Non-dimensional Coordinates Research Articles

A numerical investigation of laminar planar hydraulic jump in Herschel-Bulkley fluid

Laminar planar hydraulic jump during viscoplastic liquid flow in a horizontal channel is investigated through experiments and numerical simulation using Herschel-Bulkley (HB) model. The simulations are performed using the phase-field method with Papanastasiou regularization parameter and validated against experimental results. Both experiments and simulations show the free surface height to gradually increase upstream of jump and recede after the jump with a remarkable increase in free surface height and surface waviness at the jump. The model further reveals that an increase in any of the rheological parameters [yield stress (τo) flow behaviour index (n) and flow consistency index (k)] keeping the other properties constant increases film thickness. This increases jump strength and shifts jump towards the entry. However, each parameter influences free surface profile and jump characteristics in a different way. While a higher τo suppresses the development of the shear zone and results in a thicker plug zone, a higher n increases shear zone thickness and decreases the plug zone thickness. On the other hand, a higher k increases both shear and plug zone thickness. The steady state fully developed self-similar velocity profile is independent of k and depends on τo and n. Different jump types, obtained from simulations, are presented as phase diagrams in non-dimensional coordinates for a generalised approach.

Non-similar thermal transport analysis in entropy optimized magnetic nanofluids flow by considering effective Prandtl number model with melting heat transfer and Joule heating

Non-similar convection heat transfer analysis is performed for alumina (γAl2O3) - water (H2O) and alumina – ethylene glycol (C2H6O2) nanofluids flows over vertically stretching heated plate. Considerable interest has been placed on alumina metal as it is used in diverse engineering applications including electronic industry. Buoyancy in terms of temperature difference is incorporated in the momentum equation. An effective Prandtl number model is employed in the energy equation with melting heat transport and Joule heating. Non-similar mathematical modeling is employed for thermal transport and entropy analysis. Non-similar transformations comprising of non-dimensional coordinates (ξ, η) are utilized to convert the governing system into dimensionless partial differential system. Numerical simulations are performed using method of local non-similarity up-to second order of truncation via bvp4c. Effects of melting parameter (M), magnetic parameter (Mn) and volume fraction (φ) on velocity and temperature profiles are illustrated graphically. It is concluded that velocity profile increases for larger estimations of nanoparticles volume fraction. Decline in temperature profile is noticed for increasing values of melting parameter. Furthermore, entropy generation, the skin friction and heat transfer rate are investigated through graphs and tables.

Development of Turboshaft Engine Adaptive Dynamic Model: Analysis of Estimation Errors

Abstract One of the most perspective directions of aircraft engine development is related to implementing adaptive automatic electronic control systems (ACS). The significant elements of these systems are algorithms of matching of mathematical models to actual performances of the engine. These adaptive models are used directly in control algorithms and are a combination of static and dynamic sub-models. This work considers the dynamic sub-models formation using the Least Square method (LSM) on a base of the engine parameters that are measured in-flight. While implementing this function in the (ACS), the problem of checking the sufficiency of the used information for ensuring the required precision of the model arises. We must do this checking a priori (to determine a set of operation modes, the shape of the engine test impact and volume of recorded information) and a posteriori. Equations of the engine models are considered. Relations are derived that determine the precision of parameters of these models’ estimation depending on the precision of measurement, the composition of the engine power ratings, and durability of observations, at a stepwise change of fuel flow. We present these relations in non-dimensional coordinates that make them universal and ready for application to any turboshaft engine.

Interactions in droplet and particle system of near unity size ratio

In this study, we report collision interactions between a single droplet (diameter: 2.48–2.61mm) and a stationary hydrophobically coated thermally conductive particle (diameter: 3mm) in the low impact Weber number range of ∼0.9–47.1 Experiments were performed under both cold and hot state with the aid of high speed imaging to capture the interaction dynamics. Two outcomes were observed in the cold state: deposition at low Weber numbers and complete wetting of particle surface through formation of a spreading lamella of thick peripheral rim at higher Weber numbers. The complete wetting behaviour of droplet exhibited three distinct regimes of temporal variations in liquid film thickness at the apex point of particle. In non-dimensional coordinates, these regimes included – a short interval regime of linear reduction in film thickness due to initial droplet deformation, a relatively larger interval of inertia dominated regime of non-linear reduction in the film thickness and a relatively time invariant large interval of gravity draining regime with an oscillatory transition state due to capillary effect. A novel non-invasive laser based particle heating system was deployed for the non-isothermal interaction cases which showed two outcomes – rebound at lower Weber numbers and complete wetting followed by disintegration of the lamella at higher Weber numbers. Variation in particle temperature was insignificant in the rebound regime however significant temperature reduction (∼10–70°C) occurred due to increased wetted contact and nucleate boiling of secondary droplets at higher Weber numbers. Irrespective of the heat transfer effect at solid-liquid interface, the temporal variations in the film thickness followed the same trend in regime 1 and 2 as noted in the cold interaction cases. The effect of heat transfer was however uniquely characterised by the absence of regime 3 due to nucleate boiling at solid-liquid contact surface which led to rupture of the interface through eruption of vapour bubbles at the end of regime 2.

Operational Process and Characteristics of Liquid-gas Jet Pumps with the Ejected Vapor-gas Medium

In many industrial processes, the use of a liquid-gas jet pump refers to the ejection of the vapor-gas medium, i.e. a medium containing non-condensable and condensable gases in the liquid jet. Conventional calculation methods are used for jet devices that eject non-condensable gases. However, the presence of steam in the medium ejected makes a significant contribution to the operational process and, consequently, its mathematical description and device specifications. The article deals with the development of physical and mathematical models and determination of properties of liquid-gas jet pumps (LGJP) with an ejected vapor-gas medium. While calculating the performance there were used: Bernoulli equation – for spin-up of the active flow in the nozzle device; water balance equation, heat balance, conservation of momentum – for vapor condensation in the suction chamber; momentum equation and isothermal state equation of liquid-gas mixture – for mixture of media in the mixing chamber; liquid-gas mixture flow equation– for transformation of excess kinetic energy into the potential energy in the diffuser. Extremal characteristics that reflected the potential of the LGJP work were calculated on the basis of the equations. Analysis of specifications was carried out in a non-dimensional coordinates – in terms of the ejection coefficient by dry gas and compression ratio. The calculation of set of extremal characteristics with different ratios of the steam flow to the liquid showed that the presence of steam in the pumped medium degrades performance, i.e. the ejection coefficient is reduced at a constant compression ratio.

Instability of mixing zone formed at the boundary between the counter flows of two miscible liquids

The structure of the convective motion, which takes place after the vertical channel filled with two layers of miscible liquids is translated into a horizontal position, was studied experimentally. The characteristics and stability of the main flow were examined under various density differences and thicknesses of the channel. It is shown that the movement of the displacement front observed in all experiments can be described with a single curve by using certain nondimensional coordinates and time. It was found that an increase of density difference leads to instability of the main flow. This results in the appearance of a secondary flow in the form of helical convective rolls which form inside the mixing zone. Possible mechanisms responsible for the occurrence of such instability are discussed.

Experimental research on combustion dynamics of medium-scale fire whirl

The medium-scale fire whirl was extensively investigated by experimental means, in order to establish correlations of the burning rate, flame height and flame temperature of fire whirl, and to clarify the difference between fire whirls and general pool fires. Experimental observations and data confirmed that a free burning fire whirl is a highly stable burning phenomenon with large quasi-steady periods. Burning rates of fire whirls depend on pool diameter similarly to those of general pool fires; however the transition turbulent burning occurs sooner as the pool diameter increases. The lip height seems to have little effect on the burning rate of fire whirls. The correlation H ∗ = K · ( Q ˙ ∗ · Γ ∗ 2 ) m was proposed to couple the height of fire whirl to the fire release rate and ambient circulation. It correlates the data from both this work and the literature. Radial temperature profiles in the continuous region of the fire whirl were confirmed to be hump-type, implying the existence of fuel-rich inner core. The pool diameter and heat release rate do not significantly affect the radial temperature profiles in non-dimensional radial coordinates. It was found that the fire plume of fire whirl involves three distinct zones just like that of pool fire, but with different normalized ranges. Fire whirls maintain a higher ratio of continuous flame height to the overall flame height, and also higher maximum centerline excess temperature in continuous flame region, as compared to general pool fires. It was further demonstrated that the fire whirl plume at its origin behaves like a turbulent jet with moderate swirling, and then tends to become buoyancy dominated downstream, with slight swirling. With an increase in dimensionless height adjusted by the plume origin, the plume centerline excess temperature decays rapidly and approaches the theoretical value of −5/3 for free buoyancy plume.

Scaling the initial convective flow of power law fires

Scaling relations for the initial convective environment generated by power-law fires in geometrically similar spaces are derived, comprising functional relations for nondimensional temperature rise and gas velocity as functions of nondimensional time and location coordinates for arbitrary power-law behavior. Many fires initially can be modeled as t-squared fires, where heat release rate increases with the second power of time.

Vortex Interactions Between Chined Forebody and Strake

Nomenclature Cp = surface pressure coefficients, p p1 = 0:5 U 1 L = wing root chord length p1 = freestream static pressure Re = Reynolds number based on the wing root chord, UL= U1 = freestream velocity = angle of attack x = local nondimensional coordinates based on the characteristic length which is the sectional circumference for the sections S1–S8 and the curve length of the sectional upper surface for the sections S9–S19

The predictive power of zero intelligence in financial markets

Standard models in economics stress the role of intelligent agents who maximize utility. However, there may be situations where constraints imposed by market institutions dominate strategic agent behavior. We use data from the London Stock Exchange to test a simple model in which minimally intelligent agents place orders to trade at random. The model treats the statistical mechanics of order placement, price formation, and the accumulation of revealed supply and demand within the context of the continuous double auction and yields simple laws relating order-arrival rates to statistical properties of the market. We test the validity of these laws in explaining cross-sectional variation for 11 stocks. The model explains 96% of the variance of the gap between the best buying and selling prices (the spread) and 76% of the variance of the price diffusion rate, with only one free parameter. We also study the market impact function, describing the response of quoted prices to the arrival of new orders. The nondimensional coordinates dictated by the model approximately collapse data from different stocks onto a single curve. This work is important from a practical point of view, because it demonstrates the existence of simple laws relating prices to order flows and, in a broader context, suggests there are circumstances where the strategic behavior of agents may be dominated by other considerations.

Optimal free-form shapes for holes in flat plates under uniaxial and biaxial loading

This paper presents new and transferable design data defining precise optimal free-form shapes for holes in large and finite width plates under biaxial and uniaxial loading conditions respectively. An iterative two-dimensional finite element gradientless shape optimization procedure was used to obtain the optimal shapes, which offer the lowest possible stress concentration, subject to geometric constraints. Precise non-dimensional coordinates are given for a range of hole aspect ratios and are presented in tabular form that allows them to be easily used by designers. The optimal shapes were determined both with and without minimum radius of curvature constraints. Optimal shapes in large plates were found to provide up to a 28 per cent reduction in peak local stress over elliptical shapes of the same aspect ratio. For finite width plates the reduction in peak local stress was as high as 44 per cent. The results obtained are applicable to both the initial design of components with cut-outs and the shape reworking of existing components to remove damage.

Optimal notch shapes for loaded plates

This paper presents design data defining optimal freeform notch shapes for use in tensile loading conditions for a range of geometries, notably the notch aspect ratio and plate widths. Here, an iterative two-dimensional finite element gradientless shape optimization procedure was used to obtain the optimal shapes, which offer the lowest possible stress concentration, subject to the geometric constraints. Precise non-dimensional coordinates are given, for a range of notch aspect ratios and plate widths, and are presented in a tabular form that allows them to be easily used by designers. All cases have been solved with a minimum radius of curvature constraint, which results in a ‘robust’ optimal shape that has no sharp corners. The optimal shapes were found to provide up to a 26 per cent reduction in local peak stress over circular shapes of the same aspect ratio. Optimal geometries are also given for limiting notches that provide a net section stress equal to unity, at a finite notch aspect ratio. The results obtained are applicable to both the initial design of notched components and the shape reworking of existing components to remove damage.

The Predictive Power of Zero Intelligence in Financial Markets

Standard models in economics stress the role of intelligent agents who maximize utility. However, there may be situations where, for some purposes, constraints imposed by market institutions dominate intelligent agent behavior. We use data from the London Stock Exchange to test a simple model in which zero intelligence agents place orders to trade at random. The model treats the statistical mechanics of order placement, price formation, and the accumulation of revealed supply and demand within the context of the continuous double auction, and yields simple laws relating order arrival rates to statistical properties of the market. We test the validity of these laws in explaining the cross-sectional variation for eleven stocks. The model explains 96% of the variance of the bid-ask spread, and 76% of the variance of the price diffusion rate, with only one free parameter. We also study the market impact function, describing the response of quoted prices to the arrival of new orders. The non-dimensional coordinates dictated by the model approximately collapse data from different stocks onto a single curve. This work is important from a practical point of view because it demonstrates the existence of simple laws relating prices to order flows, and in a broader context, because it suggests that there are circumstances where institutions are more important than strategic considerations.

Similarity solution of thermal boundary layers for laminar narrow axisymmetric jets

In this work, the similarity equation describing the thermal boundary layers of laminar narrow axisymmetric jets is derived based on boundary layer assumptions. The equation is solved exactly. Some properties of the thermal jet are discussed. By introducing new-defined non-dimensional coordinates, the similarity solution results in a “universal” format. The results can also be applied in the boundary layer problem of species diffusion.

Shock unsteadiness in a reattaching shear layer

The origin of shock unsteadiness in a Mach 2.9 turbulent reattaching shear layer was investigated experimentally using temporally resolved flow visualization and measurements of wall pressure fluctuations. In this flow, the separation point of a turbulent boundary layer is essentially fixed at a backward-facing step, and the reattachment point is free to move along a ramp. In order to examine the influence of disturbances originating in the incoming shear layer, artificial disturbances were introduced into the flow through steady air injection in the vicinity of separation. The effect on the reattachment shock system was dramatic: the intensity of the pressure fluctuations and the amplitude of the shock motion increased substantially, and power spectra of the pressure fluctuations showed a distinct shift to lower frequency. The spectra collapsed onto a common curve in non-dimensional coordinates based on a length scale derived from two-point cross-correlations of the flow visualization data and a convection velocity derived from cross-correlations of the pressure measurements. The data were compared to a theory developed by Plotkin (1975), which is based on perturbation of a shock by random fluctuations in the incoming turbulent flow. Plotkin's model mimics the manner in which relatively broad-band perturbations in the incoming turbulent flow lead to relatively low-frequency motion of the separation bubble and its associated shock system. It is an excellent fit to separation shock motion, such as that generated in a blunt fin flow (briefly illustrated here). In the present shear layer flow, this low-frequency motion was detectable in the spectra near reattachment, but contained considerably less energy relative to the shock motions caused by direct perturbations by the incoming turbulent structures. These results indicate that the shock motion in the reattaching shear layer is primarily caused by organized structures in the incoming turbulent flow.

Guidance Algorithm Design: A Nonlinear Inverse Approach

The design of a missile guidance algorithm valid for any initial missile and target orientation is described. The limitations of proportional navigation- (PN-) or line-of-sight rate-based guidance for this problem are pointed out. Design heuristics are explained, and general guidelines for the synthesis of guidance algorithms are stated. It is shown thatthedesign of an algorithm valid forany initial missileand target orientation leadsto a nonlinearinverse problem. Standard geometric methods for solving nonlinear inverse problems are adapted to derive a new class of algorithms based on the relative heading error angle (RHEA). Examples of RHEA algorithms are then discussed and compared with PN guidance for different engagement scenarios. am = missile acceleration at = target acceleration Vm = missile speed Vt = target speed Xm, Ym = dimensional missile coordinates Xt, Yt = dimensional target coordinates xm, ym = nondimensional missile coordinates xt, yt = nondimensional target coordinates ®m = nondimensional missile acceleration ®t = nondimensional target acceleration µ = line of sight (LOS) angle µ 0 = LOS rate o = target to missile velocity ratio Ω = nondimensional distance to go ? = nondimensional time Âm = missile heading angle Ât = target heading angle A = relative heading error angle

A general mathematical model of evaporative cooling devices

In this paper a general nondimensional mathematical model for the description of all types of evaporative cooling devices in today's use (water cooling towers, evaporative condensers and evaporative fluid coolers, air washers, dehumidifying coils etc.) is presented. The system of differential equations describing nonadiabatic evaporation processes is transformed to a pure nondimensional form by the introduction of nondimensional coordinates and parameters and by the substitution of a straight air saturation line instead of the real one. In such a way, not only the description of the entire process becomes very simple, but also the overall performance of any such device can be expressed in terms of only few parameters and presented in one or few diagrams. This nondimensional model forms a basis for simple and rather accurate methods for rating various types of evaporative cooling devices. For every type of device, a unique rating procedure can be established, irrespective of the relative flow direction of the fluids involved, which is very convenient when cross-flow devices are considered. Since every particular type of evaporative device deserves considerable attention, the application of this general model to various evaporative devices, together with the resulting rating procedure and accuracy test will be presented in detail in the follow-on papers, each devoted to one particular type.

An Adhesion Map for the Contact of Elastic Spheres

Several continuum mechanics models of the adhesion between elastic spheres have found application to compliant materials such as rubber and to fine particles in the air or in colloidal suspension. More recently they are being used in connection with experimental techniques such as the surface force apparatus and the atomic force microscope. The appropriate model to use depends on the conditions: the size and elasticity of the spheres and the load to which they are subjected. To guide this choice a map has been constructed with nondimensional coordinates μ andP, where the elasticity parameter μ can be interpreted as the ratio of elastic deformation resulting from adhesion to the effective range of surface forces and the load parameterPis the ratio of the applied load to the adhesive force. A closed form solution to the problem based on a Dugdale force-separation law is outlined and used to construct the map. The errors introduced by the Dugdale approximation are assessed by comparison with numerical solutions using the Lennard–Jones force law.

Coherent structure in a combusting jet in crossflow

The flame behavior and coherent structure of the combusting propane gas jet in a cross airstream at low jet-to-wind momentum flux ratios are studied experimentally. A one-component constant-temperature hot-wire anemometer is employed to extract frequency information from the traveling coherent structures evolving from the upwind side of the jet body. The reduced traveling frequency of the coherent structure in the near burner region is correlated to the jet-to-wind momentum flux ratio. The reduced eddy traveling frequency approaches to a constant of about 0.35 at high jet-to-wind momentum flux ratios. The trajectories of the traveling coherent structures depicted by the bisector of the eddy traveling avenue (the narrow band where the coherent structures sweep over) are correlated by nondimensional coordinates

Power-law fluid flow in heated vertical ducts

A hydrodynamical ly and thermally fully developed flow in a vertical duct is investigated. It is assumed that the wall of the duct is maintained at a temperature which varies linearly with respect to the depth of the duct to model a vertical duct in the Earth. Furthermore, in order to investigate flows during the drilling of muds and the cementing process in oil wells, the fluid is assumed to be a power-law fluid and the fluid flow in concentric annuli are investigated. The conditions under which multiple solutions are obtained is studied and the effects of the Rayleigh number n, the relative thickness of the channels, and the enforced pressure gradient are investigated. Nomenclature d = half-width of duct k = thermal conductivity Nu = Nusselt number, Eq. (29) n = power-law index p = fluid pressure Ra = Rayleigh number, Eq. (8) T = temperature X, Y = nondimensional horizontal and vertical coordinates, Eq. (5) jt, y = horizontal and vertical coordinates U, V = nondimensional horizontal and vertical velocities, Eq. (5) u, v = horizontal and vertical velocities p = coefficient of thermal expansion 7 - rate of strain tensor 8^82 = nondimensional thickness of duct wall and outer duct £ = nondimensional pressure gradient, Eq. (7) 5 = nondimensional buoyancy, Eq. (5) 6 = temperature difference, T - Tw A = applied temperature gradient JJL = coefficient of viscosity p = density of fluid Subscripts D = duct region / = inner duct region n = quantities evaluated with n O = outer duct region W = wall region w = on outer wall of duct 0 = quantities evaluated at T()