AIAA Xiaolin Zhong * RECEPTIVITY OF MACH 6 FLOW OVER A FLARED CONE TO FREESTREAM DISTURBANCE ABSTRACT INTRODUCTION

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1 42nd AIAA Aerospace Sciences Meeting and Exhibit 5-8 January 2004, Reno, Nevada AIAA RECEPTIVITY OF MACH 6 FLOW OVER A FLARED CONE TO FREESTREAM DISTURBANCE Xiaolin Zhong * University of California, Los Angeles, California ABSTRACT This paper presents a numerical simulation study of the receptivity to weak freestream acoustic waves for a Mach axisymmetric flow over a flared cone with a 5' half-angle. The flow conditions and geometry are the same as those of the hypersonic boundary layer stability experiments carried out in NASA Lan ley Hypersonic Quiet Tunnel by Lachowicz et al. &? and transition experiments by Horvath et al. [51. In hypersonic boundary-layer flow over a blunt cone, the process of receptivity to freestream disturbances is altered considerably by the presence of the bow shock followed by an entropy layer. It is crucially important that the interaction of disturbance waves with the bow shock is accurately computed in numerical simulations. In the present study, both steady and unsteady flow solutions are obtained by computing the full Navier-Stokes equations with a fifth-order shock-fitting finite difference scheme, which is able to account for the effects of bow-shock/freestream-disturbance interaction accurately. Whenever possible, the current numerical results are compared with published experimental results. In addition, a normal-mode linear stability analysis is used to study the receptivity properties of the hypersonic boundary layer with the effects of the adverse pressure gradient and surface curvature. The focus of the study is on the generation of the second Mack modes in the boundary layer due to freestream acoustic disturbances, and the effects of surface curvatures of the flared cone on the receptivity process. INTRODUCTION The prediction of laminar-turbulent transition in hypersonic boundary layers is a critical part of the aerodynamic heating analyses on hypersonic vehicles. Despite decades of extensive research, the prediction of hypersonic boundary-layer transition is still based on mostly empirical correlation methods or semi-empirical en method because many physical mechanisms leading to transition are currently not well understood. Among them, the process of boundary-layer receptivity, which is the process of environmental disturbances initially *Professor, Mechanical and Aerospace Engineering Department, xiaolinoseas.ucla.edu, Associate Fellow AIAA. Copyright by, Inc. All rights reserved. entering the boundary layers to generate disturbance waves in the boundary layers, can strongly affect the locations of transition. Though incompressible boundarylayer receptivity to freestream disturbances is relatively well understood [61, the process of hypersonic boundarylayer receptivity is much more complex and is currently an area of active research l8-l4l. Most of our current knowledge of hypersonic boundary-layer stability and transition is based on the results of linear stability theory (LST) [I5], which studies the growth or decay of normal modes of linear disturbance waves in hypersonic boundary layers. Mack [I5] showed that the stability characteristics of high supersonic and hypersonic boundary layers are very different from those of subsonic boundary layers. He found that, in addition to the first-mode instability waves, there are higher acoustic instability modes in hypersonic boundary layers. The unstable modes in hypersonic boundary layers are termed Mack modes in this paper in order to distinguish them from other stable modes. Among the instability wave modes in hypersonic boundary layers, the second mode is most dangerous in term of laminarturbulent transition because it becomes the dominant instability for Mach numbers approximately larger than 4. The second mode instability is particularly important for practical hypersonic flight applications where the freestream disturbances are often very weak. Under such conditions, the second mode instability is a dominating instability before the transition of the boundary layers. The existence and dominance of the second mode instability in hypersonic boundary layers has been observed in boundary-layer stability experiments for hypersonic flows over sharp and blunt cones [l6-l9]. Most practical hypersonic vehicles have blunt noses in order to reduce heating to the vehicles. It has been generally recognized that bow shocks created by the blunt noses can strongly affect the stability and transition of the boundary layers behind them (Reshotko 1991) [71. Reshotko and Khan (1980) [201 showed that the swallowing of the entropy layer by a boundary layer has a strong effect on the boundary-layer stability. The effects of nose bluntness and the entropy layer swallowing on hypersonic boundary-layer transition have been studied in experiments (Potter & Whitfield 1962) [211. The development of the second mode waves can be affected by the entropy layer in the steady base flow. Overall, it has been found in stability and transition experiments that Copyright 2004 by the, Inc. All rights reserved.

2 slightly blunting the nose of a cone can greatly stabilize the flow in the boundary layer. But the trend reverses itself when the nose bluntness increases further, i.e., further blunting the nose destabilizes the boundary layer for large nose radius. Stetson et al. [ carried out boundary-layer stability experiments on an axisymmetric blunt cone in a Mach 7.99 freestream. The half angle of the cone was 7', the nose radius of the cone and the freestream Reynolds number based on the nose radius was about 33,449. It was found that the instability waves in the boundary layer were dominated by the second mode instability. significant super harmonic components of the second modes were observed after the second mode became dominant. Compared with similar hypersonic flow over a sharp cone, the second mode instability of the blunt cone appeared at a location much further downstream. This indicates a stabilization of the boundary layer by slight nose bluntness. They also found evidence of entropy-layer instability in the region outside the boundary layer for a test case with a blunt cone of larger nose radius. Stability experiments of hypersonic flows over sharp or blunt cones have also been carried out by other researchers. Demetriades [16p171 did extensive stability experiments on hypersonic boundary layers over axisymmetric cones. He presented detailed disturbance spectra in the boundary layers and obtained visualization of the laminar rope waves, which are the signature of the second mode waves in the hypersonic boundary layers. Recently, Maslov and his colleagues [l 241 reported their stability experiments on high speed flow. One of the test cases had similar geometry and flow conditions as those tested in Stetson et al.'s 7' blunt cone experiments. It was Mach 5.92 flow over a 7'-halfangle blunt cone. Maslov et al. measured the fluctuation spectra of the disturbance waves in the boundary layer for test cases of instability waves induced by two different mechanisms: 1) by natural disturbances in the wind tunnel, and 2) by artificial disturbances introduced into the boundary layer by high-frequency glow discharge on cone surface. Second mode waves are found in the boundary layer. But the amplitudes of the second mode disturbances in the blunt-cone boundary layer are smaller than those in a sharp cone. In addition to the straight cone models, flared cone models have also been studied in three hypersonic boundary layer stability experiments in the NASA Langley Research Center's quiet Mach tunnel [ The flared-cone model studied by Lachowicz et al. (Fig. 1) has a straight, 5' half-angle section that tangentially merges into a flared region. The cone has a loin straight portion followed by a loin flared region. The radius of curvature of the flared region is 93.07in. The flow has a freestream Mach number of 5.941, and a unit Reynolds number of 9.47 x 1O6/m. The flared cone was used instead of a straight cone in order to induce transition on the model within the quiet wind tunnel by an adverse pressure gradient. The first experiment was carried by Lachowicz, Chokani, and Wilkinson [1~2] on the growth of boundary disturbances on the flared cone at zero angle of attack. The second experiment was carried out by Doggett, Chokani, and Wilkinson LZ6], on the boundary layer stability for the flared cone with an angle of attack. The third experiment by Blanchard and Selby [251 studied the cooling effects on the growth of flow disturbances in the flared cone boundary layer. The experiments were also made for the cases of blunt flared cones. These three experiments showed the development of second mode waves in hypersonic boundary layer over a curved cone surface. The effects of angle of attack, adverse pressure gradient, and wall cooling on hypersonic boundary layer stability were documented and available for CFD code validation studies. The case of sharp flared cone at Mach was recommended as one of the three hypersonic boundary layer stability experiments for CFD code validation L27. Computational studies and linear stability analysis have also been carried out for the stability and transition of the flared cone experiments. In particular, Pruett and Chang ["I conducted direct numerical simulation of the instability and nonlinear breakdown of the Mach flow over the flared cone. In their simulation, the effects of nonzero streamwise surface curvature, adverse pressure gradient, and decreasing boundary-layer edge Mach number were taken into account. The main focus of the study was on the nonlinear breakdown of the instability waves induced by periodic forcing at the computational boundary layer. The effects of bow shock are not directly included in their simulation studies. With the recent renewed interest in hypersonic boundary layer transition, Horvath et al. [51 of NASA Langley have tested the transition on flared cone models to examine the effects of facility noise on boundary layer transition. So far, the receptivity of the boundary layer with effects of nose bluntness, entropy layer, and bow shock have not been studied. Therefore, the purpose of this paper is to conduct a numerical study on the receptivity to freestream acoustic waves for hypersonic flow with the effects of nose bluntness and the entropy layer. The specific test case is the 5' half-angle blunt flared cone in Mach flow, corres onding to the stability experiments of Lachowicz et al. E4] A blunt cone case is chosen in the current paper mainly because the receptivity to freestream noise can be studied without the complication introduced by the singularity at the sharp nose tip. The nose radius of the blunt cone is in at a zero angle of attack. The freestream unit Reynolds number is x lo6 ft-l. The effects of nose bluntness, entropy layer, bow shock, streamwise surface curvature, and adverse pressure gradient on the receptivity is accurately taken into account in the numerical study by mearfs of a high-order shockcapturing scheme developed by Zhong L2'1. In a numerical simulation study of the receptivity

3 of hypersonic boundary-layer flow over a blunt nose, it is crucially important that the interaction of disturbance waves with the bow shock is accurately computed. In this paper, both steady and unsteady flow fields between the bow shock and the boundary layer are numerically simulated by using a high-order shockfitting scheme ['], which has been extensively tested for its accuracy in accounting for the effects of bowshock/freestream-disturbance interactions in a transient flow computations. In previous papers this high-order shock-fitting method was applied to study the receptivity of hypersonic flow over a blunt wedge, a flat plate, and Stetson's Mach 8 blunt cones by direct numerical simulation. Such numerical studies lead to a better understanding of the receptivity mechanisms. In addition, an axisymmetric LST code is used to study the linear stability and receptivity properties of the axisymmetric boundary layer. The LST results are used to identify and analyze the wave structures and interactions based on the simulation results. From the analyses of the simulation results, we study the hypersonic boundary-layer receptivity mechanism, and the effects of nose bluntness and the entropy layer on the receptivity process. GOVERNING EQUATIONS AND NUMERICAL METHODS Boundary-layer receptivity to freestream disturbances for axisymmetric laminar hypersonic flow over a blunt cone at a zero angle of attack is considered. The governing equations for both steady and unsteady flow computations are briefly presented in this section. Details of the governing equations and numerical methods for two and three-dimensional flows have been described in previously papers [ The governing equations are the unsteady full three-dimensional Navier-Stokes equations written in the conservation-law form in a Cartesian coordinate system: where superscript "*" represents dimensional variables. The Cartesian coordinates (a*, y*, z*) are represented by (x;, x;, x:) in a tensor notation. In the current simulation of axisymmetric flow over the flared blunt cone in Fig. 1, x* axis is along the center line of the axisymmetric cone pointing in the same direction as the freestream velocity, and y* and z* axes are perpendicular to the center line. The origins of the Cartesian coordinate system is located at the center of the nose the spherical nose cone. Equation (1) is a vector equation with five independent flow variables, where where p* and e* are gas density and specific total energy per unit mass. The gas is assumed to be thermally and calorically perfect. The equations of states for pressure, p*, and total energy per unit volume, e*, are where the gas constant R* and the specific heats c; and ct are assumed to be constants. The flux vectors in Eq. (1) are p* uj' 0 -T* U* jk k - 4; where ~ij. is the viscous stress tensor and qj' is heat flux vector. They are given by 2, du* dt* = -6.- (6) 3 ax, 3 ax; =p* (2. $) - p,.* The viscosity coefficient, p*, and heat conductivity coefficient, IC*, are calculated using Sutherland's law together with a constant Prandtl number, Pr, i.e., and K* is determined by assuming a constant Prandtl number defined by Pr = ~*c;/k*. In numerical simulation of hypersonic flow over a blunt cone, the governing equations (1) written in the Cartesian coordinate system are transformed into a set of body-fitted curvilinear computational coordinates, (J, 77, <), in a computational domain, which is bounded by the bow shock from one side and the body surface from another. The positions and velocities of the bow shock are unknown variables to be solved together with the flow variables in the interior of the computational domain by a fifth-order shock-fitting method described by Zhong [291. The numerical methods for spatial discretization of the full Navier-Stokes equations are a fifth-order shock-fitting scheme in streamwise and wallnormal directions, and a Fourier collocation method in the periodic azimuthal direction for the case of a cone flow geometry. The spatially discretized equations are advanced in time using a Runge-Kutta scheme. The numerical method and computer code used in this paper have been extensive validated and tested in a number of two and three-dimensional viscous flow simulation discussed in our previous papers (for example, [29]). For this reason, the validation results and testing of the numerical accuracy are not presented in this paper. Because steady flow variables behind the bow shock are not constant in the curreqt study, the flow variables are nondimensionalized using the freestream conditions as characteristic variables. Specifically, we 3

4 nondimensionalize the flow velocities with respect to the freestream velocity U&, length scales with respect to the nose radius ri, density with respect to pl, pressure with respect to pk, temperature with respect to T&, time with respect to rrf/u&, vorticity with respect to U&/ri, entropy with respect to ci, wave number by l/ri, etc. The dimensionless flow variables are denoted by the same dimensional notation but without the superscript "*". FLOW CONDITIONS The flow conditions for the test case studied in this paper are the same as quiet tunnel experiments [1t2p5] on air flow over a flared blunt cone. The specific flow conditions are: Mm = 5.941, * -'6.205 x lo2 Pa, T& = K Pca - y = 1.4, Pr = 0.72, R* = Nm/kgK Freestream unit Reynolds number: Re: = x lo6 ft-' Blunt cone half angle: 0 = 5O, zero flow angle of attack. Flared Cone Surface Radius: in Spherical nose radius: ri = in Parameters in Sutherland's viscosity law: T,* = 288 K, T,S = I(, p: = x kglms The body surface is a no-slip and adiabatic wall for the steady base flow solution. The total length of the cone of the experimental model is I* = m. The corresponding Reynolds number at this length is Rer = x lo7. The cone surface consists of three distinct sections: (1) a spherical nose region, (2) a straight conical section of 5' half angle afterward, and (3) a flared cone surface with radius of curvature of in. The junctions of different sections of the cone surface are continuous up to first surface derivatives, but there is a finite jump in surface curvatures at the junction. The main objective of this paper is to study the recep tivity of the Mack modes in response to the freestream acoustic waves with the effects of surface curvature. In the stability experiments by Lachowicz et al. ['j2], the instability waves in the boundary layer were generated naturally in the wind tunnel without artificially generated forcing disturbances. Lachowicz et al. [lp2] showed detailed fluctuation spectra of disturbance waves at various surface stations. The wave spectra obtained in the experiments clearly show that the instability waves are dominated by two-dimensional second modes and their harmonics. The frequencies of the unstable second modes are in the range of 170 to 275 khz, centered on 230 khz. As the observation station moves downstream, the frequency of the dominant second mode stays relatively unchanged because the boundary layer thickness is almost constant as a result of the adverse pressure gradient over a flared cone. The most amplified second mode in the experiments has a frequency of approximately 230 khz. In order to reproduce similar flow conditions in the numerical simulation of the receptivity process, it is natural to introduce initial forcing waves to excite the instability waves in the boundary layer. The forcing waves can originate from the freestream or at the wall by surface roughness or vibrations. For the current numerical simulation of the receptivity to freestream acoustic waves, forcing disturbances are introduced in the freestream before reaching the bow shock. We simulate the stability experiment by imposing freestream acoustic disturbances of 15 frequencies, in a range of 15.6 khz to 234 khz. Since the wave fields in the experiments contain a wide range of second-mode frequencies, we simultaneously introduce disturbances of a number of frequencies near the dominant second mode waves in the simulation. The subsequent receptivity and development of the instability waves at these frequencies are computed by the numerical simulation. Meanwhile, the amplitudes of freestream acoustic disturbances are chosen to be small enough so that the receptivity process is in the linear regime. Consequently, the receptivity process of each individual frequency can be decomposed, by means of a temporal Fourier transform, from the simulation results which contain a mixture of different frequencies. In this sense, the receptivity results are independent of the amplitude distributions among the different forcing frequencies. FREESTREAM FORCING WAVES The receptivity of an axisymmetric Mach boundary layer to freestream acoustic waves for hypersonic flows past a 5' half angle blunt flared cone at a zero angle of attack is considered. The forcing waves are introduced by superimposing freestream disturbances to a steady base flow before reaching the bow shock. The subsequent unsteady flow is computed to investigate the process of freestream waves passing through the bow shock, entering the boundary layer, and finally inducing boundary-layer wave modes. In this paper, the freestream disturbances are a number of independent weak planar fast acoustic waves with wave fronts normal to the center line of the body. Other forms of freestream disturbances, including slow acoustic, entropy, and vorticity waves, and three-dimensional waves, are not studied here. They will be considered in a future study. The specific freestream acoustic waves introduced in the simulation are a mixture of N independent twodimensional planar fast acoustic waves of different frequencies in the freestream. The number N used in the current study is 15 independent forcing waves. The wave fields are represented by the perturbations of instantaneous flow variables with respect to the local steady base flow variables. For weak acoustic waves in 4

5 the freestream before reaching the bow shock, we impose a mixture of independent acoustic waves with a total of N frequencies. The freestream perturbations of an arbitrary flow variable can be written in the following form: qm(r, y, t)' = N /q'~ C A, e'in wl(t-')+(nl n=l (8) where q represents any of the flow variables, including velocity components, densities, pressures, and temperatures. In the equations above, IqllAn is the freestream wave amplitude of a given flow variable q at the n-th frequency of 1998). The numerical simulation for an unsteady hypersonic layer receptivity problem is carried out in three steps. First, a steady base flow field is computed by advancing the unsteady flow calculations to convergence with no disturbances imposed in the freestream. Second, unsteady viscous flows are computed by imposing freestream perturbations given by Eq. (8) to the numerically obtained steady base flow solution. Third, the unsteady computations are carried. out for a number of additional periods in time to record the perturbations. A temporal Fourier transform is performed on the perturbation variables to obtain the Fourier amplitudes and phase angles of the perturbations of the unsteady flow variables for each individual frequency throughout the flow field. where wl is the minimum frequency of the wave packet. In Eq. (8), c, is the wave speed in the freestream before reaching the bow shock. For fast acoustic waves in the freestream, where F represents the wave frequency with respect to a viscous flow scale. The receptivity problems are studied numerically by solving the unsteady Navier-Stokes equations using a fifth-order shock fitting scheme described in (Zhong STEADY BASE FLOW SOLUTIONS The steady base flow solution of the Navier-Stokes equations for the axisymmetric Mach flow over the blunt cone is obtained first by computing the flow to a steady state without freestream forcing waves. The simulation is carried out by using a multi-zone approach using 27 zones with a total of 3240 x 121 grid points for Since the wave components of different frequencies are linearly independent, the initial phase angles, q5,, of the axisymmetric flow field from the nose to the 420 nose radii (25in) surface station downstream. the forcing acoustic wave at frequency w, are chosen randomly. The total amplitude of the wave group is Figure 2 shows the Mach number contours of the curdetermined by setting the values of Iq'l. For fast acous- rent steady state solution. In the simulation, the bow tic waves in the freestream, perturbation amplitudes of shock shape is not known in advance and is obtained nondimensional flow variables satisfy the following dis- as the solution for the outer computational boundary. persion relations: The results show that the Mach numbers immediately behind the bow shock approach a constant value behind the shock at downstream locations. Figure 3 shows the pressure contours of the current steady state solution. The flared section of the cone introduces adverse where E is a dimensionless number representing the to- pressure gradient along the wall surface. As a result, tal amplitude of the group of N freestream fast acoustic compressive Mach waves are generated from the flared waves. In this paper, only linear receptivity is consid- surface. These compression waves converge to a weak ered by using a very small value oft. The forcing distur- shock near the bow shock at the upper right corner of bances contain N wave frequencies which are multiples the flow field. Because the weak shock is formed downof wl. The minimum frequency wl is chosen such that stream of the boundary layer of the test model, which the N frequencies span the dominant second-mode fre- has a total length of 320 nose radius. This weak shock quencies observed in the experiment. is expected to have little effects on the stability of the The flow is characterized by a freestream Mach num- boundary layer of the test model. ber, M, = 5, and a freestream Reynolds number The steady surface presure and temperature distribubased on the nose radius defined by, Re,, = tions along the cone surface for the current case of adia- P ~ " ~ " ~. P L batic wall are shown in Figs. 4 to 7. The current numeri- The forcing frequency of a freestream acoustic wave is cal simulation results are compared with the experimenrepresented by a dimensionless frequency F defined by tal measurements by Lachowicz et al. for a sharp cone. Since the nose bluntness of the simulation case is very small, these figures show good agreements between the DNS and experimental results. These figures also show that the maximum wall pressure is reached at the stagnation point. The surface pressures drop sharply as flow expands around the npse region. Since there is a discontinuity in surface curvatures at the junction of the spherical nose and straight cone afterward. The 5

6 flow experiences an overexpansion at the junction and goes through a recompression along the cone surface afterward. As a result, there is a slight adverse pressure gradient along the straight surface locations after the junction. At locations further downstream in the flared cone section, the surface pressures increase significantly due to the surface curvature. The strong adverse pressure gradient in the flared section also generates Mach waves in the flow fields and a weak shock further downstream. Figure 6 shows that the surface temperature also experiences a sharp rise in the flared cone region. The rise of surface temperature in the experimental results, shown in Fig. 7, is an indication of boundary layer transition. Further comparisons between the current simulation results and the corresponding experimental ones are shown in Figs. 8 to 10. The total temperatures and mass fluxes obtaihed by the simulation agree well with those of the experimental results. Figure 9 show that the differences of the two sets of the results are in the range of 2%. The differences are partially due the the effects of the nose bluntness of the computational model. The bow shock gradually changes from a local normal shock at the stagnation streamline to an oblique shock downstream. Consequently, flow variables in the inviscid region outside the boundary layer are not uniform. The distributions of Mach numbers and pressures immediately behind the bow shock are plotted in Figs. 11 and 12 as functions of the dimensionless x coordinate. These figures show that the Mach numbers immediately behind the bow shock are around 5.8. The actual values are not constant because the cone surface is not straight. Because of the adverse pressure gradient on the flared cone surface, the boundary-layer thickness on the flared cone surface is almost constant. Since the wave length of the second mode in a hypersonic boundary layer is about two times of the boundary layer thickness, a constant boundary layer thickness leads to a constant frequency of the dominant second mode instability waves in the boundary layer. Figures 13 to 15 show the Mach number profiles along the wall-normal directions at four surface locations in the flared cone region. The Mach number profiles show that the nondimensional boundary layer thickness in the flared cone region is about constant at a value of 1. These figures also show the development of the Mach waves, which will eventually lead to the formation of a shock wave outside of the boundary layer. RECEPTIVITY TO FREESTREAM ACOUSTIC WAVES Having obtained the steady state base solution for the Mach axisymmetric flow over the blunt flared cone, the boundary-layer receptivity to freestream acoustic waves is numerically simulated by computing the full Navier-Stokes equations. The freestream forcing waves are a mixture of 15 independent planar fast acoustic waves of different frequencies. The unsteady flow solutions are obtained by imposing the acoustic perturbations on the steady flow solution in the freestream before reaching the bow shock. The subsequent interactions of the freestream disturbances with the bow shock and the receptivity of the boundary layer over the cone surface are accurately computed by using the full Navier-Stokes equations. The fifth-order shock fitting method also accurately computes the oscillations of the bow shock, and the interactions between the bow shock and disturbances which are reflected from the wall and impinge on the shock from behind. The effects of the entropy layer and the nose bluntness on the receptivity process are also simulated accurately by the current numerical simulation of the full Navier-Stokes equations. In the current test case, the forcing acoustic waves in the freestream contain N = 15 independent frequencies, l.e., where the lowest frequency is f; = kHz. The remaining 14 frequencies are multiples of fi given by Eq. (14), where the highest frequency is ff, = kHz. The wave modes induced in the boundary layer are identified by comparing the simulation results with the LST ones. The relative wave amplitudes among different frequencies in the freestream are set to be very small such that the receptivity is in the linear regime. Specifically, the relative wave amplitudes in the freestream are set to be the same value for all frequencies at A, = The phase angles of the waves in Eq. (8) are chosen randomly. The overall wave amplitude is 6 = The unsteady calculations are carried out until the solutions reach a periodic state in time. Temporal Fourier analysis is then carried out on the local perturbations of unsteady flow variables. A Fourier transform for the perturbation field of an arbitrary flow variable q leads to: N ql(x, y, t) = P{.C n=o lq,(r, y) 1 ei [-nwlt+"(x~~)i 1 (15) where nwl is the frequency of the n-th wave mode, ql(x, y, t) represents an arbitrary perturbation variable. The Fourier transformed variables, Iqn(x, y)l and & (x, y), are spatially varying real variables representing the local perturbation amplitudes and phase angles at the wave frequency of nwl. For perturbations on the body surface, we can compute local growth rates, a,, and local wave numbers, ai, from the numerical perturbation fields by,, d4* a', = - ds 6

7 where the derivatives are taken along the natural coordinate s along the body surface. If the flow perturbations of the simulation results in a local region of the boundary layer are dominanted by a single wave mode, the parameters, a, and ai, computed by Eqs. (16) and (17) are the wave numbers and growth rates of this mode. In this case, ai and a, are smooth functions of x. On the other hand, if the simulation results contain simultaneously multiple wave modes in a local region of the boundary layer, ai and a, computed by Eqs. (16) and (17) do not represent the wave numbers and growth rates of a single wave mode. Instead, these two parameters represent a modulation of two or more wave modes. The distributions of ai and a, along the surface direction will be oscillatory functions of x. In this case, further decomposition of different wave components is required in order to obtain the growth rates and wave numbers of the individual wave modes. Characteristics of Induced Waves Figures 16 to 18 show the development of wave amplitudes of pressure perturbations for the 15 forcing frequencies on the cone surface as functions of dimensionless variable x. The lines represent 15 different frequencies of f = nfl, where f; = kHz and n = 1,2,..., 15. The pressure amplitudes are plotted in logarithmic scale because of the large range of amplitude variations. These figures show that the receptivity process leads to a complex wave structure for disturbance waves of the forcing frequencies. It can be shown that the basic receptivity mechanism in the current flow is the same as that of a hypersonic boundary layer over a flat plate and over a blunt cone, which were studied numerically by Ma and Zhong [ and theoretically by Fedorov [331. For the induced waves of a given frequency for the cases of freestream fast acoustic waves, the characteristics of the induced waves undergo graduate changes from dominantly mode I waves near the leading edge to mode I1 waves. The second Mack mode waves can only be induced in the current recep tivity process through resonance interaction with mode I wave, which in turn is induced by the forcing fast acoustic waves unsteam. The forcing acoustic waves do not directly interact with the second Mack mode. The results in the current studies are consistent with thkse conclusions. Figures 16 and 17 shows the amplitude development along the cone surface of the first ten lower frequency waves (n = 1,..., 10) ranging from khz to khz. Similar to a previous study ['*I, the induced wave growth in the boundary layer at these frequencies are not the Mack modes. They are the stable mode I waves which are induced by the forcing acoustic waves through resonant interactions between the forcing waves and the stable mode I waves. In other words, the waves developing near the leading edge in this figure are dominantly mode I waves [321. The wave amplitudes increase because of their resonant interaction with the forcing fast acoustic waves. For example, in the case of n = 5, mode I waves reach a maximum value for x m 200, and decay afterward. The higher the frequency, the earlier the peaks of mode I waves will be reached. Figure 18 shows the amplitude development along the cone surface of the next five frequencies (n = 11,..,15) ranging from khz to khz. Again, the waves developing near the leading edge are dominantly mode I waves in the region of x < 70. After the decay of the mode I waves and an extended region of low wave amplitudes, the second Mack mode waves start to grow at a later surface location. The generation of the second mode Mack waves is a results of the resonant interaction between mode I waves and the second Mack mode, which is not related to the branch I neutral stability of the second mode. In particular, the waves at the frequency of 234 khz (n = 15) have strongest second mode growth. The second mode of 234 khz reaches a peak value at the surface station of 340 nose radii. This figure also shows strong wave modulations in the second mode region, consisting of components of the second Mack mode waves and the forcing acoustic waves. The development of wave amplitudes along the cone surface of all frequencies can be shown more clearly by the frequency spectra of surface presure perturbations, as shown in Fig. 19 for a number of surface stations ranging from 291 to 365 nose radius. The figure shows the development of the second Mack mode around the frequency of 234 khz, which is close to the dominant frequency of the experiments by Lachowicz et al. [l"'] for a sharp cone of 230 khz. Figure 20 show the contours of real part of temperature perturbations in the whole flow field for the frequency of f' = 234kHz (n = 15). The figures show the development of induced waves at a fixed frequency in three local regions of the flow field. In the first local region of 15 < x < 25, the figure shows that the forcing waves from the freestream passes through the bow shock and enter the boundary layer to generate mode I waves inside the boundary layer. In the downstream region of 120 < x < 160, the wave modes are a mix of forcing waves and mode I waves. The second Mach mode at this frequency starts to develop further downstream. In the region of 340 < x < 370 shown in the figure, the boundary layer disturbances are dominated by the second mode waves. This figure shows that mode I has a distinctively different structure from the second Mack mode. The second mode has a typical "rope wave" structure with strpng perturbation at the edge of the boundary layer, while mode I has stronger wave amplitudes in the boundary layer. 7

8 Boundary-Layer Wave Mode Characteristics In previous receptivity studies of Mach 4.5 boundary layer flow [s191321, it was shown that a family of other wave modes, which are stable in a linear stability analysis, play an important role in the receptivity process. They are termed mode I, 11, 111, etc. in [32]. The stable wave modes generated by the forcing waves through resonant interactions can interact with the instability waves once they are generated. In the studies of boundary-layer stability, the following Reynolds number, R, based on the length scale of boundary-layer thickness is often used: where length scale of boundary-layer thickness is defined as r where s* is the curvature length along the wall surface measured from the leading edge. Hence, the relation between R and local Reynolds number Re, is A nondimensional frequency w and wave number a normalized by freestream velocity and local surface length L* as a characteristic length are given by The dimensionless frequency F is related to w by For a given values of F and surface location s, w can be computed by the equation above where the value of R can be calculated by using Eq. (20). The real part of wave number a, is related to phase velocity of a wave mode by where a is the nondimensional phase velocity normalized by the freestream velocity. For the current test case, a dimensional frequency is related to F by where the reference frequency for the current case is: f,' = Hz. The Reynolds number is calculated by where for the current case, Re: = 9.47 x lo6 rn-l. Ma and Zhong [321 studied, using the linear stability theory, for boundary-layer wave mode characteristics of supersonic flow over a flat plate. It was found that the distribution of phase velocities of boundary-layer wave modes is a function of the product of the local Reynolds number (R) and frequency (F). Almost the same distributions of phase velocities us R * F for different boundary-layer wave modes are obtained when F is changed while R is fixed, or when R is changed while F is fixed. A similar LST study is carried out in the current study for axisymmetric hypersonic boundary-layer flow over the blunt flared cone. Figure 21 shows the phase velocities of three discrete modes, i.e., mode I, mode I1 and the Mack modes, as functions of nondimensional frequencies at a fixed surface station of s = The phase velocities of the fast acoustic wave (1 + l/mm), entropy/vorticity wave (I), and slow acoustic wave (1- l/m,) are also shown in the figure for comparison. Both mode I and mode I1 originate with an initial phase velocity of the fast acoustic wave (1 + l/mm). Before these two modes become distinct modes, their eigenvalues merge with the continuous spectra. After these two wave modes appear, their phase velocities decrease gradually with increasing w (or increasing R). With increasing w, the phase velocity of mode I continues to decrease and passes across the phase velocity curve of Mack modes. At the intersection point (w = 0.39), mode I is synchronized with the second Mack mode, where both modes have very similar profiles of eigenfunctions. A very similar phenomena happens to mode I1 at a larger w = 1.11, which is also shown in this figure. The growth rates (a;) of different normal modes at the same surface station are plotted in Fig. 22. While the growth rates of Mack modes are continuous, there is a gap in the growth rate curve for mode I. It shows that both mode I and mode I1 are stable modes. Mack modes are slightly unstable for frequency less than 100 khz. The Mack mode in this range is the conventional first mode. In the frequency around 250 khz, the unstable Mack modes are the conventional second mode. In this range, the growth rates of the second mode change dramatically. As shown in Fig. 21 and 22, the first mode and the second mode are in fact different sections of a single mode. Here, both the first and the second mode are simply called Mack modes for convenience of dis cussion. A similar plot for the wave numbers of these modes are shown in Fig. 23. Although mode I is stable, it was found that the stable mode I waves play an important role in the receptivity process because they interact with both the forcin acoustic waves and the unstable Mack-mode wave^^^^]. Through the interactions, the stable mode I waves transfer wave energy from the forcing fast acoustic waves to the second Mack-mode waves. The linear stability theory is used to identify different 8

9 wave modes induced by fast acoustic waves. Phase velocities and structures of boundary-layer disturbances from computation of numerical simulations are compared with corresponding values from eigenvalues and eigenfunctions of the linear stability theory at the same frequency. Here, only one frequency with n = 15 ( khz) is chosen for the comparison. The phase velocities of the induced boundary-layer disturbances from numerical simulations are calculated based on pressure perturbations on the wall surface by using temporal Fourier analysis according to Eq. (17). The value calculated by using Eqs. (17) correspond to streamwise wave number (a,) of a single wave if the numerical solutions are dominated by a single discrete wave mode in a local region. If the numerical solutions contain a mixture of two or more wave modes, the values of streamwise wave number demonstrate the result of modulatia of these mixed wave modes. At each frequency, the induced boundary-layer disturbances can be identified by comparing wavenumber and wave structures from simulations with eigenvalues and eigenfunctions obtained from the LST. Figure 24 shows the distribution of the streamwise wavenumbers of the induced boundary-layer disturbances at frequency with n = 15 (f* = kHz) from the simulation. A similar plot for the distribution of the phase velocities of boundary-layer disturbances at the same frequency is shown in Fig. 25. The real part of the eigenvalues and the phase velocities of the Mack modes from the linear stability theory are also plotted in these two figures for comparison with the numerical solutions. These figures show that there is excellent agreement in the streamwise wavenumber between the induced boundar y-layer disturbances and the Mack mode waves, which indicates that Mack mode waves are generated after the synchronization point between mode I and the second Mack mode. These two figures show that the the wave modes generated by the receptivity process near the leading are stable mode I waves, not the Mack mode. The Mack second mode is generated by a resonant interaction between mode I and the second mode at the synchronization point located around R = The second mode generated by the resonant interaction grows rapidly because it is linearly unstable. The receptivity process for the current case is very similar to that of a straight blunt cone at Mach 8 studied in [14]. It shows that the wavenumber and phase velocity of mode I waves are almost same to those of fast acoustic waves near the leading edge. Due to synchronization between mode I waves and fast acoustic waves near the leading edge, mode I waves are generated by means of a resonant interaction between mode I waves inside the boundary layer and fast acoustic waves outside the boundary layer in the region upstream where wavenumber and phase velocity of mode I waves are close to those of fast acoustic waves. Though mode I is predicted to be always stable by the linear stabil- ity theory, mode I waves are strongly amplified before they reach the peak amplitude due to the resonant interaction with fast acoustic waves. The phase velocities of mode I waves decrease during the propagation downstream. When phase velocities of mode I waves decrease to a certain value and there is no more resonant interaction between mode I waves and acoustic waves, mode I waves decay due to their inherent stable properties after they reach the peak amplitude. In addition, there is a modulation between the induced wave modes with the forcing acoustic waves and other waves leads to the oscillation of phase velocities and growth rates calculated by the simulation. The growth rates for the receptivity simulation calculated by Eq. (16) are shown in Fig. 26. Because the modulation between the second mode and the forcing acoustic waves, the plot of cri shows strong oscillations. Such oscillations make it difficult to compare with the LST results of second mode growth rates. The circles in the figure are the values of the second mode growth rates obtained by LST. Though precise comparison is not possible in the current figure, the LST growth rates do located approximated at the center of the numerical growth rates curve. This indicates that the growth rates of the second mode of the simulation are close to the corresponding values of the LST for the pure second mode. Further studies are needed to do a more clear growth rate comparisons among the results of DNS, experiments, and the LST predictions. One way to obtain a more pure second mode results in the simulation is to generated the second mode by wall blow-and-suction in early surface station. The boundary layer second mode can be directly generated and compared with experimental and LST results. Such work in currently under way by the author. CONCLUSIONS In this paper, we have studied the receptivity of Mach flow over a 5 O half-angle blunt flared cone, corresponding to the quiet tunnel experiments by Lachowicz et al. Both the steady base flow solutions and one case of unsteady flow solutions have been obtained and studied. It is shown that the current steady flow solutions agree very well with the experimental results. The receptivity of the Mach flow over the blunt flared cone to freestream fast acoustic waves is simulated by solving the full Navier-Stokes equations. The simulation results show a complex development of wave modes induced by freestream acoustic waves. It is found that the receptivity mechanisms are the same as those of Mach 8 flow over a straight blunt cone studied in a previous paper [I4]. It is found that the synchronization location between mode I and the second mode plays an important role in the receptivity of the second Mack mode in the boundary layer. The results of LST calculations are used to identify the wave modes in the boundary layer in 9

10 10 the receptivity process and to study the cause of the delay of the development of the second mode waves. The wave modes induced by the freestream acoustic waves are mode I near the nose, as it propagate downstream, second mode is excited due to the mechanism of resonant interactions between different wave modes. Therefore mode I plays the most important role in the recep tivity process. The wave numbers and the growth rates of numerical solutions compare reasonably well with the LST results. Further studies are current under way to compare the simulation growth rates with those of LST and experiments. ACKNOWLEDGMENTS This work was sponsored by the Air Force Office of Scientific Research, USAF, under AFOSR Grant #F , monitored by Dr. John Schmisseur. ~he'view; and conclusions contained herein are those of the author and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the US. Government. The author also thanks T. J. Horvath of NASA Langley Research Center for discussions and assistance which led to the initiation of the current work. References Lachowicz, J. T., Chokani, N., and Wilkinson, S. P.,, "Hypersonic Boundary Layer Stability over a Flared Cone in a Quiet Tunnel," AIAA Paper , January Lachowicz, J. T., Chokani, N., and Wilkinson, S. P., "Boundary-Layer Stability Measurements in Hypersonic Quiet Tunnel," AIAA Journal, Vol. 34, No. 12, December Wilkinson, S., "A Review of Hypersonic Boundary Layer Stability Experiments in a Quite Mach 6 Wind Tunnel," AIAA Paper , Chokani, N., "Perspective: Stability Experiments at Hypersonic Speeds in a Quiet Wind Tunnel," AIAA Paper , January Horvath, T. J., Berry, S. A,, Hollis, B. R., Chang, C.-L., and Singer, B. A., "Boundary Layer ~ransition On Slender Cones In Conventional And Low Disturbance Mach 6 Wind Tunnels," AIAA Paper 2OO2-27&, June Saric, W. S., Reed, H. L., and Kerschen, E. J., "Boundary-Layer Receptivity to Freestream Disturbances," Annual Review of Fluid Mechanics, Vol. 34, 2002, pp Reshotko, E., "Hypersonic Stability and Transition," in Hypersonic Flows for Reentry Problems, Eds. J.-A. Desideri, R. Glowinski, and J. Periaux, Springer-Verlag, Vol. 1, 1991, pp [8] Fedorov, A. V. and Khokhlov, A. P., "Receptivity of Hypersonic Boundary Layer to Wall Disturbances," Theoretical and Computational Fluid Dynamics, Vol. 15, 2002, pp [9] Fedorov, A. V., "Receptivity of High Speed Boundary Layer to Acoustic Disturbances," AIAA Paper $ , June [lo] Ma, Y. and Zhong, X., "Receptivity of a supersonic boundary layer over aflat plate. Part 1: wave structures and interactions," Jornal of Fluid Mechanics, Vol. 488, 2003, pp [ll] Ma, Y. and Zhong, X., "Receptivity of a supersonic boundary layer over a flat plate. Part 2: receptivity to freestream sound,," Jornal of Fluid Mechanics, Vol. 488, 2003, pp Zhong, X., "Leading-Edge Receptivity to Free Stream Disturbance Waves for Hypersonic Flow Over A Parabola," Journal of Fluid Mechanics, Vol. 441, 2001, pp [13] Zhong, X. and Ma, Y., "Receptivity and Linear Stability of Stetson's Mach 8 Blunt Cone Stability Experiments," A IA A Paper , January [14] Zhong, X. and Ma, Y., "Numerical Simulation of Leading Edge Receptivity of Stetson's Mach 8 Blunt Cone Stability Experiments," AIAA Paper , January Mack, L. M., "Boundary Layer Linear Stability Theory," AGARD report, No. 709, 1984, pp. 3-1 to [16] Demetriades, A., "Hypersonic Viscous Flow Over A Slander Cone. Part 111: Laminar Instability and Transition," AIAA paper , [17] Demetriades, A,, "Laminar Boundary Layer Stability Measurements at Mach 7 Including Wall Temperature Effects," AFOSR- TR , Vol. November, Stetson, K. F. and Kimmel, R. L., "On Hypersonic Boundary Layer Stability," AIA A paper , [19] Maslov, A. A., Mironov, S. G., and Shiplyuk, A. A., "Hypersonic Flow Stability Experiments," AIAA Paper , [20] Reshotko, E. and Khan, N. M. S., "Stability of the Laminar Boundary Layer on a Blunt Plate in Supersonic Flow," lutam Symposium on Laminar- Turbulent Transition, edited by R. Eppler and 10