellian the
left
distribution
gas us/U
distribution
was carried
end-face
for
function
= 0.1, the particle
same
domain.
calculation
2. On the
out for the flow of air in a cylindrical
molecules
with fixed
flying
within
the
domain
values of the macroparameters,
where
1 and
a locally
the velocity
of the
no = 5 X 1014 cmm3, and the temperature TO = 288.2’K. Exactly was taken on the right end-face for the particles flying within the
function
The law of reflection
of particles
from
accommodation.
the lateral
surface
of the incident flow. the Reynolds number
For the chosen example in the incident flow the Knudsen Reo = 0.548, and the Mach number MO = 0.119. of the distribution
function
The temperature
of the tube is assumed
thermal
moments
is defined
density
fusive with complete
The
tube of radius
there
were
of the wall equalled
calculated
in the spherical
to be dif-
the temperature
number
Kno = 0.365,
coordinate
system.
For integration with respect to the polar angle 0 and with respect to the velocities [ the Gauss quadrature formula was used, the passage from the half-interval of variation of the velocity modulus [0, ~0) to the segment [-1, 11 being accomplished by the formula t = (1 - t)/(l + C;). The integration with respect to the azimuthal angle $ was performed by the trapezoidal method. Figure 2 shows the velocity vectors of the gas at the nodes of the computing mesh in the meridional plane. The step along the radius and length was taken and 16 in the velocities .$j. Figure
3 shows
the variation
as 0.2, The number of the gas velocity
of points along
was: 15 in the angle #, 10 in the angle 8, the axis of the cylinder.
The increase
in
194
V.M. Borisov and A.S. Litvinovich
velocity
at the centre
example
given the law of conservation
The calculations
of the tube
is due to the stagnation of mass and volume
were performed
of the subsonic
was satisfied
for low values of the numbers
flow at its walls.
with an accuracy
In the
of 2.5%.
Kna, and slow convergence
of the
iterations was observed [l] . In this case it is advisable to use methods of improving the convergence of the successive approximations considered in [4] . In our opinion it is also important to pass to the quadratures
realizing
in the discrete
velocity
approach
the fundamental
conservation
laws for Eq. (1).
Translated
by J. Berry
REFERENCES
1.
Numerical
methods
of rarefied gases (Chislennye
in the dynamics
No. 1, VTs Akad. Nauk SSSR, Moscow, 2.
SHAKHOV, zhennogo
3.
BORISOV,
V.M. and
transport 4.
theory.
MARCHUK, Moscow,
U.S.S.R.
Moscow,
dvizhenii
razre-
1974.
LITVINOVICH,
A.S.
A method
of numerical
Zh. @chisl. Mat. mat. Fiz., 15, 4, 1072-1077,
G.I. Methods
gazov),
of designing
nuclear
reactors
integration
for stationary
problems
of
1975.
(Metody
rascheta
yadernykh
reaktorpv),
Atomizdat,
1961.
Comput.
OP ergamon
of investigating the motions of a rarefied gas (Metod issledovaniya
E.M. A method gaza), “Nauka”,
metody v dinamike razrezhennykh
1973.
Maths
0041-5553/78/0301-0194
Math. Phys. Vol. 18, pp. 194-197
Press Ltd. 1978. Printed
THE GLOBAL
$7.5010
in Great Britain.
UNIQUENESS OF THE SOLUTION OF THE DIRECT PROBLEM FOR THE LAVAL NOZZLE* E.G. SHIFRIN Moscow
(Received
THE GLOBAL established
uniqueness
of two
in the transonic
densation
25 November
classes
1976; revised version 16 December
of solutions
approximation
of the direct
of the equations
1976)
problem
of motion
of the Lava1 nozzle
assuming
the absence
is
of con-
jumps.
Analytic
studies
of the direct
485, 486-497,
523-528,
of the solution
relative
the fundamental
was obtained
considered
contained
of the Lava1 nozzle
were carried
on the basis of the method
to the initial solution.
solution
the small”
problem
540-544)
The problem
for a linear variational
equation.
described
was formulated The proof
out in [l]
(see pp. 461-
in [2] for small variations in the hodograph
of the uniqueness
in [ 3, 41. In view of the fact that in the plane of the hodograph a finite
segment
of the line of singularity,
the proof
plane of
theorem
corresponded
“in
the domain to the case
where a solution with a curvilinear sonic line in the physical plane was chosen as fundamental. Below for two classes of flows (one of them contains a nozzle with a rectilinear sonic line) a simple proof possible
of the global solutions.
uniqueness
of the solution
Here, as in [ 1,4]
is given without
we use the simplification
assuming
infinitesimal
of the equations
closeness
of motion
of the
for transonic
flow velocities. We introduce an orthogonal coordinate system 4, $ connected with the streamlines $ = 9(x, y). We denote by A the velocity coefficient, fl is the slope of the velocity vector, and k is the adiabatic index.
l
Zh. z&&sl.
Mat. mat. Fiz., 18, 2, 509-512,
1978.
Pbys. Vol. 18, pp. 192-194
0041-5553/78/0301-0192
$7.50/O
Press Ltd. 1978. Printed in Great Britain.
A SCHEME FOR COMPUTING
STEADY VISCOUS GAS FLOWS*
V.M. BORISOV and A.S. LITVINOVICH Moscow (Received 17 February 1977)
A SCHEME for computing steady viscous gas flows based on the Krook-Bhatnagar-Gross presented. The flow of air in a cylindrical tube of finite length is considered as an example.
model is
Among the diverse approaches to the numerical solution of problems of the dynamics of a viscous gas, the mathematical formulation based on the use of Boltzmann’s integrodiiferential equation [ 1, 2) is of special interest. The fact is that by using asymptotic methods it is possible to pass from Boltzmann’s equation, for example, to the Navier-Stokes system of equations. On the other hand Boltzmann’s equation, for a fixed value of the parametric variables and with a certain method of quasi-linearization, may be regarded as a collection of isolated Cauchy problems, stable in a numberical calculation. This fact may be interpreted as the splitting at the differential level of the initial problem (the solution of the Navier-Stokes equations) into a sequence of simple, well-studied problems. The situation is even more simplified by the use of the Krook-Bhamagar-Gross model, from which the Navier-Stokes equations are also obtained, of course, with Frandtl number equal to unity [2]. In this paper the method of successive approximation [3] is used to calculate the steadystate flows of a viscous gas on the basis of this model. In dimensionless form the transfer equations are written as follows: (1)
div (of) = (fo-f) / Kn E.
is the distribution function of the gas molecules, is the locally Here f@, El@, fok Em) Maxwellian function, based on the local density of the gas n, its velocity u and temperature T. Also w is the unit vector of the direction of the velocity of the molecules, 5 is the modulus of the velocity of the molecules. The dimensionless Knudsen function Kn=p (2’)lJ/nkTRo defines the mode of the gas flow. is the thermal velocity of the molecules at the characteristic temperature To, Here U= (ZRT,,) ‘/a R,, is the characteristic dimension of the domain. The viscosity coefficient p(T) was calculated by Sutherland’s formula, k is Boltzmann’s constant, and R is the specific gas constant. All the physical variables are dedimensionalized with respect to the characteristic dimensions of length, density, temperature, and velocity. A computing mesh formed by a curvilinear coordinate system is introduced in the domain of integration. The computing template is a tetrahedron (Fig. 1). The vertex of the tetrahedron is the
4
ts 1 4
--
---
2
Fig. 1.
* 2%. vj&bisl. Mat. mat. Fiz., 18, 2, 507-508,
1978. 192
J
Short
desired
point
ing formula
where
4, and the base of the tetrahedron is obtained
is formed
of the outward
by the specified
Eq. (1) over the volume
by integrating
Si is the area of the face of the tetrahedron
unit vector
193
communications
normal
situated
points
of the tetrahedron
opposite
to it, L is the characteristic
the vertex
dimension
1, 2, 3. The computand has the form
numbered
i, Ni is the
of the tetrahedron,
equal to
Fig. 2.
the length
of the rectilinear
segment
with the base of the tetrahedron. formula.
The unknown
along the vector
The fixed
macroparameters
w
values o*,
are determined
from
point 4 to the point of its intersection
6, are chosen
by a highly accurate
by summation
quadrature
of the values off
described.
U
0.08
0.06
Fig. 3.
A numerical length
Maxwellian the
left
distribution
gas us/U
distribution
was carried
end-face
for
function
= 0.1, the particle
same
domain.
calculation
2. On the
out for the flow of air in a cylindrical
molecules
with fixed
flying
within
the
domain
values of the macroparameters,
where
1 and
a locally
the velocity
of the
no = 5 X 1014 cmm3, and the temperature TO = 288.2’K. Exactly was taken on the right end-face for the particles flying within the
function
The law of reflection
of particles
from
accommodation.
the lateral
surface
of the incident flow. the Reynolds number
For the chosen example in the incident flow the Knudsen Reo = 0.548, and the Mach number MO = 0.119. of the distribution
function
The temperature
of the tube is assumed
thermal
moments
is defined
density
fusive with complete
The
tube of radius
there
were
of the wall equalled
calculated
in the spherical
to be dif-
the temperature
number
Kno = 0.365,
coordinate
system.
For integration with respect to the polar angle 0 and with respect to the velocities [ the Gauss quadrature formula was used, the passage from the half-interval of variation of the velocity modulus [0, ~0) to the segment [-1, 11 being accomplished by the formula t = (1 - t)/(l + C;). The integration with respect to the azimuthal angle $ was performed by the trapezoidal method. Figure 2 shows the velocity vectors of the gas at the nodes of the computing mesh in the meridional plane. The step along the radius and length was taken and 16 in the velocities .$j. Figure
3 shows
the variation
as 0.2, The number of the gas velocity
of points along
was: 15 in the angle #, 10 in the angle 8, the axis of the cylinder.
The increase
in
194
V.M. Borisov and A.S. Litvinovich
velocity
at the centre
example
given the law of conservation
The calculations
of the tube
is due to the stagnation of mass and volume
were performed
of the subsonic
was satisfied
for low values of the numbers
flow at its walls.
with an accuracy
In the
of 2.5%.
Kna, and slow convergence
of the
iterations was observed [l] . In this case it is advisable to use methods of improving the convergence of the successive approximations considered in [4] . In our opinion it is also important to pass to the quadratures
realizing
in the discrete
velocity
approach
the fundamental
conservation
laws for Eq. (1).
Translated
by J. Berry
REFERENCES
1.
Numerical
methods
of rarefied gases (Chislennye
in the dynamics
No. 1, VTs Akad. Nauk SSSR, Moscow, 2.
SHAKHOV, zhennogo
3.
BORISOV,
V.M. and
transport 4.
theory.
MARCHUK, Moscow,
U.S.S.R.
Moscow,
dvizhenii
razre-
1974.
LITVINOVICH,
A.S.
A method
of numerical
Zh. @chisl. Mat. mat. Fiz., 15, 4, 1072-1077,
G.I. Methods
gazov),
of designing
nuclear
reactors
integration
for stationary
problems
of
1975.
(Metody
rascheta
yadernykh
reaktorpv),
Atomizdat,
1961.
Comput.
OP ergamon
of investigating the motions of a rarefied gas (Metod issledovaniya
E.M. A method gaza), “Nauka”,
metody v dinamike razrezhennykh
1973.
Maths
0041-5553/78/0301-0194
Math. Phys. Vol. 18, pp. 194-197
Press Ltd. 1978. Printed
THE GLOBAL
$7.5010
in Great Britain.
UNIQUENESS OF THE SOLUTION OF THE DIRECT PROBLEM FOR THE LAVAL NOZZLE* E.G. SHIFRIN Moscow
(Received
THE GLOBAL established
uniqueness
of two
in the transonic
densation
25 November
classes
1976; revised version 16 December
of solutions
approximation
of the direct
of the equations
1976)
problem
of motion
of the Lava1 nozzle
assuming
the absence
is
of con-
jumps.
Analytic
studies
of the direct
485, 486-497,
523-528,
of the solution
relative
the fundamental
was obtained
considered
contained
of the Lava1 nozzle
were carried
on the basis of the method
to the initial solution.
solution
the small”
problem
540-544)
The problem
for a linear variational
equation.
described
was formulated The proof
out in [l]
(see pp. 461-
in [2] for small variations in the hodograph
of the uniqueness
in [ 3, 41. In view of the fact that in the plane of the hodograph a finite
segment
of the line of singularity,
the proof
plane of
theorem
corresponded
“in
the domain to the case
where a solution with a curvilinear sonic line in the physical plane was chosen as fundamental. Below for two classes of flows (one of them contains a nozzle with a rectilinear sonic line) a simple proof possible
of the global solutions.
uniqueness
of the solution
Here, as in [ 1,4]
is given without
we use the simplification
assuming
infinitesimal
of the equations
closeness
of motion
of the
for transonic
flow velocities. We introduce an orthogonal coordinate system 4, $ connected with the streamlines $ = 9(x, y). We denote by A the velocity coefficient, fl is the slope of the velocity vector, and k is the adiabatic index.
l
Zh. z&&sl.
Mat. mat. Fiz., 18, 2, 509-512,
1978.