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84-397: VR Sports Powerboat Racing is a racing game where players choose from 16 craft with different engine sizes and handling. The game utilizes water physics, causing the boats' handling to be affected by turbulence and other boats' wakes . Up to four players are supported via a split screen . VR Sports Powerboat Racing met with mostly negative reviews. The PlayStation version held a 41% on

168-474: A r g m a x n ^ n ^ p d q d t ( A , p , n ^ ) . {\displaystyle \mathbf {I} (A,\mathbf {p} )={\underset {\mathbf {\hat {n}} }{\operatorname {arg\,max} }}\mathbf {\hat {n}} _{\mathbf {p} }{\frac {\mathrm {d} q}{\mathrm {d} t}}(A,\mathbf {p} ,\mathbf {\hat {n}} ).} In this case, there

252-423: A (single) scalar: j = I A , {\displaystyle j={\frac {I}{A}},} where I = lim Δ t → 0 Δ q Δ t = d q d t . {\displaystyle I=\lim _{\Delta t\to 0}{\frac {\Delta q}{\Delta t}}={\frac {\mathrm {d} q}{\mathrm {d} t}}.} In this case

336-501: A better option for a boat racing game than VR Sports Powerboat Racing . Turbulence 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

420-529: 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

504-402: A closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the divergence ). If the surface is not closed, it has an oriented curve as boundary. Stokes' theorem states that the flux of the curl of a vector field is the line integral of the vector field over this boundary. This path integral

588-525: 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

672-467: A flux according to the electromagnetism definition, the corresponding flux density , if that term is used, refers to its derivative along the surface that was integrated. By the Fundamental theorem of calculus , the corresponding flux density is a flux according to the transport definition. Given a current such as electric current—charge per time, current density would also be a flux according to

756-413: A good side to this title." While sharing some of their criticisms, GamePro saw more of a bright side, offering praise for the selection of play modes and the intensity of the races. They concluded, "All told, Powerboat works fine as a poor man's Wave Race [64] for PlayStation gamers. It's plenty fun at first, but [it] wears thin because of the flaws in graphics and controls. We're talking rental all

840-471: A key contribution of Joseph Fourier , in the analysis of heat transfer phenomena. His seminal treatise Théorie analytique de la chaleur ( The Analytical Theory of Heat ), defines fluxion as a central quantity and proceeds to derive the now well-known expressions of flux in terms of temperature differences across a slab, and then more generally in terms of temperature gradients or differentials of temperature, across other geometries. One could argue, based on

924-494: A magnetic field opposite to the change. This is the basis for inductors and many electric generators . Using this definition, the flux of the Poynting vector S over a specified surface is the rate at which electromagnetic energy flows through that surface, defined like before: The flux of the Poynting vector through a surface is the electromagnetic power , or energy per unit time , passing through that surface. This

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1008-416: 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

1092-532: A positive point charge can be visualized as a dot radiating electric field lines (sometimes also called "lines of force"). Conceptually, electric flux can be thought of as "the number of field lines" passing through a given area. Mathematically, electric flux is the integral of the normal component of the electric field over a given area. Hence, units of electric flux are, in the MKS system , newtons per coulomb times meters squared, or N m /C. (Electric flux density

1176-457: 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

1260-400: 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

1344-443: 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

1428-434: 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

1512-445: 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

1596-448: 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 ′

1680-404: Is j  cos  θ , while the component of flux passing tangential to the area is j  sin  θ , but there is no flux actually passing through the area in the tangential direction. The only component of flux passing normal to the area is the cosine component. For vector flux, the surface integral of j over a surface S , gives the proper flowing per unit of time through

1764-430: 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

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1848-466: 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

1932-417: Is also called circulation , especially in fluid dynamics. Thus the curl is the circulation density. We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas. An electric "charge," such as a single proton in space, has a magnitude defined in coulombs. Such a charge has an electric field surrounding it. In pictorial form, the electric field from

2016-405: 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

2100-648: Is being used according to the transport definition (and furthermore is a vector field rather than single vector). This is ironic because Maxwell was one of the major developers of what we now call "electric flux" and "magnetic flux" according to the electromagnetism definition. Their names in accordance with the quote (and transport definition) would be "surface integral of electric flux" and "surface integral of magnetic flux", in which case "electric flux" would instead be defined as "electric field" and "magnetic flux" defined as "magnetic field". This implies that Maxwell conceived of these fields as flows/fluxes of some sort. Given

2184-410: 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

2268-431: 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

2352-400: 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

2436-408: Is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines. Lines originate from areas of positive divergence (sources) and end at areas of negative divergence (sinks). See also the image at right: the number of red arrows passing through a unit area is the flux density, the curve encircling

2520-419: 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,

2604-473: 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

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2688-408: Is maximized across the disk that is perpendicular to it. The unit vector thus uniquely maximizes the function when it points in the "true direction" of the flow. (Strictly speaking, this is an abuse of notation because the "arg max" cannot directly compare vectors; we take the vector with the biggest norm instead.) These direct definitions, especially the last, are rather unwieldy. For example,

2772-399: Is no fixed surface we are measuring over. q is a function of a point, an area, and a direction (given by a unit vector n ^ {\displaystyle \mathbf {\hat {n}} } ), and measures the flow through the disk of area A perpendicular to that unit vector. I is defined picking the unit vector that maximizes the flow around the point, because the true flow

2856-422: 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

2940-469: Is sometimes referred to as the probability current or current density, or probability flux density. As a mathematical concept, flux is represented by the surface integral of a vector field , where F is a vector field , and d A is the vector area of the surface A , directed as the surface normal . For the second, n is the outward pointed unit normal vector to the surface. The surface has to be orientable , i.e. two sides can be distinguished:

3024-447: 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

3108-604: Is the concentration ( mol /m ) of component A. This flux has units of mol·m ·s , and fits Maxwell's original definition of flux. For dilute gases, kinetic molecular theory relates the diffusion coefficient D to the particle density n = N / V , the molecular mass m , the collision cross section σ {\displaystyle \sigma } , and the absolute temperature T by D = 2 3 n σ k T π m {\displaystyle D={\frac {2}{3n\sigma }}{\sqrt {\frac {kT}{\pi m}}}} where

3192-407: Is the vector area  – combination A = A n ^ {\displaystyle \mathbf {A} =A\mathbf {\hat {n}} } of the magnitude of the area A through which the property passes and a unit vector n ^ {\displaystyle \mathbf {\hat {n}} } normal to the area. Unlike in the second set of equations,

3276-509: Is the electric flux per unit area, and is a measure of strength of the normal component of the electric field averaged over the area of integration. Its units are N/C, the same as the electric field in MKS units.) Two forms of electric flux are used, one for the E -field: and one for the D -field (called the electric displacement ): This quantity arises in Gauss's law – which states that

3360-401: 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

3444-505: 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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3528-466: 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,

3612-530: Is zero. As mentioned above, chemical molar flux of a component A in an isothermal , isobaric system is defined in Fick's law of diffusion as: J A = − D A B ∇ c A {\displaystyle \mathbf {J} _{A}=-D_{AB}\nabla c_{A}} where the nabla symbol ∇ denotes the gradient operator, D AB is the diffusion coefficient (m ·s ) of component A diffusing through component B, c A

3696-404: 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

3780-529: The D -field flux equals the charge Q A within it. Here the expression "flux of" indicates a mathematical operation and, as can be seen, the result is not necessarily a "flow", since nothing actually flows along electric field lines. The magnetic flux density ( magnetic field ) having the unit Wb/m ( Tesla ) is denoted by B , and magnetic flux is defined analogously: with the same notation above. The quantity arises in Faraday's law of induction , where

3864-523: 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

3948-640: 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

4032-402: The review aggregation website GameRankings based on five reviews. Critics razed the visuals for the cheap-looking water spray effects, dramatic pop-up which makes obstacles and even walls invisible until the player is too close to avoid them, and most especially the upward-angled camera view, which causes the bulk of the screen to be occupied with blue sky and leaves the player little view of

4116-477: The Navier-Stokes equations, i.e. from first principles. Flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics . For transport phenomena , flux is a vector quantity, describing the magnitude and direction of

4200-493: 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

4284-541: The actual course. Next Generation speculated, "If only the game had more control over camera placement, it might have earned an extra star." Other common criticisms included oversensitive controls and an annoying voice-over commentator. The four reviewers of Electronic Gaming Monthly unanimously considered the game atrocious, with John Davison stating that it "takes the term 'bad' to completely new and previously uncharted territory." GameSpot were also firmly negative in their assessment, summing up, "There simply isn't

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4368-424: The area at an angle θ to the area normal n ^ {\displaystyle \mathbf {\hat {n}} } , then the dot product j ⋅ n ^ = j cos ⁡ θ . {\displaystyle \mathbf {j} \cdot \mathbf {\hat {n}} =j\cos \theta .} That is, the component of flux passing through the surface (i.e. normal to it)

4452-499: The area of that cross section, or the amount of sunlight energy that lands on a patch of ground each second divided by the area of the patch, are kinds of flux. Here are 3 definitions in increasing order of complexity. Each is a special case of the following. In all cases the frequent symbol j , (or J ) is used for flux, q for the physical quantity that flows, t for time, and A for area. These identifiers will be written in bold when and only when they are vectors. First, flux as

4536-406: The arg max construction is artificial from the perspective of empirical measurements, when with a weathervane or similar one can easily deduce the direction of flux at a point. Rather than defining the vector flux directly, it is often more intuitive to state some properties about it. Furthermore, from these properties the flux can uniquely be determined anyway. If the flux j passes through

4620-445: The curve ∂ A {\displaystyle \partial A} , with the sign determined by the integration direction. The time-rate of change of the magnetic flux through a loop of wire is minus the electromotive force created in that wire. The direction is such that if current is allowed to pass through the wire, the electromotive force will cause a current which "opposes" the change in magnetic field by itself producing

4704-492: 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

4788-457: The field, the flux for the sides is zero and there is an equal and opposite flux at both ends of the tube. This is a consequence of Gauss's Law applied to an inverse square field. The flux for any cross-sectional surface of the tube will be the same. The total flux for any surface surrounding a charge q is q / ε 0 . In free space the electric displacement is given by the constitutive relation D = ε 0 E , so for any bounding surface

4872-581: 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

4956-399: The flow of a substance or property. In vector calculus flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface. The word flux comes from Latin : fluxus means "flow", and fluere is "to flow". As fluxion , this term was introduced into differential calculus by Isaac Newton . The concept of heat flux was

5040-499: 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

5124-442: The flux of the electric field E out of a closed surface is proportional to the electric charge Q A enclosed in the surface (independent of how that charge is distributed), the integral form is: where ε 0 is the permittivity of free space . If one considers the flux of the electric field vector, E , for a tube near a point charge in the field of the charge but not containing it with sides formed by lines tangent to

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5208-399: 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

5292-405: The magnetic flux is time-dependent either because the boundary is time-dependent or magnetic field is time-dependent. In integral form: where d ℓ is an infinitesimal vector line element of the closed curve ∂ A {\displaystyle \partial A} , with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to

5376-412: The most common forms of flux from the transport phenomena literature are defined as follows: These fluxes are vectors at each point in space, and have a definite magnitude and direction. Also, one can take the divergence of any of these fluxes to determine the accumulation rate of the quantity in a control volume around a given point in space. For incompressible flow , the divergence of the volume flux

5460-498: 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

5544-460: 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

5628-727: The probability of finding a particle in a differential volume element d r is d P = | ψ | 2 d 3 r . {\displaystyle dP=|\psi |^{2}\,d^{3}\mathbf {r} .} Then the number of particles passing perpendicularly through unit area of a cross-section per unit time is the probability flux; J = i ℏ 2 m ( ψ ∇ ψ ∗ − ψ ∗ ∇ ψ ) . {\displaystyle \mathbf {J} ={\frac {i\hbar }{2m}}\left(\psi \nabla \psi ^{*}-\psi ^{*}\nabla \psi \right).} This

5712-421: 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

5796-408: The red arrows denotes the boundary of the surface, and the orientation of the arrows with respect to the surface denotes the sign of the inner product of the vector field with the surface normals. If the surface encloses a 3D region, usually the surface is oriented such that the influx is counted positive; the opposite is the outflux . The divergence theorem states that the net outflux through

5880-586: 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 ⟩

5964-627: The second factor is the mean free path and the square root (with the Boltzmann constant k ) is the mean velocity of the particles. In turbulent flows, the transport by eddy motion can be expressed as a grossly increased diffusion coefficient. In quantum mechanics , particles of mass m in the quantum state ψ ( r , t ) have a probability density defined as ρ = ψ ∗ ψ = | ψ | 2 . {\displaystyle \rho =\psi ^{*}\psi =|\psi |^{2}.} So

6048-400: 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

6132-407: The surface does not fold back onto itself. Also, the surface has to be actually oriented, i.e. we use a convention as to flowing which way is counted positive; flowing backward is then counted negative. The surface normal is usually directed by the right-hand rule . Conversely, one can consider the flux the more fundamental quantity and call the vector field the flux density. Often a vector field

6216-497: The surface here need not be flat. Finally, we can integrate again over the time duration t 1 to t 2 , getting the total amount of the property flowing through the surface in that time ( t 2  −  t 1 ): q = ∫ t 1 t 2 ∬ S j ⋅ d A d t . {\displaystyle q=\int _{t_{1}}^{t_{2}}\iint _{S}\mathbf {j} \cdot d\mathbf {A} \,dt.} Eight of

6300-763: The surface in which flux is being measured is fixed and has area A . The surface is assumed to be flat, and the flow is assumed to be everywhere constant with respect to position and perpendicular to the surface. Second, flux as a scalar field defined along a surface, i.e. a function of points on the surface: j ( p ) = ∂ I ∂ A ( p ) , {\displaystyle j(\mathbf {p} )={\frac {\partial I}{\partial A}}(\mathbf {p} ),} I ( A , p ) = d q d t ( A , p ) . {\displaystyle I(A,\mathbf {p} )={\frac {\mathrm {d} q}{\mathrm {d} t}}(A,\mathbf {p} ).} As before,

6384-675: The surface is assumed to be flat, and the flow is assumed to be everywhere perpendicular to it. However the flow need not be constant. q is now a function of p , a point on the surface, and A , an area. Rather than measure the total flow through the surface, q measures the flow through the disk with area A centered at p along the surface. Finally, flux as a vector field : j ( p ) = ∂ I ∂ A ( p ) , {\displaystyle \mathbf {j} (\mathbf {p} )={\frac {\partial \mathbf {I} }{\partial A}}(\mathbf {p} ),} I ( A , p ) =

6468-454: The surface. According to the transport definition, flux may be a single vector, or it may be a vector field / function of position. In the latter case flux can readily be integrated over a surface. By contrast, according to the electromagnetism definition, flux is the integral over a surface; it makes no sense to integrate a second-definition flux for one would be integrating over a surface twice. Thus, Maxwell's quote only makes sense if "flux"

6552-429: The surface: d q d t = ∬ S j ⋅ n ^ d A = ∬ S j ⋅ d A , {\displaystyle {\frac {\mathrm {d} q}{\mathrm {d} t}}=\iint _{S}\mathbf {j} \cdot \mathbf {\hat {n}} \,dA=\iint _{S}\mathbf {j} \cdot d\mathbf {A} ,} where A (and its infinitesimal)

6636-407: The term corresponds to. In transport phenomena ( heat transfer , mass transfer and fluid dynamics ), flux is defined as the rate of flow of a property per unit area, which has the dimensions [quantity]·[time] ·[area] . The area is of the surface the property is flowing "through" or "across". For example, the amount of water that flows through a cross section of a river each second divided by

6720-428: The transport definition—charge per time per area. Due to the conflicting definitions of flux , and the interchangeability of flux , flow , and current in nontechnical English, all of the terms used in this paragraph are sometimes used interchangeably and ambiguously. Concrete fluxes in the rest of this article will be used in accordance to their broad acceptance in the literature, regardless of which definition of flux

6804-403: 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

6888-400: 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

6972-470: The way here." Reviewing the Windows version, GameSpot further criticized that of the four controller options, a steering wheel is the only one which works even mildly well, and ridiculed the fact that name entry is done using a scrolling alphabet despite keyboards being a standard accessory on PCs. The reviewer concluded that even playing a car racing game and pretending it is taking place on water would be

7056-405: The work of James Clerk Maxwell , that the transport definition precedes the definition of flux used in electromagnetism . The specific quote from Maxwell is: In the case of fluxes, we have to take the integral, over a surface, of the flux through every element of the surface. The result of this operation is called the surface integral of the flux. It represents the quantity which passes through

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