On the stability of plane Couette-Poiseuille flow with uniform cross-flow

We present a detailed study of the linear stability of plane Couette-Poiseuille flow in the presence of a cross-flow. The base flow is characterised by the cross flow Reynolds number, $R_{inj}$ and the dimensionless wall velocity, $k$. Squire's transformation may be applied to the linear stability equations and we therefore consider 2D (spanwise-independent) perturbations. Corresponding to each dimensionless wall velocity, $k\in[0,1]$, two ranges of $R_{inj}$ exist where unconditional stability is observed. In the lower range of $R_{inj}$, for modest $k$ we have a stabilisation of long wavelengths leading to a cut-off $R_{inj}$. This lower cut-off results from skewing of the velocity profile away from a Poiseuille profile, shifting of the critical layers and the gradual decrease of energy production. Cross-flow stabilisation and Couette stabilisation appear to act via very similar mechanisms in this range, leading to the potential for robust compensatory design of flow stabilisation using either mechanism. As $R_{inj}$ is increased, we see first destabilisation and then stabilisation at very large $R_{inj}$. The instability is again a long wavelength mechanism. Analysis of the eigenspectrum suggests the cause of instability is due to resonant interactions of Tollmien-Schlichting waves. A linear energy analysis reveals that in this range the Reynolds stress becomes amplified, the critical layer is irrelevant and viscous dissipation is completely dominated by the energy production/negation, which approximately balances at criticality. The stabilisation at very large $R_{inj}$ appears to be due to decay in energy production, which diminishes like $R_{inj}^{-1}$. Our study is limited to two dimensional, spanwise independent perturbations.Comment: Accepted for publication in Journal of Fluid Mechanic

Bubble-Induced Entrainment at Viscoplastic-Newtonian Interfaces

The passage of single air bubbles through the horizontal interface between miscible viscoplastic and Newtonian fluids, considering various combinations of densities and viscosities for the fluid layers, is studied computationally. The primary focus is on the quantity of liquid transferred from the lower layer (Viscoplastic fluid) to the upper layer (Newtonian fluid) as a result of the bubble's ascent, a factor with significant implications for the turbidity of methane-emitting lakes and water bodies. The results show that at $Bo>1$ and moderate $Ar$, prolate-shaped bubbles crossing the interface undergo elongation in the direction of their poles. This elongation is further accentuated when the viscosity of upper layer is less than the plastic viscosity of the lower layer. The bubble is found to break up when leaving the lower layer, of a critical capillary number, $Ca_c \approx 5$. The results show a significant reduction in the volume of entrainment compared to the Newtonian counterpart. This suggests disturbances caused by the rising bubble at the interface dissipate over a smaller region. Four distinct entrainment regimes are identified, mainly indicating the height to which the entrained fluid can be transported away from the interface. In contrast to Newtonian fluids, the volume of entrainment increases by decreasing the viscosity of the upper layer. Interestingly, the heavy yield stress fluid that has been dragged up into the the light Newtonian fluid does not recede down by time

A New Three-Layer Model for Horizontal Slurry Flow

Paper presented at 2018 Canadian Society of Mechanical Engineers International Congress, 27-30 May 2018.A modified three-layer model for solid-liquid flow in horizontal pipes is developed, which overcomes the limitations of many previous models. The steady-state model predicts the pressure loss, critical velocity, concentration profile in the heterogeneous layer, mean heterogeneous layer and moving bed layer velocities, and bed layer heights for each set of parameters. We propose a new correlation for the turbulent solids diffusivity. This and the steady state model predictions show a good agreement with experimentally measured results in literature: for concentration profile in the heterogeneous layer and pressure loss, over a wide range of conditions [1]. In turbulent flow. the pressure loss vs mean velocity curve shows a characteristic minimum just before the critical velocity is attained

Clouds of bubbles in a viscoplastic fluid

Viscoplastic fluids can hold bubbles/particles stationary by balancing the buoyancy stress with the yield stress - the key parameter here is the yield number, the ratio of the yield stress to the buoyancy stress. In the present study, we investigate a suspension of bubbles in a yield-stress fluid. More precisely, we compute how much is the gas fraction that could be held trapped in a yield-stress fluid without motion. Here the goal is to shed light on how the bubbles feel their neighbours through the stress field and to compute the critical yield number for a bubble cloud beyond which the flow is suppressed. We perform two-dimensional computations in a full periodic box with randomized positions of the monosized circular bubbles. A large number of configurations are investigated to obtain statistically converged results. We intuitively expect that for higher volume fractions, the critical yield number is larger. Not only here do we establish that this is the case, but also we show that short-range interactions of bubbles increase the critical yield number even more dramatically for bubble clouds. The results show that the critical yield number is a linear function of volume fraction in the dilute regime. An algebraic expression model is given to approximate the critical yield number (semi-empirically) based on the numerical experiment in the studied range of 0 ≤ φ ≤ 0.31,, together with lower and upper estimates

Flow onset for a single bubble in a yield-stress fluid

We use computational methods to determine the minimal yield stress required in order to hold static a buoyant bubble in a yield-stress liquid. The static limit is governed by the bubble shape, the dimensionless surface tension and the ratio of the yield stress to the buoyancy stress . For a given geometry, bubbles are static for Y_c]]>, which we determine for a range of shapes. Given that surface tension is negligible, long prolate bubbles require larger yield stress to hold static compared with oblate bubbles. Non-zero increases and for large the yield-capillary number determines the static boundary. In this limit, although bubble shape is important, bubble orientation is not. Two-dimensional planar and axisymmetric bubbles are studied

Flow onset for a single bubble in a yield-stress fluid

We use computational methods to determine the minimal yield stress required in order to hold static a buoyant bubble in a yield-stress liquid. The static limit is governed by the bubble shape, the dimensionless surface tension and the ratio of the yield stress to the buoyancy stress . For a given geometry, bubbles are static for Y_c]]>, which we determine for a range of shapes. Given that surface tension is negligible, long prolate bubbles require larger yield stress to hold static compared with oblate bubbles. Non-zero increases and for large the yield-capillary number determines the static boundary. In this limit, although bubble shape is important, bubble orientation is not. Two-dimensional planar and axisymmetric bubbles are studied

Mathematical modelling of an aluminium spray process

Spray-forming is a newly developed industrial metal forming process in which a cylindrical metal billet is produced by the incremental deposition and solidification of an atomised metal spray on a moving substrate. A mathematical model is developed to describe billet growth and heat flow within spray-formed aluminium alloy billets. In the first part of the thesis, growth dynamics of the billet are considered. Conservation of mass at the billet surface yields a single first order quasi-linear partial differential equation for the movement of the billet surface; the nonlinearity arising from the possibility of surface shadowing. The existence of two distinctly different timescales, amongst the process motions governing billet growth, prompts the use of an averaging method. The resulting averaged equations permit analysis and are shown to provide a valid asymptotic approximation to the billet surface motion on the timescale 1/∊, for a suitably defined class of billet surfaces. The parameter ∊ ≪ 1 is the ratio of the two process timescales. Conditions under which the crown profile of the cylindrical billet becomes steady are analysed, through the averaged equations, and the stability of such profiles is examined. Computed examples of single and multiple steady state crown profiles are given. The averaged equations are also solved numerically to provide a model for transient billet growth on a "slow" timescale; results are presented. The second part of the thesis considers heat flow within the growing billet. Phase change is incorporated using an enthalpy formulation of the energy equation. The resulting equation is a nonlinear heat equation that must be solved in an expanding domain, the boundary of which is determined by solution of the billet growth model equations. Conduction on the billet length-scale takes place only on the slow timescale, with more rapid heat flow taking place only close to the billet surface. Accordingly, billet heat flow is analysed through the assumption that there is a thermal boundary layer close to the billet surface, which is driven by the "rapid" timescale spray deposition, with heat flow in the remainder of the billet driven by the time-averaged growth. The boundary layer equation is a one dimensional nonlinear advection-diffusion equation, with a nonlinear boundary condition that incorporates the intermittent deposition from the spray in the form of an irregular pulse. This equation is solved numerically using an implicit finite difference method. The slow-time heat flow is two dimensional, (assuming axisymmetric slow-time billet growth), and must also be solved numerically. For this an implicit predictor-corrector method is used. The predictor stage uses a "splitting" method, adapted from the fully implicit L.O.D. method to take account of the expanding domain. The method appears to be stable and consistent. Various numerical results are presented. The model provides significant new understanding of the dynamics of billet growth and succeeds in providing a useful framework within which the transient heat flow that occurs during spray deposition, on a number of different timescales and length-scales, can be understood. Comparison of computed model predictions with real sprayed billets confirms the validity of the model. The thesis is concluded with a summary of results and a look at possible future directions for research in this area.</p

A Comprehensive Study on Intermittent Operation of Horizontal Deep Borehole Heat Exchangers

Utilizing a deep Borehole Heat Exchanger (BHE) has been recognized as a clean, renewable, low-carbon-emission, and sustainable way for heating of residential buildings and greenhouses. In this study, the long-term performance of horizontal deep BHE in intermittent mode is scrutinized. In this regard, to predict the transient heat transfer process in the deep BHEs, a mathematical model is developed and then verified by using the experimental results. The effect various key parameters including flow rate of circulating fluid, undisturbed ground temperature, inlet fluid temperature, and ground thermal conductivity on the thermal performance of deep BHE in continuous and intermittent mode is studied. According to the results, increasing the flow rate of circulating fluid, undisturbed ground temperature, and ground thermal conductivity is favorable for heat extraction rate. Moreover, the effect of three specific parameters for intermittent operation including periodic time interval, flow rate ratio, and recovery period ratio on the long-term performance of horizontal deep BHE are scrutinized. Based on the results, by decreasing the periodic time interval and increasing the flow rate ratio, the mean heat extraction rate in the period of 30 years is increased and the mean borehole’s wall temperature is decreased. Furthermore, by increasing the recovery period ratio, the heat extraction rate increases significantly while the total extracted energy decreases.Applied Science, Faculty ofScience, Faculty ofMathematics, Department ofMechanical Engineering, Department ofReviewedFacult