Stability analysis of hypersonic flow over a backstep in chemical equilibrium

Transcription

1 Sonderforschungsbereich/Transregio 4 Summer Program Report Stabilit analsis of hpersonic flow over a backstep in chemical equilibrium B J.M. Perez, J.A. Franco, P. Paredes, S. Hein, V. Hannemann AND V. Theofilis Universidad Politécnica de Madrid, Madrid, Spain The purpose of this work is to shed light on the role of chemical equilibrium in the global instabilities that arise in hpersonic non-parallel flows. As an eample of separated flow, the stabilit of temporall evolving disturbances on a hpersonic flow over a backstep are eamined b solution of the pertinent BiGlobal eigenvalue problem. The flow conditions are unit Re = /m, boundar-laer edge temperature 35 K and freestream Mach=1. The step height is taken to equal to the local displacement thickness. At these conditions the two-dimensional laminar base flow is computed with the full Navier-Stokes DLR-TAU solver. Stabilit results at a given wave number of the perturbations in the homogeneous direction show that the leading linear mode located around the recirculation bubble is unstable to modal perturbations when both perfect and real gas assumption is made. Real gas effects increase the temporal growth rate of the leading mode, which corresponds to a zero-frequenc three-dimensional global linear instabilit, the analog of which in incompressible flow is well understood. The eigenvalue spectrum and the structure of the leading mode for real and perfect gas are compared. 1. Introduction Reentr vehicles are eposed to severe flow conditions as high heat loads. Therefore the require the use of thermal protection. This implies comple thermo-chemical processes that affect the heat flu on the bod surface. Due to the lack of knowledge about the origin of laminar-turbulent transition on parts of the vehicle surface, or the underling instabilities, ver often this protection is oversized, impling an unnecessar ecess weight and consequent performance reduction. Given the high temperatures reached on the bod surface, where the boundar laer develops, eperimental studies are prohibitivel epensive and one resorts to numerical simulations which provide information about the phsical process involved in laminar-turbulent transition. Turbulence leads to larger heat flues than a laminar boundar laer due to the increased convective transport and has a strong influence in the drag, thermal load and stabilit and therefore the accurate prediction of this is a critical part of the aerodnamic design of hpersonic vehicles. As conclusion, the stud of laminar-turbulent transition in supersonic and hpersonic boundar laers is important in the development of net generation of hpersonic vehicles. josemiguel.perez@upm.es DLR Göttingen, German Computational AeroSciences Branch, NASA Langle Research Center, Hampton, VA, USA

2 74 Perez, Franco, Paredes, Hein, V.Hannemann & Theofilis However, hpersonic flows have attracted much less attention in the literature with respect to laminar-turbulent flow transition, partl because of the compleit of the flow fields involved, since at high temperature conditions, chemical reactions need to be taken into account. Specificall the authors are not aware of an global instabilit stud at hpersonic flow conditions. In increasing temperature order, four possible flow regimes eist: (a) At moderatel high temperatures the air is a miture of perfect gases. (b) When the temperature increases the gas is a miture of species at chemical and thermal equilibrium. (c) The net flow regime chemical equilibrium is broken but thermal equilibrium assumption still holds. (c) At high temperatures both thermal and chemical equilibrium are not met. Efforts that have applied classic linear instabilit analsis in order to arrive at predictions on small-amplitude perturbation growth and correlate such results with transition location at hpersonic flight speeds and accounting for real gas effects are limited to one-dimensional, boundar-laer tpe of basic flows [7, 8, 14]. Using direct numerical simulations Mack [5] determined that whenever the relative flow between the mean flow and the disturbance phase velocit is supersonic, a famil of inviscid acoustic disturbances at high frequenc eists. The first of the higher frequenc modes is known in the literature as the second mode or the Mack mode. Later, Mack [6] arrived to the following results for hpersonic flat-plate boundar laer transition using linear stabilit theor: oblique three-dimensional viscous disturbances are dominant at all supersonic Mach numbers below four which is an etension to high speeds of the Tollmien-Schlichting instabilit, while a famil of trapped high frequenc acoustic modes traveling inside the boundar laer b reflections between the wall and the sonic line dominates at high Mach numbers. The acoustic instabilit discovered b Mack arises when the edge velocit is sufficientl fast that disturbances can propagate downstream at a subsonic speed relative to the boundar-laer edge velocit but supersonic relative to the wall. This second instabilit was measured in different eperiments, [], [4] or [13] and becomes more unstable that the first oblique mode for free-stream Mach numbers bigger than four. The stud of these modes was recovered later b Malik [7] and Malik and Anderson [8], who conducted the stabilit analsis of a zero-pressure-gradient flat plate boundar laer at hpersonic regime including real gas at thermal and chemical equilibrium. These authors were the first to calculate the real gas effects on hpersonic boundar laer stabilit. For this end the studied two cases at Mach numbers, 1 and 15, and found that the effect of considered real gas produce large differences in the temperature profile in the boundar laer. The show that chemical equilibrium stabilizes the first oblique mode and destabilizes the second mode. In this case, the size of the region of relative supersonic flow increases in the cooler boundar laer, due to the lower speeds of sound in that region. In addition, the second mode is shifted to lower frequencies. Later Stuckert and Reed [14] studied the effect of chemical non-equilibrium on flow over an aismmetric cone. B contrast, when the basic flow, collectivel called comple here, is such that an assumption of homogeneit in two or three spatial directions ma not be made, classic linear theor cannot be applied. Interesting flow phenomena arising either on account of geometrical compleit, e.g. flow over protrusions, corners, steps or gaps, or because of the interaction of a shock sstem with a laminar boundar laer over a flat plate or a (plane or aismmetric) wedge, all are beond the scope of classic linear stabilit theor. While the above is alread true for supersonic flow, the hpersonic regime further

3 Stabilit of Hpersonic Flow 75 compounds the problem, since as mentioned chemical reactions and their modeling need to be taken into account. To-date, when laminar-turbulent transition of a comple flow field need to be studied, one resorts to state-of-the-art direct numerical simulation [9, 18]. It is worth to mention that [7, 14] studied also the effect of finite-rate chemistr or non-equilibrium chemistr adding some equations for species conservation in the classic local stabilit analsis of flat-plate boundar laers. The present contribution discusses algorithmic developments to include real gas effects in stabilit equations without an restriction on spatial homogeneit. The phrase real gas is used in the sense of aerodnamics where it tpifies the high-temperature effects rather than in the sense of classical phsical chemistr where it has been used for a gas in which inter-molecular forces are important due to high pressures and/or low temperatures. The problem considered is the hpersonic flow over a backstep in chemical equilibrium where the five species model proposed b Park [11], which is valid up to 9 K, is considered in the last case. This model becomes invalid when the ionization of atoms start. These results are compared with the results obtained when perfect gas approimation is considered, where internal energ and enthalp are functions of the temperature onl. Before to this, verification is performed against spatial local stabilit results of Malik and Anderson [8] in a zero-pressure-gradient flat-plate boundar laer at Mach 1. In this wa, the chemical model and inflow profile for the solution of the backstep base flow are validated.. Problem description The governing equations used are the time-dependent three-dimensional Navier- Stokes equations for compressible fluids in dimensional form, ρ t + (ρ u ) =, (.1) [ ] u ρ t + (u )u = p + [λ ( u )] + [µ (( u ) + ( u ) T )], (.) [ ] e ρ t + (u )e + p u = (κ T ) + λ ( u ) + µ [( u ) + ( u ) T ], (.3) where u is the velocit vector, ρ the densit, p the pressure, T the temperature, e = c vt the internal energ, κ the thermal conductivit, µ the first coefficient of viscosit, and λ the second coefficient of viscosit. Note that using the Stoke s law λ = /3µ and the asterisk denotes dimensional quantities. The Cartesian coordinates are denoted b,, and z to represent the streamwise direction and the perpendicular plane to it, the normal and spanwise directions, respectivel. The non-dimensionalization is based on the freestream conditions. Lengths are scaled b a reference length lr defined as the Blasius length at h, velocit b Ur and temperature b Tr are defined from the boundar-laer edge at the step position. The reference pressure is p r, the free-stream sound speed is denoted b s r, and γ = c p,r/c v,r. The equation of state is given b the perfect gas relation p = ρ RT, where R is the gas constant, which depends on the chemical composition of the gas. The resulting dimensionless parameters are the Renolds number, Re = ρ rur lr/µ r,

4 76 Perez, Franco, Paredes, Hein, V.Hannemann & Theofilis FIGURE 1. Flow geometr for the backstep. the Prandtl number to P r = c p,rµ r/κ r, the Mach number M = U r /s r and the Eckert number Ec = (U r ) /(c p,rt r ) = (γ 1)M Perfect gas.1. Compressible fluid properties When a perfect gas is considered, the non-dimensionalized equation of state is p = 1 ρt (.4) γm where the constant of perfect gases is fied to γ = 1.4 and the Prandtl number P r =.7. The Sutherland s law is used for the viscosit coefficient µ = (T ) 3/ 1 + S T + S (.5) with µ s = N s/m and S = 11.4 K/T r for air in standard conditions. The dimensionless thermal conductivit of the medium is taken equal to the viscosit coefficient κ = µ..1.. Real gas At certain flight conditions, mostl as a cause of high temperature, some molecular species begin to dissociate due to aerodnamic heating and the perfect gas assumption is no longer applicable and real gas effects need to be considered. Air cannot be modeled as a perfect gas when vibrational ecitation and chemical dissociation become important. At pressure of 1 atmosphere, ogen starts dissociating at K, Nitrogen starts to dissociate at 4 K and both atomic components start ionizing at about 9 K. The equation of state for a real gas depend on its chemical composition. In addition, the internal energ, viscosit laws and thermal conductivit coefficients are not solel dependent on the temperature, but on the two thermodnamic variables considered. Here, the gas properties, p, e, κ and µ as function of ρ and T are calculated using the TAU software. Air of 5 species, namel N, O, NO, N and O, is considered net. Comparison with perfect gas is shown in Figure. The divergence of the gas properties for perfect and real gas is evident as the temperature increases. p and µ for perfect gas start to deviate from the real gas solution at about T K, while differences for e and κ are visible above T 1 K.

5 77 Stabilit of Hpersonic Flow 1 1 log(p) T log(ρ) - e (1-6) T (a) 8 T log(ρ) (c) κ T µ (1-5) -8 - (b) log(ρ) log(ρ) - (d) F IGURE. (a) log(p ) (log(kg m/s )), (b) e (J/kg), (c) µ (kg/(m s)) and (d) κ (W/(m K)). Note that (a) and (c) are plotted using a constant increment between iso-lines, while (b) and (d) use an eponential increase of intervals between iso-lines. F IGURE 3. Procedure followed in the stabilit analsis in the case of real gas calculation. As it can be seen TAU provides both the base flow and the database used in the calculation of the unknowns thermodnamic variables and coefficients.

6 78 Perez, Franco, Paredes, Hein, V.Hannemann & Theofilis In this work, gas properties provided b TAU are tabulated, and Lagrange interpolation is used to obtain the desired values for the stabilit analsis. Figure 3 shows the process followed in the stabilit analsis when real gas is considered. Basicall TAU provides the base flow and some tables that are used b the stabilit code in order to calculate the value of the thermodnamic variables and parameters for a real gas in thermal and chemical equilibrium. Once this information is available the net step requires the solution of the eigenvalue problem associated with the modal stabilit analsis. To do this, a zoom near the step is considered and the domain is splited into three domains. A C 1 condition is imposed at the connection between the different domains. 3. Hdrodnamic stabilit theor 3.1. Linearized Navier-Stokes equations The vector of fluid variables q = (ρ, u, v, w, T ) T is decomposed into a stead mean flow q and an unstead small disturbance or perturbation q: q(, t) = q() + ɛ q(, t), ɛ 1. (3.1) The amplitude functions of thermal conductivit and viscosit coefficients, internal energ and pressure are functions of temperature and densit, and thus are written as µ = µ T T + µr ρ, (3.) κ = κ T T + κr ρ, (3.3) ẽ = ē T T + ēr ρ, (3.4) p = p T T + pr ρ, (3.5) where ( ) T = ( ) T, ( ) R = ( ) ρ. Also, second order derivatives with respect to temperature and densit arise from the spatial derivatives of previous thermodnamic variables. For eample the -derivative of viscosit coefficient is written as ( ) µ = µ T T T + µ RT ρ + µ T T, (3.6) ( ) + µ RT T + µ RR ρ + µ R ρ, (3.7) where ( ) T T = ( ) T, ( ) RT = ( ) ρ T and ( ) RR = ( ) ρ. B introducing the previous decomposition of variables (3.1) into the governing equations (.1-.3) and neglecting the non-linear terms of O(ɛ ) and O(ɛ 3 ), the linearized Navier-Stokes equations (LNSE) are recovered. The above generic discussion is applicable to an linearization of the governing equations following the decomposition of flow quantities (3.1). Furthermore, in modal linear stabilit theor, the perturbation term is usuall written as the product of an amplitude function and a phase function, q = ˆqΘ. Table 1 summarizes the different instabilit approaches arranged b increasing constraints to the basic flow. However, depending on the dimensionalit of the base flow analzed, the resulting theoretical frameworks involve numerical solutions that require orders-of-magnitude different levels of computational work for their solution. The best known contet of classic or local linear stabilit theor [6] assumes a single

7 Stabilit of Hpersonic Flow 79 Assumptions Base Amplitude Ep(iΘ), Phase Function Θ TriGlobal - q(,, z) ˆq(,, z) ωt PSE-3D zq q, q q(,, z ) ˆq(,, z ) β(z )dz ωt BiGlobal zq = q(, ) ˆq(, ) βz ωt PSE q q q(, ) ˆq(, ) zq = α( )d + βz ωt Local q = zq = q() ˆq() α + βz ωt TABLE 1. Modal linear instabilit approaches arranged b increasing constraints to the basic flow in-homogeneous spatial direction in both the basic flow and the amplitude functions, indicated b in the last line of Table 1. On the other hand, the linear (and non-linear) stabilit of boundar-laer flows, in which a small but non-zero wall-normal velocit component eists in the base flow and the dependence of the latter on the streamwise coordinate,, is much weaker than that along the wall-normal,, can be studied b the Parabolized Stabilit Equations (PSE). Unlike an EVP-based solution, PSE solve a marching integration of the LNSE along the streamwise spatial direction, and is known as a non-local instabilit analsis; [3] provides an introduction to the PSE. The remaining three entries in Table 1 are collectivel known as global linear theories [16]. The ma consider base flows which are inhomogeneous in two or three (rather than one) spatial directions. The global stabilit theories are classified in BiGlobal, PSE-3D and TriGlobal analsis. The denote analsis of base flows developing in two (BiGlobal) or three (TriGlobal) inhomogeneous spatial directions. In-between, the PSE-3D concept etends the classic PSE to flows depending strongl on two and weakl on the third spatial direction. 3.. Temporal and spatial stabilit analsis The aforementioned LNSE can be written as initial-value-problem (IVP) in the form B T q t = A T q, (3.8) and solutions to this sstem of PDEs are considered. The operators A and B are associated with the spatial discretization of the LNSE and comprise the basic state, q() and its spatial derivatives and the thermodnamic properties of the gas. In case of stead basic flows, the separabilit between time and space coordinates in (3.8) permits introducing a Fourier decomposition in time through the epression q = ˆq ep( iωt), where ˆq = (ˆρ, û, ˆv, ŵ, ˆT ) T is the vector of amplitude functions, leading to the generalized matri eigenvalue problem (EVP) A T ˆq = ωb T ˆq, (3.9) in which matrices A T and B T discretize the operators A T and B T, respectivel, and incorporate the boundar conditions. This EVP represents a modal temporal stabilit

8 8 Perez, Franco, Paredes, Hein, V.Hannemann & Theofilis problem in which the sought comple eigenvalue is ω = ω r + iω i, the real part being a circular frequenc, while the imaginar part is the temporal amplification/damping rate. In certain problems, the stud of the spatial development of perturbations is considered instead. In such cases, at least one homogeneous direction is assumed and the disturbances obe the wave-like ansatz ep[i(βz ωt)]. Here, the IVP (3.8) is rewritten as B S q = A S q, (3.1) and the discretized version becomes a quadratic generalized matri EVP A S ˆq = β k B S,kˆq. (3.11) k=1 Here ω R is a real frequenc parameter, while β C is the sought eigenvalue, the real part of which is related with the periodicit length along the homogeneous spatial direction,, through β r = π/l z and the imaginar part, β i is the spatial amplification/damping rate. In order to solve numericall the quadratic EVP, it must be previousl reduced to a linear EVP. The latter problem can be converted into a (larger b a factor equal to the degree of non-linearit) linear EVP, as shown b [15], using the companion matri method [1], in which an auiliar vector ˆq = [ˆρ, û, ˆv, ŵ, ˆT, βû, βˆv, βŵ, β ˆT ] T is defined, resulting in A S ˆq = βb S ˆq. (3.1) Matrices A S and B S defined in (3.1) and (3.11) are different. 4. Spatial analsis of flat-plate boundar-laer The spatial stabilit analsis of a hpersonic flat-plate boundar-laer at Mach 1 is studied for verification purposes. This problem was selected b [8] to show the effects of considering real gas effects, assuming chemical equilibrium, in stabilit analsis. The objective is to validate the real gas calculation and the self-similarit solution that are used in the following section. The Mach 1 boundar-laer calculation is performed using either a self-similar boundar-laer code or the TAU software for the perfect gas and real gas cases, respectivel. In both cases adiabatic wall condition is set. The boundar-laer edge temperature is Tr = 35 K and the unit Renolds number is Re = /m. The Re affects the real gas properties through the reference pressure, which is implicitl calculated knowing Tr, M and Re. Boundar-laers profiles as function of the wall normal direction for perfect and real gas used for stabilit analsis are compared with those used b [8] in figure 4 at.66 (or R = ). Figure 4 shows the streamwise velocit and temperature for both perfect and real flow approaches where the lengths are dimensionlalized b using l r defined previousl. δ (boundar-laer thickness) is about 4 times l r. In both cases there is a good agreement with Malik s results [8]. As it can be seen, there is a decrease in temperature of about 45% under the consideration of chemical reactions. Related with this, for boundar-laer flow over a flat plate wall temperature increases when the freestream Mach number increases too. The following approimate equation relates these two parameters, Tad/T r = 1 + γ 1 P rm r.

9 1.8 Stabilit of Hpersonic Flow U.4 T 8 6. Perfect gas Real gas Perfect gas, MA91 Real gas, MA FIGURE 4. (a) Streamwise velocit, (b) Temperature c ph =ω/α r Perfect gas Real gas.935 Perfect gas, NOLOT Perfect gas, MA91.93 Real gas, MA (ω/r)1 4 -α i (ω/r)1 4 FIGURE 5. Spatial local stabilit results of two-dimensional waves (β = ). (a) Phase speed and (b) growth rate. This relation assumes that the temperature profile across the boundar laer is onl a function of the streamwise velocit. This is valid when the Prandtl number is one but it is a good approimation in other cases (see [1]). Stabilit results obtained b using the base flows shown in Figure 4 are compared in Figure 5 for two-dimensional waves, i.e. second and third Mack modes [6]. As can be seen the growth rate and phase velocit of the second and third modes are recovered in reasonable agreement with results of Malik & Anderson [8]. The small differences found in the real case are thought to be due to the base flow used here, which has been computed using the full Navier-Stokes solution of a complete flat plate including the leading edge and shock wave, instead of the self-similar solution used in Ref. [8]. In this case is the homogeneous direction and then α is the wavenumber considered in the ansatz. For the boundar laer on a flat plate, ω can be epressed as F R where R = U r /ν r is the Renolds number, F = ων r /(U r ) is the frequenc parameter and ω is the frequenc of the spatial perturbation. In Figure 5 the first of the higher modes, which was labeled as the second mode b Mack, has been found to be the dominant unstable mode for a zero-pressure-gradient boundar laer over a flat plate at high Mach numbers. This stud also shows that the third mode, which did not appear in the perfect gas analsis, is marginall unstable too.

10 8 Perez, Franco, Paredes, Hein, V.Hannemann & Theofilis FIGURE 6. Base flow at M = 1, Re =, T r = 35 K, L r = R/Re = m and h = δ obtained b using the approimation of perfect gas. (left) Streamwise velocit (center) wall-normal velocit (right) and densit. FIGURE 7. Base flow at M = 1, R =, T r = 35 K, L r = R/Re = m and h = δ obtained b using the approimation of real gas. (left) Streamwise velocit (center) wall-normal velocit (right) and densit. 5. Temporal analsis of backstep boundar-laer Temporal linear instabilit over the backstep boundar laer flow is considered in this section. In these simulations the self-similar boundar-laer solution obtained in the previous section (see Figure 4) is prescribed at the inflow. Boundar conditions at the wall are no-slip, non-cataltic, and adiabatic, while non-reflecting boundar conditions are used at the far-field and outflow boundaries. Figures 6 and 7 show the streamwise velocit, wall normal velocit and densit of the base flow for perfect and real gas approaches, respectivel. Figure 8 shows the recirculation bubble in both configurations. In the latter case the recirculation level is lower than in the former. This aspect is not crucial when analzing convective D and 3D modal instabilities, which ma be present even at zero-recirculation, but it should be taken into consideration when analzing global 3D instabilities of the entire recirculation zone, for which a recirculation level of 1% is known to be necessar in incompressible flow [1, 17]; analogous studies in hpersonic separated flow have to-date not been performed. A convergence stud of the base flow in the recirculation region is shown in Figure 9. The first figure shows the streamwise velocit as function of the wall normal direction obtained at different resolutions in the recirculation bubble. The other figure shows the temperature as function of the wall normal direction. In both cases there is a good agreement between the data obtained at different resolutions. Finall as can be seen in Figure 1 chemical equilibrium destabilizes the dominant leading mode which is a global mode defined in the recirculation bubble.

11 Stabilit of Hpersonic Flow FIGURE 8. Recirculation bubbles at M = 1, R =, T r = 35 K, L r = R/Re = m and h = δ. (left) Perfect gas and (right) real gas. In both cases the recirculation level, u ma / u min 1, is about 7%..8 Coarse mesh Fine mesh.8 [m]. [m]. Coarse mesh Fine mesh U [m/s] T [K] FIGURE 9. Convergence of the base flow inside the recirculation bubble (h = 1/δ ) and dimensional position of.3. Streamwise velocit and temperature as a function of the wall-normal distance. A convergence stud of the leading eigenmodes was also performed in order to ensure the good resolution of this mode. The module of the amplitude functions for the velocit and temperature are shown in Figure 11 and 1 for perfect and real gas, respectivel. The modulus of the streamwise and wall-normal velocit components and temperature of the leading mode using both approaches, i.e., perfect and real gas, are qualitativel similar with a small deformation due to the base flow modification. This identification allows to identif both modes as the same modal instabilit. References [1] T.J. Bridges and P.J. Morris. Differential eigenvalue problems in which the parameter appears nonlinearl. J. Comp. Phs., 55:437 46, [] A. Demetriades. Hpersonic viscous flow over a slender cone, part iii: laminar instabilit and transition. AIAA Astronaut. Fluid Plasma Dn. Conf., 7th, Palo Alto,

12 84 Perez, Franco, Paredes, Hein, V.Hannemann & Theofilis -.3 ω i Perfect Gas Real Gas ω r FIGURE 1. Comparison of the spectra obtained with real and perfect gas approaches at M = 1, Re =, T r = 35 K, L r = R/Re 1 = m, h = δ and dimensional position of.66. CA, AIAA Pap., page 535, [3] T. Herbert. Parabolized stabilit equations. Ann. Rev. Fluid Mech., 9:45 83, [4] M. R. Khorrami. Wind tunnel eperiments relating to supersonic and hpersonic boundar laer transition. AIAA Journal, 13(3):9 99, [5] L. Mack. Boundar laer stabilit theor. Technical report, JPL Report 9-77, Jet Propulsion Laborator, California Institute of Technolog, Pasadena, CA, USA, [6] L. M. Mack. Boundar laer linear stabilit theor. In AGARD R 79 Special course on stabilit and transition of laminar flow, pages , [7] M. R. Malik. Prediction and control of transition in hpersonic boundar laers. AIAA Paper , [8] M. R. Malik and E. C. Anderson. Real gas effects on hpersonic boundar-laer stabilit. Phs. Fluids A, 3(5):83 81, [9] O. Maren, G. Iaccarino, and E. S. G. Shaqfeh. Disturbance evolution in a mach 4.8 boundar laer with two-dimensional roughness-induced separation and shock. J. Fluid Mech., 648: , 1. [1] J. A. Masad and R. Abid. On transition in supersonic and hpersonic boundar laers. Int. J. Engng Sci., 33(13): , [11] C. Park. On convergence of chemical reacting flows. AIAA Paper, page 47, [1] D. Rodríguez and V. Theofilis. Structural changes of laminar separation bubbles induced b global linear instabilit. J. Fluid Mech., 655:8 35, 1. [13] K.F. Stetson and R.L. Kimmel. On hpersonic boundar-laer stabilit. AIAA Aerosp. Sci. Meet. Ehib., 3th, Reno, AIAA Pap., page 737, 199. [14] G. Stuckert and H. Reed. Linear disturbances in hpersonic, chemicall reacting shock laers. AIAA J., 3: , [15] V. Theofilis. Spatial stabilit of incompressible attachment-line flow. Theor. Comp. Fluid Dn., 7(3): , [16] V. Theofilis. Global linear instabilit. Annu. Rev. Fluid Mech., 43:319 35, 11. [17] V. Theofilis, S. Hein, and U.Ch. Dallmann. On the origins of unsteadiness and

13 85 Stabilit of Hpersonic Flow F IGURE 11. Amplitude of leading mode at M = 1, Re =, Tr = 35 K, Lr = R/Re1 = m, h = δ? and β = π/δ? =.1571 obtained b using the approimation of perfect gas. Module of the (left) streamwise amplitude velocit (center) wall-normal amplitude velocit (right) and amplitude temperature F IGURE 1. Amplitude of leading mode at M = 1, Re =, Tr = 35 K, Lr = R/Re1 = m, h = δ? and β = π/δ? =.1571 obtained b using the approimation of real gas. Module of the (left) streamwise amplitude velocit (center) wall-normal amplitude velocit (right) and amplitude temperature. three-dimensionalit in a laminar separation bubble. Phil. Trans. Ro. Soc. London (A), 358:39 34,. [18] X. Zhong. Additive semi-implicit runge-kutta methods for computing high speed non-equilibrium reactive flows. J. Comp. Phs., 18:19 31, 1996.