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Rarefied airfoil aerodynamics based on the generalized hydrodynamic model
Aerospace Science and Technology 92 (2019) 148–155
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Rarefied airfoil aerodynamics based on the generalized hydrodynamic model H. Xiao a,b,∗ , D.H. Wang a a b
School of Power and Energy, Northwestern Polytechnical University, Xi’an, 710072, China Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, CB3 0WA, UK
a r t i c l e
i n f o
Article history: Received 31 December 2018 Received in revised form 29 May 2019 Accepted 2 June 2019 Available online 5 June 2019 Keywords: Rarefied aerodynamics Airfoil Implicit non-Navier-Stokes-Fourier framework Generalized hydrodynamic model
a b s t r a c t Numerical studies on rarefied gas flows around an airfoil are conducted by an implicit non-Navier-StokesFourier (NSF) framework, namely, the generalized hydrodynamic (GH) model. A detailed study of the rarefied gas flow around an airfoil is performed following validations. Investigations show that all the characteristics of the rarefied effect on an airfoil observed by the particle method in a previous study are also observed in the present GH study. In addition, the constitutive relationships of the GH model provide a clear explanation regarding the distinctive features of the rarefied effect on the aerodynamic characteristics of an airfoil. Also, a recompression region is found at the trailing edge in a state involving subsonic flow and a high Knudsen (Kn) number. With increasing Kn number, the drag increases rapidly compared to the lift, and this effect results in a sharp decrease in the lift-drag ratio. Moreover, the NSF framework overpredicts the lift-drag ratio for subsonic cases and underpredicts the ratio for supersonic cases in rarefied gas flows with different angles of attack considered. Credible explanations are provided for the limitations of the NSF framework in a numerical study of the rarefied effects, including the near equilibrium state, considering the present constitutive relationships of the GH model. We show that the present GH model provides a new numerical tool for the investigation of the rarefied effect on aerodynamic characteristics. © 2019 Elsevier Masson SAS. All rights reserved.
1. Introduction Investigations of rarefied [1,2] and/or continuum gas flows [3–5] have been of great interest in the past several decades. Such investigations on rarefied airfoil aerodynamics can provide an explanation of the fundamental physics for the nonequilibrium state and are widely applied in several fields [6–8], including microturbines [9], chemical sensors [10], micro propulsion for spacecraft [11], flow control devices [12], and gaseous chromatographs [13]. Unfortunately, use of the classic continuum approaches are questionable for the simulation of rarefied gas flows, and thus, particle approaches, such as the direct simulation Monte Carlo (DSMC) method [14], DSMC-IP method [15] and Lattice Boltzmann Method (LBM) [16], have been applied to investigate rarefied effects on gas flows. Initially, the low-density aerodynamic features of the NACA0009 airfoil were investigated by T.C. Tai using the DSMC method [17]. The investigations show that both lift and drag are severely penalized, and the rarefaction effect becomes much more
*
Corresponding author at: School of Power and Energy, Northwestern Polytechnical University, Xi’an, 710072, China. E-mail addresses: [email protected], [email protected] (H. Xiao). https://doi.org/10.1016/j.ast.2019.06.002 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.
dominant in the transition region. Later, J. Fan, I.D. Boyd et al. [18] simulated four cases of rarefied gas flows around the NACA0012 airfoil using the DSMC-IP method and the classic Navier-Stokes equations. A case with a low Reynolds number demonstrated an increase in pressure at the trailing edge that was not observed in high Reynolds number flows. The same case was studied by Q.H. Sun [19], J.G. McDonald [20] and Z.H. Li [21] to validate the DSMC-IP method, the Gaussian moment closure and the gaskinetic schemes of the modelled Boltzmann equation, respectively. In addition, a microscale flat plate was studied by X.D. Niu [22] using the lattice Boltzmann method. This study shows that the flow may become compressible when the Reynolds number is lower than 10, which is different from the common idea that the flow is incompressible when the Mach number is lower than 0.3. Recently, Pekardan and Alexeenko [6] made new progress on rarefaction effects for transonic airfoil flows. This study concluded that thin airfoils with gradual changes in shape lead to minimized continuum breakdown and rarefaction effects. Although great progress has been made on rarefied airfoil aerodynamics, it is still of great interest to investigate the rarefied effect on airfoil aerodynamic characteristics, especially at a high Kn state. Solving the conservation laws is the widely used approach in
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the numerical investigation of gas flows. The conservation laws of mass, momentum, and energy can be obtained from kinetic theory such as the Boltzmann equation, which governs the gas flow for all Kn values. If we implement the Newtonian framework, which uses linear uncoupled constitutive relationships of thermal conduction (Fourier laws) and viscous stress (Newtonian laws), the conservation laws take the form of Navier-Stokes-Fourier (NSF) equations. This approach has proved to be successful in continuum and equilibrium state flows. Unfortunately, the use of the NSF model becomes questionable [23] in the present study concerning a microscale airfoil in a non-continuum transitional flow regime. We applied the GH model to investigate the rarefied effect on the aerodynamic characteristics of an airfoil for a wide range of Kn values. The GH theory was first proposed by Eu [24,25]. A significant result of GH theory was that the constitutive equations of non-conserved variables, such as heat flux and stresses, were obtained from the Boltzmann kinetic theory by considering entropy conditions. Then, the constitutive equations were simplified by omitting the high-order term and unsteady term. The equations were then called the nonlinear coupled constitutive relationships (NCCR) [26]. Later on, the unsteady term was considered by one of the authors of the present study [27]. Since all these works are based on Eu’s GH theory, we call these methods the Eu model or GH model. The GH model has been demonstrated to be capable of accounting for several nonequilibrium gas flows, including the rheological flow problems, sound wave propagation [28], and shock wave structures [29,30]. The present study attempts to demonstrate that the GH model is a new and useful computational model in the study of rarefied effects on aerodynamic characteristics. The theoretical study in this work involves a detailed investigation of the rarefied effect on aerodynamic characteristics at airfoil chord length scales in the region of transitional and rarefied gas flows. By extending previous flow cases, rarefied gas flows corresponding to Kn values from 0.01 to 0.5 are examined in significant detail.
2.1. GH model The Boltzmann-Curtiss equation for diatomic molecules with a moment of inertia I and an angular momentum j can be expressed under the assumption of no external force,
j ∂ ∂ +v·∇ + f (v, r, t ) = C [ f ] ∂t I ∂ψ
distribution function evolves as a function of macroscopic moments, but the flux dependence of distribution function is strictly dictated by the entropy production. The procedure of deriving the GH from the kinetic equation is described in previous works [24]. Finally, the GH model can be expressed in a compact form:
∂U + ∇ · Finv (U) + ∇ · Fvis (U, , , Q) = 0 ∂t ⎞ ⎛
(2)
ρ
U = ⎝ ρu ⎠ ,
ρe
⎛
Finv (U) = ⎝
⎞
ρu
ρ uu + pI ⎠ , (ρ e + p )u ⎛
Fvis (U, , , Q) = ⎝
⎞
0
⎠ + I ( + I) · u + Q
∂( ) + ∇ · ( u) + k + Z k = 0 ∂t ⎞ ⎛ = ⎝ + I ⎠ , ⎛ ⎜ k = ⎝
(3)
Q 2( p + )[∇ u](2) + 2[ · ∇ u](2) 2γ ( + I) : ∇ u +
2 3
γ p∇ · u
⎞ ⎟ ⎠,
( p + ) C p ∇ T + Q · ∇ u + · C p ∇ T ⎞ p η q(κ ) ⎜2 p ⎟ ⎟ Zk = ⎜ ⎝ 3 γ ηb q(κ ) ⎠ . pC p λ Qq(κ ) ⎛
Here, p is the pressure, ρ is the mass density, e is the total energy density and T is the gas temperature. Additionally, , Q and denote the shear stress, heat flux, and excess normal stress, respectively. Here, the factor q(κ ) appearing in the dissipative terms is defined by the Rayleigh-Onsager dissipation function [25,31],
2. GH model for investigating rarefied effects on aerodynamic characteristics
149
(1)
where the term C [ f ] represents the collision integral of the interactions among the particles. The Boltzmann-Curtiss equation is irreversible and consequently is expected to describe macroscopic processes progressing irreversibly towards equilibrium. The balance equations for mass, momentum, and energy can be derived by differentiating the statistical formulas of the three quantities with time and then substituting the Boltzmann-Curtiss equation. The balance equations do not have contributions from the Boltzmann collision integral since the three quantities are conserved variables whose molecular expressions are the collisional invariants of Boltzmann collision integral. However, the balance equations contain non-conserved variables, such as the shear stress , heat flux Q, and excess normal stress , whose molecular expressions do not yield a collisional invariant. We refer to these values as non-conserved variables or non-conserved moments. To derive the evolution equations for non-conserved variables, Eu applied the modified moment method to obtain the GH theory. The
q(κ ) =
sinhκ
κ
,
1/2 2 (mk B T )1/4 : Q·Q κ= √ + 2γ + . 2pd
2η
ηb
kT
(4)
In this expression, κ plays an important role in connecting the equilibrium and nonequilibrium forms. The detailed derivation of GH from Boltzmann kinetic theory can be found in previous work [24,25]. k B , m and d represent the Boltzmann constant, molecular mass and diameter of the molecule, respectively. C p denotes the heat capacity. γ = (5 − 3γ )/2 (γ is the specific-heat ratio). η, ηb , and k represent the Chapman-Enskog shear viscosity, bulk viscosity and heat conductivity, respectively. For the classic linear uncoupled Newtonian model in the NSF framework, the constitutive relationships are expressed as follows:
0 = −2η[∇ u](2) , 0 = −ηb ∇ · u, Q0 = −k∇ T
(5)
In the framework of NSF, the excess normal stress can be regarded as the difference between the hydrodynamic pressure and thermodynamic pressure. Additionally, the excess normal stress is neglected for non-monatomic gases. For solving the highly nonlinear equations of the GH model, the method of treating the solid boundary conditions for the nonlinear constitutive equations
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Fig. 1. Comparison of the GH method and DSMC method pressure coefficients for NACA0018 Case: Ma = 2.0 and Kn = 0.02 (left), Ma = 2.0 and Kn = 0.2 (right).
is an important issue. A mixed DG method [32] and Langmuir slip boundary conditions [33,34] have been proposed to develop a highly efficient framework for solving the GH model. 3. Validation and verification Simulations of gas flows that cover continuum and rarefied gas regions are challenging problems. It is generally accepted that the classic CFD methods based on the NSF equations become invalid for rarefied flows [35]. The DSMC method models fluid flows using simulation molecules that represent a large number of real molecules, and the particle collision integral is not solved directly. For this purpose, the DSMC method is usually used to treat nonequilibrium effects in rarefied gases. The fundamental assumption of the DSMC method is that the molecular movement and collision phases can be decoupled over time periods that are shorter than the mean collision time. Therefore, the calculations are very expensive for continuum flows. The DSMC method especially suffers from statistical noise for small Knudsen number flows. Consequently, the DSMC method is not easily applied to continuum states. The GH model presented here claims to calculate both the continuum and rarefied states. The GH has been proven to tend to the NSF theory framework in the limits of low thermodynamic forces at continuum states. Because there is not much experimental data on the rarefied problem, this work uses the DSMC method and/or experimental data to validate the GH model for rarefied states. These validation cases include classic NACA0018 and NACA0012 airfoils. 3.1. Validation on NACA0018 airfoil In the present study, the chord length is regarded as the characteristic length to define the Re and Kn. The first simulations were computed for the NACA0018 airfoil. The flow conditions are Ma = 2.0 and Kn = 0.02, 0.2, with 0◦ for angle of attack (AOA). Fig. 1 gives a comparison between the pressure coefficients, c p , of the GH model solution and DSMC solution. Because of the existing shockwave, a striking pressure peak can be found at the leading edge of the airfoil. Therefore, a drastic decrease was observed in Fig. 1. Additionally, the GH model results agree well with the DSMC solutions. 3.2. Validation and verification on NACA0012 airfoil The second set of simulations were computed for the NACA0012 airfoil. Four different grid numbers (grids in the wall-normal direction and grids in the airfoil-conforming direction) were generated to check the grid independency and were compared with the selected references, as summarized in Table 1. The numerical process
Table 1 Grid independence of Case 1: Ma = 2.0 and Kn = 0.026. Grid
20 × 20
40 × 40
80 × 80
100 × 100
Cd
0.4101
0.3951
0.3901
0.3902
Table 2 Free stream conditions for Cases 1-2. Case
Ma
Kn
α
1 2
2.0 2.0
0.026 0.26
0 20
is applied to all the grids with the same computational and convergent conditions. Next, the drag coefficient of the airfoil is obtained from the solutions. These results are shown in Table 1 to study the influence of the grid densities on the drag coefficient. The solutions predict that the drag coefficient tends to be stable when the number of grids is greater than 80 × 80. Thus, we use a set of triangle meshes with 100 × 100 in the following study. The free stream conditions of the first two cases for air gas flows around the NACA0012 airfoil are listed in Table 2 and are considered for validations. Fig. 2 presents the dimensionless velocity contours of the GH model, experimental and DSMC results. Due to many applications, the supersonic results of Allegre et al. [36] were used for numerous verification studies. All the results demonstrate a shock with a large thickness. Both the GH and DSMC models capture a contour with a value of 0.9 above the trailing edge as observed in the experiment. This feature is not obtained by using the NSF framework in the previous study [18]. Unfortunately, there are still regions of difference in the details. For instance, the experimental velocity varies from 0.75 to 0.85 in a small region, whereas the velocities of the DSMC and GH methods vary from approximately 0.75 to 0.85 in a large region. This difference possibly results from the tunnel wall in the experiment that has a diameter on the order of 10 cm. This diameter is not large enough to neglect the tunnel wall effect. Here, we should present a discussion of the uncertainty analysis of the measurements. Unfortunately, the test conditions are not given in detail [36]. Considering that the study of the gas flow around a cylinder has little to do with the present study of airfoil, we have not given a validation case of cylindrical flow in this study. For this validation and comparison with experimental results, we refer the reader to the Figure 5 of the literature [27] on the cylindrical drag coefficients as a function of the Knudsen number (Ma = 2.0, 0.005 ≤ Kn ≤ 1). Fig. 3 presents the distributions of the surface pressure coefficients given by the NSF framework and the GH, DSMC-IP, and DSMC models for Cases 1 and 2. In Case 1, the pressure coeffi-
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Fig. 2. Comparison of GH model, experiment and DSMC model velocity fields for Case 1.
Fig. 3. Comparison of the GH model, DSMC model and slip NSF framework surface pressure distributions for Case 1 (left) and Case 2 (right).
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Fig. 4. Solutions of normalized constitutive equations in one-dimensional solver (u x , 0, 0, T x ) with nonlinear viscosity and the ill-posed constitutive for Maxwellian molecules. The horizontal and vertical axes represent the strain (force) term N S and the normal stress , respectively. The gas is expanding in the range of N S < 0, whereas the gas is compressed in the range of N S > 0.
cient sharply decreases at the leading edge and smoothly decreases at the trailing edge. At the leading edge, the results of the NSF framework agree well with those of the GH and DSMC-IP models. However, at the trailing edge, the pressure coefficient is underpredicted by the NSF framework compared with those predicted by the GH and DSMC models. The GH and DSMC-IP solutions agree well compared with those of the NSF framework, except for a slight overprediction in the trailing edge. In Case 2, a high Kn state with high angle of attack is considered. The gas flows are compressed on the lower surface and expand significantly on the upper surface. These effects result in a difference in the pressure distributions. Further, on the lower surface, the pressure decreases more smoothly than that of the upper surface. The GH solution agrees very well with the DSMC solution; however, the NSF framework underpredicted the pressure along the surfaces both on the lower and upper surfaces. 3.3. Theoretical analysis of GH model For the validation case, the GH and DSMC models can obtain the flow and aerodynamic characteristics of an airfoil considering a rarefied flow. Unfortunately, the NSF solutions are noticeably different than the experimental data. The NSF framework also underpredicts the pressure distribution of the solid wall in Case 1 and overpredicts that in Case 2. We solve the same conservation equations in the frameworks of GH and NSF, and the only difference lies in the constitutive relationships of the viscous stress and thermal conduction, as shown in Equations (3) and (4). In general, the constitutive equations (Equations (3) and (4)) consist of unsteady, kinematic and dissipation terms. The only term in Equation (3) directly related the Boltzmann collision is the dissipation term. This dissipation is caused by the irreversible processes of the system, and the energy dissipates through molecular collisions at the molecular level. Therefore, it is important to treat this term carefully. If we assume that the relaxation time of the non-conserved variables is very short compared with that of the conserved variables, the unsteady term vanishes. To show the features of the GH model in the simplest manner, it is assumed that the shear stress evolution is not coupled to the evolution of and Q and only consider the one-dimensional solver (u x , 0, 0, T x ) and (0, v x , 0, 0). The constitutive relationships were plotted in a onedimensional solver (u x , 0, 0, T x ) and (0, v x , 0, 0). Cases of compression, expansion and shear flows are given in Figs. 4 and 5. As mentioned previously, the nonequilibrium state depends on Ma and Kn, which is why Cases 1 and 2 show different constitutive char-
Fig. 5. Solutions of normalized constitutive equations in one-dimensional solver (0, v x , 0, 0) with nonlinear viscosity and the constitutive equation with linear viscosity for velocity shear flow. The horizontal and vertical axes represent the strain ˆ y y , respectively. (force) term xy N S and the shear and normal stresses xy ,
Table 3 Free stream conditions for Cases 3-6. Case
Mach number
Kn number
T∞, K
Tw, K
3 4 5 6
0.8 0.8 2.0 2.0
0.02 0.2 0.02 0.2
290 290 290 290
290 290 522 522
acteristics for the compression and expansion states (see Fig. 4). For Case 2, with supersonic flow and high Kn, the viscous stress of the GH is lower than that of the NSF framework in the compression region, which results in weak compression on the leading edge and low pressure along the airfoil solid wall. However, Case 1 exhibits different trends in the NSF and GH solutions, and the viscous stress obtained by the GH model is higher than that of the NSF framework owing to the near equilibrium state generated by the subsonic flow and low Kn conditions. In the region of expansion, the NSF framework overpredicts the normal stress compared with that predicted by the GH model. At the trailing edge of the airfoil, the gas flow cannot expand efficiently in the NSF solutions. 4. Results and discussion 4.1. Rarefied effect on pressure distributions For the extensive investigation of the rarefied effect on the airfoil, we select two supersonic and two subsonic cases at Kn = 0.02 and 0.2. In the subsonic cases, the wall temperature is set to 290.0 K, and for the supersonic cases, the wall temperature is equal to the stagnation temperature (see Table 3). Figs. 6 through 9 compare the pressure distributions at Kn = 0.02 and 0.2 with the Mach numbers Ma = 0.8 and Ma = 2.0. Both cases demonstrate a sharp decrease in the pressure close to the leading edge, as expected. However, for the case of Ma = 0.8 and Kn = 0.2, the pressure increases at the trailing edge. This result is not observed in the other cases, where expansion is expected with a corresponding decrease in pressure. Further, the unique characteristic of recompression at the trailing edge in a high Kn state is not found in the NSF solutions. This result can be explained by the constitutive relationships derived using the GH and NSF solutions. In the near equilibrium state, the NSF solution overpredicts the normal stress in the compression region, which causes a large recompression in the overpredicted state. This recompression phenomenon is also found with the DSMC-IP [18] method, but an explanation could not be provided. This result indicates that conventional airfoils do
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153
Fig. 6. Surface pressure distributions for Ma = 0.8 and Kn = 0.02. Fig. 9. Surface pressure distributions for Ma = 2.0 and Kn = 0.2.
Fig. 7. Surface pressure distributions for Ma = 0.8 and Kn = 0.2.
Fig. 10. Lift and drag at Ma = 2.0 and
α = 20.
Fig. 11. Lift-drag ratio at Ma = 2.0 and
α = 20.
Fig. 8. Surface pressure distributions for Ma = 2.0 and Kn = 0.02.
not provide optimum aerodynamic characteristics in rarefied gas flows. 4.2. Rarefied gas flows around NACA0012 at the angle of attack The aerodynamic performances of airfoils are usually measured by the lift and drag coefficients C L and C d , respectively. The total normal and axial forces per unit span (ds) are obtained by integrating from the leading edge (LE) to the trailing edge (TE) [37]. Fig. 10 shows the lift and drag coefficients of the flows over the microscale NACA0012 airfoil at different Kn values with the same angle of attack, i.e., α = 20. Both the lift and drag coefficients C L and C d increase when Kn increases. Additionally, as compared with
C d , the lift coefficient increases only slightly when Kn varies. The drag increases more rapidly than the lift as the Kn continues to increase. Therefore, the lift-drag ratio decreases with a value less than 1 in high Kn states (see Fig. 11). The NSF framework overpredicts the drag and underpredicts the lift compared with those predicted by the GH model. These observations can be explained by the constitutive relationships. In supersonic gas flows with nonzero angles of attack, the compression and expansion flows dominate the generation of lift and drag. Therefore, the normal stress is dominant in the viscous stress, especially in a high Kn state. As shown in Fig. 4, during compression
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Fig. 12. C L and C L /C d at Ma = 0.8 and Kn = 0.5.
Fig. 13. C L and C L /C d at Ma = 2.0 and Kn = 0.5.
and expansion, the normal viscous stress of the NSF framework is higher than that of the GH model. This effect causes difficulties for gas compression at the bottom surface and expansion at the top surface. Further, the NSF framework underpredicts the lift of the airfoil. Moreover, in the NSF framework, the difference in the normal stress during compression and expansion is larger than that noted in the GH model. This result affects the drag prediction and is the reason for the overprediction by the NSF framework.
in lower drag. In contrast, the subsonic cases considered exhibit different trends in the GH and NSF solutions because of the near equilibrium state caused by the subsonic conditions. The viscous stress of the GH model is higher than that of the NSF framework in the region of compression, which results in strong compression at the leading edge, resulting in a large drag. Therefore, the rarefied effect on the aerodynamic characteristics of an airfoil leads to different trends in far-from-equilibrium states.
4.3. Rarefied effect on aerodynamic characteristics of airfoils
5. Conclusions
To understand the rarefied effect on the aerodynamic characteristics of an airfoil, cases involving supersonic Ma = 0.8 and subsonic conditions Ma = 2.0 at high Kn (Kn = 0.5) were extensively analyzed. Figs. 12 and 13 present the lift and drag coefficients and the lift-drag ratio in terms of the angle of attack. An increase in the angle of attack led to an increase in the lift-drag ratio. However, different trends were noted for subsonic and supersonic conditions. The NSF framework overpredicted the lift-drag ratio in subsonic conditions and underpredicted this ratio in supersonic conditions. The plots of C L and C D also demonstrate the different trends of the NSF and GH solutions. As mentioned, the nonequilibrium state depends on Kn and Ma, which is why the subsonic and supersonic cases considered present different characteristics in the constitutive relationships of the compression state (see Fig. 4). For supersonic cases, the viscous stress of the GH model is lower than that of the NSF framework in the region of compression, which causes weak compression at the leading edge, ultimately resulting
The Boltzmann equation is the governing equation for the gas flow in the full range of Ma and Kn states. The conservation laws of mass, momentum and energy can be derived from the Boltzmann equation. For the closure of the non-conserved variables such as viscous stress and heat flux in the conservation laws, the linear constitutive relationships of the Newtonian laws and Fourier laws are used in the NSF framework. This setup makes the use of the NSF framework difficult in the numerical study of rarefied gas flow. For the closure of the non-conserved variables, the GH model employs the new constitutive equations, which are derived from the Boltzmann equation, in which the entropy generation is considered. This model has been validated in a previous study considering subsonic and hypersonic rarefied gas flows. In this study, a numerical investigation of rarefied effects on the aerodynamic characteristics of an airfoil is presented. In the validations, all the characteristics corresponding to the rarefied effect on an airfoil that were observed using the DSMC method in the
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previous study were also observed in this study using GH. A critical finding was that the constitutive equations of the GH model provided a clear explanation of these unique features. Detailed analyses of rarefied gas flows around an airfoil were performed. The recompression region was found at the trailing edge in a subsonic and high Kn state. Furthermore, with increase in the Kn, the drag increased rapidly compared to the lift, which resulted in a sharp decrease in the lift-drag ratio. Moreover, the NSF framework overpredicted the lift-drag ratio for the subsonic cases and underpredicted this ratio for supersonic cases in the rarefied gas flows with different angles of attack considered. Detailed explanations were provided against the limitations of the NSF framework in the numerical study of rarefied flows including consideration of the near equilibrium state by the GH model. Declaration of Competing Interest The authors declare no conflict of interest. Acknowledgements The corresponding author expresses appreciation to Prof. R.S. Myong in Gyeongsang National University of South Korea and Prof. Ehsan Roohi in Ferdowsi University of Mashhad of Iran for their help on the NCCR and the DSMC method. References [1] A.K. Chinnappan, G. Malaikannan, R. Kumar, Insights into flow and heat transfer aspects of hypersonic rarefied flow over a blunt body with aerospike using direct simulation Monte-Carlo approach, Aerosp. Sci. Technol. 66 (2017) 119–128. [2] S. Mungiguerra, G. Zuppardi, R. Savino, Rarefied aerodynamics of a deployable re-entry capsule, Aerosp. Sci. Technol. 69 (2017) 395–403. [3] F. Ladeinde, X. Cai, R. Agarwal, A methodology for hybrid simulation of rarefield and continuum flow regimes, Aerosp. Sci. Technol. 75 (2018) 115–127. [4] K.H. Lee, Numerical simulation on thermal and mass diffusion of MMH–NTO bipropellant thruster plume flow using global kinetic reaction model, Aerosp. Sci. Technol. (2019), https://doi.org/10.1016/j.ast.2018.11.056, in press. [5] K.H. Lee, Satellite design verification study based on thruster plume flow impingement effects using parallel DSMC method, Comput. Fluids 173 (2018) 88–92. [6] C. Pekardan, A. Alexeenko, Rarefaction effects for transonic airfoil flows at low Reynolds numbers, AIAA J. (2017) 765–779. [7] N.T.P. Le, H. Xiao, R. Myong, A triangular discontinuous Galerkin method for non-Newtonian implicit constitutive models of rarefied and microscale gases, J. Comput. Phys. 273 (2014) 160–184. [8] N.T.P. Le, A. Shoja-Sani, E. Roohi, Rarefied gas flow simulations of NACA 0012 airfoil and sharp 25–55-deg biconic subject to high order nonequilibrium boundary conditions in CFD, Aerosp. Sci. Technol. 41 (2015) 274–288. [9] C.-C. Lin, Development of a Microfabricated Turbine-Driven Air Bearing Rig, Ph.D. thesis, Massachusetts Institute of Technology, 1999. [10] X. Fang, T. Zhai, U.K. Gautam, L. Li, L. Wu, Y. Bando, D. Golberg, ZnS nanostructures: from synthesis to applications, Prog. Mater. Sci. 56 (2) (2011) 175–287. [11] E.C. Stearns, Thermal Analysis of a Monopropellant Micropropulsion System for a CubeSat, Master’s thesis, Department of Mechanical Engineering, California Polytechnic State University, 2013. [12] D. Jansen, Passive Flow Separation Control on an Airfoil-Flap Model, Ph.D. thesis, Department of Mechanical Engineering, Delft University of Technology, 2012.
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[13] S. Eidelman, W. Grossmann, I. Lottati, Review of propulsion applications and numerical simulations of the pulsed detonation engine concept, J. Propuls. Power 7 (6) (1991) 857–865. [14] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation Monte Carlo of Gas Flows, Oxford Engineering Science, 1994. [15] C. Cai, I.D. Boyd, J. Fan, G.V. Candler, Direct simulation methods for low-speed microchannel flows, J. Thermophys. Heat Transf. 14 (3) (2000) 368–378. [16] S. Chen, G.D. Doolen, Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech. 30 (1) (1998) 329–364. [17] T.C. Tai, Direct simulation of low-density flow over airfoils, J. Aircr. 29 (5) (1992) 806–810. [18] J. Fan, I.D. Boyd, C.-P. Cai, K. Hennighausen, G.V. Candler, Computation of rarefied gas flows around a NACA 0012 airfoil, AIAA J. 39 (4) (2001) 618–625. [19] Q. Sun, I.D. Boyd, G.V. Candler, Numerical simulation of gas flow over microscale airfoils, J. Thermophys. Heat Transf. 16 (2) (2002) 171–179. [20] J. McDonald, C. Groth, Numerical modeling of micron-scale flows using the Gaussian moment closure, in: 35th AIAA Fluid Dynamics Conference and Exhibit, 2005, p. 5035. [21] Z. Li, A. Peng, H. Zhang, X. Deng, Numerical study on the gas-kinetic high-order schemes for solving Boltzmann model equation, Sci. China, Phys. Mech. Astron. 54 (9) (2011) 1687. [22] X.-D. Niu, S. Hyodo, K. Suga, H. Yamaguchi, Lattice Boltzmann simulation of gas flow over micro-scale airfoils, Comput. Fluids 38 (9) (2009) 1675–1681. [23] N.G. Hadjiconstantinou, The limits of Navier-Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics, Phys. Fluids 18 (11) (2006) 111301. [24] B.C. Eu, Kinetic Theory and Irreversible Thermodynamics, John Wiley and Sons, 1992. [25] B.C. Eu, Kinetic Theory of Nonequilibrium Ensembles, Irreversible Thermodynamics, and Generalized Hydrodynamics, vol. 1. Nonrelativistic Theories, Springer, 2016. [26] R.S. Myong, Impact of computational physics on multi-scale CFD and related numerical algorithms, Comput. Fluids 45 (1) (2011) 64–69. [27] H. Xiao, K. Tang, A unified framework for modeling continuum and rarefied gas flows, Sci. Rep. 7 (1) (2017) 13108. [28] B.C. Eu, Y.G. Ohr, Generalized hydrodynamics, bulk viscosity, and sound wave absorption and dispersion in dilute rigid molecular gases, Phys. Fluids 13 (3) (2001) 744–753. [29] M. Al-Ghoul, B.C. Eu, Generalized hydrodynamic theory of shock waves: Machnumber dependence of inverse shock width for nitrogen gas, Phys. Rev. Lett. 86 (19) (2001) 4294–4297. [30] M. Al-Ghoul, B.C. Eu, Generalized hydrodynamic theory of shock waves in rigid diatomic gases, Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 64 (4) (2001) 046303. [31] B.W. Strutt, B. Rayleigh, The Theory of Sound, Dover Publications, 1945. [32] F. Bassi, S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations, J. Comput. Phys. 131 (2) (1997) 267–279. [33] D.K. Bhattacharya, B.C. Eu, Nonlinear transport processes and fluid dynamics: effects of thermoviscous coupling and nonlinear transport coefficients on plane Couette flow of Lennard-Jones fluids, Phys. Rev. A 35 (2) (1987) 821–836. [34] B.C. Eu, R.E. Khayat, G.D. Billing, C. Nyeland, Nonlinear transport coefficients and plane Couette flow of a viscous, heat-conducting gas between two plates at different temperatures, Can. J. Phys. 65 (9) (1987) 1090–1103. [35] M. Pfeiffer, Particle-based fluid dynamics: comparison of different BGK models and DSMC for hypersonic flows, Phys. Fluids 30 (10) (2018) 106106. [36] J. Allegre, M. Raffin, L. Gottesdiener, Slip effects on supersonic flowfields around NACA 0012 airfoils, in: International Symposium on Rarefied Gas Dynamics, 15th, Grado, Italy, 1986, pp. 548–557. [37] J.D. Anderson Jr., Fundamentals of Aerodynamics, Tata McGraw-Hill Education, 2010.
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Aerospace Science and Technology www.elsevier.com/locate/aescte
Rarefied airfoil aerodynamics based on the generalized hydrodynamic model H. Xiao a,b,∗ , D.H. Wang a a b
School of Power and Energy, Northwestern Polytechnical University, Xi’an, 710072, China Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, CB3 0WA, UK
a r t i c l e
i n f o
Article history: Received 31 December 2018 Received in revised form 29 May 2019 Accepted 2 June 2019 Available online 5 June 2019 Keywords: Rarefied aerodynamics Airfoil Implicit non-Navier-Stokes-Fourier framework Generalized hydrodynamic model
a b s t r a c t Numerical studies on rarefied gas flows around an airfoil are conducted by an implicit non-Navier-StokesFourier (NSF) framework, namely, the generalized hydrodynamic (GH) model. A detailed study of the rarefied gas flow around an airfoil is performed following validations. Investigations show that all the characteristics of the rarefied effect on an airfoil observed by the particle method in a previous study are also observed in the present GH study. In addition, the constitutive relationships of the GH model provide a clear explanation regarding the distinctive features of the rarefied effect on the aerodynamic characteristics of an airfoil. Also, a recompression region is found at the trailing edge in a state involving subsonic flow and a high Knudsen (Kn) number. With increasing Kn number, the drag increases rapidly compared to the lift, and this effect results in a sharp decrease in the lift-drag ratio. Moreover, the NSF framework overpredicts the lift-drag ratio for subsonic cases and underpredicts the ratio for supersonic cases in rarefied gas flows with different angles of attack considered. Credible explanations are provided for the limitations of the NSF framework in a numerical study of the rarefied effects, including the near equilibrium state, considering the present constitutive relationships of the GH model. We show that the present GH model provides a new numerical tool for the investigation of the rarefied effect on aerodynamic characteristics. © 2019 Elsevier Masson SAS. All rights reserved.
1. Introduction Investigations of rarefied [1,2] and/or continuum gas flows [3–5] have been of great interest in the past several decades. Such investigations on rarefied airfoil aerodynamics can provide an explanation of the fundamental physics for the nonequilibrium state and are widely applied in several fields [6–8], including microturbines [9], chemical sensors [10], micro propulsion for spacecraft [11], flow control devices [12], and gaseous chromatographs [13]. Unfortunately, use of the classic continuum approaches are questionable for the simulation of rarefied gas flows, and thus, particle approaches, such as the direct simulation Monte Carlo (DSMC) method [14], DSMC-IP method [15] and Lattice Boltzmann Method (LBM) [16], have been applied to investigate rarefied effects on gas flows. Initially, the low-density aerodynamic features of the NACA0009 airfoil were investigated by T.C. Tai using the DSMC method [17]. The investigations show that both lift and drag are severely penalized, and the rarefaction effect becomes much more
*
Corresponding author at: School of Power and Energy, Northwestern Polytechnical University, Xi’an, 710072, China. E-mail addresses: [email protected], [email protected] (H. Xiao). https://doi.org/10.1016/j.ast.2019.06.002 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.
dominant in the transition region. Later, J. Fan, I.D. Boyd et al. [18] simulated four cases of rarefied gas flows around the NACA0012 airfoil using the DSMC-IP method and the classic Navier-Stokes equations. A case with a low Reynolds number demonstrated an increase in pressure at the trailing edge that was not observed in high Reynolds number flows. The same case was studied by Q.H. Sun [19], J.G. McDonald [20] and Z.H. Li [21] to validate the DSMC-IP method, the Gaussian moment closure and the gaskinetic schemes of the modelled Boltzmann equation, respectively. In addition, a microscale flat plate was studied by X.D. Niu [22] using the lattice Boltzmann method. This study shows that the flow may become compressible when the Reynolds number is lower than 10, which is different from the common idea that the flow is incompressible when the Mach number is lower than 0.3. Recently, Pekardan and Alexeenko [6] made new progress on rarefaction effects for transonic airfoil flows. This study concluded that thin airfoils with gradual changes in shape lead to minimized continuum breakdown and rarefaction effects. Although great progress has been made on rarefied airfoil aerodynamics, it is still of great interest to investigate the rarefied effect on airfoil aerodynamic characteristics, especially at a high Kn state. Solving the conservation laws is the widely used approach in
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the numerical investigation of gas flows. The conservation laws of mass, momentum, and energy can be obtained from kinetic theory such as the Boltzmann equation, which governs the gas flow for all Kn values. If we implement the Newtonian framework, which uses linear uncoupled constitutive relationships of thermal conduction (Fourier laws) and viscous stress (Newtonian laws), the conservation laws take the form of Navier-Stokes-Fourier (NSF) equations. This approach has proved to be successful in continuum and equilibrium state flows. Unfortunately, the use of the NSF model becomes questionable [23] in the present study concerning a microscale airfoil in a non-continuum transitional flow regime. We applied the GH model to investigate the rarefied effect on the aerodynamic characteristics of an airfoil for a wide range of Kn values. The GH theory was first proposed by Eu [24,25]. A significant result of GH theory was that the constitutive equations of non-conserved variables, such as heat flux and stresses, were obtained from the Boltzmann kinetic theory by considering entropy conditions. Then, the constitutive equations were simplified by omitting the high-order term and unsteady term. The equations were then called the nonlinear coupled constitutive relationships (NCCR) [26]. Later on, the unsteady term was considered by one of the authors of the present study [27]. Since all these works are based on Eu’s GH theory, we call these methods the Eu model or GH model. The GH model has been demonstrated to be capable of accounting for several nonequilibrium gas flows, including the rheological flow problems, sound wave propagation [28], and shock wave structures [29,30]. The present study attempts to demonstrate that the GH model is a new and useful computational model in the study of rarefied effects on aerodynamic characteristics. The theoretical study in this work involves a detailed investigation of the rarefied effect on aerodynamic characteristics at airfoil chord length scales in the region of transitional and rarefied gas flows. By extending previous flow cases, rarefied gas flows corresponding to Kn values from 0.01 to 0.5 are examined in significant detail.
2.1. GH model The Boltzmann-Curtiss equation for diatomic molecules with a moment of inertia I and an angular momentum j can be expressed under the assumption of no external force,
j ∂ ∂ +v·∇ + f (v, r, t ) = C [ f ] ∂t I ∂ψ
distribution function evolves as a function of macroscopic moments, but the flux dependence of distribution function is strictly dictated by the entropy production. The procedure of deriving the GH from the kinetic equation is described in previous works [24]. Finally, the GH model can be expressed in a compact form:
∂U + ∇ · Finv (U) + ∇ · Fvis (U, , , Q) = 0 ∂t ⎞ ⎛
(2)
ρ
U = ⎝ ρu ⎠ ,
ρe
⎛
Finv (U) = ⎝
⎞
ρu
ρ uu + pI ⎠ , (ρ e + p )u ⎛
Fvis (U, , , Q) = ⎝
⎞
0
⎠ + I ( + I) · u + Q
∂( ) + ∇ · ( u) + k + Z k = 0 ∂t ⎞ ⎛ = ⎝ + I ⎠ , ⎛ ⎜ k = ⎝
(3)
Q 2( p + )[∇ u](2) + 2[ · ∇ u](2) 2γ ( + I) : ∇ u +
2 3
γ p∇ · u
⎞ ⎟ ⎠,
( p + ) C p ∇ T + Q · ∇ u + · C p ∇ T ⎞ p η q(κ ) ⎜2 p ⎟ ⎟ Zk = ⎜ ⎝ 3 γ ηb q(κ ) ⎠ . pC p λ Qq(κ ) ⎛
Here, p is the pressure, ρ is the mass density, e is the total energy density and T is the gas temperature. Additionally, , Q and denote the shear stress, heat flux, and excess normal stress, respectively. Here, the factor q(κ ) appearing in the dissipative terms is defined by the Rayleigh-Onsager dissipation function [25,31],
2. GH model for investigating rarefied effects on aerodynamic characteristics
149
(1)
where the term C [ f ] represents the collision integral of the interactions among the particles. The Boltzmann-Curtiss equation is irreversible and consequently is expected to describe macroscopic processes progressing irreversibly towards equilibrium. The balance equations for mass, momentum, and energy can be derived by differentiating the statistical formulas of the three quantities with time and then substituting the Boltzmann-Curtiss equation. The balance equations do not have contributions from the Boltzmann collision integral since the three quantities are conserved variables whose molecular expressions are the collisional invariants of Boltzmann collision integral. However, the balance equations contain non-conserved variables, such as the shear stress , heat flux Q, and excess normal stress , whose molecular expressions do not yield a collisional invariant. We refer to these values as non-conserved variables or non-conserved moments. To derive the evolution equations for non-conserved variables, Eu applied the modified moment method to obtain the GH theory. The
q(κ ) =
sinhκ
κ
,
1/2 2 (mk B T )1/4 : Q·Q κ= √ + 2γ + . 2pd
2η
ηb
kT
(4)
In this expression, κ plays an important role in connecting the equilibrium and nonequilibrium forms. The detailed derivation of GH from Boltzmann kinetic theory can be found in previous work [24,25]. k B , m and d represent the Boltzmann constant, molecular mass and diameter of the molecule, respectively. C p denotes the heat capacity. γ = (5 − 3γ )/2 (γ is the specific-heat ratio). η, ηb , and k represent the Chapman-Enskog shear viscosity, bulk viscosity and heat conductivity, respectively. For the classic linear uncoupled Newtonian model in the NSF framework, the constitutive relationships are expressed as follows:
0 = −2η[∇ u](2) , 0 = −ηb ∇ · u, Q0 = −k∇ T
(5)
In the framework of NSF, the excess normal stress can be regarded as the difference between the hydrodynamic pressure and thermodynamic pressure. Additionally, the excess normal stress is neglected for non-monatomic gases. For solving the highly nonlinear equations of the GH model, the method of treating the solid boundary conditions for the nonlinear constitutive equations
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Fig. 1. Comparison of the GH method and DSMC method pressure coefficients for NACA0018 Case: Ma = 2.0 and Kn = 0.02 (left), Ma = 2.0 and Kn = 0.2 (right).
is an important issue. A mixed DG method [32] and Langmuir slip boundary conditions [33,34] have been proposed to develop a highly efficient framework for solving the GH model. 3. Validation and verification Simulations of gas flows that cover continuum and rarefied gas regions are challenging problems. It is generally accepted that the classic CFD methods based on the NSF equations become invalid for rarefied flows [35]. The DSMC method models fluid flows using simulation molecules that represent a large number of real molecules, and the particle collision integral is not solved directly. For this purpose, the DSMC method is usually used to treat nonequilibrium effects in rarefied gases. The fundamental assumption of the DSMC method is that the molecular movement and collision phases can be decoupled over time periods that are shorter than the mean collision time. Therefore, the calculations are very expensive for continuum flows. The DSMC method especially suffers from statistical noise for small Knudsen number flows. Consequently, the DSMC method is not easily applied to continuum states. The GH model presented here claims to calculate both the continuum and rarefied states. The GH has been proven to tend to the NSF theory framework in the limits of low thermodynamic forces at continuum states. Because there is not much experimental data on the rarefied problem, this work uses the DSMC method and/or experimental data to validate the GH model for rarefied states. These validation cases include classic NACA0018 and NACA0012 airfoils. 3.1. Validation on NACA0018 airfoil In the present study, the chord length is regarded as the characteristic length to define the Re and Kn. The first simulations were computed for the NACA0018 airfoil. The flow conditions are Ma = 2.0 and Kn = 0.02, 0.2, with 0◦ for angle of attack (AOA). Fig. 1 gives a comparison between the pressure coefficients, c p , of the GH model solution and DSMC solution. Because of the existing shockwave, a striking pressure peak can be found at the leading edge of the airfoil. Therefore, a drastic decrease was observed in Fig. 1. Additionally, the GH model results agree well with the DSMC solutions. 3.2. Validation and verification on NACA0012 airfoil The second set of simulations were computed for the NACA0012 airfoil. Four different grid numbers (grids in the wall-normal direction and grids in the airfoil-conforming direction) were generated to check the grid independency and were compared with the selected references, as summarized in Table 1. The numerical process
Table 1 Grid independence of Case 1: Ma = 2.0 and Kn = 0.026. Grid
20 × 20
40 × 40
80 × 80
100 × 100
Cd
0.4101
0.3951
0.3901
0.3902
Table 2 Free stream conditions for Cases 1-2. Case
Ma
Kn
α
1 2
2.0 2.0
0.026 0.26
0 20
is applied to all the grids with the same computational and convergent conditions. Next, the drag coefficient of the airfoil is obtained from the solutions. These results are shown in Table 1 to study the influence of the grid densities on the drag coefficient. The solutions predict that the drag coefficient tends to be stable when the number of grids is greater than 80 × 80. Thus, we use a set of triangle meshes with 100 × 100 in the following study. The free stream conditions of the first two cases for air gas flows around the NACA0012 airfoil are listed in Table 2 and are considered for validations. Fig. 2 presents the dimensionless velocity contours of the GH model, experimental and DSMC results. Due to many applications, the supersonic results of Allegre et al. [36] were used for numerous verification studies. All the results demonstrate a shock with a large thickness. Both the GH and DSMC models capture a contour with a value of 0.9 above the trailing edge as observed in the experiment. This feature is not obtained by using the NSF framework in the previous study [18]. Unfortunately, there are still regions of difference in the details. For instance, the experimental velocity varies from 0.75 to 0.85 in a small region, whereas the velocities of the DSMC and GH methods vary from approximately 0.75 to 0.85 in a large region. This difference possibly results from the tunnel wall in the experiment that has a diameter on the order of 10 cm. This diameter is not large enough to neglect the tunnel wall effect. Here, we should present a discussion of the uncertainty analysis of the measurements. Unfortunately, the test conditions are not given in detail [36]. Considering that the study of the gas flow around a cylinder has little to do with the present study of airfoil, we have not given a validation case of cylindrical flow in this study. For this validation and comparison with experimental results, we refer the reader to the Figure 5 of the literature [27] on the cylindrical drag coefficients as a function of the Knudsen number (Ma = 2.0, 0.005 ≤ Kn ≤ 1). Fig. 3 presents the distributions of the surface pressure coefficients given by the NSF framework and the GH, DSMC-IP, and DSMC models for Cases 1 and 2. In Case 1, the pressure coeffi-
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Fig. 2. Comparison of GH model, experiment and DSMC model velocity fields for Case 1.
Fig. 3. Comparison of the GH model, DSMC model and slip NSF framework surface pressure distributions for Case 1 (left) and Case 2 (right).
151
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Fig. 4. Solutions of normalized constitutive equations in one-dimensional solver (u x , 0, 0, T x ) with nonlinear viscosity and the ill-posed constitutive for Maxwellian molecules. The horizontal and vertical axes represent the strain (force) term N S and the normal stress , respectively. The gas is expanding in the range of N S < 0, whereas the gas is compressed in the range of N S > 0.
cient sharply decreases at the leading edge and smoothly decreases at the trailing edge. At the leading edge, the results of the NSF framework agree well with those of the GH and DSMC-IP models. However, at the trailing edge, the pressure coefficient is underpredicted by the NSF framework compared with those predicted by the GH and DSMC models. The GH and DSMC-IP solutions agree well compared with those of the NSF framework, except for a slight overprediction in the trailing edge. In Case 2, a high Kn state with high angle of attack is considered. The gas flows are compressed on the lower surface and expand significantly on the upper surface. These effects result in a difference in the pressure distributions. Further, on the lower surface, the pressure decreases more smoothly than that of the upper surface. The GH solution agrees very well with the DSMC solution; however, the NSF framework underpredicted the pressure along the surfaces both on the lower and upper surfaces. 3.3. Theoretical analysis of GH model For the validation case, the GH and DSMC models can obtain the flow and aerodynamic characteristics of an airfoil considering a rarefied flow. Unfortunately, the NSF solutions are noticeably different than the experimental data. The NSF framework also underpredicts the pressure distribution of the solid wall in Case 1 and overpredicts that in Case 2. We solve the same conservation equations in the frameworks of GH and NSF, and the only difference lies in the constitutive relationships of the viscous stress and thermal conduction, as shown in Equations (3) and (4). In general, the constitutive equations (Equations (3) and (4)) consist of unsteady, kinematic and dissipation terms. The only term in Equation (3) directly related the Boltzmann collision is the dissipation term. This dissipation is caused by the irreversible processes of the system, and the energy dissipates through molecular collisions at the molecular level. Therefore, it is important to treat this term carefully. If we assume that the relaxation time of the non-conserved variables is very short compared with that of the conserved variables, the unsteady term vanishes. To show the features of the GH model in the simplest manner, it is assumed that the shear stress evolution is not coupled to the evolution of and Q and only consider the one-dimensional solver (u x , 0, 0, T x ) and (0, v x , 0, 0). The constitutive relationships were plotted in a onedimensional solver (u x , 0, 0, T x ) and (0, v x , 0, 0). Cases of compression, expansion and shear flows are given in Figs. 4 and 5. As mentioned previously, the nonequilibrium state depends on Ma and Kn, which is why Cases 1 and 2 show different constitutive char-
Fig. 5. Solutions of normalized constitutive equations in one-dimensional solver (0, v x , 0, 0) with nonlinear viscosity and the constitutive equation with linear viscosity for velocity shear flow. The horizontal and vertical axes represent the strain ˆ y y , respectively. (force) term xy N S and the shear and normal stresses xy ,
Table 3 Free stream conditions for Cases 3-6. Case
Mach number
Kn number
T∞, K
Tw, K
3 4 5 6
0.8 0.8 2.0 2.0
0.02 0.2 0.02 0.2
290 290 290 290
290 290 522 522
acteristics for the compression and expansion states (see Fig. 4). For Case 2, with supersonic flow and high Kn, the viscous stress of the GH is lower than that of the NSF framework in the compression region, which results in weak compression on the leading edge and low pressure along the airfoil solid wall. However, Case 1 exhibits different trends in the NSF and GH solutions, and the viscous stress obtained by the GH model is higher than that of the NSF framework owing to the near equilibrium state generated by the subsonic flow and low Kn conditions. In the region of expansion, the NSF framework overpredicts the normal stress compared with that predicted by the GH model. At the trailing edge of the airfoil, the gas flow cannot expand efficiently in the NSF solutions. 4. Results and discussion 4.1. Rarefied effect on pressure distributions For the extensive investigation of the rarefied effect on the airfoil, we select two supersonic and two subsonic cases at Kn = 0.02 and 0.2. In the subsonic cases, the wall temperature is set to 290.0 K, and for the supersonic cases, the wall temperature is equal to the stagnation temperature (see Table 3). Figs. 6 through 9 compare the pressure distributions at Kn = 0.02 and 0.2 with the Mach numbers Ma = 0.8 and Ma = 2.0. Both cases demonstrate a sharp decrease in the pressure close to the leading edge, as expected. However, for the case of Ma = 0.8 and Kn = 0.2, the pressure increases at the trailing edge. This result is not observed in the other cases, where expansion is expected with a corresponding decrease in pressure. Further, the unique characteristic of recompression at the trailing edge in a high Kn state is not found in the NSF solutions. This result can be explained by the constitutive relationships derived using the GH and NSF solutions. In the near equilibrium state, the NSF solution overpredicts the normal stress in the compression region, which causes a large recompression in the overpredicted state. This recompression phenomenon is also found with the DSMC-IP [18] method, but an explanation could not be provided. This result indicates that conventional airfoils do
H. Xiao, D.H. Wang / Aerospace Science and Technology 92 (2019) 148–155
153
Fig. 6. Surface pressure distributions for Ma = 0.8 and Kn = 0.02. Fig. 9. Surface pressure distributions for Ma = 2.0 and Kn = 0.2.
Fig. 7. Surface pressure distributions for Ma = 0.8 and Kn = 0.2.
Fig. 10. Lift and drag at Ma = 2.0 and
α = 20.
Fig. 11. Lift-drag ratio at Ma = 2.0 and
α = 20.
Fig. 8. Surface pressure distributions for Ma = 2.0 and Kn = 0.02.
not provide optimum aerodynamic characteristics in rarefied gas flows. 4.2. Rarefied gas flows around NACA0012 at the angle of attack The aerodynamic performances of airfoils are usually measured by the lift and drag coefficients C L and C d , respectively. The total normal and axial forces per unit span (ds) are obtained by integrating from the leading edge (LE) to the trailing edge (TE) [37]. Fig. 10 shows the lift and drag coefficients of the flows over the microscale NACA0012 airfoil at different Kn values with the same angle of attack, i.e., α = 20. Both the lift and drag coefficients C L and C d increase when Kn increases. Additionally, as compared with
C d , the lift coefficient increases only slightly when Kn varies. The drag increases more rapidly than the lift as the Kn continues to increase. Therefore, the lift-drag ratio decreases with a value less than 1 in high Kn states (see Fig. 11). The NSF framework overpredicts the drag and underpredicts the lift compared with those predicted by the GH model. These observations can be explained by the constitutive relationships. In supersonic gas flows with nonzero angles of attack, the compression and expansion flows dominate the generation of lift and drag. Therefore, the normal stress is dominant in the viscous stress, especially in a high Kn state. As shown in Fig. 4, during compression
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Fig. 12. C L and C L /C d at Ma = 0.8 and Kn = 0.5.
Fig. 13. C L and C L /C d at Ma = 2.0 and Kn = 0.5.
and expansion, the normal viscous stress of the NSF framework is higher than that of the GH model. This effect causes difficulties for gas compression at the bottom surface and expansion at the top surface. Further, the NSF framework underpredicts the lift of the airfoil. Moreover, in the NSF framework, the difference in the normal stress during compression and expansion is larger than that noted in the GH model. This result affects the drag prediction and is the reason for the overprediction by the NSF framework.
in lower drag. In contrast, the subsonic cases considered exhibit different trends in the GH and NSF solutions because of the near equilibrium state caused by the subsonic conditions. The viscous stress of the GH model is higher than that of the NSF framework in the region of compression, which results in strong compression at the leading edge, resulting in a large drag. Therefore, the rarefied effect on the aerodynamic characteristics of an airfoil leads to different trends in far-from-equilibrium states.
4.3. Rarefied effect on aerodynamic characteristics of airfoils
5. Conclusions
To understand the rarefied effect on the aerodynamic characteristics of an airfoil, cases involving supersonic Ma = 0.8 and subsonic conditions Ma = 2.0 at high Kn (Kn = 0.5) were extensively analyzed. Figs. 12 and 13 present the lift and drag coefficients and the lift-drag ratio in terms of the angle of attack. An increase in the angle of attack led to an increase in the lift-drag ratio. However, different trends were noted for subsonic and supersonic conditions. The NSF framework overpredicted the lift-drag ratio in subsonic conditions and underpredicted this ratio in supersonic conditions. The plots of C L and C D also demonstrate the different trends of the NSF and GH solutions. As mentioned, the nonequilibrium state depends on Kn and Ma, which is why the subsonic and supersonic cases considered present different characteristics in the constitutive relationships of the compression state (see Fig. 4). For supersonic cases, the viscous stress of the GH model is lower than that of the NSF framework in the region of compression, which causes weak compression at the leading edge, ultimately resulting
The Boltzmann equation is the governing equation for the gas flow in the full range of Ma and Kn states. The conservation laws of mass, momentum and energy can be derived from the Boltzmann equation. For the closure of the non-conserved variables such as viscous stress and heat flux in the conservation laws, the linear constitutive relationships of the Newtonian laws and Fourier laws are used in the NSF framework. This setup makes the use of the NSF framework difficult in the numerical study of rarefied gas flow. For the closure of the non-conserved variables, the GH model employs the new constitutive equations, which are derived from the Boltzmann equation, in which the entropy generation is considered. This model has been validated in a previous study considering subsonic and hypersonic rarefied gas flows. In this study, a numerical investigation of rarefied effects on the aerodynamic characteristics of an airfoil is presented. In the validations, all the characteristics corresponding to the rarefied effect on an airfoil that were observed using the DSMC method in the
H. Xiao, D.H. Wang / Aerospace Science and Technology 92 (2019) 148–155
previous study were also observed in this study using GH. A critical finding was that the constitutive equations of the GH model provided a clear explanation of these unique features. Detailed analyses of rarefied gas flows around an airfoil were performed. The recompression region was found at the trailing edge in a subsonic and high Kn state. Furthermore, with increase in the Kn, the drag increased rapidly compared to the lift, which resulted in a sharp decrease in the lift-drag ratio. Moreover, the NSF framework overpredicted the lift-drag ratio for the subsonic cases and underpredicted this ratio for supersonic cases in the rarefied gas flows with different angles of attack considered. Detailed explanations were provided against the limitations of the NSF framework in the numerical study of rarefied flows including consideration of the near equilibrium state by the GH model. Declaration of Competing Interest The authors declare no conflict of interest. Acknowledgements The corresponding author expresses appreciation to Prof. R.S. Myong in Gyeongsang National University of South Korea and Prof. Ehsan Roohi in Ferdowsi University of Mashhad of Iran for their help on the NCCR and the DSMC method. References [1] A.K. Chinnappan, G. Malaikannan, R. Kumar, Insights into flow and heat transfer aspects of hypersonic rarefied flow over a blunt body with aerospike using direct simulation Monte-Carlo approach, Aerosp. Sci. Technol. 66 (2017) 119–128. [2] S. Mungiguerra, G. Zuppardi, R. Savino, Rarefied aerodynamics of a deployable re-entry capsule, Aerosp. Sci. Technol. 69 (2017) 395–403. [3] F. Ladeinde, X. Cai, R. Agarwal, A methodology for hybrid simulation of rarefield and continuum flow regimes, Aerosp. Sci. Technol. 75 (2018) 115–127. [4] K.H. Lee, Numerical simulation on thermal and mass diffusion of MMH–NTO bipropellant thruster plume flow using global kinetic reaction model, Aerosp. Sci. Technol. (2019), https://doi.org/10.1016/j.ast.2018.11.056, in press. [5] K.H. Lee, Satellite design verification study based on thruster plume flow impingement effects using parallel DSMC method, Comput. Fluids 173 (2018) 88–92. [6] C. Pekardan, A. Alexeenko, Rarefaction effects for transonic airfoil flows at low Reynolds numbers, AIAA J. (2017) 765–779. [7] N.T.P. Le, H. Xiao, R. Myong, A triangular discontinuous Galerkin method for non-Newtonian implicit constitutive models of rarefied and microscale gases, J. Comput. Phys. 273 (2014) 160–184. [8] N.T.P. Le, A. Shoja-Sani, E. Roohi, Rarefied gas flow simulations of NACA 0012 airfoil and sharp 25–55-deg biconic subject to high order nonequilibrium boundary conditions in CFD, Aerosp. Sci. Technol. 41 (2015) 274–288. [9] C.-C. Lin, Development of a Microfabricated Turbine-Driven Air Bearing Rig, Ph.D. thesis, Massachusetts Institute of Technology, 1999. [10] X. Fang, T. Zhai, U.K. Gautam, L. Li, L. Wu, Y. Bando, D. Golberg, ZnS nanostructures: from synthesis to applications, Prog. Mater. Sci. 56 (2) (2011) 175–287. [11] E.C. Stearns, Thermal Analysis of a Monopropellant Micropropulsion System for a CubeSat, Master’s thesis, Department of Mechanical Engineering, California Polytechnic State University, 2013. [12] D. Jansen, Passive Flow Separation Control on an Airfoil-Flap Model, Ph.D. thesis, Department of Mechanical Engineering, Delft University of Technology, 2012.
155
[13] S. Eidelman, W. Grossmann, I. Lottati, Review of propulsion applications and numerical simulations of the pulsed detonation engine concept, J. Propuls. Power 7 (6) (1991) 857–865. [14] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation Monte Carlo of Gas Flows, Oxford Engineering Science, 1994. [15] C. Cai, I.D. Boyd, J. Fan, G.V. Candler, Direct simulation methods for low-speed microchannel flows, J. Thermophys. Heat Transf. 14 (3) (2000) 368–378. [16] S. Chen, G.D. Doolen, Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech. 30 (1) (1998) 329–364. [17] T.C. Tai, Direct simulation of low-density flow over airfoils, J. Aircr. 29 (5) (1992) 806–810. [18] J. Fan, I.D. Boyd, C.-P. Cai, K. Hennighausen, G.V. Candler, Computation of rarefied gas flows around a NACA 0012 airfoil, AIAA J. 39 (4) (2001) 618–625. [19] Q. Sun, I.D. Boyd, G.V. Candler, Numerical simulation of gas flow over microscale airfoils, J. Thermophys. Heat Transf. 16 (2) (2002) 171–179. [20] J. McDonald, C. Groth, Numerical modeling of micron-scale flows using the Gaussian moment closure, in: 35th AIAA Fluid Dynamics Conference and Exhibit, 2005, p. 5035. [21] Z. Li, A. Peng, H. Zhang, X. Deng, Numerical study on the gas-kinetic high-order schemes for solving Boltzmann model equation, Sci. China, Phys. Mech. Astron. 54 (9) (2011) 1687. [22] X.-D. Niu, S. Hyodo, K. Suga, H. Yamaguchi, Lattice Boltzmann simulation of gas flow over micro-scale airfoils, Comput. Fluids 38 (9) (2009) 1675–1681. [23] N.G. Hadjiconstantinou, The limits of Navier-Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics, Phys. Fluids 18 (11) (2006) 111301. [24] B.C. Eu, Kinetic Theory and Irreversible Thermodynamics, John Wiley and Sons, 1992. [25] B.C. Eu, Kinetic Theory of Nonequilibrium Ensembles, Irreversible Thermodynamics, and Generalized Hydrodynamics, vol. 1. Nonrelativistic Theories, Springer, 2016. [26] R.S. Myong, Impact of computational physics on multi-scale CFD and related numerical algorithms, Comput. Fluids 45 (1) (2011) 64–69. [27] H. Xiao, K. Tang, A unified framework for modeling continuum and rarefied gas flows, Sci. Rep. 7 (1) (2017) 13108. [28] B.C. Eu, Y.G. Ohr, Generalized hydrodynamics, bulk viscosity, and sound wave absorption and dispersion in dilute rigid molecular gases, Phys. Fluids 13 (3) (2001) 744–753. [29] M. Al-Ghoul, B.C. Eu, Generalized hydrodynamic theory of shock waves: Machnumber dependence of inverse shock width for nitrogen gas, Phys. Rev. Lett. 86 (19) (2001) 4294–4297. [30] M. Al-Ghoul, B.C. Eu, Generalized hydrodynamic theory of shock waves in rigid diatomic gases, Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 64 (4) (2001) 046303. [31] B.W. Strutt, B. Rayleigh, The Theory of Sound, Dover Publications, 1945. [32] F. Bassi, S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations, J. Comput. Phys. 131 (2) (1997) 267–279. [33] D.K. Bhattacharya, B.C. Eu, Nonlinear transport processes and fluid dynamics: effects of thermoviscous coupling and nonlinear transport coefficients on plane Couette flow of Lennard-Jones fluids, Phys. Rev. A 35 (2) (1987) 821–836. [34] B.C. Eu, R.E. Khayat, G.D. Billing, C. Nyeland, Nonlinear transport coefficients and plane Couette flow of a viscous, heat-conducting gas between two plates at different temperatures, Can. J. Phys. 65 (9) (1987) 1090–1103. [35] M. Pfeiffer, Particle-based fluid dynamics: comparison of different BGK models and DSMC for hypersonic flows, Phys. Fluids 30 (10) (2018) 106106. [36] J. Allegre, M. Raffin, L. Gottesdiener, Slip effects on supersonic flowfields around NACA 0012 airfoils, in: International Symposium on Rarefied Gas Dynamics, 15th, Grado, Italy, 1986, pp. 548–557. [37] J.D. Anderson Jr., Fundamentals of Aerodynamics, Tata McGraw-Hill Education, 2010.