Optimal Disturbances in Compressible Boundary Layers

Tumin, Anatoli; Reshotko, Eli

doi:10.2514/6.2003-792

Cited by 26 publications

(67 citation statements)

References 8 publications

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“…The governing equations for steady, three-dimensional disturbances in the supersonic flow past a sharp cone are derived from the linearized Navier-Stokes equations, in the same fashion as in [10,11,19,20].…”

Section: Governing Equationsmentioning

confidence: 99%

“…In the compressible case, previous works [10,11,19,20] maximized Mack's energy norm [24] of the perturbation kinetic energy, density, and temperature (or simply the part containing u and T) at the outlet plane,…”

Section: Formulation Of the Optimization Problemmentioning

confidence: 99%

“…A model for transient growth including nonparallel effects in the compressible boundary layer past a flat plate has also been developed [20] and then extended to the compressible boundary layer past a sphere [10,11,19]. In [10,11] compressible optimal perturbations were calculated by including surface curvature effects and nonparallel growth of the boundary layer.…”

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confidence: 99%

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Optimal Disturbances in the Supersonic Boundary Layer Past a Sharp Cone

Zuccher

Shalaev

Tumin³

et al. 2007

AIAA Journal

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Optimal disturbances for the supersonic flow past a sharp cone are computed to assess the effects due to flow divergence. This geometry is chosen because previously published studies on compressible optimal perturbations for flat plate and sphere could not isolate the influence of divergence alone, as many factors characterized the growth of disturbances on the sphere (flow divergence, pressure gradient, centrifugal forces, and dependence of the edge parameters on the local Mach number). Flow-divergence effects result in the presence of an optimal distance from the cone tip for which the optimal gain is the largest possible, showing that divergence effects are stronger in the proximity of the cone tip. By properly rescaling the gain, wave number, and streamwise coordinate due to the fact that the boundary-layer thickness on the sharp cone is 3 p thinner than the one over the flat plate, it is found that both the gain and the wave number compare fairly well. Moreover, results for the sharp cone collapse into those for the flat plate when the initial location for the computation tends to the final one and when the azimuthal wave number is very large. Results show also that a cold wall enhances transient growth. Nomenclature A, B 0 , B 1 , B 2 , C, D, H 1 , H 2 , M,M = 5 5 matrices E = perturbation energy f = vector of perturbation unknowns G = energy ratio G E out =E in H = wall-normal characteristic length i = 1 p J = objective function L = streamwise characteristic length L = augmented functional M = Mach number m = azimuthal index m = azimuthal wave number (m m) n = nth streamwise step Pr = Prandtl number p = perturbation pressure p = vector of adjoint variables Re = Reynolds number (Re UL ) T = temperature U = base-flow streamwise velocity component u = perturbation streamwise velocity component V = base-flow wall-normal velocity component v = perturbation wall-normal velocity component w = perturbation spanwise velocity component x = streamwise coordinate y = wall-normal coordinate = spanwise wave number for flat plate = specific heat ratio x = streamwise interval (x x out x in ) = small parameter ( H ref =L ref ) = half-angle of cone tip = kinematic viscosity = density = azimuthal coordinate Subscript ad = adiabatic conditions in = inlet conditions loc = local (edge) conditions out = outlet conditions ref = reference conditions s = basic state w = wall conditions 1 = upstream conditions Superscript T = transpose

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Section: Governing Equationsmentioning

confidence: 99%

Section: Formulation Of the Optimization Problemmentioning

confidence: 99%

mentioning

confidence: 99%

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Optimal Disturbances in the Supersonic Boundary Layer Past a Sharp Cone

Zuccher

Shalaev

Tumin³

et al. 2007

AIAA Journal

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“…As elaborated on by Mack, 3 the contribution of this compression work to the total perturbation energy should vanish due to the conservative nature of the compression work. Note that the Mach energy norm ͑21͒ has also been used in the "spatial" transient growth analyses of compressible flows, 15 Let G͑t , ␣ , ␤ ;Re, M͒ be the amplification of the initial energy density maximized over all possible initial conditions, i.e.,…”

Section: Transient Energy Growthmentioning

confidence: 99%

“…[12][13][14][15] For compressible boundary layers, Hanifi, Schmid and Henningson 12 have found the maximum transient energy growth ͑G max ͒ increases with both Reynolds number ͑Re͒ and Mach number ͑M͒. They have used an energy norm such that the pressurerelated energy transfer rate is equal to zero.…”

Section: Introductionmentioning

confidence: 97%

Nonmodal energy growth and optimal perturbations in compressible plane Couette flow

2006

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Nonmodal transient growth studies and an estimation of optimal perturbations have been made for the compressible plane Couette flow with three-dimensional disturbances. The steady mean flow is characterized by a nonuniform shear rate and a varying temperature across the wall-normal direction for an appropriate perfect gas model. The maximum amplification of perturbation energy over time, G max , is found to increase with increasing Reynolds number Re, but decreases with increasing Mach number M. More specifically, the optimal energy amplification G opt ͑the supremum of G max over both the streamwise and spanwise wave numbers͒ is maximum in the incompressible limit and decreases monotonically as M increases. The corresponding optimal streamwise wave number, ␣ opt , is nonzero at M = 0, increases with increasing M, reaching a maximum for some value of M and then decreases, eventually becoming zero at high Mach numbers. While the pure streamwise vortices are the optimal patterns at high Mach numbers ͑in contrast to incompressible Couette flow͒, the modulated streamwise vortices are the optimal patterns for low-to-moderate values of the Mach number. Unlike in incompressible shear flows, the streamwise-independent modes in the present flow do not follow the scaling law G͑t /Re͒ϳRe 2 , the reasons for which are shown to be tied to the dominance of some terms ͑related to density and temperature fluctuations͒ in the linear stability operator. Based on a detailed nonmodal energy analysis, we show that the transient energy growth occurs due to the transfer of energy from the mean flow to perturbations via an inviscid algebraic instability. The decrease of transient growth with an increasing Mach number is also shown to be tied to the decrease in the energy transferred from the mean flow ͑Ė 1 ͒ in the same limit. The sharp decay of the viscous eigenfunctions with increasing Mach number is responsible for the decrease of Ė 1 for the present mean flow.

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Special issue on the fluid mechanics of hypersonic flight

Theofilis

Pirozzoli

Martin

2022

Theor. Comput. Fluid Dyn.

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Flow phenomena on maneuverable vehicles that fly at hypersonic speeds and altitudes that reach the Earth's upper stratosphere and lower mesosphere pose challenges that collectively form one of the present-day research frontiers of Fluid Mechanics. Research agencies around the world have realized the multiple benefits derived from transitioning a deeper understanding of hypersonic flow phenomena to actual flying platforms, and support for hypersonic research has increased substantially in the last decade. The result can be seen in Fig. 1, showing the distribution over years and originating countries of a total of 26,300 peer-reviewed research papers that have been published since the word hypersonic first appeared in the literature in the early 1940s and have this word in their title. Early peak activity subsided after the Moon landing, but soon picked up in the early 80s and again around the turn of the century, growing monotonically to the present day. As also shown in Fig. 1, a breakdown by country of origin reveals that two-thirds of the total number of hypersonic publications originate from the USA and China, giving rise to concerns raised by strategic think-tanks [57] and policy advisers [61] regarding potential misuse of hypersonic technology.Hypersonic research has not always come to the limelight on account of potential strategic conflicts. The high plateau in number of papers published around the 1990s, shown in the left plate of Fig. 1, corresponds to intense activity that took place on the crest of the success of the early Space Shuttle flights, relating to the new Orient Express [47] of hypersonic travel. Large (and largely drawing board) projects involving substantial research into hypersonic flow physics, namely the National Aerospace Plane in the USA, Buran in the final years of the Soviet Union, HOTOL in the UK, Hermes in France and Sänger in Germany, alongside lower-scale efforts in India and Brazil, all testified to a time of optimism regarding hypersonic travel in a civilian transport V. Theofilis

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Optimal Disturbances in Compressible Boundary Layers

Cited by 26 publications

References 8 publications

Optimal Disturbances in the Supersonic Boundary Layer Past a Sharp Cone

Optimal Disturbances in the Supersonic Boundary Layer Past a Sharp Cone

Nonmodal energy growth and optimal perturbations in compressible plane Couette flow

Special issue on the fluid mechanics of hypersonic flight

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