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Spatial variation in regional flows within the heart, skeletal muscle, and in other organs, and temporal variations in local arteriolar velocities and flows is measurable even with low resolution techniques. A problem in the assessment of the importance of such variations has been that the observed variance increases with increasing spatial or temporal resolution in the measurements. This resolution-dependent variance is now shown to be described by the fractal dimension, D. For example, the relative dispersion (RD = SD/mean) of the spatial distribution of flows for a given spatial resolution, is given by: RD(m) = RD(mref).[m/mref]1-Ds where m is the mass of the pieces of tissue in grams, and the reference level of dispersion, RD(mref), is taken arbitrarily to be the RD found using pieces of mass mref, which is chosen to be 1 g. Thus, the variation in regional flow within an organ can be described with two parameters, RD(mref) and the slope of the logarithmic relationship defined by the spatial fractal dimension Ds. In the heart, this relation has been found to hold over a wide range of piece sizes, the fractal Ds being about 1.2 and the correlation coefficient 0.99. A Ds of 1.2 suggests moderately strong correlation between local flows; a Ds = 1.0 indicates uniform flow and a Ds = 1.5 indicates complete randomness.

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The grid generation capability built into the numerical simulator TOUGH for multi-phase fluid and heat flow through geologic media can create one-column grids with linear or radial geometry, corresponding to one-dimensional or two-dimensional radial flow, respectively. The integral-finite-difference-method that TOUGH employs for spatial discretization makes it very simple to generalize the grid-generation algorithm from integer to non-integer (fractal) flow dimension. Here the grid-generation algorithm is generalized to create one-column grids with fractal flow dimension ranging from less than 1 to 3. The fractal grid generation method is verified by comparing numerical simulation results to an analytical solution for a generalized Theis solution for integer and non-integer flow dimensions between 0.4 and 3. It is then applied to examine gas production decline curves from hydraulically fractured shale that is modeled as a fractal-dimensioned fracture network with flow dimensions between 0.25 and 3. Grids with fractal flow dimension are useful for representing flow through fracture networks or highly heterogeneous geologic media with fractal geometry, and may be particularly useful for inverse methods.

Boming Yu graduated in Engineering Physics at Tsinghua University (Beijing, China) in 1970, and graduated in Nuclear Engineering at Tsinghua University (Beijing, China) in 1981 and received Master degree there. He began his academic career at the Wuhan Steel and Iron Institute (Wuhan, China) in 1983. In 1986, he joined the Huazhong University of Science and Technology (Wuhan, China), where he was promoted to Full Professor in 1996. His research interests cover a wide range of topics such as fractals, transport in porous media, nanofluids, boiling heat transfer, and complex networks. So far, he has published more than 100 journal papers, and he also serves as a pioneer referee for more than ten international journals.

The flow in porous media has received a great deal of attention due to its importance and many unresolved problems in science and engineering such as geophysics, soil science, underground water resources, petroleum engineering, fibrous composite manufacturing, biophysics (tissues and organs), etc. It has been shown that natural and some synthetic porous media are fractals, and these media may be called fractal porous media. The flow and transport properties such as flow resistance and permeability for fractal porous media have steadily attracted much attention in the past decades. This review article intends to summarize the theories, methods, mathematical models, achievements, and open questions in the area of flow in fractal porous media by applying the fractal geometry theory and technique. The emphases are placed on the theoretical analysis based on the fractal geometry applied to fractal porous media. This review article shows that fractal geometry and technique have the potentials in analysis of flow and transport properties in fractal porous media. A few remarks are made with respect to the theoretical studies that should further be made in this area in the future. This article contains 220 references.

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Fluid transportations are fundamentally crucial in numerous scientific and technological problems, including but not limited to biophysics1,2, process engineering3,4 and thermal management5,6. The most common type of fluid flow found in nature and has been predominantly exploited in a variety of engineering applications is turbulence7,8. Such flow dynamic is highly chaotic yet diffusive, which contributes significantly to momentum, heat and mass transfer in flows of practical interest. More importantly, the capability of turbulent motion to facilitate the continuous reshuffling of flow and thermal boundary layers is vital to potently enhance forced convective thermal dissipation9,10,11,12. Such desirable flow characteristics have drawn much attention in deciphering the chaotic nature of turbulence13,14,15,16,17,18,19. Multiple turbulence generators continue to be invented and explored to effectively produce specific fluid flow perturbation. To date, the most well-known passive turbulators are the 2D planar space-filling biplane square grid and fractal insert. Fractal is a geometrical structure having self-similarity shapes at all scales that form an intricate pattern of different number of iterations20. Recently, it has become the focus of multitudinous investigations owing to its exciting performance in producing multilength-scales turbulence, paving the way to tune fluid flow dynamically. Interestingly, the turbulent flow generated using 2D planar space-filling fractal grids embodies more intermittent or more clustered vorticity field and higher turbulence intensities as compared to fluid flow undulations induced via the conventional regular grids of equal or higher blockage ratio21,22,23,24.

The understanding of turbulence is indispensable for the development or optimization of technological applications and industrial processes. Heating, Ventilation and Air Conditioning (HVAC) systems have been the subject of intensive studies in recent years25,26,27,28. HVAC systems provide good indoor air quality and desirable thermal environments, yet it consumes about 40% of the total energy in a commercial building29. The manifestation of the foremost heat exchanging element in HVAC systems, the finned-tube heat exchanger, contributes significantly to energy consumption. An enormous number of engineering design and research are being invested in promoting the efficiency of heat exchangers by intensifying the air-side heat transfer coefficient. One of the renowned methods is to introduce effective fluid flow perturbations across finned-tube heat exchanger30,31,32,33,34. The turbulent fluid flow fluctuations allow the restructuring of flow-boundary layers within the highly compacted finned-tube heat exchangers to promote forced convection. Cafiero et al.35,36 experimentally reported that fractal grid is able to significantly augment circular impinging jets forced convection as compared to a regular grid of equivalent blockage ratio and pumping power due to the high turbulence intensity and high magnitude and persistence streamwise vorticity immediate leeward of the fractal grid. Furthermore, several investigations have revealed the fact that vortex shedding, as well as turbulence integral length scale, play critical roles in thermal dissipation12,37,38. Although many studies have unravelled the undertaking of turbulence integral length scale upon forced convective heat transfer, the basic correlation of both remains contentious12,37,38,39,40,41.

PIV is considered the most renowned optical velocimetry that is specifically designed for airflow measurements. Numerous developments have been conducted since its first appearance in Adrian49 in the 1980s47,50,51,52,53,54,55. It provides two-dimensional simultaneous velocity vectors of multiple tracer particles by averaging the motion of particles in subsections. This velocity computation approach smooths the velocity field within the interrogation areas causing PIV to be impractical for measuring flow dynamics with large velocity gradients56,57. On the other hand, PTV is a Lagrangian oriented concurrent tracer particle tracking, which follows and tracks individual particle26. Thus, it is not surprising to perceive the increasingly reinvigorated interest in the applications of PTV to secure fluid flow observation with higher temporal and spatial resolutions, and more importantly, longer trajectories26,58,59,60,61,62. The downside of PTV, however, is that it requires a tremendously complex tracking system to thoroughly follow the motion of all seeding particles56. More importantly, a significant number of computer processors are required to effectively support the parallelized tracking algorithm63,64,65,66. Thus, a balance between HWA, PIV and PTV is of great importance in understanding the efficacy of fractal grid-induced turbulence on passive heat sink forced convection. In this study, a practical, non-intrusive and low-computational cost in-house stationary-particle tracking velocimetry (SPTV) is carefully designed, set up, validated and developed with the key objective of unveiling the fundamental correlation between fractal-generated turbulence and plate-fin array forced convective heat transfer augmentation. Evidently, in-depth comprehension of fractal-induced turbulence and its implications on thermal enhancement may shed light on the research and development of efficient heat exchangers that can pave the way towards long-term energy sustainability. 006ab0faaa