An energy analysis of convective instabilities of the Bödewadt and Ekman boundary layers over rough surfaces

Alveroğlu, Burhan; Segalini, Antonio; Garrett, S. D.

doi:10.1016/j.euromechflu.2016.09.006

Cited by 7 publications

(8 citation statements)

References 22 publications

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“…This paper should be viewed as an extension to the study of the von Kármán flow over rough disks modeled with YHP method by Garrett et al [13] and the work Alveroglu et al [8] who investigated stabilizing mechanism of surface roughness modeled by MW model within the BEK family of flows. Our presented results for Ekman flow are consistent with those of Alveroglu et al [8,30].…”

Section: Resultssupporting

confidence: 93%

“…In contrast, the earlier onset of Type II mode with increased roughness parameters is interpreted as a destabilization . We extended the study of Cooper et al [7] for the complete BEK family boundary layer flows using the MW model for roughness [8] and predicted similar roughness effects on these boundary layer flows [8, 30]. However, no known theoretical research has focused on the instability properties of the Ekman flow over a rough disk modeled with the YHP model.…”

Section: Introductionmentioning

confidence: 78%

“…An integral energy analysis for three‐dimensional disturbances

false(\overset{u}{true} \overset{v}{true} \overset{w}{true} \overset{p}{true} false)

to the undisturbed steady mean‐flow profiles ( U , V , W , P ) helps to understand the physical mechanisms of the conclusions derived in the convective instability part. This method was extensively used in the literature for the BEK system flows [7, 9, 30] and full details for the derivation of the equations can be found there. Here, it is sufficient to say that the energy balance calculation is performed for a particular eigenmode, and it is determined by [9]

\begin{array}{l} - 2 α_{i} = \underset{I}{\underset{true⏟}{(P_{1} + P_{2} + P_{3})}} + \underset{II}{\underset{true⏟}{D}} + \underset{III}{\underset{true⏟}{({PW}_{1} + {PW}_{2})}} \\ \underset{IV}{\underset{true⏟}{(S_{1} + S_{2} + S_{3})}} + \underset{V}{\underset{true⏟}{(G_{1} + G_{2} + G_{3})}} . \end{array}

…”

Section: Convective Instability Analysismentioning

confidence: 99%

“…The mathematical formulation of these terms for the general BEK system flow is provided in our previous study [30] and is given by

\begin{array}{l} - 2 α_{i} = \underset{I}{\underset{true⏟}{\int_{0}^{normal∞} [(- \overset{true‾}{\overset{true^}{u} \overset{true^}{w}} \frac{\partial U}{\partial z}) + (- \overset{true‾}{\overset{true^}{v} \overset{true^}{w}} \frac{\partial V}{\partial z}) + (\overset{true‾}{{\overset{true^}{w}}^{2}} \frac{italicRo}{italicRe} \frac{\partial W}{\partial z})] dz}} \\ - \underset{II}{\underset{true⏟}{\int_{0}^{normal∞} (\overset{σ_{ij} \frac{\partial {\overset{true^}{u}}_{j}}{\partial x_{i}}}{true}) dz}} + \underset{III}{\underset{true⏟}{\int_{0}^{normal∞} (\frac{italicRo}{italicRe} \overset{true‾}{\overset{true^}{u} \overset{true^}{p}}) dz + {(\overset{true‾}{\overset{true^}{w} \overset{true^}{p}})}_{w}}} \\ - \underset{IV}{\underset{true⏟}{{false[\overset{true‾}{\overset{true^}{u} σ_{31}} + \overset{true‾}{\overset{true^}{v} σ_{32}} + \overset{true‾}{\overset{true^}{w} σ_{33}} false]}_{w}}} \\ + \int_{0}^{normal∞} \frac{italicRo}{italicRe} \frac{\partial \overset{true‾}{K}}{\partial z} Wdz + \int_{0}^{normal∞} \frac{italicRo}{italicRe} \end{array}

…”

Section: Convective Instability Analysismentioning

confidence: 99%

“…These terms can be physically interpreted as: The mathematical formulation of these terms for the general BEK system flow is provided in our previous study [30] and is given by…”

Section: Energy Analysismentioning

confidence: 99%

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On the instability of Ekman boundary‐layer flow over a rough disk with a wavy surface profile

Alveroğlu

2020

Numerical Methods Partial

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We apply the method of lines and the backward difference method to solve highly nonlinear governing mean flow equations of the Ekman boundary‐layer flow over a rough disk with a wavy surface profile. These equations are derived using the YHP roughness model that imposes a wavy surface profile in terms of the radial position r, assuming rotational symmetry. Our main interest is to determine the effect of roughness on the local linear convective instability of the Ekman flow, and we solve the local linear stability equations of the disturbances stationary relative to the disk using the Chebyshev polynomials. The crossflow and viscous types of instability modes, the Type I and Type II modes respectively, are analyzed. Our theoretical findings prompts that an adequately designed rough disk can be a useful tool for a passive flow‐control method for the Ekman flow. The underlying physical mechanisms are investigated applying a fundamental energy analysis.

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Section: Resultssupporting

confidence: 93%

Section: Introductionmentioning

confidence: 78%

“…An integral energy analysis for three‐dimensional disturbances

false(\overset{u}{true} \overset{v}{true} \overset{w}{true} \overset{p}{true} false)

\begin{array}{l} - 2 α_{i} = \underset{I}{\underset{true⏟}{(P_{1} + P_{2} + P_{3})}} + \underset{II}{\underset{true⏟}{D}} + \underset{III}{\underset{true⏟}{({PW}_{1} + {PW}_{2})}} \\ \underset{IV}{\underset{true⏟}{(S_{1} + S_{2} + S_{3})}} + \underset{V}{\underset{true⏟}{(G_{1} + G_{2} + G_{3})}} . \end{array}

…”

Section: Convective Instability Analysismentioning

confidence: 99%

“…The mathematical formulation of these terms for the general BEK system flow is provided in our previous study [30] and is given by

\begin{array}{l} - 2 α_{i} = \underset{I}{\underset{true⏟}{\int_{0}^{normal∞} [(- \overset{true‾}{\overset{true^}{u} \overset{true^}{w}} \frac{\partial U}{\partial z}) + (- \overset{true‾}{\overset{true^}{v} \overset{true^}{w}} \frac{\partial V}{\partial z}) + (\overset{true‾}{{\overset{true^}{w}}^{2}} \frac{italicRo}{italicRe} \frac{\partial W}{\partial z})] dz}} \\ - \underset{II}{\underset{true⏟}{\int_{0}^{normal∞} (\overset{σ_{ij} \frac{\partial {\overset{true^}{u}}_{j}}{\partial x_{i}}}{true}) dz}} + \underset{III}{\underset{true⏟}{\int_{0}^{normal∞} (\frac{italicRo}{italicRe} \overset{true‾}{\overset{true^}{u} \overset{true^}{p}}) dz + {(\overset{true‾}{\overset{true^}{w} \overset{true^}{p}})}_{w}}} \\ - \underset{IV}{\underset{true⏟}{{false[\overset{true‾}{\overset{true^}{u} σ_{31}} + \overset{true‾}{\overset{true^}{v} σ_{32}} + \overset{true‾}{\overset{true^}{w} σ_{33}} false]}_{w}}} \\ + \int_{0}^{normal∞} \frac{italicRo}{italicRe} \frac{\partial \overset{true‾}{K}}{\partial z} Wdz + \int_{0}^{normal∞} \frac{italicRo}{italicRe} \end{array}

…”

Section: Convective Instability Analysismentioning

confidence: 99%

“…These terms can be physically interpreted as: The mathematical formulation of these terms for the general BEK system flow is provided in our previous study [30] and is given by…”

Section: Energy Analysismentioning

confidence: 99%

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On the instability of Ekman boundary‐layer flow over a rough disk with a wavy surface profile

Alveroğlu

2020

Numerical Methods Partial

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The instability of non-Newtonian boundary-layer flows over rough rotating disks

Alqarni

Alveroğlu

Griffiths

et al. 2019

Journal of Non-Newtonian Fluid Mechanics

Self Cite

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Stability and control of an annular rotor/stator cavity limit cycle

2020

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Rotating cavity flows have been widely studied for years because of many implications that these have on industrial applications. These flows can indeed generate, under specific conditions, self-sustained oscillations that can be noisy or even dangerous for the integrity of a system. The coherent structures or flow modes composing this unsteady phenomenon usually called “pressure band phenomenon” are misunderstood and therefore difficult to control. In the present study, the dynamics of an annular rotor/stator cavity is investigated to shed some light on the flow organization and identify control strategies based on reliable theory and analysis to stabilize the observed undesired flow modes. No specific tool is known today to control a multi-frequency phenomenon. To address this first issue, the mode dominance and interactions appearing in this multi-frequency problem are investigated, thanks to dynamic mode tracking and control [M. Queguineur et al., “Dynamic mode tracking and control with a relaxation method,” Phys. Fluids 31, 034101 (2019)]. The benefit of this method is to be able to follow in time several modes while controlling them one by one and observe mode dominance and interactions. This purely numerical controller shows that, here, the dominant mode of the annular cavity is at the source of another low frequency mode. Based on this information and to develop a physically relevant control strategy, the global linear stability framework previously used by Queguineur et al. [“Large eddy simulations and global stability analyses of an annular and cylindrical rotor/stator cavity limit cycles,” Phys. Fluids 31, 104109 (2019)] is further developed to make use of the sensitivity to a base flow modification theory. This specific analysis indeed enables us to point out the exact location where the base flow should be modified to shift the dominant mode frequency and/or growth rate. In this context, passive controller positioning is identified for the studied annular cavity flow. Such strategies are then validated through new large eddy simulations of a controlled cavity using low amplitude injection/suction demonstrating the adequacy of the analysis and control strategy.

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An energy analysis of convective instabilities of the Bödewadt and Ekman boundary layers over rough surfaces

Cited by 7 publications

References 22 publications

On the instability of Ekman boundary‐layer flow over a rough disk with a wavy surface profile

On the instability of Ekman boundary‐layer flow over a rough disk with a wavy surface profile

The instability of non-Newtonian boundary-layer flows over rough rotating disks

Stability and control of an annular rotor/stator cavity limit cycle

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