ASCECP_200_40856_195_1

Hydraulic Design of a Lazy River

Bruce M. McEnroe1

1Department of Civil, Environmental and Architectural Engineering, University ofKansas, 1530 W. 15th St., Lawrence, KS 66045; PH (785)864-2925; email:[email protected]

Abstract

Lazy rivers are popular attractions at modern aquatic centers and water parks. Theserecreational water channels carry patrons on floating tubes around a meandering loop.The current is maintained by pumps that withdraw a small fraction of the flow fromthe channel through large bottom grates and return it to the channel in jets directeddownstream. This paper presents the hydraulic relationships needed for lazy-riverdesign. These relationships account for the propulsion of the flow by the water jetsand the resistance to flow resulting from friction, bends and drag forces on standingpersons. The total pump output power is minimized when the downstreamcomponent of the jet velocity equals twice the desired current speed. However, otherpractical considerations generally favor a higher jet speed. Field tests on three lazyrivers indicate that a Manning n value of 0.015 is sufficient to account for boundaryfriction, bend losses and other local losses. Persons standing in the flow cause addeddrag, which can reduce the current speed substantially. A design example illustratesthe practical application of the relationships and experimental findings.

The investigations presented in this paper were conducted for Water’s Edge AquaticDesign LLC of Lenexa, Kansas. Water’s Edge Aquatic Design is a leading designfirm specializing in aquatic centers and water parks. The relationships in this paperhave been applied successfully to numerous lazy-river design projects.

Introduction

Lazy rivers are popular attractions at modern aquatic centers and water parks. Theserecreational water channels carry patrons on floating tubes around a meandering loop.Current speeds are typically 3 ft/s or lower. The current is maintained by pumps thatwithdraw a small fraction of the flow from the channel through large grates and returnit to the channel in jets directed downstream. Only a small fraction of the channelflow can be withdrawn and pumped at a single location without disturbing the flowexcessively, so multiple pumping stations may be needed. The main variables in thehydraulic design of the propulsion system are the number of pumping stations and thenumber, diameter, angle and speed of the jets. The objective is to find a workablecombination of these variables that will yield the desired current speed.

Copyright ASCE 2006 World Environmental and Water Resources Congress 2006Downloaded 01 Jun 2011 to 87.194.144.48. Redistribution subject to ASCE license or copyright. Visithttp://www.ascelibrary.org

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This paper presents the hydraulic relationships needed for lazy-river design. Energyaspects of lazy-river design are also analyzed. Results from field tests on three lazyrivers provide guidance on Manning n values to account for bend losses as well asfriction. An example illustrates the practical application of the hydraulic relationshipsand experimental findings.

Figure 1. Summer Fun in a Lazy River

Hydraulic Relationships for Lazy-River Design

The flow in the lazy river can be analyzed with the momentum equation for one-dimensional steady flow. The control volume for the analysis is the entire volume ofwater in the channel. Water exits the control volume through grates on the bottom orsides of the channel, and re-enters the control volume through downstream-directedjets on the bottom or sides of the channel. The downstream component of the exitvelocity is assumed to equal the current speed in the channel. The momentumequation, applied in the direction of flow, is

b p j c j jF F Q V Q V cos+ = ρ −ρ θ (1)


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in which Fb is the drag exerted on the flow by boundary friction and bend losses, Fp isthe drag exerted on the flow by persons standing in the channel, ρ is the density of thewater, Qj is the combined discharge of all jets, Vc is the current speed in the channel,Vj is the jet speed, and θ is the angle of the jet discharge relative to the direction offlow in the channel. The drag on floating persons is relatively small and is neglectedhere.

The boundary drag force can be expressed as

bF P L= −τ (2)

in which τ is the average shear stress at the boundary, P is the wetted perimeter, and Lis the length of the channel measured along the centerline. The average shear stress isrelated to the friction slope by the equation

fR Sτ = γ (3)

in which γ is the specific weight of the water, R is the hydraulic radius, and Sf is thefriction slope. The friction slope is the head loss per unit distance of flow. In thisanalysis, the friction slope accounts for head losses due to bends as well as boundaryfriction. The substitution of γRSf for τ in Eq. 3 leads to

b fF US=− γ (4)

in which U is the volume of water in the channel. The friction slope can be computedwith the Manning equation,

2

c ef 2 / 3

V nS

C R

=

(5)

in which Vc is the average velocity in the channel, ne is the effective value of theManning resistance coefficient and C is a units-conversion constant (C = 1 for R inmeters and Vc in m/s; C = 1.49 for R in feet and Vc in ft/s). The effective Manning naccounts for bend losses as well as boundary friction. Replacing Sf in Eq. 4 with theright-hand side of Eq. 5 leads to

22e

b c2 4 / 3

U nF V

C R

γ= − (6)

Persons standing in the flow can exert considerable drag on the flow. This drag forcecan be expressed as


4

2c

p D fV

F mC A2

= − ρ (7)

in which Fp is the total drag force from standing persons, m is the number of standingpersons, CD is the drag coefficient, and Af is the frontal area of a standing person(normal to the direction of flow, below the water surface).

The sum of the boundary forces and the drag forces on standing persons can beexpressed in the form

2b p cF F K V+ = − ρ (8)

in which K is a combined drag constant defined as

2e

D f2 4 / 3

g n U 1K m C A

2C R= + (9)

The substitution of the right-hand side of Eq. 8 for Fb + Fp in Eq. 1 leads to

2c j j cK V Q (V cos V )= θ− (10)

Eq. 10 can be rearranged algebraically to obtain explicit equations for Qj and for Vc.These equations are

2c

jj c

K VQ

V cos V=

θ−(11)

2j j j j

c

Q Q 4K Q V cosV

2 K

− + + θ= (12)

Eq. 11 is useful for preliminary design, and Eq. 12 is useful for analysis of proposeddesigns.

Energy Aspects of Lazy-River Design

A desired current speed, Vc, can be achieved by different combinations of jetdischarge and jet speed. The smaller the jet discharge, the higher the jet speed neededto obtain the desired current speed. However, different combinations of jet dischargeand jet speed require different amounts of power for pumping. It is useful to find thecombination of jet discharge and jet speed that requires the least power. The pumppower, P, equals γQjhp, in which hp is the pump head. The pump head is proportional


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to the square of the jet speed; therefore, the power requirement is proportional toQjVj

2:

2j jP Q V∝ (13)

The right-hand side of Eq. 11 can be substituted into Eq. 13 to obtain

2 2c j

j c

K V VP

V cos V∝

θ− (14)

The jet speed that requires the least power for a given current speed can be found bydifferentiating Eq. 14 with respect to Vj, setting the result equal to zero, and solvingfor Vj. The result is

j cV cos 2 Vθ = (15)

This result indicates that power requirement is minimized when the downstreamcomponent of the jet velocity equals twice the desired current speed; i.e., when theratio (Vj cos θ)/Vc equals 2. Other values of (Vj cos θ)/Vc require more power. Theratio of the actual power requirement to the minimum power requirement, Pmin, varieswith the ratio (Vj cos θ)/Vc as follows:

2j

c

jmin

c

V cos

VP 1V cosP 4

1V

θ =

θ −

(16)

Figure 2 shows this relationship graphically. The power requirement is only mildlysensitive to (Vj cos θ)/Vc in the vicinity of the optimum value; the power requirementis within 20% of the minimum value for (Vj cos θ)/Vc values between 1.4 and 3.4.

Although pump output power is minimized by a jet speed equal to 2Vc/(cos θ), otherpractical considerations generally dictate a much higher jet speed. The lower the jetvelocity, the greater the jet discharge needed to maintain the desired current speed. Alazy river designed for minimum power would typically require a jet discharge equalto a large fraction of the channel flow. A large number of pumping stations, eachwithdrawing a small fraction of the flow, would be needed to avoid excessivedisturbance of the flow. Such a system would cost much more to construct andmaintain than a system with a fewer pumping stations and a higher jet speed. Safetyconsiderations dictate an upper limit on jet speed. Jet speeds higher than 20 ft/sshould be avoided.


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0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10

P/P

min

(Vj cos θ) / Vc

Figure 2. Relative Power Requirements for a Fixed Current Speed

Effective Manning n from Field Tests

Bend losses in lazy rivers are clearly significant but difficult to estimate. Hydraulicengineering textbooks and handbooks provide approximate head-loss coefficients forisolated bends. However, summing the loss coefficients for individual bends leads tooverestimation of losses for layouts with connected or closely spaced bends. In ourformulation, the effective Manning resistance coefficient, ne, accounts for flowresistance from both friction and bends.

We conducted field tests on three lazy rivers to determine the effective Manning nvalues from measured current speeds and jet speeds. We tested lazy rivers atmunicipal aquatic centers in Marshalltown, Clive, and West Des Moines, Iowa.These facilities were all designed by Water’s Edge Aquatic Design LLC. Thechannels are all 10 feet wide with walls of smooth painted concrete and a water depthof 3.5 feet. Channel lengths range from 310 to 610 ft and bend radii range from 30 ftto 45 ft. The currents are driven by 1.5-in. bottom jets directed 22.5º above thehorizontal. The three lazy rivers all have side openings with steps for entry and exitthat disturb the flow and cause local energy losses.

The tests were performed with the lazy rivers running at reduced current speeds withsome jets shut off or throttled. Current speeds ranged from 1.5 ft/s to 2.4 ft/s, and jetspeeds ranged from 8.7 ft/s to 19.3 ft/s.


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Effective Manning n values for the three tests were computed from the equation

1/ 22 4 / 3 n nj 2

e j,i c j,i2i 1 i 1c

C R an cos V V V

g U V = =

= θ −

∑ ∑ (16)

in which aj is the cross-sectional area of a single jet, i is a jet numbering index, andVj,i is the velocity for jet i. Eq. 16 was obtained by modifying Eq. 10 to account forthe different velocities of individual jets, replacing K with the right-hand side of Eq. 9(with m = 0), and solving for ne. The field data yielded ne values of 0.015 forMarshalltown, 0.013 for Clive, and 0.015 for West Des Moines. Based on thesefield-test results, an effective Manning n value of 0.015 appears to be appropriate forthe design of a typical lazy river.

Example Application

The following example illustrates the practical application of the hydraulicrelationships for lazy river design.

Problem: A lazy river is to be 600 ft long and 10 feet wide with a water depth of 3.5ft. The flow will be propelled by rows of water jets in the channel bottom. Each rowwill contain ten 1.5-in.-diameter jets, directed 22.5º above the horizontal in thedownstream direction.

The desired maximum current speed with no persons standing in the channel is 3 ft/s,and the desired maximum jet speed is 20 ft/s. A single pumping station shouldwithdraw no more than 5% of the flow in the channel. Determine the total dischargethat must be pumped and the number of pumping stations needed. Also find thecurrent speed for this design with ten persons standing in the channel.

Solution: At a depth of 3.5 ft, the channel has a cross-sectional area of 35 ft2, awetted perimeter of 17 feet and a hydraulic radius of 2.059 ft. The total volume ofwater in the 600-ft-long channel is 17,500 ft3. The effective Manning n is set to0.015, based on the findings from the three field tests. The drag constant, K, with nostanding persons is computed with Eq. 9:

2 22e

2 4 / 3 2 4 / 3

g n U 32.2(0.015) (17,500)K 21.80 ft

C R (1.49) (2.059)= = =

The required jet discharge for a current speed of 3 ft/s and a jet speed of 20 ft/s iscomputed with Eq. 11:

2 2c

jj c

K V 21.80(3)Q 12.68 cfs

V cos V 20cos 22.5 3= = =

θ− −o


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At a jet speed of 20 ft/s, a discharge of 12.68 cfs requires a total jet area of 0.634 ft2,or 52 jets of 1.5-in. diameter. Therefore, a design with sixty jets arranged in six rowsof ten is selected.

At the desired current speed of 3 ft/s, the discharge in the channel is 105 cfs. Thetotal jet discharge of 12.68 cfs is 12.1% of the discharge in the channel. Because asingle pumping station should withdraw no more than 5% of the channel flow, threepumping stations are needed. Each pumping station will supply two rows of jets.

At the maximum jet speed of 20 ft/s, the total discharge from sixty jets is 14.73 cfs.The corresponding current speed is computed with Eq. 12:

2j j j j

c

2

Q Q 4 K Q V cosV

2K

14.73 (14.73) 4(21.80) (14.73) (20cos 22.5 ) ft3.21

2(21.80) s

− + + θ=

− + += =

o

The effect of ten persons standing in the channel can be estimated roughly byassuming a drag coefficient of 1.2 and a frontal area of 3 ft2 (below the water surface)for each standing person. The combined drag constant for these conditions iscomputed with Eq. 9:

2 22e

D f2 4 / 3 2 4 / 3

g n U 1 32.2(0.015) (17,500) 1K m C A (10) (1.2) (3.0) 39.80 ft

2 2C R (1.49) (2.059)= + = + =

The corresponding current speed, computed with Eq. 12, is 2.44 ft/s. This resultshows that the added drag from standing persons can cause a substantial reduction incurrent speed.

Summary

Some fairly simple relationships based on momentum and resistance principlesprovide a sound basis for the hydraulic design of lazy rivers. Energy consumption isminimized when the downstream component of the jet velocity equals twice thedesired current speed. However, other practical considerations generally favor ahigher jet speed. Field tests on three lazy rivers indicate that a Manning n value of0.015 is sufficient to account for boundary friction, bend losses and other local losses.Persons standing in the flow cause added drag, which can reduce the current speedsubstantially.


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Acknowledgment

The research presented in this paper was sponsored by Water’s Edge Aquatic DesignLLC of Lenexa, Kansas. The author sincerely appreciates this support. Jeff Bartley,P.E., and Katie Schultz of Water’s Edge Aquatic Design assisted with the field tests.


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