6

Turbulent velocity profiles

Most sediment-laden flows are characterized by irregular velocity fluctua­tions indicating turbulence. The turbulent fluctuation superimposed on the principal motion is complex and remains difficult to treat mathemat­ically. This chapter outlines the fundamentals of turbulence with empha­sis on turbulent velocity profiles (Section 6.1), turbulent flow along rough and smooth boundaries (Sections 6.2 and 6.3), departure from logarith­mic velocity profiles (Section 6.4), and open-channel flow measurements (Section 6.5).

In describing flow in mathematical terms, it is convenient to separate the mean motion (notation with overbar) from the fluctuation (notation with superscript +) as sketched in Figure 6.1. Denoting a fluctuating pa­rameter Dx of time-averaged value iix and fluctuation vj", the pressure and the velocity components can be rewritten as

A Yz

T Vz

p = p+p+

l..._ _____ _ Time

Figure 6.1. Time velocity measurements

91

(6:!a>

(6.lb)

92 Turbulent velocity profi/es

Vy = Dy+ v/

Vz = Dz + vt The time-averaged values ata fixed point in space are given by

(6.lc)

í6.ld)

(6.2)

Taking the mean values overa sufficiently long time interval ti> the time­averaged values of the fluctuations equal zero; thus, uf = vJ == vt = p+ =O.

Likewise, the time-averaged values of the derivatives of velocity fluctu­ations, such as ovjlox, o2vifox2, oDxvflox2, also vanish, owing to Equa­tion (6.2). The quadratic terms arising from the products of cross·velocity fluctuations such as vjvf, vjv/, ovfv/lox, however, do ::lot vanish. The overbar of simple time-averaged parameters is omitted for notational convenience.

It is seen that both the time-averaged velocity components and the fluc­tuating components satisfy the equation of continuity. Thus, for incom­pressible fluids,

OVx + OVy + OVz = O ax ay az (6.3a)

(6.3b)

The velocity and pressure terms from Equation (6.1) are substituted into the Navier-Stokes equations (Table 5.1) to give the following accel­eration terms:

OVx iJv, iJv, iJv, 1 iJp 2 [ªv:v: ov;:v: iJv,+v: l (6.4a) -+v,-+v -+v,-=g,---+vmV v,- --+---·+--¡¡¡ iJx Y iJy iJz Pm iJx iJx iJy iJz ..

avy avy avy avy 1 ap 2

[av:v: év_;v} "hv:vr¡· -+vx-+Vy-+v,-=gy---+vmV v,- ---+---+--- (6.4b) iJt iJx iJy iJz Pm iJy . íJx iJy iJ¡; .

iJv, iJv, iJv, iJu, 1 iJp 2 [ªv:u: a~:v: a~;'v:] -+v,-+v,-+v,-=g,---+vmV u,---+--·+---¡¡¡ iJx · iJy élz Pm élz élx ély Jz -,- ! '-,--J '-,--J '----v---__J

(6.4c)

local convectlveo arav- pressure \'iscous turbulenl fluctu1tions lla- gradlent

tlonal

In addition to the terms found in the Navier-Stokes ec¡uatior1s, three cross-products of velocity fluctuations are obtained from the co n.•1ective acceleration terms on the left-hand side of Equation (6.4). These turbulent

Logarithmic velocity pro.files 93

acceleration terms provide additional stresses called Reynolds stresses, which are usually added to the right-hand side of Equation (6.4). Gener­ally speaking, these turbulent acceleration terms far outweigh the viscous components in turbulent flow.

6.1 Logarithmic velocity profiles

Since the fluid <loes not slip at solid boundaries, ali turbulent components must vanish at the walls and remain very small in their immediate neigh­borhood. It follows that, near the boundary, all turbulent acceleration terms in Equation (6.4) become smaller than the viscous acceleration terms of the Navier-Stokes equations. In turbulent flows, laminar motion must therefore persist in a very thin layer next to the boundary. This is known as the laminar sublayer.

Consider a thin flat plate set parallel to the main flow direction x. We are interested in describing the time-averaged velocity profile Vx as a func­tion of the distance z away from the plate. Drawing an analogy with the mean free path in the kinetic theory of gases, Prandtl imagined the mixing­length concept, which implies that the transverse velocity fluctuation vt is of the same order of magnitude as vi. He hypothesized that the aver­age of the absolute value of velocity fluctuations is proportional to the velocity gradient in the form

- - dvx lvfl- lvtl- lm dz (6.5)

in which the proportionality constant lm denotes the Prandtl mixing length. The average products of velocity fluctuations were then formu­lated in terms of the mixing length with the aid of equation (6.5):

(6.6a)

v+v+ - -12 -2'.. (dv )2

X Z m dz (6.6b)

The corresponding shear stress of a mixture can then be written as

dvx 2 (dvx)2

Tzx = llm dz +Pmlm dz (6.7)

~ '------v-----J vlscous turbulent

In turbulent flows, the turbulent shear stress far outweighs the viscous shear stress. The converse is true in the laminar sublayer. The turbulent

94 Turbulent velocity projiles

shear stress can be written alternatively as a function of the Boussinesq eddy viscosity Em:

(6.8)

It is further assumed that the mixing Iength lm is proportional to the dis­tance z from the boundary,

(6.9)

in which K is the von Kármán constant (K::::: 0.4). The shear stress Tzx in the region clase to the wall remains virtually

equal to the boundary shear stress r0 =pu;; thus, for the turbulent region near the boundary one obtains the following equation for the shear ve­locity u. from Equations (6.8) and (6.9) when Tzx =To= Pmu;:

g =U* = KZ( ~;) (6.10)

Since u. is constant, the variables v_, and z can be separated and inte­grated to yield the logarithmic average velocity distribution for steady turbulent flow near a flat boundary,

Vx J - = - lnz+c0 U* K

(6.11)

in which c0 is an integration constant evaluated at a distance z0 from the flat boundary, where the logarithmic velocity Vxo hypothetically equals zero. Hence,

.!2.=!ln~ U* K Zo

(6.12)

Two types of boundary conditions are recognized depending on the relative magnitude of the grain size d, and the laminar sublayer thickness 5, examined in Section 6.3. Conceptually, the boundary is said to be hydraulically smooth when 5 >> d5 and, conversely, hydraulically rough when d5 >> 5 (Section 6.2). A transition zone is also recognized, as shown in Figure 6.2.

6.2 Rough plane boundary

Consider steady turbulent flow over a plane surface of coarse salid par­ticles of grain roughncss height k;. The flow is described as turbulent over

Rough plane boundary

Re.<4

Smooth

fü3 < d1 <68 4 <Re.< 70

Transition

d1 >68

Re.> 70

Rough

Figure 6.2. Hydraulically smooth and rough boundaries

95

a rough boundary when k; >> li. Gravel-bed and cobble-bed streams are considered hydraulically rough. Early experiments in pipes indicated that in such a case the distance z0 = k;/30 and the corresponding velocity pro­file is

vt 2.3 ( z) (30z) u~ = -K- log k; + 8.5 = 5. 75 log k; (6.13) •

which yields the depth-averaged velocity Vx for wide rectangular channels after integration of Equation (6.13) over the entire flow depth h:

Vx = ~ log(~)+6.25 = 5.75log(12·:h)

u,. K ks k 5

{6.14) •

Note that the integration of Equation (6.13) strlctly yields an integration constant of 6.0 in Equation (6.14), while the given value 6.25 is commonly referenced.

The total resistance to flow can be described in terms of the Chézy co­efficient C, the Darcy-Weisbach friction factor f, or the Manning coeffi­cient n. The following identity between these three factors has been es­tablished:

l.49Rll6 . E 1· h . = (m .ng 1s umts) n

(6.15a) +

where R 11 is the hydraulic radius and g the gravitational acceleration. The fundamental dimensions are as follows: C is in L 112!T, f is dimension­less, and nis in T!L 113

•

96 Turbu/ent velocity pro.files

In plane bed channels, the total resistance is composed solely of grain resistance (indicated by a prime), and the corresponding grain resistance parameters are (l) the grain Chézy coefficient C'; (2) the Darcy-Weisbach grain friction factor j'; and (3) the grain Manning coefficicnt n'. The iden­tity between these grain resistance parameters is

l.49Rl(6 . E l' h . = (m ng 1s umts) n'

(6.15b)

In plane bed channels, the total resistance equals the grain resistance, and To= TÓ, u.= u~. C= C', f =f', and n = n'.

For plane bed channels, depth-averaged velocity relationships [e.g., Eq. (6.14)] directly express grain resistance to flow in terms of the Darcy­Weisbach grain friction factor f' through the following iJentities:

ar

I - f' v2 To= g-Pm x (6.l6a)

(6.l6b)

The carresponding depth-averaged velacity relatianships and tl::eir range af applicability are campiled in Table 6.1. Far sand-bed channel1;, Kamp­huis (1974) recammended k: = 2d90 • Far gravel-bed rivers, Bray (1982) found k; = 3.ld90 ; k; = 3.5d84 ; k; = 5.2d65 ; k; = 6.8d50 . The relat~onship k; = 3d90 appears to be a reasanably good approxinatian.

The resulting Darcy-Weisbach grain friction factor f' and í he grain Chézy coefficient C' are cammonly approximated by

/si 2.3 ( 12.2Rh) I 4h) C' =-\} f' = -K-y'glag 3d90 = 5.75yglog(,d90 (6.17)

It follows that grain resistance (/', C', or n') far turbulent flo-.v overa rough baundary can be obtained from (1) the depth-averaged velocity Vx

and grain shear stress TÓ from Equatian (6.16a); (2) the gram shear ve­locity u~ and the depth-averaged velocity Vx fram Equation (6.16b); or (3) the flow depth h and the grain size d90 fram Equation (6.17).

The grain resistance equation in lagarithmic forro can be transformed into an equivalent power form in which the exponent b varies with rela­tive submergence hlds,

Rough plane boundary 97

Table 6.1. Grain resistance and velocity formulationsfor turbulent jlow over hydraulically rough plane boundaries (C = C' and f = f')

Formulation Range Resistance parameter

Chézy hld,-+ 00 C = ~ constant

Manning hld,> 100 C :: a .....!!. ;;= _h_ (SI) ( R )''6 R110 d, n

Logarithmic ..f..= f! = 5 75 log(l2.2Rh) Vg ~7 . k;

Velocity•

V= !R~1is:12 (SI) n

n = 0.062dJ¡j6 (dio in m) n = 0.046dJ{6 (dn in m) n = 0.03Sd.;¿6 (d90 in m)

( 12.ÍRh) ~ V= 5.75log---¡;- vgRhSr

k; = 3d90 k; =: 3.5d84

k; = 5.2d6l k;;;= 6.Sdso

ª The hydraulic radius Rh = AIP is used, where A is the cross-sectional area and Pis the wetted perimeter; the friction slope Sr is the slope of the energy grade line.

1 ~ = a - e a In -lo ( h )b A [ bh] '\J f' d5 d5

(6.18)

/

under the transformation that imposes the value and the first derivative to be identical:

and

A (d )b a=% hs

b= 1 ln(hhld5)

(6.19a)

(6.l9b)

The values of the exponent b are plotted in Figure 6.3 as a function of relative submergence h/d5 • It is shown that b gradually decreases to zero as h/d5 --+ oo, which implies that the Darcy-Weisbach grain friction factor f' and the grain Chézy coefficient C' are constant for very large values of hld5 • At values of hld5 > 100, the exponent bis roughly compa­rable to 116, which corresponds to the Manning-Strickler approximation (n - di 16 ). At lower values of the relative submergence hld5 < 100, the

98 Turbulent velocity pro.files

1.0

._, ___ T!!''~··.!> = "'

Chézy, b •O o

2 5 10 hld,

20 50 100

Figure 6.3. Exponent b of the grain resistance equation, h = 12.2

exponent b of the power form varies with hlds and the logarithmic for­mulation is preferred.

6.3 Smooth plane boundary

Generally speaking, aplane bed surface is hydraulically smooth for grain sizes finer than medium sand (ds < 0.25 mm or d.< 5), as the grain rough­ness height k; becomes very small compared with the laminar sublayer thickness. For turbulent flows overa smooth boundary, the distance z0 is proportional to the ratio vmlu., and experiments show that z0 = vm19u. in smooth pipes. Substituting into the velocity profile relationship [Eq. (6.12)], one obtains

- = 5.75log - +5.5 Vx (u•z) U* Vm

(6.20) •

This relationship (plotted in Fig. 6.4) is valid for steady turbulent flow near a smooth plane boundary.

Smooth p/ane boundary

Laminar Buffer zone--o.i.---lTurbulent zona 30 .__~zo~n~ª---+-------1------+-~'-'----1

~ 20L------J.. ___ __,.__-L-____ c..._-J.. ____ ~ u.

99

O'--~~~~-'-"'--..;.~~~..._~~~~-'-~~~~--'

1 10 102 103

104

u.z Tni'

Figure 6.4. Velocity profiles for the law of the wall

The depth-averaged velocity Vx is then obtained from the integration of Equation (6.20) over the entire flow depth h. By definition of the Darcy­Weisbach grain friction factor/', from ró = (f'l8)pmV} the mean flow velocity Vx is given by

fs Vx (u"h) '\f F =U. = 5. 75 log --;;; + 3.25 (6.21) •

A closer look at Equation (6.7) given Equation (6.9) indicates that, in ali turbulent flows, a thin !ayer must exist very clase to the boundary where the viscous shear stress overcomes the turbulent shear stress, be/ cause lm-+ O as z-+ O. This indicates that the flow becomes laminar in a !ayer of thickness ó adjacent to the wall called the laminar sub/ayer. Con­sidering a thin laminar sublayer of thickness ó where the outermost ve­locity is Vxó• the boundary shear stress To= Pmu; is, from Equation (5.1), given as:

(6.22a)

The corresponding di'llensionless velocity from Equation (6.22a) is

(6.22b)

Because the velocity profile is continuous, the velocity Vx = Vxó at the innermost point of the turbulent velocity profile over a smooth boundary [Eq. (6.20)] must equal the velocity Vxó at the outermost point of the lam­inar sublayer [Eq. (6.22b)]. The thickness ó of the laminar sublayer can

100 Turbulent ve/ocity pro.files

therefore be determined by simultaneously solving Equations (6.22b) ar.d (6.20) for ó; hence,

ó = ll.6vm u.

(6.23) •

Consider a grain shear Reynolds number Re. defined as the product of shear velocity and grain size over the kinematic viscosity of the mix­ture "m• Re. = u.dsl"m· Turbulent ftows are called hydraulically smooth as long as the height of the boundary roughness character::zed by the :>edi­ment size ds remains much smaller than the laminar sublayer thickness [3ds < ó, which from Eq. (6.23) corresponds to Re*= u*d5 /vm < 4]. Like­wise, turbulent ftows are called hydraulically rough when the gra~n dze d5

far exceeds the laminar sublayer thickness (d5 > 6ó, or Re. = u. d5 1vm > 70). A transition zone exists where ó/3 < d5 < 6ó, or 4 < Re~ < 70. Ex­ample 6.1 details typical calculation procedures for turbt:lent ftows.

Example 6.1 Application to a turbulent velocity profile. Consider the given measured velocity profile for steady uniform turbu1ent ftcw in a wide rectangular channel, Rh = h (Fig. E6.1.1). Consider two points 1 and 2 near the bed,

3.0

2.0

u. Z¡ V¡=-ln­

K k;

1.0 lt/s

u = 0.85 lt/s Z2• 1.St-----------+

1.0

o ---.,,. o 0.5 1.0

Velocity (lt/s)

Figure E6.l.1. Measured velocity profile

Smooth plane boundory

and estimate the following parameters:

(a) Shear velocity:

u. = K(V2 - V1) = 0.4(0.85 -0.55) = O.ll Q_ ln(z2/z1) ln(l.5/0.5) s

(b) Boundary shear stress:

= 2 = 1.92 slugs (0 11)2 ft2 =O 023 ~ ro PmU• ft3 . s2 . ft3

(e) Laminar sublayer thickness:

Ó _ 11.6vm _ 11.6X1X10-s ft2s _

0 001 f _

0 32 ---- - . t- . mm u. s x0.11 ft

101

The flow is hydraulically smooth if the bed material d5 is finer than about 0.1 mm or hydraulically rough if d5 > 1.8 mm; the transition zone roughly corresponds to sand fractions 0.1 < d5 < 1.8 mm.

(d) Mean flow velocity:

V: 0.85 ft/s

(e) Froude number:

V V 0.85 ft/s Fr=--=--= =0.078

./iR;. ..fiii .../32.2 X 3. 7 ft2/s2

(f) Friction slope:

Sr=~= _!E_= 0.023 lb X ft3

= 9.96X10-s 'YmRh 'Ymh ft2 X 62.4 lb X 3. 7 ft

(g) Darcy-Weisbach factor:

f = 8Sr = 8 X 9.96 X 10-5

=o 13 Fr2 0.0782 .

(h) Manning coefficient:

n= l.i9 Rfi13SV2= ~::~(3.7)2/3(9.96x10-s)112=0.Ó42 ft~/3

(i) Chézy coefficient:

102 Turbulent velocity pro.files

e= /si" = ~ = 44.5 ft112

"1 f "1 ü.13 s

(j) Momentum correction factor [Eq. (E3.5.l)]:

{3 _1¡2 _1"2 m - AV2 VxdA = hV 2 ~Vx¡dh¡

X A X 1

2 f 3 s 2 2 2 2 t_ X 2 2

(0.55 +0.85 +l.0 +(l.l X0.7))-2

-l.074 (0.85) ft s

(k) Energy correction factor [Eq. (E3.6.2)]:

_lf3 _l"3 O:e - AV3 Vx dA = hV3 ~ Vxi dh;

X A X 1

l a""--e- 3.7ft

3 f 4 s 3 3 3 3 t_ X 3 3

(0.55 +0.85 +l.0 +(l.l X0.7))-3

-1.194 (0.85) ft s

6.4 Deviation from logarithmic velocity profiles

Two types of deviation to the logarithmic velocity profiles are considered: beyond the law of the wall (Section 6.4.l) and in narrow channels (Sec­tion 6.4.2).

6.4.1 Wake ftow function

Departure from logarithmic velocity profiles is observed as the distance from the boundary increases (see dashed line u.zlvm > 1,000 in Fig. 6.4). The reason for this is essentially related to the invalidity of the following assumptions: (l) constant shear stress throughout the fluid and (2) mixing length approximation lm = KZ.

A more complete description of the velocity distribution Vx, including the law of the wake for steady turbulent open-channel ftow, has been sug­gested by Coles (1956):

Deviation from logarithmic velocity pro.files

Vx -[2.3 I (U•Z) 5 5] llvx + 2Ilw . 2(1l'Z) ---og--+. --- --sm -U* K Pm U• K 2h

'--.,---) '-----.,--J

l•w of lh• w1U rouahness wake ftow functlon functlon

where h is the total ftow depth.

103

(6.24)

The terms in brackets depict the original logarithmic law of the wall for smooth boundaries from Equation (6.20). The last two terms have been added to describe the entire boundary layer velocity profile out­side of the thin laminar sublayer. The term !lvxlu. is the channel rough­ness velocity reduction function. The last term describes the velocity in­crease in the wake region as described by the wake strength coefficient

Ilw• The wake ftow function equals zero near the boundary and increases

gradually toward 2IIwlK at the upper surface (Z = h). With Vx = Vxm at z = h, the upper limit of the velocity profile is

Vxm 2.3 l (u•h) llvx 2Ilw -=- og - +5.5--+--u. K Pm U* K

(6.25)

The velocity defect law obtained after subtracting Equation (6.24) from Equation (6.25) gives

_Vx_m_-_v_x = [-2_II_w -[-2._3 log ~])- _2II_w sin2(_11'_Z) U* K K h J K 2h

(6.26).

/

In this form, the term in brackets is the original velocity defect equa-tion for the logarithmic law. The wake ftow term vanishes as z approaches zero, and the velocity defect asymptotically reaches the term in braces in Equation (6.26) as z/h diminishes. This means that the van Kármán con­stant K must be defined from the slope of the logarithmic part in the lower portian of the velocity profile. The wake strength coefficient Ilw is then determined by projecting a straight line, fit in the lower portian of the velocity profile, to z/h = 1 and calculating Ilw from

Ilw=- atzlh=l K [Vxm-Vx] 2 u.

(6.27)

This procedure, illustrated in Figure 6.5, generally shows that Ilw increases with sediment concentration, while the van Kármán K remains constant around 0.4.

104 Turbulent velocity pro.files

41--~~~~~--l~--'~~~~__,,.,.-,

2nw '"lC

o~~~~~~--~~~~-----_J_ 10·2 10·1

Zlh

Figure 6.5. Evaluation of K and Ilw from the velocity defect law

6.4.2 Sidewall correction method

Consider steady uniform ftow in a narrow open channel &.t a ¿¡se harge Q measured from a calibrated orifice and friction slope Sr. Ir, smooth-waHed Iaboratory ftumes where the ftume width W is less than five times tlli! ftcw depth h, the sidewall resistance is different from the bed resista:Jce. T3e Vanoni-Brooks correction method can be applied to determine the b•!d shear stress Tb = PmU;b· For a rectangular channel, the hydrauLc radius Rh = Wh/W + 2h and the Reynolds number Re= 4VRhlv,, are cz.lculated given the average velocity V= Q/Wh. The shear velocity u.= ./gRhS"¡ computed from the slope Sr is then used to calculate the Da;:-cy-Weisbach friction factor f =Su; IV 2• The wall friction factor fw for turbulent ftow over a smooth boundary, 105 < Relf < 108

, can be calculated from

Íw = 0.0026[1og( ~e) r-0.0428 log( ~e)+ 0.1884 (6.28a)

The bed friction factor f b is then obtained from

(6.28b)

The hydraulic radius related to the bed Rb = Rhfblf is then used to calculate the bed shear stress Tb from Tb = 'YmRbSr. Example 6.2 provides the details of the calculation procedure.

Deviation from logarithmic velocity profiles 105

Example 6.2 Application of the sidewall correction method. Consider a discharge Q = 1.2 ft3/s in a 4-ft-wide flume inclined at a 0.001 slope. Calculate the bed shear stress rb given the normal flow depth of 0.27 ft. The measured water temperature is 70°F.

Step 1

Q = 1.2 ft3/s, S0 =Sr= 0.001, hn = 0.27 ft,

Step 2. Hydraulic radius:

R _ Whn = 4 ft X 0.27 ft = 0.238 ft h- W+2hn (4+2x0.27)ft

Step 3. Flow velocity:

V= _g_ = 1.2 ft3 = 1.11 !!_ Whn s X 4 ft X 0.27 ft s

Step 4. Reynolds number:

4VR Re= __ h = 1.06x105

l'm

Step 5. Shear velocity:

u. = .JgRhSr = .,/32.2 ft/s 2 X 0.238 ft X 0.001 = 0.087 ft/s

Step 6. Darcy-Weisbach factor:

f = su: = 8(0.087)2

ft 2 52 = 0.049

V 2 (1.11)2 ft 2 s2

Step 7. Wall f riction factor:

[ Re] 2 [ Re] fw = 0:0026 log f -0.0428 log f +0.1884

fw = 0.021

Step 8. Bed friction factor:

f 2hn

b=/+ W (f-fw)

Íb = 0.049 + 2 X º·27 ft (0.049-0.021) = 0.0527

4 ft

Step 9. Bed hydraulic radius:

106 Turbulent velocity pro.files

R = Íb R = 0.0527 X 0.238 ft = 0.255 ft

b f h 0.049

Step JO. Bed shear stress:

62.3 lb lb Tb = 'YmRbSr = 3 X 0.255 ft X 0.001 = 0.016 - 2 ft ft

6.5 Open-channel ftow measurements

Open-channel tlow measurements normally include stage and tlow veloc­ity measurements.

6.5.1 Stage measurements

Stage measurements determine water surface elevation with reference to a datum such as the mean sea leve), a local datum related to project activ­ity, a reference elevation plane, a benchmark, or an arbitrary datum be­low the elevation of zero tlow. Simple nonrecording gages require fre­quent readings to develop continuous water leve! records. Nonrecording gages are either directly read or provide measurements of the water sur­face elevation at a fixed point.

Stalf gages are usually vertical boards or rods precisely graduated with reference to a datum.

Point gages consist of mechanical devices to locate and measure the water surface elevation. Measurements can be taken from a graduated rod, drum, or steel tape housed in a small box mounted on a rigid struc­ture (such as a bridge) directly above the water surface.

Float gages are used primarily with an analog water stage recorder. The gage consists of a tloat and counterweight connected by a graduated steel tape, which passes over a pulley assembly. A relatively large tloat and counterweight are required for stability, sensitivity, and accuracy - such as a 10-in. copper float anda 2-lb lead counterweight.

Pressure-type gages use water pressure transmitted through a tube to a manometer inside a gage shelter to measure stage. Stage can also be mea­sured by gas bubbling freely into a stream from a submerged tube set ata fixed elevation; the gage pressure in the tube equals the piezometric head at the open end of the tube.

Crest-stage gages measure maximum flood stage from granulated cork floats as the water rises in the pipe. When the water recedes, the cork ad­heres to the pipe, marking the crest stage.

Open-channel jfow measurements 107

A recording stage gage produces a punched, printed, traced, analog or digital record of water surface elevation with respect to time. The gage height recording is usually activated by either a float mechanism or a pressure-sensing device. The strip-chart records show an uninterrupted recording of water-level fluctuations with time. Digital recorders store, punch, or print out gage heights at preselected time intervals.

Analog recorders provide a continuous visual record of stage useful for graphical presentation.

Digital recorders provide data in digitally coded form suitable for digi­tal computer processing. Because digital systems record gage height only at preselected time intervals, maximum and mínimum peak gage heights of a flashy stream cannot be accurately measured.

Telemetering systems using telephone, radio, or satellite communica­tion are desirable when current information on stage is frequently needed from remote locations. Sorne telemetering sysems continuously indicate or record stage at a given site; others report instantaneous gage readings on request.

6.5.2 Velocity measurements

Velocity is measured with such devices as floats, drag bodies, tracers, rotating-element current meters, and deflection vanes, and by means of optical, laser, ultrasonic, and velocity-head methods.

Rotating current meters are based on the proportionality between the angular velocity of the rotation device and the flow velocity_.,.By counting the number of revolutions of the rotor in a measured time interval, point velocity is determined. A vertical-axis current meter measures the differ­ential drag on two sides of cups in relative motion in a fluid. The rotation speed of these devices, such as Price current meters, is calibrated against the fluid velocity. Horizontal-axis current meters act as propellers in a moving fluid. Common horizontal-axis meters include the Ott and the Neyrpic current meters.

Mechanical meters are limited in both high and low velocities; electro­magnetic meters are less so. They are easier to use, have no moving parts, are generally more accurate, indicate both velocity and direction, and pro­vide electronic readout with averaging.

Acoustic (ultrasonic) velocity meters measure velocity by determining the travel time of sound pulses moving in both directions along a diagonal path between transducers mounted near each bank of the stream. The water velocity is the average velocity component parallel to the acoustic

108 Turbulent velocity profiles

path line between the two transducers. Acoustic velocity meter systems operate satisfactorily in the laboratory, but they are usually complex and expensive.

Ultrasonic Doppler velocimeters measure the phase shift between the signa! emitted along the upstream path and the scattered signa! received in the opposite direction.

Hot film and hot wire anemometers are electrically heated sensor3 being cooled by advection. The heat loss being a function of the flow velocity, this laboratory instrument is calibrated to measure fiuctuating ve'.ocities with high spatial resolution and high-frequency response.

Laser Doppler anemometers measure the Doppler frequer..cy shift of light-scattering particles moving with the fluid. The frequency shift pro­vides very accurate flow velocity measurements in the laboratory without fiow disturbance.

Electromagnetic flow meters are based on Faraday's induction law stat­ing that voltage is induced by the motion of a conductor (fluid) perpen­dicular to a magnetic field.

The depth-averaged velocity is normally obtained from a measured ve­locity profile. The following approximate methods for turbulent flows can be used to determine the depth-averaged fiow velocity from point velocity measurements at one, two, or three points:

1. The one-point method (at 6007o of the total depth measured down from the water surface) uses the observed velocity at 0.6 has the mean velocity in the vertical. lt gives reliable results in uniform cross sections without large irregularities.

2. The two-point method (at 200Jo and 80% of the total depth mea­sured down from the water surface) averages the veiocities at the two depths, and the average fairly approximates the mean veloc­ity along the vertical.

3. The three point method (at 200Jo, 60%, and ~OOfo of the tot~l depth measured down from the water surface) combilíles the one-point and two-point methods. Velocities at 0.2 and 0.8 h are av~raged, and the value is then averaged with the 0.6 dep·ch vdocity mea­surement to obtain fairly accurate values of depth-aven .. ged fiow velocity.

4. The surface method, which has only limited use, assurnes a co­efficient (usually about O. 85) to convert the surface velo~ity mea­sured with a fioat to the depth-averaged velocity. Th:s mctt.oc is not very accurate.

Prob/ems 109

Exercises

*6.1. Substitute Equations (6.la-d) into the Navier-Stokes equations (Table 5.1) to obtain Equation (6.4a).

*6.2. Demonstrate that Equation (6.13) is obtained from Equation (6.12) when zo = k~/30.

*6.3. Demonstrate that Equation (6.20) is obtained from Equation (6.12) when Zo = vm19u •.

*6.4. Derive Equation (6.23) from Equations (6.22b) and (6.20) at z=ó.

**Problem 6.1

Consider the clear-water and sediment-laden velocity profiles measured in a smooth laboratory flume ata constant discharge by Coleman (1986). Notice the changes in the velocity profiles dueto the presence of sediments. Determine the von Kármán constant K from Equation (6.12) for the two velocity profiles in the following tabulation, given u. = 0.041 mis, d5 = 0.105 mm, Q = 0.064 m3/s, h =: 0.17 m, Sr= 0.002, and W = 0.356 m.

Sediment-Clear-water la den

Elevationº flow velocity velocity Concentration (mm) (m/s) (mis) by volume

6 0.709 0.576 2.1X10-2

12 0.773 0.649 1.2 X 10-2 ,/

18 0.823 0.743 7.7x10-1

24 0.849 0.798 5.9 X 10-3

30 0.884 0.838 4.8 X 10-3

46 0.927 0.916 3.2x 10-1

69 0.981 0.976 2.5 X 10-3

91 1.026 1.047 1.6 X 10-3

122 1.054 1.07 8.0x 10-• 137 1.053 1.07 152 1.048 1.057 162 1.039 1.048

• Elevation above the bed.

Answer: The von Kármán constant K remains close to 0.4 when the lowest portion of both velocity profiles is considered. When the main portion of the velocity profiles is considered, K becomes significantly smaller for sediment-laden flow.

11 O Turbulent velocity pro.files

*Problem 6.2

(a) In turbulent ftows, determine the elevation at which the local ve­locity Vx is equal to the depth-averaged velocity Vx. (Hin!: Vx = ll(h-zo) fz~ Vx dz.)

(b) Determine the elevation at which the local velocity Vx equals the shear velocity u*.

**Problem 6.3

Determine the Darcy-Weisbach friction factor f from the data in Prob­lem 6.1.

Answer

f = ~; = 0.012

**Problem 6.4

(a) Calculate the laminar sublayer thickness o in Problem 6.1. (b) Estimate the range of laminar sublayer thicknesses for bed slopes

1x10-5 < S0 < 0.01 and ftow depths 0.5 m < h < 5 m.

**Problem 6.5

From turbulent velocity measurements at two elevations (v1 at z1 and v2

at z2 ) in a wide rectangular channel, use Equation (6.12) to determine the shear velocity u*; the boundary shear stress r 0 ; and the laminar sublayer thickness o.

Answer

Problems 111

* Problem 6.6

(a) With reference to Problem 6.1, evaluate the parameters K and Ilw from the velocity defect formulation in Equation (6.26). Com­pare the value of K with the value obtained previously (Problem 6.1) from Equation (6.12).

(b) Compare the experimental velocity profiles from Problem 6.1 with the velocity profiles calculated from Equations (6.13) and (6.20).

**Problem 6.7

For the velocity profile given in Problem 6.1, calculate the depth-averaged velocity from (a) the velocity profile; (b) the one-point method; (e) the two­point method; (d) the three-point method; and (e) the surface method.