The limiting contraction ratio revisited, open channel hydraulics, F. M. Henderson, Nicole R. Nuccitelli, Victor M. Ponce, San Diego State University

The Limiting Contraction Ratio
Revisited

Nicole R. Nuccitelli and Victor M. Ponce

April 2014

ABSTRACT

Henderson's formulations of the energy-based and momentum-based limiting contraction ratios are reviewed (Henderson 1966). Henderson's explicit energy-based equation is found to be correct, however, his implicit momentum-based equation is found to be incorrect. A new explicit momentum-based equation is derived, rendering the implicit formulation unnecessary. An online calculator enables the calculation of the limiting contraction ratio for both energy and momentum formulations.

1. THE LIMITING CONTRACTION RATIO

The limiting contraction ratio σ is the ratio of contraction width b_c to upstream channel width b₁ (σ = b_c /b₁), or, alternatively, downstream channel width b₃ (σ = b_c /b₃), that produces critical flow at the contraction in a rectangular channel. The ratio has an upper limit σ = 1 when the upstream or downstream flow is critical.

According to Henderson (1966), the ratio σ may be calculated in two ways: (1) using an energy balance between the upstream and critical contraction sections, or (2) using a momentum balance between the critical contraction and downstream sections. Henderson has argued that the momentum balance is more likely to be correct because it does not depend on an assumption of energy conservation. However, Henderson's implicit formulation of the momentum-based limiting contraction ratio appears to be incorrect. These propositions are now substantiated.

2. ENERGY-BASED EQUATION FOR LIMITING CONTRACTION RATIO

The energy-based equation is derived by equating the specific energy at the upstream flow section (Section 1) with the specific energy at critical flow at the contraction (Section 2, or Section c) (Fig. 1):

E₁ = E₂ = E_c (1)

The specific energy of the upstream flow is:

v₁²
E₁ = y₁ + ^_______
2 g (2)

The Froude number at the upstream section is defined as follows:

v₁²
F₁² = ^_______
g y₁ (3)

Thus, Eq. 2 reduces to:

F₁²
E₁ = y₁ [ 1 + ^______ ]
2 (4)

By definition, the critical depth is:

F₁²
y_c = (2/3) E₁ = (2/3) y₁ [ 1 + ^______ ]
2 (5)

Thus:

y_c F₁²
^_____ = (2/3) [ 1 + ^______ ]
y₁ 2 (6)

Or, alternatively:

y_c
^_____ = (1/3) [ 2 + F₁² ]
y₁ (7)

From continuity:

v₁ y₁ b₁ = v_c y_c b_c (8)

The limiting contraction ratio σ is:

b_c v₁ y₁
σ = ^_______ = ^________
b₁ v_c y_c (9)

Or:

v₁
σ = ^{_____________}
v_c (y_c/y₁) (10)

Or:

v₁
σ = ^{___________________}
(g y_c)^1/2 (y_c/y₁) (11)

Or:

v₁
σ = ^{______________________}
(g y₁)^1/2 (y_c/y₁)^3/2 (12)

Replacing Eq. 3 in Eq. 12:

F₁
σ = ^{______________}
(y_c/y₁)^3/2 (13)

Replacing Eq. 7 in Eq. 13:

F₁
σ = ^{________________________}
(1/3)^3/2 (2 + F₁²)^3/2 (14)

Simplifying Eq. 14:

3^3/2 F₁
σ = ^{_________________}
(2 + F₁²)^3/2 (15)

Equation 15 is the energy-based limiting contraction ratio.

Squaring Eq. 15:

3³ F₁²
σ² = ^{________________}
(2 + F₁²)³ (16)

Equation 16 is the same as that presented by Henderson for the limiting contraction ratio (op. cit., Eq. 7-35, p. 267).

Fig. 1 Definition sketch.

3. MOMENTUM-BASED EQUATION FOR LIMITING CONTRACTION RATIO

The momentum-based equation is derived by equating the specific force at critical flow at the contraction (Section 2, or Section c) with the specific force at the downstream flow section (Section 3).

M₃ = M₂ = M_c (17)

The specific force at the downstream flow section is:

q₃² y₃²
M₃ = ^_______ + ^_______
g y₃ 2 (18)

Replacing the discharge Q:

Q² y₃²
M₃ = ^___________ + ^______
g y₃ b₃² 2 (19)

From continuity:

Q = v₃ y₃ b₃ = v_c y_c b_c (20)

Thus:

v₃² y₃² y₃²
M₃ = ^__________ + ^______
g y₃ 2 (21)

The Froude number at the downstream section is defined as follows:

v₃²
F₃² = ^_______
g y₃ (22)

Thus, Eq. 21 reduces to:

1
M₃ = y₃² ( ^____ + F₃² )
2 (23)

The specific force at the contraction is:

q_c² y_c²
M_c = ^_______ + ^_______
g y_c 2 (24)

Replacing the discharge Q:

Q² y_c²
M_c = ^___________ + ^______
g y_c b_c² 2 (25)

From continuity:

v₃² y₃² b₃² y_c²
M_c = ^{_______________} + ^______
g y_c b_c² 2 (26)

Or:

v₃² y₃³ b₃² y_c²
M_c = ^{_______________} + ^______
g y_c b_c² y₃ 2 (27)

Replacing Eq. 22 in Eq. 27:

F₃² y₃³ y_c²
M_c = ^___________ + ^______
σ² y_c 2 (28)

By definition, the critical depth is:

q²
y_c = ( ^____ ) ^1/3
g (29)

Or:

Q²
y_c = ( ^________ ) ^1/3
b_c²g (30)

From continuity:

v₃² y₃² b₃²
y_c = ( ^{_____________} ) ^1/3
b_c²g (31)

Or:

v₃² y₃³ b₃²
y_c = ( ^{_____________} ) ^1/3
b_c²g y₃ (32)

Replacing Eq. 22 in Eq. 32:

F₃² y₃³
y_c = ( ^_________ ) ^1/3
σ² (33)

Replacing Eq. 33 in Eq. 28:

F₃² y₃² σ^2/3 F₃² y₃³
M_c = ^{______________} + (1/2) ( ^_________ )^2/3
σ² F₃^2/3 σ² (34)

Reducing terms:

F₃^4/3
M_c = (3/2) ( ^________ ) y₃²
σ^4/3 (35)

Equating Eqs. 23 and 35:

1 F₃^4/3
( ^___ + F₃² ) = (3/2) ( ^________ )
2 σ^4/3 (36)

Reducing:

3^3/4 F₃
σ = ^{_____________________}
( 1 + 2 F₃² ) ^3/4 (37)

Equation 37 is the explicit formulation of the momentum-based limiting contraction ratio.

4. IMPLICIT MOMENTUM-BASED FORMULATION

To derive an implicit formulation, Eq. 36 is expressed as follows:

3 F₃^4/3
σ^4/3 = ^{_______________}
1 + 2 F₃² (38)

Or:

(3/σ) F₃^4/3
σ^1/3 = ^{_______________}
1 + 2 F₃² (39)

Reducing:

(3/σ)³ F₃⁴
σ = ^{_________________}
(1 + 2 F₃²)³ (40)

Equation 40 differs slightly from Henderson's Eq. 7-36, page 267, repeated here:

(2 + 1/σ)³ F₃⁴
σ = ^{__________________}
(1 + 2 F₃²)³ (41)

Thus, Henderson's implicit equation for the momentum-based limiting contraction ratio (Eq. 41) appears to be incorrect.

5. SUMMARY

Two explicit equations for the limiting contraction ratio, the first based on an energy balance, and the second based on a momentum balance, are derived following Henderson (1966). An implicit equation based on momentum balance is also derived, to parallel Henderson's treatment.

The explicit energy-based equation, which is shown to be same as Henderson's, is:

3^3/2 F₁
σ = ^{_________________}
(2 + F₁²)^3/2 (15)

The explicit momentum-based equation, not presented by Henderson, is:

3^3/4 F₃
σ = ^{__________________}
(1 + 2 F₃²) ^3/4 (37)

Note the great resemblance between Eq. 15 (energy-based) and Eq. 37 (momentum-based).

The implicit momentum-based equation derived here (Eq. 40) differs slightly from that of Henderson (Eq. 41), suggesting that the latter is incorrect.

The online calculator ONLINE_LIMITING_CONTRACTION calculates the energy-based and momentum-based limiting contraction ratios for a pair of Froude numbers F₁ and F₃. Additionally, the online calculator ONLINE_LIMITING_CONTRACTION_SET calculates the set of energy-based and momentum-based limiting contraction ratios for a given range of Froude numbers and Froude number interval.

REFERENCES

Henderson, F. M., 1966. Open channel flow. MacMillan Publishing Co., Inc., New York.

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