Dimensional Analysis Theory Research Articles

Development of empirical models for the modular limit of trapezoidal and triangular throat flumes

One of the essential challenges of flow measurement using the flume structure is whether the flow is free or submerged. Therefore, determining the free-submerged flow boundary is a vital topic. This study experimentally investigates the modular limit/threshold submergence of two kinds of flumes: trapezoidal throat flume (TraTF) and triangular throat flume (TriTF). To this end, the general form of empirical models of the flumes' modular limit index was presented based on dimensional analysis and incomplete self-similarity theory. Then, a set of experimental data was collected. In the next step, empirical relationships were developed using non-linear multivariate regression. The experimental modeling was done in a channel that was 20 m long, 0.6 m wide, and 0.5 m high. A total of 271 tests were conducted, examining nine flumes with trapezoidal throats and four with triangular throats at various flow rates. The TraTF formula has been developed, with 98% of its predictions having an error of less than 10% and an average error of 3%. For the TriTF formula, 96% of its predictions have an error of less than 10%, and the average error is 3.6%.

Flow measurement mechanism and hydraulic characteristics analysis of the airfoil weir-orifice facility in a rectangular channel

ABSTRACT Experimental studies and numerical simulations are conducted on the airfoil weir-orifice facilities to find a flow measurement and control device with a stable combined outflow form and good hydraulic performance. The orifice height under the weir is changed by adjusting the chord length and rotation angle of the weir. This paper uses three airfoil weir-orifice facilities for comparative study. The facilities have three flow forms under different rotation angles and flow conditions. Orifice discharge in combined outflow form is related to weir crest height but not upstream water depth. Flow formulas for three discharge forms are derived using dimensional analysis and incomplete self-similarity theory. The results show that the relative error of the formula is within ±6.34%. The Froude number is less than 0.4 under all working conditions; the critical submerged range is between 0.7 and 0.95; and under the premise of the unified weir crest height, the head loss decreases with the increased orifice height. The hydraulic jump location analysis yields the airfoil weir-orifice facility under the combined outflow as a submerged orifice outflow, and the hydraulic jump energy cancelation efficiency under the same flow condition is consistent with the upstream water depth change trend.

Data-based models to investigate protective piles effects on the scour depth about oblong-shaped bridge pier

The primary objective of this study is to examine the suitability of employing a machine learning models (MLMs) comprising Support Vector Machine (SVM), Gene Expression Programming (GEP), and Artificial Neural Network (ANN) algorithms to emulate the influence of geometric and geotechnical parameters of the piles on the scour depth reduction about the oblong-shaped bridge pier. A dataset consisting of 90 laboratory observations was utilized, allocating 70 % of the data for the training and 30 % for the testing the MLMs. An oblong-shaped bridge pier of diameter D, was positioned and subjected to a rectangular flume featuring an erodible bed comprising sand sediment particles of D50 = 1.7 mm under clear water and subcritical flow conditions. The spacing between the piles, LD, the angle of pile orientation with respect to the flow, θ, and the Froude number, Fr, were identified as independent parameters through the dimensional analysis and Buckingham-Π theory. The MLMs’ performance were assessed employing root mean square error (RMSE), mean absolute error (MAE), coefficient of determination (R2), and developed discrepancy ratio (DDR). The outcomes, affirming the predictive capacity of all MLMs included, revealed that the GEP model employing a three-gene configuration exhibited superior accuracy compared to the other two MLMs. Specifically, the performance indicators (RMSE, MAE, R2, DDR) during the training and testing phases were (0.0433, 0.033, 0.981, 6.8) and (0.054, 0.045, 0.958, 4.81) respectively. The sensitivity analysis elucidated the hierarchy of influential variables as follows: Fr, LD and θ.

On Augmented Dimensional Analysis

We present an innovative approach to dimensional analysis, referred to as augmented dimensional analysis and based on a representation theorem for complete quantity functions with a scaling-covariant scalar representation. This new theorem, grounded in a purely algebraic theory of quantity spaces, allows the classical π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\pi $\\end{document} theorem to be restated in an explicit and precise form and its prerequisites to be clarified and relaxed. Augmented dimensional analysis, in contrast to classical dimensional analysis, is guaranteed to take into account all relations among the quantities involved. Several examples are given to show that the information thus gained, together with symmetry assumptions, can lead to new or stronger results. We also explore the connection between dimensional analysis and matroid theory, elucidating combinatorial aspects of dimensional analysis. It is emphasized that dimensional analysis rests on a principle of covariance.

Comment on “Hydraulic characteristics of labyrinth sluice gates” by T. Hashem, A.Y. Mohammed, T.J. Alfatlawi

In this paper, the stage-discharge relationship of a labyrinth sluice gate is theoretically deduced by the Π-Theorem of dimensional analysis and the self-similarity condition. This approach was then compared with the classical approach using the discharge coefficient Cd proposed by Hashem et al. (2024). The classical stage-discharge equation (Eq. 4), with the discharge coefficient estimated by the empirical relationship (Eq. (1)) proposed by Hashem et al. (2024), resulted in errors E that are less than or equal to ±5 % for 74.3 % of the investigated cases and a mean square error MSE of 0.0078. The new stage-discharge relationship, deduced by dimensional analysis and self-similarity theory, was calibrated using the dataset by Hashem et al. (2024). The obtained stage-discharge equation (Eq. 20) is characterized by errors in the discharge estimate E that are less than or equal to ±5 % for 82.2 % of the investigated cases and a mean square error MSE of 0.0051. In conclusion, the stage-discharge relationship deduced by dimensional analysis allows for the best performance in the discharge estimate, which can be obtained without knowledge of the empirical discharge coefficient.

Оценка технологического потенциала деталей

The article describes a methodology for assessing the technological potential of parts, based on the sociophysical theory of potentials and dimensional analysis of the technological process. The technological potential of a part acts as a system characteristic of a complex technical system. The program A'PROPOS was chosen as a tool for calculating the technological potential. A hypothesis has been put forward that the technological potential of the part is associated with the complexity of the technological process of manufacturing this part. It is possible to evaluate it using dimensional analysis using CAD tools. Methods and materials. The emergence and creation of fundamentally new technologies is considered a historical process of development of the designed technical system, in this case, the DSE. The technological potential of the part is formed as the technological process is implemented. The value and structure of the potential are determined and consistent with the properties of the designed part. Based on the results of the analysis of the technological process of manufacturing a part, the construction of a sketch of a part in A'PROPOS., the technological potential is calculated using a mathematical model. The potential is calculated on the basis of the initial data of the enterprise for the object of study. Result. Based on the constructed model, the technological process is optimized. The result is a new machining plan containing the compositions and accuracy of the surfaces machined in the stages and the minimum allowance. Based on the model with variations of dimensional relationships, the dimensional structure is improved to expand the tolerance fields with guaranteed maintenance of technological parameters. Based on this, the deviation in the recommended labor intensity is calculated compared to the real one. Next, the accumulated technological potential of the part was calculated. Conclusions. The system makes it possible to iteratively perform technological calculations, as well as integration with CAD systems in the enterprise. When calculating the technological potential, instead of using the initial data of the enterprise, it is interpreted by the adequacy of the model.

Distribution grid fault diagnosis based on fault indicator and random matrix

The analysis and utilization of massive recording data of distribution grid fault indicator is beneficial to improve the effect of distribution grid fault diagnosis. In this paper, the distribution grid monitoring are realized by the random matrix theory (RMT) of high dimensional statistical analysis. The fault diagnosis method based on RMT has the advantages of no need for detailed distribution grid topology, comprehensive utilization of wide-area spatiotemporal data, and observation from a muti-dimensional view. By explaining the application principle of limit spectrum distribution function, the linear eigenvalue statistics (LES) is proposed as the state monitoring index. Distribution grid fault diagnosis based on fault indicator and RMT provides a new data-driven method for distribution grid fault diagnosis while efficiently utilizing massive fault indicator data.

Development of a soil wetting pattern estimation model for drip irrigation

Abstract Accurately estimating the soil wetting pattern that closely reflects the measured value can improve the water use efficiency for drip irrigation. Ignoring the effect of the initial soil water content on the soil wetting pattern affects the accuracy of the estimated results to a certain extent. This research aimed to develop a soil wetting pattern estimation model for drip irrigation that included four easily measurable parameters (i.e., initial soil water content, saturated hydraulic conductivity, total volume of applied water, and emitter discharge rate) based on dimensional analysis theory. In this study, the wetting front advance data of 12 typical soil textures were obtained in Hydrus-2D/3D. The estimated values were then compared with measured or simulated wetting front advance values. For different experiments, the mean absolute error, root mean square error, and mean relative error varied from 2.77 to 4.69 cm, 6.20 to 10.61 cm, and 5.61% to 10.51%, respectively. Compared with the existing models, the proposed model was more consistent between the measured and simulated values. Therefore, the proposed model of this study is efficient and simple, which can help accurately estimate the soil wetting pattern of drip irrigation with a variety of soil textures.

Data-driven discovery of dimensionless numbers and governing laws from scarce measurements

Dimensionless numbers and scaling laws provide elegant insights into the characteristic properties of physical systems. Classical dimensional analysis and similitude theory fail to identify a set of unique dimensionless numbers for a highly multi-variable system with incomplete governing equations. This paper introduces a mechanistic data-driven approach that embeds the principle of dimensional invariance into a two-level machine learning scheme to automatically discover dominant dimensionless numbers and governing laws (including scaling laws and differential equations) from scarce measurement data. The proposed methodology, called dimensionless learning, is a physics-based dimension reduction technique. It can reduce high-dimensional parameter spaces to descriptions involving only a few physically interpretable dimensionless parameters, greatly simplifying complex process design and system optimization. We demonstrate the algorithm by solving several challenging engineering problems with noisy experimental measurements (not synthetic data) collected from the literature. Examples include turbulent Rayleigh-Bénard convection, vapor depression dynamics in laser melting of metals, and porosity formation in 3D printing. Lastly, we show that the proposed approach can identify dimensionally homogeneous differential equations with dimensionless number(s) by leveraging sparsity-promoting techniques.

Experimental study and numerical simulation of flow over the inclined airfoil-shaped weir in rectangular channel

In order to find an integrated flow measurement and control facility with stable outflow form, a series of experiments and numerical simulations were performed for the inclined airfoil-shaped weir. Based on the experimental and simulated data, the stage-discharge relationship was deduced by using dimensional analysis and incomplete self-similarity theory. And two flow measurement formulas with inclination angle of 0°–45° were obtained through different fitting schemes. The variation law of discharge ratio with inclination angle and the accuracy comparison of flow measurement formulas were analyzed. Moreover, the Froude number, critical submergence degree, head loss and velocity distribution of the flow field over the weir were studied. The results show that streamlined airfoil improves the stability of flow behind the weir. The average relative error of the free water surface line data between experiments and simulations is 5.05%, which indicates the reliability of the numerical simulation. The discharge ratio proposed in the fitting scheme I takes the 25° inclination angle as the limit, which increases slightly at first and then decreases gradually, and the flow measurement relative errors of the two fitting formulas are all within ±7%. The average Froude number of the upstream reference section at each inclination angle is less than 0.25, the critical submergence range is between 0.87 and 0.96, and the relative head loss ranges from 3.54% to 12.16%, so as to ensure the accuracy of flow measurement.

Dimensioned Algebra: Mathematics with Physical Quantities

A rigorous mathematical theory of dimensional analysis, systematically accounting for the use of physical quantities in science and engineering, perhaps surprisingly, was not developed until relatively recently. We claim that this has shaped current mathematical models of theoretical physics, which generally lack any explicit reference to units of measurement, and we propose a novel mathematical framework to alleviate this. Our proposal is a generalization of the usual categories of algebraic structures used to formulate physical theories (groups, rings, vector spaces...), herein dubbed dimensioned, that can naturally articulate the structure of physical dimension. Our goal in the present work is not so much to define an algebraic theory of physical quantities—this has already been done—but to define a theory of algebra informed by how physical quantities are used in practice. We conclude by studying the dimensioned analogue of Poisson algebras in some detail due to their relevance in Jacobi geometry and classical mechanics.

Physical simulation and feasibility evaluation for construction of salt cavern energy storage with recycled light brine under gas blanket

The high cost and serious pollution of salt cavern construction (SCC) with fresh water (FW) under oil blanket (OB) poses a major challenge to the development of salt cavern energy storage, and it is of great significance for clean brine mining and efficient SCC to replace FW with light brine (LB) as well as OB with gas blanket (GB). However, the expansion and control of cavern shape after replacement remains unclear, and it also needs to be clarified of the feasibility of SCC with LB. Therefore, the experimental and theoretical research on the physical simulation of SCC with LB under GB was carried out. Firstly, the dissolution test of salt rock in brine was performed, and the effect of brine concentration on the dissolution rate of salt rock was investigated, which provides foundation for the brine preparation of the physical simulation and on-site SCC. Then, based on dimensional analysis and similarity theory, a visual physical simulation platform of SCC under GB was established and five groups of physical simulation experiments were carried out. The effects of tube lifting method and circulation mode on the concentration of withdrawn brine and cavern shape were explored. The relationship between lateral dissolution angle and the effective volume of salt cavern energy storage were analyzed, and the measures and suggestions of controlling lateral dissolution angle and GB were proposed. Finally, the techno-economic analysis of SCC with FW and LB under GB was carried out, which shows that SCC with LB under GB possesses lower cost and better environmental protection. This research can serve as a basis for promoting clean and efficient brine mining and SCC.

On the compressive strength of brittle lattice metamaterials

Lightweight architected materials made of a brittle parent solid have great potential for use in environments where extreme conditions occur due to their excellent strength-to-weight ratio and temperature resistance. In this study we combine experimental and modeling efforts to correlate the strength of three-dimensional lattice metamaterials to topological and microstructural characteristics, as well as the mechanical behavior of the base material. Three regular periodic lattices with varying rigidity, owing to their node connectivity, are additively manufactured using a brittle photopolymer and then tested to failure under compressive loads. Micromechanical models are developed to elucidate the experimental results and uncouple the effects of microstructure and strut failure on the resulting macroscopic strength. Our results indicate that stress concentrations at the nodes, where struts intersect, are significantly affecting brittle failure and need to be accounted for in predictive modeling frameworks. Towards this goal, both beam- and solid-based finite element models are considered and their efficacy in capturing the response and critical stresses is examined in detail. We finally show the dependence of lattice strength to relative density and compare the trends with well-known scaling formulas based on dimensional analysis and beam theory.

Calculation Method of the Blasting Throwing Energy and Its Variation Affected by the Burden

Precise control of casting velocity and effective throwing kinetic energy conversion efficiency in blasting engineering are challenges. To provide a theoretical basis and reference for the implementation plan and fine construction of the cast blasting project, we study the problems of casting velocity and energy consumption ratio of broken rock under the impact load of explosions in this manuscript. The calculation methods of casting velocity and throwing energy of broken rock under two blasting modes of spherical charge and cylindrical charge are established by using the theory of dimensional analysis and rock breaking by blasting. A large number of model tests are carried out by using high-speed photography. The results indicate that the casting velocity of broken rock after explosive initiation has two evident stages: instantaneous acceleration to a certain value and subsequent fluctuation; the velocity presents an ordinary distribution law with the step height, and the fitting correlation of high-speed photography results is more than 91%. With the minimum burden increasing from 0.12 m to 0.2 m, the energy consumption decreases from 1306.88 J to 747.49 J and the proportion of energy consumption decreases from 14.77% to 8.45%.

A Study of The Design Method and Similitude for A Small-Scale Test Drilling Rig (Part 1): An Application of The Geometrically Distorted Scaled Modeling Method

Drillstrings often vibrate severely and tend to twist off during hard rock drilling. Therefore, dynamic testing is crucial in the design of drilling systems. Designers tend to employ the most powerful analytical tools, using the most elaborate electronic computers, however, actual testing is required to the designed system function optimally. In cases of enormous drilling systems, complex dynamic tests are often performed on a smaller-scale replica of the system, referred to as the model, which is more convenient, cost-effective, and time-effective. This study, therefore, describes the establishment of similar conditions among structural systems, with the main objective of studying the similitude theory’s applicability in establishing the necessary similar conditions for designing scaled-down models to predict the drillstring’s vibration behavior. The scaling laws for all the relevant parameters regarding the scaled drillstring model, as well as the full-size drillstring system, were derived from the respective equations of motion. The scaling factors for all relevant parameters are determined using the theory of dimensional analysis. In addition, the geometry distorted similitude theory is revisited and employed to overcome the physical limitation and develop the necessary similar conditions for dynamic testing of the scaled drillstring. Meanwhile, the similitude relationship between the prototype and the model was validated with a case study using lumped segments bond graph modeling and simulation software.

DIMENSIONLESS PHYSICAL-MATHEMATICAL MODELING OF TURBULENT NATURAL CONVECTION

Natural convection heat transfer is present in the most diverse applications of Thermal Engineering, such as in electronic equipment, transmission lines, cooling coils, biological systems, etc. The correct physical-mathematical modeling of this phenomenon is crucial in the applied understanding of its fundamentals and the design of thermal systems and related technologies. Dimensionless analyses can be applied in the study of flows to reduce geometric and experimental dependence and facilitate the modeling process and understanding of the main influence physical parameters; besides being used in creating models and prototypes. This work presents a methodology for dimensionless physical-mathematical modeling of natural convection turbulent flows over isothermal plates, located in an “infinite” open environment. A consolidated dimensionless physical-mathematical model was defined for the studied problem situation. The physical influence of the dimensionless numbers of Grashof, Prandtl, and Turbulent Prandtl was demonstrated. The use of the Theory of Dimensional Analysis and Similarity and its application as a tool and numerical device in the process of building and simplifying CFD simulations were discussed.

Application of first‐order finite similitude in structural mechanics and earthquake engineering

AbstractAn important experimental approach for the testing of earthquake‐resistant structures is scaled experimentation with experimental designs impacted upon by the similitude theory of dimensional analysis. Unfortunately, the type of similitude provided by dimensional analysis seldom applies to complex structures, which is particularly problematic when scaling ratios are large. The issue is one of scale effects where the behaviour of the scaled version of any full‐size structure can be markedly different. Recently however a new theory of scaling called finite similitude has emerged in the open literature that confirms that the similitude offered by dimensional analysis is just one of a countable infinite number of alternative possibilities. The new theory of scaling raises the possibility that buildings and structures can be designed and tested in new ways and this aspect is the focus of this paper. Similitude rules for single and two scaled experiments are examined to illustrate the benefits provided by alternative forms of similitude. The two types of similitude examined are termed zeroth order and first order finite similitude, which are shown to be two forms in an infinite number of alternative possibilities efficiently defined using a recursive relationship. The theory of scaling is founded on the metaphysical concept of space scaling yet provides the means to establish all scale dependencies for structural components and high‐rise steel buildings along with buildings equipped with nonlinear‐fluid viscous dampers for resisting earthquake loading conditions. It is shown through case‐studies of increasing complexity how the new theory can be applied to reconstruct full‐scale behaviours but also revealed are some of the limitations of the new approach.

Prediction method for ground shock parameters of explosion in concrete

The ground shock caused by the explosion is the main load causing the destruction of underground engineering. To study the propagation law of ground shock parameters in concrete, free-field tests of explosions at seven different scaled depth of burial (SDOB) have been carried out by using self-constructed concrete blocks with large-sizes. The results indicate that when the SDOB ranges from −0.25 to 1.2 m/kg1/3, the pressure and acceleration at the same scaled distance from the explosion center increase with the SDOB. When the SDOB reaches 1.2 m/kg1/3, the explosion energy is completely enclosed into underground, and the coupling coefficient reaches 1.0. Combined with the dimensional analysis theory, it is found that when the SDOB of an explosion is determined, the propagation law of pressure and acceleration presents a power function attenuation with the increase of the scaled distance from the explosion center. For the three different strength grades of C30, C50, C70 concrete under semi-buried explosion, the attenuation gradient of the pressure or acceleration is inconsistent with an increase in the scaled distance from the explosion center. The higher the concrete strength, the slower the gradient changes. By introducing the depth of burial (DOB) coefficient of equivalent energy, the prediction method of ground shock parameters at different SDOB of explosion and explosions with different mass of charge can be obtained, and it has high accuracy for calculating the ground shock parameters of explosion with large mass charge which verified by numerical simulation.

Finite similitude in fracture mechanics

Scaled experimentation can potentially provide significant benefits such as reduced costs materials and time in testing but is afflicted by the phenomena of scale effects, where the behaviour at scale can be markedly different to that at full size. The design of scaled experiments is presently predominantly founded on the theory of dimensional analysis, which itself is grounded on the invariance of dimensionless governing equations with scale. The reality of fracture mechanics however is not invariant equations, but significant deviations and it is evident that scaled-fracture related experiments are presently limited by this affliction.This paper examines an advance in a new approach to scaled experimentation called finite similitude. The key question addressed here is whether it is possible to overcome the affliction of scale effects by performing not one but two scaled experiments at different scales. It is shown that finite similitude (unlike dimensional analysis) is able to capture all forms of scale dependency, which opens up the possibility of selecting alternative forms of scale invariance and consequently alternative forms of similitude.First-order finite similitude is investigated in the paper and applied to cracked compact-tension and three-point bending test specimens along with a cracked pressure vessel to illustrate the new concepts. These case studies reveal the veracity and potential of the new approach and highlight possibilities that hitherto would have been deemed impossible with the circumvention of scale effects (as traditionally defined).

A first order finite similitude approach to scaled aseismic structures

For many decades the designs of earthquake-resistant (aseismic) structures have been influenced by scaled experiments, underpinned by the theory of dimensional analysis. Although scaled experiments still play an important role, they are recognised to suffer shortcomings, which are particularly severe when scaling ratios are pronounced. The issue is one of scale effects and the inability of dimensional analysis to offer any solution in their presence.This paper is concerned with a new theory for the analysis of aseismic structures that is founded on the metaphysical concept of space scaling, where beams, substructures or buildings etc. are contracted through the mechanism of space contraction. Although space contraction is evidently practically impossible the theory describes the effects of such a process on the underpinning governing mechanics involved. Unlike dimensional analysis the approach which is termed finite similitude embraces scale effects and accounts for them by linking experiments at more than one scale.It is demonstrated in this work how it is possible to reconstruct full-scale behaviour by means of two scaled experiments of a selected beam, column and multi-storey structure when subjected to dynamic loading conditions.