--- title: "Specific force and conjugate depths" subject: "Hydraulics" chapter: 7 key: "semester_4/hydraulics:specific-force-curve-conjugate-depths" source: https://www.bctengineer.com/q/semester_4/hydraulics/specific-force-curve-conjugate-depths site: BCT Engineering --- # Specific force and conjugate depths Hydraulics · Chapter 7: Energy and Momentum Principles in Open channel flow - Asked in: 2077 Chaitra, 4 marks; 2070 Bhadra, 1+2 marks (asked 2 times) - Video: [Specific Force Diagram | Open Channel Flow | Hydraulics and Fluid Mechanics](https://www.youtube.com/watch?v=1SP4CDxAdj0) — The Content Paradox - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/hydraulics/specific-force-curve-conjugate-depths - Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-7 ## Question Develop the expression for specific force and explain the concept of conjugate depths using the specific force curve (sketch the specific force curve showing conjugate depths and the zones of subcritical, critical and supercritical flow). ## Short answer Specific force F = Q²/(gA) + Aȳ, or per unit width F = q²/(g y) + y²/2. Conjugate depths are the two depths having equal specific force, one supercritical and one subcritical, before and after a hydraulic jump. The specific force curve has a lower supercritical limb, an upper subcritical limb, and a minimum at critical depth. ## Answer #### Specific force (momentum function) Specific force $F$ at a section is the sum of the momentum flux and the hydrostatic pressure force per unit weight of water: $$ F=\frac{Q^2}{gA}+A\bar y $$ where $\bar y$ = depth of the centroid of the flow area below the water surface. #### Development (from the momentum equation) For a short horizontal reach (friction and the weight component negligible), the net force equals the change in momentum flux: $$ \gamma A_1\bar y_1-\gamma A_2\bar y_2=\rho Q(V_2-V_1)=\rho Q^2\left(\frac{1}{A_2}-\frac{1}{A_1}\right) $$ $$ \frac{Q^2}{gA_1}+A_1\bar y_1=\frac{Q^2}{gA_2}+A_2\bar y_2\;\Rightarrow\;F_1=F_2 $$ For a rectangular channel of width $b$ ($q=Q/b$, $A\bar y=by^2/2$), the specific force per unit width is $$ F=\frac{q^2}{gy}+\frac{y^2}{2} $$ #### Conjugate (sequent) depths For a given $Q$ and a value of $F>F_{min}$, the equation $F(y)=$ const has two positive roots, $y_1$ (supercritical) and $y_2$ (subcritical). These are the **conjugate depths**: they have the same specific force. They are the depths before and after a hydraulic jump, where energy is lost but specific force is conserved. For a rectangular channel: $$ \frac{y_2}{y_1}=\frac12\left(\sqrt{1+8Fr_1^2}-1\right) $$ #### Specific force curve ```text y ^ | . upper limb: subcritical y2 |----------------* conjugate depth y2 | . | . yc |-------* <- F_min, critical depth | . | . y1 |----------------* conjugate depth y1 | . lower limb: supercritical +-------+--------+-------> F F_min F1 = F2 ``` - Lower limb ($yy_c$): subcritical flow. - A vertical line at a given $F$ cuts the curve at the two conjugate depths. The difference in specific energy between these depths is the energy loss in the jump. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Specific force per unit weight at a section is A. Q^2/(gA) + A*ybar B. Q^2/(2gA^2) + y C. Q/A + ybar D. gA*ybar + Q Answer: A. F is the sum of momentum flux and hydrostatic pressure force, per unit weight: F = Q^2/(gA) + A*ybar. (Key point: Specific force expression) 2. Which principle gives F1 = F2 between two sections of a short horizontal reach? A. Continuity equation B. Bernoulli equation with no loss C. Manning's equation D. Momentum equation, with friction and weight component neglected Answer: D. Net force (pressure difference) equals change in momentum flux, which rearranges to equal specific force at both sections. (Key point: Origin of F1 = F2) 3. What are conjugate depths? A. Two depths with the same specific energy B. Two depths with the same specific force, supercritical y1 and subcritical y2, before and after a hydraulic jump C. Critical depth and normal depth D. Depths on either side of a hump Answer: B. Conjugate depths share the same F for a given Q; energy is lost across the jump but specific force is conserved. (Key point: Conjugate depths) 4. On the specific force curve, the nose (minimum F) corresponds to A. Supercritical flow with Fr > 1 B. Subcritical flow on the upper limb C. Zero depth D. Critical depth, where Q^2T/(gA^3) = 1 Answer: D. At F = Fmin the depth is yc and Fr = 1; the lower limb is supercritical, the upper limb subcritical. 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