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Hydraulic Design of Energy Dissipators for Culverts and Channels
Hydraulic Engineering Circular Number 14, Third Edition

Chapter 4: Flow Transitions

A flow transition is a change of open channel flow cross section designed to be accomplished in a short distance with a minimum amount of flow disturbance. Five types of transitions are shown in Figure 4.1: cylindrical quadrant, straight line, square end, warped, and wedge. Expansion transitions are illustrated, but contraction transitions would have similar geometry.

Figure 4.1. Transition Types

Illustration of the cylindrical quadrant, straight line, warped, wedge, and square end transition types.

The most common flow transitions are the square end expansion (headwall) and the straight-line (wingwall) transitions. Both of these transitions are considered abrupt expansions and are discussed in Section 4.1. Procedures are provided for determining the velocity and depth exiting these standard headwall and wingwall configurations. An apron, which is an integral part of these transitions, protects the channel bottom at the culvert outlet from erosion.

Specially designed open channel flow inlet transitions (contractions) are normally not required for highway culverts. The economical culvert is designed to operate with an upstream headwater pool that dissipates the channel approach velocity and, therefore, negates the need for an approach flow transition. Side and slope tapered culvert inlets are designed as submerged transitions and do not fall within the intended limits of open channel transitions discussed in this chapter (see Normann, et al., 2001). Special inlet transitions are useful when the conservation of energy is essential because of allowable headwater considerations such as an irrigation structure in subcritical flow (see Section 4.2) or where it is desirable to maintain a small cross section with supercritical flow in a steep channel (see Section 4.3). Section 4.4 addresses supercritical flow expansions.

Expansions/transitions upstream of stilling basins are designed to decrease depth, increase velocity, and, therefore, increase Froude number. These supercritical expansions include design of a chute and determination of the needed depression below the streambed to force an efficient hydraulic jump. This topic is addressed in detail in Section 8.1.

4.1 Abrupt Expansion

As a jet of water, which is not laterally constrained, leaves a culvert flowing in outlet control, the water surface plunges or drops very rapidly (see Figure 4.2). As the water surface drops and the flow spreads out, the potential energy stored as depth is converted to kinetic energy or velocity. Therefore, the velocity leaving the wingwall apron can be higher than the culvert outlet velocity and must be considered in determining outlet protection. The straight-line transition may also be considered an abrupt transition if the tanθ is greater than 1/(3Fr), where θ is the angle between the wingwall and culvert axis.

Figure 4.2. Dimensionless Water Surface Contours (Watts, 1968)

Description of the abrupt expansion of flow in terms of depth. Longitudinal section view shows depth decreasing with distance from culvert outlet. End section view shows decreasing depth away from the flow centerline.

A reasonable estimate of transition exit velocity can be obtained by using the energy equation and assuming the losses to be negligible. By neglecting friction losses, a higher velocity than actually occurs is predicted making the error on the conservative side.

A more accurate way to determine transition exit flow conditions was developed by Watts (1968). Watts' experimental data has been converted to Equation 4.1 (for boxes) and Equation 4.2 (for circular pipes) for determining VA/Vo.

(4.1)

V sub A divided by V sub o equals 1.65 minus 0.3 times Fr

(4.2)

V sub A divided by V sub o equals 1.65 minus 0.45 times Q divided by the square root of (g times D to the 5th power)

where,

VA = average velocity on the apron, m/s (ft/s)

Vo = velocity at the culvert outlet, m/s (ft/s)

Also based on Watts' work, Figures 4.3 and 4.4 relate Froude number (Fr) orQ/(gD5)0.5 to the average depth/brink depth ratio (yA/yo). These equations and curves were developed for Fr from 1 to 3, which are applicable for most abrupt culvert outlet transitions. Normally, low tailwater is encountered at the culvert outlet and flow is supercritical on the outlet apron.

Figure 4.3. Average Depth for Abrupt Expansion Below Rectangular Culvert Outlet

Graph with x-axis of Froude number and y-axis of Y sub A divided by Y sub o showing a family of curves for L values ranging from 0.5 times B to 3.0 times B.

Figure 4.4. Average Depth for Abrupt Expansion Below Circular Culvert Outlet

Graph with x-axis of Froude number and y-axis of Y sub A divided by Y sub o showing a family of curves for L values ranging from 0.5 times B to 3.0 times B.

Water cannot completely expand to fill the setion between the wingwalls in an abrupt expansion. The majority of the flow will stay within an area whose boundaries are defined by:

(4.3)

Theta equals inverse tangent of (1/(3Fr))

where,

Θ = optimum flare angle

The downstream width of the apron, W2, is given by:

(4.4)

W sub 2 equals W sub o plus 2 times L times tangent of theta sub w

where,

W2 = width of apron at length, L, downstream from the culvert outlet, m (ft)

L = distance downstream from culvert outlet, m (ft)

θw = wingwall flare angle

If θw > θ then the designer should consider reducing θw to θ. As shown in Figure 4.2 flaring the wingwall more than 1/(3Fr) (for example 45°) provides unused space which is not completely filled with water.

The design procedure for an abrupt expansion may be summarized in the following steps:

Step 1. Determine the flow conditions at the culvert outlet: Vo and yo (see Chapter 3).

Step 2. Calculate the Froude number: Fr = Vo /(g yo)0.5 at the culvert outlet.

Step 3. Find the optimum flare angle, θ, using Equation 4.3. If the chosen wingwall flare, θw, is greater than θ, consider reducing θw to θ.

Step 4. Find the average depth on the apron. For boxes, use Figure 4.3. For pipes, use Figure 4.4. The ratio yA/yo is obtained knowing the Froude number (Fr) and the desired distance downstream, L.

Step 5. Find average velocity on the apron, VA, using Equation 4.1 or Equation 4.2. VA = V2.

Step 6. Calculate the downstream width, W2, using Equation 4.4.

Step 7. Calculate downstream depth, y2.

If θ was used in Equation 4.4, calculate y2 = Q/(VAW2). This depth will be larger than yA since the flow prism is now laterally confined.

If θw was used in Equation 4.4, calculate y2 = yA. However, estimate the average flow width, WA, = Q/(VAyA). Check that WA < W2. If it is not, then y2 = Q/(VA W2).

Design Example: Abrupt Expansion Transition (SI)

Find the flow conditions (y2 and V2) at end of a 3.1 m apron. Assume negligible tailwater. Given:

  • RCB = 1524 mm x 1524 mm
  • Wingwall flare θw = 45°
  • Culvert lentgth = 61 m
  • So = 0.002 m/m
  • Q = 7.65 m3/s
  • yc = 1.37 m

Solution

Step 1. Find culvert outlet velocity from Figure 3.3 with TW/D ≈ 0.

Need quantity1.811 Q/(BD3/2) = 1.811(7.65)/(1.524(1.524)3/2) = 4.83

yo /D = 0.68

yo = 0.68(1.524) = 1.036 m

Vo = Q/A = 7.65/(1.036 (1.524)) = 4.84 m/s

Step 2. Find culvert outlet Froude number.

Fr = Vo /(g yo)0.5 = 4.84/(9.81(1.036))0.5 = 1.52

Step 3. Find θ

tan θ = 1/(3Fr) = 1/(3(1.52)) = 0.22

θ = 12.37°

Step 4. Estimate average depth.

Apron Length/Diameter = 3.1/1.524 = 2 (Convert apron length to multiple of culvert diameter.)

Use Figure 4.3 for yA/yo = 0.26

yA = 0.26(1.036) = 0.269 m

Step 5. Find average velocity, VA, using Equation 4.1.

VA/Vo = 1.65 - 0.3Fr = 1.65 - 0.3(1.52) = 1.2

VA = 4.84(1.2) = 5.82 m/s

VA = V2 = 5.82 m/s

Step 6. Calculate downstream width using Equation 4.4.

θw > θ use θw

W2 = Wo + 2L tan ( θw) = 1.524 + 2(3.1)(1.0) = 7.72 m

Step 7. Calculate downstream depth.

θw was used, therefore,

y2 = yA = 0.269 m

Check WA = Q/(VA yA) = 7.65/((5.82)(0.269))

WA = 4.89 m < 7.72 m

Compare the above solution with two alternatives using the energy equation.

Alternative 1. Assume W2 = full width between wingwalls at the end of the apron.

W2 = Wo + 2L tan 45o = 7.72 m

A2 = W2 y2 = 7.62 y2

V2 = Q/A2 = 7.65/(7.72y2) = 0.99/y2

The energy balance between flow at the culvert outlet and the apron is given by:

zo + yo + Vo2 /(2g) = z2 + y2 + V22 /(2g) + Hf

Assuming Hf = 0 and zo = z2

1.036 + (4.84)2 /(2(9.81)) = y2 + (0.99/y2)2 /(2(9.81))

1.036 + 1.194 = y2 + 0.050/y22

2.230 = y2 + 0.050/y22

y2 = 0.157 m, which is 41% lower than the original solution.

V2 = 0.99/0.157 = 6.31 m/s which is 8% higher than the original solution.

Alternative 2. Assume W2 is based on θ where tanθ = 1/(3Fr).

W2 = Wo + 2L tan 12.41o = 1.524 + 6.2(0.22) = 2.89 m

A2 = 2.89 y2 and V2 = 7.65/(2.89y2) = 2.65/y2

2.230 = y2 + 0.360/y22

y2 = 0.45 meters, which is 68% higher than the original solution

V2 = 2.65/0.45 = 5.89 m/s, which is 2% higher than the original solution.

Design Example: Abrupt Expansion Transition (CU)

Find the flow condition (y2 and V2) at end of a 10 ft apron. Assume negligible tailwater. Given:

  • RCB = 5 ft x 5 ft
  • Wingwall flare θw = 45°
  • Culvert length = 200 ft
  • So = 0.002 ft/ft
  • Q = 270 ft3/s
  • yc = 4.5 ft

Solution

Step 1. Find culvert outlet velocity from Figure 3.3 with TW/D ≈ 0

Need quantity Q/(BD3/2) = 270/(5(5)3/2) = 4.83

yo /D = 0.68

yo = 0.68(5) = 3.4 ft

Vo = Q/A = 270/ ((5) 3.4) = 15.9 ft/s

Step 2. Find culvert outlet Froude number.

Fr = Vo /(g yo)0.5 = 15.9/(32.2(3.4))0.5 = 1.52

Step 3. Find θ

tan θ = 1/(3Fr) = 1/(3(1.52)) = 0.22

θ = 12.37°

Step 4. Estimate average depth.

Apron Length/Diameter = 10/5 = 2 (Convert apron length to multiple of culvert diameter.)

Use Figure 4.3 for yA/yo = 0.26

yA = 0.26(3.4) = 0.88 ft

Step 5. Find average velocity, VA, using Equation 4.1.

VA/Vo = 1.65 - 0.3Fr = 1.65 - 0.3(1.52) = 1.2

VA = 15.9(1.2) = 19.1 ft/s

VA = V2 = 19.1 ft/s

Step 6. Calculate downstream width using Equation 4.4.

θw > θ use θw

W2 = Wo + 2L tan ( θw) = 5 + 2(10)(1.0) = 25 ft

Step 7. Calculate downstream depth.

θw was used:

y2 = yA = 0.88 ft

Check WA = Q/(VA yA) = 270/((19.1)(0.88))

WA = 16.1 ft < 25 ft

Compare the above solution with two alternatives using the energy equation.

Alternative 1. Assume W2 = full width between wingwalls at the end of the apron.

W2 = Wo + 2L tan 45o = 25 ft

A2 = W2 y2 = 25 y2

V2 = Q/A2 = 270/(25y2) = 10.8/y2

The energy balance between flow at the culvert outlet and the apron is given by:

zo + yo + Vo2 /(2g) = z2 + y2 + V22 /(2g) + Hf

Assuming Hf = 0 and zo = z2,

3.4 + (15.9)2 /(2(32.2)) = y2 + (10.8/y2)2 /(2(32.2))

3.4 + 3.92 = y2 + 1.81/y22

7.32 = y2 + 1.81/y22

y2 = 0.52 ft, which is 41% lower than the original solution.

V2 = 10.8/0.52 = 20.8 ft/s which is 10% higher than the original solution.

Alternative 2. Assume W2 is based on θ where tanθ = 1/(3Fr).

W2 = Wo + 2L tan 12.41o = 5 + 20(0.22) = 9.4 ft

A2 = 9.4 y2 and V2 = 270/(9.4y2) = 28.7/y2

7.32 = y2 + 12.8/y22

y2 = 1.48 ft, which is 68% higher than the original solution

V2 = 28.7/1.48 = 19.4 ft/s, which is 2% higher than the original solution.

4.2 Subcritical Flow Transition

Subcritical flow can be transitioned into and out of highway structures without causing adverse effect if subcritical flow is maintained throughout the structure. The flow cannot approach or pass through critical depth, yc. The range of depths to avoid is 0.9yc to 1.1yc. In this range, slight changes in specific energy are reflected in large changes in depth, i.e., wave problems develop. The straight line or wedge transition should be used if conservation of flow energy is required, for example, for an irrigation canal structure that traverses a highway. Warped and cylindrical transitions are more efficient, but the additional construction cost can only be justified for structures where backwater is critical.

Figure 4.5 illustrates the design problem. Starting upstream of section 1 where some backwater exists due to the culvert, the flow is transitioned from a canal into and then out of the highway culvert. The flare angle, θw, should be 12.5°, (1:4.5 (lateral:longitudinal) or smaller) according to Hinds (1928). This criterion provides a gradually varied transition that can be analyzed using the energy equation.

Figure 4.5. Subcritical Flow Transition

Plan and profile sketch showing the energy gradeline and hydraulic gradeline at four locations: 1) approach section, 2) entrance to a narrowed section, 3) exit of the narrowed section, and 4) downstream widened section.

As the flow transitions into the culvert, the water surface approaches yc. To minimize waves, y2 should be equal to or greater than 1.1yc. In the culvert, the depth will increase and will reach yn if the culvert is long enough. In the expansion, the depth increases to yn of the downstream channel, Section 4. Associated with both transitions are energy losses that are proportional to the change in velocity head in the transitions. The energy loss in the contraction, HLc, and in the expansion, HLe, are:

(4.5)

H sub Lc equals C sub c times (V sub 2 squared divided by 2 times g minus V sub 1 squared divided by 2 times g)

(4.6)

H sub Le equals C sub e times (V sub 3 squared divided by 2 times g minus V sub 4 squared divided by 2 times g)

where Cc and Ce are found from Table 4.1.

Table 4.1. Transition Loss Coefficients (USACE, 1994)
Transistion TypeContraction CcExpansion Ce
Warped0.100.20
Cylindrical Quadrant0.150.25
Wedge0.300.50
Straight Line0.300.50
Square End0.300.75

The depth in the culvert, y3, can be found by trial and error using the energy equation with y4 = yn in the downstream channel and assuming hf2 = 0 (see Figure 4.5). The streambed elevation is equal to z. Writing the energy equation between sections 3 and 4 yields:

z4 + y4 + V42 /(2g) + HLe + Hf2 = z3 + y3 + V32 /(2g)

Assuming Hf2 ≈ 0, V3 = Q/(W3 y3), V4 = Q/(W4 y4)

Substituting Equation 4.6,

(4.7)

z4 + y4 +V42 /(2g) +Ce (V32 /(2g) - V42 /(2g)) = z3 + y3 + V32 /(2g)

z4 + y4 + (1 - Ce) V42 /(2g) = z3 + y3 + (1 - Ce) v32 /(2g)

z4 - z3 + y4 + (1 - Ce) (Q/(W4 y4))2 /(2g) = y3 + (1 - Ce) (Q/(W3 y3))2 /(2g)

After known values are substituted, Equation 4.7 reduces to

C1 = y3 + C2 /y32

which can be solved by trial and error.

In a similar manner, y1 can be determined by assuming y2 = y3 and hf1 = 0.

(4.8)

z2 + y2 + V22 /(2g) + HLC + Hf1 = z1 + y1 + V12 /(2g)

z2 + y2 +V22 /(2g) +Cc (V22 /(2g) - V12 /(2g)) = z1 + y1 + V12 /(2g)

z2 + y2 + (1 + Cc) V22 /(2g) = z1 + y1 + (1 +Cc) V12 /(2g)

z2 - z1 + y2 + (1 + Cc) (Q/(W2 y2))2 /(2g) = y1 + (1 + Cc) (Q/(W1 y1))2 /(2g)

These depths are approximate because friction loss was neglected. They should be checked by computing the water surface profile using a standard step method computer program, like HEC-RAS (USACE, 2002).

4.3 Supercritical Flow Contraction

The design of transitions for supercritical flow is difficult to manage without causing a hydraulic jump or other surface irregularity. Therefore, the full flow area should be maintained if at all possible. A smooth transition of supercritical flow requires a structure longer than typical wingwalls and should not be attempted unless the structure is of primary importance. A model study should be used to determine transition geometry where a hydraulic jump is not desired. If a hydraulic jump is acceptable, the inlet structure can be designed as shown in Figure 4.6. This design, which must be accomplished in a rectangular channel, yields a long transition. The design approach outlined below is from USACE (1994) and Ippen (1951).

Figure 4.6. Supercritical Inlet Transition for Rectangular Channel (USACE, 1994)

Profile and plan showing water surface elevation increasing in an inlet transition with supercritical approach flow.

The length, L, is defined by the channel contraction, W1 -W3, and the wall deflection angle, θw.

(4.9)

L = (W1-W2)/ (2Tan θw)

To minimize surface disturbances, L should also equal L1 + L2 where

(4.10)

L1 = W1/(2Tan β1)

(4.11)

L2 = W3/ (2Tan (β2- θw))

(4.12)

tangent theta sub w equals (tangent beta sub 1 times (the square root of (1 plus 8 times Fr sub 1 squared times (sin of beta sub 1) squared) minus 3)) divided by (2 times (tangent of beta sub 1) squared plus square root of (1 plus 8 times Fr sub 1 squared times (sin of beta sub 1) squared) minus 1)

The transition design requires a trial θw that fixes L as defined by Equation 4.9. This length is then checked by finding L1 + L2. To determine L1, β1 is found from Equation 4.12 by trial and error and then substituted into Equation 4.10. L2 is calculated from Equation 4.11 with β2 determined from Equation 4.12 by substituting β2 for β1 and Fr1 for Fr2. To find Fr2 first calculate:

(4.13)

y sub 2 divided by y sub 1 equals one-half times (square root of (1 plus 8 times Fr sub 1 squared times (sin of beta sub 1) squared) minus 1)

Then:

(4.14)

Fr sub 2 squared equals y sub 1 divided by y sub 2 times (Fr sub 1 squared minus (y sub 1 divided by (2 times y sub 2)) times (y sub 2 divided by y sub 1 minus 1) times (y sub 2 divided by y sub 1 plus 1)squared)

If the trial θw was chosen correctly the total length, L, will equal the sum of L1 and L2. If not, choose another trial θw and repeat the process until the lengths match. The depth, y3, and Fr3 in the culvert can now be calculated using Equation 4.13 and Equation 4.14 if the subscripts are increased by 1; i.e., y2/y1 is now y3/y2. The above design approach assumes that the width of the channel (W1) and the width of the culvert (W3) are known and L is found by trial and error. If W3 has to be determined, the design problem is complicated by another trial and error process.

4.4 Supercritical Flow Expansion

Supercritical flow expansion design has, in part, been discussed in Section 4.1. The procedure outlined in Section 4.1 should be used to determine apron or expansion flow conditions if the culvert exit Froude number, Fr, is less than 3, if the location where the flow conditions are desired is within 3 culvert diameters of the outlet and So is less than 10%. For expansions outside these limits, the energy equation can be used to determine flow conditions leaving the transition. Normally, these parameters would then be used as the input values for a basin design.

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Cynthia Nurmi
Resource Center (Atlanta)
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cynthia.nurmi@dot.gov

Updated: 10/03/2012
 

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