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90-400: In fluid dynamics , turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity . It is in contrast to laminar flow , which occurs when a fluid flows in parallel layers with no disruption between those layers. Turbulence is commonly observed in everyday phenomena such as surf , fast flowing rivers, billowing storm clouds, or smoke from

180-477: A Poisson distribution are inaccurate, and networks designed without taking self-similarity into account are likely to function in unexpected ways. Similarly, stock market movements are described as displaying self-affinity , i.e. they appear self-similar when transformed via an appropriate affine transformation for the level of detail being shown. Andrew Lo describes stock market log return self-similarity in econometrics . Finite subdivision rules are

270-442: A self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines , are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals . Scale invariance is an exact form of self-similarity where at any magnification there

360-485: A white noise contribution obtained from the fluctuation-dissipation theorem of statistical mechanics is added to the viscous stress tensor and heat flux . The concept of pressure is central to the study of both fluid statics and fluid dynamics. A pressure can be identified for every point in a body of fluid, regardless of whether the fluid is in motion or not. Pressure can be measured using an aneroid, Bourdon tube, mercury column, or various other methods. Some of

450-526: A chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason, turbulence is commonly realized in low viscosity fluids. In general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other, consequently drag due to friction effects increases. The onset of turbulence can be predicted by

540-525: A constitutive relation between a turbulent flux and the gradient of a mean variable similar to the relation between flux and gradient that exists for molecular transport. In the best case, this assumption is only an approximation. Nevertheless, the turbulent diffusivity is the simplest approach for quantitative analysis of turbulent flows, and many models have been postulated to calculate it. For instance, in large bodies of water like oceans this coefficient can be found using Richardson 's four-third power law and

630-455: A continuum, do not contain ionized species, and have flow velocities that are small in relation to the speed of light, the momentum equations for Newtonian fluids are the Navier–Stokes equations —which is a non-linear set of differential equations that describes the flow of a fluid whose stress depends linearly on flow velocity gradients and pressure. The unsimplified equations do not have

720-525: A figure are small replicas of the whole, then the figure is called self-similar ....A figure is strictly self-similar if the figure can be decomposed into parts which are exact replicas of the whole. Any arbitrary part contains an exact replica of the whole figure. Since mathematically, a fractal may show self-similarity under indefinite magnification, it is impossible to recreate this physically. Peitgen et al. suggest studying self-similarity using approximations: In order to give an operational meaning to

810-440: A function of the fluid velocity and have different values in frames of reference with different motion. To avoid potential ambiguity when referring to the properties of the fluid associated with the state of the fluid rather than its motion, the prefix "static" is commonly used (such as static temperature and static enthalpy). Where there is no prefix, the fluid property is the static condition (so "density" and "static density" mean

900-405: A general closed-form solution , so they are primarily of use in computational fluid dynamics . The equations can be simplified in several ways, all of which make them easier to solve. Some of the simplifications allow some simple fluid dynamics problems to be solved in closed form. In addition to the mass, momentum, and energy conservation equations, a thermodynamic equation of state that gives

990-416: A mean value: and similarly for temperature ( T = T + T′ ) and pressure ( P = P + P′ ), where the primed quantities denote fluctuations superposed to the mean. This decomposition of a flow variable into a mean value and a turbulent fluctuation was originally proposed by Osborne Reynolds in 1895, and is considered to be the beginning of the systematic mathematical analysis of turbulent flow, as

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1080-536: A model of the effects of the turbulent flow. Such a modelling mainly provides the additional momentum transfer by the Reynolds stresses , although the turbulence also enhances the heat and mass transfer . Another promising methodology is large eddy simulation (LES), especially in the form of detached eddy simulation (DES) — a combination of LES and RANS turbulence modelling. There are a large number of other possible approximations to fluid dynamic problems. Some of

1170-443: A point in a flow. All fluids are compressible to an extent; that is, changes in pressure or temperature cause changes in density. However, in many situations the changes in pressure and temperature are sufficiently small that the changes in density are negligible. In this case the flow can be modelled as an incompressible flow . Otherwise the more general compressible flow equations must be used. Mathematically, incompressibility

1260-652: A powerful technique for building self-similar sets, including the Cantor set and the Sierpinski triangle . The viable system model of Stafford Beer is an organizational model with an affine self-similar hierarchy, where a given viable system is one element of the System One of a viable system one recursive level higher up, and for whom the elements of its System One are viable systems one recursive level lower down. Self-similarity can be found in nature, as well. To

1350-419: A region of the flow called a control volume . A control volume is a discrete volume in space through which fluid is assumed to flow. The integral formulations of the conservation laws are used to describe the change of mass, momentum, or energy within the control volume. Differential formulations of the conservation laws apply Stokes' theorem to yield an expression that may be interpreted as the integral form of

1440-457: A statistical description is needed. The Russian mathematician Andrey Kolmogorov proposed the first statistical theory of turbulence, based on the aforementioned notion of the energy cascade (an idea originally introduced by Richardson ) and the concept of self-similarity . As a result, the Kolmogorov microscales were named after him. It is now known that the self-similarity is broken so

1530-400: A sub-field of fluid dynamics. While the mean values are taken as predictable variables determined by dynamics laws, the turbulent fluctuations are regarded as stochastic variables. The heat flux and momentum transfer (represented by the shear stress τ ) in the direction normal to the flow for a given time are where c P is the heat capacity at constant pressure, ρ is the density of

1620-443: A third hypothesis of Kolmogorov was that at very high Reynolds number the statistics of scales in the range η ≪ r ≪ L are universally and uniquely determined by the scale r and the rate of energy dissipation ε . The way in which the kinetic energy is distributed over the multiplicity of scales is a fundamental characterization of a turbulent flow. For homogeneous turbulence (i.e., statistically invariant under translations of

1710-430: A universal constant. This is one of the most famous results of Kolmogorov 1941 theory, describing transport of energy through scale space without any loss or gain. The Kolmogorov five-thirds law was first observed in a tidal channel, and considerable experimental evidence has since accumulated that supports it. Outside of the inertial area, one can find the formula below : In spite of this success, Kolmogorov theory

1800-445: A vector r (since the turbulence is assumed isotropic, the flow velocity increment depends only on the modulus of r ). Flow velocity increments are useful because they emphasize the effects of scales of the order of the separation r when statistics are computed. The statistical scale-invariance without intermittency implies that the scaling of flow velocity increments should occur with a unique scaling exponent β , so that when r

1890-448: A wide range of length scales and the hierarchy can be described by the energy spectrum that measures the energy in flow velocity fluctuations for each length scale ( wavenumber ). The scales in the energy cascade are generally uncontrollable and highly non-symmetric. Nevertheless, based on these length scales these eddies can be divided into three categories. The integral time scale for a Lagrangian flow can be defined as: where u ′

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1980-399: Is Self-affinity . The Mandelbrot set is also self-similar around Misiurewicz points . Self-similarity has important consequences for the design of computer networks, as typical network traffic has self-similar properties. For example, in teletraffic engineering , packet switched data traffic patterns seem to be statistically self-similar. This property means that simple models using

2070-429: Is a factor in developing turbulent flow. Counteracting this effect is the viscosity of the fluid, which as it increases, progressively inhibits turbulence, as more kinetic energy is absorbed by a more viscous fluid. The Reynolds number quantifies the relative importance of these two types of forces for given flow conditions, and is a guide to when turbulent flow will occur in a particular situation. This ability to predict

2160-466: Is a range of scales (each one with its own characteristic length r ) that has formed at the expense of the energy of the large ones. These scales are very large compared with the Kolmogorov length, but still very small compared with the large scale of the flow (i.e. η ≪ r ≪ L ). Since eddies in this range are much larger than the dissipative eddies that exist at Kolmogorov scales, kinetic energy

2250-466: Is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample , whereas any portion of a straight line may resemble

2340-405: Is at present under revision. This theory implicitly assumes that the turbulence is statistically self-similar at different scales. This essentially means that the statistics are scale-invariant and non-intermittent in the inertial range. A usual way of studying turbulent flow velocity fields is by means of flow velocity increments: that is, the difference in flow velocity between points separated by

2430-410: Is characterized by a hierarchy of scales through which the energy cascade takes place. Dissipation of kinetic energy takes place at scales of the order of Kolmogorov length η , while the input of energy into the cascade comes from the decay of the large scales, of order L . These two scales at the extremes of the cascade can differ by several orders of magnitude at high Reynolds numbers. In between there

2520-431: Is characterized by the following features: Turbulent diffusion is usually described by a turbulent diffusion coefficient . This turbulent diffusion coefficient is defined in a phenomenological sense, by analogy with the molecular diffusivities, but it does not have a true physical meaning, being dependent on the flow conditions, and not a property of the fluid itself. In addition, the turbulent diffusivity concept assumes

2610-400: Is considerable evidence that turbulent flows deviate from this behavior. The scaling exponents deviate from the ⁠ n / 3 ⁠ value predicted by the theory, becoming a non-linear function of the order n of the structure function. The universality of the constants have also been questioned. For low orders the discrepancy with the Kolmogorov ⁠ n / 3 ⁠ value

2700-419: Is essentially not dissipated in this range, and it is merely transferred to smaller scales until viscous effects become important as the order of the Kolmogorov scale is approached. Within this range inertial effects are still much larger than viscous effects, and it is possible to assume that viscosity does not play a role in their internal dynamics (for this reason this range is called "inertial range"). Hence,

2790-466: Is expressed by saying that the density ρ of a fluid parcel does not change as it moves in the flow field, that is, where ⁠ D / D t ⁠ is the material derivative , which is the sum of local and convective derivatives . This additional constraint simplifies the governing equations, especially in the case when the fluid has a uniform density. For flow of gases, to determine whether to use compressible or incompressible fluid dynamics,

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2880-543: Is given a special name—a stagnation point . The static pressure at the stagnation point is of special significance and is given its own name— stagnation pressure . In incompressible flows, the stagnation pressure at a stagnation point is equal to the total pressure throughout the flow field. In a compressible fluid, it is convenient to define the total conditions (also called stagnation conditions) for all thermodynamic state properties (such as total temperature, total enthalpy, total speed of sound). These total flow conditions are

2970-473: Is governed by the random walk principle. In rivers and large ocean currents, the diffusion coefficient is given by variations of Elder's formula. Via this energy cascade , turbulent flow can be realized as a superposition of a spectrum of flow velocity fluctuations and eddies upon a mean flow . The eddies are loosely defined as coherent patterns of flow velocity, vorticity and pressure. Turbulent flows may be viewed as made of an entire hierarchy of eddies over

3060-517: Is known as unsteady (also called transient ). Whether a particular flow is steady or unsteady, can depend on the chosen frame of reference. For instance, laminar flow over a sphere is steady in the frame of reference that is stationary with respect to the sphere. In a frame of reference that is stationary with respect to a background flow, the flow is unsteady. Turbulent flows are unsteady by definition. A turbulent flow can, however, be statistically stationary . The random velocity field U ( x , t )

3150-494: Is often represented via a Reynolds decomposition , in which the flow is broken down into the sum of an average component and a perturbation component. It is believed that turbulent flows can be described well through the use of the Navier–Stokes equations . Direct numerical simulation (DNS), based on the Navier–Stokes equations, makes it possible to simulate turbulent flows at moderate Reynolds numbers. Restrictions depend on

3240-422: Is scaled by a factor λ , should have the same statistical distribution as with β independent of the scale r . From this fact, and other results of Kolmogorov 1941 theory, it follows that the statistical moments of the flow velocity increments (known as structure functions in turbulence) should scale as where the brackets denote the statistical average, and the C n would be universal constants. There

3330-429: Is statistically stationary if all statistics are invariant under a shift in time. This roughly means that all statistical properties are constant in time. Often, the mean field is the object of interest, and this is constant too in a statistically stationary flow. Steady flows are often more tractable than otherwise similar unsteady flows. The governing equations of a steady problem have one dimension fewer (time) than

3420-447: Is sufficiently high. Thus, Kolmogorov introduced a second hypothesis: for very high Reynolds numbers the statistics of small scales are universally and uniquely determined by the kinematic viscosity ν and the rate of energy dissipation ε . With only these two parameters, the unique length that can be formed by dimensional analysis is This is today known as the Kolmogorov length scale (see Kolmogorov microscales ). A turbulent flow

3510-401: Is the mean turbulent kinetic energy of the flow. The wavenumber k corresponding to length scale r is k = ⁠ 2π / r ⁠ . Therefore, by dimensional analysis, the only possible form for the energy spectrum function according with the third Kolmogorov's hypothesis is where K 0 ≈ 1.5 {\displaystyle K_{0}\approx 1.5} would be

3600-474: Is the only non-empty subset of Y such that the equation above holds for { f s : s ∈ S } {\displaystyle \{f_{s}:s\in S\}} . We call a self-similar structure . The homeomorphisms may be iterated , resulting in an iterated function system . The composition of functions creates the algebraic structure of a monoid . When the set S has only two elements,

3690-449: Is the velocity fluctuation, and τ {\displaystyle \tau } is the time lag between measurements. Although it is possible to find some particular solutions of the Navier–Stokes equations governing fluid motion, all such solutions are unstable to finite perturbations at large Reynolds numbers. Sensitive dependence on the initial and boundary conditions makes fluid flow irregular both in time and in space so that

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3780-492: Is to use two flow models: the Euler equations away from the body, and boundary layer equations in a region close to the body. The two solutions can then be matched with each other, using the method of matched asymptotic expansions . A flow that is not a function of time is called steady flow . Steady-state flow refers to the condition where the fluid properties at a point in the system do not change over time. Time dependent flow

3870-462: Is treated separately. Reactive flows are flows that are chemically reactive, which finds its applications in many areas, including combustion ( IC engine ), propulsion devices ( rockets , jet engines , and so on), detonations , fire and safety hazards, and astrophysics. In addition to conservation of mass, momentum and energy, conservation of individual species (for example, mass fraction of methane in methane combustion) need to be derived, where

3960-466: Is very small, which explain the success of Kolmogorov theory in regards to low order statistical moments. In particular, it can be shown that when the energy spectrum follows a power law with 1 < p < 3 , the second order structure function has also a power law, with the form Since the experimental values obtained for the second order structure function only deviate slightly from the ⁠ 2 / 3 ⁠ value predicted by Kolmogorov theory,

4050-401: Is well beyond the limit of DNS simulation ( Re = 4 million). Transport aircraft wings (such as on an Airbus A300 or Boeing 747 ) have Reynolds numbers of 40 million (based on the wing chord dimension). Solving these real-life flow problems requires turbulence models for the foreseeable future. Reynolds-averaged Navier–Stokes equations (RANS) combined with turbulence modelling provides

4140-403: The C n constants, are related with the phenomenon of intermittency in turbulence and can be related to the non-trivial scaling behavior of the dissipation rate averaged over scale r . This is an important area of research in this field, and a major goal of the modern theory of turbulence is to understand what is universal in the inertial range, and how to deduce intermittency properties from

4230-589: The Mach number of the flow is evaluated. As a rough guide, compressible effects can be ignored at Mach numbers below approximately 0.3. For liquids, whether the incompressible assumption is valid depends on the fluid properties (specifically the critical pressure and temperature of the fluid) and the flow conditions (how close to the critical pressure the actual flow pressure becomes). Acoustic problems always require allowing compressibility, since sound waves are compression waves involving changes in pressure and density of

4320-549: The Mach numbers , which describe as ratios the relative magnitude of fluid and physical system characteristics, such as density , viscosity , speed of sound , and flow speed . The concepts of total pressure and dynamic pressure arise from Bernoulli's equation and are significant in the study of all fluid flows. (These two pressures are not pressures in the usual sense—they cannot be measured using an aneroid, Bourdon tube or mercury column.) To avoid potential ambiguity when referring to pressure in fluid dynamics, many authors use

4410-468: The Reynolds number , which is the ratio of inertial forces to viscous forces within a fluid which is subject to relative internal movement due to different fluid velocities, in what is known as a boundary layer in the case of a bounding surface such as the interior of a pipe. A similar effect is created by the introduction of a stream of higher velocity fluid, such as the hot gases from a flame in air. This relative movement generates fluid friction, which

4500-593: The conservation laws , specifically, conservation of mass , conservation of linear momentum , and conservation of energy (also known as the First Law of Thermodynamics ). These are based on classical mechanics and are modified in quantum mechanics and general relativity . They are expressed using the Reynolds transport theorem . In addition to the above, fluids are assumed to obey the continuum assumption . At small scale, all fluids are composed of molecules that collide with one another and solid objects. However,

4590-639: The kinetic energy is significantly absorbed due to the action of fluid molecular viscosity gives rise to a laminar flow regime. For this the dimensionless quantity the Reynolds number ( Re ) is used as a guide. With respect to laminar and turbulent flow regimes: The Reynolds number is defined as where: While there is no theorem directly relating the non-dimensional Reynolds number to turbulence, flows at Reynolds numbers larger than 5000 are typically (but not necessarily) turbulent, while those at low Reynolds numbers usually remain laminar. In Poiseuille flow , for example, turbulence can first be sustained if

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4680-441: The no-slip condition generates a thin region of large strain rate, the boundary layer , in which viscosity effects dominate and which thus generates vorticity . Therefore, to calculate net forces on bodies (such as wings), viscous flow equations must be used: inviscid flow theory fails to predict drag forces , a limitation known as the d'Alembert's paradox . A commonly used model, especially in computational fluid dynamics ,

4770-405: The stress due to these viscous forces is linearly related to the strain rate. Such fluids are called Newtonian fluids . The coefficient of proportionality is called the fluid's viscosity; for Newtonian fluids, it is a fluid property that is independent of the strain rate. Non-Newtonian fluids have a more complicated, non-linear stress-strain behaviour. The sub-discipline of rheology describes

4860-464: The Navier-Stokes equations, i.e. from first principles. Fluid dynamics Fluid dynamics offers a systematic structure—which underlies these practical disciplines —that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of

4950-492: The Reynolds number is larger than a critical value of about 2040; moreover, the turbulence is generally interspersed with laminar flow until a larger Reynolds number of about 4000. The transition occurs if the size of the object is gradually increased, or the viscosity of the fluid is decreased, or if the density of the fluid is increased. When flow is turbulent, particles exhibit additional transverse motion which enhances

5040-409: The continuum assumption assumes that fluids are continuous, rather than discrete. Consequently, it is assumed that properties such as density, pressure, temperature, and flow velocity are well-defined at infinitesimally small points in space and vary continuously from one point to another. The fact that the fluid is made up of discrete molecules is ignored. For fluids that are sufficiently dense to be

5130-487: The dimensionless Reynolds number , the ratio of kinetic energy to viscous damping in a fluid flow. However, turbulence has long resisted detailed physical analysis, and the interactions within turbulence create a very complex phenomenon. Physicist Richard Feynman described turbulence as the most important unsolved problem in classical physics. The turbulence intensity affects many fields, for examples fish ecology, air pollution, precipitation, and climate change. Turbulence

5220-577: The first." A similar witticism has been attributed to Horace Lamb in a speech to the British Association for the Advancement of Science : "I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather more optimistic." The onset of turbulence can be, to some extent, predicted by

5310-471: The flow is irrotational everywhere, Bernoulli's equation can completely describe the flow everywhere. Such flows are called potential flows , because the velocity field may be expressed as the gradient of a potential energy expression. This idea can work fairly well when the Reynolds number is high. However, problems such as those involving solid boundaries may require that the viscosity be included. Viscosity cannot be neglected near solid boundaries because

5400-498: The fluid, μ turb is the coefficient of turbulent viscosity and k turb is the turbulent thermal conductivity . Richardson's notion of turbulence was that a turbulent flow is composed by "eddies" of different sizes. The sizes define a characteristic length scale for the eddies, which are also characterized by flow velocity scales and time scales (turnover time) dependent on the length scale. The large eddies are unstable and eventually break up originating smaller eddies, and

5490-408: The fluid, such as flow velocity , pressure , density , and temperature , as functions of space and time. Before the twentieth century, "hydrodynamics" was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability , both of which can also be applied to gases. The foundational axioms of fluid dynamics are

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5580-436: The governing equations of the same problem without taking advantage of the steadiness of the flow field. Turbulence is flow characterized by recirculation, eddies , and apparent randomness . Flow in which turbulence is not exhibited is called laminar . The presence of eddies or recirculation alone does not necessarily indicate turbulent flow—these phenomena may be present in laminar flow as well. Mathematically, turbulent flow

5670-399: The kinetic energy of the initial large eddy is divided into the smaller eddies that stemmed from it. These smaller eddies undergo the same process, giving rise to even smaller eddies which inherit the energy of their predecessor eddy, and so on. In this way, the energy is passed down from the large scales of the motion to smaller scales until reaching a sufficiently small length scale such that

5760-403: The law applied to an infinitesimally small volume (at a point) within the flow. In the above integral formulation of this equation, the term on the left is the net change of momentum within the volume. The first term on the right is the net rate at which momentum is convected into the volume. The second term on the right is the force due to pressure on the volume's surfaces. The first two terms on

5850-580: The macroscopic and microscopic fluid motion at large velocities comparable to the velocity of light . This branch of fluid dynamics accounts for the relativistic effects both from the special theory of relativity and the general theory of relativity . The governing equations are derived in Riemannian geometry for Minkowski spacetime . This branch of fluid dynamics augments the standard hydrodynamic equations with stochastic fluxes that model thermal fluctuations. As formulated by Landau and Lifshitz ,

5940-404: The magnitude of inertial effects compared to the magnitude of viscous effects. A low Reynolds number ( Re ≪ 1 ) indicates that viscous forces are very strong compared to inertial forces. In such cases, inertial forces are sometimes neglected; this flow regime is called Stokes or creeping flow . In contrast, high Reynolds numbers ( Re ≫ 1 ) indicate that the inertial effects have more effect on

6030-403: The medium through which they propagate. All fluids, except superfluids , are viscous, meaning that they exert some resistance to deformation: neighbouring parcels of fluid moving at different velocities exert viscous forces on each other. The velocity gradient is referred to as a strain rate ; it has dimensions T . Isaac Newton showed that for many familiar fluids such as water and air ,

6120-405: The monoid is known as the dyadic monoid . The dyadic monoid can be visualized as an infinite binary tree ; more generally, if the set S has p elements, then the monoid may be represented as a p-adic tree. The automorphisms of the dyadic monoid is the modular group ; the automorphisms can be pictured as hyperbolic rotations of the binary tree. A more general notion than self-similarity

6210-566: The more commonly used are listed below. While many flows (such as flow of water through a pipe) occur at low Mach numbers ( subsonic flows), many flows of practical interest in aerodynamics or in turbomachines occur at high fractions of M = 1 ( transonic flows ) or in excess of it ( supersonic or even hypersonic flows ). New phenomena occur at these regimes such as instabilities in transonic flow, shock waves for supersonic flow, or non-equilibrium chemical behaviour due to ionization in hypersonic flows. In practice, each of those flow regimes

6300-498: The onset of turbulent flow is an important design tool for equipment such as piping systems or aircraft wings, but the Reynolds number is also used in scaling of fluid dynamics problems, and is used to determine dynamic similitude between two different cases of fluid flow, such as between a model aircraft, and its full size version. Such scaling is not always linear and the application of Reynolds numbers to both situations allows scaling factors to be developed. A flow situation in which

6390-460: The particular geometrical features of the boundaries (the size characterizing the large scales will be denoted as L ). Kolmogorov's idea was that in the Richardson's energy cascade this geometrical and directional information is lost, while the scale is reduced, so that the statistics of the small scales has a universal character: they are the same for all turbulent flows when the Reynolds number

6480-453: The power of the computer used and the efficiency of the solution algorithm. The results of DNS have been found to agree well with experimental data for some flows. Most flows of interest have Reynolds numbers much too high for DNS to be a viable option, given the state of computational power for the next few decades. Any flight vehicle large enough to carry a human ( L > 3 m), moving faster than 20 m/s (72 km/h; 45 mph)

6570-551: The pressure as a function of other thermodynamic variables is required to completely describe the problem. An example of this would be the perfect gas equation of state : where p is pressure , ρ is density , and T is the absolute temperature , while R u is the gas constant and M is molar mass for a particular gas. A constitutive relation may also be useful. Three conservation laws are used to solve fluid dynamics problems, and may be written in integral or differential form. The conservation laws may be applied to

6660-467: The production/depletion rate of any species are obtained by simultaneously solving the equations of chemical kinetics . Magnetohydrodynamics is the multidisciplinary study of the flow of electrically conducting fluids in electromagnetic fields. Examples of such fluids include plasmas , liquid metals, and salt water . The fluid flow equations are solved simultaneously with Maxwell's equations of electromagnetism. Relativistic fluid dynamics studies

6750-459: The property of self-similarity, we are necessarily restricted to dealing with finite approximations of the limit figure. This is done using the method which we will call box self-similarity where measurements are made on finite stages of the figure using grids of various sizes. This vocabulary was introduced by Benoit Mandelbrot in 1964. In mathematics , self-affinity is a feature of a fractal whose pieces are scaled by different amounts in

6840-430: The quantity f ( x , t ) {\displaystyle f(x,t)} exhibits dynamic scaling . The idea is just an extension of the idea of similarity of two triangles. Note that two triangles are similar if the numerical values of their sides are different however the corresponding dimensionless quantities, such as their angles, coincide. Peitgen et al. explain the concept as such: If parts of

6930-421: The rate of energy and momentum exchange between them thus increasing the heat transfer and the friction coefficient. Assume for a two-dimensional turbulent flow that one was able to locate a specific point in the fluid and measure the actual flow velocity v = ( v x , v y ) of every particle that passed through that point at any given time. Then one would find the actual flow velocity fluctuating about

7020-586: The reference frame) this is usually done by means of the energy spectrum function E ( k ) , where k is the modulus of the wavevector corresponding to some harmonics in a Fourier representation of the flow velocity field u ( x ) : where û ( k ) is the Fourier transform of the flow velocity field. Thus, E ( k ) d k represents the contribution to the kinetic energy from all the Fourier modes with k < | k | < k + d k , and therefore, where ⁠ 1 / 2 ⁠ ⟨ u i u i ⟩

7110-432: The right are negated since momentum entering the system is accounted as positive, and the normal is opposite the direction of the velocity u and pressure forces. The third term on the right is the net acceleration of the mass within the volume due to any body forces (here represented by f body ). Surface forces , such as viscous forces, are represented by F surf , the net force due to shear forces acting on

7200-405: The same thing). The static conditions are independent of the frame of reference. Because the total flow conditions are defined by isentropically bringing the fluid to rest, there is no need to distinguish between total entropy and static entropy as they are always equal by definition. As such, entropy is most commonly referred to as simply "entropy". Self-similarity In mathematics ,

7290-399: The statistical description is presently modified. A complete description of turbulence is one of the unsolved problems in physics . According to an apocryphal story, Werner Heisenberg was asked what he would ask God , given the opportunity. His reply was: "When I meet God, I am going to ask him two questions: Why relativity ? And why turbulence? I really believe he will have an answer for

7380-440: The stress-strain behaviours of such fluids, which include emulsions and slurries , some viscoelastic materials such as blood and some polymers , and sticky liquids such as latex , honey and lubricants . The dynamic of fluid parcels is described with the help of Newton's second law . An accelerating parcel of fluid is subject to inertial effects. The Reynolds number is a dimensionless quantity which characterises

7470-405: The term static pressure to distinguish it from total pressure and dynamic pressure. Static pressure is identical to pressure and can be identified for every point in a fluid flow field. A point in a fluid flow where the flow has come to rest (that is to say, speed is equal to zero adjacent to some solid body immersed in the fluid flow) is of special significance. It is of such importance that it

7560-414: The terminology that is necessary in the study of fluid dynamics is not found in other similar areas of study. In particular, some of the terminology used in fluid dynamics is not used in fluid statics . Dimensionless numbers (or characteristic numbers ) have an important role in analyzing the behavior of fluids and their flow as well as in other transport phenomena . They include the Reynolds and

7650-402: The value for p is very near to ⁠ 5 / 3 ⁠ (differences are about 2%). Thus the "Kolmogorov − ⁠ 5 / 3 ⁠ spectrum" is generally observed in turbulence. However, for high order structure functions, the difference with the Kolmogorov scaling is significant, and the breakdown of the statistical self-similarity is clear. This behavior, and the lack of universality of

7740-497: The velocity field than the viscous (friction) effects. In high Reynolds number flows, the flow is often modeled as an inviscid flow , an approximation in which viscosity is completely neglected. Eliminating viscosity allows the Navier–Stokes equations to be simplified into the Euler equations . The integration of the Euler equations along a streamline in an inviscid flow yields Bernoulli's equation . When, in addition to being inviscid,

7830-400: The viscosity of the fluid can effectively dissipate the kinetic energy into internal energy. In his original theory of 1941, Kolmogorov postulated that for very high Reynolds numbers , the small-scale turbulent motions are statistically isotropic (i.e. no preferential spatial direction could be discerned). In general, the large scales of a flow are not isotropic, since they are determined by

7920-423: The volume surface. The momentum balance can also be written for a moving control volume. The following is the differential form of the momentum conservation equation. Here, the volume is reduced to an infinitesimally small point, and both surface and body forces are accounted for in one total force, F . For example, F may be expanded into an expression for the frictional and gravitational forces acting at

8010-454: The whole, further detail is not revealed. A time developing phenomenon is said to exhibit self-similarity if the numerical value of certain observable quantity f ( x , t ) {\displaystyle f(x,t)} measured at different times are different but the corresponding dimensionless quantity at given value of x / t z {\displaystyle x/t^{z}} remain invariant. It happens if

8100-546: The x- and y-directions. This means that to appreciate the self similarity of these fractal objects, they have to be rescaled using an anisotropic affine transformation . A compact topological space X is self-similar if there exists a finite set S indexing a set of non- surjective homeomorphisms { f s : s ∈ S } {\displaystyle \{f_{s}:s\in S\}} for which If X ⊂ Y {\displaystyle X\subset Y} , we call X self-similar if it

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