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104-466: It is a meandering river, with a catchment area of 3,887 km (1,501 sq mi). It is 71 meters wide near its mouth, with a slope gradient of 0.7 m/km. The average diameter of the bed material is 0.37 mm ( sand ). The river carries annually 22,185 tonnes of bedload and 7.6 million tonnes of suspended sediment to Lake Tana . This article related to a river in Ethiopia

208-400: A laminar boundary layer. This results in a lower skin friction due to the characteristic velocity profile of laminar flow. However, the boundary layer inevitably thickens and becomes less stable as the flow develops along the body, and eventually becomes turbulent , the process known as boundary layer transition . One way of dealing with this problem is to suck the boundary layer away through

312-415: A porous surface (see Boundary layer suction ). This can reduce drag, but is usually impractical due to its mechanical complexity and the power required to move the air and dispose of it. Natural laminar flow (NLF) techniques push the boundary layer transition aft by reshaping the airfoil or fuselage so that its thickest point is more aft and less thick. This reduces the velocities in the leading part and

416-430: A dependent variable τ = μ ∂ u / ∂ y {\displaystyle \tau =\mu \partial u/\partial y} (shear stress) instead of u {\displaystyle u} . The boundary layer equation then becomes The original coordinate is recovered from The treatment of turbulent boundary layers is far more difficult due to the time-dependent variation of

520-452: A maximum at the apex to zero at a crossing point (straight line), also called an inflection, because the curvature changes direction in that vicinity. The radius of the loop is the straight line perpendicular to the down-valley axis intersecting the sinuous axis at the apex. As the loop is not ideal, additional information is needed to characterize it. The orientation angle is the angle between sinuous axis and down-valley axis at any point on

624-405: A meander because helicoidal flow of water keeps the bank washed clean of loose sand, silt, and sediment and subjects it to constant erosion. As a result, the meander erodes and migrates in the direction of the outside bend, forming the cut bank. As the cut bank is undermined by erosion, it commonly collapses as slumps into the river channel. The slumped sediment, having been broken up by slumping,

728-411: A meander is part of an entrenched river or part of a freely meandering river within a floodplain, the term slip-off slope can refer to two different fluvial landforms that comprise the inner, convex, bank of a meander loop. In case of a freely meandering river on a floodplain, a slip-off slope is the inside, gently sloping bank of a meander on which sediments episodically accumulate to form a point bar as

832-431: A meandering watercourse is termed meander geometry or meander planform geometry. It is characterized as an irregular waveform . Ideal waveforms, such as a sine wave , are one line thick, but in the case of a stream the width must be taken into consideration. The bankfull width is the distance across the bed at an average cross-section at the full-stream level, typically estimated by the line of lowest vegetation. As

936-405: A modified version of Lévêque's profile, This results in a very good approximation, even for low P r {\displaystyle Pr} numbers, so that only liquid metals with P r {\displaystyle Pr} much less than 1 cannot be treated this way. In 1962, Kestin and Persen published a paper describing solutions for heat transfer when the thermal boundary layer

1040-409: A non-mathematical utility as well. Streams can be placed in categories arranged by it; for example, when the index is between 1 and 1.5 the river is sinuous, but if between 1.5 and 4, then meandering. The index is a measure also of stream velocity and sediment load, those quantities being maximized at an index of 1 (straight). Boundary layer In physics and fluid mechanics , a boundary layer

1144-428: A river meanders. This type of slip-off slope is located opposite the cutbank. This term can also be applied to the inside, sloping bank of a meandering tidal channel. In case of an entrenched river, a slip-off slope is a gently sloping bedrock surface that rises from the inside, concave bank of an asymmetrically entrenched river. This type of slip-off slope is often covered by a thin, discontinuous layer of alluvium. It

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1248-413: A self-intensifying process...in which greater curvature results in more erosion of the bank, which results in greater curvature..." The cross-current along the floor of the channel is part of the secondary flow and sweeps dense eroded material towards the inside of the bend. The cross-current then rises to the surface near the inside and flows towards the outside, forming the helical flow . The greater

1352-407: A similar order-of-magnitude analysis, the above equations can be reduced to leading order terms. By choosing length scales δ {\displaystyle \delta } for changes in the transverse-direction, and L {\displaystyle L} for changes in the streamwise-direction, with δ << L {\displaystyle \delta <<L} ,

1456-434: A symmetrical valley sides. He argues that the symmetrical valley sides are the direct result of rapid down-cutting of a watercourse into bedrock. In addition, as proposed by Rich, Thornbury argues that incised valleys with a pronounced asymmetry of cross section, which he called ingrown meanders , are the result of the lateral migration and incision of a meander during a period of slower channel downcutting . Regardless,

1560-605: A transformation which takes x {\displaystyle x} and ψ {\displaystyle \psi } ( stream function ) as independent variables instead of x {\displaystyle x} and y {\displaystyle y} and uses a dependent variable χ = U 2 − u 2 {\displaystyle \chi =U^{2}-u^{2}} instead of u {\displaystyle u} . The boundary layer equation then become The original variables are recovered from This transformation

1664-413: A waveform the meandering stream follows the down-valley axis, a straight line fitted to the curve such that the sum of all the amplitudes measured from it is zero. This axis represents the overall direction of the stream. At any cross-section the flow is following the sinuous axis, the centerline of the bed. Two consecutive crossing points of sinuous and down-valley axes define a meander loop. The meander

1768-430: Is a stub . You can help Misplaced Pages by expanding it . Meander A meander is one of a series of regular sinuous curves in the channel of a river or other watercourse . It is produced as a watercourse erodes the sediments of an outer, concave bank ( cut bank or river cliff ) and deposits sediments on an inner, convex bank which is typically a point bar . The result of this coupled erosion and sedimentation

1872-412: Is an alternative definition stating that the boundary layer represents a deficit in mass flow compared to inviscid flow with slip at the wall. It is the distance by which the wall would have to be displaced in the inviscid case to give the same total mass flow as the viscous case. The no-slip condition requires the flow velocity at the surface of a solid object be zero and the fluid temperature be equal to

1976-449: Is applicable to other fluids (besides air) with moderate to low viscosity such as water. For the case where there is a temperature difference between the surface and the bulk fluid, it is found that the majority of the heat transfer to and from a body takes place in the vicinity of the velocity boundary layer. This again allows the equations to be simplified in the flow field outside the boundary layer. The pressure distribution throughout

2080-479: Is arbitrary. Since the solution is not unique from mathematical perspective, to the solution can be added any one of an infinite set of eigenfunctions as shown by Stewartson and Paul A. Libby . Von Kármán derived the integral equation by integrating the boundary layer equation across the boundary layer in 1921. The equation is where The energy integral was derived by Wieghardt . where For steady two-dimensional boundary layers, von Mises introduced

2184-419: Is called lateral accretion. Lateral accretion occurs mostly during high water or floods when the point bar is submerged. Typically, the sediment consists of either sand, gravel, or a combination of both. The sediment comprising some point bars might grade downstream into silty sediments. Because of the decreasing velocity and strength of current from the thalweg of the channel to the upper surface of point bar when

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2288-419: Is contained entirely within the momentum layer and for various wall temperature distributions. For the problem of a flat plate with a temperature jump at x = x 0 {\displaystyle x=x_{0}} , they propose a substitution that reduces the parabolic thermal boundary-layer equation to an ordinary differential equation. The solution to this equation, the temperature at any point in

2392-622: Is for any instantaneous laminar or turbulent boundary layer, but is used mainly in laminar flow studies since the mean flow is also the instantaneous flow because there are no velocity fluctuations present. This simplified equation is a parabolic PDE and can be solved using a similarity solution often referred to as the Blasius boundary layer . Prandtl observed that from any solution u ( x , y , t ) ,   v ( x , y , t ) {\displaystyle u(x,y,t),\ v(x,y,t)} which satisfies

2496-491: Is governed by an easier to solve PDE . The continuity and Navier–Stokes equations for a two-dimensional steady incompressible flow in Cartesian coordinates are given by where u {\displaystyle u} and υ {\displaystyle \upsilon } are the velocity components, ρ {\displaystyle \rho } is the density, p {\displaystyle p}

2600-432: Is later extended to compressible boundary layer by von Kármán and HS Tsien . For steady two-dimensional compressible boundary layer, Luigi Crocco introduced a transformation which takes x {\displaystyle x} and u {\displaystyle u} as independent variables instead of x {\displaystyle x} and y {\displaystyle y} and uses

2704-424: Is produced by the gradual outward migration of the meander as a river cuts downward into bedrock. A terrace on the slip-off slope of a meander spur, known as slip-off slope terrace , can be formed by a brief halt during the irregular incision by an actively meandering river. The meander ratio or sinuosity index is a means of quantifying how much a river or stream meanders (how much its course deviates from

2808-458: Is readily eroded and carried toward the middle of the channel. The sediment eroded from a cut bank tends to be deposited on the point bar of the next downstream meander, and not on the point bar opposite it. This can be seen in areas where trees grow on the banks of rivers; on the inside of meanders, trees, such as willows, are often far from the bank, whilst on the outside of the bend, the tree roots are often exposed and undercut, eventually leading

2912-536: Is so exceedingly winding that everything winding is called meandering.’ The Meander River is south of Izmir, east of the ancient Greek town of Miletus , now Milet, Turkey. It flows through series of three graben in the Menderes Massif, but has a flood plain much wider than the meander zone in its lower reach. Its modern Turkish name is the Büyük Menderes River . Meanders are a result of

3016-412: Is the boundary layer. There are two different types of boundary layer flow: laminar and turbulent. Laminar boundary layer flow The laminar boundary is a very smooth flow, while the turbulent boundary layer contains swirls or "eddies." The laminar flow creates less skin friction drag than the turbulent flow, but is less stable. Boundary layer flow over a wing surface begins as a smooth laminar flow. As

3120-507: Is the formation of a sinuous course as the channel migrates back and forth across the axis of a floodplain . The zone within which a meandering stream periodically shifts its channel is known as a meander belt . It typically ranges from 15 to 18 times the width of the channel. Over time, meanders migrate downstream, sometimes in such a short time as to create civil engineering challenges for local municipalities attempting to maintain stable roads and bridges. The degree of meandering of

3224-478: Is the length along the centerline. Once a channel begins to follow a sinusoidal path, the amplitude and concavity of the loops increase dramatically. This is due to the effect of helical flow which sweeps dense eroded material towards the inside of the bend, and leaves the outside of the bend unprotected and vulnerable to accelerated erosion. This establishes a positive feedback loop . In the words of Elizabeth A. Wood: "...this process of making meanders seems to be

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3328-405: Is the most common type of fluvial lake, is a crescent-shaped lake that derives its name from its distinctive curved shape. Oxbow lakes are also known as cutoff lakes . Such lakes form regularly in undisturbed floodplains as a result of the normal process of fluvial meandering. Either a river or stream forms a sinuous channel as the outer side of its bends are eroded away and sediments accumulate on

3432-610: Is the pressure, and ν {\displaystyle \nu } is the kinematic viscosity of the fluid at a point. The approximation states that, for a sufficiently high Reynolds number the flow over a surface can be divided into an outer region of inviscid flow unaffected by viscosity (the majority of the flow), and a region close to the surface where viscosity is important (the boundary layer). Let u {\displaystyle u} and υ {\displaystyle \upsilon } be streamwise and transverse (wall normal) velocities respectively inside

3536-473: Is the tangent of the Poiseuille parabola intersecting the wall. Although Lévêque's solution was specific to heat transfer into a Poiseuille flow, his insight helped lead other scientists to an exact solution of the thermal boundary-layer problem. Schuh observed that in a boundary-layer, u is again a linear function of y , but that in this case, the wall tangent is a function of x . He expressed this with

3640-417: Is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condition (zero velocity at the wall). The flow velocity then monotonically increases above the surface until it returns to the bulk flow velocity. The thin layer consisting of fluid whose velocity has not yet returned to

3744-430: Is two consecutive loops pointing in opposite transverse directions. The distance of one meander along the down-valley axis is the meander length or wavelength . The maximum distance from the down-valley axis to the sinuous axis of a loop is the meander width or amplitude . The course at that point is the apex. In contrast to sine waves, the loops of a meandering stream are more nearly circular. The curvature varies from

3848-576: Is well known from several textbooks, heat transfer tends to decrease with the increase in the boundary layer. Recently, it was observed on a practical and large scale that wind flowing through a photovoltaic generator tends to "trap" heat in the PV panels under a turbulent regime due to the decrease in heat transfer. Despite being frequently assumed to be inherently turbulent, this accidental observation demonstrates that natural wind behaves in practice very close to an ideal fluid, at least in an observation resembling

3952-542: The Ancient Greeks as Μαίανδρος Maiandros ( Latin : Maeander ), characterised by a very convoluted path along the lower reach. As a result, even in Classical Greece (and in later Greek thought) the name of the river had become a common noun meaning anything convoluted and winding, such as decorative patterns or speech and ideas, as well as the geomorphological feature. Strabo said: ‘...its course

4056-465: The Coriolis effect (rather than convective inertia), an Ekman layer forms. In the theory of heat transfer, a thermal boundary layer occurs. A surface can have multiple types of boundary layer simultaneously. The viscous nature of airflow reduces the local velocities on a surface and is responsible for skin friction. The layer of air over the wing's surface that is slowed down or stopped by viscosity,

4160-483: The Ozark Plateau . As noted above, it was initially either argued or presumed that an incised meander is characteristic of an antecedent stream or river that had incised its channel into underlying strata . An antecedent stream or river is one that maintains its original course and pattern during incision despite the changes in underlying rock topography and rock types. However, later geologists argue that

4264-449: The bedrock are known as either incised , intrenched , entrenched , inclosed or ingrown meanders . Some Earth scientists recognize and use a finer subdivision of incised meanders. Thornbury argues that incised or inclosed meanders are synonyms that are appropriate to describe any meander incised downward into bedrock and defines enclosed or entrenched meanders as a subtype of incised meanders (inclosed meanders) characterized by

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4368-435: The absence of secondary flow we would expect low fluid velocity at the outside bend and high fluid velocity at the inside bend. This classic fluid mechanics result is irrotational vortex flow. In the context of meandering rivers, its effects are dominated by those of secondary flow. Secondary flow : A force balance exists between pressure forces pointing to the inside bend of the river and centrifugal forces pointing to

4472-411: The average fullbank channel width. The length of the stream is measured by channel, or thalweg, length over the reach, while the bottom value of the ratio is the downvalley length or air distance of the stream between two points on it defining the reach. The sinuosity index plays a part in mathematical descriptions of streams. The index may require elaboration, because the valley may meander as well—i.e.,

4576-405: The behavior of the boundary layer to minimize drag. Two effects have to be considered. First, the boundary layer adds to the effective thickness of the body, through the displacement thickness , hence increasing the pressure drag. Secondly, the shear forces at the surface of the wing create skin friction drag . At high Reynolds numbers , typical of full-sized aircraft, it is desirable to have

4680-554: The bottom from the outside to the inside. The flow is supplied by a counter-flow across the surface from the inside to the outside. This entire situation is very similar to the Tea leaf paradox . This secondary flow carries sediment from the outside of the bend to the inside making the river more meandering. As to why streams of any size become sinuous in the first place, there are a number of theories, not necessarily mutually exclusive. The stochastic theory can take many forms but one of

4784-401: The boundary layer equations, further solution u ∗ ( x , y , t ) ,   v ∗ ( x , y , t ) {\displaystyle u^{*}(x,y,t),\ v^{*}(x,y,t)} , which is also satisfying the boundary layer equations, can be constructed by writing where f ( x ) {\displaystyle f(x)}

4888-437: The boundary layer in the direction normal to the surface (such as an airfoil ) remains relatively constant throughout the boundary layer, and is the same as on the surface itself. The thickness of the velocity boundary layer is normally defined as the distance from the solid body to the point at which the viscous flow velocity is 99% of the freestream velocity (the surface velocity of an inviscid flow). Displacement thickness

4992-429: The boundary layer. Therefore, within the boundary layer, pressure force dominates and fluid moves along the bottom of the river from the outside bend to the inside bend. This initiates helicoidal flow: Along the river bed, fluid roughly follows the curve of the channel but is also forced toward the inside bend; away from the river bed, fluid also roughly follows the curve of the channel but is forced, to some extent, from

5096-415: The boundary layer. Using scale analysis , it can be shown that the above equations of motion reduce within the boundary layer to become and if the fluid is incompressible (as liquids are under standard conditions): The order of magnitude analysis assumes the streamwise length scale significantly larger than the transverse length scale inside the boundary layer. It follows that variations in properties in

5200-405: The boundary layer. Notably, the characteristic of the partial differential equations (PDE) becomes parabolic, rather than the elliptical form of the full Navier–Stokes equations. This greatly simplifies the solution of the equations. By making the boundary layer approximation, the flow is divided into an inviscid portion (which is easy to solve by a number of methods) and the boundary layer, which

5304-406: The bulk flow velocity is called the velocity boundary layer. The air next to a human is heated, resulting in gravity-induced convective airflow, which results in both a velocity and thermal boundary layer. A breeze disrupts the boundary layer, and hair and clothing protect it, making the human feel cooler or warmer. On an aircraft wing , the velocity boundary layer is the part of the flow close to

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5408-559: The case of the Anderson Bottom Rincon, incised meanders that have either steep-sided, often vertical walls, are often, but not always, known as rincons in the southwest United States . Rincon in English is a nontechnical word in the southwest United States for either a small secluded valley, an alcove or angular recess in a cliff, or a bend in a river. The meanders of a stream or river that has cut its bed down into

5512-470: The channel of a river, stream, or other watercourse is measured by its sinuosity . The sinuosity of a watercourse is the ratio of the length of the channel to the straight line down-valley distance. Streams or rivers with a single channel and sinuosities of 1.5 or more are defined as meandering streams or rivers. The term derives from the winding river Menderes located in Asia-Minor and known to

5616-428: The circumstances under which they are created. The thin shear layer which develops on an oscillating body is an example of a Stokes boundary layer , while the Blasius boundary layer refers to the well-known similarity solution near an attached flat plate held in an oncoming unidirectional flow and Falkner–Skan boundary layer , a generalization of Blasius profile. When a fluid rotates and viscous forces are balanced by

5720-408: The curvature of the bend, and the faster the flow, the stronger is the cross-current and the sweeping. Due to the conservation of angular momentum the speed on the inside of the bend is faster than on the outside. Since the flow velocity is diminished, so is the centrifugal pressure. The pressure of the super-elevated column prevails, developing an unbalanced gradient that moves water back across

5824-405: The downvalley length is not identical to the reach. In that case the valley index is the meander ratio of the valley while the channel index is the meander ratio of the channel. The channel sinuosity index is the channel length divided by the valley length and the standard sinuosity index is the channel index divided by the valley index. Distinctions may become even more subtle. Sinuosity Index has

5928-489: The edge of the boundary layer is the pressure throughout the boundary layer at a given streamwise position. The external pressure may be obtained through an application of Bernoulli's equation . Let U {\displaystyle U} be the fluid velocity outside the boundary layer, where u {\displaystyle u} and U {\displaystyle U} are both parallel. This gives upon substituting for p {\displaystyle p}

6032-407: The flow continues back from the leading edge, the laminar boundary layer increases in thickness. Turbulent boundary layer flow At some distance back from the leading edge, the smooth laminar flow breaks down and transitions to a turbulent flow. From a drag standpoint, it is advisable to have the transition from laminar to turbulent flow as far aft on the wing as possible, or have a large amount of

6136-431: The flow field into two areas: one inside the boundary layer, dominated by viscosity and creating the majority of drag experienced by the boundary body; and one outside the boundary layer, where viscosity can be neglected without significant effects on the solution. This allows a closed-form solution for the flow in both areas by making significant simplifications of the full Navier–Stokes equations . The same hypothesis

6240-451: The flow properties. One of the most widely used techniques in which turbulent flows are tackled is to apply Reynolds decomposition . Here the instantaneous flow properties are decomposed into a mean and fluctuating component with the assumption that the mean of the fluctuating component is always zero. Applying this technique to the boundary layer equations gives the full turbulent boundary layer equations not often given in literature: Using

6344-461: The flow, it is necessary to asymptotically match the solutions from both regions of the flow. Such analysis will yield either the so-called log-law or power-law . Similar approaches to the above analysis has also been applied for thermal boundary layers, using the energy equation in compressible flows. The additional term u ′ v ′ ¯ {\displaystyle {\overline {u'v'}}} in

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6448-415: The fluid, can be expressed as an incomplete gamma function . Schlichting proposed an equivalent substitution that reduces the thermal boundary-layer equation to an ordinary differential equation whose solution is the same incomplete gamma function. Analytic solutions can be derived with the time-dependent self-similar Ansatz for the incompressible boundary layer equations including heat conduction. As

6552-415: The following result For a flow in which the static pressure p {\displaystyle p} also does not change in the direction of the flow so U {\displaystyle U} remains constant. Therefore, the equation of motion simplifies to become These approximations are used in a variety of practical flow problems of scientific and engineering interest. The above analysis

6656-565: The formation of both entrenched meanders and ingrown meanders is thought to require that base level falls as a result of either relative change in mean sea level , isostatic or tectonic uplift, the breach of an ice or landslide dam, or regional tilting. Classic examples of incised meanders are associated with rivers in the Colorado Plateau , the Kentucky River Palisades in central Kentucky , and streams in

6760-428: The full force of the flood. After a cutoff meander is formed, river water flows into its end from the river builds small delta-like feature into either end of it during floods. These delta-like features block either end of the cutoff meander to form a stagnant oxbow lake that is separated from the flow of the fluvial channel and independent of the river. During floods, the flood waters deposit fine-grained sediment into

6864-404: The fullbank channel width and 3 to 5 times, with an average of 4.7 times, the radius of curvature at the apex. This radius is 2–3 times the channel width. A meander has a depth pattern as well. The cross-overs are marked by riffles , or shallow beds, while at the apices are pools. In a pool direction of flow is downward, scouring the bed material. The major volume, however, flows more slowly on

6968-416: The inner side, which forms a meandering horseshoe-shaped bend. Eventually as the result of its meandering, the fluvial channel cuts through the narrow neck of the meander and forms a cutoff meander. The final break-through of the neck, which is called a neck cutoff , often occurs during a major flood because that is when the watercourse is out of its banks and can flow directly across the neck and erode it with

7072-443: The inside bank of a river bend. On the inside bend, this sediment and debris is eventually deposited on the slip-off slope of a point bar. Scroll-bars are a result of continuous lateral migration of a meander loop that creates an asymmetrical ridge and swale topography on the inside of the bends. The topography is generally parallel to the meander, and is related to migrating bar forms and back bar chutes, which carve sediment from

7176-406: The inside of the bend where, due to decreased velocity, it deposits sediment. The line of maximum depth, or channel, is the thalweg or thalweg line. It is typically designated the borderline when rivers are used as political borders. The thalweg hugs the outer banks and returns to center over the riffles. The meander arc length is the distance along the thalweg over one meander. The river length

7280-450: The inside to the outside bend. The higher velocities at the outside bend lead to higher shear stresses and therefore result in erosion. Similarly, lower velocities at the inside bend cause lower shear stresses and deposition occurs. Thus meander bends erode at the outside bend, causing the river to becoming increasingly sinuous (until cutoff events occur). Deposition at the inside bend occurs such that for most natural meandering rivers,

7384-404: The interaction of water flowing through a curved channel with the underlying river bed. This produces helicoidal flow , in which water moves from the outer to the inner bank along the river bed, then flows back to the outer bank near the surface of the river. This in turn increases carrying capacity for sediments on the outer bank and reduces it on the inner bank, so that sediments are eroded from

7488-556: The leading order momentum equation for this "inner boundary layer" is given by: In the limit of infinite Reynolds number, the pressure gradient term can be shown to have no effect on the inner region of the turbulent boundary layer. The new "inner length scale" η {\displaystyle \eta } is a viscous length scale, and is of order ν u ∗ {\displaystyle {\frac {\nu }{u_{*}}}} , with u ∗ {\displaystyle u_{*}} being

7592-420: The meanders are fixed. Various mathematical formulae relate the variables of the meander geometry. As it turns out some numerical parameters can be established, which appear in the formulae. The waveform depends ultimately on the characteristics of the flow but the parameters are independent of it and apparently are caused by geologic factors. In general the meander length is 10–14 times, with an average 11 times,

7696-418: The more heterogeneous braided river deposits. There are two distinct patterns of scroll-bar depositions; the eddy accretion scroll bar pattern and the point-bar scroll pattern. When looking down the river valley they can be distinguished because the point-bar scroll patterns are convex and the eddy accretion scroll bar patterns are concave. Scroll bars often look lighter at the tops of the ridges and darker in

7800-504: The most general statements is that of Scheidegger: "The meander train is assumed to be the result of the stochastic fluctuations of the direction of flow due to the random presence of direction-changing obstacles in the river path." Given a flat, smooth, tilted artificial surface, rainfall runs off it in sheets, but even in that case adhesion of water to the surface and cohesion of drops produce rivulets at random. Natural surfaces are rough and erodible to different degrees. The result of all

7904-511: The near wall region. Due to the damping of the vertical velocity fluctuations near the wall, the Reynolds stress term will become negligible and we find that a linear velocity profile exists. This is only true for the very near wall region . In 1928, the French engineer André Lévêque observed that convective heat transfer in a flowing fluid is affected only by the velocity values very close to

8008-514: The outer bank and redeposited on the inner bank of the next downstream meander. When a fluid is introduced to an initially straight channel which then bends, the sidewalls induce a pressure gradient that causes the fluid to alter course and follow the bend. From here, two opposing processes occur: (1) irrotational flow and (2) secondary flow . For a river to meander, secondary flow must dominate. Irrotational flow : From Bernoulli's equations, high pressure results in low velocity. Therefore, in

8112-403: The outside bend of the river. In the context of meandering rivers, a boundary layer exists within the thin layer of fluid that interacts with the river bed. Inside that layer and following standard boundary-layer theory, the velocity of the fluid is effectively zero. Centrifugal force, which depends on velocity, is also therefore effectively zero. Pressure force, however, remains unaffected by

8216-529: The outside of the curve and deposit sediment in the slower flowing water on the inside of the loop, in a process called lateral accretion. Scroll-bar sediments are characterized by cross-bedding and a pattern of fining upward. These characteristics are a result of the dynamic river system, where larger grains are transported during high energy flood events and then gradually die down, depositing smaller material with time (Batty 2006). Deposits for meandering rivers are generally homogeneous and laterally extensive unlike

8320-415: The oxbow lake. As a result, oxbow lakes tend to become filled in with fine-grained, organic-rich sediments over time. A point bar , which is also known as a meander bar , is a fluvial bar that is formed by the slow, often episodic, addition of individual accretions of noncohesive sediment on the inside bank of a meander by the accompanying migration of the channel toward its outer bank. This process

8424-475: The physical factors acting at random is channels that are not straight, which then progressively become sinuous. Even channels that appear straight have a sinuous thalweg that leads eventually to a sinuous channel. In the equilibrium theory, meanders decrease the stream gradient until an equilibrium between the erodibility of the terrain and the transport capacity of the stream is reached. A mass of water descending must give up potential energy , which, given

8528-468: The pressure drag from flow separation and skin friction from induced turbulence. When using half-models in wind tunnels, a peniche is sometimes used to reduce or eliminate the effect of the boundary layer. The deduction of the boundary layer equations was one of the most important advances in fluid dynamics. Using an order of magnitude analysis , the well-known governing Navier–Stokes equations of viscous fluid flow can be greatly simplified within

8632-439: The rear part of the wing chord, a laminar boundary layer will tend to separate from the surface. Such flow separation causes a large increase in the pressure drag , since it greatly increases the effective size of the wing section. In these cases, it can be advantageous to deliberately trip the boundary layer into turbulence at a point prior to the location of laminar separation, using a turbulator . The fuller velocity profile of

8736-636: The river width remains nearly constant, even as the river evolves. In a speech before the Prussian Academy of Sciences in 1926, Albert Einstein suggested that because the Coriolis force of the earth can cause a small imbalance in velocity distribution, such that velocity on one bank is higher than on the other, it could trigger the erosion on one bank and deposition of sediment on the other that produces meanders However, Coriolis forces are likely insignificant compared with other forces acting to produce river meanders. The technical description of

8840-432: The same Reynolds number is achieved with a greater length. At lower Reynolds numbers , such as those seen with model aircraft, it is relatively easy to maintain laminar flow. This gives low skin friction, which is desirable. However, the same velocity profile which gives the laminar boundary layer its low skin friction also causes it to be badly affected by adverse pressure gradients . As the pressure begins to recover over

8944-460: The same thickness. If the Prandtl number is greater than 1, the thermal boundary layer is thinner than the velocity boundary layer. If the Prandtl number is less than 1, which is the case for air at standard conditions, the thermal boundary layer is thicker than the velocity boundary layer. In high-performance designs, such as gliders and commercial aircraft, much attention is paid to controlling

9048-416: The same velocity at the end of the drop as at the beginning, is removed by interaction with the material of the stream bed. The shortest distance; that is, a straight channel, results in the highest energy per unit of length, disrupting the banks more, creating more sediment and aggrading the stream. The presence of meanders allows the stream to adjust the length to an equilibrium energy per unit length in which

9152-417: The sediment is deposited the vertical sequence of sediments comprising a point bar becomes finer upward within an individual point bar. For example, it is typical for point bars to fine upward from gravel at the base to fine sands at the top. The source of the sediment is typically upstream cut banks from which sand, rocks and debris has been eroded, swept, and rolled across the bed of the river and downstream to

9256-505: The shape of an incised meander is not always, if ever, "inherited", e.g., strictly from an antecedent meandering stream where its meander pattern could freely develop on a level floodplain. Instead, they argue that as fluvial incision of bedrock proceeds, the stream course is significantly modified by variations in rock type and fractures , faults , and other geological structures into either lithologically conditioned meanders or structurally controlled meanders . The oxbow lake , which

9360-445: The shortest possible path). It is calculated as the length of the stream divided by the length of the valley . A perfectly straight river would have a meander ratio of 1 (it would be the same length as its valley), while the higher this ratio is above 1, the more the river meanders. Sinuosity indices are calculated from the map or from an aerial photograph measured over a distance called the reach , which should be at least 20 times

9464-411: The sinuous axis. A loop at the apex has an outer or concave bank and an inner or convex bank. The meander belt is defined by an average meander width measured from outer bank to outer bank instead of from centerline to centerline. If there is a flood plain , it extends beyond the meander belt. The meander is then said to be free—it can be found anywhere in the flood plain. If there is no flood plain,

9568-419: The stream carries away all the sediment that it produces. Geomorphic refers to the surface structure of the terrain. Morphotectonic means having to do with the deeper, or tectonic (plate) structure of the rock. The features included under these categories are not random and guide streams into non-random paths. They are predictable obstacles that instigate meander formation by deflecting the stream. For example,

9672-403: The stream might be guided into a fault line (morphotectonic). A cut bank is an often vertical bank or cliff that forms where the outside, concave bank of a meander cuts into the floodplain or valley wall of a river or stream. A cutbank is also known either as a river-cut cliff , river cliff , or a bluff and spelled as cutbank . Erosion that forms a cut bank occurs at the outside bank of

9776-478: The streamwise direction are generally much lower than those in the wall normal direction. Apply this to the continuity equation shows that υ {\displaystyle \upsilon } , the wall normal velocity, is small compared with u {\displaystyle u} the streamwise velocity. Since the static pressure p {\displaystyle p} is independent of y {\displaystyle y} , then pressure at

9880-480: The surface. For flows of large Prandtl number, the temperature/mass transition from surface to freestream temperature takes place across a very thin region close to the surface. Therefore, the most important fluid velocities are those inside this very thin region in which the change in velocity can be considered linear with normal distance from the surface. In this way, for when y → 0 {\displaystyle y\rightarrow 0} , then where θ

9984-416: The swales. This is because the tops can be shaped by wind, either adding fine grains or by keeping the area unvegetated, while the darkness in the swales can be attributed to silts and clays washing in during high water periods. This added sediment in addition to water that catches in the swales is in turn is a favorable environment for vegetation that will also accumulate in the swales. Depending upon whether

10088-476: The temperature of the surface. The flow velocity will then increase rapidly within the boundary layer, governed by the boundary layer equations, below. The thermal boundary layer thickness is similarly the distance from the body at which the temperature is 99% of the freestream temperature. The ratio of the two thicknesses is governed by the Prandtl number . If the Prandtl number is 1, the two boundary layers are

10192-406: The trees to fall into the river. A meander cutoff , also known as either a cutoff meander or abandoned meander , is a meander that has been abandoned by its stream after the formation of a neck cutoff. A lake that occupies a cutoff meander is known as an oxbow lake . Cutoff meanders that have cut downward into the underlying bedrock are known in general as incised cutoff meanders . As in

10296-454: The turbulent boundary layer allows it to sustain the adverse pressure gradient without separating. Thus, although the skin friction is increased, overall drag is decreased. This is the principle behind the dimpling on golf balls, as well as vortex generators on aircraft. Special wing sections have also been designed which tailor the pressure recovery so laminar separation is reduced or even eliminated. This represents an optimum compromise between

10400-502: The turbulent boundary layer equations is known as the Reynolds shear stress and is unknown a priori . The solution of the turbulent boundary layer equations therefore necessitates the use of a turbulence model , which aims to express the Reynolds shear stress in terms of known flow variables or derivatives. The lack of accuracy and generality of such models is a major obstacle in the successful prediction of turbulent flow properties in modern fluid dynamics. A constant stress layer exists in

10504-405: The velocity scale of the turbulent fluctuations, in this case a friction velocity . Unlike the laminar boundary layer equations, the presence of two regimes governed by different sets of flow scales (i.e. the inner and outer scaling) has made finding a universal similarity solution for the turbulent boundary layer difficult and controversial. To find a similarity solution that spans both regions of

10608-478: The wing surface within the laminar portion of the boundary layer. The low energy laminar flow, however, tends to break down more suddenly than the turbulent layer. The aerodynamic boundary layer was first hypothesized by Ludwig Prandtl in a paper presented on August 12, 1904, at the third International Congress of Mathematicians in Heidelberg, Germany . It simplifies the equations of fluid flow by dividing

10712-471: The wing, where viscous forces distort the surrounding non-viscous flow. In the Earth's atmosphere , the atmospheric boundary layer is the air layer (~ 1 km) near the ground. It is affected by the surface; day-night heat flows caused by the sun heating the ground, moisture, or momentum transfer to or from the surface. Laminar boundary layers can be loosely classified according to their structure and

10816-404: The x-momentum equation simplifies to: This equation does not satisfy the no-slip condition at the wall. Like Prandtl did for his boundary layer equations, a new, smaller length scale must be used to allow the viscous term to become leading order in the momentum equation. By choosing η << δ {\displaystyle \eta <<\delta } as the y -scale,

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