Criminale, W.O.

De Santi, Francesca

Scarsoglio, Stefania

Tordella, Daniela

DE SANTI, FRANCESCA

SCARSOGLIO, STEFANIA

W. O. Criminale

TORDELLA, Daniela

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

04 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEPerturbed cross-flow boundary layer: nontrivial effects of the obliquity angle at small and high reynolds numbers / F. DeSanti; S. Scarsoglio; W. O. Criminale; D. Tordella. - ELETTRONICO. - (2013). ((Intervento presentato al convegno 14thEuropean Turbulence Conference tenutosi a Lyon (F) nel 1-4 settembre 2013.OriginalPerturbed cross-flow boundary layer: nontrivial effects of the obliquity angle at small and high reynoldsnumbersPublisher:PublishedDOI:Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2507764 since:European Mechanics Society14TH EUROPEAN TURBULENCE CONFERENCE, 1–4 SEPTEMBER 2013, LYON, FRANCEPERTURBED CROSS-FLOW BOUNDARY LAYER: NONTRIVIAL EFFECTS OF THEOBLIQUITY ANGLE AT SMALL AND HIGH REYNOLDS NUMBERSFrancesca De Santi1, Stefania Scarsoglio1, William O. Criminale2 & Daniela Tordella11Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Torino, Italy 101292Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-2420, USAAbstract We carry out an exploratory linear study on the transient dynamics of arbitrary three-dimensional perturbations acting on theFalkner-Skan-Cooke cross-flow boundary layer. The analysis is in particular focused on the effects of the direction of the perturbationsat low and high Reynolds numbers. We show evidence of a non-trivial dependency of the transient and asymptotic behavior on thetravelling waves obliquity with respect to the base flow.The Falkner-Skan-Cooke cross flow boundary layer stability has been analyzed in literature using modal theory [1], inthe context of receptivity and transient optimal perturbations [2, 3] and experimentally [4]. Here we treat the full linearthree-dimensional perturbation problem by paying particular attention to the role of the obliquity of the perturbation withrespect to base flow direction. In literature there are few contributions with a specific focus on the effect of the angle ofobliquity on the perturbation transient life. Works on optimal perturbations [3] usually consider obliquity angles in theneighborhood of a fixed value. For instance, Breuer and Kuraishi [2] investigated the direction of the waves, but varyingsimultaneously φ and k. The exploratory analysis here proposed highlights a rich and, for certain aspects, counterintuitivescenario on the role of the perturbation direction. We observed that the "instability" of the waves does not depend in atrivial way from their obliquity: there are in fact configurations where both the longitudinal and orthogonal waves areunstable while oblique ones in-between are stable.Figure 1. (a) Reference system and sketch of the cross-flow boundary layer: θ is the angle of cross-flow, φ is the angle of obliquity. (b)Components of the base flow, U(y), W (y), β = −0.1988, 1 and θ = pi/6, pi/4, pi/3.We consider sub- and supercritical flow configurations (the Reynolds number, based on the displacement thickness δ∗, isequal to 100 and 5000), subject to adverse and favorable pressure gradients (Hartree parameter, β, equal to -0.1988 and1, respectively). The cross flow angle, θ, between the streamwise direction and the chordwise direction, is taken equal topi/6, pi/4, pi/3 (see Figure 1). Concerning the direction and wavelength of the perturbation, we vary both the obliquityangle with respect to the streamwise direction and the polar wavenumber, φ and k. The transient and asymptotic behaviorof the perturbative waves is observed through the amplification factor, G(t) (defined as the kinetic energy density, E(t),normalized over its initial value), the temporal growth rate, r = log(E(t))/2t, and the frequency, ω. This last is definedas the temporal derivative of the phase at a fixed distance from the wall.Fig. 2 shows the effect of the obliquity angle over two very different values of the Reynolds number. The amplificationfactor, G(t), is shown in the top panels, the frequency, ω(t), in the bottom panels. Panel (a) contains the first result thatwe wish to highlight here. One can see that even at a Re value as low as 100, in the presence of a sufficiently highpressure gradient, a small range of unstable longitudinal wavenumbers still exist: the longitudinal wave is unstable, whilethe oblique and the orthogonal waves have an initial decay followed by a transient growth and a final decay. Panels (b) and(c) present the case with the higher Reynolds number, 5000. Here, both a favorable and an adverse pressure gradient areconsidered together with a cross flow angle of pi/6. In the favorable situation, the orthogonal wave is unstable while thelongitudinal and oblique perturbations are asymptotically stable, but have a growth-decay phase in the early transient. Inpanel (c), the most unstable configuration achievable with our choice of parameters, we see that the longitudinal and theorthogonal waves are unstable but oblique waves in between are not, and this is unexpected. Because, usually, if one seesFigure 2. Role of the obliquity angle, φ = 0.pi/4, pi/2. Temporal evolution of the amplification factor, G(t) (panels a, b, c), and thefrequency, ω(t) (panels d, e, f). (a-d) Re = 100, β = −0.1988, θ = pi/4, k = 0.2. (b-e) Re = 5000, β = 1, θ = pi/6, k = 0.4. (c-f)Re = 5000, β = −0.1988, θ = pi/6, k = 0.4.Figure 3. Role of θ: Temporal evolution of the amplification factor,G(t) (panel a), and the temporal growth rate r(k) in the asymptoticlimit (panel b), β = −0.1988, Re = 5000, φ = 0. Pinch points results by Taylor and Peake [1] are added for a qualitative comparison.instability in the longitudinal direction, one then sees a progressive tendency to stability toward the orthogonal direction,and viceversa. Instead, here intermediate angles, as φ = pi/4, have an intense initial growth and then become stable.Independently of the stable or unstable asymptotic behavior of the perturbation, the frequency along the transient presentsudden impulsive variations, which we may describe as frequency jumps (see panels d-e-f). We interpret these jumps asthe transition between the early part of the transient and the beginning of an intermediate term that show up for times largeenough for the influence of the fine details of the initial condition to disappear.The combined effect of β and θ has been investigated by Taylor and Peake [1]. They found that asymptotically the pinchpoints with an adverse pressure gradient flow are more unstable at lower cross flow angles, while for negative pressuregradient the opposite is true. In Fig. 3, we show that this result is valid not only for the pinch point formulation but,more in general, for an arbitrary variation of the perturbation wavelength. A comparison with available data from [1] ispresented in panel (b). Even if the Reynolds number is different (1000 versus 5000) the comparison is good.We have obtained these results by adopting the initial-value problem formulation in the velocity-vorticity formulation.We perform a Laplace-Fourier transformation of the governing disturbance equations in the streamwise and spanwisedirections, and then numerically solve the resulting partial differential equations [5, 6]. This approach offers an alternativemeans for which arbitrary initial conditions are imposed and their full temporal behavior, including both the early transientand the long-time asymptotics, can be observed. As initial condition we consider a Gaussian function, v(t = 0, y) =y2 exp(−y.2), ωy(t = 0, y) = 0. Such initial condition concentrates the energy of the wave in the shear region [7].References[1] Tayor, M. J., Peake, N. 1997, J. Fluid Mech. 355, 359–381.[2] Breuer, K. S., Kuraishi, T. 1994, Phys. Fluids 6, 1983-1994.[3] Corbett, P., Bottaro, A. 2000, J. Fluid Mech. 435, 1-23.[4] Saric, W. S., Reed, H. L., White, E. B. 2003, Annu. Rev. Fluid Mech 35, 413–440.[5] Scarsoglio, S., Tordella, D., Criminale, W. O. 2009, Stud. Applied Math. 123, 153–173.[6] Scarsoglio, S., Tordella, D., Criminale, W. O. 2010, Phys. Rev. E 81, 036326/1-9.[7] Lasseigne, D. G., Joslin, R. D., Jackson, T. L. & Criminale, W. O. 1999, J. Fluid Mech. 381, 89–119.

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Perturbed cross-flow boundary layer: nontrivial effects of the obliquity angle at small and high reynolds numbers

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[Proceeding] Perturbed cross-flow boundary layer: nontrivial effects of the
obliquity angle at small and high reynolds numbers
Original Citation:
F. De Santi, S. Scarsoglio, W.O. Criminale, D. Tordella (2013). Perturbed cross-flow boundary layer:
nontrivial effects of the obliquity angle at small and high reynolds numbers. In: 14th European
Turbulence Conference, Lyon (F), 1-4 settembre 2013.
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14TH EUROPEAN TURBULENCE CONFERENCE, 1–4 SEPTEMBER 2013, LYON, FRANCE
PERTURBED CROSS-FLOW BOUNDARY LAYER: NONTRIVIAL EFFECTS OF THE
OBLIQUITY ANGLE AT SMALL AND HIGH REYNOLDS NUMBERS
Francesca De Santi1, Stefania Scarsoglio1, William O. Criminale2 & Daniela Tordella1
1Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Torino, Italy 10129
2Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-2420, USA
Abstract We carry out an exploratory linear study on the transient dynamics of arbitrary three-dimensional perturbations acting on the
Falkner-Skan-Cooke cross-flow boundary layer. The analysis is in particular focused on the effects of the direction of the perturbations
at low and high Reynolds numbers. We show evidence of a non-trivial dependency of the transient and asymptotic behavior on the
travelling waves obliquity with respect to the base flow.
The Falkner-Skan-Cooke cross flow boundary layer stability has been analyzed in literature using modal theory [1], in
the context of receptivity and transient optimal perturbations [2, 3] and experimentally [4]. Here we treat the full linear
three-dimensional perturbation problem by paying particular attention to the role of the obliquity of the perturbation with
respect to base flow direction. In literature there are few contributions with a specific focus on the effect of the angle of
obliquity on the perturbation transient life. Works on optimal perturbations [3] usually consider obliquity angles in the
neighborhood of a fixed value. For instance, Breuer and Kuraishi [2] investigated the direction of the waves, but varying
simultaneously φ and k. The exploratory analysis here proposed highlights a rich and, for certain aspects, counterintuitive
scenario on the role of the perturbation direction. We observed that the "instability" of the waves does not depend in a
trivial way from their obliquity: there are in fact configurations where both the longitudinal and orthogonal waves are
unstable while oblique ones in-between are stable.
Figure 1. (a) Reference system and sketch of the cross-flow boundary layer: θ is the angle of cross-flow, φ is the angle of obliquity. (b)
Components of the base flow, U(y), W (y), β = −0.1988, 1 and θ = pi/6, pi/4, pi/3.
We consider sub- and supercritical flow configurations (the Reynolds number, based on the displacement thickness δ∗, is
equal to 100 and 5000), subject to adverse and favorable pressure gradients (Hartree parameter, β, equal to -0.1988 and
1, respectively). The cross flow angle, θ, between the streamwise direction and the chordwise direction, is taken equal to
pi/6, pi/4, pi/3 (see Figure 1). Concerning the direction and wavelength of the perturbation, we vary both the obliquity
angle with respect to the streamwise direction and the polar wavenumber, φ and k. The transient and asymptotic behavior
of the perturbative waves is observed through the amplification factor, G(t) (defined as the kinetic energy density, E(t),
normalized over its initial value), the temporal growth rate, r = log(E(t))/2t, and the frequency, ω. This last is defined
as the temporal derivative of the phase at a fixed distance from the wall.
Fig. 2 shows the effect of the obliquity angle over two very different values of the Reynolds number. The amplification
factor, G(t), is shown in the top panels, the frequency, ω(t), in the bottom panels. Panel (a) contains the first result that
we wish to highlight here. One can see that even at a Re value as low as 100, in the presence of a sufficiently high
pressure gradient, a small range of unstable longitudinal wavenumbers still exist: the longitudinal wave is unstable, while
the oblique and the orthogonal waves have an initial decay followed by a transient growth and a final decay. Panels (b) and
(c) present the case with the higher Reynolds number, 5000. Here, both a favorable and an adverse pressure gradient are
considered together with a cross flow angle of pi/6. In the favorable situation, the orthogonal wave is unstable while the
longitudinal and oblique perturbations are asymptotically stable, but have a growth-decay phase in the early transient. In
panel (c), the most unstable configuration achievable with our choice of parameters, we see that the longitudinal and the
orthogonal waves are unstable but oblique waves in between are not, and this is unexpected. Because, usually, if one sees
Figure 2. Role of the obliquity angle, φ = 0.pi/4, pi/2. Temporal evolution of the amplification factor, G(t) (panels a, b, c), and the
frequency, ω(t) (panels d, e, f). (a-d) Re = 100, β = −0.1988, θ = pi/4, k = 0.2. (b-e) Re = 5000, β = 1, θ = pi/6, k = 0.4. (c-f)
Re = 5000, β = −0.1988, θ = pi/6, k = 0.4.
Figure 3. Role of θ: Temporal evolution of the amplification factor,G(t) (panel a), and the temporal growth rate r(k) in the asymptotic
limit (panel b), β = −0.1988, Re = 5000, φ = 0. Pinch points results by Taylor and Peake [1] are added for a qualitative comparison.
instability in the longitudinal direction, one then sees a progressive tendency to stability toward the orthogonal direction,
and viceversa. Instead, here intermediate angles, as φ = pi/4, have an intense initial growth and then become stable.
Independently of the stable or unstable asymptotic behavior of the perturbation, the frequency along the transient present
sudden impulsive variations, which we may describe as frequency jumps (see panels d-e-f). We interpret these jumps as
the transition between the early part of the transient and the beginning of an intermediate term that show up for times large
enough for the influence of the fine details of the initial condition to disappear.
The combined effect of β and θ has been investigated by Taylor and Peake [1]. They found that asymptotically the pinch
points with an adverse pressure gradient flow are more unstable at lower cross flow angles, while for negative pressure
gradient the opposite is true. In Fig. 3, we show that this result is valid not only for the pinch point formulation but,
more in general, for an arbitrary variation of the perturbation wavelength. A comparison with available data from [1] is
presented in panel (b). Even if the Reynolds number is different (1000 versus 5000) the comparison is good.
We have obtained these results by adopting the initial-value problem formulation in the velocity-vorticity formulation.
We perform a Laplace-Fourier transformation of the governing disturbance equations in the streamwise and spanwise
directions, and then numerically solve the resulting partial differential equations [5, 6]. This approach offers an alternative
means for which arbitrary initial conditions are imposed and their full temporal behavior, including both the early transient
and the long-time asymptotics, can be observed. As initial condition we consider a Gaussian function, v(t = 0, y) =
y2 exp(−y.2), ωy(t = 0, y) = 0. Such initial condition concentrates the energy of the wave in the shear region [7].
References
[1] Tayor, M. J., Peake, N. 1997, J. Fluid Mech. 355, 359–381.
[2] Breuer, K. S., Kuraishi, T. 1994, Phys. Fluids 6, 1983-1994.
[3] Corbett, P., Bottaro, A. 2000, J. Fluid Mech. 435, 1-23.
[4] Saric, W. S., Reed, H. L., White, E. B. 2003, Annu. Rev. Fluid Mech 35, 413–440.
[5] Scarsoglio, S., Tordella, D., Criminale, W. O. 2009, Stud. Applied Math. 123, 153–173.
[6] Scarsoglio, S., Tordella, D., Criminale, W. O. 2010, Phys. Rev. E 81, 036326/1-9.
[7] Lasseigne, D. G., Joslin, R. D., Jackson, T. L. & Criminale, W. O. 1999, J. Fluid Mech. 381, 89–119.


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Perturbed cross-flow boundary layer: nontrivial effects of the obliquity angle at small and high reynolds numbers

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