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72-447: The physics of a bouncing ball concerns the physical behaviour of bouncing balls , particularly its motion before, during, and after impact against the surface of another body . Several aspects of a bouncing ball's behaviour serve as an introduction to mechanics in high school or undergraduate level physics courses. However, the exact modelling of the behaviour is complex and of interest in sports engineering . The motion of

144-466: A , v , and r denote the acceleration, velocity, and position of the ball, and v 0 and r 0 are the initial velocity and position of the ball, respectively. More specifically, if the ball is bounced at an angle θ with the ground, the motion in the x - and y -axes (representing horizontal and vertical motion, respectively) is described by The equations imply that the maximum height ( H ) and range ( R ) and time of flight ( T ) of

216-467: A ball bouncing on a flat surface are given by Further refinements to the motion of the ball can be made by taking into account air resistance (and related effects such as drag and wind ), the Magnus effect , and buoyancy . Because lighter balls accelerate more readily, their motion tends to be affected more by such forces. Air flow around the ball can be either laminar or turbulent depending on

288-464: A ball is generally described by projectile motion (which can be affected by gravity , drag , the Magnus effect , and buoyancy ), while its impact is usually characterized through the coefficient of restitution (which can be affected by the nature of the ball, the nature of the impacting surface, the impact velocity, rotation, and local conditions such as temperature and pressure ). To ensure fair play , many sports governing bodies set limits on

360-572: A certain length of flow, a laminar boundary layer will become unstable and turbulent. This instability occurs across different scales and with different fluids, usually when Re x ≈ 5 × 10 , where x is the distance from the leading edge of the flat plate, and the flow velocity is the freestream velocity of the fluid outside the boundary layer. For flow in a pipe of diameter D , experimental observations show that for "fully developed" flow, laminar flow occurs when Re D < 2300 and turbulent flow occurs when Re D > 2900. At

432-418: A component parallel to the surface, which would contribute to friction, and thus contribute to rotation. In racquet sports such as table tennis or racquetball , skilled players will use spin (including sidespin ) to suddenly alter the ball's direction when it impacts surface, such as the ground or their opponent's racquet . Similarly, in cricket , there are various methods of spin bowling that can make

504-436: A density times an acceleration. Each term is thus dependent on the exact measurements of a flow. When one renders the equation nondimensional, that is when we multiply it by a factor with inverse units of the base equation, we obtain a form that does not depend directly on the physical sizes. One possible way to obtain a nondimensional equation is to multiply the whole equation by the factor where If we now set we can rewrite

576-416: A larger pipe. The larger pipe was glass so the behaviour of the layer of the dyed stream could be observed. At the end of this pipe, there was a flow control valve used to vary the water velocity inside the tube. When the velocity was low, the dyed layer remained distinct throughout the entire length of the large tube. When the velocity was increased, the layer broke up at a given point and diffused throughout

648-465: A matter of convention in some circumstances, notably stirred vessels. In practice, matching the Reynolds number is not on its own sufficient to guarantee similitude. Fluid flow is generally chaotic, and very small changes to shape and surface roughness of bounding surfaces can result in very different flows. Nevertheless, Reynolds numbers are a very important guide and are widely used. If we know that

720-494: A speed advantage by pumping a polymer solution such as low molecular weight polyoxyethylene in water, over the wetted surface of the hull. It is, however, a problem for mixing polymers, because turbulence is needed to distribute fine filler (for example) through the material. Inventions such as the "cavity transfer mixer" have been developed to produce multiple folds into a moving melt so as to improve mixing efficiency. The device can be fitted onto extruders to aid mixing. For

792-428: A sphere in a fluid, the characteristic length-scale is the diameter of the sphere and the characteristic velocity is that of the sphere relative to the fluid some distance away from the sphere, such that the motion of the sphere does not disturb that reference parcel of fluid. The density and viscosity are those belonging to the fluid. Note that purely laminar flow only exists up to Re = 10 under this definition. Under

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864-682: A velocity and a characteristic length or characteristic dimension (L in the above equation). This dimension is a matter of convention—for example radius and diameter are equally valid to describe spheres or circles, but one is chosen by convention. For aircraft or ships, the length or width can be used. For flow in a pipe, or for a sphere moving in a fluid, the internal diameter is generally used today. Other shapes such as rectangular pipes or non-spherical objects have an equivalent diameter defined. For fluids of variable density such as compressible gases or fluids of variable viscosity such as non-Newtonian fluids , special rules apply. The velocity may also be

936-547: A wind tunnel and the full-size version. The predictions of the onset of turbulence and the ability to calculate scaling effects can be used to help predict fluid behavior on a larger scale, such as in local or global air or water movement, and thereby the associated meteorological and climatological effects. The concept was introduced by George Stokes in 1851, but the Reynolds number was named by Arnold Sommerfeld in 1908 after Osborne Reynolds (1842–1912), who popularized its use in 1883. (cf. this list ) The Reynolds number

1008-414: A wing, where chord is still used. The Reynolds number for an object moving in a fluid, called the particle Reynolds number and often denoted Re p , characterizes the nature of the surrounding flow and its fall velocity. Where the viscosity is naturally high, such as polymer solutions and polymer melts, flow is normally laminar. The Reynolds number is very small and Stokes' law can be used to measure

1080-436: Is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow , while at high Reynolds numbers, flows tend to be turbulent . The turbulence results from differences in the fluid's speed and direction, which may sometimes intersect or even move counter to

1152-456: Is a consequence of the theory of thermodynamics , where, for every such interaction, some energy must be converted into alternative forms of energy or is absorbed by the deformation of the objects involved in the collision. This scattering –related article is a stub . You can help Misplaced Pages by expanding it . Reynolds number In fluid dynamics , the Reynolds number ( Re )

1224-702: Is a function of ρ u L μ − 1 {\displaystyle \rho uL\mu ^{-1}} , the Reynolds number. Alternatively, we can take the incompressible Navier–Stokes equations (convective form) : ∂ u ∂ t + ( u ⋅ ∇ ) u − ν ∇ 2 u = − 1 ρ ∇ p + g {\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} -\nu \,\nabla ^{2}\mathbf {u} =-{\frac {1}{\rho }}\nabla p+\mathbf {g} } Remove

1296-412: Is better described by where ρ is the density of air, C L the lift coefficient , A the cross-sectional area of the ball, and v the velocity of the ball relative to air. The lift coefficient is a complex factor which depends amongst other things on the ratio rω / v , the Reynolds number, and surface roughness . In certain conditions, the lift coefficient can even be negative, changing

1368-437: Is defined as: R e = u L ν = ρ u L μ {\displaystyle \mathrm {Re} ={\frac {uL}{\nu }}={\frac {\rho uL}{\mu }}} where: The Reynolds number can be defined for several different situations where a fluid is in relative motion to a surface. These definitions generally include the fluid properties of density and viscosity, plus

1440-403: Is different for every geometry. For a fluid moving between two plane parallel surfaces—where the width is much greater than the space between the plates—then the characteristic dimension is equal to the distance between the plates. This is consistent with the annular duct and rectangular duct cases above, taken to a limiting aspect ratio. For calculating the flow of liquid with a free surface,

1512-399: Is equal to the weight of the fluid displaced by the object. In the case of a sphere, this force is equal to The buoyant force is usually small compared to the drag and Magnus forces and can often be neglected. However, in the case of a basketball, the buoyant force can amount to about 1.5% of the ball's weight. Since buoyancy is directed upwards, it will act to increase the range and height of

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1584-476: Is even possible for the COR to be greater than 1, or for the forward velocity of the ball to increase upon impact. A popular demonstration involves the bounce of multiple stacked balls. If a tennis ball is stacked on top of a basketball, and the two of them are dropped at the same time, the tennis ball will bounce much higher than it would have if dropped on its own, even exceeding its original release height. The result

1656-452: Is impractical, sieve diameters are used instead as the characteristic particle length-scale. Both approximations alter the values of the critical Reynolds number. The particle Reynolds number is important in determining the fall velocity of a particle. When the particle Reynolds number indicates laminar flow, Stokes' law can be used to calculate its fall velocity or settling velocity. When the particle Reynolds number indicates turbulent flow,

1728-429: Is involved. For a straight drop on the ground with no rotation, with only the force of gravity acting on the ball, the COR can be related to several other quantities by: Here, K and U denote the kinetic and potential energy of the ball, H is the maximum height of the ball, and T is the time of flight of the ball. The 'i' and 'f' subscript refer to the initial (before impact) and final (after impact) states of

1800-407: Is surprising as it apparently violates conservation of energy. However, upon closer inspection, the basketball does not bounce as high as it would have if the tennis ball had not been on top of it, and transferred some of its energy into the tennis ball, propelling it to a greater height. The usual explanation involves considering two separate impacts: the basketball impacting with the floor, and then

1872-403: Is the ratio of inertial forces to viscous forces within a fluid that is subjected to relative internal movement due to different fluid velocities. A region where these forces change behavior is known as a boundary layer , such as the bounding surface in the interior of a pipe. A similar effect is created by the introduction of a stream of high-velocity fluid into a low-velocity fluid, such as

1944-418: Is the wetted perimeter . The wetted perimeter for a channel is the total perimeter of all channel walls that are in contact with the flow. This means that the length of the channel exposed to air is not included in the wetted perimeter. For a circular pipe, the hydraulic diameter is exactly equal to the inside pipe diameter: For an annular duct, such as the outer channel in a tube-in-tube heat exchanger ,

2016-402: Is the radius of the ball. This force acts in opposition to the ball's direction (in the direction of − v ^ {\displaystyle \textstyle -{\hat {\mathbf {v} }}} ). For most sports balls, however, the Reynolds number will be between 10 and 10 and Stokes' law does not apply. At these higher values of the Reynolds number, the drag force on

2088-453: Is theoretically possible, but would indicate that the ball went through the surface ( e < 0 ), or that the surface was not "relaxed" when the ball impacted it ( e > 1 ), like in the case of a ball landing on spring-loaded platform. To analyze the vertical and horizontal components of the motion, the COR is sometimes split up into a normal COR ( e y ), and tangential COR ( e x ), defined as where r and ω denote

2160-431: The Reynolds number (Re), defined as: where ρ is the density of air , μ the dynamic viscosity of air, D the diameter of the ball, and v the velocity of the ball through air. At a temperature of 20 °C , ρ = 1.2 kg/m and μ = 1.8 × 10 Pa·s . If the Reynolds number is very low (Re < 1), the drag force on the ball is described by Stokes' law : where r

2232-481: The Reynolds-averaged Navier–Stokes equations . For flow in a pipe or tube, the Reynolds number is generally defined as where For shapes such as squares, rectangular or annular ducts where the height and width are comparable, the characteristic dimension for internal-flow situations is taken to be the hydraulic diameter , D H , defined as where A is the cross-sectional area, and P

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2304-424: The chord Reynolds number R = Vc / ν , where V is the flight speed, c is the chord length, and ν is the kinematic viscosity of the fluid in which the airfoil operates, which is 1.460 × 10  m /s for the atmosphere at sea level . In some special studies a characteristic length other than chord may be used; rare is the "span Reynolds number", which is not to be confused with span-wise stations on

2376-434: The hydraulic radius must be determined. This is the cross-sectional area of the channel divided by the wetted perimeter. For a semi-circular channel, it is a quarter of the diameter (in case of full pipe flow). For a rectangular channel, the hydraulic radius is the cross-sectional area divided by the wetted perimeter. Some texts then use a characteristic dimension that is four times the hydraulic radius, chosen because it gives

2448-410: The viscosity of the fluid. Spheres are allowed to fall through the fluid and they reach the terminal velocity quickly, from which the viscosity can be determined. The laminar flow of polymer solutions is exploited by animals such as fish and dolphins, who exude viscous solutions from their skin to aid flow over their bodies while swimming. It has been used in yacht racing by owners who want to gain

2520-496: The Navier–Stokes equation without dimensions: where the term ⁠ μ / ρLV ⁠ = ⁠ 1 / Re ⁠ . Finally, dropping the primes for ease of reading: This is why mathematically all Newtonian, incompressible flows with the same Reynolds number are comparable. Notice also that in the above equation, the viscous terms vanish for Re → ∞ . Thus flows with high Reynolds numbers are approximately inviscid in

2592-519: The Reynolds number. This argument is written out in detail on the Scallop theorem page. The Reynolds number can be obtained when one uses the nondimensional form of the incompressible Navier–Stokes equations for a newtonian fluid expressed in terms of the Lagrangian derivative : Each term in the above equation has the units of a "body force" (force per unit volume) with the same dimensions of

2664-405: The ball deviate significantly off the pitch . The bounce of an oval-shaped ball (such as those used in gridiron football or rugby football ) is in general much less predictable than the bounce of a spherical ball. Depending on the ball's alignment at impact, the normal force can act ahead or behind the centre of mass of the ball, and friction from the ground will depend on the alignment of

2736-408: The ball is instead described by the drag equation : where C d is the drag coefficient , and A the cross-sectional area of the ball. Drag will cause the ball to lose mechanical energy during its flight, and will reduce its range and height, while crosswinds will deflect it from its original path. Both effects have to be taken into account by players in sports such as golf. The spin of

2808-407: The ball to the left . Additionally, if the ball is spinning at impact, friction will have a "rotational" component in the direction opposite to the ball's rotation. On the figure, the ball is spinning clockwise, and the point impacting the ground is moving to the left with respect to the ball's center of mass . The rotational component of friction is therefore pushing the ball to the right . Unlike

2880-519: The ball will affect its trajectory through the Magnus effect . According to the Kutta–Joukowski theorem , for a spinning sphere with an inviscid flow of air, the Magnus force is equal to where r is the radius of the ball, ω the angular velocity (or spin rate) of the ball, ρ the density of air, and v the velocity of the ball relative to air. This force is directed perpendicular to

2952-399: The ball's spin ( F M ), and the buoyant force ( F B ). In general, one has to use Newton's second law taking all forces into account to analyze the ball's motion: where m is the ball's mass. Here, a , v , r represent the ball's acceleration , velocity , and position over time t . The gravitational force is directed downwards and is equal to where m is the mass of

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3024-428: The ball, and g is the gravitational acceleration , which on Earth varies between 9.764  m/s and 9.834 m/s . Because the other forces are usually small, the motion is often idealized as being only under the influence of gravity. If only the force of gravity acts on the ball, the mechanical energy will be conserved during its flight. In this idealized case, the equations of motion are given by where

3096-445: The ball, as well as its rotation, spin, and impact velocity. Where the forces act with respect to the centre of mass of the ball changes as the ball rolls on the ground, and all forces can exert a torque on the ball, including the normal force and the force of gravity. This can cause the ball to bounce forward, bounce back, or sideways. Because it is possible to transfer some rotational kinetic energy into translational kinetic energy, it

3168-477: The ball. When a ball impacts a surface, the surface recoils and vibrates , as does the ball, creating both sound and heat , and the ball loses kinetic energy . Additionally, the impact can impart some rotation to the ball, transferring some of its translational kinetic energy into rotational kinetic energy . This energy loss is usually characterized (indirectly) through the coefficient of restitution (or COR, denoted e ): where v f and v i are

3240-480: The ball. Likewise, the energy loss at impact can be related to the COR by The COR of a ball can be affected by several things, mainly External conditions such as temperature can change the properties of the impacting surface or of the ball, making them either more flexible or more rigid. This will, in turn, affect the COR. In general, the ball will deform more at higher impact velocities and will accordingly lose more of its energy, decreasing its COR. Upon impacting

3312-423: The basketball impacting with the tennis ball. Assuming perfectly elastic collisions , the basketball impacting the floor at 1 m/s would rebound at 1 m/s. The tennis ball going at 1 m/s would then have a relative impact velocity of 2 m/s, which means it would rebound at 2 m/s relative to the basketball, or 3 m/s relative to the floor, and triple its rebound velocity compared to impacting

3384-460: The bounciness of their ball and forbid tampering with the ball's aerodynamic properties. The bounciness of balls has been a feature of sports as ancient as the Mesoamerican ballgame . The motion of a bouncing ball obeys projectile motion . Many forces act on a real ball, namely the gravitational force ( F G ), the drag force due to air resistance ( F D ), the Magnus force due to

3456-453: The center of the deflategate controversy. Some sports do not regulate the bouncing properties of balls directly, but instead specify a construction method. In baseball , the introduction of a cork-based ball helped to end the dead-ball era and trigger the live-ball era . Deflection (physics) An object's deflective efficiency can never equal or surpass 100%, for example: This transfer of some energy into heat or other radiation

3528-410: The condition of low Re , the relationship between force and speed of motion is given by Stokes' law . At higher Reynolds numbers the drag on a sphere depends on surface roughness. Thus, for example, adding dimples on the surface of a golf ball causes the boundary layer on the upstream side of the ball to transition from laminar to turbulent. The turbulent boundary layer is able to remain attached to

3600-432: The direction of the Magnus force ( reverse Magnus effect ). In sports like tennis or volleyball , the player can use the Magnus effect to control the ball's trajectory (e.g. via topspin or backspin ) during flight. In golf , the effect is responsible for slicing and hooking which are usually a detriment to the golfer, but also helps with increasing the range of a drive and other shots. In baseball , pitchers use

3672-569: The effect to create curveballs and other special pitches . Ball tampering is often illegal, and is often at the centre of cricket controversies such as the one between England and Pakistan in August 2006 . In baseball, the term ' spitball ' refers to the illegal coating of the ball with spit or other substances to alter the aerodynamics of the ball. Any object immersed in a fluid such as water or air will experience an upwards buoyancy . According to Archimedes' principle , this buoyant force

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3744-626: The fast-moving center of the pipe while slower-moving turbulent flow dominates near the wall. As the Reynolds number increases, the continuous turbulent-flow moves closer to the inlet and the intermittency in between increases, until the flow becomes fully turbulent at Re D > 2900. This result is generalized to non-circular channels using the hydraulic diameter , allowing a transition Reynolds number to be calculated for other shapes of channel. These transition Reynolds numbers are also called critical Reynolds numbers , and were studied by Osborne Reynolds around 1895. The critical Reynolds number

3816-413: The final and initial velocities of the ball, and u f and u i are the final and initial velocities impacting surface, respectively. In the specific case where a ball impacts on an immovable surface, the COR simplifies to For a ball dropped against a floor, the COR will therefore vary between 0 (no bounce, total loss of energy) and 1 (perfectly bouncy, no energy loss). A COR value below 0 or above 1

3888-424: The floor on its own. This implies that the ball would bounce to 9 times its original height. In reality, due to inelastic collisions , the tennis ball will increase its velocity and rebound height by a smaller factor, but still will bounce faster and higher than it would have on its own. While the assumptions of separate impacts is not actually valid (the balls remain in close contact with each other during most of

3960-524: The fluid's cross-section. The point at which this happened was the transition point from laminar to turbulent flow. From these experiments came the dimensionless Reynolds number for dynamic similarity—the ratio of inertial forces to viscous forces. Reynolds also proposed what is now known as the Reynolds averaging of turbulent flows, where quantities such as velocity are expressed as the sum of mean and fluctuating components. Such averaging allows for 'bulk' description of turbulent flow, for example using

4032-432: The free stream. Osborne Reynolds famously studied the conditions in which the flow of fluid in pipes transitioned from laminar flow to turbulent flow . In his 1883 paper Reynolds described the transition from laminar to turbulent flow in a classic experiment in which he examined the behaviour of water flow under different flow velocities using a small stream of dyed water introduced into the centre of clear water flow in

4104-522: The gravity term g {\displaystyle {\mathbf {g}}} , then the left side consists of inertial force ∂ u ∂ t + ( u ⋅ ∇ ) u {\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} } , and viscous force ν ∇ 2 u {\displaystyle \nu \,\nabla ^{2}\mathbf {u} } . Their ratio has

4176-424: The ground, some translational kinetic energy can be converted to rotational kinetic energy and vice versa depending on the ball's impact angle and angular velocity. If the ball moves horizontally at impact, friction will have a "translational" component in the direction opposite to the ball's motion. In the figure, the ball is moving to the right , and thus it will have a translational component of friction pushing

4248-446: The hot gases emitted from a flame in air. This relative movement generates fluid friction, which is a factor in developing turbulent flow. Counteracting this effect is the viscosity of the fluid, which tends to inhibit turbulence. 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

4320-425: The hydraulic diameter can be shown algebraically to reduce to where For calculation involving flow in non-circular ducts, the hydraulic diameter can be substituted for the diameter of a circular duct, with reasonable accuracy, if the aspect ratio AR of the duct cross-section remains in the range ⁠ 1 / 4 ⁠ < AR < 4. In boundary layer flow over a flat plate, experiments confirm that, after

4392-399: The impact), this model will nonetheless reproduce experimental results with good agreement, and is often used to understand more complex phenomena such as the core collapse of supernovae , or gravitational slingshot manoeuvres . Several sports governing bodies regulate the bounciness of a ball through various ways, some direct, some indirect. The pressure of an American football was at

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4464-491: The lower end of this range, a continuous turbulent-flow will form, but only at a very long distance from the inlet of the pipe. The flow in between will begin to transition from laminar to turbulent and then back to laminar at irregular intervals, called intermittent flow. This is due to the different speeds and conditions of the fluid in different areas of the pipe's cross-section, depending on other factors such as pipe roughness and flow uniformity. Laminar flow tends to dominate in

4536-400: The motion and perpendicular to the axis of rotation (in the direction of ω ^ × v ^ {\displaystyle \textstyle {\hat {\mathbf {\omega } }}\times {\hat {\mathbf {v} }}} ). The force is directed upwards for backspin and downwards for topspin. In reality, flow is never inviscid, and the Magnus lift

4608-405: The normal force and the force of gravity, these frictional forces will exert a torque on the ball, and change its angular velocity ( ω ). Three situations can arise: If the surface is inclined by some amount θ , the entire diagram would be rotated by θ , but the force of gravity would remain pointing downwards (forming an angle θ with the surface). Gravity would then have

4680-543: 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 linear and the application of Reynolds numbers to both situations allows scaling factors to be developed. With respect to laminar and turbulent flow regimes: The Reynolds number

4752-453: The order of ( u ⋅ ∇ ) u ν ∇ 2 u ∼ u 2 / L ν u / L 2 = u L ν {\displaystyle {\frac {(\mathbf {u} \cdot \nabla )\mathbf {u} }{\nu \,\nabla ^{2}\mathbf {u} }}\sim {\frac {u^{2}/L}{\nu u/L^{2}}}={\frac {uL}{\nu }}} ,

4824-493: The overall direction of the flow ( eddy currents ). These eddy currents begin to churn the flow, using up energy in the process, which for liquids increases the chances of cavitation . The Reynolds number has wide applications, ranging from liquid flow in a pipe to the passage of air over an aircraft wing. It is used to predict the transition from laminar to turbulent flow and is used in the scaling of similar but different-sized flow situations, such as between an aircraft model in

4896-407: The radius and angular velocity of the ball, while R and Ω denote the radius and angular velocity the impacting surface (such as a baseball bat). In particular rω is the tangential velocity of the ball's surface, while RΩ is the tangential velocity of the impacting surface. These are especially of interest when the ball impacts the surface at an oblique angle , or when rotation

4968-893: The relevant physical quantities in a physical system are only ρ , u , L , μ {\displaystyle \rho ,u,L,\mu } , then the Reynolds number is essentially fixed by the Buckingham π theorem . In detail, since there are 4 quantities ρ , u , L , μ {\displaystyle \rho ,u,L,\mu } , but they have only 3 dimensions (length, time, mass), we can consider ρ x 1 u x 2 L x 3 μ x 4 {\displaystyle \rho ^{x_{1}}u^{x_{2}}L^{x_{3}}\mu ^{x_{4}}} , where x 1 , . . . , x 4 {\displaystyle x_{1},...,x_{4}} are real numbers. Setting

5040-440: The same value of Re for the onset of turbulence as in pipe flow, while others use the hydraulic radius as the characteristic length-scale with consequently different values of Re for transition and turbulent flow. Reynolds numbers are used in airfoil design to (among other things) manage "scale effect" when computing/comparing characteristics (a tiny wing, scaled to be huge, will perform differently). Fluid dynamicists define

5112-559: The surface of the ball much longer than a laminar boundary and so creates a narrower low-pressure wake and hence less pressure drag. The reduction in pressure drag causes the ball to travel farther. The equation for a rectangular object is identical to that of a sphere, with the object being approximated as an ellipsoid and the axis of length being chosen as the characteristic length scale. Such considerations are important in natural streams, for example, where there are few perfectly spherical grains. For grains in which measurement of each axis

5184-631: The three dimensions of ρ x 1 u x 2 L x 3 μ x 4 {\displaystyle \rho ^{x_{1}}u^{x_{2}}L^{x_{3}}\mu ^{x_{4}}} to zero, we obtain 3 independent linear constraints, so the solution space has 1 dimension, and it is spanned by the vector ( 1 , 1 , 1 , − 1 ) {\displaystyle (1,1,1,-1)} . Thus, any dimensionless quantity constructed out of ρ , u , L , μ {\displaystyle \rho ,u,L,\mu }

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