A frequency domain analysis of compressible linearizedNavier-Stokes equations in a hypersonic compression ramp flow

Anubhav Dwivedi, Graham V. Candler, and Mihailo R. Jovanović

Abstract— We utilize the frequency response analysis ofthe linearized Navier-Stokes equations to quantify amplifi-cation of exogenous disturbances in a hypersonic flow overa compression ramp. Using the spatial structure of thedominant response to time-periodic inputs, we explain theorigin of steady reattachment streaks. Our analysis of thelaminar shock/boundary layer interaction reveals that thestreaks arise from a preferential amplification of upstreamcounter-rotating vortical perturbations with a specific span-wise wavelength. These streaks are associated with heat fluxstriations at the wall near flow reattachment and they cantrigger transition to turbulence. The streak wavelength pre-dicted by our analysis compares favorably with observationsfrom two different hypersonic compression ramp experiments.Furthermore, we utilize the dominant response to analyzethe physical effects in the linearized dynamical system re-sponsible for amplification of disturbances. We show thatflow compressibility that arises from base flow decelerationcontributes to the amplification of streamwise velocity andthat the baroclinic effects are responsible for the productionof streamwise vorticity. Both these effects contribute to theappearance of temperature streaks observed in experimentsand are critically important for the development of control-oriented models for transition to turbulence in hypersonicflows.

I. IntroductionHypersonic flows over complex surfaces are governed

by the compressible Navier-Stokes (NS) equations. Atthese conditions, where the fluid flows at least five timesfaster than the speed of sound, aerodynamic heating ofthe vehicle surface becomes a major concern: it is aboutfive times larger in turbulent than in laminar flow [1]and can lead to structural failure. The dynamics at suchextreme flow conditions are poorly understood and devel-opment of control-oriented models is critically importantfor vehicle performance and survivability. In this paper,we analyze dynamical properties of the compressibleNS equations in the presence of spatially distributedand temporally varying external disturbances. Thesedisturbances are considered as inputs, while variouscombinations of velocities, temperature, and pressure areconsidered as outputs. In principle, input-output analysis

Financial support from the Air Force Office of Scientific Researchunder award FA9550-18-1-0422 and the Office of Naval Researchunder award N00014-19-1-2037 is gratefully acknowledged.

A. Dwivedi and G. V. Candler are with the Department ofAerospace Engineering and Mechanics, University of Minnesota,Minneapolis, MN 55455, USA. M. R. Jovanović is with the MingHsieh Department of Electrical and Computer Engineering, Univer-sity of Southern California, Los Angeles, CA 90089, USA. Emails:dwive016@umn.edu, candler@umn.edu, mihailo@usc.edu

using the non-linear models based on the compressibleNS equations can be carried out. However, the imple-mentation involves large number of degrees of freedomand prohibitively expensive direct numerical simulations.Furthermore, it is difficult to obtain useful results insuch a generality. Thus, in the present work, we conductinput-output analysis of the compressible linearized NSequations.

To demonstrate the usefulness of input-output frame-work, we consider the hypersonic flow over a compressionramp as a canonical flow configuration over a controlsurface. At such high speeds, this inevitably resultsin shock/boundary layer interaction (SBLI) [2] thatinvolves separation and reattachment of flow close to thesurface (the boundary layer) and discontinuity associatedwith a shock system. Even though the compressionramp geometry is homogeneous in the spanwise direction,experiments [3] show that the flow over it exhibits three-dimensionality in the form of streamwise streaks nearreattachment. The streaks are associated with persistentlarge local peaks of heat transfer; they can destabilizethe boundary layer and cause transition to turbulence [2],[3].

Recently, there has been an increased interest inexperimental investigation of hypersonic compressionramp flows [3], [4]. Techniques such as temperaturesensitive paint (TSP) and infrared (IR) imaging wereemployed to study the formation of streamwise streaksand reattachment heat flux patterns. Previous studies [2],[4], [5] indicated that these streaks might originate fromupstream perturbations to the SBLI setup. However,developing physical insights responsible for the devel-opment of these three-dimensional structures by relyingsolely on the experimental and numerical observationscan be misleading [6]. In this context, we demonstratehow physical mechanisms responsible for the develop-ment of these streaks can be discovered by examining theinput-output dynamics of the linearized NS equations.

The linearized NS equations have been widely usedfor carrying out the modal and non-modal stabilityanalysis of idealized low-speed setups involving parallel(spatially inhomogeneous in one direction) and non-parallel (spatially inhomogeneous in multiple direction)flows [7]. In particular, an input-output framework thatevaluates the response (outputs) of a the linearized NSequations to external perturbation sources (inputs) hasbeen successfully employed to quantify the amplification

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and study transition mechanisms in low speed chan-nels [8], [9], boundary layers [10]–[12], and jets [13].However, applications to the high-speed flows in absenceof any simplifying geometrical and physical assumptionsis still an open challenge. In this paper, we utilizethe I/O analysis to demonstrate that the hypersonicshock/boundary layer interaction over a compressionramp strongly amplifies low-frequency upstream dis-turbances with a specific spanwise length scale. Thedominant I/O pair resulting from our analysis is usedto explain the emergence of reattachment streaks and tocompare our results with experiments.

Our presentation is organized as follows. In Section II,we present the linearized model and provide a briefsummary of the I/O formulation. Section III providesdetails about the flow geometry and computationalsetup. In Section IV, we evaluate the frequency responseof 2D laminar hypersonic base flow on a compressionramp to 3D upstream disturbances and illustrate thatthe dominant output field appears in the form of steadystreamwise streaks near reattachment. In Section V, weexamine the physical mechanisms responsible for streakamplification by analyzing the spatial structure of thedominant response. This is used to identify amplificationmechanisms active in the different regions of the flow fieldand demonstrate the utility of the input-output analysisin analyzing complex flow phenomenon. We conclude ourpresentation in Section VI.

II. Linearized model for compressible flowsThe compressible NS equations for perfect gas in

conservative form are given by∂U

∂t+

∂Fj

∂xj= 0, (1)

where Fj(U) is the flux vector and U =(ρ, ρu, E

)is the vector of conserved variables representing mass,three components of momentum, and total energy perunit volume of the gas [14]. We decompose the statevector U(x, t) into a steady base component U(x) and atime-varying perturbation component U′(x, t), U(x, t) =U(x) +U′(x, t). The evolution of small perturbations isthen governed by the linearized flow equations,

∂

∂tU′(x, t) = A(U)U′(x, t), (2)

where A(U) represents the compressible NS operatorresulting from linearization of (1) around the base flowU. We assume that (ρu)′ satisfies homogeneous Dirichletboundary conditions while ρ′ and E′ satisfy homogeneousNeumann boundary conditions at the wall. Along therest of the boundaries, all the perturbations are assumedto satisfy homogeneous Dirichlet boundary conditions.A second order central finite volume discretization (asdescribed in [15]) is used to obtain the finite dimensionalapproximation of Eq. (2),

d

dtq = Aq, (3)

which describes the dynamics of the spatially discretizedperturbation vector q. In general, for a finite volumediscretization with nel cells, the discrete state q ∈R5nel×1 and system matrix A ∈ R5nel×5nel.

In this paper, we are interested in quantifying theamplification of exogenous disturbances in boundarylayer flows [7], [9]. To accomplish this objective, weaugment the evolution model (3) with external excitationsources

d

dtq = Aq + Bd,

ϕ = Cq,(4)

where d is a spatially distributed and temporally varyingdisturbance source (input) and ϕ = (ρ′,u′, T ′) is thequantity of interest (output), where T ′ denotes temper-ature perturbations. In Eq. (4) the matrix B specifieshow the input enters into the state equation, while thematrix C extracts the output from the state q.

In boundary layer flows, the linearized flow systemis globally stable. Thus, for a time-periodic input withfrequency ω, d(t) = d(ω)eiωt, the steady-state output ofa stable system (4) is given by ϕ(t) = ϕ(ω)eiωt, whereϕ(ω) = H(iω)d(ω) and H(iω) is the frequency response

H(iω) = C(iωI − A)−1B. (5)

At any ω, the singular value decomposition of H(iω)can be used to quantify amplification of time-periodicinputs [7], [8],

H(iω) = Φ(iω)Σ(iω)D∗(iω). (6)

Here, (·)∗ denotes the complex-conjugate transpose, Φ

and D are unitary matrices, and Σ is the rectangular di-agonal matrix of the singular values σi(ω). The columnsdi of the matrix D represent the input forcing directionsthat are mapped through the frequency response Hto the corresponding columns ϕi of the matrix Φ;for d = di, the output ϕ is in the direction ϕi andthe amplification is determined by the correspondingsingular value σi. For a given temporal frequency ω,

G(ω) := σ1(ω) =∥H(iω)d1(ω)∥E

∥d1(ω)∥E=

∥ϕ1(ω)∥E∥d1(ω)∥E

,

(7)denotes the largest induced gain with respect to Chu’scompressible energy norm [16] which is defined as,

∥ · ∥2E =

∫Ω

ρ

2|u′|2 + p

2ρ2ρ′2 +

ρCv

2TT ′2 dΩ, (8)

where (p, ρ, T ) denote the base flow pressure, densityand temperature, and Cv is the specific heat at constantvolume in the domain Ω. Consistent with this energynorm, we evaluate the eigenvalue decomposition of theself-adjoint system H†H using a parallel implementationof the power method [17] to obtain the spatial structureof the dominant I/O pair (d1(ω),ϕ1(ω)) and gain G.

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TABLE IFree-stream conditions for experiments [4] and [3]

ReL p∞ T∞ U∞ ρ∞3.7× 105 355 Pa 55 K 1190 m/s 0.022 Kg/m3

2.0× 105 164 Pa 55 K 1188 m/s 0.010 Kg/m3

III. Flow over a hypersonic compression rampStreamwise streaks in wall temperature are often

observed in compression ramp experiments. Althoughtheir appearance is typically attributed to amplificationthat arises near reattachment from centrifugal [4] effects,quantifying amplification in the presence of a recircula-tion bubble is an open challenge. Herein, we employ theI/O framework to study the amplification of infinitesimalspanwise periodic upstream disturbances in hypersoniccompression ramp flow and explain origin of the heatstreaks at reattachment.

Recently, [3] and [4] reported multiple hypersoniccompression ramp experiments in two different facilitieswith matched free-stream Mach and Reynolds num-bers. Temperature sensitive paint (TSP) and infraredthermography measurements of reattachment heat-fluxwall patterns revealed quantitatively similar streaks. Theeffects of free-stream Reynolds number Re and leadingedge radius on the spanwise wavelength λz of the streakswere also reported. Our objective is to identify thestreak wavelength λz that is selected by the linearizedcompressible NS equations in the shock/boundary layerinteraction.

We consider the experiments performed in the UT-1MLudwig tube [4] at Mach 8. As illustrated in figure 1(a),the geometry consists of an L = 50mm isothermal flatplate with a sharp leading edge and wall temperatureTw = 293K, followed by an inclined ramp at 15. Thestreamwise domain extends from x/L = 0 to x/L = 1.65.Table I summarizes the two free-stream conditions whichdetermine the 2D base flows considered in our study.For the first free-stream condition, figure 1(b) providescomparison of the experimental schlieren image withthe 2D base flow density gradient magnitude field thatwe computed using the finite volume compressible flowsolver US3D [14]. Our 2D simulations correctly capturethe presence of both the separation and reattachmentshocks.

The Stanton number St is a non-dimensional parame-ter that determines the wall heat-transfer coefficient [4].In experiments, the Stanton number can be inferredfrom TSP and infrared thermography measurements.Figure 1(c) compares experimental values of St tothose predicted by our 2D simulations at different gridresolutions. We see that the computed flow captures theheat flux trends correctly except near the separationand the post-reattachment regions. In experiments, theseregions display significant spanwise variation in St andthey are marked by the grey band in figure 1(c).

Since the flow is globally stable with respect to 3D per-turbations [15], we conjecture that spanwise variationsarise from non-modal amplification of 3D perturbationsaround the 2D base flow. To verify our hypothesis, weemploy global I/O analysis to quantify the amplificationof exogenous disturbances and uncover mechanisms thatcan trigger the early stages of transition in a hypersoniccompression ramp flow.

IV. Frequency response analysisWe utilize frequency response analysis to investigate

the amplification of infinitesimal upstream perturbationsin a hypersonic compression ramp flow. This choice ismotivated by the experimental studies [3], [4] where vari-ation in the properties of the incoming boundary layerwere found to have profound effects on the downstreamstreaks. We model these upstream disturbances by re-stricting the inputs to the domain prior to separation(i.e., xs/L < 0.5). This is implemented by selecting thematrix B := Iny×ny ⊗ Λnx×nx, where Λ is a diagonalmatrix with tanh[(x − xs)/Lϵ] along the diagonal (ϵ =1e−6). Furthermore, we choose the perturbation field inthe entire domain as the output, ϕ = q, by settingC = I. The I/O analysis is conducted on a grid with412 cells in the streamwise and 249 cells in the wall-normal direction (labeled as G3 in figure 1(c)). Numericalsponge boundary conditions [15] (to no reflection fromthe inflow and outflow) are applied near the leading edge(x/L < 0.02) and the outflow (x/L > 1.6).

The left plot in figure 2 shows the input-outputamplification G(ω), defined in Eq. (7), in a flow withhigh Reynolds number (ReL = 3.7 × 105) for differentspanwise wavelengths λz. Here, λz := λ∗

z/δsep andω := ω∗δsep/U∞ denote the non-dimensional spanwisewavelength and temporal frequency, respectively, λ∗

z andω∗ are the corresponding quantities in physical units,whereas δsep represents the displacement boundary layerthickness at separation. We observe the low-pass featureof the amplification curve: G achieves its largest value atω = 0, it decreases slowly for low frequencies, and it ex-periences a rapid decay after the roll-off frequency (ω ≈0.01). The visualization of the dominant input-outputdirections d1 and ϕ1 in figures 2(a) and 2(b) reveals thatthe flat region of the amplification curve corresponds toincoming streamwise vortical disturbances (as inputs)that generate streak-like downstream perturbations (asoutputs). In contrast, figure 2(c) demonstrates that, athigh temporal frequency (ω = 0.1), dominant input-output pairs exhibit streamwise periodicity and takethe form of oblique waves. It should be noted thatthe low-pass frequency response features as well as theresulting changes in the response shape (from streaks tooblique waves) were also observed in canonical channeland boundary layer flows [8], [12].

The impact of the spanwise wavelengths λz on theamplification G for steady perturbations (i.e., at ω = 0)is shown in figure 3(a). For both Reynolds numbers, the

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Fig. 1. (a) Flow geometry and 2D steady streamwise velocity at ReL = 3.7 × 105; (b) comparison to experimental schlieren; and (c)variation of Stanton number (St) with x/L. Curves G1-G4 denote computational grids at varying resolution (nξ×nη), with G1 (577×349),G2 (495 × 300), G3 (412 × 249), and G4 (330 × 200) where nξ and nη denote the number of streamwise and wall normal grid cells,respectively. The shaded grey region denotes envelope of spanwise variation of St measured in experiments.

Fig. 2. Left: the ω-dependence of the largest induced gain with respect to the compressible energy norm, G(ω), for unsteady inputs withthe spanwise wavelengths λz = 1, 3, 6, 10. Right: isosurfaces of streamwise vorticity corresponding to the input d1 and temperaturecorresponding to the output ϕ1 for (a) ω = 0; (b) ω = 0.02; and (c) ω = 0.1 for λz = 3.

amplification curve achieves its maximum for a particularvalue of λz. This indicates that SBLI preferentially am-plifies upstream perturbations with a specific spanwisewavelength. The experimental estimates of λz resultingfrom the observed spanwise modulations in the TSPimages in figure 3(b) agree well with the predictions ofour I/O analysis. This shows that the compression rampflow strongly amplifies steady upstream disturbanceswith a preferential spanwise length scale.

V. Amplification of steady reattachment streaks:physical mechanism

As demonstrated in the previous section, the hyper-sonic flow over a compression ramp selectively amplifiessmall upstream perturbations of a specific spanwisewavelength. In order to gain insights into the structure ofthe most amplified perturbations we analyze the velocityand vorticity components (u′

s, ω′s). These components of

the dominant output ϕ1 quantify the kinetic energy ofthe spanwise periodic streamwise vortices along the baseflow streamlines [6]. We utilize the (s, n, z) coordinatesystem which is locally aligned with the streamlines ofthe base flow

(us, 0, 0

)to simplify our analysis. Here,

s denotes direction along streamlines and n denotesdirection normal to it. This is shown in figure 4, where wealso show the wall-aligned coordinate system, with ξ andη denoting the directions parallel and normal to the wall,

respectively. In the flow with ReL = 3.7×105, figure 4(b)illustrates the output components corresponding to λz =3 near reattachment in the (η, z) plane. We note that themost amplified perturbations are given by alternatingregions of high and low velocities with counter rotatingvortices between them and that u′

s and ω′s are 90 out

of phase in the spanwise direction.To investigate the physical mechanisms responsible

for the amplification of 3D reattachment streaks weanalyze the dominant terms in the system matrix A.In particular, we examine the spatial structure of thestreamwise vorticity, velocity, and temperature com-ponents associated with the dominant output ϕ1 andidentify amplification mechanisms that result from theinteractions of 3D flow perturbations with 2D base flowgradients.

A. Spatial evolution of streamwise vorticity

In order to quantify the development of streaks, weconsider the spatial evolution of streamwise vorticity ω′

s.After comparing relative magnitude of various terms anddropping terms (see [6] for details) with small magnitudewe obtain,

us∂sω′s ≈ ∂nρ

ρ2iβp′ (9)

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Fig. 3. (a) The λz-dependence of the amplification map G for steady inputs (i.e., ω = 0); and (b) comparison of experiments anddominant output ϕ1 at reattachment. The vertical dashed line in (b) denotes the approximate reattachment line in experiments and 2Dsimulations.

Fig. 4. (a) Schematic of the different coordinate systems for analyzing the perturbation evolution; and (b) color plots of streamwisevelocity u′

s and contour lines of streamwise vorticity ω′s at streamwise location x/L = 1.4 (post-reattachment) corresponding to the

dominant output ϕ1 at λz = 3. Solid lines denote positive values and dashed lines denote negative values of ω′s.

Fig. 5. (a) Illustration of the baroclinic term ∇ρ × ∇p′ for thedominant output with λz = 3; and (b) corresponding quantitiesnear the reattachment plane at x/L = 1.36.

The term on the left quantifies evolution of ω′s as

we go along the base streamlines. The right-hand sideappears due to the baroclinic effect, which arises frommisalignment of pressure and density gradients andaccounts for differential acceleration caused by variableinertia [18].

We illustrate the linear baroclinic mechanism in fig-ure 5(a) by showing three quantities: (i) the base flowdensity ρ in the (x, y) plane using an orange colormap;(ii) the spanwise gradient of the pressure perturbations p′in the (y, z) plane near reattachment using the red-white-blue colormap; and (iii) the isosurfaces of streamwise vor-ticity ω′

s using a grey-black colormap. Since the linearized

baroclinic torque that is active in the steady responseis associated with ∇ρ × ∇p′, we focus on examiningthe gradients of ρ and p′ shown in figure 5(b). Nearreattachment, the density gradient is aligned with thewall-normal direction η. This is because the ρ colormapbecomes darker as we move away from the wall in thedirection of increasing η. At the same x location, thegradient of p′ is orthogonal to the (x, y) plane; it achievesits largest value midway between the blue and the redlobes and it points in the direction from the center ofthe blue to the center of red lobes. As illustrated infigures 5(a) and (b), the resulting linearized baroclinictorque ∇ρ×∇p′ aligns with the streamwise vorticity ω′

s,thereby leading to its production.

B. Spatial evolution of streamwise velocity and temper-ature

Recent experimental [19] and numerical studies [20]demonstrated that streamwise velocity perturbationscontribute most to the kinetic energy in SBLI. Thespatial evolution of streamwise velocity u′

s is governedby,

us ∂su′s ≈ −∂sus u

′s (10)

This equation demonstrates that the growth of u′s

appears because of the streamwise deceleration of thebase flow (where ∂sus < 0). This is a unique feature ofcompressible flows and demonstrates the utility of theI/O approach in identifying new physical phenomenonpresent in complex fluid flows.

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C. Spatial evolution of temperature perturbationsTo understand the formation of heat streaks near

reattachment, we consider the spatial amplification oftemperature perturbations T ′ as they are transportedby the base flow. For the most amplified output pertur-bations, we retain the terms with significant contributionto the inviscid transport equation for T ′2,

us ∂s(T′2/2) ≈ − ∂sT (u′

sT′) − ∂nT (u′

nT′)

− (γ − 1) (∇ · u)T ′2.(11)

The first term on the right-hand-side leads to productionof temperature perturbations due to streamwise gradi-ent ∂sT near reattachment, whereas the second termaccounts for the transport from ω′

s due to the wall-normal thermal base flow gradients in the boundarylayer. Therefore, both u′

s and ω′s contribute to production

of T ′ at reattachment. The third term quantifies the baseflow dilatation in the reattachment shock where ∇ · utakes large negative values. All of these three physicaleffects significantly contribute to the amplification of T ′

near reattachment.VI. Concluding remarks

We have employed an input-output analysis to in-vestigate the dynamics of compressible linearized NSequations and to study amplification of disturbances incompressible boundary layer flows. Our approach utilizesglobal linearized dynamics to study the growth of flowperturbations and identify the spatial structure of thedominant response.

In an effort to explain the heat streaks observed in thehypersonic flows, we have examined the experimentallyobserved reattachment streaks in shock/boundary layerinteraction on Mach 8 flow over 15 compression ramp.The I/O analysis predicts large amplification of incomingsteady streamwise vortical disturbances with a specificspanwise length scale. The dominant output takes theform of steady streamwise streaks near reattachment.

We have also uncovered physical mechanisms respon-sible for amplification of steady reattachment streaks byevaluating the dominant terms in the system matrix.Physically, this quantifies the contribution of the baseflow gradients to the production of the perturbationquantities. Using our approach we demonstrate that theappearance of the temperature streaks near reattach-ment is triggered by the compressible flow effects asso-ciated with streamwise deceleration in the recirculationbubble and the baroclinic effects near reattachment.

The I/O approach provides a useful computationalframework to quantify the growth of external perturba-tions in complex high-speed flows. Improved understand-ing of amplification mechanisms can provide importantphysical insights about transition to turbulence. Weexpect that our work will motivate additional studiesthat explore nonlinear aspects of transition and pave theway for the development of predictive transition modelsand effective flow control strategies.

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