Papers
High-speed boundary-layer transition induced by a discrete roughness element
- Prahladh S. Iyer, Krishnan Mahesh
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- Published online by Cambridge University Press:
- 24 July 2013, pp. 524-562
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Direct numerical simulation (DNS) is used to study laminar to turbulent transition induced by a discrete hemispherical roughness element in a high-speed laminar boundary layer. The simulations are performed under conditions matching the experiments of Danehy et al. (AIAA Paper 2009–394, 2009) for free-stream Mach numbers of 3.37, 5.26 and 8.23. It is observed that the Mach 8.23 flow remains laminar downstream of the roughness, while the lower Mach numbers undergo transition. The Mach 3.37 flow undergoes transition closer to the bump when compared with Mach 5.26, in agreement with experimental observations. Transition is accompanied by an increase in ${C}_{f} $ and ${C}_{h} $ (Stanton number). Even for the case that did not undergo transition (Mach 8.23), streamwise vortices induced by the roughness cause a significant rise in ${C}_{f} $ until 20$D$ downstream. The mean van Driest transformed velocity and Reynolds stress for Mach 3.37 and 5.26 show good agreement with available data. Temporal spectra of pressure for Mach 3.37 show that frequencies in the range of 10–1000 kHz are dominant. The transition process involves the following key elements: upon interaction with the roughness element, the boundary layer separates to form a series of spanwise vortices upstream of the roughness and a separation shear layer. The system of spanwise vortices wrap around the roughness element in the form of horseshoe/necklace vortices to yield a system of counter-rotating streamwise vortices downstream of the element. These vortices are located beneath the separation shear layer and perturb it, which results in the formation of trains of hairpin-shaped vortices further downstream of the roughness for the cases that undergo transition. These hairpins spread in the span with increasing downstream distance and the flow increasingly resembles a fully developed turbulent boundary layer. A local Reynolds number based on the wall properties is seen to correlate with the onset of transition for the cases considered.
Interaction modes of multiple flexible flags in a uniform flow
- Emad Uddin, Wei-Xi Huang, Hyung Jin Sung
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- 24 July 2013, pp. 563-583
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Fish schooling is not merely a social behaviour; it also improves the efficiency of movement within a fluid environment. Inspired by the hydrodynamics of schooling, a group of flexible bodies was modelled as a collection of individuals arranged in a combination of tandem and side-by-side formations. The downstream bodies were found to be strongly influenced by the vortices shed from an upstream body, as revealed in the vortex–vortex and vortex–body interactions. To investigate the interactions between flexible bodies and vortices, the present study examined flexible flags in a viscous flow by using an improved version of the immersed boundary method. Three different flag formations were modelled to cover the basic structures involved in fish schooling: triangular, diamond and conical formations. The drag coefficients of the downstream flags could be decreased below the value for a single flag by adjusting the streamwise and spanwise gap distances and the flag bending coefficient. The drag variations were influenced by the interactions between vortices shed from the upstream flexible flags and those surrounding the downstream flags. The interactions between the flexible flags were investigated as a function of both the gap distance between the flags and the bending coefficients. The maximum drag reduction and the trailing flag position were calculated for different sets of conditions. Single-frequency and multifrequency modes were identified and were found to correspond to constructive and destructive vortex interaction modes, which explained the variations in the drag on the downstream flags.
Numerical calculations of two-dimensional large Prandtl number convection in a box
- J. A. Whitehead, A. Cotel, S. Hart, C. Lithgow-Bertelloni, W. Newsome
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- 24 July 2013, pp. 584-602
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Convection from an isolated heat source in a chamber has been previously studied numerically, experimentally and analytically. These have not covered long time spans for wide ranges of Rayleigh number Ra and Prandtl number Pr. Numerical calculations of constant viscosity convection partially fill the gap in the ranges $\mathit{Ra}= 1{0}^{3} {{\unicode{x2013}}}1{0}^{6} $ and $\mathit{Pr}= 1, 10, 100, 1000$ and $\infty $. Calculations begin with cold fluid everywhere and localized hot temperature at the centre of the bottom of a square two-dimensional chamber. For $\mathit{Ra}\gt 20\hspace{0.167em} 000$, temperature increases above the hot bottom and forms a rising plume head. The head has small internal recirculation and minor outward conduction of heat during ascent. The head approaches the top, flattens, splits and the two remnants are swept to the sidewalls and diffused away. The maximum velocity and the top centre heat flux climb to maxima during head ascent and then adjust toward constant values. Two steady cells are separated by a vertical thermal conduit. This sequence is followed for every value of $Pr$ number, although lower Pr convection lags in time. For $\mathit{Ra}\lt 20\hspace{0.167em} 000$ there is no plume head, and no streamfunction and heat flux maxima with time. For sufficiently large Ra and all values of Pr, an oscillation develops at roughly $t= 0. 2$, with the two cells alternately strengthening and weakening. This changes to a steady flow with two unequal cells that at roughly $t= 0. 5$ develops a second oscillation.
DNS of a turbulent boundary layer with surface roughness
- James Cardillo, Yi Chen, Guillermo Araya, Jensen Newman, Kenneth Jansen, Luciano Castillo
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- Published online by Cambridge University Press:
- 24 July 2013, pp. 603-637
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A pioneer direct numerical simulation (DNS) of a turbulent boundary layer at $R{e}_{\theta } = 2077{{\unicode{x2013}}}2439$, was performed, on a rough surface and with a zero pressure gradient (ZPG). The boundary layer was subjected to transitional, 24-grit sandpaper surface roughness, with a roughness parameter of ${k}^{+ } \simeq 11$. The computational method involves a synergy of the dynamic multi-scale approach devised by Araya et al. (2011) for prescribing inlet turbulent boundary conditions and a new methodology for mapping high-resolution topographical surface data into a computational fluid dynamics (CFD) environment. It is shown here that the dynamic multi-scale approach can be successfully extended to simulations which incorporate surface roughness. The DNS results demonstrate good agreement with the laser Doppler anemometry (LDA) measurements performed by Brzek et al. (2008) and Schultz & Flack (2003) under similar conditions in terms of mean velocity profiles, Reynolds stresses and flow parameters, such as the skin friction coefficient, boundary and momentum thicknesses. Further, it is demonstrated that the effects of the surface roughness on the Reynolds stresses, at the values of $R{e}_{\theta } = 2077{{\unicode{x2013}}}2439$, are scale-dependent. Roughness effects were mainly manifested up to $y/ \delta \approx 0. 1$. Generally speaking, it was observed that inner peak values of Reynolds stresses increased when considering outer units. However, decreases were seen in inner units. In the outer region, the most significant differences between the present DNS smooth and rough cases were computed in the wall-normal component $\langle {v}^{\prime 2} \rangle $ of the Reynolds stresses and in the Reynolds shear stresses $\langle {u}^{\prime } {v}^{\prime } \rangle $ in outer units. From the resulting flow fields a proper orthogonal decomposition (POD) analysis is performed and the effects of the surface roughness are distinctly observed in the most energetic POD modes. The POD analysis shows that the surface roughness causes a redistribution of the kinetic energy amongst the POD modes with energy being shifted from low-order to high-order modes in the rough case versus the smooth case. Also, the roughness causes a marked decrease in the characteristic wavelengths observed in the POD modes, particularly in the streamwise component of the velocity field. Low-order modes of the streamwise component demonstrated characteristic wavelengths of the order of $3\delta $ in the smooth case, whereas the same modes for the rough case demonstrated characteristic wavelengths of only $\delta $.
Stability of film flow over inclined topography based on a long-wave nonlinear model
- D. Tseluiko, M. G. Blyth, D. T. Papageorgiou
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- Published online by Cambridge University Press:
- 24 July 2013, pp. 638-671
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The stability of a viscous liquid film flowing under gravity down an inclined wall with periodic corrugations is investigated. A long-wave model equation valid at near-critical Reynolds numbers is used to study the film dynamics, and calculations are performed for either sinusoidal or rectangular wall corrugations assuming either a fixed flow rate in the film or a fixed volume of fluid within each wall period. Under the two different flow assumptions, steady solution branches are delineated including subharmonic branches, for which the period of the free surface is an integer multiple of the wall period, and the existence of quasi-periodic branches is demonstrated. Floquet–Bloch theory is used to determine the linear stability of steady, periodic solutions and the nature of any instability is analysed using the method of exponentially weighted spaces. Under certain conditions, and depending on the wall period, the flow may be convectively unstable for small wall amplitudes but undergo transition to absolute instability as the wall amplitude increases, a novel theoretical finding for this class of flows; in other cases, the flow may be convectively unstable for small wall amplitudes but stable for larger wall amplitudes. Solutions with the same spatial period as the wall become unstable at a critical Reynolds number, which is strongly dependent on the period size. For sufficiently small wall periods, the corrugations have a destabilizing effect by lowering the critical Reynolds number above which instability occurs. For slightly larger wall periods, small-amplitude corrugations are destabilizing but sufficiently large-amplitude corrugations are stabilizing. For even larger wall periods, the opposite behaviour is found. For sufficiently large wall periods, the corrugations are destabilizing irrespective of their amplitude. The predictions of the linear theory are corroborated by time-dependent simulations of the model equation, and the presence of absolute instability under certain conditions is confirmed. Boundary element simulations on an inverted substrate reveal that wall corrugations can have a stabilizing effect at zero Reynolds number.
Localization of flow structures using $\infty $-norm optimization
- D. P. G. Foures, C. P. Caulfield, P. J. Schmid
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- 24 July 2013, pp. 672-701
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Stability theory based on a variational principle and finite-time direct-adjoint optimization commonly relies on the kinetic perturbation energy density ${E}_{1} (t)= (1/ {V}_{\Omega } )\int \nolimits _{\Omega } e(\boldsymbol{x}, t)\hspace{0.167em} \mathrm{d} \Omega $ (where $e(\boldsymbol{x}, t)= \vert \boldsymbol{u}{\vert }^{2} / 2$) as a measure of disturbance size. This type of optimization typically yields optimal perturbations that are global in the fluid domain $\Omega $ of volume ${V}_{\Omega } $. This paper explores the use of $p$-norms in determining optimal perturbations for ‘energy’ growth over prescribed time intervals of length $T$. For $p= 1$ the traditional energy-based stability analysis is recovered, while for large $p\gg 1$, localization of the optimal perturbations is observed which identifies confined regions, or ‘hotspots’, in the domain where significant energy growth can be expected. In addition, the $p$-norm optimization yields insight into the role and significance of various regions of the flow regarding the overall energy dynamics. As a canonical example, we choose to solve the $\infty $-norm optimal perturbation problem for the simple case of two-dimensional channel flow. For such a configuration, several solutions branches emerge, each of them identifying a different energy production zone in the flow: either the centre or the walls of the domain. We study several scenarios (involving centre or wall perturbations) leading to localized energy production for different optimization time intervals. Our investigation reveals that even for this simple two-dimensional channel flow, the mechanism for the production of a highly energetic and localized perturbation is not unique in time. We show that wall perturbations are optimal (with respect to the $\infty $-norm) for relatively short and long times, while the centre perturbations are preferred for very short and intermediate times. The developed $p$-norm framework is intended to facilitate worst-case analysis of shear flows and to identify localized regions supporting dominant energy growth.
Generation of steady longitudinal vortices in hypersonic boundary layer
- A. I. Ruban, M. A. Kravtsova
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- Published online by Cambridge University Press:
- 24 July 2013, pp. 702-731
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In this paper we study the three-dimensional perturbations produced in a hypersonic boundary layer by a small wall roughness. The flow analysis is performed under the assumption that the Reynolds number, $R{e}_{0} = {\rho }_{\infty } {V}_{\infty } L/ {\mu }_{0} $, and Mach number, ${M}_{\infty } = {V}_{\infty } / {a}_{\infty } $, are large, but the hypersonic interaction parameter, $\chi = { M}_{\infty }^{2} R{ e}_{0}^{- 1/ 2} $, is small. Here ${V}_{\infty } $, ${\rho }_{\infty } $ and ${a}_{\infty } $ are the flow velocity, gas density and speed of sound in the free stream, ${\mu }_{0} $ is the dynamic viscosity coefficient at the ‘stagnation temperature’, and $L$ is the characteristic distance the boundary layer develops along the body surface before encountering a roughness. We choose the longitudinal and spanwise dimensions of the roughness to be $O({\chi }^{3/ 4} )$ quantities. In this case the flow field around the roughness may be described in the framework of the hypersonic viscous–inviscid interaction theory, also known as the triple-deck model. Our main interest in this paper is the nonlinear behaviour of the perturbations. We study these by means of numerical solution of the triple-deck equations, for which purpose a modification of the ‘skewed shear’ technique suggested by Smith (United Technologies Research Center Tech. Rep. 83-46, 1983) has been used. The technique requires global iterations to adjust the viscous and inviscid parts of the flow. Convergence of such iterations is known to be a major problem in viscous–inviscid calculations. In order to achieve improved stability of the method, both the momentum equation for the viscous part of the flow, and the equations describing the interaction with the flow outside the boundary layer, are treated implicitly in this study. The calculations confirm the fact that in this sort of flow the perturbations are capable of propagating upstream in the boundary layer, resulting in a perturbation field which surrounds the roughness on all sides. We found that the perturbations decay rather fast with the distance from the roughness everywhere except in the wake behind the roughness. We found that if the height of the roughness is small, then the perturbations also decay in the wake, though much more slowly than outside the wake. However, if the roughness height exceeds some critical value, then two symmetric counter-rotating vortices form in the wake. They appear to support themselves and grow as the distance from the roughness increases.
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Energy cascade and scaling in supersonic isothermal turbulence
- Alexei G. Kritsuk, Rick Wagner, Michael L. Norman
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- 22 July 2013, R1
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Supersonic turbulence plays an important role in a number of extreme astrophysical and terrestrial environments, yet its understanding remains rudimentary. We use data from a three-dimensional simulation of supersonic isothermal turbulence to reconstruct an exact fourth-order relation derived analytically from the Navier–Stokes equations (Galtier & Banerjee, Phys. Rev. Lett., vol. 107, 2011, p. 134501). Our analysis supports a Kolmogorov-like inertial energy cascade in supersonic turbulence previously discussed on a phenomenological level. We show that two compressible analogues of the four-fifths law exist describing fifth- and fourth-order correlations, but only the fourth-order relation remains ‘universal’ in a wide range of Mach numbers from incompressible to highly compressible regimes. A new approximate relation valid in the strongly supersonic regime is derived and verified. We also briefly discuss the origin of bottleneck bumps in simulations of compressible turbulence.
Bounds for Euler from vorticity moments and line divergence
- Robert M. Kerr
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- 24 July 2013, R2
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The inviscid growth of a range of vorticity moments is compared using Euler calculations of anti-parallel vortices with a new initial condition. The primary goal is to understand the role of nonlinearity in the generation of a new hierarchy of rescaled vorticity moments in Navier–Stokes calculations where the rescaled moments obey ${D}_{m} \geq {D}_{m+ 1} $, the reverse of the usual ${\Omega }_{m+ 1} \geq {\Omega }_{m} $ Hölder ordering of the original moments. Two temporal phases have been identified for the Euler calculations. In the first phase the $1\lt m\lt \infty $ vorticity moments are ordered in a manner consistent with the new Navier–Stokes hierarchy and grow in a manner that skirts the lower edge of possible singular growth with ${ D}_{m}^{2} \rightarrow \sup \vert \boldsymbol{\omega} \vert \sim A_{m}{({T}_{c} - t)}^{- 1} $ where the ${A}_{m} $ are nearly independent of $m$. In the second phase, the new ${D}_{m} $ ordering breaks down as the ${\Omega }_{m} $ converge towards the same super-exponential growth for all $m$. The transition is identified using new inequalities for the upper bounds for the $- \mathrm{d} { D}_{m}^{- 2} / \mathrm{d} t$ that are based solely upon the ratios ${D}_{m+ 1} / {D}_{m} $, and the convergent super-exponential growth is shown by plotting $\log (\mathrm{d} \log {\Omega }_{m} / \mathrm{d} t)$. Three-dimensional graphics show significant divergence of the vortex lines during the second phase, which could be what inhibits the initial power-law growth.
Available potential energy in Rayleigh–Bénard convection
- Graham O. Hughes, Bishakhdatta Gayen, Ross W. Griffiths
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- Published online by Cambridge University Press:
- 09 August 2013, R3
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The mechanical energy budget for thermally equilibrated Rayleigh–Bénard convection is developed theoretically, with explicit consideration of the role of available potential energy, this being the form in which all the mechanical energy for the flow is supplied. The analysis allows derivation for the first time of a closed analytical expression relating the rate of mixing in symmetric fully developed convection to the rate at which available potential energy is supplied by the thermal forcing. Only about half this supplied energy is dissipated viscously. The remainder is consumed by mixing acting to homogenize the density field. This finding is expected to apply over a wide range of Rayleigh and Prandtl numbers for which the Nusselt number is significantly greater than unity. Thus convection at large Rayleigh number involves energetically efficient mixing of density variations. In contrast to conventional approaches to Rayleigh–Bénard convection, the dissipation of temperature or density variance is shown not to be of direct relevance to the mechanical energy budget. Thus, explicit recognition of available potential energy as the source of mechanical energy for convection, and of both mixing and viscous dissipation as the sinks of this energy, could be of further use in understanding the physics.
Front Cover (OFC, IFC) and matter
FLM volume 729 issue 1 Cover and Front matter
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- Published online by Cambridge University Press:
- 09 August 2013, pp. f1-f2
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Back Cover (IBC, OBC) and matter
FLM volume 729 issue 1 Cover and Back matter
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- Published online by Cambridge University Press:
- 09 August 2013, pp. b1-b2
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