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From three-dimensional to quasi-two-dimensional: transient growth in magnetohydrodynamic duct flows
- Oliver G. W. Cassells, Tony Vo, Alban Pothérat, Gregory J. Sheard
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- Journal:
- Journal of Fluid Mechanics / Volume 861 / 25 February 2019
- Published online by Cambridge University Press:
- 19 December 2018, pp. 382-406
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This study seeks to elucidate the linear transient growth mechanisms in a uniform duct with square cross-section applicable to flows of electrically conducting fluids under the influence of an external magnetic field. A particular focus is given to the question of whether at high magnetic fields purely two-dimensional mechanisms exist, and whether these can be described by a computationally inexpensive quasi-two-dimensional model. Two Reynolds numbers of $5000$ and $15\,000$ and an extensive range of Hartmann numbers $0\leqslant \mathit{Ha}\leqslant 800$ were investigated. Three broad regimes are identified in which optimal mode topology and non-modal growth mechanisms are distinct. These regimes, corresponding to low, moderate and high magnetic field strengths, are found to be governed by the independent parameters; Hartmann number, Reynolds number based on the Hartmann layer thickness $R_{H}$ and Reynolds number built upon the Shercliff layer thickness $R_{S}$, respectively. Transition between regimes respectively occurs at $\mathit{Ha}\approx 2$ and no lower than $R_{H}\approx 33.\dot{3}$. Notably for the high Hartmann number regime, quasi-two-dimensional magnetohydrodynamic models are shown to be excellent predictors of not only transient growth magnitudes, but also the fundamental growth mechanisms of linear disturbances. This paves the way for a precise analysis of transition to quasi-two-dimensional turbulence at much higher Hartmann numbers than is currently achievable.
Pressure-driven flow past spheres moving in a circular tube
- G. J. SHEARD, K. RYAN
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- Journal:
- Journal of Fluid Mechanics / Volume 592 / 10 December 2007
- Published online by Cambridge University Press:
- 14 November 2007, pp. 233-262
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A computational investigation, supported by a theoretical analysis, is performed to investigate a pressure-driven flow around a line of equispaced spheres moving at a prescribed velocity along the axis of a circular tube. This fundamental study underpins a range of applications including physiological circulation research. A spectral-element formulation in cylindrical coordinates is employed to solve for the incompressible fluid flow past the spheres, and the flows are computed in the reference frame of the translating spheres.
Both the volume flow rate relative to the spheres and the forces acting on each sphere are computed for specific sphere-to-tube diameter ratios and sphere spacing ratios. Conditions at which zero axial force on the spheres are identified, and a region of unsteady flow is detected at higher Reynolds numbers (based on tube diameter and sphere velocity). A regular perturbation analysis and the reciprocal theorem are employed to predict flow rate and drag coefficient trends at low Reynolds numbers. Importantly, the zero drag condition is well-described by theory, and states that at this condition, the sphere velocity is proportional to the applied pressure gradient. This result was verified for a range of spacing and diameter ratios. Theoretical approximations agree with computational results for Reynolds numbers up to O(100).
The geometry dependence of the zero axial force condition is examined, and for a particular choice of the applied dimensionless pressure gradient, it is found that this condition occurs at increasing Reynolds numbers with increasing diameter ratio, and decreasing Reynolds number with increasing sphere spacing.
Three-dimensional simulations and predictions of a Floquet linear stability analysis independently elucidate the bifurcation scenario with increasing Reynolds number for a specific diameter ratio and sphere spacing. The steady axisymmetric flow first experiences a small region of time-dependent non-axisymmetric instability, before undergoing a regular bifurcation to a steady non-axisymmetric state with azimuthal symmetry m = 1. Landau modelling verifies that both the regular non-axisymmetric mode and the axisymmetric Hopf transition occur through a supercritical (non-hysteretic) bifurcation.
Manipulating meat quality and composition
- J. D. Wood, M. Enser, A. V. Fisher, G. R. Nute, R. I. Richardson, P. R. Sheard
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- Journal:
- Proceedings of the Nutrition Society / Volume 58 / Issue 2 / May 1999
- Published online by Cambridge University Press:
- 28 February 2007, pp. 363-370
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Meat quality describes the attractiveness of meat to consumers. The present paper focuses on two major aspects of meat quality, tenderness and flavour. Both aspects of quality can be influenced by nutrition, principally through its effects on the amount and type of fat in meat. In several countries, high levels of intramuscular fat (marbling fat), i.e. above 30 g/kg muscle weight in longissimus, are deemed necessary for optimum tenderness, although poor relationships between fat content and tenderness have generally been found in European studies, where fat levels are often very low, e.g. below 10 g/kg in UK pigs. Muscle lipid may be a marker for red oxidative (type 1) muscle fibres which are found at higher concentrations in tender muscles and carcasses. Nutritional treatment can be used to manipulate the fatty acid content of muscle to improve nutritional balance, i.e. increase the polyunsaturated (PUFA) : saturated fatty acid value and reduce the n−6 : n−3 PUFA value. Increasing PUFA levels may also change flavour because of their greater susceptibility to oxidative breakdown and the generation of abnormal volatile compounds during cooking. This situation particularly applies to the n−3 PUFA which are the most unsaturated meat lipids. In pigs, a concentration of 3 mg α-linolenic acid (18 : 3)/100 mg in muscle and fat tissue fatty acids can easily be achieved by including whole linseed in the diet. This level has led to abnormal odours and flavours in some studies, but not in others. In cattle and sheep, feeding whole linseed raised 18 : 3 concentrations in muscle fatty acids from about 0.7 mg/100 mg to > 1 mg/100 mg. As with pigs, this diet also increased levels of long-chain n−3 PUFA formed from 18 : 3, including eicosapentaenoic acid (20 : 5). Although this increase led to greater oxidative breakdown of lipids during storage and the generation of large quantities of lipid-derived volatile compounds during cooking, there were no deleterious effects on odour or flavour. When 18 : 3 levels are raised in lamb and beef because of grass feeding, the intensity of the flavours increases in comparison with grain-fed animals which consume and deposit relatively more linoleic acid (18 : 2). In ruminants, very high levels of 18 : 2 produced by feeding protected oil supplements cause the cooked beef to be described as oily, bland or pork-like.
The evolution of a subharmonic mode in a vortex street
- G. J. SHEARD, M. C. THOMPSON, K. HOURIGAN, T. LEWEKE
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- Journal:
- Journal of Fluid Mechanics / Volume 534 / 10 July 2005
- Published online by Cambridge University Press:
- 21 June 2005, pp. 23-38
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The development of a subharmonic three-dimensional instability mode in a vortex street is investigated both numerically and experimentally. The flow past a ring is considered as a test case, as a previous stability analysis has predicted that for a range of aspect ratios, the first-occurring instability of the vortex street is subharmonic. For the flow past a circular cylinder, the development of three-dimensional flow in the vortex street is known to lead to turbulent flow through the development of spatio-temporal chaos, whereas subharmonic instabilities have been shown to cause a route to chaos through the development of a period-doubling cascade. The three-dimensional vortex street in the flow past a ring is analysed to determine if a subharmonic instability can alter the route to turbulence for a vortex street.
A linear stability analysis and non-axisymmetric computations are employed to compute the flow past a ring with an aspect ratio ${\sc ar}\,{=}\,5$, and comparisons with experimental dye visualizations are included to verify the existence of a subharmonic mode in the wake. Computations at higher Reynolds numbers confirm that the subharmonic instability does not initiate a period-doubling cascade in the wake.
Computations of the drag coefficients for low-Reynolds-number flow past rings
- G. J. SHEARD, K. HOURIGAN, M. C. THOMPSON
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- Journal:
- Journal of Fluid Mechanics / Volume 526 / 10 March 2005
- Published online by Cambridge University Press:
- 25 February 2005, pp. 257-275
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The variation in the drag coefficient for low-Reynolds-number flow past rings orientated normal to the direction of flow is investigated numerically. An aspect ratio parameter is used for a ring, which describes at its limits a sphere and a circular cylinder. This enables a continuous range of bodies between a sphere and a circular cylinder to be studied.
The computed drag coefficients for the flow past rings at the minimum and maximum aspect ratio limits are compared with the measured and computed drag coefficients reported for the sphere and the circular cylinder. Some interesting features of the behaviour of the drag coefficients with variation of Reynolds number and aspect ratio emerge from the study. These include the decrease in the aspect ratio at which the minimum drag coefficient occurs as the Reynolds number is increased, from $\hbox{\it Ar} \,{\approx}\, 5$ at $\hbox{\it Re} \,{=}\, 1$ to $\hbox{\it Ar} \,{\approx}\, 1$ at $\hbox{\it Re} \,{=}\, 200$. In addition, a substantial decrease in the pressure component of the drag coefficient is observed after the onset of three-dimensional flow while the viscous contribution is similar to that for flow with imposed axisymmetry. Typically, the sudden reduction in drag caused by transition to Mode A shedding is 6%, which is consistent with the behaviour for flow past a circular cylinder. Power-law fits to the drag coefficient for $\hbox{\it Re} \,{\lesssim}\, 100$ are provided, which are accurate within approximately 2%.
From spheres to circular cylinders: non-axisymmetric transitions in the flow past rings
- G. J. SHEARD, M. C. THOMPSON, K. HOURIGAN
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- Journal:
- Journal of Fluid Mechanics / Volume 506 / 10 May 2004
- Published online by Cambridge University Press:
- 28 April 2004, pp. 45-78
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Non-axisymmetric simulations verify and extend the results from a previous linear Floquet stability analysis of the wakes of rings. The wakes corresponding to the saturated state of each predicted non-axisymmetric instability mode over the entire aspect ratio range are successfully computed, and isosurface plots are presented elucidating the vortical wake structures. The existence of three non-axisymmetric flow regimes (Modes I, II and III) for the flow past rings with aspect ratios $\textit{Ar\/} \lesssim 3.9$ is verified, as is the existence of non-axisymmetric instabilities of vortex streets in the flow past rings with $\textit{Ar\/} \gtrsim 3.9$. Wakes are computed which correspond to the Mode A and B instabilities found in the flow past a circular cylinder, and a wake is computed which develops from a subharmonic Mode C instability. This wake features an azimuthal wavelength of approximately 1.7 ring cross-section diameters, which is between the azimuthal wavelengths of the Mode A and B instabilities. This mode cannot occur, at least in a pure state, in the flow past a circular cylinder.
Nonlinear transition characteristics are predicted by evaluating coefficients of the truncated Landau equation, and transition hysteresis is verified by studying the mode amplitude variation with Reynolds number in the vicinity of the transitions. The regular Mode I and Mode III transitions are found to occur through supercritical and subcritical bifurcations, respectively, and the secondary Hopf bifurcations to these transitions, as well as the Mode II Hopf transition, are found to be supercritical. We verify that the Mode A and Mode B transitions are subcritical and supercritical, respectively, and we determine that the nature of the Mode C transition is dependent on aspect ratio. Landau constants are evaluated for the Hopf transitions throughout the aspect ratio range studied.
From spheres to circular cylinders: the stability and flow structures of bluff ring wakes
- G. J. SHEARD, M. C. THOMPSON, K. HOURIGAN
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- Journal:
- Journal of Fluid Mechanics / Volume 492 / 10 October 2003
- Published online by Cambridge University Press:
- 16 September 2003, pp. 147-180
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The low-Reynolds-number wake dynamics and stability of the flow past toroids placed normal to the flow direction are studied numerically. This bluff body has the attractive feature of behaving like the sphere at small aspect ratios, and locally like the straight circular cylinder at large aspect ratios. Importantly, the geometry of the ring is described by a single parameter, the aspect ratio ($\hbox{\it Ar}$), defined as a ratio of the torus diameter to the cross-sectional diameter of the ring. A rich diversity of wake topologies and flow transitions can therefore be investigated by varying the aspect ratio. Studying this geometry allows our understanding to be developed as to why the wake transitions leading to turbulence for the sphere and circular cylinder differ so greatly. Strouhal–Reynolds-number profiles are determined for a range of ring aspect ratios, as are critical Reynolds numbers for the onset of flow separation, unsteady flow and asymmetry. Results are compared with experimental findings from the literature. Calculated Strouhal–Reynolds-number profiles show that ring wakes shed at frequencies progressively closer to that of the straight circular cylinder wake as aspect ratio is increased from $\hbox{\it Ar}\,{ =}\, 3$. For $\hbox{\it Ar} \,{>}\, 8$, the initial asymmetric transition is structurally analogous to the mode A transition for the circular cylinder, with a discontinuity present in the Strouhal–Reynolds-number profile. The present numerical study reveals a shedding-frequency decrease with decreasing aspect ratio for ring wakes, and an increase in the critical Reynolds numbers for flow separation and the unsteady flow transition. A Floquet stability analysis has revealed the existence of three modes of asymmetric vortex shedding in the wake of larger rings. Two of these modes are analogous to mode A and mode B of the circular cylinder wake, and the third mode, mode C, is analogous to the intermediate wavelength mode found in the wake of square section cylinders and circular cylinder wakes perturbed by a tripwire. Furthermore, three distinct asymmetric transition modes have been identified in the wake of small aspect ratio bluff rings. Fully developed asymmetric simulations have verified the unsteady transition for rings that exhibit a steady asymmetric wake.