Discrete- and Finite-Bandwidth-Frequency Distributions in Nonlinear Stability Applications

Transcription

1 Discrete- and Finite-Bandwidth-Frequency Distributions in Nonlinear Stability Applications 1, a) Joseph J. Kuehl Baylor University (Dated: 16 January 2017) A new wave packet formulation of the parabolized stability equations method is presented. This method accounts for the influence of finite-bandwidth-frequency distributions on nonlinear stability calculations. The methodology is motivated by convolution integrals and is found to appropriately represent nonlinear energy transfer between primary modes and harmonics, in particular nonlinear feedback, via a nonlinear coupling coefficient. It is found that traditional discrete mode formulations overestimate nonlinear feedback by approximately 70%. This results in smaller maximum disturbance amplitudes than those observed experimentally. The new formulation corrects this overestimation, accounts for the generation of side lobes responsible for spectral broadening and results in disturbance representation more consistent with experiment than traditional formulations. A Mach 6 flared-cone example is presented. PACS numbers: A-, Fe, Ib, Ki Keywords: Hypersonic instability, Boundary layer stability and and transition, Nonlinear parabolized stability equations a) Electronic mail: Joe Kuehl@Baylor.edu 1

2 I. INTRODUCTION In low-disturbance environments such as flight, boundary layer transition to turbulence is generally acknowledged to occur through a multi-step process. Disturbances in the freestream, such as sound or vorticity, enter the boundary layer as steady and/or unsteady fluctuations of the basic state. This part of the process is called receptivity (Morkovin 1969) and it establishes the initial conditions of disturbance amplitude, frequency, and phase for the breakdown of laminar flow. A number of different instabilities can occur independently or together and the appearance of any particular type of instability depends on Reynolds number, Mach number, wall curvature, sweep, roughness, and initial conditions, and can be modulated by pressure gradients, surface mass transfer, temperature gradients, and so forth. This work will follow the modal growth scenario. That is, receptivity processes initiate small amplitude disturbances in the boundary layer which are governed by linear dynamics. As the disturbance amplitude grows, nonlinear interactions and secondary instabilities lead to breakdown. Comprehensive reviews are given by Mack (1984), Morkovin et al. (1994), Reed et al. (1996), Schmid & Henningson 2001, Saric et al. (2002, 2003), Reshotko (2008), Fedorov (2011), and Theofilis (2011). Nonlinear effects have been modeled by techniques such as nonlinear parabolized stability equations [e.g. Herbert (1997), Haynes & Reed (2000), Bertolotti et al. (1992), Li & Malik (1996), Chang (2004), Kuehl et al. (2012)], and direct numerical simulations [e.g. Zhong & Wang (2012), Sivasubramania & Fasel (2013, 2014)]. One feature of these nonlinear computations is that they often assume discrete-frequency primary modes and harmonics without consideration of disturbance bandwidth. However, physically a disturbance will consist of some finite-bandwidth distribution of frequencies. Thus, frequency bandwidth is inherent to the problem and its consideration is required to appropriately determine disturbance amplitude. Chokani (2005) presents a very nice experimental investigation in which disturbance bandwidth is quantified in the context of nonlinear energy exchange. Figure 1, from an experimental investigation of the acoustic Mack mode instability prevalent in hypersonic flows (Mack 1984), shows normalized fast-pressure-sensor data for the Purdue compression cone and illustrates finite-bandwidth frequency content (Chynoweth 2015 and McKiernan et al. 2015). This data was generously provided by Prof. Steven Schneider and Brandon Chynoweth of Purdue University with nominal run conditions of Mach 6 and unit 2

3 FIG. 1. Normalized PCB fast-pressure-sensor experimental data of the Purdue compression cone illustrating finite-width frequency-band content. Re = 9.3x10 6 /m on the roughness insert Purdue compression cone. The present work considers the effect of distributed frequency disturbances on the boundary layer transition problem and provides a basis for their treatment. In particular, the nonlinear coupling of modes resulting from finite-width frequency distributions is discussed. II. PARABOLIZED STABILITY EQUATIONS: Originally identified by Herbert and Bertolotti (1987), during a critical review of Gaster s (1974) early nonparallel work, the parabolized stability equations (PSE) have been developed as an efficient and powerful tool for studying the stability of advection-dominated laminar flows. Excellent introductions to the PSE method and summary of its early development were provided by Herbert (1994; 1997). During the early stages of both linear (LPSE) 3

4 and nonlinear (NPSE) development of this technique, much was established related to basic marching procedures, curvature, normalization conditions and numerical stability of the method itself (Bertolotti 1991; Chang et al. 1991; Joslin et al. 1992; Li & Malik 1996; Haynes & Reed 2000). In a relatively short time, the field rapidly expanded to include complex geometries, compressible flow, and finite-rate thermodynamics (Stuckert & Reed 1994; Chang et al. 1997; Johnson et al. 1998; Haynes & Reed 2000; Malik 2003; Chang 2004; Johnson & Candler 2005; Li et al. 2010; Theofilis 2011; Paredes et al. 2011; Kuehl et al. 2012; Kocian et al. 2013; Perez et al. 2012). One universal feature of these works is that they assume discrete frequency primary modes and harmonics. The goal of the present work is to provide a rigorous basis for the treatment of nonlinearities when distributed frequency (finite-bandwidth) disturbances are present in hypersonic boundary-layers. The stability analysis begins by considering a generic disturbance and decomposing the flow variables into a steady basic state plus a disturbance (1) and assuming the basic state is a solution to the governing equations of motion, which represents the flow that exists in the absence of any environmental disturbances or forcing. Assuming, without loss of generality, a 2D basic state and 2D disturbances, substitution into the governing set of equations yields the equations describing the evolution of the disturbances. φ(x, y, t) = φ(x, y) + φ (x, y, t) }{{}}{{} basic state disturbance φ = [u, v, T, ρ] T (1) Here x, y are the wall-parallel and -normal coordinates, t is time, u, v are the wall-parallel and -normal velocity components, ρ is the density and T is the temperature. Different stability methods solve the disturbance equations under different sets of assumptions. Fedorov and Tumin (2011) have highlighted the important difference between Linear Stability Theory (LST) and the PSE method. Formally, LST imposes the assumptions of infinitesimal (linear) disturbances and parallel flow upon which separable solutions may be found. That is, the basic state is assumed to be only a function of y and φ = X(x)Y (y)t (t) = ˆφ(y)e i(αx ωt). Here α and ω are the streamwise wavenumber and frequency, respectively, of the disturbance. Substitution of this form into the disturbance equations yields a generalized eigenvalue problem. Thus, LST describes the solution to the disturbance equations at a given location 4

5 as a spectrum (continuous or discrete) of linearly independent eigenmodes, which have no relation to neighboring locations. The Fourier/Laplace transform method is another approach to solving the disturbance equations with less restrictive assumptions. The Laplace transform is performed along the dimension on which stability is of interest, x for spatial stability and t for temporal stability. With φ representing the disturbance state and φ F its Fourier transform, the methods may be shown to be equivalent with the following restrictions (for spatial stability analysis): a) lim t ± φ = 0 b) φ F can not grow faster than e αx c) φ F x can not grow faster than eαx d) φ F (0, y, ω) = 0 e) φ F (0, y, ω) = 0. x Restriction a is a result of the temporal Fourier transform while restrictions b e result from the spatial Laplace transform. Restrictions d, e state that the initial disturbance amplitude must be zero (i.e. not considering finite-amplitude disturbances) and the parallel-flow assumption be made, respectively. Notice that restrictions a c are required for the existence of the Fourier/Laplace transforms, but restrictions d and e may be lifted by calculation of the inhomogeneous terms. Thus, the Fourier/Laplace transform method results in an initial-value problem capable of handling a distribution of dependent modes. PSE analysis is similar to the Fourier/Laplace transform in that it considers an initialvalue problem. However, an assumption is made for PSE based on the observation that the basic flow state is slowly varying in the streamwise direction, an assumption the Laplace transform is not capable of accommodating. Instead a method-of-multiple-scales approach (Nayfeh 2004) is taken in which a disturbance is assumed of the form F [φ ] = φ( x, y) Φ(x, t) }{{}}{{} shape wave 5

6 where the wave part satisfies Φ = iα( x)φ x (2) Φ = iωφ t (3) Notice the multiple-scales introduction of a slow variable x = x Re into the shape function and streamwise (x) complex exponential α( x), where Re = Ueδr ν e is a Reynolds number based on characteristic values of edge velocity (U e ), edge kinematic viscosity (ν e ), and reference boundary-layer length scale (δ r ). Formally, equation 3 is the result of a Fourier transform. Thus, PSE considers disturbances of the form φ = φ( x, y, ω) A( x, ω)e iωt dω (4) }{{}}{{} shape wave where A( x, ω) = e i α( x,ω)dx and the dependence of the shape function ( φ) and amplitude function (A) on ω has been made explicit. The shape and amplitude functions are essentially the Fourier transform of the disturbance. Upon expansion of the streamwise derivatives φ x = 2 φ x 2 = ( ( 1 φ ) + iα φ Ae iωt dω Re x 1 2 φ Re 2 x + 2iα 2 Re it is found that the second spatial derivative φ x + i φ ) α Re x α2 φ Ae iωt dω, 2 φ x 2 is of highest order and a perturbation expansion may be consistently truncated resulting in the neglect of this term. This leaves the disturbance equation nearly parabolized (Li and Malik 1996) and an efficient marching solution may be sought. III. BANDWIDTH REPRESENTATION A. Properties of Stability Formulations As indicated above, different stability methods solve the disturbance equations under different sets of assumptions. However, it is common practice for numerical methods to be performed in the frequency domain and, inherent to numerical computation, discrete modes 6

7 are selected from what is experimentally observed as a continuous frequency spectrum of disturbances. Generically, experimental observation provides a shape function [ φ(x, y, ω)] and an amplitude spectral density function [A(x, ω), x variation of which represents downstream, possibly frequency dependent, growth], eqn. 4 The goal is to represent an experimentally observed finite-bandwidth of disturbances by a single (or multiple) discrete shape function (φ k ) that oscillates at frequency ω k with amplitude A k, in a physically consistent way. As both experimental observations and numerical methods are inherently discrete, it is assumed that the disturbance can be represented as a Fourier series φ = k c k e i kπ L t (5) c k = 1 L φ e i kπ L t dt. (6) 2L L Defining ω k = kπ/l and ω = ω k+1 ω k = π/l, then taking the limit as L results in c k = 1 φ e iωkt dt = 1 2L 2π F [φ ] k ω, (7) and a Fourier series representation of the disturbance with the form φ = 1 2π ωf [φ ] k e iωkt, (8) k where ωf [φ ] k represents the total area under the spectrum at frequency ω k, in this case a top hat function of width ω and amplitude F [φ ] k. In terms of the PSE formulation this represents the shape and amplitude functions, ωf [φ ] k = φ(x, y; ω k )A(x, ω; ω k )dω. Note that this representation has placed all functional dependence of the spectrum on frequency into the amplitude function. Further, let us assume the amplitude function is separable into a frequency independent amplitude coefficient and a weighting function which describes the frequency content/bandwidth, A(x, ω; ω k ) = A k (x; ω k )W (ω; ω k ). For convenience of explanation, let us introduce the following notation: shape function φ(x, y; ω k ) = φ k, amplitude coefficient A(x, ω k ) = A k and weighting function W (ω; ω k ) = W k. 7

8 1. Discrete Disturbances. Traditionally, to obtain a discrete mode for numerical calculation, a delta function spectrum, W k = δ(ω ω k ), is assumed which leads one to represent a continuous bandwidth disturbance discretely as φ = k φ k A δ e iω kt. (9) It is clear that the amplitude coefficient must represent the area under the spectrum that the kth mode is meant to represent. Thus, to appropriately represent a continuous spectrum discretely, the amplitude of the discretized disturbance(s) must preserve the area under the spectrum. This discrete mode case has been used to approximate experimental observations by considering multiple discrete primary modes as in Kuehl et al Continuous Disturbances. The delta function can be thought of as the limit of a integral-normalized Gaussian function, lim 1 σ 0 2πσ e ω2 2σ 2 = δ(ω). To represent a continuous bandwidth of disturbances, consider the weighting function to be an integral-normalized Gaussian function centered at some frequency, W 0 = 1 σ 0 2π e (ω ω 0 ) 2 2σ 0 2, and the amplitude coefficient represents the area under the Gaussian. In this case, the total disturbance is represented as φ = k φ k A k W k e iω kt. (10) Equivalently, one could consider an amplitude-normalized Gaussian with the amplitude coefficient being interpreted as the peak amplitude of the Gaussian instead of the total area under the Gaussian. Note, the function φ k represents a single shape function and is used to approximate that of the entire spectral peak. Also note, this representation is similar to that of Gaster (1975) and Gaster & Grant (1975), however in this case we will focus on nonlinear interactions. 8

9 3. Nonlinear Terms. In the small amplitude (linear) case, the disturbance amplitudes cancel in the equations of motion. However, as the disturbance amplitude increases at the onset of nonlinearity, the subtle details of disturbance representation (as described above) become important. Nonlinear interactions may be understood by considering the convolution property of the Fourier transform (f g) ( ) = f (ω) g ( ω) dω, and quadratic nonlinear terms may be calculated as follows F [ 1 F [ φ iφ j]] [ = F 1 F [φ i] F [ ]] φ j [( ) ( )] = F 1 φi A i W i φj A j W j = φ i φj A i A j (W i W j ) e iωt dω. (11) In the final step of the numerical treatment, the nonlinear terms are included as forcing for the linear terms, φ 0 A 0 W 0 e iωt dω φ 1 φ2 A 1 A 2 (W 1 W 2 ) e iωt dω, (12) which is often accomplished via the concept of harmonic balancing. Harmonic balancing states that temporal exponentials must balance, otherwise the interactions will average out to zero. This results in a separate equation being written for each frequency (ω k ) of interest. The nonlinear terms are treated as a forcing on the linear equations and iteratively converged. Note that, while quadratic nonlinearities have been explicitly shown, higherorder nonlinearities follow as a straightforward extension. Consider the following two options for disturbance representation: 1. Delta function weighting (traditional discrete mode representation) W 0 = δ(ω ω 0 ) (13) (W 1 W 2 ) = δ(ω (ω 1 + ω 2 )) (14) 9

10 The resulting balance between linear and nonlinear terms is φ 0 A 0 W 0 e iω 0t dω φ 1 φ2 A 1 A 2 e i(ω 1+ω 2 )t, (15) and the traditional harmonic balancing condition of ω 0 = ω 1 + ω 2 results. 2. Integral-Normalized Gaussian weighting with standard deviation σ (modeling a continuous-frequency spectrum) W 0 = 1 e (ω ω 0) 2πσ σ 2 0 (16) (W 1 W 2 ) = 1 2π (σ σ 2 2) [ω (ω1+ω2)] 2 e 2(σ 2 1 +σ2 2) (17) As is typical in wavepacket analysis, the procedure is to Taylor expand the integrals about a central frequency. Expanding the left hand side of relationship 12 (i.e. equation 16) about ω 0 (i.e. ω = ω 0 + δω) yields 1 φ 0 A 0 e (ω ω 2 0) 2σ 0 2 e iωt dω φ 1 0 A 0 e iω 0t 2πσ 2 0 2πσ 2 0 e (δω) 2 2σ 0 2 e iδωt d(δω) (18) and similarly the right hand side of relationship 12 (i.e. equation 17) about ω 1 + ω 2 (i.e. ω = ω 1 + ω 2 + δω) yields 1 φ 1 A 1 φ2 A 2 2π (σ σ2) 2 e i(ω 1+ω 2 )t e (δω) 2 2(σ 2 1 +σ2 2 ) e iδωt d(δω), (19) it is found that the standard harmonic balancing condition again appears (ω 0 = ω 1 + ω 2 ) as well as a second condition: σ 2 0 = σ σ 2 2. This second condition provides information about energy transfer bandwidth. IV. IMPLEMENTATION Implementation of the delta function weighting (Case a) is a straightforward application of harmonic balancing. Assuming a primary mode of frequency, ω 0, harmonics are described as ω i = (i + 1)ω 0. That is, for the first harmonic (i = 1) ω 1 = 2ω 0, and so on. Recalling that complex conjugates and the mean flow distortion (MFD; zero frequency mode) must also be 10

11 accounted for, the nonlinear forcing terms for the primary mode are found by evaluating all the nonlinear terms of those combinations of modes with frequencies equal to the primary mode. i.e. ω 0 = ω 1 ω 0 = 2ω 0 ω 0 and so on. Implementation of the integral-normalized gaussian weighting (Case b), according to eqns. 18 and 19, must account for balancing of both frequency, ω, and disturbance bandwidth, σ. To do this, one must first make some statement about the bandwidths of harmonics relative to the primary disturbance. Consider the generic case of a growing nondispersive primary wave packet that drives harmonic growth (i.e. no external noise is forcing growth). In this case, it is reasonable to assume a typical energy cascade (which follows from eqns. 18 and 19): given a primary disturbance bandwidth (ω 0, σ 0 ), the first harmonic will be driven at a central frequency of ω 1 = ω 0 + ω 0 = 2ω 0 with a bandwidth of σ1 2 = σ0 2 + σ0 2 or σ 1 = 2σ 0. It follows that ω i = (i + 1)ω 0 and σ i = i + 1σ 0. If one now considers the standard forcing sequence of primary mode forcing first harmonics, the primary and first harmonic mode combinations forcing second harmonics and so on, then the leading coefficients in eqns. 18 and 19 cancel and there is no apparent difference between the delta function weighting formulation and the integral-normalized gaussian function. With the exception that the harmonics must be interpreted with the appropriate σ i. However, we have not yet considered the feedback of the harmonics onto the primary mode (so-called nonlinear saturations, which is distinct from nonlinear detuning, Kuehl et al. 2014; Kuehl and Paredes 2016, Kuehl 2017). Consider the example above where the primary mode is forced by the primary complex conjugate and the first harmonic, ω 0 = ω 1 ω 0. The bandwidths have been set by the initial energy cascade and do not balance in this case. Instead, consideration of the coefficients in eqns. 18 (1/ 2πσ0) 2 and 19 (1/ 2π (σ1 2 + σ0) 2 = 1/ 2π3σ0), 2 result in a nonlinear coupling coefficient of 1/ 3. That is, the traditional delta function weighting approach has over predicted the amplitude of the leading order nonlinear feedback terms onto the primary mode by approximately 70%. Graphically, this situation is illustrated in figure 2. Notice that the nonlinear feedback term is forcing a larger bandwidth than that of the primary mode. Not all of the forcing is applied directly to the primary mode, but instead the additional 70% is forcing side lobes. These side lobes are fundamentally different from the primary disturbance as they are forced modes which are not extracting their energy from the mean flow. Physically, 11

12 Amplitude Frequency Distributions in Nonlinear Stability Applications 1 Primary Feedback Side Lobe Side Lobe Frequency (khz) FIG. 2. A schematic representation of finite-bandwidth nonlinear feedback forcing of a primary mode (lighter color) from harmonics (darker color). these side lobes represent spectral broadening which is often observed in experiment (figure 1) and is fundamental to the modal growth transition to turbulence. V. EXAMPLE: HYPERSONIC PURDUE FLARED CONE A. Method To illustrate both the importance of and implications of clearly identifying and appropriately representing frequency bandwidth in nonlinear numerical computations, the Mach 6 Purdue flared cone at zero yaw conditions will be modeled. Prior experimental and numerical investigations (Wheaton et al. 2009, Chynoweth 2015, and McKiernan et al Purdue flared-cone experiment; Hofferth et al flared cone experiment; Kocian et al. 2013, Perez et al Purdue flared-cone LST, PSE; Huang and Zhong 2012, Fasel et al Purdue flared-cone direct numerical simulation [DNS]) find appreciable Mackmode energy in frequency bands of approximately 50 khz width. The model used has a nose bluntness of meters and a radius of curvature of 3 meters for the flare. The total length 12

13 of the model is 0.49 meters, with a base diameter of meters. The test conditions for the computations correspond to a freestream Mach number M = 6, zero angle of attack, freestream static temperature T = 52.8 K, and freestream static pressure P = Pa abs, which results in a unit Reynolds number Re = per meter. A wall boundary condition of T wall = 300 K is imposed. The steady, laminar, basic-state solution is computed using GASP (General Aerodynamic Solver Program), with the mesh created in Pointwise. Perez et al. (2012), with more thorough discussion given by Reed et al. (2014), confirmed that the grid is converged by showing the invariance of boundary-layer stability results for different architectures. The Purdue cone is a good candidate for studying the Mack mode instability, in that it is specifically designed so that the boundary-layer height remains fairly constant along the length of the cone, thus isolating Mack mode disturbances whose frequencies tend to tune to the local boundary-layer thickness. To solve the disturbance equations, the nonlinear parabolized stability equations (NPSE) method (Herbert 1997) will be implemented via an in-house extension of the original JoKHeR code (Kuehl et al. 2012). This code is written for compressible flow in a general orthogonal curvilinear coordinate system and uses the primitive variable formulation. Currently, an ideal gas is considered, with Sutherland s law and Keyes formula representation of viscosity and thermal conductivity, respectively. The code is capable of running quasi 3-D NPSE; that is, it calculates first- and second-mode and crossflow stability in a 3-D boundary layer, but is restricted to march along a predetermined path. This marching is accomplished through application of a normalization condition to account for the wave number variations in the marching direction (Kuehl et al. 2014), however the wave number normal to the marching path must be specified. The curvature of both a generic 3-D geometry as well as that of the marching trajectory on the 3-D geometry are accounted for analytically by locally fitting a torus to the surface of interest. This is particularly convenient for bodies of revolution such as cones. B. Impact of New Formulation NPSE calculation where carried out on the above described Purdue flared-cone basic state calculation. In each case, the disturbances were initiated with linear stability eigen-modes of amplitude 2.5e 8 based on the non-dimensional temperature disturbance, slightly upstream 13

14 of the neutral point. This amplitude was given to the primary mode as well as the side lobes, while the amplitudes of harmonics and MFD are determined by the nonlinear interactions. The initial amplitude was chosen as it results in saturation near the end of the cone. The bandwidth was chosen based on the bandwidth of unstable Mack-modes as determined by LST calculations. Note, the actual value of the bandwidth does not affect the calculations. It is simply the fact that a bandwidth exists, that necessitates the wave packet formulation over the traditional formulation. The chosen bandwidth does however control the frequency at which side lobes are specified. In this case, side lobes are placed at 265 and 305 khz where natural instability is no longer present. This is important, as the side lobes represent forced modes which would otherwise not be amplified or are only weakly amplified compared to the primary mode. Here only two side lobes are included, but more can be added if desired. For comparison, calculations include: 1) A single primary mode with harmonics using the traditional delta function weighting. 2) Three primary modes with harmonics using the traditional delta function weighting. 3) Three primary modes with harmonics using the new Gaussian weighting (nonlinear coupling coefficient incorporated for the leading order, quadratic, nonlinear feedback terms). Summarized in figure 3, are the key results of this work. In particular, the maximum saturation amplitude reached by the 285 khz primary Mack-mode, the MFD peak amplitude and the peak side lobe amplitudes. Note, the total initial disturbance energy in cases 2 and 3 are identical. Case 1 has 1/3 the total initial energy of cases 2 or 3. The most physically relevant comparison is between cases 2 and 3 but case 1 in considered for completeness. It is found that the new formulation: 1) results in a 34% increase in primary mode streamwise velocity disturbance amplitude. 2) results in a 74% increase in MFD streamwise velocity disturbance amplitude. 3) results in a 4300% increase in side lobe streamwise velocity disturbance amplitude. This ultimately results in a saturation amplitude of approximately 13% total streamwise velocity disturbance amplitude (Primary mode plus side lobes, neglecting the MFD contribution) which represent a 90% increase in saturation amplitude prediction over the traditional formulation. This is a significant improvement in both saturation amplitude prediction as well as physical disturbance representation (via accounting for spectral broadening). Please note, in general the boundary layer stability and transition problem has a multitude of open questions associated with the physics of the transition problem. This 14

15 manuscript address one of the open questions, but many others remain. The experiments of Chynoweth (2015) and McKiernan et al. (2015) suggest saturation amplitude of 25% based on wall pressure measurements. Plotted in figure 4 are overlaid Gaussian and discrete methods for both maximum RMS disturbance velocity and RMS disturbance wall pressure as well as total disturbance wall pressure (primary mode plus side lobes) compared with experimental data. Note, the experimental data has been visually extracted from Chynoweth (2015, figure 3.26) and are shifted by 1e 6 in Re x. Physically, this shift is consistent with larger experimental initial disturbance amplitudes than applied here numerically and hence earlier disturbance growth. The calculations provided here are not 1-to-1 with experimental data and should not be taken as such. They simply illustrate that the new formulation more accurately represents the physics of the transition problem and ultimately more accurately represent the experimental data. As an example of other relevant open questions, Kuehl and Parades (2016) have recently shown that the presence of spanwise roll-type vorticies fundamentally changes the behavior of acoustic disturbances in hypersonic boundary layers. i.e. the acoustic disturbances appear to behave as they are trapped in a 2D waveguide rather than a 1D waveguide. This calls into significant question the appropriateness of a traditional PSE approach when spanwise disturbances are present, which may be the case in the experimental data provided. VI. CONCLUSION A new PSE formulation (wave packet formulation) has been presented which accounts for finite-bandwidth disturbance affects on hypersonic boundary layer laminar-turbulent transition. In contrast to traditional discrete mode calculations, this formulation identifies a new nonlinear coupling coefficient which follows from the wave packet representation and which owes its existence to the finite nature of disturbance bandwidth. It is found that the traditional discrete mode calculations over estimate the nonlinear feedback by approximately 70%. This over estimation results in too much nonlinear saturation and ultimately a reduction in peak disturbance amplitude reached. The wave packet formulation accounts for the division of nonlinear feedback terms into primary mode forcing and side lobe forcing. Thus, spectral broadening is accounted for which yield a more physically consistent disturbance representation. 15

16 RMS(u' max ) Frequency Distributions in Nonlinear Stability Applications Linear 285 khz MFD Ampitude e -7 # khz khz 265 khz 305 khz MFD khz 265 khz 305 khz MFD Frequency (khz) Streamwise Distance (m) FIG. 3. NPSE simulations of a Mach 6 Purdue flared-cone. Shown are maximum RMS streamwise velocity disturbance amplitudes (right) and disturbance representation (left). Top: single discrete primary mode. Middle: three discrete primary modes. Bottom: three Gaussian primary modes. Acknowledgements: Dr. Kuehl gratefully acknowledges support from the AFOSR Young Investigator Program via Grant FA Dr. Steve Schneider and Brandon Chynoweth are thanked for fruitful discussions and access to experiment data. Also note that this work stems from ideas develop under the AFOSR/NASA National Center for Hypersonic Research in Laminar-Turbulent Transition through Grant FA , with Dr. Helen Reed at Texas A&M University. REFERENCES 1 Bertolotti, F.P. (1991) Compressible Boundary Layer Stability Analyzed with the PSE Equations. AIAA

17 RMS(u' RMS ) RMS(P' wall ) RMS(P' wall ) Frequency Distributions in Nonlinear Stability Applications G 265 G 305 G MFD G 285 D 265 D 305 D MFD D P total exp Streamwise Distance (m) Streamwise Distance (m) Re x #10 6 FIG. 4. NPSE simulations of a Mach 6 Purdue flared-cone. Shown are: Left) overlaid Gaussian and discrete maximum RMS of streamwise velocity disturbance amplitudes. Middle) overlaid Gaussian and discrete total RMS of wall pressure disturbance amplitudes. Right) Total RMS of wall pressure disturbance amplitudes with experimental data. Note, the experimental data has been shifted by 1e 6 in Re x. 2 Bertolotti, F. P., Herbert, Th. & Spalart, P. R. (1992) Linear and nonlinear stability of a blasius boundary layer. Journal of Fluid Mechanics, 242, Chang, C.L., Malik, M.R., Erlebacher, G., Hussaini, M.Y. (1991) Compressible stability of growing boundary layers using parabolized stability equations. AIAA Chang, C.L., Vinh, H., Malik, M.R. (1997) Hypersonic boundary-layer stability with chemical reactions using PSE. AIAA Chang, C.-L. (2004) The Langley Stability and Transition Analysis Code (LASTRAC) Version 1.2 User Manual. NASA/TM

18 6 Chynoweth, B. (2015) A new roughness array for controlling the nonlinear breakdown of second-mode waves at Mach 6. Masters Thesis, Purdue University. 7 Fasel, H. F., Sivasubramanian, J. & Laible, A. (2015) Numerical Investigation of Transition in a Flared Cone Boundary Layer at Mach 6. Procedia IUTAM Vol. 14, pp Fedorov, A., & Tumin, A., (2011) High-Speed Boundary-Layer Instability: Old Terminology and a New Framework. AIAA Journal, 49, No. 8, Fedorov, A., (2011) Transition and Stability of High-Speed Boundary Layers. Annual Review of Fluid Mechanics, 43, Gaster, M. (1974) On the effects of boundary-layer growth on flow stability. Journal of Fluid Mechanics, Vol. 66, 3, pp Gaster, M. (1975) A theoretical mode for a wave packet in the boundary layer on a flat plate. Proc. R. Soc. Lond. A Vol. 347, doi: /rspa Gaster, M. and I. Grant (1975) An Experimental Investigation of the Formation and Development of a Wave Packet in a Laminar Boundary Layer. Proc. R. Soc. Lond. A, Vol. 347, doi: /rspa Haynes, T.S., and Reed, H.L., (2000) Simulation of Swept-Wing Vortices Using Nonlinear Parabolized Stability Equations. Journal of Fluid Mechanics, Vol. 405, pp Herbert, T., and F. P. Bertolotti, (1987) Stability analysis of nonparallel boundary layers. Bulletin of American Physical Society, 32, Herbert, Th. (1994) Parabolized stability equations. AGARD Rep. No. 793 (Special course on progress in transition modelling), Von Karmon Inst., Rhode-St.-Genese, Belg. 16 Herbert, T., (1997) Parabolized Stability Equations, Annual Review of Fluid Dynamics, Vol. 29, pp Hofferth, J. W., S. W. Saric, J. Kuehl, E. Perez, T. Kocian, H. L. Reed, (2013) Boundary- Layer Instability & Transition on a Flared Cone in a Mach 6 Quiet Wind Tunnel, International Journal of Engineering Systems Modelling and Simulation, Vol. 5, 1/2/3. 18 Huang, Y., and Zhong, X., (2012) Numerical study of boundary-layer receptivity on blunt compression cones in Mach-6 flow with localized freestream hot-spot perturbations, RTO- MP-AVT-200 Hypersonic Laminar-Turbulent Transition, San Diego, Paper Johnson, H.B., Seipp, T.G., Candler, G.V., (1998) Numerical Study of Hypersonic Reacting Boundary Layer Transition on Cones. Physics of Fluids, 10 (10):

19 20 Johnson, H.B., Candler, G.V. (2005) Hypersonic Boundary Layer Stability Analysis using PSE-Chem. AIAA Joslin, R.D., Streett, C.L., Chang, C.L. (1992) 3-D incompressible spatial direct numerical simulation code validation study: a comparison with linear stability and parabolic stability equation theories in boundary-layer transition on a flat plate. NASA TP Kocian, T. S., E. Perez, N. B. Oliviero, J. Kuehl, H. L. Reed, (2013) Hypersonic Stability Analysis of a Flared Cone. AIAA Kuehl, J., E. Perez, and H. L. Reed (2012) JoKHeR: NPSE Simulations of Hypersonic Crossflow Instability. AIAA Kuehl, J., Reed, H. L., Kocian, T. S., & Oliviero, N. B. (2014). Bandwidth Effects on Mack-Mode Instability. AIAA Kuehl, J. & Paredes (2016) Görtler modified Mack-modes on a hypersonic flared cone. AIAA Kuehl, J. (2017) Nonlinear Saturation versus Nonlinear Detuning: Quantification on a Mach 6 Flared Cone, AIAA Li, F., and M. R. Malik, (1996) On the Nature of PSE Approximation Theoretical and Computational Fluid Dynamics, 8: Chokani, N. (2005) Nonlinear evolution of Mack modes in a hypersonic boundary layer. Physics of Fluids, 17, ; doi: / Li, F., Choudhari, M., Chang, C.-L., and White, J., (2010) Analysis of Instabilities in Non- Axisymmetric Hypersonic Boundary Layers over Cones. 10th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, 28 June-1 July 2010, Chicago, Illinois. 30 Mack, L.M., (1984) Boundary Layer Stability Theory. In AGARD Special Course on Stability and Transition of Laminar Flow, pp Malik, M.R., (2003) Hypersonic Flight Transition Data Analysis using Parabolized Stability Equations with Chemistry Effects, Journal of Spacecraft and Rockets, Vol. 40, No. 3, pp McKiernan, G., Chynoweth, B. & Schneider, S. P. (2015) Boundary-Layer Transition Experiments in the Boeing/AFOSR Mach 6 Quiet Tunnel. AIAA Morkovin, M. V. (1969) Critical evaluations of transition from laminar to turbulent shear layers with emphasis on hypersonically traveling bodies. Tech. Rep. Research Institute for Advances Studies (AFFDL-TR ). 19

20 34 Morkovin, M., Reshotko, E., and Herbert, T. (1994) Transition in Open Flow Systems: A Reassessment. Bulletin of the American Physical Society, 39, Nayfeh, A. H. (2004) Perturbation Methods. Wiley-VCH Verlag GmbH & Co. KGaA. 36 Paredes, P., Theofilis, V., Rodriguez, D., Tendero, J. A. (2011) The PSE-3D instability analysis methodology for flows depending strongly on two and weakly on the third spatial dimension. AIAA Perez, E., T. S. Kocian, J. Kuehl, H. L. Reed, (2012) Stability of Hypersonic Compression Cones. AIAA Reed, H. L., Saric, W. S. & Arnal, D. (1996) Linear stability theory applied to boundary layers. Annual Review of Fluid Mechanics, 28, Reed, H. L., Perez, E., Kuehl, J., Kocian, T., & Oliviero, N. (2014) Verification and Validation Issues in Hypersonic Stability and Transition Prediction. Journal of Spacecraft and Rockets, 52 pp Reshotko, E., (2008) Transition Prediction: Supersonic and Hypersonic Flows. NATO RTO-EN-AVT-151: Advances in Laminar-Turbulent Transition Modelling, Lecture Series held at the von Karman Institute, Rhode St. Gense, Belgium, Saric, W., Reed, H., & Kerschen, E., (2002) Boundary-Layer Receptivity. Annual Review of Fluid Mechanics, 34, Saric, S. W., Reed, H. L. & White, E. B. (2003) Stability and transition of threedimensional boundary layers. Annual Review of Fluid Mechanics, 35, Schmid, P., & Henningson, D. (2001) Stability and Transition in Shear Flows. Springer- Verlag, New York. 44 Sivasubramania, J. & Fasel, H. F. (2013) Direct Numerical Simulation of Controlled Transition In a Boundary Layer on a Sharp Cone at Mach 6. AIAA Sivasubramanian, J. & Fasel, H. F. (2014) Numerical investigation of the development of three-dimensional wavepackets in a sharp cone boundary layer at Mach 6. Journal of Fluid Mechanics, 756, pp doi: /jfm Stuckert, G., and Reed, H., (1994) Linear Disturbances in Hypersonic, Chemically Reacting Shock Layers. AIAA Journal, 32, No. 7, Theofilis, V. (2011) Global linear instability. Annual Review of Fluid Mechanics, 43,

21 48 Wheaton, B. M., T. J. Juliano, D. C. Berridge, A. Chou, P. L. Gilbert, K. M. Casper, L. E. Steen, S. P. Schneider, (2009) Instability and transition measurements in the Mach-6 quiet tunnel. AIAA Zhong, X., and Wang, X., (2012) Direct Numerical Simulation on the Receptivity, Instability, and Transition of Hypersonic Boundary Layers. Annual Review of Fluid Mechanics, Vol. 44, No. 1, pp. 527?561. doi: /annurev-fluid

22

23 Amplitude 1 Primary Feedback Side Lobe Side Lobe Frequency (khz)

24 RMS(u' max ) Linear 285 khz MFD Ampitude e -7 # khz khz 265 khz 305 khz MFD khz 265 khz 305 khz MFD Frequency (khz) Streamwise Distance (m)

25 RMS(u' RMS ) RMS(P' wall ) RMS(P' wall ) 285 G 265 G 305 G MFD G 285 D 265 D 305 D MFD D P total exp Streamwise Distance (m) Streamwise Distance (m) Re x #10 6