Because the flow is in the continuum– transition regime Kn 0.1, the Navier–Stokes equations become inaccurate, and the differences between the Navier–Stokes and the augmented Burnett so
Trang 1The no-slip-/no-temperature-jump boundary conditions are employed at the wall when solving the
contin-uum Navier–Stokes equations for Kn 0.001 In the contincontin-uum–transition regimes, the
non-slip-boundary conditions are no longer correct First-order slip/temperature-jump non-slip-boundary conditionsshould be applied to both the Navier–Stokes equations and Burnett equations in the range
0.001 Kn 0.1 The transition regime spans the range 0.01 Kn 10; the second-order
slip/temperature-jump conditions should be used in this regime with the Navier–Stokes as well as the Burnett equations The
Navier–Stokes equations are first-order accurate in Kn, while the Burnett equations are second-order accurate in Kn Both first- and second-order Maxwell–Smoluchowski slip/temperature-jump boundary
conditions are generally employed on the body surface when solving the Burnett equations
The first-order Maxwell–Smoluchowski slip-boundary conditions in Cartesian coordinates are[Smoluchowski, 1898]:
and
The subscript s denotes the flow variables on the solid surface of the body First-order Maxwell–
Smoluchowski slip-boundary conditions can be derived by considering the momentum and energy-fluxbalance on the wall surface The reflection coefficient –σand the accommodation coefficient α– are assumed
to be equal to unity (for complete accommodation) in the calculations presented in this chapter
Beskok’s slip-boundary condition [Beskok et al., 1996] is the second-order extension of the Maxwell’sslip-velocity-boundary condition excluding the thermal creep terms, given as:
where b is the slip coefficient determined analytically in the slip flow regime and empirically in
transi-tional and free molecular regimes
Langmuir’s slip-boundary condition has also been employed in the literature [Myong, 1999].Langmuir’s slip-boundary condition is based on the theory of adsorption phenomena at the solid wall.Gas molecules do not in general rebound elastically but condense on the surface, being held by the field
of force of the surface atoms These molecules may subsequently evaporate from the surface resulting insome time lag Slip is the direct result of this time lag The slip velocity at the wall is given as:
2γ
γ 1
2 –α–α
2 –σ–σ
Trang 2In this chapter, these slip boundary conditions are applied and compared to determine their influence
on the solution
Bobylev (1982) showed that the conventional Burnett equations are not stable to small wavelength turbances; hence, the solutions to conventional Burnett equations tend to diverge when the mesh size ismade progressively finer Balakrishnan and Agarwal (1999) performed the linearized stability
dis-of one-dimensional original Burnett equations, conventional Burnett equations, augmented Burnettequations, and the BGK–Burnett equations They considered the response of a uniform gas subjected tosmall one-dimensional periodic perturbationsρ , u , and T for density, velocity, and temperature respec-
tively Burnett equations were linearized by neglecting products and powers of small perturbations, and a
linearized set of equations for small perturbation variables V [ρ , u , T ] Twas obtained They assumedthat the solution is of the form:
whereφ α iβ, andαandβdenote the attenuation and dispersion coefficients respectively For stability,
α 0 as the Knudsen number increases Substitution of Equation (8.31) in the equations for small
per-turbation quantities V results in a characteristic equation, |F(φ,ω)| 0 The trajectory of the roots of thischaracteristic equation is plotted in a complex plane on which the real axis denotes the attenuation coefficientand the imaginary axis denotes the dispersion coefficient For stability, the roots must lie to the left of theimaginary axis as the Knudsen number increases Figures 8.2 to8.5 show the trajectory of the three roots
of the characteristic equations as the Knudsen number increases The plots show that the Navier–Stokes tions, the augmented Burnett equations, and the BGK–Burnett equations (withγ 1.667) are stable, butthe conventional Burnett equations are unstable Euler equations are employed to approximate the materialderivatives in all three types of Burnett equations The BGK–Burnett equations, however, become unstableforγ 1.4 On the other hand, if the material derivatives are approximated using the Navier–Stokes equa-tions, then the conventional, augmented, and BGK–Burnett equations are all stable to small wavelengthdisturbances
equa-Based on these observations, we have employed the Navier–Stokes equations to approximate the materialderivatives in the conventional, augmented, and BGK–Burnett equations presented in Section 8.3 For thedetailed analysis behind Figures 8.2 to 8.5, see Balakrishnan and Agarwal (1999) The linearized stability
0
− 1
− 0.5 0.5
FIGURE 8.2 Characteristic trajectories of the one-dimensional Navier–Stokes equations
Trang 3analysis of conventional, augmented, and super-Burnett equations has also been performed in threedimensions with similar conclusions [Yun and Agarwal, 2000].
An explicit finite-difference scheme is employed to solve the governing equations of Section 8.3 TheSteger–Warming flux-vector splitting method [Steger and Warming, 1981] is applied to the inviscid-fluxterms The second-order, central-differencing scheme is applied to discretize the stress tensor and heat-flux terms Converged solutions were obtained with a reduction in residuals of six orders of magnitude
FIGURE 8.3 Characteristic trajectories of the one-dimensional augmented Burnett equations (γ 1.667); Euler
equations are used to express the material derivatives D( )/Dt in terms of spatial derivatives.
− 40 − 35 − 30 − 25 − 20 − 15 − 10 − 5 0
− 150
− 100
− 50 0 50 100 150
Attenuation coefficient ()
FIGURE 8.4 Characteristic trajectories of the one-dimensional BGK–Burnett equations (γ 1.667); Euler
equa-tions are used to express the material derivatives D( )/Dt in terms of spatial derivatives.
Trang 4All the calculations were performed on a sequence of successively refined grids to assure grid ence of the solutions.
Numerical simulations have been performed for both the hypersonic flows and microscale flows in thecontinuum–transition regime Hypersonic flow calculations include one-dimensional shock structure,two-dimensional and axisymmetric blunt bodies, and a space shuttle re-entry condition Microscale flowsinclude the subsonic flow and supersonic flow in a microchannel
The hypersonic shock for argon was computed using the BGK–Burnett equations The upstream flowconditions were specified and the downstream conditions were determined from the Rankine–Hugoniotrelations For purposes of comparison, the same flow conditions as in Fiscko and Chapman (1988) wereused in the computations The parameters used were
T∞ 300 K, P∞ 1.01323 105N/m2, γ
argon 1.667, µ
argon 22.7 106kg/sec mThe Navier–Stokes solution was taken as the initial value This initial Navier–Stokes spatial distribu-tion of variables was imposed on a mesh that encloses the shock The length of the control volume enclos-ing the shock was chosen to be 1000 λ∞where the mean free path based on the freestream parameters
is given by the expressionλ∞ 16µ/(5ρ∞兹2π苶R苶T苶∞苶) This is the mean free path that would exist in theunshocked region if the gas were composed of hard elastic spheres and had the same viscosity, density,and temperature as the gas being considered The solution was marched in time until the observed devi-ations were smaller than a preset convergence criterion
A set of computational experiments was carried out to compare the BGK–Burnett solutions with theBurnett solutions of Fiscko and Chapman (1988) Tests were conducted at Mach 20 and Mach 35 In order totest for instabilities to small wavelength disturbances, the grid points were increased from 101 to 501 points
Figures 8.6 and 8.7 show variations of specific entropy across the shock wave The BGK–Burnett equations
− 5 0 5
FIGURE 8.5 Characteristic trajectories of the one-dimensional conventional Burnett equations; Euler equations are
used to express the material derivatives D( )/Dt in terms of spatial derivatives.
Trang 5show a positive entropy change throughout the flow field, while the conventional Burnett equations giverise to a negative entropy spike just ahead of the shock as the number of grid points is increased This spikeincreases in magnitude until the conventional Burnett equations break down completely The BGK–Burnettequations did not exhibit any instabilities for the range of grid points considered.Figure 8.8 shows thevariation of reciprocal density thickness with Mach number BGK–Burnett calculations compare well tothose of Woods and simplified Woods equations [Reese et al., 1995] and the experimental data ofAlsmeyer (1976) Extensive calculations for one-dimensional hypersonic shock structure using varioushigher order kinetic formulations are given in Balakrishnan (1999).
0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6
X
Burnett (F&C) BGK−Burnett
0 2 4 6 8
− 2
FIGURE 8.6 Specific entropy variation across a Mach 20 normal shock in a monatomic gas (argon), ∆x/λ∞ 4.0and γ 1.667; F & C⬅ Fiscko and Chapman (1988)
0 2 4 6 8 10
0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6
X
Navier−Stokes Burnett (F&C) BGK−Burnett
FIGURE 8.7 Specific entropy variation across a Mach 35 normal shock in a monatomic gas (argon), ∆x/λ∞ 4.0and γ 1.667; F & C⬅ Fiscko and Chapman (1988)
Trang 68.7.2 Application to Two-Dimensional Hypersonic Blunt Body Flow
The two-dimensional augmented Burnett code was employed to compute the hypersonic flow over acylindrical leading edge with a nose radius of 0.02 m in the continuum–transition regime The grid sys-tem (50 82 mesh) used in the computations is shown in Figure 8.9 The results were compared withthose of Zhong (1991)
The flow conditions for this case are as follows:
M∞ 10, Kn∞ 0.1, Re∞ 167.9,
P∞ 2.3881 N/m2, T∞ 208.4 K, T w 1000.0 KThe viscosity is calculated by Sutherland’s law,µ c1T1.5/(T c2) The coefficients c1and c2for air are1.458 106kg/(sec m K1/2) and 110.4 K, respectively Other constants used in this computation for airareγ 1.4, Pr 0.72 and R 287.04 m2/(sec2K)
The comparisons of density, velocity, and temperature distributions along the stagnation streamline areshown in Figures 8.10,8.11, and8.12respectively The results agree well with those of Zhong (1991) forboth the Navier–Stokes and the augmented Burnett computations Because the flow is in the continuum–
transition regime (Kn 0.1), the Navier–Stokes equations become inaccurate, and the differences between
the Navier–Stokes and the augmented Burnett solutions are obvious In particular, the difference between theNavier–Stokes and Burnett solutions for the temperature distribution is significant across the shock.Temperature and Mach number contours of the Navier–Stokes solutions and the augmented Burnettsolutions are compared in Figures 8.13 and 8.14 respectively The shock structure of the augmentedBurnett solutions agrees well with that of Zhong (1991) The shock layer of the augmented Burnett solutions
is thicker, and the shock location starts upstream of that of the Navier–Stokes solutions However, becausethe local Knudsen number decreases and the flow tends toward equilibrium as it approaches the wall surface,the differences between the Navier–Stokes and augmented Burnett solutions become negligible near the
BGK−Burnett
FIGURE 8.8 Plot showing the variation of reciprocal density thickness with Mach number, obtained with theNavier–Stokes, Woods and Simplified Woods [Reese et al., 1995], and BGK–Burnett equations for a monatomic gas(argon) Experimental data were obtained from Alsmeyer (1976) (Reprinted with permission from Alsmeyer, H.(1976) “Density Profiles in Argon and Nitrogen Shock Waves Measured by the Absorption of an Electron Beam,”
J Fluid Mech 74, pp 497–513.)
Trang 7wall, especially near the stagnation point Thus, the Maxwell–Smoluchowski slip boundary conditions can beapplied for both the Navier–Stokes and the augmented Burnett calculations for the hypersonic blunt body.
The results of the axisymmetric augmented Burnett computations are compared with the DSMC resultsobtained by Vogenitz and Takara (1971) for the axisymmetric hemispherical nose The computed resultsare also compared with Zhong and Furumoto’s (1995) axisymmetric augmented Burnett solutions Theflow conditions for this case are
Trang 8The comparisons of density and temperature distributions along the stagnation streamline among thecurrent axisymmetric augmented Burnett solutions, the axisymmetric augmented Burnett solutions ofZhong and Furumoto, and the DSMC results are shown in Figures 8.15 and 8.16, respectively The corre-sponding Navier–Stokes solutions are also compared in these figures The axisymmetric augmentedBurnett solutions agree well with Zhong and Furumoto’s axisymmetric augmented Burnett solutions inboth density and temperature The density distributions for both the Navier–Stokes and augmented
0 5 10 15 20
Augmented Burnett (Zhong)
FIGURE 8.10 Density distributions along stagnation streamline for blunt body flow: air, M∞ 10, and Kn∞ 0.1
0
2500 3000
Augmented Burnett (Zhong)
x/r n
FIGURE 8.11 Velocity distributions along stagnation streamline for blunt body flow: air, M∞ 10, and Kn∞ 0.1
Trang 9Burnett equations show little differences from the DSMC results The temperature distributions, however,show that the DSMC method predicts a thicker shock than the augmented Burnett equations The max-imum temperature of the DSMC results is slightly higher than those of the augmented Burnett solutions.However, the augmented Burnett solutions show much closer agreement with the DSMC results than theNavier–Stokes solutions.
Augmented Burnett (Zhong)
FIGURE 8.12 Temperature distributions along stagnation streamline for blunt body flow: air, M∞ 10, and
Kn∞ 0.1
Augmented Burnett Navier−Stokes
FIGURE 8.13 Comparison of temperature contours for blunt body flow: air, M∞ 10, and Kn∞ 0.1
Trang 10Overall, the axisymmetric augmented Burnett solutions presented here agree well with Zhong andFurumoto’s (1995) axisymmetric augmented Burnett solutions and describe the shock structure closer tothe DSMC results than the Navier–Stokes solutions.
As another application to the hypersonic blunt body, the augmented Burnett equations are applied tocompute the hypersonic flow field at re-entry condition encountered by the nose region of the space shuttle
20
10
− 0.7 − 0.6 − 0.5 − 0.4 − 0.3 − 0.2 − 0.1
Augmented Burnett Augmented Burnett (Zhong and Furumoto) DSMC (Vogenitz and Takara)
Trang 11The computations are compared with the DSMC results of Moss and Bird (1984) The DSMC methodaccounts for the translational, rotational, vibrational, and chemical nonequilibrium effects.
An equivalent axisymmetric body concept [Moss and Bird, 1984] is applied to model the windward
cen-terline of the space shuttle at a given angle of attack A hyperboloid with nose radius of 1.362 m andasymptotic half angle of 42.5° is employed as the equivalent axisymmetric body to simulate the nose ofthe shuttle.Figure 8.17 shows the side view of the grid (61 100 mesh) around the hyperboloid Thefreestream conditions at an altitude of 104.93 km as given in Moss and Bird (1984) are
M∞ 25.3, Kn∞ 0.227, Re∞ 163.8,
ρ∞ 2.475 107kg/m3, T∞ 223 K, T w 560 K
The viscosity is calculated by the power law The reference viscosityµ
r and the reference temperature T r
are taken as 1.47 105kg/(sec m) and 223 K, respectively
Figures 8.18 and 8.19 show comparisons of the density and temperature distributions along the tion streamline between the Navier–Stokes solutions, the augmented Burnett solutions, and the DSMCresults The differences between the augmented Burnett solutions and the DSMC results are significant inboth density and temperature distributions In Figure 8.18, the density distribution of the DSMC results islower and smoother than that of the augmented Burnett solutions In Figure 8.19, the DSMC method pre-dicts about 30% thicker shock layer and 9% lower maximum temperature than the augmented Burnettequations The DSMC results can be considered to be more accurate than the augmented Burnett solutions
stagna-as the DSMC method accounts for all the effects of translational, rotational, vibrational, and chemical equilibrium, while the augmented Burnett equations do not However, the augmented Burnett solutionsagree much better with the DSMC results than the Navier–Stokes computations The difference betweenthe Navier–Stokes solutions and the augmented Burnett solutions in temperature distributions is very sig-nificant The shock layer of the augmented Burnett solutions is almost two times thicker than theNavier–Stokes solutions The augmented Burnett solutions predict about 11% less maximum temperaturethan the Navier–Stokes solutions
non-0 5 10 15
25
20
30
Augmented Burnett (Zhong and Furumoto) Augmented Burnett
DSMC (Vogenitz and Takara)
Trang 128.7.4 Application to NACA 0012 Airfoil
The Navier–Stokes equations are applied to compute the rarefied subsonic flow over a NACA 0012 airfoilwith chord length of 0.04 m The grid system in the physical domain is shown in Figure 8.20.The flowconditions are
M∞ 0.8, Re∞ 73, ρ∞ 1.116 104kg/m3, T∞ 257 K, and Kn∞ 0.014
Various constants used in the calculation for air areγ 1.4, Pr 0.72, and R 287.04 m2/(sec2K)
Figure 8.21 shows the density contours of the Navier–Stokes solution with the first-order Maxwell–Smoluchowski slip-boundary conditions These contours using the continuum approach agree well withthose of Sun et al (2000) using the information preservation (IP) particle method At these Mach andKnudsen numbers, the contours from the DSMC calculations are not smooth due to the statistical scatter.The comparison of pressure distribution along the surface between our Navier–Stokes solution with a slip-boundary condition and the DSMC calculation [Sun et al., 2000] is shown in Figure 8.22; the agreementbetween the solutions is good.Figure 8.23compares the surface slip velocity from the DSMC, IP, andNavier–Stokes methods as calculated by Sun et al and by our Navier–Stokes calculation The slip velocity
= 0°
r n
FIGURE 8.17 Side view of the grid (61 100 mesh) around a hyperboloid nose of radius r n 1.362 m
Trang 13distribution from our Navier–Stokes calculation shows good agreement with that obtained from theDSMC and IP methods except near the trailing edge However, our Navier–Stokes results disagree consid-erably with those reported in Sun et al (2000) This calculation again demonstrates that Navier–Stokes
equations with slip-boundary conditions can provide accurate flow simulation 0.001 Kn 0.1.
The augmented Burnett equations are employed for computation of subsonic flow in a microchannel
with a ratio of channel length to height of 20 (L/H 20) For the wall boundary conditions, Beskok’s and
15000 20000
10000
5000
DSMC (Moss and Bird) Augmented Burnett Navier−Stokes
Trang 141 1.1
− 0.1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
1 1.1
1.08 1.04 1.12
0.96
1.00
FIGURE 8.21 Density contours for NACA 0012 airfoil: air, M∞ 0.8 and Kn∞ 0.014; Navier–Stokes solution withfirst-order slip boundary condition
Trang 150 1
IP (Sun et al.) DSMC (Sun et al.)
... 8.21 shows the density contours of the Navier–Stokes solution with the first-order Maxwell–Smoluchowski slip-boundary conditions These contours using the continuum approach agree well withthose... encloses the shock The length of the control volume enclos-ing the shock was chosen to be 1000 λ∞where the mean free path based on the freestream parametersis given by the expressionλ∞... stability, the roots must lie to the left of theimaginary axis as the Knudsen number increases Figures 8.2 to8.5 show the trajectory of the three roots
of the characteristic equations as the