Applied Computing Research Report series
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Item An extension to the theory of steady selective withdrawal for a two layer fluid(Lincoln University. Applied Computing, Mathematics and Statistics Group., 2000-05) Wood, Ian R; Choo, KennethMost reservoirs contain stratified fluid and selective withdrawal is used to obtain water of the desired properties. Initially we review the case with an infinite upper layer with a sharp interface. When the total discharge is specified, then the ratio of the discharge from each layer is determined by the criteria of smoothness at the virtual control (i.e. the critical point). At this point, the long wave velocity on the interface is zero. For the case when the upper layer depth is large, we show that the control is in the valve and the virtual control (which determines the ratio of the discharge in each layer) moves further from the source as the total discharge increases. When there is a finite upper layer, a portion of the flow is in the duct and a portion of the flow is in the free surface. We derive the criteria for the virtual control in the free surface flow and show that the duct control occurs first. If we then assume that the flow is not over-specified, we determine the necessary conditions for a smooth transition between the duct and the free surface flow. This enables us to determine the minimum ratio of the upper layer depth to the lower layer depth for the steady duct solution to be valid. This contrasts with the conclusions of Bryant and Wood (1976).Item Solute dispersion by 1D stepped velocity fluctuations(Lincoln University. Applied Computing, Mathematics and Statistics Group., 2003) Verwoerd, W; Kulasiri, DonThe effect of fluctuations in the drift velocity on dispersion by a porous medium is investigated. An analytical model is developed which represents the effect of a single discrete step in the velocity of a 1 dimensional flow as a multiplicative factor that modifies the underlying linear growth in solute variance predicted by the standard advection-dispersion equation. The algebraic structure of the model identifies two variable combinations ∆ and α that characterize the step and the rate of stochastic dispersion respectively, in terms of which a simple formula for the downstream effect of the step on dispersion is obtained. This formalism is next applied to a sequence of 3 steps representing a velocity fluctuation, and it is shown that while kinetic compression effects cancel out across such a fluctuation, the stochastic dispersion increases for any plausible combination of ∆ and α. This implies that a dispersion enhancement factor ƒ is associated with a fluctuation, and a simple formula is obtained for this in terms of variables that describe the fluctuation length and amplitude. Moreover, the algebra leads to the definition of a natural length scale Λ related to the Peelet number of the flow. Repeated application of this formula is used to find the cumulative dispersion enhancement by a sequence of identical fluctuations, leading to an expression for dispersivity as a function of the distance traversed by a solute plume. Key features of the model are that the dispersivity behaves differently for traversal lengths above and below Λ, and that above this transition it is proportional to a fractional power of the traversal length. These features are in agreement with experimental observation of scale-dependent dispersivity, but quantitatively the observed growth in dispersivity over several orders of magnitude is not obtained for any reasonable choice of parameter values.Item Scale-dependent dispersivity: a velocity fluctuation model(Applied Management and Computing Division, 2003-08) Verwoerd, Wynand S.In the previous paper, (03/2001) it was shown that the cumulative effect of multiple one-dimensional velocity fluctuations can explain qualitative features of the observed scale dependent dispersivity in natural aquifers, but not the magnitude of the effect. It is plausible that in real systems the enhancement of dispersion caused by a single fluctuation may be larger than that derived for the 1 dimensional stepped fluctuation, because for example there are additional enhancement mechanisms in 2 and 3 dimensional systems. However this paper shows that to achieve the observed magnitude, it is not enough to increase the size of enhancement factor but in addition the rate at which the effect of a single fluctuation changes with fluctuation length and with position along the fluctuation sequence need to be modified. Several variations are explored. Simple assumptions are shown to lead to dispersivity formulas in terms of purely algebraic power laws, while more elaborate assumptions yield expressions that are still analytic but contain non-elementary functions. In either case it is possible to find the required variation of the dispersivity over 3 or more orders of magnitude and with curve shapes that are consistent with historical observations. Moreover, this is achieved with plausible parameter values, leading for example to the conjecture that in the observed systems the porous medium could not have been homogeneous on a scale of more than centimeters. The model presented is schematic in the sense that it contains some detail assumptions not derived from first principles, but is believed to capture the essentials of the mechanism that causes scale dependent dispersivity. It sets some boundaries for viable detail models, but within those boundaries the final predictions are not very sensitive to the detail assumptions. A key merit of the treatment is that it identifies crucial variables that need to be measured or controlled in experimental studies.Item Determination of fat content in retail ready meat samples using image analysis(Lincoln University. Applied Computing, Mathematics and Statistics Group, 2003-08) Chandraratne, Meegalla R.; Samarasinghe, Sandhya; Kulasiri, Gamalathge D.; Isherwood, Peter; Bekhit, A. E. D.; Bickerstaffe, RoyAs a result of constantly growing consumer expectations for meat quality, the meat industry is placing more and more emphasis on quality assurance issues. Fat content in meat influences some important meat quality parameters and meat marketability. Visible fat includes marbling (intramuscular) and intermuscular fat. Chemical analysis is currently used to determine the fat percentage in meat. However, this is a tedious, expensive and time-consuming method. Some measurements, like the number, size distribution and spatial distribution of marbling, are totally impossible by chemical analysis. For the meat industry, it is very useful to have an accurate, reliable, cost effective, fast and nondestructive technique to determine the fat content. Computer vision has enormous potential for evaluating meat quality because image processing and analysis techniques can quantitatively and consistently characterize complex geometric, colour and textural properties. The objectives of the present study were: a) to apply image processing techniques to quantify fat content of beef and lamb steaks; b) to develop a relationship between the chemical fat content and the fat content measured by image analysis.Item Constrained visualization using the Shepard Interpolation Family(Lincoln University. Applied Computing, Mathematics and Statistics Group, 2003-12) Brodlie, K. W.; Asim, M. R.; Unsworth, KeithThis paper discusses the problem of visualizing data where there are underlying constraints that must be preserved. For example, we may know that the data is inherently positive. We show how the Modified Quadratic Shepard method, which interpolates scattered data of any dimensionality, can be constrained to preserve positivity. We do this by forcing the quadratic basis functions to be positive. The method can be extended to handle other types of contraints, including lower bound of 0 and upper bound of 1, as occurs with fractional data. A further extension allows general range restrictions, creating an interpolant that lies between any two specified functions as the lower and upper bounds.