Similarity Theories Research Articles

Analysis of Fractal Properties of Atmospheric Turbulence and the Practical Applications

Atmospheric turbulence, recognized as a quintessential space–time chaotic system, can be characterized by its fractal properties. The characteristics of the time series of multiple orders of fractal dimensions, together with their relationships with stability parameters, are examined using the data from an observational station in Horqin Sandy Land to explore how the diurnal variation, synoptic process, and stratification conditions can affect the fractal characteristics. The findings reveal that different stratification conditions can disrupt the quasi-three-dimensional state of atmospheric turbulence in different manners within different scales of motion. Two aspects of practical applications of fractal dimensions are explored. Firstly, fractal properties can be employed to refine similarity relationships, thereby offering prospects for revealing more information and expanding the scope of application of similarity theories. Secondly, utilizing different orders of fractal dimensions, a systematic algorithm is developed. This algorithm distinguishes and eliminates non-turbulent motions from observational data, which are shown to exhibit slow-changing features and result in a universal overestimation of turbulent fluxes. This overestimation correlates positively with the boundary frequency between turbulent and non-turbulent motions. The evaluation of these two aspects of applications confirms that fractal properties hold promise for practical studies on atmospheric turbulence.

Local Similarity Theory as the Invariant Solution of the Governing Equations

The present paper shows that local similarity theories, proposed for the strongly-stratified boundary layers, can be derived as invariant solutions defined under the Lie-group theory. A system truncated to the mean momentum and buoyancy equations is considered for this purpose. The study further suggests how similarity functions for the mean profiles are determined from the vertical fluxes, with a potential dependence on a measure of the anisotropy of the system. A time scale that is likely to characterize the transiency of a system is also identified as a non-dimensionalization factor.

Enhancing Ensemble Clustering with Adaptive High-Order Topological Weights

Ensemble clustering learns more accurate consensus results from a set of weak base clustering results. This technique is more challenging than other clustering algorithms due to the base clustering result set's randomness and the inaccessibility of data features. Existing ensemble clustering methods rely on the Co-association (CA) matrix quality but lack the capability to handle missing connections in base clustering. Inspired by the neighborhood high-order and topological similarity theories, this paper proposes a topological ensemble model based on high-order information. Specifically, this paper compensates for missing connections by mining neighborhood high-order connection information in the CA matrix and learning optimal connections with adaptive weights. Afterward, the learned excellent connections are embedded into topology learning to capture the topology of the base clustering. Finally, we incorporate adaptive high-order connection representation and topology learning into a unified learning framework. To our knowledge, this is the first ensemble clustering work based on topological similarity and high-order connectivity relations. Extensive experiments on multiple datasets demonstrate the effectiveness of the proposed method. The source code of the proposed approach is available at https://github.com/ltyong/awec.

Analytical closed-form solution of the Navier–Stokes equations for the aerodynamic near-field and sonic boom from axisymmetric bodies

An analytical closed-form solution is presented for the aerodynamic near-field and ground signature from an axisymmetric body with a low thickness ratio. The Navier–Stokes equations are formulated as a boundary value problem that incorporates the incoming ambient flow-field and the aerodynamic properties on the body surface. The shape of the aerodynamic body is defined as a product of generalized functions. A direct solution for the density of the aerodynamic near-field, represented as a function of both space and time, is proposed through the integration of the Navier–Stokes equations in a generalized functional form. Pressure, temperature, velocity, and Mach number are then derived in the near-field. The methodology, being fully nonlinear, surpasses the traditional F-function, impulse, and hypersonic similarity theories originally developed for near-field prediction. The presentation outlines the major steps in deriving the analytical solution and provides predictions from an aerodynamic body in the near-field, along with the associated ground signature. The methodology is focused on aerodynamic bodies operating at high-speeds, ranging from the supersonic to the hypersonic regime. [Work supported by the Defense Advanced Research Project Agency, under Grant No. W911NF-21-1-0342.]

Physics-based scintillations for outdoor sound auralization.

The sound propagating in a turbulent atmosphere fluctuates in amplitude and phase. This phenomenon, known as acoustic scintillation, is caused by random fluctuations in the acoustic refractive index of the air induced by atmospheric turbulence. Auralization techniques should consider this phenomenon to increase the realism of the synthetic sound. This paper proposes a physics-based formulation to model sequences of log-amplitude and phase fluctuations of a sound propagating in a turbulent atmosphere. This method applies to slanted and vertical propagation of the sound, which is useful for simulating elevated noise sources such as aircraft, drones, and wind turbines. The theoretical framework is based on the spatial correlation functions for the log-amplitude and phase fluctuations for spherical waves, the von Kármán spectrum, and similarity theories to model atmospheric turbulence. Two applications with audio files are presented to demonstrate the applicability of this method to tonal and broadband noise.

Influence of Stability and Surface Roughness on Turbulence during the Stable Atmospheric Variability and Transport (SAVANT) Field Campaign

Abstract Harvesting of crops in a weakly sloping Midwestern field during the Stable Atmospheric Variability and Transport (SAVANT) observation campaign allowed for a systematic investigation of the influence of surface roughness and static stability magnitude on the applicability of the Monin–Obukhov similarity (MOST) and hockey-stick transition (HOST) theories during stable boundary layer periods. We analyze momentum flux and turbulent velocity scale VTKE in three regimes, defined using the gradient Richardson number Ri and flux Richardson number Rif as regime 1 (0 < Ri ≤ 0.1 and 0 < Rif ≤ 0.1), regime 2 (0.1 < Ri ≤ 0.23 and 0.1 < Rif ≤ 0.23), and regime 3 (both Ri and Rif > 0.23). After harvest, in regime 1, stability varied from near-neutral to weakly stable and both MOST and HOST were applicable to estimate the momentum fluxes and VTKE as a function of mean wind speed. In regime 2, the momentum flux deviated from the MOST linear relationship as stability increased. In regimes 1 and 2, a HOST-defined threshold wind speed Vs was identified beyond which VTKE increased linearly with wind speed at a rate of 0.26 for all observation heights. Below this threshold wind speed, VTKE behaved independent of mean wind and observation heights. Alternatively, for preharvest periods, MOST was applicable in regimes 1 and 2 for all heights and HOST was applicable with reduced Vs for heights above the crop layer. Regime 3 during pre- and postharvest consisted of strongly stable periods and very weak to weak winds, where MOST was found to be invalid and VTKE remained low and independent of wind speed. The results suggest that roughness due to crops enhances the turbulence generation at lower wind speeds.

Local Similarity Theory of Convective Turbulent Layer Using “Spectral” Prandtl Mixing Length and Second Moment of Vertical Velocity

Abstract Approximations of the turbulent moments of the atmospheric convective boundary layer are constructed based on a variant of the local similarity theory. As the basic parameters of this theory, the second moment of vertical velocity and the “spectral” Prandtl mixing length are used. This specific choice of the basic parameters allows us to consider the coefficient of turbulent transfer and the dissipation of kinetic energy of the Prandtl turbulence theory as the forms of the local similarity. Therefore, the obtained approximations of the turbulent moments should be considered as natural complementation to the semiempirical turbulence theory. Moreover, within the atmospheric surface layer, the approximations of the new local similarity theory are identical to the relations of the Monin–Obukhov similarity theory (MOST). Therefore, the proposed approximations should be considered as a direct generalization of the MOST under free-convection conditions. The new approximations are compared with the relations of the known local similarity theories. The advantages and limitations of the new theory are discussed. The comparison of the approximations of the new local similarity theory with the field and laboratory experimental data indicates the high effectiveness of the proposed approach.

A one – source model to estimate sensible heat flux in agricultural landscapes

A one-source model type that combines surface renewal (SR) and similarity theories to estimate the sensible heat flux (H) involving the land surface temperature (LST) and low-frequency measurements, HSR-LST, was tested over agricultural surfaces with fractions of ground canopy cover that ranged from 39% up to 97% and canopy heights ranging from 0.2 to 1.8 m. The performance was tested under clear sky conditions and for stable and unstable cases during the day. The sensible heat flux was estimated using in situ LST measurements over a homogenous canopy (wheat) under stable conditions. Unstable conditions were met for two growing crops (rain-fed soybean and corn) where HSR-LST was tested using LST retrieved from Landsat 5 to 7. The sensible heat flux taken as a reference for soybean and corn was determined using the eddy covariance (EC) method and for wheat it was determined using the residual method (the evapotranspiration was measured using a large weighing lysimeter). For wheat HSR-LST performed slightly better than a calibrated one-source model. For soybean and corn, it performed similarly to one-source and two-source model types. The one-source models included two types, one requiring calibration and the other one self-calibrating, using ‘hot’ and ‘cold’ spots in the surroundings. Further testing for sparse canopies is suggested prior to the development of a two-source model in the framework of SR-LST because HSR-LST performed similarly to the other models. For corn, HSR-LST had the closest performance to HEC and for soybean the two-source model (the best) was slightly closer to HEC than HSR-LST. The root mean square differences (RMSD) comparing HSR-LST against the reference were in the range 26 (corn) – 46 (wheat) Wm−2. Therefore, HSR-LST is an alternative to consider for estimating H because it was competitive and simpler to apply than other thermal remote sensing-based methods.

Study on propellant management device in plate surface tension tanks

Surface tension tanks exploit the liquid surface tension in liquid transport and gas–liquid separation to provide gas-free propellant for the thruster. High transport efficiency of deflectors can ensure liquid relocation in the microgravity environment within a short period of time, which is helpful for providing gas-free propellant. This paper explores transport efficiency of deflectors through drop tower experiments and numerical simulations with the volume of fluid method. The experimental and numerical results show that both the liquid flow speed and the mass flow rate along deflectors increase as the widths of deflectors increase. Besides, a new type of propellant management device, which consists of four deflectors and four anti-sloshing baffles, is verified with numerical simulations and drop tower experiments. Moreover, long-term conditions are predicted by using similarity theories. These results can provide important references for the design of plate tanks.

Scaling Turbulent Combustion Fields in Explosions

We considered the topic of explosions from spherical high-explosive (HE) charges. We studied how the turbulent combustion fields scale. On the basis of theories of dimensional analysis by Bridgman and similarity theories of Sedov and Barenblatt, we found that all fields scaled with the explosion length scale r0. This included the blast wave, the mean and root mean squared (RMS) profiles of thermodynamic variables, combustion variables, velocities, vorticity, and turbulent Reynolds stresses. This was a consequence of the formulation of the problem and our numerical method, which both satisfied the similarity conditions of Sedov. We performed numerical simulations of 1 g charges and 1 kg charges; the solutions were identical (within roundoff error) when plotted in scaled variables. We also explored scaling laws related to three-phase pyrotechnic explosions. We show that although the scaling formally broke down, the fireball still essentially scaled with the explosion length scale r0. However, the discrete Lagrange particles (DLP) (phase 2) and the heterogeneous continuum model (HCM) of the DLP wakes (phase 3) did not scale with r0, and mean and RMS profiles could differ by a factor of 10 in some regions. This was because the DLP particles and wakes introduced an additional scale that broke the similarity conditions.

Experimental Investigation on Mechanical Characteristics of Waterproof System for Near-Sea Tunnel: A Case Study of the Gongbei Tunnel

The water around thenear-sea tunnels is supplied infinitely, and mechanical characteristics of the lining and movement joint are inevitably affected by waterproof methods. The research on the mechanical characteristics of the waterproof system is immature. As a case study of the Gongbei tunnel, a scale model was established in this study, and the stratum, pipe curtain, grouting circle, lining, waterproof board, and movement joint were simulated based on the similarity theories. By changing the externally applied water pressure and drainage discharge, the variation and distribution of the water pressure and strain on the lining with the fully wrapped waterproof (FWW) method, the lining with the partially wrapped waterproof (PWW) method, and the movement joint were investigated. Furthermore, several suggestions on the selection of the waterproof method were presented. The results indicate that the PWW method can reduce the water pressure and strain on the lining under the drained state. Under the state of free drainage, the strain on the lining with the PWW method may get a discount of about 30%. More attention could be paid to the waterproof of the movement joints in the construction process, especially the invert. The research results may offer some valuable insights into the waterproof design of similar near-sea tunnels.

Development and application of a new similar material for fluid–solid coupling model test

Correct selection of similar materials is key to geomechanical model tests. According to similarity theories, a novel similar material for the fluid–solid coupling model test was developed to solve problems of low similarity and poor stability. A similar material is mixed with calcium carbonate, white cement, paraffin, quartz sand, silicone oil, talc, and iron powder. Quartz sand and talc powder are the main materials; white cement, calcium carbonate, and paraffin are glue; silicone oil and iron powder are regulators. The effects of different mixing ratios on the compressive strength, permeability coefficient, and specific gravity were studied using a single factor analysis method. The main components controlling the material properties were determined through numerous laboratory tests. The experimental results show that the new material strength is controlled by cement, paraffin, and calcium carbonate. Its permeability coefficient can be adjusted by altering the ratio of silicone oil and paraffin. Its specific gravity is mainly affected by iron powder. The new material can simulate low-strength and medium-strength rock materials with different permeabilities, and it can effectively solve the problem that mechanical properties and nonhydrophilic properties cannot be satisfied simultaneously. The material was successfully applied in geological model tests of fault water inrush.

Surface eddy fluxes and friction velocity estimates taking measurements at the canopy top

Agricultural engineers, crop physiologists and hydrologists are often forced to deploy instrumentation close to the canopy top to estimate surface eddy fluxes. This is a widely recognized problem either at farm-level or research-level. The eddy covariance (EC) method is grounded on hypothesis, simplifications and relationships valid in the inertial sublayer and alternative methods may require either calibration or inputs that are not commonly available. The present study enables a method that combines similarity, transilient and surface renewal theories in conjunction with the analysis of small eddies (Surface renewal of small eddies, SRSE, method) to estimate the friction velocity and surface fluxes of different scalars (sensible heat, latent heat and carbon dioxide) taking measurements at the canopy top. SRSE minimizes estimation of canopy parameters (only the zero-plane displacement is involved), allows implementing a primary flux quality control and to guide instrumentation deployment. For a mature maize crop, SRSE and EC surface energy balances indicated that EC was slightly more reliable than SRSE during daytime. For unstable cases, the normalized mean absolute error (MAE) to the available net surface energy (ANSE), MAE/ANSE, obtained by the EC and SRSE methods were 9% and 13 %, respectively. For stable cases during the day, 10 % and 11 %, respectively. For stable cases at night SRSE performed better than EC, MAE/ANSE were 23 % and 16 %, respectively. Regardless of the half-hourly eddy flux (including the friction velocity), the EC and SRSE were highly correlated (the minimum coefficient of determination was 0.91) and, in average (for all data) flux of scalars nearly matched including water-use-efficiency. To avoid acquisition of expensive and stringent instrumentation, the closure of the surface energy balance equation was forced to derive affordable approaches to estimate the latent heat flux for different stability cases.

Transposition of the mechanical behavior from model to prototype of Francis turbines

Hydropower is nowadays essential for balancing the electrical grid providing flexibility and fast response. The role of hydraulic turbines has changed from working at their Best Efficiency Point (BEP) to work in the whole operating range when demanded. Francis turbines working at off-design conditions suffer from dynamic problems that affect the useful life of their components, especially of the runner.Reduced scale physical models of Francis turbines are widely used to determine their hydraulic behavior. Most of the hydraulic parameters obtained in model tests are transposable to prototype by using the similarity theories as long as they meet similitude requirements. However, from the mechanical behavior point of view, both model and prototype structures are not always similar and the mechanical behavior of the model is often not considered for transposition.In this paper, a transposition method for the mechanical behavior of Francis turbines models is presented and the results are compared with the real size prototype. The modal behavior of the runner, its stress and fatigue life estimation for different operating conditions are experimentally compared for both model and prototype. Results present the possibilities and limitations of transposing the mechanical parameters from model to prototype.

РАЗВИТИЕ СИСТЕМНОГО МОДЕЛИРОВАНИЯ И МЕТОДЫ ФОРМАЛИЗОВАННОГО ПРЕДСТАВЛЕНИЯ СИСТЕМ

Modern science constantly improves approaches and methods of formalization of certain regularities using the latest achievements in mathematics, computer science and other exact Sciences. The methodological basis for research and modeling of systems is often considered to be the theory and practice of mathematical modeling. In the classical system approach, as a rule, system modeling is based on the use of similarity theories and scientific experiment, as well as mathematical statistics, algorithm theory, and a number of other fundamental classical theories. However, focusing only on mathematical modeling ignores the system nature of objects and their models, and also complicates the formalization of the problem, its translation from the verbal form of description to the formal one. The need to understand the relationships between the elements of the system, its structure and functions, combined with the description of system patterns, led to the development of system modeling, which aims not only to gain knowledge about the system, but also to optimize it.

Correction for semi-infinite assumption in the theories of natural convection and determination of average Nusselt number for finite inclined plates

It is established that large errors would be incurred if the average Nusselt number for natural convection above heated horizontal plates of finite size is determined by the similarity or integral theories which assume the plate is semi-infinite. Recognizing that boundary layers grow from both edges of a finite plate, correction factors are mathematically deduced for the isothermal case (Ft=22/5) and for the case of constant heat flux (Fq=21/3). Correction factors are also developed for near-horizontal orientations of the heated plate. The semi-infinite assumption does not alter the computation of the overall heat transfer rate for inclination angles at which the vertical mechanism of natural convection is dominant since then, even in the case of a finite plate, a single boundary layer covers the entire length of the plate. The theoretical rationalization developed in the paper brings the predictions of explicit algebraic expressions for average Nusselt number in line with those determined accurately by computational fluid dynamics (with extensive experimental verification), and provides physical explanation of the subtle thermo-fluid-dynamics of natural convection established by the present CFD simulations.

A universal wind profile for the inversion‐capped neutral atmospheric boundary layer

Wind profiles above the atmospheric surface layer are not accurately described by classic similarity theories. Far from the surface, the underlying assumptions of such surface‐layer theories break down due to the stronger influence of buoyancy forces induced by the temperature inversion that caps the atmospheric boundary layer (ABL), as well as the Coriolis force. This paper examines the influence of these forces on the mean flow and presents a new similarity theory to predict mean wind profiles in and above the surface layer for an ABL with zero surface heat flux and capped by an inversion of potential temperature, that is, the conditionally neutral ABL. The analysis here is based on the results of 17 large‐eddy simulations (LES) over a flat homogeneous rough surface, which leads to and supports the new similarity theory. The development is based on two applications of the Buckingham Π theorem. A first application allows determination of the entrainment‐induced heat flux profile through the ABL and into the surface layer, which is then used within a second dimensional argument for the vertical shear of mean wind speed. We subsequently find a new dimensionless group (Π2) depending on the capping inversion strength, the Coriolis parameter, the surface stress and the ABL depth, which is correlated to the dimensionless shear (Π1) through a universal function β. Integrating the functional relation between Π1 and Π2 produces an equation for the mean wind speed profile; it effectively includes an additive “correction” to the log‐law in terms of Π2, analogous to the Monin–Obukhov profile correction function. Unlike surface‐layer similarity, the new form accounts for the influences of both the surface and the ABL top. Relative to the LES, the new profile form exhibits errors in mean wind speed below 5% for heights below 90% of the ABL depth; this is relevant for applications above the surface layer (e.g., wind energy).

Methods for the Simulation of the Aerodynamic Heating Conditions of the Structural Elements of Space Shuttles

The paper considers the problem of creating and operating products of reusable space-rocket hardware, in particular with respect to ensuring the integrity of structural elements and safe return to Earth under aerodynamic heating conditions. This problem has two aspects, which differ in specificity of the approaches to designing the elements of space shuttle systems (SSS). The first aspect is associated with ensuring the reliable functioning of the most thermally stressed elements of the re-entry glider, which are structures with large angles of attack, such as fuselage nose and leading edge, wing loading edge, elevators and air intake edges. The second aspect mainly concerns ensuring the allowable temperature level of the spacecraft pressure shell on all flight path segments, especially during re-entry in the Earth’s atmosphere. In view of this, one of the main goals in creating SSSs is to develop a reliable heat shield having acceptable size and weight parameters, and cost. The successful solution of these problems is determined in many respects by the optimal choice of appropriate classes of materials: special high-temperature alloys, structural ceramics, highmelting-point metallic and polymeric composite materials. For the modern structures of SSSs, the use of structural metallic materials is worth-while in many respects. Rig test procedures are proposed. A complex of gas-dynamic test rigs was used as the basic equipment, whose fundamental design features and methodological solutions ensure the complete rig test cycle for the solution of problems in both directions. The methodological basis of rig tests is a set of specialized procedures for the simulation of thermally stressed states of the material and the intesity of the external action of the environment, which provide the equivalence of material damage processes and the limit state of the structural element under investigation under model and full-scale conditions. The fundamental basis of these approaches is the classical similarity and dimensional theories, the main postulates of which have been transformed and adapted to the problems of the study of the strength of materials and damageability of structural elements under thermal cyclic loading in corrosive environments. The developed procedures and experimental means allowed the modeling of aerodynamic heating processes of the structural elements of space shuttles. It has been shown that the implemented methods enable the evaluation of functional characteristics, determination of a set of properties and refinement of the technology for the formation of structural elements of aerospace vehicles operating under the conditions of aerodynamic heating to extremely high temperatures.

Vortex scaling ranges in two-dimensional turbulence

We survey the role of coherent vortices in two-dimensional turbulence, including formation mechanisms, implications for classical similarity and inertial range theories, and characteristics of the vortex populations. We review early work on the spatial and temporal scaling properties of vortices in freely evolving turbulence and more recent developments, including a spatiotemporal scaling theory for vortices in the forced inverse energy cascade. We emphasize that Kraichnan-Batchelor similarity theories and vortex scaling theories are best viewed as complementary and together provide a more complete description of two-dimensional turbulence. In particular, similarity theory has a continued role in describing the weak filamentary sea between the vortices. Moreover, we locate both classical inertial and vortex scaling ranges within the broader framework of scaling in far-from-equilibrium systems, which generically exhibit multiple fixed point solutions with distinct scaling behaviour. We describe how stationary transport in a range of scales comoving with the dilatation of flow features, as measured by the growth in vortex area, constrains the vortex number density in both freely evolving and forced two-dimensional turbulence. The new theories for coherent vortices reveal previously hidden nontrivial scaling, point to new dynamical understanding, and provide a novel exciting window into two-dimensional turbulence.

Enhanced similarity-based metamodel updating strategy for reliability-based design optimization

Metamodels are becoming increasingly popular for handling large-scale optimization problems in product development. Metamodel-based reliability-based design optimization (RBDO) helps to improve the computational efficiency and reliability of optimal design. However, a metamodel in engineering applications is an approximation of a high-fidelity computer-aided engineering model and it frequently suffers from a significant loss of predictive accuracy. This issue must be appropriately addressed before the metamodels are ready to be applied in RBDO. In this article, an enhanced strategy with metamodel selection and bias correction is proposed to improve the predictive capability of metamodels. A similarity-based assessment for metamodel selection (SAMS) is derived from the cross-validation and similarity theories. The selected metamodel is then improved by Bayesian inference-based bias correction. The proposed strategy is illustrated through an analytical example and further demonstrated with a lightweight vehicle design problem. The results show its potential in handling real-world engineering problems.