Instability Analysis of the Separated Boundary Layer in Shock Tubes

Transcription

1 AIAA SciTech Forum 9 - January 7, Grapevine, Texas 55th AIAA Aerospace Sciences Meeting.54/ Instability Analysis of the Separated Boundary Layer in Shock Tubes Kevin Grogan and Matthias Ihme Department of Mechanical Engineering, Stanford University, Stanford, CA 945 The instabilities manifesting from the shock-separated boundary layer in shock tube reaction-kinetic experiments are examined using linear stability analysis. The shockseparated boundary layer is modeled using a polynomial profile with matching conditions across the shock, and the compressible Rayleigh equation is employed to determine the most hydrodynamically unstable wavelengths and growth rates. An instability time scale is identified to characterize the mixing of the boundary layer fluid with the test gas using the length scales and time scales determined from the stability analysis. From this analysis, the shock-separated boundary layer is found to be unstable for all Mach numbers examined. Furthermore, an argon-diluted mixture is found to yield a more stable boundary layer when compared to a nitrogen-diluted mixture. From this, it is concluded that the dilution of the test gas with argon would yield less mixing of the boundary layer fluid with the core flow and therefore, would reduce the likelihood of a non-ideal ignition in a shock tube reaction-kinetic experiment for the low-temperature regime analyzed. Introduction Shock tubes are a primary tool for chemical kinetic measurements, providing ignition delay times, species time-histories, and elementary reaction rates. They replicate zero-dimensional reactors to high accuracy for test times up to the millisecond-range allowing for idealized modeling of the combustion process at well-defined temperature and pressure conditions. A shock tube consists of a driver and a driven section, and an x-t diagram illustrating the relevant regions is shown in Fig.. A shock tube experiment is initiated by the rupture of a diaphragm separating the driver and driven sections, which are denoted as 4 and, respectively, in Fig.. The incident shock propagates towards the end wall with the shocked gas in region immediately following. The incident shock reflects off the end wall and intersects the shocked gas in region to produce the twice-shocked test gas in region 5; the test gas in region 5 is ideally stagnant due to the no-penetration boundary condition at the wall. A contact surface, separating the driver and driven gas and traveling at the velocity of the fluid, trails the incident shock and interacts with the reflected shock wave. Figure illustrates a condition where the driver and driven gas are the same; however, experiments with a tailored driver gas extend the test time, τ TEST, from that shown by modifying how the contact surface interacts with the reflected shock. The expansion fan initially propagates into region 4 as a procession of Mach waves traveling at the local speed of sound; the expansion fan reflects off the driver section s wall in a complex fashion with interaction between the reflected part of the fan and that still propagating towards the wall. The ideal combustion mode for shock tube experiments is strong ignition, which is characterized by the uniform ignition of the test gas beginning at the end wall. However, inhomogeneous or weak ignition has been found in shock tube systems under certain conditions; 4 8 weak ignition can occur when pockets of reactive gas ignite prior to the bulk gas, and this is primarily due to a confluence of chemical sensitivities and gasdynamic fluctuations. There has been increased interest in recent years for chemical kinetics measurements in shock tubes within the negative temperature coefficient (NTC) region due to advancements in extending shock tube test times., 9 In the NTC regime, the ignition time decreases with temperature, which could lead to the preignition of the cool boundary layer fluid near the shock tube walls; furthermore, the interaction of Graduate Student, AIAA member Assistant Professor, AIAA member of Copyright 7 by the, American Inc. All Institute rights reserved. of Aeronautics and Astronautics

2 7 6 t 5 TEST 4 x l DRIVER l DRIVEN Figure. Diagram illustrating a typical shock tube experiment. Regions are labeled in accordance with standard convention. Shock waves are shown by solid RED lines, the contact surface is depicted by a dashed GREEN line, and the expansion fan is represented by thin BLUE lines Figure. Detailed simulations of SBLI in a shock tube. Snapshots of the temperature field are taken subsequent to shock-reflection from the end wall. The nominal conditions behind the reflected shock are constant while the incident shock Mach number, M s, is varied. A 78.% nitrogen-diluted n-heptane/oxygen mixture is simulated. The specific heat ratio is γ =.. the reflected shock wave with the boundary layer could yield enhanced mixing of the boundary layer fluid with the core gas leading to weak ignition. Shock boundary layer interaction (SBLI) demonstrates a regime-dependence based on the incident shock Mach number and the specific heat ratio; canonical SBLI regimes include incipient separation and shock bifurcation. Previous detailed simulations of SBLI in shock tube experiments are shown in Fig.. As shown in this figure, the same nominal test conditions for a shock tube experiment (i.e., T 5, p 5, and composition) can produce drastically differing flow fields when the incident shock Mach number, M s, is varied. Of interest to this work is the intermediate case (M s =.47) between the quiescent simulation (M s =.5) and the simulation showing bifurcation (M s =.79). This intermediate case shows the separation of the incident boundary layer by the reflected shock wave and the ejection of boundary layer fluid into the bulk gas. Hence, it is proposed that the instabilities resultant from the SBLI can produce of

3 u, + U s x s 5 U s y p, p 5, x 5 Figure. Illustration showing the development of instabilities in a separated shear layer immediately subsequent to the reflected shock. Velocities are given in the frame of the reflected shock. inhomogeneities in temperature that could lead to non-ideal ignition phenomenon for mixtures in the NTC regime. Therefore, it is the objective of this paper to analyze the intermediate SBLI shown in Fig. and to develop a relevant time scale for comparison with shock tube experiments. From this, insight will be gleaned about a source of inhomogeneity that could give rise to non-ideal ignition in shock tubes. This paper proceeds with the mathematical model in Sec. I, where the theory and numerical method are described; subsequently, Sec. II includes the results from the application of the model; finally, a discussion of the conclusions drawn from this work is contained in Sec. III. I. Mathematical Model When the reflected shock wave is sufficiently strong, the flow reverses direction in the laboratory frame, and a separated shear layer forms. An illustration of this process is shown in Fig.. The separated shear layer is proposed to engender instabilities. After a certain time scale related to the growth of the linear instability, instabilities in the shear layer become non-linear, and vortices are formed. Furthermore, the instabilities formed by the SBLI could transition the flow into turbulence. Hence, a characteristic time scale for the ejection of nominal vortices into the test gas, τ INST, is developed using the length scales and time scales of the linear instability as well as physical dimensions of the boundary layer. The stability of the separated shear layer is inferred from the eigenvalues of the compressible Rayleigh equation: ˆp + [ T 5 ũ 5 T 5 (Ũs + ũ 5 ) c ] ˆp α T 5 { T 5 M [(Ũs + ũ 5 ) c] }ˆp =, () where for an arbitrary variable, χ, the tilde, χ, indicates a non-dimensional quantity (excluding α, c, and ω); the hat, ˆχ, is used to denote a quantity for which the normal modes hypothesis has been applied (i.e., χ = ˆχ(ỹ) exp ( iα x iω t ) ); and the prime, χ, indicates a derivative in y. The pressure fluctuation is normalized by the pressure in the free stream: p = p/p 5, ; the coordinates, x, y, are non-dimensionalized by the post-shock boundary layer thickness (e.g., x = x/δ 5 ); the velocities (i.e., Ũs, and ũ 5 ) are normalized by the speed of the reflected shock, U s ; the temperature is normalized with respect to the free stream value, T 5 = T 5 /T 5, ; M = U s /a 5, is the free stream Mach number; and the phase speed is related to the growth rate and the wave number by c = ω/α C. In the following, a temporal stability analysis is undergone using the wave speed, c, as the eigenvalue. From the application of Eq., it is assumed that the flow is parallel and that the instability mechanism for this flow is primarily inviscid. A cubic profile is assumed for the incident momentum and thermal boundary layers: 4 u = T = [ (y/δ ) (y/δ ) ] u,, for y < δ (a) u,, for y δ [ T + (T, T ) (y/δ ) (y/δ ) ], for y < δ (b) T,, for y δ, where the thicknesses of the boundary layers have been presumed to be similar due to the Prantl number of

4 of gases analyzed being close to unity. Additionally, the wall is assumed to be isothermal and at a value equal to the initial gas temperature (i.e., K). Other functions may be used to describe the intercepted boundary layer such as the similarity solution of Mirels; 5 however, a simplistic profile is thought to be sufficient to capture the shape of the separated profile while making the exploration of the parameter space more computationally efficient. A quartic profile is assumed for the separated boundary layer: 4 i= u 5 = a i (y/δ 5 ) i, for y < δ 5 (a), for y δ 5 [ T + (T 5, T ) T 5 = (y/δ 5) (y/δ 5) ], for y < δ 5 (b) T 5,, for y δ, wherein the thermal boundary layer is proposed to adjust with the momentum boundary layer. As before, the wall is assumed to be isothermal. The quartic equation has six parameters that need to be accounted for (i.e., a i for i =,..., 4, and δ 5 ). Hence, the six proposed conditions on Eq. are. No slip condition on wall (u 5 y= = ).. Inflection point on the wall (d u 5 /dy y= = ).. Matching condition in free stream (u 5 y=δ5 = ). 4. Stress free condition in free stream (du 5 /dy y=δ5 = ). 5. Conservation of mass across the reflected shock. 6. Balance of momentum across the reflected shock. Mass conservation and momentum balance across the shock yield δ δ ρ (u + U s )dy + ρ, (u, + U s )(δ 5 δ ) = ρ (u + U s ) dy + ρ, (u, + U s ) (δ 5 δ ) + p δ 5 = δ5 δ5 ρ 5 (u 5 + U s )dy, ρ 5 (u 5 + U s ) dy + p 5 δ 5, where the free-stream component of the mass conservation and momentum balance cancel through the Rankine-Hugoniot relations, and the post-shock boundary layer is assumed to thicken. The density profile is found via the assumed thermal boundary layer and the ideal gas equation of state. Since the model utilizes invicid, parallel flows, the pressure is constant across the shear layer. Also, the shear stress at the wall is neglected over the interaction region since its sign necessarily changes for separated flows, which likely makes the integral value of the shear stress on the wall small. Using Eq., the cubic eigenvalue problem is rewritten in the following form: (4a) (4b) where the operators, L k, are given by c L ˆp + c L ˆp + cl ˆp + L ˆp =, (5) [ L = α M (Ũs + ũ 5 ) α T5 (Ũs + ũ 5 ) (Ũs + ũ 5 ) + T ] 5 ũ T 5 T d 5 5 dỹ + (Ũs + ũ 5 ) d dỹ, L = α M (Ũs + ũ 5 ) + α T5 T 5 d T 5 T 5 dỹ d dỹ, L = α M (Ũs + ũ 5 ) T 5, (6c) L = α M T 5. (6a) (6b) (6d) 4 of

5 Equation 5 is discretized using a second-order, centered scheme on a non-uniform grid (i.e., ˆp p and L k L k, where p C N, L k R N N, and N is the number of grid points). The non-uniform grid points are distributed using an arctangent profile, and points are found to be sufficient for convergence in the eigenvalue of the most unstable mode. Finally, the cubic eigenvalue problem is solved by rewriting Eq. 5 as a generalized eigenvalue problem, Aπ = cbπ, where in block-matrix form, A = L L L I I, B = L I I, π = p cp. (7) c p The identity and zero matrices in Eq. 7 are given by I and, respectively. The solution of the discrete eigenvalue problem yields the eigenvalue, c, and the eigenvector, π. Additionally, a Neumann boundary condition is imposed on the pressure fluctuation in the free stream and at the wall. The effect of the instability of the separated boundary layer is the introduction of inhomogeneity into the test gas. A quantification of the inhomogeneity added to the flow within a given timespan can be made using the length scales and time scales obtained from the solution of Eq.. A metric for the inhomogeneity in the flow is postulated to be V BL /V, where V BL is the volume of the fluid ejected into the test gas from the boundary layer, and V is the total volume of fluid behind the reflected shock at a given time; the total volume is taken to be V = U s τ INST πd /4 where τ INST refers to the characteristic time scale of the instability, and D is the diameter of the shock tube. Thus, the volume of fluid ejected by the boundary layer is given by V BL = N VORT j= V VORT,j, (8) where the volume of the j th nominal ejected vortex is denoted by V VORT,j. The volume of this nominal vortex is assumed to have a toroidal shape: V VORT,j = πd(πl INST,j /4), where l INST,j is the characteristic diameter of the j th nominal ejected vortex. The characteristic length of the instability taken to be twice the wave number at the most unstable mode: l INST,j = πδ 5,j /α max. Hence, the volume of the fluid ejected into the test gas is approximated by V VORT,j = π4 Dδ5,j αmax. (9) Similarly, the frequency of the ejection of nominal vortices into the test gas is assumed to be related to the linear growth rate by a constant: f INST,j = b f ω i,max U s /(πδ 5,j ), where b f is a proportionality constant. Hence, the period of the ejected vortex is given by τ j = f INST,j. Equation 8 can now be rewritten as V BL = N VORT j= τinst V VORT,j f INST,j τ j, V VORT f INST dτ, where the approximation of the sum by an integral should hold for large N VORT (i.e., the regime of interest with this model). The value of δ 5 is known from the separated shear layer model and Mirel s similarity solution, where the boundary layer thickness has the functional form ν w (U s + U s )τ δ = (U s u, ) g(m s, γ, Pr), () () where the Prandtl number is denoted by Pr, and the function g is found from the similarity solution of Mirels. 5 By positing that there exists a critical threshold for the ratio of the injected boundary layer fluid to test gas, (V BL /V ) crit, where the quality of ignition changes (i.e., strong to weak), the time scale of the instability can be obtained from Eq. 8 as ( ) α τ INST = β ( max δ ω i,max δ 5 ) [ ] ( D D δ (τ = D/U s ) U s ), () 5 of

6 .5 γ =. (n-heptane/o/n) γ =.47 (n-heptane/o/ar) M s =..5 γ =. (n-heptane/o/n) γ =.47 (n-heptane/o/ar) y/δ.5 y/δ.5 M s =..5 M s =..5 M s = u 5 /U s T 5 /T 5, Figure 4. Comparison of the modeled separated boundary layer profiles for a nitrogen-diluted and an argondiluted n-heptane/o mixture. Both mixtures are diluted at 78.% and have an equivalence ratio of φ =.5. Normalized velocities are given in the laboratory frame. where the non-dimensional constant is given by β = (V BL /V ) crit /(4π b f ), and both α max and ω i,max are found from the solution of Eq.. It is supposed that β is mostly a fuel-dependent parameter that accounts for the sensitivity of the mixture to fluctuations. II. Results The modeled profiles of the separated shear layers are shown in Fig 4. The boundary layer profiles of two n-heptane mixtures are shown for comparison due to the fuel s prominent NTC characteristics. As shown in the figure, the monatomic diluent, argon, exhibits a much thinner profile relative to the incident boundary layer. This is likely due to monatomic gasses being stiffer than diatomic gasses and therefore, less compliant to pressure gradients. Additionally, the variation of the profiles with respect to Mach number is shown to decrease at higher Mach numbers. The most unstable wavelengths and growth rates are shown in Fig. 5; these are found by applying Eq. to the model boundary layers shown in Fig. 4. For all the incident Mach numbers examined, the boundary layer is found to be unstable. This indicates that there are only regimes in which the separated boundary layer is quasi-stable; that is, the manifestation of the instability is much slower than the chemistry. A jump in the most unstable wavelength, α max, is shown for both mixtures between an incident shock Mach number of.5 to.7. This occurs because the growth rate of the lower wavelength mode becomes negligible at higher Mach numbers. Interestingly, the comparative size of the maximum growth rate, ω i,max, between the two gases switches as the most unstable mode changes. The growth rates of the most unstable mode as a function of the wavelength is shown in Fig. 6. At the lower Mach number, two local maxima are demonstrated, and at the higher Mach number, a single global maximum exists. The two maxima at the lower Mach number correspond to distinct modes, and a single mode is tracked at the higher Mach number until approximately α = 4.. Figure 7 shows the eigenfunctions of the separated boundary layer at the most unstable mode for two different Mach numbers. While the pressure fluctuation, ˆp, is found from the solution of Eq., the normal velocity fluctuation, ˆv, is determined using the y-momentum equation: T 5 ˆp ˆv = () iα[(ũs + ũ 5 ) c]γm As shown in Fig. 7, the eigenfunctions demonstrate the highest variability within the boundary layer (i.e., y/δ 5 < ). The normal velocity fluctuations are shown to decay in the free stream and at the wall for all modes; due to the Neumann boundary condition applied to Eq. 7, the derivative of the pressure fluctuation is zero at the wall and in the free stream. Also, the eigenfunctions at the higher Mach number 6 of

7 .5.5 ω i,max α max M s Figure 5. Comparison of the most unstable wavenumber, α max, and the corresponding growth rate, ω i,max. RED lines denote an argon-diluted n-heptane mixture, and BLUE lines denote a nitrogen-diluted n-heptane mixture. Both mixtures are diluted at 78.% and have an equivalence ratio of φ =.5. ωi,max M s =. M s =.5 M s =.8 M s =. 4 5 α (a) N -Dilution, γ =. ωi,max M s =. M s =.5 M s =.8 M s =. 4 5 α (b) Ar-Dilution, γ =.47 Figure 6. Comparison of the growth rates of the most unstable modes for two selected Mach numbers. Both mixtures are diluted at 78.% and have an equivalence ratio of φ =.5. and, correspondingly, higher wavelength are shown to have increased variation. For both modes and gases, the normal velocity fluctuation demonstrates a peak near the outer edge of the separated shear layer; hence, these mode shapes could give rise to vortex formation at this location and entrainment of boundary layer fluid. A comparison of the instability time scale with the ignition delay of n-heptane is shown in Fig. 8. The parameters in Eq. are set to be b f = and (V BL /V ) CRIT =.. The initial temperature is presumed to be T = K. The ignition delay time is taken from an adiabatic, isobaric (HP) reactor using the time at which the concentration of the OH radical has peaked. The chemistry is modeled with a 94-species reduced n-heptane mechanism derived from the detailed mechanism due to Mehl et al. 6 using the procedure of Stagni et al. 7 It is shown in Fig. 8 that the chemical time scale, τ IGN, is much larger than the instability time scale at low temperatures (i.e, T 5 < K for argon dilution and T 5 < 8 K for nitrogen dilution); this indicates that at low temperatures, inhomogeneities emanating from the separated boundary layer may become prevalent and lead to weak ignition. The dimensional time scale indicates that the argon mixture, while less stable than the nitrogen mixture, 7 of

8 .5 M s =. M s =..5 M s =. M s =. y/δ5.5 y/δ y/δ ˆv (a) N -Dilution, γ =. M s =. M s = ˆp (c) N -Dilution, γ =. y/δ ˆv (b) Ar-Dilution, γ =.47 M s =. M s = ˆp (d) Ar-Dilution, γ =.47 Figure 7. Comparison of the eigenfunctions at the most unstable wavelength for two diluted n-heptane mixtures. Both mixtures are diluted at 78.% and have an equivalence ratio of φ =.5. would likely induce less mixing of the cool boundary layer fluid with the bulk gas. Since the dynamic viscosities of the argon-diluted and nitrogen-diluted mixtures are similar, it is concluded that the discrepancy between the two time scales is due to the differing specific heat ratios of the mixtures. Disparate specific heat ratios can yield different shock velocities, kinematic viscosities, and sound speeds, which as shown by Eq., can cause the observed discrepancy in the instability time scale. The lower wavelength mode is shown to produce a smaller instability timescale at low temperatures, which indicates that more boundary layer fluid may become entrained within the test gas in a shorter period of time. Hence, the instability mode may change for low temperature shock tube experiments in a fashion that has a higher propensity for a non-ideal ignition. Also, preheating the test gas would yield a leftward shift in the instability timescale curves due to a lower requisite T 5 /T temperature ratio and correspondingly, a lower Mach number. Hence, the model implies that preheating the test gas would reduce the entrainment of boundary layer fluid and therefore, would lead to a decreased likelihood of weak ignition. III. Conclusions A low-order model is developed to describe the temperature and velocity profiles of a shock-separated boundary layer, and the hydrodynamic stability of the model boundary layer is analyzed using the compressible Rayleigh equation. The profile is found to be unstable for all incident Mach numbers examined. 8 of

9 τ IGN τ INST, Ar-diluted τ INST, N -diluted τ [s] /T 5 [K ] Figure 8. Comparison of the instability time scale with that of the ignition delay curve of n-heptane using a HP reactor model with a test pressure of p 5 =.7 bar. The mixtures are diluted at 78.% and have an equivalence ratio of φ =.5. Additionally, this boundary layer instability is proposed to cause inhomogeneities in the core gas, which could potentially yield weak ignition. An instability time scale is developed for comparison to the ignition time of a test mixture. It is found that while an argon-diluted mixture is more unstable due to a thinner shear layer profile, a nitrogen-diluted mixture has increased mixing of the boundary layer fluid with the core gas. Hence, our analysis suggests that argon dilution is preferable to combat inhomogeneous ignition in shock tube experiments. Acknowledgments The authors gratefully acknowledge financial support through the Air Force Office of Scientific Research under Award No. FA with Dr. Chiping Li as program manager. References R. K. Hanson and D. F. Davidson. Recent advances in laser absorption and shock tube methods for studies of combustion chemistry. Prog. Energy Combust. Sci., 44: 4, 4. O. Trass and D. Mackay. Contact surface tailoring in a chemical shock tube. AIAA J., (9):6 6, 96. G. Ben-Dor, O. Igra, and T. Elpherin, editors. Handbook of Shock Waves. Elsevier, New York,. 4 V. V. Voevodsky and R. I. Soloukhin. On the mechanism and explosion limits of hydrogen-oxygen chain self-ignition in shock waves. Proc. Combust. Inst., :79 8, J. W. Meyer and A. K. Oppenheim. On the shock-induced ignition of explosive gases. Proc. Combust. Inst., :5 64, D. J. Vermeer, J. W. Meyer, and A. K. Oppenheim. Auto-ignition of hydrocarbons behind reflected shock waves. Combust. Flame, 8:7 6, H. Yamashita, J. Kasahara, Y. Sugiyama, and A. Matsuo. Visualization study of ignition modes behind bifurcatedreflected shock waves. Combust. Flame, 59: ,. 8 K. P. Grogan and M. Ihme. Weak and strong ignition of hydrogen/oxygen mixtures in shock-tube systems. Proc. Combust. Inst., 5():8 89, 5. 9 M. F. Campbell, S. Wang, C.S. Goldenstein, R. M. Spearrin, A. M. Tulgestke, L. T. Zaczek, D. F. Davidson, and R. K. Hanson. Constrained reaction volume shock tube study of n-heptane oxidation: Ignition delay times and time-histories of multiple species and temperature. Proc. Combust. Inst., 5: 9, 5. K. P. Grogan and M. Ihme. Regimes describing shock boundary layer interaction and ignition in shock tubes. Proc. Combust. Inst., 6, 7. In Press. D. R. Chapman, D. M. Kuehn, and H. K. Larson. Investigation of separated flows in supersonic and subsonic streams with emphasis on the effect of transition. NACA-TN-56, Ames Aeronautical Lab, Moffett Field, CA, of

10 H. Mark. The interaction of a reflected shock wave with the boundary layer in a shock tube. J. Aeronaut. Sci., 4:4 6, 957. W. Criminale, T. Jackson, and R. Joslin. Theory and Computation of Hydrodynamic Stability. Cambridge University Press, Cambridge,. 4 W. Kays, M. Crawford, and B. Weigand. Convective Heat and Mass Transfer. McGraw-Hill, 5. 5 H. Mirels. Laminar boundary layer behind shock advancing into stationary fluid. NACA TN 4, Lewis Flight Propulsion Laboratory, Cleveland, OH, M. Mehl, W. J. Pitz, C. K. Westbrook, and H. J. Curran. Kinetic modeling of gasoline surrogate components and mixtures under engine conditions. Proc. Combust. Inst., :9,. 7 A. Stagni, A. Frassoldati, A. Cuoci, T. Faravelli, and E. Ranzi. Skeletal mechanism reduction through species-targeted sensitivity analysis. Combust. Flame, 6:8 9, 6. of