Toward robust prediction of crossflow-wave instability in hypersonic boundary layers

Accepted Manuscript

Toward robust prediction of crossflow-wave instability in hypersonic boundary layers Matthew T. Lakebrink, Pedro Paredes, Matthew P. Borg PII: DOI: Reference:

S0045-7930(16)30373-5 10.1016/j.compfluid.2016.11.016 CAF 3337

To appear in:

Computers and Fluids

Received date: Revised date: Accepted date:

11 February 2016 6 August 2016 26 November 2016

Please cite this article as: Matthew T. Lakebrink, Pedro Paredes, Matthew P. Borg, Toward robust prediction of crossflow-wave instability in hypersonic boundary layers, Computers and Fluids (2016), doi: 10.1016/j.compfluid.2016.11.016

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Highlights • Prediction techniques for crossflow waves are studied on an elliptic cone

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at Mach 6

• Growth rate is sensitive to analysis technique, wave angle and phase speed are not

• Wave properties computed using LASTRAC are validated against test

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data

• Traditional stability analysis techniques are compared to Spatial BiGlobal analysis

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and costs less

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• Surface marching PSE is in excellent agreement with Spatial BiGlobal

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Toward robust prediction of crossflow-wave instability in hypersonic boundary layers

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Matthew T. Lakebrinka , Pedro Paredesb , Matthew P. Borgc a

b

Boeing Research and Technology, Saint Louis, MO 63042, USA School of Aeronautics, Universidad Polit´ecnica de Madrid, Madrid 28040, Spain c Air Force Research Laboratory, WPAFB, OH 45433, USA

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Abstract

Traveling and stationary crossflow-wave instabilities in a laminar Mach 6 boundary layer are investigated on a 38.1% scale model of the Fifth Hypersonic International Flight Research Experiment (HIFiRE-5) elliptic cone at

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zero angle of attack and yaw. The Langley Stability and Transition Analysis Code (LASTRAC) was used to analyze the crossflow dominated boundary

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layer in the mid-span region near the downstream end of the model. Disturbance growth rates, wave angles, and phase speeds are computed with

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LASTRAC using quasi-parallel Linear Stability Theory (LST), Linear Parabolized Stability Equations (LPSE), and two-plane or surface marching LPSE

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(2pLPSE). The predicted wave angles and phase speeds are validated using experimental data, and are found to be in better agreement than previous computations. Further numerical analysis is conducted using the Spatial

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BiGlobal technique (SBG), which simultaneously accounts for wall-normal and spanwise gradients in the mean boundary layer at a particular axial station. For the first time in the literature, a comparison is made between Email address: [email protected] (Matthew T. Lakebrink) Preprint submitted to Computers & Fluids

December 1, 2016

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crossflow wave growth rates computed using LST, LPSE, and 2pLPSE and those computed using SBG, accounting for curvature and geometric diver-

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gence of the elliptic cone. The agreement between LST, LPSE, and SBG is fair at best, but excellent agreement is realized between 2pLPSE and SBG.

This result constitutes a co-verification of the LASTRAC and SBG stability

codes, and provides evidence that 2pLPSE accurately models the physics of

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the traveling-crossflow instability.

1. Introduction

Computational boundary layer stability analysis has been steadily evolving over the last 50 years. Early examples of stability analysis featured local

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solutions of the Orr-Sommerfeld equations for incompressible boundary layers on simple geometries such as the flat plate or channel; see Schmid and

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Henningson (2001) for a review.

The advent of LPSE (Herbert, 1997) allowed analysis to efficiently account for boundary layer growth and non-parallel effects via marching. This

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was particularly important for boundary layers undergoing rapid streamwise variation, e.g. near the leading edge of a lifting surface.

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Another step away from local and toward global stability analysis (The-

ofilis, 2011) was taken with the introduction of the Spatial BiGlobal and

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LPSE-3D methods (Paredes et al., 2011; De Tullio et al., 2013). Rather than solving for eigenvalues by using the boundary layer profiles at a single location (e.g. classical LST), SBG takes into account the flowfield within an entire crosswise slice of the mean flow. This is particularly important in regions where the mean flow exhibits strong spanwise or azimuthal variation, 3

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such as in the vicinity of the large-scale counter-rotating vortex pair that develops along the centerline of an elliptic cone or along the leeward ray of a

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circular cone at non-zero angle of attack. The LPSE-3D technique extends the classical line-marching LPSE for base flows with a single strongly inhomogeneous direction, namely the wall-normal direction, to base flows with a mild variation in the streamwise coordinate and strong gradients in the

other two spatial direction, i.e., the wall-normal and spanwise directions in

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boundary layer problems.

Development of Non-Linear PSE (NLPSE) provided a means for studying the interactions of different Fourier modes and their tendency to distort the mean flow. This helped to further the understanding of how disturbances behave just prior to transition, as their amplitudes become very large. NLPSE

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has been used to study the highly non-linear nature of crossflow waves, and has been successfully compared to experimental data (see Haynes and Reed,

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2000; Reed and Saric, 2003) when accurate initial conditions have been available. NLPSE-3D (Paredes et al., 2015b) has also emerged as a novel approach

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for extending NLPSE to complex three-dimensional flows. Laminar-to-turbulent boundary layer transition dominated by the cross-

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flow instability can play a significant role in the performance and survivability of hypersonic air vehicles. A relevant model problem for studying this phe-

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nomenon is the HIFiRE-5 (Dolvin, 2008) elliptic cone forebody. The HIFiRE5, which showcases several instabilities including attachment and centerline modes, second Mack modes (Mack, 1984), and crossflow, was flight tested with partial success in April 2012 (Juliano et al., 2015). Several wind tunnel tests (see Juliano and Schneider, 2010; Borg, 2013) focusing on HIFiRE-5 4

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have shown that, over the acreage between the attachment line and centerline, crossflow waves are the dominant mechanism contributing to transition.

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Stability analyses (see Choudhari et al., 2009; Gosse et al., 2014; Paredes et al., 2015a) of varying fidelity have achieved good qualitative agreement with the wind tunnel tests by predicting highly amplified crossflow distur-

bances over the acreage. In a numerical study performed by Li et al. (2012), the peak frequency for crossflow disturbances computed using LST was found

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to be in good agreement with the peak from experimental (Borg et al., 2011)

power spectral density of wall pressure fluctuations. A comparison presented by Borg (2013) (see also Borg et al. (2015)) exhibits reasonable agreement between experimental results and LST for crossflow wave angles and phase speeds.

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While significant steps have been taken toward a more complete understanding of crossflow instability in hypersonic flow, major deficiencies still

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exist in the validation of numerical methods. Owing partially to the nonlinear nature of large-scale crossflow vorticies, these deficiencies are most

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evident in the lower order linear methods of stability analysis (e.g. LST and LPSE). In light of the myriad computational tools available for stability

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analysis, it is essential to obtain a firm understanding of not only the range of applicability for each tool, but also which tool is most appropriate for

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predicting a given phenomenon. As the modeling techniques change and improve, it is important that

the best practices associated with their usage are developed and remain up to date. This can be accomplished through code verification and validation against high quality experimental datasets. The purpose of the present 5

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study is to provide comparisons between four numerical techniques and experimental data. These comparisons serve as partial validation of several of

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the numerical techniques, and simultaneously verify the two codes used for the numerical analysis.

The following sections provide information about the approaches used for analysis, and the results they produced. Section 2 discusses various aspects

of the numerical analysis, including details about the laminar mean flow, and

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differences between the examined stability analysis techniques. Section 3 focuses on the experiment, and provides information about the test article, the instrumentation used for data acquisition, and the data reduction techniques.

Comparisons and examination of the numerical and experimental results are

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2. Numerical methods

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provided in section 4, and concluding remarks are given in section 5.

Modal analysis of boundary layer stability takes place primarily in two stages. The first consists of developing a high quality computational grid,

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and then using it along with a flow solver to obtain an approximate solution to the laminar Navier-Stokes equations. The quality of this mean flow is

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paramount because the boundary layer profiles (velocity, density, and temperature) along with their first two derivatives are input directly into the

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linearized Navier-Stokes (LNS) equations solved by the stability code. The second stage is to superimpose disturbances of given frequencies onto the aforementioned boundary layer profiles from the mean flow solution. In the present study, the sensitivity of stability analysis results with respect to the governing disturbance equations is investigated. Four sets of governing equa6

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tions are studied which correspond to the quasi-parallel LST, LPSE, 2pLPSE, and SBG analysis techniques.

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2.1. Laminar mean flow computation

The computational grid used in the present study was shock-fitted and

composed entirely of hexahedral elements. Shock-fitting was achieved via

an iterative adaptation process. In this process, the Boeing Computational Fluid Dynamics (BCFD) flow solver (Mani et al., 2004) was used to obtain

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second order accurate solutions to the Navier-Stokes equations. The Modular

Aerodynamic Design Computational Analysis Process (MADCAP) was then used to fit the grid to the shock surface described by the BCFD solution. Once fitting to the shock was complete, the solution was fully converged so

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that it could be used as input for stability analysis.

The conditions simulated for the mean flow were chosen to correspond to

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an experimental run performed in the Purdue Mach 6 quiet tunnel, and are presented in table 1. The number of grid points in the axial, wall-normal, and azimuthal directions for one quadrant of the geometry are 908, 400, and

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500 respectively. Owing to the doubly symmetric nature of the problem, two symmetry boundaries were prescribed on planes containing the semi-minor

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and semi-major axes.

The gas was modeled as calorically perfect air with γ = 1.4 and R = 287

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m2 /s2 /K, where γ and R are the ratio of specific heats and specific gas constant respectively. Sutherland’s law was used to compute viscosity from the local temperature, using a Sutherland’s temperature of 291.15 K. The thermal diffusivity was then determined by assuming a Prandtl number of 0.72. Second order spatial accuracy was achieved using the Roe flux-difference 7

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M∞

α [◦ ]

β [◦ ]

U∞ [m/s]

P∞ [Pa]

T∞ [K]

Twall [K]

Re/m

6.0

0

0

850.4

436.3

50

300

8.1 × 106

From left to right:

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Table 1: Run conditions for the laminar mean flow simulation.

freestream Mach number, angle of attack, angle of yaw, freestream velocity, freestream

pressure, freestream temperature, wall temperature, and freestream unit Reynolds num-

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ber.

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Figure 1: Five inviscid streamline paths used for marching during stability analysis.

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scheme (Roe, 1981) in conjunction with the Koren flux limiter. Numerical dissipation was controlled via a grid resolution study, which revealed that the dimensions reported in table 1 were sufficient to achieve grid indepen-

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dent mean flow profiles and stability analysis results.

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2.2. Stability analysis

Hydrodynamic stability analysis theory studies the behavior of unper-

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turbed flow fields upon the introduction of small-amplitude perturbations. The development in time and space of small-amplitude perturbations superimposed upon a given flow can be described by the LNS equations. Linearization of the equations of motion is performed around a laminar steady flow, ¯ = (¯ here denoted as base flow, q ρ, u¯, v¯, w, ¯ T¯)T . The small-amplitude pertur8

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Figure 2: Mach-number contours and surface-limiting streamlines depicting the Mach 6 laminar-mean flow over the HIFiRE-5 elliptic cone.

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˜ (x, y, z, t) = bations superimposed on the base flow are denoted as the vector q (˜ ρ, u˜, v˜, w, ˜ T˜)T , comprising the perturbation functions of density, velocity

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components and temperature. Therefore, flow quantities, q(x, t), are decomposed according to

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¯ (x) + ε˜ q(x, t) = q q(x, t),

ε  1.

(1)

Linear stability analysis for the HIFiRE-5 laminar mean flow is performed

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using the LASTRAC code (Chang, 2004) and an in-house developed code for multi-dimensional stability analysis (Paredes, 2014; Paredes et al., 2013).

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The effects of both compressibility and wall-curvature are accounted for in the stability analysis, except for the case of quasi-parallel LST where wall-curvature is neglected. The mean flow was analyzed using LST, LPSE, 2pLPSE, and SBG, and these approaches are briefly introduced here for the purpose of completeness. 9

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LST analysis assumes the base flow to be locally homogeneous along the streamwise and spanwise directions. Therefore, the disturbance quantities

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are written as

˜ (x, y, z, t) = q ˆ (y) exp[i(αx + βz − ωt)] + c.c., q

(2)

where, in the present spatial analysis framework, ω denotes the angular fre-

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quency, β is a real value denoting the spanwise wavenumber, and α is a complex value, the real part of which is the streamwise wavenumber and the imaginary part is the growth/damping rate. A negative value of αi in˜ in space, while αi > 0 denotes decay of q ˜ dicates exponential growth of q in space. Upon introduction of the LST disturbance equation (2) into the

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LNS equations, an ODE-based generalized eigenvalue problem (EVP) is recovered. The resulting EVP is nonlinear on eigenvalue α, but it is converted

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into a linear eigenvalue problem larger in size by using the companion matrix method (Theofilis, 1995; Bridges and Morris, 1984). This method consists in

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ˆ + = [ˆ the introduction of an auxiliary vector, q ρ, uˆ, vˆ, w, ˆ Tˆ, αˆ u, αˆ v , αw, ˆ αTˆ]T . The complex eigenvalues, α, and the related complex eigenvectors, i.e., the

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ˆ (y), are sought for a given frequency one dimensional amplitude functions, q and position.

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On the other hand, the stability of weakly non-parallel flows, in which

the dependence of the base flow on the streamwise coordinate, x, is much weaker than that along the wall-normal direction, y, can be studied by the

LPSE. Unlike the LST problem based on the solution of an EVP, LPSE solve a marching integration of the LNS equations along the streamwise spatial 10

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direction. In the LPSE context, the disturbance quantities are written as Z ˜ (x, y, z, t) = q ˆ (x, y) exp[i( α(x0 ) dx0 + βz − ωt)] + c.c., q (3) x

0

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where the superscript denotes the integrated variable.

The 2pLPSE, or surface-marching approach additionally takes into ac-

count the mild spanwise variations of three-dimensional boundary layers.

Details on the development of this method within LASTRAC, how it differs

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from LST and LPSE, and its application to a swept wing subject to transonic and supersonic flows are presented by Chang (2004). Similar to the distur-

bance form of the LPSE (3), a complex spanwise wave number is introduced along with functional dependence of wave numbers and shape functions on

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the spanwise coordinate, resulting in Z Z 0 0 ˜ (x, y, z, t) = q ˆ (x, y, z) exp[i( α(x , z) dx + β(x, z 0 ) dz 0 − ωt)] + c.c.. (4) q x

z

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The LST, LPSE, and 2pLPSE methods implemented in LASTRAC are solved along wall-normal grid lines. For the present study, the wall-normal grids were generated by LASTRAC using 81 points, with clustering at the

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wall and near the edge of the boundary layer. The base-flow solution was then interpolated onto the wall-normal grid, and the governing disturbances

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equations were solved. Further details on the discretization methods used by LASTRAC are presented in Chang (2004).

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LPSE and 2pLPSE analyses are initialized at the stability neutral point

using a solution obtained from the LST analysis. Non-parallel solutions are then obtained at the neutral point and subsequently marched downstream. Several viable marching path options exist. Chang (2004) explored marching along inviscid streamlines and group velocity lines, and found both ap11

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proaches to be comparable. Oliviero et al. (2015) demonstrated the use of a generalized inflection point method for computing marching paths for

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stationary-crossflow waves. This approach produced paths that were in good agreement with those computed from Direct Numerical Simulation (DNS).

Li et al. (2012) showed that marching along the inviscid streamline compared well with experiment, and so the inviscid streamline is used here. Analysis is

performed by marching along inviscid streamlines chosen such that they cov-

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ered the crossflow-dominated mid-span region of the model. Figure 1 shows a plan-view of the geometry overlaid with five of the marching paths used for stability analysis. The inboard curving of the streamlines is due to the highly three-dimensional mean flow, illustrated in Figure 2. The SBG stability analysis approach extends the spatial quasi-parallel

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LST for base flows with a single strongly inhomogeneous direction to base flows with two strongly inhomogeneous directions, while the third direction,

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i.e. streamwise direction in boundary layer problems, is considered locally homogeneous or quasi-parallel. The disturbance quantities are then written

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as

˜ (x, y, z, t) = q ˆ (y, z) exp[i(αx − ωt)] + c.c.. q

(5)

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A two-dimensional PDE-based generalized EVP is obtained upon introduction of the disturbance equation (5) into the LNS equations. The resulting

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two-dimensional partial PDE-based generalized EVP is nonlinear on eigenvalue α, but, as in the LST method, it is converted into a linear eigenvalue problem larger in size by using the companion matrix method (Theofilis, 1995; Bridges and Morris, 1984). In the spatial context, the complex eigenvalues, α, and the related complex eigenvectors, i.e., the two-dimensional ampli12

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ˆ (y, z), are sought for a given frequency ω and axial position tude functions, q x. Note that the corresponding α (axial wavenumber and growth/damping

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rate) does not depend on z, contrary to the other methods. SBG analysis simultaneously resolves the two inhomogeneous spatial directions on a plane. Therefore, a coordinate transformation is needed to map the computational coordinate system (ξ, η, ζ) into the desired physical coor-

dinate system (x, y, z). A modified analytical confocal elliptic transformation

x = ξ,

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is used as explained in Paredes and Theofilis (2015) and is written as follows, y = cξ sinh(η0 + sp(ζ)η) sin ζ,

z = cξ cosh(η0 + sp(ζ)η) cos ζ, (6)

where c sets the half angle of the cone minor-axis, θ, by c = tan θ/ sinh η0 ,

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sp(ζ) is a fitting polynomial function chosen to truncate the domain below the shock layer, and η0 is a parameter controlling the aspect ratio (AR) of the

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cone, η0 = atanh (1/AR). The (η, ζ) directions are discretized in a coupled manner as discussed by Paredes et al. (2013); Paredes and Theofilis (2015). The base-flow variables are interpolated from the DNS solution into this grid.

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The elliptic BiGlobal EVP problem is complemented with adequate boundary conditions for the disturbance variables. Dealing first with the azimuthal

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direction, ζ, the symmetries of the problem at hand, namely zero angle of attack and yaw, are exploited in order to reduce the computational require-

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ments and only one quarter of the cone is discretized. Symmetric or antisymmetric boundary conditions can be imposed at ζ = π/2 and ζ = 0 (Paredes and Theofilis, 2015; Paredes et al., 2015a). This work focuses on crossflow instabilities, whose amplitude function variables decay far from the minoraxis or major-axis meridian. Consequently, the results are insensitive to the 13

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selection of symmetric or antisymmetric boundary conditions. In this work, symmetric boundary conditions are used. The four methods, i.e., LST, LPSE,

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2pLPSE, and SBG, follow the same strategy for the wall-normal direction. Specifically, the perturbations are forced to decay through the imposition of homogeneous Dirichlet boundary conditions are used at η = 1. No boundary condition needs to be imposed for the density amplitude function at the wall, because the linearized continuity equation is satisfied at η = 0.

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Given recent evidence (Lakebrink and Borg, 2016) of the potential usefulness of linear theory for estimating crossflow induced transition, further investigation of the sensitivity to various linear analysis techniques is warranted. Furthermore, knowledge of the merits and shortcomings for each technique is valuable because of their varying computational costs. Specif-

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ically, the cost of executing an analysis using LST is slightly less expensive than LPSE, which is again slightly less expensive than 2pLPSE. As might be

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expected, however, the increase in cost between these three methods buys an increase in the amount of physics included in the governing equations. The

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greatest increase in cost comes with SBG, because it solves a 2D PDE-based problem, while the rest are based on the solution of a 1D ODE-based linear

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system.

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3. Experimental approach A 38.1% scale model of the HIFiRE-5 flight vehicle was used as the test

article for the experiments. This geometry is a 2:1 elliptic cone with a base semimajor axis of 82.3 mm. The cone half angle in the minor axis plane is 7.00◦ . The nosetip cross section in the minor axis plane is a 0.95 mm circular 14

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arc that preserves tangency in the minor axis plane. The nose maintains an elliptic cross section to the tip. The model angle of attack and yaw were held

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at 0.0◦ ± 0.1◦ .

The experiments, the details of which are more thoroughly reported by Borg et al. (2015), were performed at Mach 6 in the Boeing/AFOSR Mach-6

Quiet Tunnel (BAM6QT). The BAM6QT provides flow with freestream noise levels comparable to those experienced in atmospheric flight for stagnation

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pressures up to about 1.2 MPa. The low noise levels are achieved by maintaining a laminar boundary layer along the tunnel wall (Schneider, 1998).

The ratio of the model wall temperature to the adiabatic wall temperature was about 0.8.

The model was instrumented with 3 pressure transducers, Kulite models

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XCE-062-15A and XCQ-062-15A. The sensors were mounted flush with the model surface at (x,z) locations of (310.1,41.2), (312.6,39.8), and (318.1,39.8) mm.

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These sensors have been shown to have a flat frequency response up to 80– 110 kHz (Beresh et al., 2010).

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Two sensor pairs were needed to calculate the properties of the traveling crossflow waves. The cross spectrum was calculated for both sensor pairs.

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The arctangent of the ratio of the imaginary and real components of the cross spectrum yielded the phase. Utilizing the phase and the physical locations

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of the sensors then allowed the phase speed and wave angle to be calculated as a function of frequency. A more detailed description of this technique, including additional experimental results, can be found in Ref. Borg et al. (2015). The experimental test campaign included a sweep through Re/m val15

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ues ranging from 6.3e6–12.2e6. The current work focuses on the experiment for which Re/m=8.1e6. This condition gave the maximum power-spectral-

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density amplitude of the traveling crossflow waves prior to the onset of transition. Higher values of Re/m yielded effects such as spectral broadening and

wideband increases in power levels, indications that the boundary layer was beginning to break down.

Eleven repeat runs at the same conditions were used to estimate the

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uncertainty in wave angle and phase speed. The uncertainty in the cross-

spectrum phase was taken to be twice the maximum standard deviation of the cross-spectrum phase over a frequency range of 25–75 kHz for the 11 repeated runs. This cross-spectrum phase uncertainty and the uncertainty in the relative sensor positions were propagated through the equations for wave

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angle and phase speed. The uncertainties for these values were estimated to

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be 2.4◦ and 0.038 (as normalized by the computed edge velocity), respectively.

4. Results and Discussion

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The stability analysis is performed by marching along inviscid streamlines chosen such that they cover the crossflow-dominated mid-span region of the

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model. To compare with the SBG analysis, streamwise growth rates are queried on each streamline near x = 0.3 m, where x is the axial coordinate

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as indicated by figure 1. In order to compare with the experimental results, wave angles and phase speeds are computed near x = 0.312 m. The wave angles and phase speeds computed using LST, LPSE, and

2pLPSE are compared to the experimentally measured values for the flow conditions reported in table 1. To make the comparison as consistent as 16

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possible, the stability analysis is performed for disturbances with spanwise wavelengths matching those measured in the test (Table 2), and a frequency

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of 45 kHz. Measured wavelengths in the wave-angle direction were found by dividing the phase speed by the frequency. Dividing the resulting quantity

by the sine of the wave angle then gave the spanwise wavelength. Figure 3(a)

shows traveling crossflow wave angles as a function of disturbance frequency.

Wave angle is defined as the inverse tangent of the ratio of streamwise wave-

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length to spanwise wavelength. The disturbance wave angle varies inversely

with the frequency in a linear fashion, ranging from 72◦ to 51◦ . The predicted values fall within the uncertainty of the experimental data across the spectrum. This indicates that the traveling crossflow waves were behaving linearly at the point of measurement for Re/m = 8.1 × 106 . It is also ob-

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served that while the LPSE and 2pLPSE predictions are slightly closer to the experiment, there is very little difference between the wave angles predicted

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by LST, LPSE, and 2pLPSE. This suggests that the boundary layer is fairly uniform in the streamwise and spanwise directions, and based on the mean

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flow solution, this is indeed the case in the vicinity of the probes. The variation of disturbance phase speed normalized by boundary layer

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edge velocity is plotted in figure 3(b). The phase speed exhibits a direct linear variation with frequency, ranging from 0.3 to 0.58. As with the wave

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angles, the predicted phase speeds lie within the experimental uncertainty of the measured data across all frequencies. The insensitivity between LST, LPSE, and 2pLPSE is once again observed in the phase speed prediction, and to an even greater degree than with the wave angles. These comparisons for wave angle and phase speed extend previous LST validation (see Li et al., 17

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7.35

40.3

6.48

45.2

6.69

50.0

6.40

54.9

6.35

59.8

6.58

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35.4

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F requency [kHz] Spanwise W avelength [mm]

64.7

6.85

69.6

7.13

74.5

6.84

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Table 2: Spanwise wavelengths used to compute wave angle for each frequency plotted in

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Figure 3(a).

60 55

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50 35

0.6

Phase Speed

65

(b) 0.7

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70

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Wave angle [deg]

(a) 75

Experiment LST LPSE 2pLPSE

40

45

0.5

Experiment LST LPSE 2pLPSE

0.4 0.3

50

55

60

65

70

0.2 35

75

f [kHz]

40

45

50

55

60

65

70

75

f [kHz]

Figure 3: Comparison of measured and computed (a) wave angles and (b) phase speeds for traveling crossflow waves. Bars plotted on the experimental data points depict uncertainties of ±2.4◦ and ±0.0376 for wave angle and phase speed respectively.

18

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2012; Borg et al., 2015) for linear traveling crossflow waves. Although the wave angle and phase speed predictions do not exhibit sig-

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nificant sensitivity to the various modeling techniques, other wave properties such as growth rate do. To illustrate this, crossflow wave growth rates in the streamwise direction (i.e. along the inviscid streamline) were computed

for six frequencies ranging from 50 kHz to 75 kHz using LST, LPSE, and

2pLPSE. These computation were performed at a location near the attach-

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ment line (x = 0.043 m) in addition to the downstream station on the same

streamline (x = 0.312 m). The computations at the upstream location were for a spanwise wavelength of 1.3 mm, and the computations at the downstream location were for a spanwise wavelength of 7.5 mm.

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Figure 4(a), which compares non-dimensional growth rates near the attachment line, shows that the growth rate is very sensitive to prediction

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technique, not only in terms of magnitude, but also slope. The LST growth rates are a factor of 3-4 less than LPSE and 2pLPSE, but increase more rapidly with frequency. The LPSE and 2pLPSE results exhibit similar vari-

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ation with frequency, but the 2pLPSE growth rates are between 13% and 19% greater across the spectrum. At the downstream location, figure 4(b)

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indicates that the growth rate is significantly less sensitive to prediction technique, with differences between LST and LPSE ranging from 19% to 35%.

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This is in contrast to the upstream location, where variations between LST and 2pLPSE were between 281% and 434%. Like the upstream location, LST predicts the lowest growth rate across the spectrum. Unlike the upstream location, the downstream LPSE growth rates are greater than those predicted by 2pLPSE, and the growth rate slopes for all three techniques are 19

×10−3 4.5 4 3.5 3 LST 2.5 LPSE 2 2pLPSE 1.5 1 0.5 0 55 45 50

(b) Streamwise Growth Rate

Streamwise Growth Rate

(a)

60

65

70

75

80

×10−3 4.5 4 3.5 3 LST 2.5 LPSE 2 2pLPSE 1.5 1 0.5 0 55 45 50

f [kHz]

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60

65

70

75

80

f [kHz]

Figure 4: Comparison of crossflow wave growth rates computed using LST, LPSE, and

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2pLPSE at (a) x = 0.043 m and (b) x = 0.312 m.

very similar.

The difference in sensitivity to prediction technique, between the upstream and downstream locations, is caused by differences in the amount of

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spatial variation of the boundary layer. At the upstream location near the attachment line, the boundary layer is changing very rapidly in all three spa-

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tial directions, so large differences between LST and the non-parallel PSE approaches is to be expected. The insensitivity at the downstream location is consistent with the state of the boundary layer, which is slowly varying in

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both the streamwise and spanwise directions. As a result of the insensitivity of predicted wave properties between LPSE

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and 2pLPSE at the downstream location, it is reasonable to hypothesize that, in the vicinity of the probes, the spanwise gradients in the mean boundary

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layer are not major contributors to the evolution of traveling crossflow waves. To verify this, a SBG analysis was conducted on the same mean flow used to generate the foregoing LASTRAC results. The SBG solution was obtained for disturbances with a frequency of 45 kHz on a slice taken at x = 0.3 m. The resulting spectrum of eigenfunctions was then studied to determine 20

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(b)

(c)

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(a)

Figure 5: Temperature eigenfunctions at x = 0.3 m for (a) centerline, (b) crossflow, and (c) mixed modes. The left edge of each slice is the semi-minor axis symmetry boundary.

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which modes were crossflow. The non-spurious modes generally fell into three

categories: centerline modes, crossflow modes, and mixed modes. Also, no attachment line or second Mack modes were observed because the frequencies amplified along the attachment line are on the order of 600 kHz. The

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centerline modes amplify most heavily near the outer edges of the large-scale counter-rotating vortex pair that rolls up along the semi-minor axis merid-

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ian. The crossflow modes are most amplified near the middle of the boundary layer in the mid-span region. Finally, the mixed modes are amplified in the region between the centerline and the crossflow modes, and exhibit a mix of

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qualities found in each. Examples of these three types of modes are depicted in figure 5 using the magnitude of the temperature eigenfunction. Several

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instances of each mode type are present in the complete eigenvalue spectrum. The eigenvalue spectrum for the x = 0.3 m plane is presented in figure 6.

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The modes labeled (a), (b), and (c) correspond to the eigenfunctions plotted in figure 5. The most, and second-most unstable modes plotted are for centerline and mixed modes, respectively.

After all of the crossflow modes

were identified, the eigenvalue spectrum was used to select the most unstable crossflow mode for comparison with the LASTRAC results. The real part 21

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15

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Growth Rate [1/m]

20

(a) 10

(b) (c)

5

0 0

100

200

300

400

500

600

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Wave Number [1/m]

Figure 6: Eigenvalue spectrum for a 45 Figure 7: Measurements of spanwise wavekHz disturbance at x = 0.3 m.

The length at x = 0.3 m taken from the real part

modes identified as (a), (b), and (c) cor- of the SBG temperature eigenfunction for a 45 respond to the centerline, crossflow, and kHz crossflow wave.

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mixed modes plotted in figure 5.

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of the temperature eigenfunction for the most unstable mode was plotted in order to determine the spanwise wavelength of the disturbance. The periodic wave structure depicted in the real part of the temperature eigenfunction 7,

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corresponds to the off-surface streak shown by the eigenfunction magnitude in Figure 5(b). Based on the measurements presented in figure 7, the span-

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wise wavelength for the most unstable 45 kHz crossflow wave is somewhere between 2.31 mm and 2.44 mm. For the sake of discussion, the spanwise

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wavelength as predicted by the SBG analysis is taken to be 2.4 mm. The corresponding streamwise wavelength taken from the eigenvalue was approximately 10.5 mm, which results in a wave angle around 77◦ . LASTRAC was used to compute growth rates at x = 0.3 m for several

spanwise locations around the peak in temperature eigenfunction predicted 22

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by SBG. The spanwise distributions of streamwise growth rates predicted by LST and LPSE (using the integral disturbance kinetic energy growth rate) are

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plotted in figures 8(a) and 8(b) for several spanwise wavelengths. The SBG growth rate is given as the solid red line. Only one growth rate is obtained for the SBG solution, which gives the false impression of invariance with span. The LST and LPSE distributions are compared to the SBG growth

rate at the location corresponding to the peak amplitude in the temperature

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eigenfunction. This point of comparison is indicated by the black vertical dashed line in figures 8(a) and 8(b). The LST growth rates were computed for constant spanwise wavelengths ranging from 2.5 mm to 5.0 mm. Quasi-

parallel growth rates for disturbances with spanwise wavelengths less than 3.5 mm could not be found at (x, z) = (0.3, 0.036) m (where x and z are

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the coordinate directions indicated by figure 1), which is why the shorter

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wavelength distributions truncate closer and closer to the attachment line. The growth rates based on the integral-disturbance-kinetic energy predicted by LPSE exhibit two major differences from those predicted by LST.

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First, the LPSE distributions have greater growth rates than LST for a given spanwise wavelength. This is not surprising since it is well known that non-

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parallel growth rates are typically greater than quasi-parallel growth rates. Second, the LPSE distributions at shorter spanwise wavelengths do not suffer

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the same truncation issue as the LST distributions. This allows for the LPSE distribution to extend to the point of comparison, which indicates that a disturbance with a spanwise wavelength slightly less than 2.6 mm matches the SBG data. The agreement between the LPSE and SBG spanwise wavelengths is in fair agreement with the value of 2.4 mm predicted by SBG. 23

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Disturbance growth rates (based on integral-kinetic energy) obtained using the 2pLPSE approach are compared with the SBG growth rate in fig-

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ure 8(c). For a given spanwise wavelength, the growth rates predicted by 2pLPSE are greater than those predicted by LPSE (figure 8b). Another dif-

ference between LPSE and 2pLPSE is that the spanwise growth rate distribution flattens out closer to the attachment line, before suddenly decreasing. This is caused by the spanwise modulation of the boundary layer induced

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by the system of centerline vortices. Figure 9(a) shows the Mach number distribution of the base flow at x = 0.3m, overlaid with key spanwise stations from Figure 8c. Figure 9(b) plots the boundary-layer thickness as a function of spanwise location, which clearly illustrates the thinning that occurs at stations farther from the centerline. The most interesting part of

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the 2pLPSE results is that for a disturbance with spanwise wavelength 2.43 mm, the growth rates match very closely with the SBG result. It seems in-

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tuitive that 2pLPSE would provide the best agreement with SBG because both techniques account for spanwise variation in the meanflow. In addi-

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tion to the excellent agreement for growth rate, the wave angles between the two methods agree fairly well. As mentioned before, the wave angle for

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the 45 kHz crossflow disturbance was predicted to be approximately 77◦ by SBG. Both LPSE and 2pLPSE analysis with LASTRAC predict the wave angle for the same disturbance to be approximately 77.6◦ .

The foregoing

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results demonstrated that linear stability analysis can be used to accurately predict traveling crossflow wave properties at a point on the geometry. To demonstrate the applicability of LST and 2pLPSE for predicting crossflow waves over a broader extent of the elliptic cone, a qualitative comparison of

24

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31 26

(b)

SBG LST (5mm) LST (4mm) LST (3.5mm) LST (3mm) LST (2.75mm) LST (2.6mm) LST (2.5mm)

21 16 11 6 0.025

0.03

0.035

0.04

0.045

41 36 31

(5mm) (4mm) (2.6mm) (2.5mm)

26 21 16 11 6 0.025

0.05

SBG LPSE LPSE LPSE LPSE

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36

Growth Rate [1/m]

Growth Rate [1/m]

(a) 41

0.03

0.035

Z [m]

0.045

0.05

41 36

SBG 2pLPSE (2.6mm) 2pLPSE (2.5mm) 2pLPSE (2.43mm)

31 26 21 16 11 6 0.025

0.03

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Growth Rate [1/m]

(c)

0.04

Z [m]

0.035

0.04

0.045

0.05

Z [m]

Figure 8: Comparison of SBG and (a) LST, (b) LPSE, and (c) 2pLPSE growth rates. LPSE

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and 2pLPSE growth rates are based on disturbance kinetic energy. Spanwise wavelengths for each curve are given parenthetically in the legend. The vertical dashed line represents

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the spanwise location of the peak in temperature eigenfunction as predicted by SBG.

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(b) 2.2

2.1

Delta [mm]

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(a)

2 1.9 1.8 1.7 1.6 1.5 0.025

0.03

0.035

0.04

Z [m]

0.045

0.05

Figure 9: Centerline vortices causing spanwise variation in boundary-layer thickness (a), and the boundary-thickness based on 84% freestream total pressure as a function of spanwise location (b).

25

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stability analysis and experimental infrared thermography (IRT) images is presented in figure 10. The growth rates used to generate the N-factor con-

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tours were computed by marching along 250 inviscid streamlines spanning the surface from the attachment line to the centerline. For LST, the spanwise

wavelength at each station is chosen such that the streamwise growth rate is maximized. The N-factors are then computed by integrated these max-

imized growth rates for a given frequency. This procedure is repeated for

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several frequencies, and the envelope formed by these frequencies is plotted

in figure 10(a). In the case of 2pLPSE, the growth rates are computed by choosing a disturbance spanwise wavelength at the neutral point, solving for the associated non-parallel solution, and then marching that solution along the inviscid streamline. This procedure is repeated for a matrix of frequen-

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plotted in figure 10(c).

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cies and initial spanwise wavelengths, and the resulting N-factor envelope is

Figure 10(a) shows the N-factor envelopes computed by LST for stationary waves on the top half, and traveling waves on the bottom half. Frequen-

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cies included in the envelope range from 0 kHz to 80 kHz in increments of 5 kHz. Results are similar to those presented by Li et al. (2012). Experimen-

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tal IRT results are plotted in figure 10(b), and show streaks of elevated heat flux in regions corresponding to the high N-factors in figure 10(a). Using a

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subroutine called QCALC, IR data were reduced to heat flux by solving the transient one-dimensional heat equation on a pixel-by-pixel basis. QCALC uses second order Euler-explicit finite differences to solve for the temperature distribution through the model. Heat flux is calculated from a second-order approximation of the derivative of the temperature profile at the surface 26

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(Boyd and Howell, 1994). The result of this analysis is a time history of the heat flux over the model surface. The data shown in Figure 10b are

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thus a good approximation of the instantaneous heat flux. Finally, N-factors obtained by integrating 2pLPSE growth rates (based on the integral dis-

turbance kinetic energy) are plotted in figure 10(c). The envelope plotted in figure 10(c) represents frequencies from 0 kHz to 80 kHz in 5 kHz increments,

and spanwise wavelengths from 2 mm to 10 mm in 1 mm increments. The N-

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factors predicted from 2pLPSE, consistent with the growth rate distributions in figure 8, are greater than the LST N-factors. Near the attachment line, the 2pLPSE N-factors are observed to be greater than the LST N-factors.

This is consistent with the significant difference in growth rate in this region, as illustrated in Figure 4(a). Farther downstream, the 2pLPSE N-factors

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are only slightly greater than those computed with LST, which is consistent with the relative difference in downstream growth rates presented in Fig-

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ure 4(b). Overall, however, the N-factor distributions predicted by LST and 2pLPSE are roughly similar, and are in good qualitative agreement with the

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experimental IRT. One exception to this is near the centerline, where the traveling-crossflow computations indicate an elevated N-factor streak, which

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is inconsistent with the experimental heat flux. This is due to the presence of the centerline-vortex system, which violates the assumption made by LST

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and 2pLPSE that the mean flow varies slowly along the surface.

5. Concluding remarks Stability analysis was performed using the LST, LPSE, and 2pLPSE approaches in LASTRAC. The wave angles and phase speeds of traveling cross27

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(b)

(c)

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(a)

Figure 10: Qualitative comparison of stability analysis and experiment. (a): Comparison of stationary (top) and traveling (bottom) crossflow wave LST N-factor envelopes. (b):

Infrared thermography at Re/m=11.8e6 revealing hot streaks and transition under quiet

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flow. (c): Comparison of stationary (top) and traveling (bottom) crossflow wave 2pLPSE N-factor envelopes.

flow waves predicted using these methods were compared with data measured in the Purdue Mach 6 quiet tunnel. The agreement between the computation

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and experiment provided validation of the predictive ability of LASTRAC for linear traveling crossflow wave angles and phase speeds. Additional anal-

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ysis was performed using the SBG approach, and the resulting growth rate and wave angle for a 45 kHz disturbance was compared with the LASTRAC analysis. Agreement between LST and SBG was poor. The slightly better

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agreement between LPSE and SBG indicated the importance of accounting for the non-parallel effects in the boundary layer. The excellent agreement

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between 2pLPSE and SBG provided further verification that both codes correctly solve the governing equations. It is also revealed that the computa-

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tionally less expensive 2pLPSE approach is sufficient for use in predicting the behavior of linear traveling crossflow waves. This result is significant because of the relative cost between the 2pLPSE and SBG methods. Furthermore, this is the first time that a comparison has been made between crossflow wave growth rates computed using LST, LPSE, and 2pLPSE and those computed 28

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using SBG, which accounts for the divergence and curvature of the elliptic

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cone geometry.

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