FLUVIAL GEOMORPH ESS 426
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LECTURE 2 FLUID MECHANICS Introduction Conservation of mass and momentum General types of ow Laminar vs turbulent ow Shear Stress Reachaverage shear stress Bed roughness and reach average ow velocity Shear stress partitioning Local shear stress Laminar velocity pro le Turbulent velocity profile Determining u and Z0 Laminar sublayer Smooth bed Rough bed Flow Energy Forms of stream energy Bernouilli equation NavierStokes Equation Derivation Simpli cations Reynolds number Froude number Hydraulic scaling Geology 412 Spring 2002 Introduction Water flowing in a channel is subject to two principal forces gravity and friction Gravity drives the flow and friction resists it The balance between these forces determines the ability of flowing water to transport and erode sediment In addition we expect mass and momentum to be conserved at cross sections 1 Z n unless mass or energy are added in between Cmsewatzon ofMass Q Alu1 AzuZ Anun 1 Cmsewatzm ofMomentum leu1 pQZuZ anun 2 note Q discharge A X sectional area u velocity so these equations are in volume terms We will use these two basic principles to derive the shear stress that acts on the channel bed and that transports sediment the velocity profile in a river and the equations governing channel flow General Types of Flow steady velocity constant with time unsteady velocity variable with time uniform velocity constant with position nonuniforrn velocity variable with position Simple mathematical models of flow in channels can be constructed only if flow is uniform and steady Although flow in natural rivers is characteristically nonuniforrn and unsteady most models rely upon the steady uniform flow assumption E55 426 21 Spring 2006 Laminar vs Turbulent Flow velocity La velocity v y gt N 395 K1 392 ix TURBULENT FLOW LAMINAR FLOW Note that water is assumed to be stuck to the boundary the noislip assumption E88 426 22 Spring 2006 Reach Average Shear Stress Natural rivers have local irregularities in bed and bank topography that introduce significant local convergence and divergence of flow that can impose large local gradients in flow velocity and shear stress We use a reach average view of channels in order to make for a solvable analytical model First consider the force balance on the volume of water in an entire reach of length L and slope e Assume that acceleration of flow in the reach is negligible and that the bed is not moving so there must be a balance between 1 the i i 1 force 1 i the water and 2 the frictional resistance to flow caused by the boundary Which slows the fluid velocity to zero at the bed and banks and therefore causes internal deformation Of the HOW ESS 426 23 Spring 2006 Within the reach the downstream component of gravitational force is AL pgsine 3 The total boundary resistance which is also a force ie stress 39 area for the reach equals rb L P 4 where Tb is the average drag force per unit area shear on the boundary Equating the force moving the ow 3 with the force resisting ow 4 since we are assuming no additional energy inputs we get IbLPALpgsin9 5 Rearranging terms and dividing by L yields Tb A p g sin 9 6 If we de ne the hydraulic radius as R E A then this simpli es to the standard expression for the reach average shear stress IbRpgsin9 7 Note that for wide channels AH D and for small 9 sin 9 tan 9 S E55 426 24 Spring 2006 Hence the reach average basal shear stress is approximated by the quotdepthslopequot product TbpgDsin6 8 The force exerted by ow on the channel bed is proportional to ow depth and slope E55 426 25 Spring 2006 Bed Roughness 639 Reach Averaged Flow Velocity Prediction of flow velocity is a fundamental problem in uvial geomorphology that is important for such problems as ood prediction and the drag force exerted on objects in the flow We39ve now established that the basal shear stress is related to the depthslope product but how do we get at flow velocity Chezy 1775 first applied mathematical analysis to the mechanics of uniform flow He made 2 assumptions 1 Exact balance between force driving flow downslope component of the weight of water and the total force of bed resistance ie the same assumption we made in writing equation 5 2 The force resisting the flow per unit bed area ie Tb varies as the square of velocity rb k 112 9 where k is a roughness coefficient Recall that we can express Driving force Weight of water x sine of bed slope pgALsinG 10 Resisting force Total bed area x bed shear stress P L rb 11 Assuming no acceleration Chezy s assumption 1 above then these forces balance and rb p g AP sine 12 E55 426 26 Spring 2006 This is the same as equation 6 Substituting equation 9 Chezy s assumption 2 yields kuz pgA sine 13 Rearranging in terms of velocity yields u2 p gk AA sine 14 Recalling that R E AP and sin 9 tan 9 S then uz PgMRS 15 and hence u c R S 05 16 Where c p g k 5 17 Equation 16 is called the Chezy equation and C is called the Chezy Coef cient Hence if both of Chezy s assumptions are correct the average velocity in a channel should increase with the square root of the gradient the square root of the hydraulic radius which for Wide shallow channels is equal to the average depth and a coefficient that re ects the smoothness of the channel ie the inverse of channel roughness E55 426 27 Spring 2006 Later empirical investigations ie simultaneous measurements of u R and S in experimental flumes indicated that C varied slightly with R in any given channel C x R16 and so a new proportionality was de ned c RW n 18 where n is the Manning roughness coefficient another empirical coefficient Substitution of this result into the Chezy equation eqn 16 produced the famous Manning Equation u k1RZ3S1Zn 19 Manning s n is a roughness coefficient that depends on channel margin irregularity and the grain size of the bed material The scaling of n has been chosen so that constant k1 1 in SI units and 149 in English units fps This has been known to cause confusion Manning s n reflects the net effect of all the factors causing flow resistance in a fluid of a given viscosity because of the temperature effect on viscosity a channel s n varies slightly throughout the year The third common roughness equation is the Darcy Weisbach equation for frictional losses in circular pipes which can be modified for open channel flow ff8gRSuz 20 E55 426 28 Spring 2006 We will omit the derivation for this equation but it too has its advocates because the Darcv Weisbach friction factor has the advantage of being dimensionless and hence the units don39t matter The three common roughness coefficients are all interrelated c ROW11 ff 8 gnZR13 ff 8 gCZ 21 E55 426 29 Spring 2006 Shear Stress Partitioning The force available for transporting sediment is that component of the basal shear stress that is not dissipated by flow roughness which can be viewed as the sum of 1 Grain or quotskinquot resistance or quotroughnessquot due to the presence of small distributed irregularities such as bed for39ming material 2 Form resistance due to the largerscale internal deformation in the flow field imposed by channel bed irregularities such as bedfor39ms eg dunes bars pools etc and by variations in the plan form of the river e g meanders 3 Spill resistance due to surface waves generated by large obstacles protruding from banks steps in the channel bed pro le or other obstacles such as logs and boulders The reach average basal shear stress Tb is often considered to be composed of linearly additive components of shear stress attributable to these different aspects of flow resistance Ib 1g be 15 I 22 Where Eg is the grain roughness be is the roughness due to bedforms 15 is the net effect of other sources of roughness eg logs and so 1y is the effective shear stress available for sediment transport Rearranging 22 yields IV Tb Tg W Is 23 E55 426 210 Spring 2006 Consider 1339 as the force left over to move stuff after accounting for the forms of roughness that impede ow through the channel Bedform roughness be can account for 10 70 of the total roughness in channels with well developed macroscale bedforms In forest landscapes roughness attributable to inchannel wood debris Is also can account for a substantial portion of the reach average basal shear stress Accounting for these various forms of roughness is a major challenge for predicting ow velocity and sediment transport and it is done in 3 typical ways 1 analogy tables books pictures 2 theory simplify and predict 3 measurement eld determinations of ow parameters to back calculate roughness You will have an opportunity to practice all three on the rst eld trip E55 426 211 Spring 2006 Local Shear Stress The View at a Point Within a Channel Reach Imagine any point within the channel at which the flow can be reasonably viewed as one dimensional and parallel to the bed three dimensional complexities add a lot of mathematics which is ignored in the following The shear stress on any surface at height 2 above the bed is caused by the downslope gravitational stress of the water above the plane ie by the downslope component of the weight of the uid between 2 and the water surface at height Hence the shear stress at any point within the fluid will be given by I p g H z sin 9 24 Equation 24 indicates that the shear stress decreases linearly with height above the bed surfac ZS H Tb bottom 1 Note also that for the case ofz 0 ie at the channel bed I Tb and so equation 24 reduces to IbpgDsin9 25 which we39ve seen before as equations 7 and 12 E55 426 212 Spring 2006 Laminar Velocity Profile Water is a viscous uid that cannot resist a shear stress however small It deforms or strains Newton found by experiment that for laminar ow I u dudz 26 where I is shear stress 11 viscosity and dudz is the strain rate Or strain rate shear stressviscosity dudz I 11 Or The more your push the faster it goesquot Combining 26 with 24 above shear stress distribution in the flow I ududz p g H z sin 9 27 Rearranging yields du pgsinGuHdz pgsinGuzdz 28 Integrating upgsmeuHzgt ltpgsineugtltzZ2gtC lt29 Combining terms and using the boundary condition that u 0 when 2 O which inspection of 29 shows implies that C 0 yields u pgsinGu HZ ZZ2 30 This equation defines the parabolic velocity Z H HOW direc on profile of laminar flow which describes the velocity in many debris flows or very close to the bed of a river dz the laminar sublayer Farther from the bed in most rivers the ow paths du of water parcels in the turbulent flow become erratic and develop into eddies in which velocity components in x y and 2 directions uctuate randomly about a mean value Bed U average for a given depth ESS 426 213 Spring 2006 Turbulent Velocig Pro le Turbulent flow mixing between adjacent layers in the flow involves transfer of momentum via large scale eddies which impart an extra quoteddy viscosityquot term 8 that can be considered analogous to momentum transfer by conventional viscosity 1 p S dudz s dudz 31 This works because typically 8 gtgt p and hence turbulent flow is slower than laminar flow at the same shear stress This is because the drag from the bed is transferred more ef ciently into the body of the flow by eddies than by viscosity alone It is extremely dif cult to determine the eddy viscosity but Prandtl proposed that the eddies would have a length scale a distance across which they could exchange momentum between layers in a unit of time that was proportional to the distance away from the solid uid boundary eddying would be suppressed near the boundary He also proposed that S depended on the velocity gradient dudz Thus he developed an expression for the eddy viscosity 8 p l2 dudz 32 where p is the density of water and l is Prandtl39s mixing length which depends on proximity to the boundary and was experimentally determined as lKz 33 where K 04 Equation 33 can be substituted back into 32 and then 31 to yield I K2 22 p dwsz 34 E55 426 214 Spring 2006 Prandtl then introduced the concept of the quotshear velocityquot u which is not really a velocity but has the dimensions of velocity ie L t It is assumed to be constant near the bed Where E was also assumed to be constant and equal to Tb 11 Tb PO395 EHSO395 35 For E Tb incorporating 35 back into 34 yields u KZ dudz 36 Rearranging 36 yields du uK dzz 37 Integrating and re arranging terms yields uuKlnzC 38 If we impose the boundary condition that u zero at some elevation 20 just above the bed then 0 uKlnzOC 39 and therefore C u K ln 2O 40 Hence 38 becomes u ultKlnz ultKlnzO 41 E55 426 215 Spring 2006 which can be simpli ed to u uglt K In 220 42 This is the quotLaw of the Wallquot ie the equation for turbulent velocity distribution away from but close to a fixed boundary such that E H tb V Flow direcTion surface 7 Z H In 2 gl Z Z0 bottom u z 20 Z 0 Bed U The quotLaw of the Wallquot predicts a logarithmic velocity profile that begins at a roughness length scale that defines the height above the bed of 20 Below this height flow is must be assumed to be laminar because it is indeterminate under our turbulent assumptions since u O at z 20 Note that K in equations 33 42 is called von Karman39s constant and 04 E88 426 216 Spring 2006 Reiterating The solution for the velocity pro le in a turbulent river assumes 1 Newton39s viscous ow law applies as modi ed in 31 to include an eddy viscosity 2 l K2 in the neighborhood of the boundary ie turbulent mixing is scaled by distance to the bed 3 E Tb is constant close to the boundary Farther from the boundary E 72 11b and perhaps at such points in the interior of the uid the eddy viscosity will depend not on the local distance from the bed 2 but rather the on total ow depth If so it will be constant across this interior ow Mathematically this is equivalent to m equation 30 ie a constant viscosity only in this case it s an eddy viscosity As a result the velocity pro le in the interior of the ow will also be parabolic see equation 30 although with a different viscosity than in the laminar sublayer E55 426 217 Spring 2006 Determining 14 and 20 Since the slope of the velocity pro le is a measure of u the shear velocity and since I luz the slope of the velocity pro le on a semilog plot can be used to measure the local shear stress particularly near the channel bed either over bedfor39ms or if the velocity pro le can be de ned suf ciently close to the bed over the grains themselves To obtain u and 20 in equation 42 measure u at various heights 2 above the bed If you take the natural logarithm of the 2 values then if the points conform to 42 they will plot as a straight line where the X axis is velocity and the yaXis is ln 2 because 42 would be written as u uK ln 2 ln 20 43 Hence u can be calculated from either the best t line through paired values of u and ln 2 data or by reading pairs of data and using the equation for the slope of a line If you plot the logarithm of ow depth on the yaxis and velocity on the X axis then the slope of the line is given by Ku 111 21 1n 22 111 112 44 Hence if you take a linear regression of ln 2 the natural logarithm of the ow depth at which each velocity measurement was made versus the ow velocity u then in the slopeintercept form of the expression y mX b the slope of that line m is given by Ku and the intercept of that line b is equal to ln 20 So 20 eh And you can calculate u as E55 426 218 Spring 2006 Because the theory tries to specify conditions only close to the solid boundary it is strictly a reasonable approximation only close to the boundary and has therefore become known as quotthe Law of the Wal quot Farther away from the bed the mixing length becomes constant at an empirically determined fraction of the total depth and the velocity pro le becomes parabolic above that depth Log and parabolic pro les predict the same velocity at 02H which is the presumed level of this transition However the difference between the computed logarithmic and upper parabolic profiles in most streams is negligible and so for many applications a logarithmic profile can be assumed throughout E55 426 219 Spring 2006 Laminar Sublayer Very close to the bed velocity is low and turbulence is suppressed so the ow is laminar Above this Hlaminar sublayerH also sometimes called the viscous sublayerquot the turbulent velocity pro le with its apparent 20 begins The thickness of the sublayer depends on the near bed shear velocity By dimensional analysis it should have a thickness proportional to llpu by experiment the generally accepted equation for the sublayer thickness is CV 116vu 45 Where V is the kinematic viscosity llp Recall that ugtxlt Tb p05 Note that V 1 x 10Z cmZs 1 centistoke or 1 x 10 6 mZs at 20 C lNTERIOR OF FLOW Laminar sublayer V 7 So What is the scale of CV for ow in a typical gravel bed river with a depth ofl m and sin 9 00057 about 005 mm but Work it out yourselfl What is the scale of CV for ow in a typical gravel bed river with a depth of 2 m and sin 9 00357 high estimates 2 001 mm What is the scale of CV for ow in a typical gravel bed river with a depth of 05 m and sin 9 00017 low estimates x 02 mm Hence the length scale onV is about the diameter of silt to ne sand grains ESS 426 220 Spring 2006 Smooth Bed If the laminar sublayer is much thicker than the size of roughness elements on the bed ks the surface is considered smooth What size of bed material would allow hydraulically smooth ow where the turbulence doesn39t interact with the bed roughness7 We can already expect that ks must be muc quot less than 116 V u INTERIOR OF FLOW Lominor sublayer HYDRAULICALLY SMOOTH FLOW Note that we can de ne a dimensionless ratio of the laminar sublayer thickness to the roughness elements on the bed This has been termed the Roughness Reynolds numberquot and for dimensional homogeneity and linear dependence of CV on V and u Re k5 u v 46 From 45 we know that this ratio must be muc quot less than 116 because ks must be muc quot less than CV for hydraulically smooth ow to occur but only experiments can determine just how much less The answer is 3 Thus for hydraulically smooth ow 3 2 k5 ugtxlt V 4 7 For hydraulically smooth ow measured Velocity pro les in the oVerrunning turbulent ow indicate an apparent 20 of Zn CV 100 48 Combining 45 and 48 yields 2 x V 9 u ie Very small 49 ESS 426 221 Spring 2006 Rough Bed If the bed roughness elements are large relative to DV ie gt sand or ne gravel then the laminar sublayer will rise and fall over the protuberances and the grains will begin contributing addition form drag in addition to ordinary surface friction INTERIOR OF FLOW lg Laminar sublgyer k5 I 39 HYDRAU LICALLY ROUGH FLOW Consequently turbulence interacts directly with the roughness elements causing 20 to be scaled by their size We know that ks must be muc quot greater than CV and thus that Regtxlt must be muc quot greater than 116 but once again experiments were required to determine just how much Nikuradse39s experiments for such quothydraulically rough owquot showed that it occurred when k5 u v 2100 50 He also anticipated that the Value of 2 would depend on k5 by further experiment 2 1ltS 30 51 Substitution of 51 into the quotLaw of the Wallquot yields u ult K In 30 zks 52 Field measurements have shown D54 to provide a reasonable measure of k5 although Whiting and Dietrich 1991 reported eld measured 26 values that were about 3 times larger than predicted by equation 51 ESS 426 222 Spring 2006 Flow Energy Precipitation over a landscape results in downslope movement of water causing erosion and energy expenditure that forms and maintains channels The frequency and magnitude of precipitation and the topographic relief onto which it falls provide the source of this potential energy For the simple case of spatiallyuniform rainfall the potential energy in a catchment is equal to the integral of the product of water mass m gravitational acceleration g and elevation z 13 f m gdz 53 Initially the total energy of the system consists of potential energy mgz Downslope movement of water converts this potential energy into kinetic energy muz 2 pressure energy ng and energy dissipated by friction and turbulence Conservation of energy implies that AB 0 and hence this dissipative system is charcterized by AB o Amgz AmuZZ Ang F 54 Where u and D are respectively the flow velocity and depth The loss of potential energy is compensated by increased flow velocity increased flow depth andor greater frictional energy dissipation Thus F Amgz Amu2 Z Ang 55 E55 426 223 Spring 2006 Combining the bed elevation z and the flow depth D into a water surface elevation allows recasting 55 as F AmgH AmuZZ 56 Assuming that change in the downstream flow velocity is small ie Amu2 2 z 0 then the rate of frictional energy dissipation is related to the fall in the water surface per unit channel length L M mg AHA 57 The frictional energy dissipation per unit channel length effectively scales the channel roughness R Noting that AHA is the water surface slope S implies that R X S In general changes in slope dominate flow depth changes Leopold et al 1964 Since channels tend to be steep in their headwaters and decrease in slope downstream this implies that channel roughness generally decreases downstream This leads to the rather counterintuitive result that steep headwater channels flow slower than their lowland counterparts For many years geologists simply asserted that steep headwater channels obviously flowed faster than their lowland counterparts In 1953 Luna Leopold showed that this conventional wisdom was incorrect by having the audacity to actually go out and measure stream velocity at many points down a channel network This effect is due to the greater roughness of steeper channels low gradient rivers can be deceptively fast E55 426 224 Spring 2006 Bernoulli Equatizm The Bernoulli equation describes the interrelation of stream slope water surface depth and ow Velocity based on conservation of energy g d cos 0 pressure neon IVI 2 eyntse r Total energy ofa unit Volume of ow 2g ujer L g furnace sloDe a vz 29 077 hiswi rf 2 2 potential energy p g h H d 11 streamline d 2 H pressure energy p g d cos 9 bed slope 5 II I kinetic energy puz 2 2 5 10quot 8 5m 0 HUMANquot K dX gtl E pgh pgdcos9pu22 58 For small slopes d x d cos 9 and thus 58 can be re expressed as E Pglh du22gl 59 The term in parentheses is the total head and flow is driven from high to low head Note that H is now a distance above the datum not the total flow depth as before H h d uZZg 60 This is the Bernoulli equation which describes conservation of energy from reach to reach Consider two reaches designated with subscript 1 and 2 The head loss between the reaches AH will be equal to H1 7 H2 and hence h1 d1u122g hz d2 rill2g AH 61 Note that the energy water surface and bed slopes are not necessarily parallel ESS 426 225 Spring 2006 Navier Stokes Equation Derivation of the full equations of uid motion Up to this point we have made implicit assumptions about the flow particularly its steady and uniform nature It is instructive however to reconstruct our derivations by starting with the full equations of fluid motion in order to remember what we ultimately must leave out and to understand where some of our most useful flow parameters actually come from The basic principle is Newton s Second Law F m a 62 This can be stated in words that the rate of change of momentum of a body is equal to the forces acting on that body or particle or in nitesimal element of material or whatever Recall that momentum is equal to mass m times velocity u and acceleration a is the first derivative of velocity with respect to time ie the rate of change Because we do not expect mass to change with time dmu dt m dudt m a 63 This becomes complicated only because we need to address both body forces gravity is the most common of these and surface forces also called tractions and because if we are being complete then we must deal with them in all 3 dimensions The notation for Newton s second law in 3 dimensions with body and surface forces called out separately expressed per unit volume is d N N N upgpV39T 64 This is Cauchy s first law and it applies to any material since we have only made the assumption that it behaves in accord with Newton s second law It says in tensor notation ie vectors in 3 dimensions indicated by the symbol over the 3D variables that the change in momentum per unit volume equals the sum of the body force gravity only no magnetic fields allowed and the surface tractions c more about them later E55 426 226 Spring 2006 We can expand and rearrange this equation slightly do N pE VpVr pg 65 We have separated the surface forces into those that apply a shear 5 and those that act isotropically p which we normally call the pressure It is de ned as where I is the notation whereby the force in question is acting on the face perpendicular to the nithn thy 3 axis and is applied in the direction parallel to the j xis With equation 66 we are stuck until the nonisotropic also called the deviatoric part can be expanded To do this we need a constitutive equation that relates strain deformation or movement of the material to the applied stress which by de nition is a force per unit area This requires experimentation Fortunately there is a large class of common materials that behave rather simply their strain gradient is proportional to the applied shear stress In 3D tensor notation this can be written as du L 0c 139 67 dx U z The proportionality constant For these materials called Newtonian fluids that constant which will vary for different substances but which is the same value in any direction and under any applied stress regime is called the viscosity u We ve done this already but we came at it then with a less explicit set of simplifying assumptions see equation 26 If we add the additional requirements that the material is incompressible and isotropic Cauchy s first law equation 65 becomes do N N p V p qu p g 68gt This is the Navier Stokes equation for incompressible isotropic Newtonian fluids E55 426 227 Spring 2006 Now what This still cannot be solved analytically So let s radically simplify things 1 Steady ow so all the 66t terms go to zero N ZD ow so all the 66y terms go to zero no crossstream variations k Uniform ow so all the 66X terms go to zero except the downstream pressure gradient KipBX otherwise this is just an exercise in staticsl These simpli cations applied to equation 68 yield two equations for the X and 2 directions zapax a 1362 69 61962 pg 70 Integrating 70 yields pgz C 71 and since the pressure equals 0 atmospheric at the water surface where z H we can de ne h as the distance down from the level 2 H p pgh 72 This is the hydrostatic equation Substitute 72 into 69 pg BhBX a 1362 73 and integrate this equation with respect to z 113 pgH z BhBX 73 or 3953 pgh tan 9 74 E55 426 228 Spring 2006 This one is pretty familiar too Note that if h is measured perpendicular to the bed instead of vertically the equation is czx pgh sin 9 as in equation 25 Finally we can add our Newtonian constitutive relationship equation 67 for our 2D flow 113 u BU62 75 and solve for u u p g sin 911 Hz 22 2 76 This is also equation 30 from our earlier discussion Note that this holds strictly for steady 2 d139mensz390nal flow This rules out turbulence Even the simple hydrostatic equation was built from these same assumptions and so strictly speaking it too applies only for nonturbulent conditions We can evaluate whether we need to worry about turbulence and we can also figure out what to do about it using two different approaches First we can just pretend that it doesn t matter and make some experimental measurements From these we find that the basic equations derived from the NavierStokes equation ie equations 72 and 74 work pretty well virtually all of the time So we ll continue to use them For the velocity distribution 76 however results are not so friendly We already found that where turbulence is important 1 p s dudz m s dudz 31 and the form of the eddy viscosity 8 leads to a logarithmic as opposed to a parabolic velocity profile wherever that viscosity depends on height above the bed E55 426 229 Spring 2006 Reynolds Number To decide if turbulent ow is likely the dimensionless Reynolds number was de ned The Reynolds number distinguishes laminar and turbulent ow on the basis of the ratio between inertial and viscous forces It is named after the Irish engineer Osborne Reynolds 184271912 who rst showed that the transition from laminar to turbulent ow generally takes place at a critical value of the Reynolds number Re Inertial Force p u D density X velocity X flow depth Viscous Force p viscosity de nes relation between applied stress and the strain rate or the resistance of the material to deformation Re puDu 77 Laminar flow Re lt 500 Viscous forces large relative to inertial forces as evidenced by little vertical mixing Transitional flow 500 lt Re lt 2000 Fully turbulent flow Re gt 2000 Inertial forces gt gt viscous forces as evidenced by chaotic streamlines Velocity components of turbulent flow consist at any point of a time average mean velocity ie fix and flucutating velocity components ie ux39 uX ix ux39 uY iiy uy39 78 uZ iiz u239 E55 426 230 Spring 2006 The mean values of ux39 uy39 and uzy are zero but their standard deviations are nonzero and scale the intensity of turbulence It It 112 uy39z uz39z 3 105 Mix 79 Froude Number The qude numbeT FT named for the English engineer William Froude 1810 1879 is important because it is the ratio between the velocity of stream ow u and that of a shallow gravity wave gdo395 or the ratio of inertial forces to gravity forces as simpli ed as follows Pr u gd0395 80 Think of the Froude number as a measure of whether flow can outrun its own wake Subcn39tical ow F1r lt 1 Flow is tranquil and the wave speed exceeds the flow velocity so that ripples on the water surface are able to travel upstream SupeTcmical ow F1r gt 1 Flow is rapid and gravity waves cannot migrate upstream Surface waves are unstable and may break which greatly increases resistance to flow The Reynolds and Froude numbers are independent of the scale of the river and hence provide dimensionless ways to characterize flow They also have distinct physical manifestations in the behavior of flow in a channel E55 426 231 Spring 2006