4 results
The effect of an unsteady flow incident on an array of circular cylinders
- C. A. Klettner, I. Eames, J. C. R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 872 / 10 August 2019
- Published online by Cambridge University Press:
- 13 June 2019, pp. 560-593
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In this paper we investigate the effect of an inhomogeneous and unsteady velocity field incident on an array of rigid circular cylinders arranged within a circular perimeter (diameter
$D_{G}$) of varying solid fraction
$\unicode[STIX]{x1D719}$, where the unsteady flow is generated by placing a cylinder (diameter
$D_{G}$) upwind of the array. Unsteady two-dimensional viscous simulations at a moderate Reynolds number (
$Re=2100$) and also, as a means of extrapolating to a flow with a very high Reynolds number, inviscid rapid distortion theory (RDT) calculations were carried out. These novel RDT calculations required the circulation around each cylinder to be zero which was enforced using an iterative method. The two main differences which were highlighted was that the RDT calculations indicated that the tangential velocity component is amplified, both, at the front and sides of the array. For the unsteady viscous simulations this result did not occur as the two-dimensional vortices (of similar size to the array) are deflected away from the boundary and do not penetrate into the boundary layer. Secondly, the amplification is greater for the RDT calculations as for the unsteady finite Reynolds number calculations. For the two highest solid fraction arrays, the mean flow field has two recirculation regions in the near wake of the array, with closed streamlines that penetrate into the array which will have important implications for scalar transport. The increased bleed through the array at the lower solid fraction results in this recirculation region being displaced further downstream. The effect of inviscid blocking and viscous drag on the upstream streamwise velocity and strain field is investigated as it directly influences the ability of the large coherent structures to penetrate into the array and the subsequent forces exerted on the cylinders in the array. The average total force on the array was found to increase monotonically with increasing solid fraction. For high solid fraction
$\unicode[STIX]{x1D719}$, although the fluctuating forces on the individual cylinders is lower than for low
$\unicode[STIX]{x1D719}$, these forces are more correlated due to the proximity of the cylinders. The result is that for mid to high solid fraction arrays the fluctuating force on the array is insensitive to
$\unicode[STIX]{x1D719}$. For low
$\unicode[STIX]{x1D719}$, where the interaction of the cylinders is weak, the force statistics on the individual cylinders can be accurately estimated from the local slip velocity that occurs if the cylinders were removed.
Flow and passive transport in planar multipolar flows
- M. A. Zouache, I. Eames, C. A. Klettner, P. J. Luthert
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- Journal:
- Journal of Fluid Mechanics / Volume 858 / 10 January 2019
- Published online by Cambridge University Press:
- 02 November 2018, pp. 184-227
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We study the flow and transport of heat or mass, modelled as passive scalars, within a basic geometrical unit of a three-dimensional multipolar flow – a triangular prism – characterised by a side length
$L$, a normalised thickness
$0.01\leqslant \unicode[STIX]{x1D700}\leqslant 0.1$ and an apex angle
$0<\unicode[STIX]{x1D6FC}<\unicode[STIX]{x03C0}$, and connected to inlet and outlet pipes of equal normalised radius
$0.01\leqslant \unicode[STIX]{x1D6FF}\leqslant 0.1$ perpendicularly to the plane of the flow. The flow and scalar fields are investigated over the range
$0.1\leqslant Re_{p}\leqslant 10$ and
$0.1\leqslant Pe_{p}\leqslant 1000$, where
$Re_{p}$ and
$Pe_{p}$ are respectively the Reynolds and Péclet numbers imposed at the inlet pipe when either a Dirichlet (
$\text{D}$) or a Neumann (
$\text{N}$) scalar boundary condition is imposed at the wall unattached to the inlets and outlets. A scalar no-flux boundary condition is imposed at all the other walls. An axisymmetric model is applied to understand the flow and scalar transport in the inlet and outlet regions, which consist of a turning region close to the pipe centreline and a channel region away from it. A separate two-dimensional model is then developed for the channel region by solving the integral form of the momentum and scalar advection–diffusion equations. Analytical relations between geometrical, flow and scalar transport parameters based on similarity and integral methods are generated and agree closely with numerical solutions. Finally, three-dimensional numerical calculations are undertaken to test the validity of the axisymmetric and depth-averaged analyses. Dominant flow and scalar transport features vary dramatically across the flow domain. In the turning region, the flow is a largely irrotational straining flow when
$\unicode[STIX]{x1D700}\geqslant \unicode[STIX]{x1D6FF}$ and a dominantly viscous straining flow when
$\unicode[STIX]{x1D700}\ll \unicode[STIX]{x1D6FF}$. The thickness of the scalar boundary layer scales to the local Péclet number to the power
$1/3$. The diffusive flux
$j_{d}$ and the scalar
$C_{s}$ at the wall where (
$\text{D}$) or (
$\text{N}$) is imposed, respectively, are constant. In the channel region, the flow is parabolic and dominated by a source flow near the inlet and an irrotational straining flow away from it. When
$(\text{D})$ is imposed the scalar decreases exponentially with distance from the inlet and the normalised scalar transfer coefficient converges to
$\unicode[STIX]{x1D6EC}_{\infty }=2.5694$. When
$(\text{N})$ is imposed,
$C_{s}$ varies proportionally to surface area. Transport in the straining region downstream of the inlet is diffusion-limited, and
$j_{d}$ and
$C_{s}$ are functions of the geometrical parameters
$L$,
$\unicode[STIX]{x1D700}$,
$\unicode[STIX]{x1D6FC}$ and
$\unicode[STIX]{x1D6FF}$. In addition to describing the fundamental properties of the flow and passive transport in multipolar configurations, the present work demonstrates how geometrical and flow parameters should be set to control transfers in the different regions of the flow domain.
The effect of a uniform through-surface flow on a cylinder and sphere
- C. A. Klettner, I. Eames, S. Semsarzadeh, A. Nicolle
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- Journal:
- Journal of Fluid Mechanics / Volume 793 / 25 April 2016
- Published online by Cambridge University Press:
- 23 March 2016, pp. 798-839
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The effect of a uniform through-surface flow (velocity
$U_{b}$) on a rigid and stationary cylinder and sphere (radius
$a$) fixed in a free stream (velocity
$U_{\infty }$) is analysed analytically and numerically. The flow is characterised by a dimensionless blow velocity
${\it\Lambda}\,(=U_{b}/U_{\infty })$ and Reynolds number
$Re\,(=2aU_{\infty }/{\it\nu}$, where
${\it\nu}$ is the kinematic viscosity). High resolution numerical calculations are compared against theoretical predictions over the range
$-3\leqslant {\it\Lambda}\leqslant 3$ and
$Re=1,10,100$ for planar flow past a cylinder and axisymmetric flow past a sphere. For
$-{\it\Lambda}\gg 1$, the flow is viscously dominated in a thin boundary layer of thickness
${\it\nu}/|U_{b}|$ adjacent to the rigid surface which develops in a time
${\it\nu}/U_{b}^{2}$; the surface vorticity scales as
$Re|{\it\Lambda}|U_{\infty }/a$ for a cylinder and sphere. A boundary layer analysis is developed to analyse the unsteady viscous forces. Numerical results show that the surface pressure and vorticity distribution within the boundary layer agrees with a steady state analysis. The flow downstream of the body is irrotational so the wake volume flux,
$Q_{w}$, is zero and the drag force is
$F_{D}=-{\it\rho}U_{\infty }Q_{b}$, where
${\it\rho}$ is the density of the fluid and
$Q_{b}$ is the normal flux through the body surface. The drag coefficient is therefore
$-2{\rm\pi}{\it\Lambda}$ or
$-8{\it\Lambda}$ for a cylinder or sphere, respectively. A dissipation argument is applied to analyse the drag force; the rate of working of the drag force is balanced by viscous dissipation, flux of stagnation pressure and rate of work by viscous stresses due to sucking. At large
$Re|{\it\Lambda}|$, the drag force is largely determined by viscous dissipation for a cylinder, with a weak contribution by the normal viscous stresses, while for a sphere, only
$3/4$ of the drag force is determined by viscous dissipation with the remaining
$1/4$ due to the flux of stagnation pressure through the sphere surface. When
${\it\Lambda}\gg 1$, the boundary layer thickness initially grows linearly with time as vorticity is blown away from the rigid surface. The vorticity in the boundary layer is weakly dependent on viscous effects and scales as
$U_{\infty }/a{\it\Lambda}$ or
$U_{\infty }/a{\it\Lambda}^{3/2}$ for a cylinder and sphere, respectively. For large blow velocity, the vorticity is swept into two well-separated shear layers and the maximum vorticity decreases due to diffusion. The drag force is related to the vorticity distribution on the body surface and an approximate expression can be derived by considering the first term of a Fourier expansion in the surface vorticity. It is found that the drag coefficient
$C_{D}$ for a cylinder (corrected for flow boundedness) is weakly dependent on
${\it\Lambda}$ while for a sphere,
$C_{D}$ decreases with
${\it\Lambda}$.
Low-Reynolds-number flow past a cylinder with uniform blowing or sucking
- C. A. Klettner, I. Eames
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- Journal:
- Journal of Fluid Mechanics / Volume 780 / 10 October 2015
- Published online by Cambridge University Press:
- 09 September 2015, R2
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We analyse the low-Reynolds-number flow generated by a cylinder (of radius
$a$) in a stream (of velocity
$U_{\infty }$) which has a uniform through-surface blowing component (of velocity
$U_{b}$). The flow is characterized in terms of the Reynolds number
$Re$ (
$=2aU_{\infty }/{\it\nu}$, where
${\it\nu}$ is the kinematic viscosity of the fluid) and the dimensionless blow velocity
${\it\Lambda}$ (
$=U_{b}/U_{\infty }$). We seek the leading-order symmetric solution of the vorticity field which satisfies the near- and far-field boundary conditions. The drag coefficient is then determined from the vorticity field. For the no-blow case Lamb’s (Phil. Mag., vol. 21, 1911, pp. 112–121) expression is retrieved for
$Re\rightarrow 0$. For the strong-sucking case, the asymptotic limit,
$C_{D}\approx -2{\rm\pi}{\it\Lambda}$, is confirmed. The blowing solution is valid for
${\it\Lambda}<4/Re$, after which the flow is unsymmetrical about
${\it\theta}={\rm\pi}/2$. The analytical results are compared with full numerical solutions for the drag coefficient
$C_{D}$ and the fraction of drag due to viscous stresses. The predictions show good agreement for
$Re=0.1$ and
${\it\Lambda}=-5,0,5$.
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