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Geometric modeling of the microstructure of the surface based on the
theory
Tashkent State Transport University
Mamurova Feruza Islomovna
Annotation.
The microstructure of the surface is an important characteristic
associated with properties such as microhardness, wear resistance, fatigue strength,
coefficient of friction, thermal and power loads, aero and gas dynamic resistances, etc.
Solving problems of gas dynamics in the absence of data on the real structure of the
surface with which gas molecules interact makes it difficult to formulate boundary
conditions for calculation the transfer of energy and momentum during the flow of
bodies .
Keywords:
surface, resistance, problems, structure, calculation, nano, coatings,
modeling, geometry.
Mathematical models of surfaces of metals, alloys, and nano-coatings used in
modeling physical processes such as scattering of light fields, laminar-turbulent
transition, or interaction of gas molecules with a surface, contain in most cases strong
simplifications. The assumption of surface smoothness leads to a number of physically
unfounded conclusions, because it has been experimentally proven that the
microstructure of the surface at the nanometer and atomic levels has a significant impact
on the calculation results. As studies in recent years have shown, an effective way to
model rough (non-differentiable) surfaces for solving problems of gas dynamics is to
use fractal geometry methods that take into account roughness at the micro and nano
levels and are based on the validity of the statement that the structure of a natural surface
is equally fractal at all levels.
The task of constructing a geometric model of a microsurface has both theoretical
and applied aspects. As you know, the physical processes that occur when atoms and
molecules of a gas interact with a surface are very complex. Therefore, a theoretically
justified interpretation of the results of ground-based experimental testing of fragments
of aircraft is required. The applied value is determined by the need to optimize flow
diagnostics tools in high-enthalpy installations in which thermal loading acting on
aircraft during flight occurs, as well as technological processes for creating heat-
shielding materials and coatings for rocket and space technology (RCT) products.
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1. The choice of the fractal dimension parameter as an effective characteristic of
the development of the micro- and nanostructure of a rough surface is justified. Fractal
dimension equivalently replaces a whole range of amplitude and step characteristics of
surface roughness. The use of the fractal dimension parameter is proposed in relation
to the problems of heat transfer in high-enthalpy nonequilibrium flows
2. Improved algorithms for modeling the microstructure of the surface by
introducing a geometrically ordering component, which made it possible to effectively
approximate the roughness of the surface with irregularities at the micro and nano levels
simultaneously. The advantage of constructing fractal surfaces over classical models of
non-smooth shapes is the completeness of the description of the surface geometry.
3. The possibility of using fractal surface models to determine the properties of
heat-shielding materials of aviation and rocket and space technology products is
investigated. It has been experimentally confirmed that fractal models make it possible
to simplify and reduce the cost of calculating the interaction of gas with a wall by
eliminating the stage of direct investigation of the geometric characteristics of a real
surface.
The fractal surface model obtained using the proposed construction algorithms is
designed to refine experimental data on the catalytic activity of thermal protection
materials and transfer these data to the conditions of full-scale aircraft flight in the
atmosphere. The practical significance of the study is the results obtained by modeling
non-differentiable shapes for the needs of gas dynamics, including to reduce the
measurement error of the enthalpy of gas deceleration in high-temperature installations
and stands through the use of highly catalytic (metal) coatings with a highly developed
structure of the surface layer. It has been experimentally proved that physical quantities
such as the measured heat flux and the specular reflection coefficient are influenced by
the micro- and nano geometry of the surface.
The practical application of the work was the introduction of the research results
into the process of experimental testing of the structural elements of the docking unit
of the apparatus in terms of interpreting the effect of reducing convective heat flows on
protruding elements coated with high-temperature paint. Validity and reliability. The
research was carried out taking into account the current state of knowledge of the
problem of the influence of the geometric properties of micro- and nano-roughness of
materials on the physico-chemical processes of their flow with a high-enthalpy gas. The
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construction of a mathematical model of a rough surface using fractal principles is
theoretically justified, experimentally confirmed and is the basis for the study of the
influence of the geometry of the micro- and nanostructure of the surface on the physical
properties of the materials under study. Data on the geometry of the micro- and
nanostructure of the surface of the studied materials were obtained using a scanning
tunneling microscope, for calibration of which a test surface with known relief
parameters was used.
A review of the geometric characteristics of the surface that affect the results of
ground-based testing of heat-protective materials is carried out. It is shown that the
scanning tunneling microscope (STM) most fully satisfies the conditions of non-contact
investigation of the spatial microstructure of the surface layer. The theoretical and
applied issues of studying the geometry of the surface at the micro and nano levels and
the problem of taking into account the roughness of the surface with irregularities at the
micro and nano levels at the same time are studied. It has been found out that in recent
years there has been a tendency to switch from working with a two-dimensional model
to a three-dimensional model with a simultaneous reduction in the number of
parameters taken into account without loss in their information completeness.
It is established that the fractal dimension parameter replaces the set of classical
roughness parameters. It is proved that the fractal dimension does not depend on the
magnification scale. This indicates that the geometry of real surfaces is close to fractal.
The second chapter examines existing approaches to creating models of non-
differentiable shapes and describes improved algorithms based on fractal theory for
modeling surface microstructure. In his work, de Carpentier describes the creation of a
surface model using Brownian motion, the geometric meaning of which is to project
sections of a linear surface onto a plane. At each iteration, the elevation of the resulting
"fragments" changes.
However, in this work it was found that when constructing a microsurface model
in this way, after 48 iterations and adding filtering in accordance with the normal law
of probability distribution, the value of the fractal dimension corresponds more to the
dimension of the plane than to the "highly developed" surface. This led to the need to
improve the algorithm by using a cyclic surface instead of a linear surface with a
parallelism plane.
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The design task was to obtain a fractal surface similar to the surface of a copper
alloy having D = 2.47. Thus, the required value of D was achieved only at 186
iterations. Volumetric visualization of the image of the micro surface of the copper
alloy.
However, the most interesting modifications of the method were algorithms
based on the use of projections of the torus section and the torus node (Fig. 1)
Fig. 1
The results of the algorithms for projecting torus sections and torus node sections
also lead to the construction of a fractal surface with dimension D≈2.47.
It should be noted that the number of iterations to achieve the fractal dimension
D≈2.47 for each algorithm is a quantitative characteristic.
The shaded area indicates the optimal ratio of the number of variables taking a
random value in an iteration to ensure a minimum of the iterations themselves. To verify
the simulation results and calculate the fractal dimension, the algorithms of construction
and calculation were tested. This revealed a correspondence between the theoretical
and calculated values.
The verification of the calculation method based on the calculation of the fractal
dimension of the surface for simple fractal surfaces has a good correspondence with the
theoretical results. It should be noted that a comparison of the real micro-surface of a
copper alloy and an artificial fractal surface obtained using an improved Brownian
motion algorithm only on the basis of equality of their fractal dimension D would look
incorrect. Therefore, a comprehensive calculation of the roughness characteristics was
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carried out in accordance with: – the height characteristics of the profile (Ra, Rz,
Rmax); – the step characteristics of the profile (Sm, S); – the parameter of the relative
reference length of the profile (tp).
It is established that the fractal dimension parameter, calculated for a local
surface area by the cubic dimension method, replaces a whole complex of amplitude
and step parameters of the surface profile. Fractal dimension, as a measure of the
development of the structure of the surface layer, 17 correctly describes the geometric
characteristics of roughness and harmoniously fits into the calculations of heat transfer
in high-enthalpy nonequilibrium flows, which allows us to solve the problem of the
influence of microstructure on the speed of heterogeneous processes occurring on the
surface of a heat-shielding material.
Algorithms have been improved to build a model approximating the roughness
parameters at the micro and nano levels simultaneously. The introduction of a
geometrically ordering component in the iterative process in the form of linear and
cyclic surfaces with a plane of parallelism made it possible to reduce the number of
iterations to achieve the necessary values of fractional dimension by almost half. By
means of verification of modeling methods and calculation of fractal dimension, it is
established that the fractal model qualitatively describes the roughness parameters of a
real surface.
Numerical simulation of the interaction of an incoming atom with atoms of the
crystal lattice of the surface by the PSM method shows that the average number of
collisions is proportional to the surface area. A formula is derived to determine the
relationship between the fractal dimension of the surface and the average number of
multiple reflections of a molecule from the surface.
It has been experimentally confirmed that the heat flow in chemically
nonequilibrium dissociated gases is influenced by the catalytic properties of the surface,
which depend both on the chemical composition of the material and on the structure of
the surface layer.
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