Dimensionless Representation Research Articles

Analysis of the scaling laws for the turbulence driven panel responses

The high computational costs, associated to the numerical solution of the fluctuating pressure field generated at the wall by the turbulent boundary layer and of the induced structural response, push for the exploration of alternative methodologies of analysis. Wall pressure fluctuations spectra are often modeled using semi-empirical expressions based on the experimental evidence and on the identification of universal scaling laws. In this work the possibility to adopt a dimensionless representation, able to provide a universal expression for the structural response of plates under turbulent boundary layer excitations, is investigated with the help of pressure fluctuations and acceleration experimental data sets. The test article is a plane thin plate wetted by a fluid over one face, the boundary layer is fully developed and pressure gradient effects are negligible. The attention is devoted to the investigation and the definition of a normalization of the required axes: the excitation frequency and the power spectral density of the structural response. The analysis is initially based on analytical models for the structural response under turbulent boundary layer excitations. The proposed scaling laws are successively and successfully applied to four data sets measured in different conditions both in wind tunnels and in a towing tank.

Calculation of free energy landscapes: A histogram reweighted metadynamics approach

We present an efficient method for the calculation of free energy landscapes. Our approach involves a history-dependent bias potential, which is evaluated on a grid. The corresponding free energy landscape is constructed via a histogram reweighting procedure a posteriori. Because of the presence of the bias potential, it can be also used to accelerate rare events. In addition, the calculated free energy landscape is not restricted to the actual choice of collective variables and can in principle be extended to auxiliary variables of interest without further numerical effort. The applicability is shown for several examples. We present numerical results for the alanine dipeptide and the Met-Enkephalin in explicit solution to illustrate our approach. Furthermore, we derive an empirical formula that allows the prediction of the computational cost for the ordinary metadynamics variant in comparison with our approach, which is validated by a dimensionless representation.

Computational algorithms to evaluate design solutions using Space Syntax

In the past, conventional computer-aided architectural design (CAAD) systems could not manage semantic information on building components and spaces but only graphical and geometric information. However, since the advent of Building Information Modeling (BIM), which has been used for managing semantic building information, determining spatial relationships as well as quantities and properties of building components in CAAD systems has become easier. It is necessary to make current CAAD systems capable of performing spatial analysis functions using BIM because they can easily recognize building components and spaces. Accordingly, this study aims to develop the computational algorithms to evaluate design solutions using Space Syntax during the process of computer-aided architectural designing. To extract topological information from design solutions, this study proposes algorithms to recognize building information produced in the form of Industry Foundation Classes (IFC), deduce the necessary topological information, and store the information in the form of matrices. The Space Syntax theory is employed to evaluate the solutions based on social properties of spaces in a building and examine the potential for adding a spatial analysis function into CAAD applications. The developed algorithms calculate the integration value for each space from spatial connectivity based on J-graphs. To validate the proposed algorithms, a program named J-Studio for Architectural Planning (J-SAP) was developed to evaluate design solutions easily and quickly. The validation results are as follows: (1) the topological information extracted from building information was decoded into a dimensionless representation and legible J-graph, (2) mathematical analyses for choosing a better design solution during computer-aided architectural designing were presented, and (3) the examination of the privacy level of each space in a building through Space Syntax analysis was discussed. Thus, this study demonstrates the possibility of determining the social properties and accessibility of spaces during the process of computer-aided architectural designing to meet client requirements by extracting topological information from building information model and performing Space Syntax analysis for evaluating alternatives using the information.

An analytical solution of the convection–dispersion–reaction equation for a finite region with a pulse boundary condition

Transport of particles through porous media is commonly described by the convection–dispersion–reaction equation. Although experimental studies are obviously performed in finite domains, a comparison with analytical solutions is problematic because the latter are available for semi-infinite regions or are subject to unrealistic boundary conditions. In the present study, an analytical solution of the convection–dispersion–reaction equation is obtained for a finite one-dimensional region. A pulse boundary condition, widely used in the experiments, is applied at the inlet. Thus, although Danckwerts’ boundary condition is still used at the outlet, the present formulation is closer to the reality than those existing in the literature. The problem is solved using Laplace transform, with the inverse transform based on the complex formulation and residue theory. The effect of various parameters, including the dispersion coefficient, approach velocity and attachment coefficient, on the breakthrough curve is shown. These parameters are taken from typical experimental data. A dimensionless representation of the solution makes it possible to define the range where it is physically meaningful. It is shown how the Péclet number, based on the approach velocity, domain length and dispersion coefficient, affects this range. A comparison with experimental data is also presented. It is shown that the suggested solution may be used for a broad variety of experimental conditions.

A new perturbation solution to the Michaelis‐Menten problem

AbstractWe develop a new dimensionless representation of the time‐dependent mass balances for the Michaelis–Menten (MM) reaction mechanism; we identify several dimensionless parameters that control the fundamental nature of the solution; and we solve the scaled equations with a combined regular and singular perturbation expansion. Unlike several approximate solutions to the MM problem offered previously in the literature, each of which is valid only for some limited range of conditions, the new solution converges accurately for any combination of initial substrate concentration, initial enzyme concentration, and kinetic rate constants. We discuss the physical significance and interdependence of the dimensionless parameters that emerge from our scaling analysis; we use these parameters to categorize previous approximations for the MM problem and to delimit their accuracy; and we verify the accuracy of our solution via comparisons to an exact numerical solution and various approximations offered previously by others. © 2008 American Institute of Chemical Engineers AIChE J, 2008

A dimensionless number analysis of the hybridization process in diffusion- and convection-driven DNA microarray systems

The present theoretical analysis aims at providing a general understanding of the combined effect the many different process variables have on the hybridization rate in diffusion- and convection-driven DNA microarray systems. It is shown that all process variables can be grouped into only four different dimensionless numbers (the Damkohler number Da, the dimensionless association constant kappa(A), the dimensionless initial concentration C'(0) and a geometrical ratio alpha). These four numbers have a straightforward physical meaning and only contain easily measurable parameters. Reducing the solution space from 7D to 4D, the dimensionless number representation greatly facilitates the insight in the conditions leading to the occurrence of diffusion-limited hybridization rates in both diffusion- and convection-driven DNA microarray systems. This in turn simplifies their design and the interpretation of the experimental results that are obtained with these systems.

Dimensionless robust control with application to vehicles

The selection of a nominal plant model and an uncertainty representation is a central design choice for a robust controller design. In this brief, we examine the choice of both a plant model and an uncertainty representation from a dimensionless viewpoint. We develop a dimensionless representation of a linear bicycle model for vehicle dynamics that is suitable for a generalized vehicle analysis. Within such a dimensionless framework, the average of vehicle parameters becomes well defined thereby giving a nominal vehicle. Additionally, it becomes easy to develop perturbations of the vehicle model about the nominal one that reasonably encompass all production vehicles. These uncertainty bounds are then used to generate a robust controller suitable for any vehicle. Tighter uncertainty bounds result from the dimensionless analysis versus a dimensioned one and therefore provide less conservative controllers. For the purposes of demonstration, the focus of this brief is a lateral-positioning control task. The resulting control design approach is demonstrated on a scaled experimental vehicle as well as through simulations.

Comparison of shear-induced crystallization behavior of PB-1 samples with different molecular weight distribution

Shear-induced crystallization behavior of three grades of PB-1 having different molecular weights and breadths of molecular weight distribution was investigated by means of rotational rheometry covering a shear rate range of 0.0001 up to 0.3 s−1 and temperatures between 99 °C and 107 °C. The up-shoot in viscosity after a certain time is used to define an onset time ton for both quasiquiescent and shear-induced crystallization. The plot of ton versus the shear rate can be converted into a temperature-invariant curve by dividing the y axis by quiescent onset time ton,q and multiplying the x axis by the square root of ton,q. The shape and location of the T-invariant curves for the three PB-1 samples are compared with respect to their molecular parameters. An additional normalization step by multiplying the abscissa by the square root of the characteristic retardation time of the melt yields a dimensionless representation of the T-invariant curve. Furthermore, by this procedure the three curves are shifted into one region, where they have the same threshold value, indicating the onset of shear-induced crystallization. A connection of this T-invariant curve to the morphology of the crystallites formed are addressed.

Mathematical modeling of a direct current electric arc: Part II. Dimensionless representation of a direct current arc

A mathematical model presented in Part I of this work is used to describe fluid-flow, heat-transfer, and electromagnetic phenomena in the arc region of a direct current electric arc furnace (DC EAF). From the analysis of the system, the effects of the arc current and the arc length on the arc characteristics and arc bath interactions can be represented analytically in a rather generalized way, based on the finding that the arc shape is unique (defined with the location of the 10,000 K isotherm) when plotted in the appropriate dimensionless form. This dimensionless arc expansion (only restricted to arcs burned between graphite electrodes in air and outside the region where the arc jet impinges on the bath surface) gives the possibility to derive a unique arc description regardless of the values of the arc current and arc length. Dimensionless correlations representing the relationships among important arc characteristics and arc-bath interactions are derived in this work. These expressions may be used to estimate process variables without the need to run complex numerical calculation.

Shear rates investigation in a scraped surface heat exchanger

The electrodiffusion technique was performed in order to investigate the shear rate on a scraped surface heat exchanger. Microelectrodes were placed inside: the walls of the outer cylinder; the inlet and outlet bowls; the rotor and the blades. Highly viscous Newtonian fluid (Emkarox HV45 solutions) and non-Newtonian model fluid (aqueous solutions of CMC) were used. The electrodiffusion method allowed us to measure wall shear rates. Maximum shear rate was observed at the scraping surface and caused by blades scraping, high shear rate was also measured on the leading edge of the blades. In the other parts of the exchanger, shear rate remained low but the development of Taylor vortices completely modified the scraped surface heat exchangers behaviour inside the surface of the bowls. A dimensionless representation of the friction factor was established for the inner and outer wall surface of the exchanger.

High heat flux removal using water subcooled flow boiling in a single-side heated circular channel

High heat flux removal from plasma-facing components and electronic heat sinks involves conjugate heat transfer analysis of the applicable substrate and flowing fluid. For the present case of subcooled flow boiling inside a single-side heated circular channel, the dimensional results show the significant radial, circumferential and axial variations in all thermal quantities for the present radial aspect ratio ( R o=outside radius to inside radius) of 3.0. A unified, dimensionless representation of the two-dimensional inside wall heat flux, and the dimensional inside wall heat flux ( q i( φ, z)) and temperature ( T i( φ, z)) data was found and used to collapse the data for all circumferential locations. Finally, 2-D boiling curves are presented and are among the first full set of 2-D boiling data presented for a single-side heated circular configuration.

Determination of Coagulation Coefficients and Aggregation Kinetics for Charged Aerosols

A new method has been developed which allows determination of the coagulation coefficient of two oppositely charged particles experimentally. For this purpose, quasi-monodisperse particles of different sizes and morphology were used to study the influence of different parameters on the coagulation coefficient. A good agreement between experimental results and the classic Fuchs' theory was obtained when including a method accounting for particle nonsphericity. In experiments with polydisperse bipolarly charged aerosols, no principal differences to uncharged aerosols were found when a dimensionless representation was used. Changes in particle number concentration and geometric mean diameter can be described by simple empirical expressions.

Scaling of the pedestal density in type-I ELMy H-mode discharges and the impact of upper and lower triangularity in JET and ASDEX Upgrade

Experiments have been performed to disentangle the individual role of upper and lower triangularity on density buildup of lower single null, type-I ELMy H-mode discharges in JET. Comparison with corresponding data from ASDEX Upgrade allowed us to determine a dimensionless representation of the relation between the main chamber recycling and core density and to widen the triangularity variation in the data base. To incorporate the recycling flux density Γ in a dimensionless form, an effective scrape-off layer density ne,SOL∝Γ0.5 is introduced allowing us to parametrize and scale the core density by the density rise factor n̄e/ne,SOL. The scaling uses edge-specific definitions of normalized Larmor radius, collisionality and beta. Rewritten in dimensional form, a behaviour is found which is very similar to energy confinement scalings for the ELM-averaged conditions considered here: the density rise factor exhibits an almost linear dependence on plasma current, a weak negative toroidal field dependence as well as power degradation. While a pronounced positive dependence of density buildup on the upper triangularity is observed, no significant correlation with the lower triangularity is found. In particular the dependences on plasma current and upper triangularity emphasize the importance of transport physics for the density buildup.

Wall shear rate in the Taylor–Couette–Poiseuille flow at low axial Reynolds number

An experimental investigation was done by the use of visual observations and the electrochemical technique in order to study the appearance of hydrodynamic instabilities at low Re axg in the gap between two coaxial cylinders, with radius ratio R r/ R s=0.615 and aspect ratio L/ e=24. A motor drove the inner cylinder and the outer cylinder was fixed. A Newtonian fluid (Emkarox HV45) and two non-Newtonian fluids (aqueous solutions of guar and CMC) have been used. The analysis of the evolution of the size of the Taylor vortices is carried out for Re axg<4.0. For Re axg<2 we showed that the vortices underwent a contraction phase up to Ta g≈80, and a stretching phase for Ta g>80. For higher Re axg, the contraction phase vanished and the vortices progressively stretched with increasing Ta g. A dimensionless representation was also proposed for wall friction generalized to non-Newtonian fluids following the Ostwald law.

Anharmonic force field for methanol

An ab initio quartic anharmonic force field for methanol has been calculated at the equilibrium position using the CCSD(T) method for the structure and the harmonic potential energy surface, and the MP4(SDQ) method for the anharmonic part of the surface. A triple zeta basis set was employed with symmetrized curvilinear internal valence coordinates in all calculations. The internal coordinate force field constants have been transformed into force constants in the dimensionless normal coordinate representation for various isotopomers. Vibrational term values for CH3OH, CH3OD, CD3OH, and CD3OD have been obtained using second order perturbation theory. Particular care has been devoted to the inclusion of Fermi resonance interactions between different vibrational states. A good accuracy has been achieved in the calculation of the fundamentals for all the isotopomers, the mean absolute error being 5.8 cm−1.

The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek–Brown failure criterion

Abstract This paper is intended to illustrate the relationship between the Hoek–Brown parameters describing the strength of rock masses and the mechanical response of underground openings. A formulation of the elasto-plastic behavior of rock in terms of the Hoek–Brown criterion is presented. The analysis assumes that the joint system present in the rock mass has no preferred orientation so that the medium can be considered to behave as an isotropic continuum. It is shown that appropriate scaling of the Hoek–Brown parameters leads to considerable simplification in defining the elasto-plastic response of the rock mass. The classical case in which the excavation process is treated as a uniform reduction of internal pressure in symmetrically loaded cylindrical and spherical cavities is considered. Closed-form expressions are given for the extent of plastic behavior and the related stress and displacement fields. A dimensionless graphical representation of these solutions is provided that allows accurate estimates of the response of excavations in Hoek–Brown materials to be made quickly and easily. Examples are given to illustrate the use of the graphs. Illustrative applications of the derived closed-form solutions are also described. The construction of ground reaction curves for the design of cylindrical tunnels according to the convergence–confinement method and a case study of stability analysis of spherical cavities produced by underground nuclear explosions in French Polynesian atolls are discussed.

Modeling the Capacity of Pin-Ended Slender Reinforced Concrete Columns Using Neural Networks

This study demonstrates the feasibility of using multilayer feedforward neural networks to model the complicated nonlinear relationship between the various input parameters associated with reinforced concrete columns and the actual ultimate capacity of the column. The neural network models were constructed directly from a fairly comprehensive set of experimental results and were found to be tolerant of certain levels of errors in the original testing results. Comparison with the original testing data and theoretical model showed that the ultimate capacity of reinforced concrete columns predicted by the neural network models is reasonably accurate. Parametric analysis indicates that the neural network model has reasonably captured the behavior of reinforced concrete columns. Numerical studies are conducted to investigate modeling issues such as different data scaling schemes and dimensionless representation schemes. Nonlinear transformation of the output values resulted in an overall improvement in the generalization capabilities of the neural network model. Preliminary studies using a limited data set of 54 test results on high strength concrete columns also showed promising results. The neural network model can be useful in checking routine designs because it provides instantaneous results once it is properly trained and tested.

Jet-wave amplification in organ pipes

An envelope-based method to estimate both the jet-wave amplification factor and the mouth-field strength in organ pipes is developed by using flow visualization of a smoked jet with a high-speed digital video camera. A theoretical envelope of wave growth, which is approximated using a negative displacement model of the jet drive, is compared with an experimental envelope derived from superposing many instantaneous shapes of jet deflection in the steady-state oscillation. The estimation results are presented in dimensional terms with respect to two particular models, where the flue-to-edge distances are, respectively, 15.8 and 10.2 mm, with a common flue thickness of 2.2 mm. In our experiment the jet velocity ranges from 7 to 33 m/s, the Reynolds number from 1000 to 5000, and the sounding frequency from 130 to 580 Hz. The amplification factor of organ pipe jets, estimated to lie in the 0.18–0.26-mm−1 range, tends to decrease and saturate with increasing blowing velocity in each oscillation mode; the mouth-field strength defined as the acoustic displacement amplitude, roughly estimated to be 0.5–1.5 mm, tends to increase and saturate with increasing blowing velocity. A hot-wire anemometer is then used to measure the mouth-field strength, whose value shows a good agreement with the estimated one. This result confirms the validity of our envelope-based method. A dimensionless representation of the experimental data is used to compare wave characteristics between an organ pipe jet and an acoustically perturbed free jet. The applicability of the spatial and temporal theories of jet instability is discussed to analyze them. If we can assume a Poiseuille flow at the flue exit and a subsequent Bickley jet, the spatial theory seems to be relevant to our organ pipe jets. However, for lack of a reliable experimental measurement of the jet half-thickness we cannot draw a definite conclusion about the wave characteristics of organ pipe jets.

Power consumption of a dual turbine–helical ribbon impeller mixer in ungassed conditions

Ungassed power measurements in a dual coaxial mixer composed of a helical ribbon and a Rushton turbine were carried out in laminar mixing conditions for Newtonian and non-Newtonian shear thinning fluids. For the Newtonian case, the power draw constant K p for the hybrid geometry was not the sum of the individual impellers. This was explained by considering the radial discharge flow in the turbine region as well as the top-to-bottom circulation pattern of the helical ribbon impeller. For the non-Newtonian fluids, the results showed that, at a given Reynolds number, power consumption decreases as the shear thinning behaviour increases. A dimensionless and unique representation of the power draw data was obtained by shifting the non-Newtonian power draw results to the Newtonian curve. This was carried out with a K s function defined from the K p( n) data. The predictions for K s were found to be in good agreement with those obtained using the classical method of A.B. Metzner and R.E. Otto (AIChE J., 3 (1957) 3–10). It was observed that pseudoplasticity tends to shift the upper limit of the laminar region toward Reynolds numbers higher than 10.

Sound production in recorderlike instruments. I. Dimensionless amplitude of the internal acoustic field

Data on the internal acoustic pressure signals in an experimental recorderlike flue organ pipe are presented. A dimensionless representation appears to be a powerful basis for the analysis of these data. The dimensionless amplitude of the fundamental is, for a given geometry, a function of the Strouhal number only. For the first hydrodynamic mode of the jet, this amplitude is independent of the acoustic mode involved. The dimensionless amplitude of the second harmonic displays two different behaviors, depending on whether the jet is laminar or turbulent. This specific Strouhal number dependency implies a strong influence of the distance W from the flue exit to the labium on the timbre of recorderlike instruments. In recorders the ratio W/h, where h is the height of the flue exit, is adjusted by craftsmen to a value close to 4. This specific ratio ensures an optimal harmonic content to noise ratio in the produced sound. Increasing the distance W yields a more powerful sound but at the expense of additional turbulence noise. For these large ratios W/h, transitions between different hydrodynamic modes of the jet are observed at low blowing pressures. This phenomenon is avoided by the choice of a recorderlike geometry.