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Energy Loss for Fully Developed Turbulent Boundary Layer Flow

작성자 Uploader : ezcivil 작성일 Upload Date: 2017-02-06변경일 Update Date: 2017-02-16조회수 View : 675

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Energy Loss for Fully Developed Turbulent Boundary Layer Flow

a. General

Methods for determining the energy loss related to boundary roughness (friction) have been developed by various investigators. The most notable and widely used methods are the Darcy-Weisbach equation, the Chezy equation, and the Manning equation. The Darcy-Weisbach equation involves the direct use of a known effective roughness value, k , from which a boundary resistance (friction) coefficient, f , can be derived for use in the energy loss computation. The Darcy-Weisbach equation is applicable to all fully turbulent flow conditions. The Chezy equation is essentially similar to the Darcy-Weisbach equation in that it involves the direct use of a known effective roughness value and is applicable to all flow conditions. The Manning equation, probably the most commonly used, involves use of an empirically derived resistance coefficient, n , and is considered only applicable to fully turbulent flow. Some investigators such as Strickler have attempted to correlate the Manning′s n value to a measured effective roughness value; others have equated the Manning equation to the Darcy-Weisbach equation and to the Chezy equation in order to take advantage of the effective roughness parameter used in those equations. These modifications to the Manning equation have all been accomplished in order to establish some degree of confidence for an otherwise empirically derived roughness coefficient.

b. Darcy-Weisbach Equation.

The Darcy-Weisbach equation expresses the energy loss due to boundary roughness in terms of a resistance coefficient, f , as:

hf = ( fL / 4R )( V^2 / 2g ) ----------------------------- (2-5)

where hf is the energy loss due to friction through a length of channel L having an average hydraulic radius R and an average velocity V .

The energy loss has a length dimension (ft-lb/lb) and is usually expressed in feet of water. The resistance coefficient, f , is a dimensionless parameter which can be determined for fully turbulent flow conditions by a form of the Colebrook-White equation

f = [ 1 / ( 2log(12.8R/k) ) ]^2 -------------------------- (2-6)

or by the Strickler-Manning equation

f = 0.113(k/R)^(1/3) ------------------------------------- (2-7)

which may more accurately derive the resistance coefficient for R/k > 100 . In both equations 2-6 and 2-7, k is the effective roughness value and R is the hydraulic radius. Both equations 2-6 and 2-7 are valid only for fully turbulent flow defined by the relationship:

Re > 200 / f^(1/2) (4R/k) -------------------------------- (2-8)

where Re is the Reynolds number. The actual Reynolds number of the flow condition is defined as:

Re = 4RV / ν --------------------------------------------- (2-9)

where ν is the kinematic viscosity of the water. Resistance coefficients throughout the entire range of flow conditions can be obtained through the use of Plate 2-1.

c. Chezy Equation

The Chezy equation defines velocity in terms of the hydraulic radius, the slope S , and the Chezy resistance coefficient C in the form of

V = C(RS)^(1/2) ------------------------------------------ (2-10)

By equating S to hf/L and rearranging terms in equation 2-10, the Chezy equation expresses the energy loss due to boundary roughness as

hf = L/R (V/C)^(2) --------------------------------------- (2-11)

The resistance coefficient, C , is dependent upon the Reynolds number and the effective roughness value. The C value can be determined through the use of Plate 2-l or by equation 2-12:

C = 32.6 log(12.2R/k) ------------------------------------ (2-12)

for fully turbulent flow conditions as defined by the relationship:

Re > 50CR/k ---------------------------------------------- (2-13)

Chezy′s C can also be determined through the use of the Darcy-Weisbach resistance coefficient, f , by equation 2-14:

C = ( 8g / f )^(1/2) ------------------------------------- (2-14)

The characteristics of f in circular pipe flow have been thoroughly investigated by Nikuradse and Colebrook and White; however, a similar complete investigation of the characteristics of C in open channel flow have not been made due to the extra variables and wide range of surface roughness involved. However, reasonably accurate results should be obtained through the use of the Chezy equation.

d. Manning Equation.

The Manning equation 2-15 defines velocity in terms of the hydraulic radius and slope, in a similar manner to the Chezy equation; however, the resistance coefficient is defined by the Manning′s n value.

V = ( 1.486 R^(2/3) S^(1/2) ) / n ------------------------ (2-15)

The constant 1.486 converts the metric equation to foot-second units. By equating S = hf/L and rearranging terms in equation 2-15, the Manning equation expresses the energy loss due to boundary roughness as

hf = ( V^(2) n^(2) L ) / ( 2.21 R^(4/3) ) ---------------- (2-16)

The Manning′s resistance coefficient n , reported in numerous hydraulic publications, is founded on empiricism. It does not address the degree of turbulence or the interaction between the flow and boundary. The empiricism of this coefficient limits its accuracy when applied to conditions somewhat different from those from which it is derived. However, Manning′s method is widely used due mainly to the large volume of reference data available to correlate resistance coefficients with boundary conditions and the ease in which the method can be used. When the design involves a significant amount of surface roughness energy loss resulting from fully turbulent flow, such as
with a long spillway chute, the Manning′s resistance coefficient may be calculated to account for the relative roughness effect by the use of

n = ( f^(1/2) R^(1/6) ) / 10.8 -------------------------- (2-17)

or

n = 1.486 R^(1/6) / C ----------------------------------- (2-18)

and the procedures described for equation 2-6 or 2-7. A review of energy loss computation using the Manning equation 2-16 modified to account for relative roughness by equations 2-6 or 2-7 and 2-17 or 2-18 will show that, if the effect of relative roughness is required, the Darcy-Weisbach or the Chezy method provides a more direct and simpler procedure.

e. Roughness Values.

Values of effective roughness k normally are based on prototype measurements of flow over various boundary materials rather than physically measured values. Essentially all hydraulic textbooks provide extensive data of Chezy′s C and Manning′s n values; however, data are somewhat limited on effective roughness values k . Some suggested roughness values for various spillway surfaces are provided in the following tabulation:

Surface : Effective Roughness ( k, feet )
------------------------------------------
Concrete
For discharge design : 0.007
For velocity design : 0.002

Excavated rock
Smooth and uniform : 0.025-0.25
Jagged and irregular : 0.15 -0.55

Natural vegetation
Short grass : 0.025-0.15
Long grass : 0.10 -0.55
Scattered brush and weeds : 0.15 -1.0

Due to the inability to predict the roughness that will be constructed, the designer should use maximum values in computing flow profiles and minimum values in computing energy losses required for terminal structure design.

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USACE, EM 1110-2-1603, Engineering and Design, HYDRAULIC DESIGN OF SPILLWAYS

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Energy Loss for Fully Developed Turbulent Boundary Layer Flow

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