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Indian Journal of Theoretical Physics, Vol. 62, No. 1-2. 2014Some Aspects of Turbulent Flow Study

N. C. GhoshE. K. T. P. Phase – IV, No. C -34/4

Kolkata - 700107Email : ghosnarayan@gmail.com

A B S T R A C T

Turbulence is seen as one of the last outstanding unsolved problems in classical physics. In the last century, great minds viz, Heisenberg, von Weizs"acker, Kolmogorov, Prandtl and G.I. Taylor had worked on it. Einstein put his last postdoc Bob Kraichnan on the subject of Turbulence. Rapid development of experimental and numerical techniques in this area and the growth of computing power created a lot of activities on turbulence research. Citeing turbulent associated real life phenomena here author has elaborated methodologies of turbulent study.

INTRODUCTIONBig whirls have little whirls that feed on their velocity:little whirls have lesser whirls, and so on to viscosity.

Thus E.G. Rechardson expressed the energy cascading principle in turbulence. Perhaps considering complexities of analysing turbulence one scientist wrote: If there is a god and I can meet him, I will ask him something about willy-lilly behaviours of turbulent phenomena.

Turbulence is one of the oldest and most difficult open problems in physics. Applied Mathematicians deals it very carefully from the mathematical stand point of view.

The story is told of many giants of modern physics, but most plausibly of Heisenberg, on his death-bed, he remarked that the two great unsolved problems were reconciling quantum mechanics and general relativity, and turbulence. "Now, I'm optimistic about gravity..."Knowledge of the study of Theory Turbulence is essential

Turbulence is a notoriously difficult subject. The goal is for investigation of such complex phenomena is to debate with some fundamental questions related to this that are wide ranging, from the initiation of turbulence through to its asymptotic state at high Reynolds number, including the effects of rotation and stratification, and the addition of different phases, such as bubbles, particles and polymers.

Knowledge of the study of Theory Turbulence is essential for analyzing many real life situation viz,For analyzing atmospheric aspects

Continuously moving air in the atmospheric region is a burning example of turbulence. Without the knowledge of turbulent characteristics of air no way can say a single word perfectly regarding the atmospheric region.For Weather forecasting

Weather forecasting in a rudimentary form is probably one of the oldest sciences. Back in 650 BC, the Babylonians were able to predict the weather by observing cloud patterns. Aristotle has described weather patterns in his Meteorologica. Chinese weather prediction traditions date back to 300 BC. India also has a long history of weather prediction. Early philosophical writings of around 3000 BC, such as the Upnishadas, contain discussions about the processes of cloud formation and rain and the cycle of seasons. Kautilya’s Arthashastra contains records of scientific measurements of rainfall and its application to the country’s revenue and relief work. Kalidasa has mentioned the onset of monsoon over central India in Meghdoot.

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The simplest method of weather forecasting is the use of today’s conditions to forecast tomorrow’s weather. This method of forecasting strongly depends upon the presence of stagnant weather pattern, like during the summer season in tropics. Numerical weather prediction or NWP is a set of mathematical equations that governs the behaviour of atmosphere. These are combined in a complex mathematical model, and this is applied to observations of the atmosphere – a turbulent phenomena. There are several variables, which decide the outcome of these predictions. Parameters like the earth’s size, its rotation rate, geography, daily and seasonal variations of incoming solar radiation are more or less constant. However, other parameters like surface reflectivity, melting, evaporation, cloud, rain, friction and sea temperatures vary through the period of a forecast. All these parameters are most complicated due to their turbulent character.

Experts believe that to-days numerical weather models are still too broad-brush to permit truly ‘local’ forecasts to be made. Given the sheer number of variables that affect local weather, mathematicians believe that absolutely accurate forecasts beyond the short term will perhaps remain a chimera as theories for analyzing turbulence is still far from its goal. For understanding aspects of Cyclone :

Cyclone is a storm accompanied by high speed whistling and howling winds. In meteorology, a cyclone is an area of low atmospheric pressure characterized by inward spiraling winds that rotate counter clockwise in the northern hemisphere and clockwise in the southern hemisphere of the Earth. Since the generic term covers a wide variety of meteorological phenomena, such as tropical cyclones, extratropical cyclones, and tornadoes, meteorologists rarely use it without additional qualification.

Cold-core cyclones (most cyclone varieties) form due to the nearby presence of an upper level trough, which increases divergence aloft over an area that induces upward motion and surface low pressure. Warm-core cyclones (such as tropical cyclones and many mesocyclones) can have their initial start due to a nearby upper trough, but after formation of the initial disturbance, depend upon a storm-relative upper level high to maintain or increase their strength.

It is often difficult to predict where a cyclone will strike. When it starts moving from oceans towards the land area, a cyclone can change track and hit areas other than those anticipated earlier.

Tropical Cyclones, a cyclones of special category, are low pressure systems in the tropics that, in the Southern Hemisphere, have well defined clockwise wind circulations with a region surrounding the centre with gale force winds (sustained winds of 63 km/h or greater with gusts in excess of 90 km/h).The gale force winds can extend hundreds of kilometres from the cyclone centre. If the sustained winds around the centre reach 119 km/h (gusts in excess 170 km/h). then the system is called a severe tropical cyclone. These are referred to as hurricanes or typhoons in other countries.

Tropical Cyclones, an example of atmospheric turbulence, derive their energy from the warm tropical oceans and do not form unless the sea-surface temperature is above 26.5°C, although, once formed, they can persist over lower sea-surface temperatures. Tropical cyclones can persist for many days and may follow quite erratic paths. They usually dissipate over land or colder oceans.

Structure of a Cyclone (Schematic)

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Tropical Cyclones are dangerous because they produce destructive winds, heavy rainfall with flooding and damaging storm surges that can cause inundation of low-lying coastal areas. Research has shown that cyclones in the Australian region exhibit more erratic paths than cyclones in other parts of the world. A tropical cyclone can last for afew days or up to two or three weeks. Movement in any direction is possible including sharp turns and even loops. To analyse turbulent condition of fluid during

Tsunami : A tsunami (pronounced tsoo-nah-mee) is a wave train, or series of waves, generated in a body of water by an impulsive disturbance that vertically displaces the water column. Earthquakes, landslides, volcanic eruptions, explosions, and even the impact of cosmic bodies, such as meteorites, can generate tsunamis. Tsunamis can savagely attack coastlines, causing devastating property damage and loss of life.

Tsunamis are unlike wind-generated waves, which many of us may have observed on a local lake or at a coastal beach, in that they are characterized as shallow-water waves, with long periods and wavelengths. The wind-generated swell one sees at a California beach, for example, spawned by a storm out in the Pacific and rhythmically rolling in, one wave after another, might have a period of about 10 seconds and a wave length of 150 m. A tsunami, on the other hand, can have a wavelength in excess of 100 km and period on the order of one hour.

As a result of their long wavelengths, tsunamis behave as shallow-water waves. A wave becomes a shallow-water wave when the ratio between the water depth and its wavelength gets very small. Shallow-water wave move at a speed that is equal to the square root of the product of the acceleration of gravity (9.8 m/s/s) and the water depth - let's see what this implies: In the Pacific Ocean, where the typical water depth is about 4000 m, a tsunami travels at about 200 m/s, or over 700 km/hr. Because the rate at which a wave loses its energy is inversely related to its wave length, tsunamis not only propagate at high speeds – turbulent in nature, they can also travel great, transoceanic distances with limited energy losses.

Turbulent sea surface Ocean TurbulenceAir preasure, due to temperature variation of air over ocean water are not uniform, are in continuous movement in and over ocean water generating ripple, wave, bollow and then crest of waves to turbulence. Rapid ocean waves breaking generate more and more complex turbulence. This process gets momentum with the speed and variation of air pressure and velocity over water and temperature variation process. As near coastal area such variations are high and boundary layer causes resistance in water flow and therefore streaks and vortices interact to generate turbulence there. In deep sea, though fluid is under high pressure

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due to roughness of sea surface, influence of fluid velocity in upper layers and continuous bursting a regular feature in sea bed water is turbulence there.Turbulence in Gas turbine In modern society use of gas turbine in industries/engine is all most a common feature. Strong turbulent condition of fluid inside operates gas turbine.

Turbulence in Hydel plantAt high water velocities and because of edge effects and surface roughness of structures, given that

water is a viscous fluid, flows in a hydropower turbine system are turbulent. The tendency of water molecules to resist shear forces, due to the presence of viscosity, causes irregular water movement. The shear stresses within a flow field tear the fluid into highly energetic, irregular, and three-dimensional eddies, with scales ranging from the size of the flow passage down to unity. These eddies exist randomly in space and time in turbulent shear flows. Turbulent flow occurs when fluid particles move in a highly irregular manner, even if the fluid as a whole is traveling in a single direction. That is, there are intense, small-scale motions present in directions other than that of the main, large-scale flow. Unlike laminar flow, which is most easily described by linear equations, turbulent flow can only be defined statistically; descriptions of the overall motion within turbulent flows cannot be taken as describing the paths of individual particles. For perfect visualization of turbulence in plume of chimney

Use of chimney, through which gasses are been ejected, is a common in human life. Plume of gas comes out from chimney. The plume is a good example of turbulent character of fluid.

In the stable atmosphere case (producing a fanning plume), there is horizontal dispersion at a right angle to the

wind due to turbulence and diffusion. In the vertical, dispersion is suppressed by the stability of the atmosphere, so pollution does not spread toward the ground. This results in very low pollution concentrations at the ground.

In unstable air, the plume will whip up and down as the atmosphere mixes around (whenever an air parcel goes up, there must be air going down someplace else to maintain continuity, and the plume follows these air currents). This gives the plume the appearance that it is looping around. An inversion aloft will trap pollutants underneath it, since the stable inversion prevents vertical dispersion. Pollution released underneath the inversion layer will fumigate the mixed layer. Note that if the smokestack was high enough to release the pollution within the inversion layer, the plume would fan because the plume occurs within stable air. For artificial raining :

Artificial raining is sometimes called cloud seeding. For cloud seeding chemicals like silver iodide is used. Recently solid carbon dioxide, liquid organic compound propel, salt like chemicals are also used. Cloud in the upper atmosphere cannot be solidified due to turbulence in the air of upper sky. So if there is no rain from clouded sky then for raining chemicals are been spayed in the clouded zone.

For raining cold water, though liquid but its temperature is less than 00C is needed. In natural condition liquid becomes solid when temperature is less than 00C, but in turbulent condition of air in high

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altitude remains in liquid condition at below 00C temperature. In such situation silver iodide like chemicals turns liquid water in air to ice, which later becomes drops of rainwater. For analysing Turbulence in stars

Differential rotation in solar-type stars is assumed as due to the influence of the global rotation upon anisotropic turbulence ('the Lambda-effect'). The effect, however, is quenched by the dynamo-induced large-scale magnetic fields. The resulting reduction of the differential rotation feeds back on the dynamo itself. Many of the newly discovered explants are apparently gas giants in close proximity to their parent stars. They therefore raise tides on their host stars and (if similar to Jupiter) will likely have substantial magnetospheres that can interact with stellar fields. This phenomenae is known as stellar turbulence.Turbulent concepts in Stenosis Problem

In human body blood flow is a continuous process. Under certain conditions if blood pressure become abnormally high then in inflow tube near the heart blood flow may be turbulent causing heart attack. Similarly due to an aneurysm, a sac-like protrusion of an artery caused by a weakened area within the vessel wall cerebral (brain) aneurysm ruptures, there are chances for turbulent nature blood flow in major arteries of the brain, escaping blood with high pressure within the brain may cause severe neurologic complications or death. Recently some attempts have been taken to visualize cardiac attack and cerebral (brain) aneurysm ruptures in  computer simulating data with help of mathematical formulae.Salient Points to know Turbulence

Today turbulence are been understand well enough so that airplanes don't fall from the sky and liquid fuel igniting inside a rocket engine produces enough thrust to blast payloads into space. One even may exploit turbulence to make golf balls fly farther (ever noticed the dimples?) and stir milk thoroughly into a cup of tea.

Turbulence is the seemingly chaotic motion of fluid collectively gases and liquids under certain

conditions, such as the smoke rising from a lit cigar or an oil film blown by the wind while floating on a puddle of water.

Water flow experiments through pipes, conducted by Osbourne Reynolds at the turn of the 20th century, gave rise to a basic understanding of the laminar (smooth flow) to turbulent transition through the Reynolds number. The governing partial differential equations for fluid flow, incorporating turbulence, were proposed even earlier by Navier and Stokes. The Navier-Stokes equations assume fluids to be a flexible continuum rather than a collection of molecules. Yet even with this simplification the Navier-Stokes equations remained stubbornly difficult to solve until the advent of computers and numerical analysis.In recent times direct numerical simulation of the Navier-Stokes equations on the most powerful parallel computers has lifted the veil still further on turbulence. However, engineers have simulated turbulent flows since the beginning of the computer age to make safer, faster and more efficient machines. Exploiting cunning models of the Navier-Stokes equations within the field commonly referred to as Computational Fluid Dynamics (CFD). The cunning involves recognizing that modeling the statistically significant

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turbulence (turbulence modeling) captures the majority of engineering flows relatively accurately. This makes such modeling possible on a laptop PC.

Yet for all progress in simulating turbulent flows are still limited to inaccurate five-day weather forecasts. The accuracy of our forecast models (though not perfect) is not the problem; it is the measurement of the initial conditions fed into those models. The well-known butterfly effect is at work or more formally the nature of the system (and therefore any model of it) is chaotic, which means acute sensitivity to initial conditions. Essentially weather forecasts and many environmentally driven simulations (e.g. pollution plumes from power stations) are hindered by inability to provide accurate initial conditions.At the center of the planet, about 4,000 miles down, sits a solid ball of iron the size of the moon. The earth is standing on about 1,800 miles of rock, forming Earth’s crust and mantle. But in between the mantle and the iron ball there is a churning ocean of liquid of some sort, but scientists aren’t certain what it’s made of or how it reacts to the stuff around it. There is a lot of iron in this ocean. Also there are some other thing, based on what researchers understand about the pressure, temperature, and density of materials down there, some maintain that the core also contains lots of hydrogen and sulfur. Raymond Jeanloz of UC Berkeley believes that another component is oxygen, which comes from rocks in the part of the mantle that borders the liquid core.

Knowing more about the molten concoction would give scientists clues about how Earth formed and how heat and convection affect plate tectonics. More information could help solve another mystery, too: whether, as many researchers suspect, the inner core is growing. If so, it could eventually overtake the molten metal surrounding it, throwing off Earth’s magnetic field. Physics of Turbulence

Laminar flow converting to turbulent flow

Laminar flows have a very high degree of order liness. They can be completely solved by solving a set of known differential equations, analytically or numerically. The only diffusivity is by molecular transport, and not very large. The stability against disturbances increases with frequency. Turbulent flows are chaotic, have high mixing and diffusivity. The flows are unstable and are affected by disturbances. The disturbances can cause fluctuations in one component, which is amplified in the other components. Laminar flows are usually very hard to maintain above Reynolds Numbers of about 4000. Laminar boundary layers grow as x1/2. This introduces a new length scale, the boundary layer thickness, into the problem. Turbulent boundary layers grow much faster than laminar boundary layers. δ/x ~ x-1/5- for moderately high Reynolds numbers (Blasius) and δ/x ~ (log Rex)-2.58 for high Reynolds numbers (from Nikuradse).

Hence, turbulent boundary layers generate more drag and transport more heat than laminar boundary layers. Turbulence arises from the inherent instability in laminar flows.

Laminar flows are very stable. Turbulent flows are very much unstable. So turbulent flows need a constant source of energy to be sustained. The most important source of energy is shear in the mean flow. Stirring and Buoyancy also generate turbulence. Because of the chaotic nature of turbulence, analytical solutions for the velocities are difficult.

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Ivestigators resort to a Statistical approach which uses average values to predict the effects of turbulent flow. Determinstic study, Stuctural Study, Deterministic Study are other approaches of turbulent study. Interaction between observations, theoretical ideas and modelling may be illustrated as figure No. I

The term ‘observation’ here means not only empirical data from observation of the physical (real) world, but also empirical data from observation of computer simulations representing solutions of the full Navier-Stokes equation or other suitable solutions. The term “theoretical ideas’ means the realm in which observations are transformed into (normally mathematical) idealisation or conceptualisation. Conceptualising or theorising is vital for the study of turbulence, as it is to the study of any scientific discipline. Finally, the term ‘modeling’ is used to denote the realm in which theoretical ideas are placed within a formal system by means of mathematics.

Mathematical Concepts for Analysing TurbulenceObservation of turbulence is as old as recorded history. In all the epics there are several references of

natural turbulence. The sketch Circa (1500) drawn by Lionardo de Vince indicates that he was intrigued by turbulence. During 40’s of this century A.N. Kolmogorov attempted to predict the properties of fully-developed turbulence. But one will wonder how his work can be reconciled with Leonardo’s half a millennium old drawings of eddy motion in the study for the elimination of rapids in the river Arno. Now one may say from experimental fact that turbulence, once generated, decays quite slowly. This may actually have been the very first scientific observation ever made about turbulent flow ; Leonardo’s note reveals his observation on this phenomena

Where the turbulence of water is generatedWhere the turbulence of water maintains for longWhere the turbulence of water comes to rest.Also a cartoon drawn in 1977 by the astronomer Philippe Delache, a penetrating observer of the

turbulence community needs to be mentioned before the end of the observational history of the mysterious phenomena.

The Modern scientific study of turbulence started during late 1800’s with the work of Osborne Reynolds. From starting to the present these studies may be characterised into three distinct parts, earliest of which has a strong nondeterministic flavour may referred to as the statistical study; the next predominantly observational - may be referred to as the structural study, and the most recent one is called the deterministic study.

For the small scales in turbulence mixture of hypothesis, theory and experiment gives some unity by the phenomenological picture established by Richardson, Taylor, Kolmogorov, and Rudy- the famous researchers of turbulence study. Phenomenology paints a picture of cascades of energy and information from large-scale eddies down to small, and of universal features of these cascades, provided the Reynolds numbers is large enough. In some

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sense this vision has worked well, providing a convenient conceptual framework within which many empirical observations can be rationalised. However, it was clear from the outset that this was too simplistic a point of view and half a century later there remain many fundamental unanswered questions. For example, exactly what do we mean by an eddy or a cascade, and how should we interpret cascade-like arguments in terms of the evolving morphology of the vorticity field? Indeed, what is the spatial structure of the vorticity field and how does this relate to the observed energy spectra?

Human understanding of turbulent boundary-layers, and of turbulence in rotating-stratified fluids, is equally uncertain. For example, the log-law of the wall represents an early milestone in turbulence theory. It is based on the hypothesis that the near-wall eddies are immune from the remote eddies in the core flow. However, we have always known that the near-wall eddies cannot be independent of the larger far-field vortices, so why does the log-law work so well? There are many other controversies in shear flows. For example, it has been known for over thirty years that turbulence near a wall is dominated by streaks of low-speed fluid and by long, stream-wise vortices. It is now generally agreed that these streaks and vortices interact in some kind of quasi-periodic cycle, yet the nature of this cycle, and its possible relationship to the structures seen in transition studies, is still a matter of debate. The situation is little better in rotating-stratified turbulence, where there is a subtle interaction between waves and turbulence. While all agree that, in such flows, the large vortices acquire a distinctive shape, reminiscent of cigars or pancakes, few can agree on the mechanisms by which these structures form. Evidently, there is much to debate.

Modern theoretical approaches to turbulence, in which the problem can be seen as a branch of statistical field theory, and where the treatment has been strongly influenced by analogies with the quantum many-body problem. The dominant themes treated are the development of renormalized perturbation theories and, more recently, of renormalization group methods.

Fluid dynamics is part of the physics curriculum introducing some background concepts in fluid dynamics, followed by a skeleton treatment of the phenomenology of turbulence taking flow through a straight pipe or a plane channel as a representative example one proceed to understand most complex turbulence phenomanae. The general statistical formulation of the problem may be given, leading to a moment closure problem, which is analogous to the well known BBGKY hierarchy, and to the Kolmogorov -5/3 power law, which is a consequence of dimensional analysis. Also one may show how RPT have been used to tackle the moment closure problem, distinguishing between those which are compatible with the Kolmogorov spectrum and those which are not. Further one may discuss the use of RG to reduce the number of degrees of freedom in the numerical simulation of the turbulent equations of motion, while giving a clear statement of the technical problems which lie in the way of doing this. Lastly one basis of the theories developed their ability to meet the stated goals may be assessed by numerical computation and comparison with experiment.

Fluid flow, it should be said, is in one sense very well understood; since the early 1800s there's been a fine, non-linear, Newtonian equation for the velocity field that seems to work, the Navier-Stokes equation. (Like Newton's law of gravitation, it should be branded on to anyone who babbles that non-linear physics is "new" or "non-Newtonian".) One of its properties is that it's invariant so long as the Reynolds number stays the same. This is why wind-tunnels work: the model in the tunnel is shorter than the original, but the mean speed is higher, so the flows are equivalent. When the Reynolds number is small, the equation is mathematically nice, the non-linearities are small, and one can solve the equation. The stream-lines --- the paths followed by small tracer particles dropped into the fluid --- form nice layers around the boundaries of the flow, which is why the flow is called laminar, and these laminæ are stable.

With turn up the Reynolds number, the non-linearities become important, and the flow gets uglier --- it is no longer steady, but erratic (probably chaotic in the strict sense), and the nice regular stream-lines and their laminæ get snarled and then completely confused; eddies and vortices form and spin and dissolve

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without much obvious pattern, and the develop their own eddies in turn; odd structures with names like "von Kármán streets" appear. Turbulence --- yea, "fully developed turbulence", even --- is when this decay into confusion is complete, when there are eddies and motions on all length scales, from the largest possible in the fluid on down to the so-called "dissipation scale," which is (roughly!) the minimum eddy size, as set by the mechanical properties of the fluid (its viscosity and the like). When faced with this confusion, if not well before, one give up and turn to statistics; Thus one can make some nice observations, and even come up with two well-confirmed empirical laws about these statistics, and endless graphs.

A solution to the problem of turbulence would be, more or less, a valid derivation from the Navier-Stokes equation (and statements about the appropriate conditions) of our measured statistics. Physicists are very far from this at present. Current closest approach stems from the work of Kolmogorov, who, by means of some statistical hypotheses about small-scale motion, was able to account for the empirical laws. Unfortunately, no one has managed to coax the hypotheses from the Navier-Stokes equation and the hypotheses hold exactly only in the limit of infinite Reynolds number, i.e. they are not true of any actual fluid.

All sorts of things, including more or less direct simulations of flows by cousins of cellular automata called "lattice gasses". One approach uses the vorticity (the curl of the velocity field, which tells us about how the fluid swirls), since it turns out to be possible to identify some (more or less) simple objects in the flow, called vortex lines or vortex tubes, work out how they interact, and then use statistical mechanics to calculate various emergent properties and tolerate negative temperatures (which are not impossible, and actually hotter than infinity) gives the Kolmogorov laws. This could've been custom-tailored for philosophical and methodological biases as do all the leaps in the approximation scheme used.

Considering velocity, pressure and density correlation in a homogeneous and isotropic flow field; investigation may be carried out for consequent local changes of velocity, temperature, pressure, density etc. For this the use of relevant structure function will be taken into consideration. Further, different aspects of cross correlation between different field variables may be studied at length.Reynolds-Averaged Navier-Stokes Equations

Most people are intuitively aware that the fluid flow of both gases and liquids is inherently transient (unsteady) by nature. As an example, just ponder the movement of leaves on a tree in a breeze or the flow of water from a tap. Less intuitive is that even a relatively steady flow in a wind tunnel has transient velocities that vary at scales and frequencies that our human senses are unable to discern. Such variations are known as turbulence. The treatment of turbulence in the Reynolds-Averaged Navier-Stokes (RANS) equations lies at the heart of most practical Computational Fluid Dynamics (CFD) approaches. Affordable RANS Analysis available now.

Navier-Stokes EquationsThe Navier-Stokes equations accurately describe fluid flow for a

remarkably large class of problems, by assuming that fluid behaves as a continuum rather than as discrete particles. However, inherent in these equations is the representation of velocity scales and variations that make solving them nearly impossible on present-day computers except for the simplest of flows using Direct Numerical Simulation (DNS). The Navier-Stokes equations are also inherently unsteady (varying with time), which means averaging multiple solutions at a series of time steps is required to produce engineering quantities such as lift and drag from a pressure solution (or field). The Navier-Stokes equations accurately describe fluid flow for a

Cyclone Flow Simulation: remarkably large class of problems, by assuming that fluid behaves as a

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Solving the RANS equations continuum rather than as discrete particles. However, inherent in these equations is the representation of velocity scales and variations that make solving them nearly impossible on present-day computers except for the simplest of flows using Direct Numerical Simulation (DNS). The Navier-Stokes equations are also inherently unsteady (varying with time), which means averaging multiple solutions at a series of time steps is required to produce engineering quantities such as lift and drag from a pressure solution (or field).Reynolds-Averaged Navier-Stokes Equations

Decomposing the Navier-Stokes equations into the RANS equations makes it possible to simulate practical engineering flows, such as the airflow over an airplane. The assumption (known as the Reynolds decomposition) behind the RANS equations is that the time-dependent turbulent (chaotic) velocity fluctuations can be separated from the mean flow velocity. This transform then introduces a set of unknowns called the Reynolds stresses, which are functions of the velocity fluctuations, and which require a turbulence model (e.g., the two-equation k-epsilon model) to produce a closed system of solvable equations. The reduced computational requirements for the RANS equations, while still significant, are orders of magnitude less than that required for the original Navier-Stokes equations. Another advantage of using the RANS equations for steady fluid flow simulation is that the mean flow velocity is calculated as a direct result without the need to average the instantaneous velocity over a series of time steps.

It is a relatively straightforward process to produce an unsteady variant of the RANS equations (sometimes referred to as URANS) for transient flows, while still solving for the mean flow velocity separately from the turbulent velocity fluctuations.Computational Fluid Dynamics

Combining the RANS equations with assumptions that enforce the conservation of mass and energy produces the mainstream approach used within CFD to simulate a wide variety of practical fluid flows. CFD breaks down a fluid domain into discrete cells (a mesh) and then solves the RANS and conservation laws in each cell. The accuracy of a typical CFD simulation is primarily determined by the mesh resolution (usually the higher the better, but at the cost of more computing resources and slower turnaround times) and the turbulence model.

Turbulence Model TuningThe requirement of a turbulence model in the RANS

equations is an inherent weakness. Turbulence models are typically tuned empirically for specific cases, such as attached flow on an airfoil, or the massively detached flow behind a bluff body, or the high swirl encountered in a cyclone. These problems have driven the development of the Large Eddy Simulation (LES) method as an alternative to the RANS equations, resulting in a massive increase in the required computing resources - but that's a whole other story...

Application of Wavelets Analysis to Turbulence Study: The concept of ‘wavelets’ and ‘ondeletts’ started to appear in literature only in early 1980’s. This new concept can be viewed as a synthesis of various ideas Surface

Mesh Elements originated from different disciplines viz. mathematics, physics and engineering. In 1982, Jean Morlet, a French geophysical engineer, first introduced the idea of wavelet transform as a new mathematical tool for seismic signal analysis. French theoretical physicist Alex Grossmann quickly recognised the importance of the Morlet wavelet transform which something similar to

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coherent states formalism in quantum mechanics. He developed an exact inversion formula for the wavelet transform. Many researchers have done lot of works to develop wavelet concept. Daubechies paper had a tremendous positive impact on the study of wavelets and their diverse applications.

Very recently it is suggested that wavelet analysis could be an effective method for an accurate and deeper understanding of turbulent flows. Several authors Farge et al, Meneveau and Fröhlich and Schneider have developed wavelet analysis to investigate the structures and dynamics of turbulent flows. These studies reveal that the strongest modes of the wavelet transform of two dimensional turbulent flows represent the coherent structures, while weaker mode describes the unorganised background flow. Study of Wickerhauser et al shows that coherent vortices can be well described by only a very few wavelet modes.

With the help of Wavelet transform of space-scale energy density function can be defined. Farge et al has used wavelet transform to define the local intermittency as the ratio of the local energy density and the space averaged energy density. This shows a striking contrast with the Fourier transform analysis. When Fourier transform analysis can describe a signal in terms of wave numbers only, but cannot give any local information.

Farge and Rabreau, Frage, Mandeveau have employed wavelets to study homogenous turbulent flows in different configuration. Starting from a random voriticity distribution with a k-3 energy spectrum they showed that during the flow evaluation the small scales of the vorticity become increasingly localised in physical scale. From their investigations it is also revealed that the energy in the two-dimensional turbulent flows. Meneveau first measured the local energy spectra and then carried out direct numerical simulations of turbulent shear flows. His investigation reveals that the mean spatial values of the turbulent shear flow agree with their corresponding results in Fourier space, but their spatial variations at each scale is found to be very large and the local energy flux associated with very small scales exhibits large spatial intermittency. His computational analysis of the spatial fluctuations of shows that the average value of it is positive for all small scales and negative for large scales to small scales so that energy is eventually dissipated by viscosity, Striking phenomena of his study is the energy cascade is reversed in the sense that energy transfer takes place from small scales to large scales in many places in the flow field. Perrier et al confirms that the mean wavelet spectrum.

This result gives the correct Fourier exponent for a power-law of the Fourier energy spectrum provided the associated wavelet has at least n > ½(p-1) vanishing moments. Based on a recent wavelet analysis of a numerically calculated two-dimensional homogeneous turbulent flow, Benzi and Vergossola confirmed the existence of coherent structures with negative exponents.

Form the studies it is revealed that the wavelet transform analysis has ability - not only to give more precise local description, but also to determine and characterise singularities of turbulent flows. Argoul et al and Everson et al have showed that the wavelet analysis has the ability to reveal cantor like fractal structure of the Richardson cascade of turbulent eddies. Ghosh has derived energy cascading process, in very simplified case, in the upper ocean with the help of wavelet analysis.Application of Fractal Analysis to Turbulence Study : Turbulence, fractals in nature, i.e., the turbulent structures (instantaneous velocity fields) are similar at different special resolutions, until one reaches the molecular scale. To accurately model a turbulent flow one need to resolve all these turbulent length scales (Direct Numerical Simulation), but it is extremely resource intensive (massive element counts) and impractical for all but the simplest problems. Most practical CFD approaches use Reynolds-Average Navier-Stokes (RANS) with turbulence models to represent the turbulence indirectly and avoid resolving all the fractal turbulence length scales. In recent times Large Eddy Simulation (LES) has become practical for some flows (typically requires large element counts) by representing larger (though still small) turbulent eddies directly and modeling smaller eddies. So in answer to the question the turbulence model is the primary means by which the fractal nature of turbulence is addressed.

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Turbulence models still require relatively fine mesh elements along walls to resolve boundary layers in order to capture viscous effects, such as flow separation and skin friction. Without a well resolved model the best turbulence model won't provide accurate results.Application of Fuzzy Analysis to Turbulence Study: Considering complexity of turbulence attempt has been made to model an ideal turbulence in terms of an fuzzy dynamical system (FDS). Till now the model is not idealistic. In this model the generation of vortices as fuzzy systems has been treated.

Usual description of turbulence starts with the Navier-Stokes equations describing the motion of the underlying body of the fluid, but for fuzzy turbulence Navier-Stokes equations are not considered. To start with a definition it has been stated that ‘Turbulence is chaotic occurrence of vortices in a dynamic fluid.It is a common practice to use a discrete distribution of point vortices in the analysis of two and three dimensional flows of fluid. In Venkatesan’s work an elliptic function (doubly periodic function in the complex plane) has been used to describe the distribution of vortices in a two-dimensional channel flow. It is also a very common practice to describe the distribution of such vortices by probability distribution function. For simplicity the distribution functions are taken two dimensional, but for more realistic model it need be extended to three-dimensional. To incorporate time into it the model will be a four-dimensional space-time model. In this sense it is the most general representation of fuzzy turbulence. One may say this in the following words :Since simulated vortex in a dynamic fluid thus resolved by solving the FDI (4) may be termed as fuzzy vortex and chaotic occurrence of such fuzzy vortices in a dynamic fluid may be called fuzzy turbulence.

ConclusionTurbulence study has been declared as thrust area of research in the 21st century. In the present

global frame for further technological development knowledge about turbulent phenomena is most essential. In this context knowing full well about the complexity of Turbulence in present day investigation attempts need be taken to understand its tend to consist of an uneasy mix of plausible but uncertain hypotheses, deterministic but highly simplified cartoons, and vast, complex data sets. For the small scales in turbulence this mixture of hypothesis, theory and experiment is given some unity by the phenomenological picture established by Richardson, Kolmogorov, Taylor, Sen, Narasimha, Hossain and Rudy. This phenomenology paints a picture of cascades of energy and information from large-scale eddies down to small, and of universal features of these cascades, provided the Reynolds numbers is large enough. In some sense this vision has worked well, providing a convenient conceptual framework within which many empirical observations can be rationalised.

However, it was clear from the outset that this was too simplistic a point of view and half a century later there remain many fundamental unanswered questions. For example, what exactly the term eddy or a cascade mean, and how cascade-like arguments in terms of the evolving morphology of the vorticity field are been interpreted evolving morphology of the vorticity field? Indeed, what is the spatial structure of the vorticity field and how does this relate to the observed energy spectra ? Moreover understanding of turbulent boundary-layers, and of turbulence in rotating-stratified fluids, is equally uncertain. For example, the log-law of the wall represents an early milestone in turbulence theory. It is based on the hypothesis that the near-wall eddies are immune from the remote eddies in the core flow. However, it is always known that the near-wall eddies cannot be independent of the larger far-field vortices, so why does the log-law work so well ? There are many other controversies in shear flows. For example, it has been known for over thirty years that turbulence near a wall is dominated by streaks of low-speed fluid and by long, stream-wise vortices. It is now generally agreed that these streaks and vortices interact in some kind of quasi-periodic cycle, yet the nature of this cycle, and its possible relationship to the structures seen in transition studies, is

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still a matter of debate. The situation is little better in rotating-stratified turbulence, where there is a subtle interaction between waves and turbulence. While all agree that, in such flows, the large vortices acquire a distinctive shape, reminiscent of cigars or pancakes, few can agree on the mechanisms by which these structures form. Evidently, there is much to debate.

Turbulent study need to analyse first the ideas of leading experts across the world to debate these fundamental questions. The discussion should be wide ranging, from the initiation of turbulence through to its asymptotic state at high Reynolds number, including the effects of rotation and stratification, and the addition of different phases, such as bubbles, particles and polymers.

In continuation of first discussion in the works initially velocity, pressure and density correlation may be considered in a homogeneous and isotropic flow field; and thereby investigation will be carried out for consequent local changes of velocity, temperature, pressure, density etc. For this the use of relevant structure function will be taken into consideration. For further study, different aspects of cross correlation between different field variables need be studied at length.

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