Numerical and Experimental Investigations of Thermal
Stress Effect on Nonlinear Thermomolecular Pressure
Difference
Victor Yu. Alexandrov, Oscar G. Friedlander, Yurii V. Nikolsky
Central Aerohydrodynamic Institute (TsAGI)
1, Zhukovsky str., 140180 Zhukovsky, Russia
Abstract. Investigation of slow nonisothermal gas flow is continued. The problem of nonlinear thermomolecular pressure
difference at the ends of capillary is considered. This pressure difference originates from temperature difference at the
ends of capillary under any flow regime. Gas flow in cylindrical capillary is numerically investigated in continuum
regime. In this case the slow nonisothermal flow equations and Navier-Stokes equations are used. Slow nonisothermal
flow equations are the generalization of Navier-Stokes equations, taking into account temperature stress action in the bulk
of a gas. Numerical investigation are carried out for two dependence of viscosity coefficient on temperature, - linear and
square root. These dependences agree with soft potential (Maxwell molecules) and stiff potential (hard sphere molecules)
of molecular interaction. Temperature boundary distribution corresponds to experimental conditions. The correlation
parameter is found. The numerical investigation of flow in plane channel is carried out for kinetic (transition) regime. The
limits of applicability (on small values of Knudsen number) for slow nonisothermal flow equations are numerically
determined. Experimental investigation of differential nonlinear thermomolecular pressure difference is carried out on
special facility. The flows of monatomic gases (helium, argon) and diatomic gases (nitrogen, air) are investigated.
Qualitative correlation between theoretical and experimental data is established. Correlation of the data for monatomic
gases is possible by Knudsen number. Inapplicability of Navier-Stokes equations for investigation of slow gas flows
under strong heat transfer is confirmed. The quantitative agreement of numerical and experimental data for helium is
shown.
INTRODUCTION.
The goal of this research is the investigation of the thermal stress effects in the problem, which is available for
numerical and experimental methods. The interest in these effects is explained by the following circumstances. The
theoretical study of the slow non-isothermal flows of gas as a continuum has highlighted several unusual effects:
- inapplicability of Navier-Stokes equations for investigation of slow flows under strong heat transfer [1-3],
- thermal stresses cause the motion of gas even in the absence of any mass forces and external pressure
difference in the thermally non-uniform gases (except the cases when the flow boundaries are absolutely
symmetric) [1-3],
- heated or cooled particles with the uniform surface temperature mutually repulse [3,4],
- the particle is subjected not to the drag , but to the action of accelerating force in Stokes flow around a strongly
heated spherical particle with the uniform temperature of the surface [5,6].
Reviews of these researches is given in [3,7,8]. Various variants of derivation of slow nonisothermal (SNIT) flow
equations by Chapmen-Enskog and Hilbert methods are proposed in [1,2,9-11]. Of course, it lead to the same SNIT
flows equations with thermal creep slip boundary condition for velocity. All of these effects up to now are
inaccessible ones for experimental detection. This is the sequent of small values of velocity and pressure difference
which are originated from thermal stresses. The velocity is of the order of c Kn, where c –thermal velocity, and
pressure difference is of the order of pKn2. In [12-14] the nonlinear differential thermo-molecular pressure difference
problem was offered for experimental research. In this problem the main effect at the small Knudsen number is
originated from the thermal creep boundary condition. It leads to linear and nonlinear effects. In opposite to thermal
creep thermal stress action is only nonlinear in temperature difference. To separate the action of thermal stress in the
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
250
TW
δp/∆p1d
0.1
T2
NS
0.0
T1
SNIT
0
∆L
1
∆L0
2
∆L/D
-0.1
0.0
0.4
0.8
1.2
FIGURE 1. Left: temperature profile of capillary surface, which is necessary for detecting of thermal stress action,
∆L -zone with high temperature gradient. Right: DNTPD, equal to (p0-p2), divided by one-dimensional TPD, equal
to ( p 2 − p1 )(∆L → ∞) , see Eq.(2); in accordance wits SNIT flow equation or Navier-Stokes equations. T2/T1=2.5,
rhomb –Maxwell molecules, circles – Hard sphere molecules.
total effect is difficult. Therefore it was proposed to compensate linear effect by pure design efforts. It can be done if
along the capillary to assign temperature profile, shown on Fig.1. Temperature distribution consists of two zones
with temperature gradients: with small gradient ( ∆L0 ) and with high gradient ( ∆L ). In ∆L0 nonlinear effects are
not become apparent, or pressure difference on this zone is determined only linear effect of thermal creep. Because
the sign of thermal creep is opposite in zones ∆L , ∆L0 , then linear part of thermomolecular pressure difference is
compensated. The resulting nonlinear part of thermomolecular pressure difference named here DNTPD, differential nonlinear thermomolecular pressure difference. As it was numerically shown in [13], the nonlinear effect
of thermal creep leads to increase of pressure difference (which is determined by linear approximation the
temperature slip velocity and Stokes equations). There was shown also, that thermal stress action leads to alteration
of DNTPD sign. This circumstance gives the opportunity to detect experimentally thermal stress action by the sign
of measured DNTPD. The special facility was designed. It realized the temperature profile, shown on Fig.1.
In [15] the first results were reported. It shows that the sign of DNTPD is opposite to determined by NavierStokes equations. But, numerical and experimental investigations were carried out with high errors. The limitation of
this experimental research consists in the measurement only with air as a working gas. The air is a mixture, so the
diffusion stresses may exist. Additionally, the Burnett temperature coefficients are not known for gas with diatom
molecule, the first investigation on this problem published in this year [16]. The numerical solution of continuum
equations was only approximate ones. So the interpretation of the results of experiment was difficult. In this research
we tried to extract these defects.
The experimental facility, gauges were tested once more. The new numerical codes for continuum flow and for
flow in transition regime give the opportunity to calculate the interesting values with very small checked and tested
error. Experiments with several gases (helium, argon, nitrogen, air) in continuum and transition regime were carried
out.
NUMERICAL INVESTIGATION
The gas flow in cylindrical capillary is investigated. The temperature distribution on the tube surface is the same
as on the Fig.1. On the short zone the high temperature gradient is exist The distribution of temperature on this part
is linear in accordance to experimental design. All parameters will be related to its value on the cold part of
251
capillary. Undisturbed gas supposed to be at z → −∞ (zone 1, Fig.1), so pressure and temperature of a gas believed
to be p ∞ , T∞ = T1 .
Continuum flow regime
As it was mentioned above the investigation of DNTPD demands to take into account the thermal stress action in
equation of impulse. In the boundary conditions must be taken into account the thermal creep velocity.
Problem statement
The investigation of monatomic one component gas flow through capillary with high temperature gradients the
SNIT flow equations and Navier-Stokes equations are used. The difference between the results on p, achieved
bythese calculations, is due to thermal stress action. The sum of nonlinear effect of thermo-creep velocity and
thermal stress action must be measured experimentally. The dependence of viscosity coefficient on temperature
adopted to be power law, corresponding to power law of molecular interaction. Burnett transport coefficients are
known only for Maxwell molecules and for hard sphere molecules. So the index s=1 and s=0.5 are used. Lower the
SNIT flow equations in non-dimensional form are shown.
1
∇ i (u i ) = 0
T
u ∇ T
1
1
1
∇ j ( u i u j ) + ∇ i Π = ∇ j T s [∇ i u j ] + A1 ( 2sK 3 − K 5 ){− T 2 s − 2 (∇ k T ⋅ ∇ k T ) + T s −1 2 Pr k k }∇ i T
T
T
2
2
(1)
1
s
∇ i ui =
∇ iT ∇ iT ,
2 Pr
1
1
ρ = , A1 =
T
4
p
2µ 1 RT1
, density to 1 , temperature to T1, modified variable part of
Here coordinates related to D, velocity to
p1 D
RT1
pressure to
2µ 12 RT1
p1 D
2
. Besides Pr =
µC p
2
λ = 3 , constants K3, K5 depends on molecular potential. For Maxwell
molecules s = 1, K 3 = K 5 = 3 , for hard sphere molecules s = 1 , K 3 = 2.418 , K 5 = 0.219 .Equations (1) must be
2
completed by the thermal creep slip
∂T
uwi ⋅ eτ i = κ ⋅ T s ⋅
⋅ eτ i
∂xi
Here s = 1, κ = 0.5625 for Maxwell molecules and s = 1 , κ = 0.6463 for hard sphere molecules. Values of A are
2
equal to 1/4 for SNIT flow equations and 0 for Navier-Stokes equations.
Numerical method
The problem was solved in cylindrical system of coordinate with symmetry conditions on the axes, and
undisturbed flow condition on the ends of the flow. The method of time-dependent solution was used. The
conservative finite difference scheme was applied. The set 200x80 or 300x40 was used.
Results
In the consideration of DNTPD we use the ratio of δ p = p 0 − p 2 , pressure difference on sensible element with
temperature distribution, that is shown on Fig.1(left). On Fig. 1 (right) this pressure difference is divided by onedimensional approximation for the case of small temperature gradients (linear approximation), ∆p1d = p 2 − p1 . This
value is equal
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∆pT/∆pT1d
1.0
∆pT/∆pT1d
0.1
0.5
T2 / T1
∆L/D
0.0
0.0
0.0
0.5
1
1.0
2
3
4
5
6
FIGURE 2. Termo-molecular pressure difference due to temperature stress action, divided by estimated value of its action in
one-dimensional approximation. Left: dependence on the length of zone with high temperature gradient; Right: dependence on
temperature ratio on this zone. Circle Maxwell molecules, rhomb –hard sphere molecules.
∆p1d = 0.5625 ⋅
32
⋅ (T23 − T13 )
3
(2)
for Maxwell molecules and
∆p1d = 0.6463 ⋅ 16 ⋅ (T22 − T12 )
for hard sphere molecules (the values of thermal creep coefficient [16] were used). The nonlinear effect under
Navier-Stokes equations leads to increase the value of TPD, determined in linear approximation. Thermal stress
action (it differs from zero only in nonlinear approximation on temperature difference) leads to opposite sign of
DNTPD (or δ p )
As it is seen from Fig.1 the difference for different viscosity law is not large.
On Fig.2 the more detailed analysis of the thermal stress action is shown. Here value
∆p T = δ p ( SNIT ) − δ p( Navier − Stokes) divided by one-dimensional estimated value of stress action
∞
∆pT = − A1 ⋅
∫
−∞
(
dTW 3
K
) ⋅ ( sK 3 − 5 ) ⋅ TW2 s − 2 dz
dz
2
For Maxwell molecules
3 (T − T ) ⋅ D 3
∆p T = − ⋅ ( 2 1
) ⋅ (∆L ⋅ D −1 ) ,
8
∆L
hard sphere molecules for
1
(T − T ) ⋅ D 2
T
∆pT = − ⋅ 1.0995 ⋅ ( 2 1
) ⋅ ln( 2 ) .
4
∆L
T1
All variables are in non-dimension forms. From data shown on Fig. 3 it is seen, that influence of molecular
models is correlated by the ∆p T .
Kinetic Regime
The determination of applicability limits for SNIT flow equations at nonzero values of Knudsen number
was carried out by numerical investigation. The problem of DNTPD was solved in kinetic (transition) flow regime.
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∆p/∆p1d
1.0
δp/∆p1d
0.2
NS
SNIT
NS
0.1
0.0
0.5
SNIT
-0.1
0.0
0.00 0.05 0.10 0.15 0.20
0.00
FIGURE 3. Left:
∆p
0.05
0.10
0.15
α
-0.2
0.00 0.05 0.10 0.15 0.20
Kn
0.00
0.05
0.10
0.15
Kn
- pressure difference on zones with temperature gradient devided by one-dimensional TPD in continuum
regime (see Eq.(2)). Triangle – TPD on ∆L0 → ∞ ., square – TPD on ∆L . Cross – results of SNIT flow equations solution for
plane channel, star – results of NS equations solution for plane channel. Right: DNTPD, p 0 − p 2 , devided by one dimensional
TPD. Circle- kinetic solution; Cross, star –see above.
Problem statement
The difficalties, connected with solution of kinetic equation for small Knudsen number lead us to application
models kinetic equations (BGK model and S-model, derived by E.Shahov). Last model give in continuum limit true
value of Prandtl number. Flow in plane channel (nor in cylindrical capillary) was investigated for simplicity. If we
relate coordinate to distance between plates, velocity to ( 2 RT1 )1 2 , pressure and temperature to p1 and T1, then
kinetic S-model equation has form
∂f α +
ξi
= (f − f)
∂xi τ
Here α =
p∗ D
12
. Parameter α connected with Knudsen number by relation Kn =
5
π α −1 . Boundary
8
µ∗ (2 RT∗ )
condition was complete diffusive one with temperature distribution as on Fig.1. The kinetic equation was solved by
the finite difference scheme. Nonuniform grid was used. In the velocity space the Hermitian nodes were used (with
weighted Maxwellian function under condition T = T1 = 1 ).
Results
On Fig.3, left, it is shown dependence the pressure difference at zone with high surface temperature gradient
∆p 01 = p 0 − p1 or at zone with small gradient ∆p 21 = p 2 − p1 upon Knudsen number. There are results of
continuum equations (SNIT and Navier-Stokes) solutions for plane channel and gas with linear dependence of
254
-10δp, µm Hg
-δp, arbitrary units
28
2
He
26
24
1
SNIT
22
20
p, µm Hg
t, min0
18
0
2
4
6
8
10
10
100
1000
FIDURE 4. Left – method of processing of differential pressure gauge signal. Cross and Cross in the circle –
various connection of battery to pressure gauge, circle – zero signal.
Right DNTPD in helium. Various symbol relate to various runs.
viscosity coefficient on temperature also. It is seen, that small difference between ∆p 01 and ∆p 21 tends to zero if
Knudsen number is more than 0.1. Additionally, there is no possibility for determination the limit if Knudsen
number tends to zero (solutions of SNIT or Navier-Stokes equations). But if we determine δ p = ∆p 01 − ∆p 21 , then
the answer on question “what equations we can use for mathematical modeling DNTPD?” is evident (see Fig.3,
right). If Knudsen number is less 0.02, then the solution of kinetic equation is close to SNIT flow equations solution.
Solution of Navier-Stokes equations leads to wrong result.
EXPERIMENTAL INVESTIGATION
The results of experiments, which were carried out with air as working gas [15], show, that it is possible to
measure DNTPD in monatomic gases. Application monatomic gases is important, because in the air, besides thermal
stresses may act diffusion (concentration) stresses. Moreover, for comparison of experimental data with numerical
ones it is necessary to know Burnett transport coefficients. Investigation on the values of Burnett coefficients for
diatomic molecules is at the beginning.
Test Bed
Principal scheme of experimental design and its realization was reported in detail in [15]. The main features of
this scheme are such ones:
1. The base of design is sensitive element with temperature distribution on its surface, corresponding to profile
on Fig.1. Length of zone with high temperature gradient is ∆L = 0.15D . Length of zone with small
temperature gradient is ∆L0 = D . Temperature of cold and hot zones at sensitive elements are
80 K ± 3K and 187 K ± 4 K . It is used cryogenic cooling by liquid nitrogen. Temperature of hot zone
stabilized by free convection in work chamber of vacuum aerodynamic tube.
2. The temperatures of both ends of capillary tube are equal ones. This property give us the possibility to use
battery of elements instead one element. In this case pressure difference is multiplied by number of elements
in battery. In facility we use 10 elements.
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3. The test bed was placed in work chamber of vacuum aerodynamic tube VAT-2M TsAGI to avoid a leakage.
4. The horizontal snake position of elements was chosen to diminish free convection influence on pressure
difference.
Average pressure in facility was in the range 20 – 2000 µ mHg , and typical pressure difference was
approximately in the range 0.1 – 2.0 µ mHg .
The design for all gases distribution was assembled. All gauges were tested and calibrated, new methods of
processing signals were used. For example, method of determination of δ p is shown on Fig.4. Value of δ p was
determined by half sum of signals.
-10δp, µm Hg
2
-10δp, µm Hg
2
Ar
N2 , air
1
1
SNIT
0
0
10
100
10
1000
p, µm Hg
100
1000
p, µm Hg
FIGURE 5. Experimental data on DNTPD in argon and in nitrogen, air. Various symbols relate to various runs.
1
(-δp)/(-δp)max
He
Ar
0.5
0
0.001
0.01
FIGURE 6. Correlation of data on DNTPD for monatomic gases by Knudsen number on cold zone Kn D,c
256
0.1
KnD,c
Results
Experimental data on DNTPD for monatomic (helium, argon) and diatomic (nitrogen, air) gases are shown on
Fig. 4,5. On these figures there are results of SNIT flow equations solution also. The results of experiment show
following features.
1. Measurements with light gas (helium) indicate qualitative (sign) and quantitative (40% difference)
accordance in numerical and experimental data. Numerical solution of Navier-Stokes equations leads to
opposite sign of pressure difference (DNTPD).
2. Results of experiments with heavy monatomic gas (argon) show big difference (up to 10 times) in numerical
and experimental data.
3. Results for diatomic gases show that data for nitrogen and air are practically coincide. This effect means that
diffusion stresses (if exist) are small in these experiments.
The correlation of DNTPD experimental data for helium and argon by Knudsen number is shown on Fig.6. As it
is seen the maximum values of DNTPD are realized at the equal values of Knudsen number ( Kn D,c ≈ 0.01 )
.
ACKNOWLEDGEMENTS
The research was supported by Russian Foundation for Basic Research (Grant 02-01-00597) and Program of
State Support for the Leading Scientific Groups (Grant 00-15-96069).
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