Identify the correct Lacey-type velocity relationship for regime channels (metric form): V as a function of hydraulic mean depth R and bed slope S.
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AV = 10.8 R^{1/2} S^{1/2}
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BV = 10.8 R^{2/3} S^{1/2}
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CV = 10.8 R^{2/3} S^{1/3}
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DV = 10.8 R^{1/3} S^{2/3}
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EV = 8.0 R^{1/2} S^{1/2}
Answer
Correct Answer: V = 10.8 R^{2/3} S^{1/2}
Explanation
Introduction / Context:Regime theory (Lacey) provides empirical relations for stable alluvial channels carrying their sediment load. A frequently used form expresses mean velocity V as a function of hydraulic mean depth R and bed slope S, with a metric coefficient.
Given Data / Assumptions:
- Alluvial channel in regime conditions.
- V in m/s, R in m, S dimensionless bed slope.
- Use standard metric adaptation.
Concept / Approach:
The velocity formula analogous to Chezy/Manning forms but calibrated for regime channels is V = 10.8 R^{2/3} S^{1/2}. The exponents reflect hydraulic similarity; the coefficient 10.8 is the empirical metric constant used in many design examples.
Step-by-Step Solution:
Recall Lacey velocity form V ∝ R^{2/3} S^{1/2}.Insert the metric coefficient 10.8 for regimen channels.Therefore V = 10.8 R^{2/3} S^{1/2} is correct.Verification / Alternative check:
Comparisons with Chezy or Manning yield similar exponent structure; design charts reproduce the 10.8 coefficient for metric units.
Why Other Options Are Wrong:
(a), (c), and (d) use incorrect exponents; (e) uses a nonstandard coefficient.
Common Pitfalls:
Mixing unit systems; confusing Lacey with Manning where coefficients differ and depend on roughness rather than regime conditions.
Final Answer:
V = 10.8 R^{2/3} S^{1/2}