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Edmund Chadwick - One of the best experts on this subject based on the ideXlab platform.
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experimental verification of an oseen flow Slender Body theory
Journal of Fluid Mechanics, 2010Co-Authors: Edmund Chadwick, H M Khan, Mojtaba Moatamedi, M H Mappin, M A PenneyAbstract:Consider uniform flow past four Slender bodies with elliptical cross-section of constant ellipticity along the length of 0, 0.125, 0.25 and 0.375, respectively, for each Body. Here, ellipticity is defined as the ratio of the semiminor axis of the ellipse to the semimajor axis. The bodies have a pointed nose which gradually increases in cross-section with a radius of curvature 419mm to a mid-section which then remains constant up to a blunt end section with semimajor axis diameter 160 mm, the total length of all bodies being 800 mm. The bodies are side-mounted within a low-speed wind tunnel with an operational wind speed of the order 30ms−1. The side force (or lift) is measured within an angle of attack range of −3◦ to 3◦ such that the Body is rotated about the major axis of the ellipse cross-section. The lift slope is determined for each Body, and how it varies with ellipticity. It is found that this variance follows a straight line which steadily increases with increasing ellipticity. It is shown that this result is predicted by a recently developed Oseen flow Slender Body theory, and cannot be predicted by either inviscid flow Slender Body theory or viscous crossflow theories based upon the Allen and Perkins method.
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Slender Body expansions in potential theory along a finite straight line
Zeitschrift für angewandte Mathematik und Physik, 2010Co-Authors: Edmund Chadwick, Ali HatamAbstract:Consider a distribution of singularities in a potential field along a finite straight line such that the potential satisfies the Laplace equation. An example is a distribution of sources representing a ship or missile moving with forward velocity in a potential inviscid flow field. Such bodies are often truncated or bluff at the ends, and so the strength of the resulting distributions may not gradually tend to zero close to these ends and may instead be non-zero finite. A near-field expansion is obtained which accounts for this using the Slender Body theory integral splitting method. All terms in the expansion are obtained, and the coefficient of each term in the infinite sequence is given in terms of differentials of the distribution strength. Hence an exact separation of variables solution (separating the axial distance from the cross-sectional distances) is obtained for the potential. This is different from previous representations in that it represents a distribution over a finite length, and the resulting expansion is a simple single summation expression that is straightforward to apply. The resulting numerical scheme is discussed, in particular the evaluation close to the ends and also a comparison between the presented Slender Body theory and existing numerical methods.
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A Slender Body theory in oseen flow obtained by expanding the Oseenlets in the Green's integral representation
Fluid Dynamics Research, 2009Co-Authors: Edmund ChadwickAbstract:Consider steady, uniform flow at a small angle of incidence to a Slender Body. Let us assume that Oseen flow, the far field approximation obtained by linearizing the velocity about the uniform flow velocity, also holds in a near-field region that continues right up to the Body boundary layer where the impermeability (slip) boundary condition is applied. Let us assume that in this near field the standard Slender Body approximation also holds, that changes along the Body length are much smaller than in the transverse plane. A Green's integral representation is then given over a near-field surface that encloses the Body. The Taylor series expansion is then further applied to give a representation along a line axis within the Slender Body. An interesting consequence of the analysis, a jump in lift towards the rear of the Slender Body, is presented and discussed.
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a Slender Body theory in oseen flow
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2002Co-Authors: Edmund ChadwickAbstract:Consider steady flow generated by a uniform velocity field past a fixed closed Slender Body whose major axis is aligned closely to the uniform stream direction. Let us assume Oseen flow with the slip boundary condition. A Slender-Body theory is presented. In the near field, let us assume that the second-derivative changes in the velocity and pressure fields are of lower order in the axial direction of the Slender Body than in the transverse plane. The hydrodynamic forces are then related to the Body shape by using matched asymptotics.
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A Slender–Body theory in Oseen flow
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2002Co-Authors: Edmund ChadwickAbstract:Consider steady flow generated by a uniform velocity field past a fixed closed Slender Body whose major axis is aligned closely to the uniform stream direction. Let us assume Oseen flow with the slip boundary condition. A Slender-Body theory is presented. In the near field, let us assume that the second-derivative changes in the velocity and pressure fields are of lower order in the axial direction of the Slender Body than in the transverse plane. The hydrodynamic forces are then related to the Body shape by using matched asymptotics.
Yoichiro Mori - One of the best experts on this subject based on the ideXlab platform.
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an error bound for the Slender Body approximation of a thin rigid fiber sedimenting in stokes flow
Research in the Mathematical Sciences, 2020Co-Authors: Yoichiro MoriAbstract:We investigate the motion of a thin rigid Body in Stokes flow and the corresponding Slender Body approximation used to model sedimenting fibers. In particular, we derive a rigorous error bound comparing the rigid Slender Body approximation to the classical PDE for rigid motion in the case of a closed loop with constant radius. Our main tool is the Slender Body PDE framework established by the authors and D. Spirn in [18,19], which we adapt to the rigid setting.
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Theoretical Justification and Error Analysis for Slender Body Theory with Free Ends
Archive for Rational Mechanics and Analysis, 2019Co-Authors: Yoichiro Mori, Laurel Ohm, Daniel SpirnAbstract:Slender Body theory is a commonly used approximation in computational models of thin fibers in viscous fluids, especially in simulating the motion of cilia or flagella in swimming microorganisms. In Mori et al. (Commun Pure Appl Math, 2018. arXiv:1807.00178 ), we developed a PDE framework for analyzing the error introduced by the Slender Body approximation for closed-loop fibers with constant radius $$\varepsilon $$ ε , and showed that the difference between our closed-loop PDE solution and the Slender Body approximation is bounded by an expression proportional to $$\varepsilon |\log \varepsilon |$$ ε | log ε | . Here we extend the Slender Body PDE framework to the free endpoint setting, which is more physically relevant from a modeling standpoint but more technically demanding than the closed loop analysis. The main new difficulties arising in the free endpoint setting are defining the endpoint geometry, identifying the extent of the 1D Slender Body force density, and determining how the well-posedness constants depend on the non-constant fiber radius. Given a Slender fiber satisfying certain geometric constraints at the filament endpoints and a one-dimensional force density satisfying an endpoint decay condition, we show a bound for the difference between the solution to the Slender Body PDE and the Slender Body approximation in the free endpoint setting. The bound is a sum of the same $$\varepsilon |\log \varepsilon |$$ ε | log ε | term appearing in the closed loop setting and an endpoint term proportional to $$\varepsilon $$ ε , where $$\varepsilon $$ ε is now the maximum fiber radius.
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theoretical justification and error analysis for Slender Body theory
arXiv: Analysis of PDEs, 2018Co-Authors: Yoichiro Mori, Daniel SpirnAbstract:Slender Body theory facilitates computational simulations of thin fibers immersed in a viscous fluid by approximating each fiber using only the geometry of the fiber centerline curve and the line force density along it. However, it has been unclear how well Slender Body theory actually approximates Stokes flow about a thin but truly three-dimensional fiber, in part due to the fact that simply prescribing data along a one-dimensional curve does not result in a well-posed boundary value problem for the Stokes equations in $\mathbb{R}^3$. Here, we introduce a PDE problem to which Slender Body theory (SBT) provides an approximation, thereby placing SBT on firm theoretical footing. The Slender Body PDE is a new type of boundary value problem for Stokes flow where partial Dirichlet and partial Neumann conditions are specified everywhere along the fiber surface. Given only a 1D force density along a closed fiber, we show that the flow field exterior to the thin fiber is uniquely determined by imposing a fiber integrity condition: the surface velocity field on the fiber must be constant along cross sections orthogonal to the fiber centerline. Furthermore, a careful estimation of the residual, together with stability estimates provided by the PDE well-posedness framework, allow us to establish error estimates between the Slender Body approximation and the exact solution to the above problem. The error is bounded by an expression proportional to the fiber radius (up to logarithmic corrections) under mild regularity assumptions on the 1D force density and fiber centerline geometry.
Daniel Spirn - One of the best experts on this subject based on the ideXlab platform.
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Theoretical Justification and Error Analysis for Slender Body Theory with Free Ends
Archive for Rational Mechanics and Analysis, 2019Co-Authors: Yoichiro Mori, Laurel Ohm, Daniel SpirnAbstract:Slender Body theory is a commonly used approximation in computational models of thin fibers in viscous fluids, especially in simulating the motion of cilia or flagella in swimming microorganisms. In Mori et al. (Commun Pure Appl Math, 2018. arXiv:1807.00178 ), we developed a PDE framework for analyzing the error introduced by the Slender Body approximation for closed-loop fibers with constant radius $$\varepsilon $$ ε , and showed that the difference between our closed-loop PDE solution and the Slender Body approximation is bounded by an expression proportional to $$\varepsilon |\log \varepsilon |$$ ε | log ε | . Here we extend the Slender Body PDE framework to the free endpoint setting, which is more physically relevant from a modeling standpoint but more technically demanding than the closed loop analysis. The main new difficulties arising in the free endpoint setting are defining the endpoint geometry, identifying the extent of the 1D Slender Body force density, and determining how the well-posedness constants depend on the non-constant fiber radius. Given a Slender fiber satisfying certain geometric constraints at the filament endpoints and a one-dimensional force density satisfying an endpoint decay condition, we show a bound for the difference between the solution to the Slender Body PDE and the Slender Body approximation in the free endpoint setting. The bound is a sum of the same $$\varepsilon |\log \varepsilon |$$ ε | log ε | term appearing in the closed loop setting and an endpoint term proportional to $$\varepsilon $$ ε , where $$\varepsilon $$ ε is now the maximum fiber radius.
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theoretical justification and error analysis for Slender Body theory
arXiv: Analysis of PDEs, 2018Co-Authors: Yoichiro Mori, Daniel SpirnAbstract:Slender Body theory facilitates computational simulations of thin fibers immersed in a viscous fluid by approximating each fiber using only the geometry of the fiber centerline curve and the line force density along it. However, it has been unclear how well Slender Body theory actually approximates Stokes flow about a thin but truly three-dimensional fiber, in part due to the fact that simply prescribing data along a one-dimensional curve does not result in a well-posed boundary value problem for the Stokes equations in $\mathbb{R}^3$. Here, we introduce a PDE problem to which Slender Body theory (SBT) provides an approximation, thereby placing SBT on firm theoretical footing. The Slender Body PDE is a new type of boundary value problem for Stokes flow where partial Dirichlet and partial Neumann conditions are specified everywhere along the fiber surface. Given only a 1D force density along a closed fiber, we show that the flow field exterior to the thin fiber is uniquely determined by imposing a fiber integrity condition: the surface velocity field on the fiber must be constant along cross sections orthogonal to the fiber centerline. Furthermore, a careful estimation of the residual, together with stability estimates provided by the PDE well-posedness framework, allow us to establish error estimates between the Slender Body approximation and the exact solution to the above problem. The error is bounded by an expression proportional to the fiber radius (up to logarithmic corrections) under mild regularity assumptions on the 1D force density and fiber centerline geometry.
Haecheon Choi - One of the best experts on this subject based on the ideXlab platform.
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a discrete forcing immersed boundary method for the fluid structure interaction of an elastic Slender Body
Journal of Computational Physics, 2015Co-Authors: Haecheon ChoiAbstract:We present an immersed boundary (IB) method for the simulation of flow around an elastic Slender Body. The present method is based on the discrete-forcing IB method for a stationary, rigid Body proposed by Kim, Kim and Choi (2001) 25. The discrete-forcing approach is used to relieve the limitation on the computational time step size. The incompressible Navier-Stokes equations are implicitly coupled with the dynamic equation for an elastic Slender Body motion. The first is solved in the Eulerian coordinate and the latter is described in the Lagrangian coordinate. The elastic Slender Body is modeled as a thin and flexible solid and is segmented by finite number of thin blocks. Each block is moved by external and internal forces such as the hydrodynamic, elastic and buoyancy forces, where the hydrodynamic force is obtained directly from the discrete forcing used in the IB method. All the spatial derivative terms are discretized with the second-order central difference scheme. The present method is applied to three different fluid-structure interaction problems: flows around a flexible filament, a flapping flag in a free stream, and a flexible flapping wing in normal hovering, respectively. Computations are performed at maximum CFL numbers of 0.75-1. The results obtained agree very well with those from previous studies.
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a discrete forcing immersed boundary method for the fluid structure interaction of an elastic Slender Body
ASME-JSME-KSME 2011 Joint Fluids Engineering Conference: Volume 1 Symposia – Parts A B C and D, 2011Co-Authors: Haecheon ChoiAbstract:In the present study, a new immersed boundary method for the simulation of flow around an elastic Slender Body is suggested. The present method is based on the discrete-forcing immersed boundary method by Kim et al. (J. Comput. Phys., 2001) and is fully coupled with the elastic Slender Body motion. The incompressible Navier-Stokes equations are solved in an Eulerian coordinate and the elastic Slender Body motion is described in a Lagrangian coordinate, respectively. The elastic Slender Body is modeled as a thin flexible beam and is segmented by finite number of blocks. Each block is then moved by the external and internal forces such as the hydrodynamic, tension, bending, and buoyancy forces. With the proposed method, we simulate several flow problems including flows over a flexible filament, an oscillating insect wing, and a flapping flag. We show that the present method does not impose any severe limitation on the size of computational time step. The results obtained agree very well with those from previous studies.Copyright © 2011 by KSME
Eliot Fried - One of the best experts on this subject based on the ideXlab platform.
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Slender Body theory for viscous flow via dimensional reduction and hyperviscous regularization
Meccanica, 2014Co-Authors: Giulio G Giusteri, Eliot FriedAbstract:A new Slender-Body theory for viscous flow, based on the concepts of dimensional reduction and hyperviscous regularization, is presented. The geometry of flat, elongated, or point-like rigid bodies immersed in a viscous fluid is approximated by lower-dimensional objects, and a hyperviscous term is added to the flow equation. The hyperviscosity is given by the product of the ordinary viscosity with the square of a length that is shown to play the role of effective thickness of any lower-dimensional object. Explicit solutions of simple problems illustrate how the proposed method is able to represent with good approximation both the velocity field and the drag forces generated by rigid motions of the immersed bodies, in analogy with classical Slender-Body theories. This approach has the potential to open up the way to more effective computational techniques, since geometrical complexities can be significantly reduced. This, however, is achieved at the expense of involving higher-order derivatives of the velocity field. Importantly, both the dimensional reduction and the hyperviscous regularization, combined with suitable numerical schemes, can be used also in situations where inertia is not negligible.