Civil Engineering 2023 Paper I 50 marks Solve

Q6

A flow of 9·0 m³/s occurs in a long rectangular channel of 3·0 m width with 1·5 m depth of water flow. There is a smooth constriction in the channel to 2·0 m width in the downstream direction. Answer the following : (i) What depths are to be expected in and just upstream of the constriction, if losses are neglected ? (ii) Classify the gradually varied flow profile upstream of the constriction, with proper justification. 15 A two-dimensional incompressible flow field is given by V = 2xy î + (x² – y²) ĵ , where î and ĵ are the unit vectors along x and y axes, respectively. Answer the following : (i) Determine the magnitude and the angle the velocity vector makes with x-axis at x = 3 m and y = 1 m. (ii) Is the flow physically possible ? If so, determine an expression for stream function. (iii) What is the discharge between the streamlines passing through (1, 0) and (0, 1) ? (iv) Is the flow irrotational ? Justify your answer with appropriate reasons. 15 A retaining wall is shown in the figure below : Layer ① γ = 17 kN/m³ φ' = 28° C = 0 Ground Water Table Layer ② γsat = 20 kN/m³ φ' = 35° C = 0 Assuming that the wall can yield sufficiently, determine the Rankine active force per unit length of the wall and also determine the location of the resultant line of action.

हिंदी में प्रश्न पढ़ें

3·0 m चौड़ी एक लंबी आयताकार वाहिका में 1·5 m की जल प्रवाह की गहराई पर 9·0 m³/s का एक प्रवाह होता है । वाहिका में, अनुप्रवाह की दिशा में 2·0 m की चौड़ाई तक का एक मसृण संकुचन है । निम्नलिखित के उत्तर दीजिए : (i) यदि हानियाँ नगण्य हैं, तो संकुचन में और संकुचन के ठीक प्रतिप्रवाह पर प्रत्याशित गहराइयाँ क्या हैं ? (ii) संकुचन के प्रतिप्रवाह पर क्रमशः-परिवर्ती प्रवाह प्रोफाइल का वर्गीकरण उचित औचित्य देते हुए कीजिए । 15 एक द्वि-विमीय असंपीड्य प्रवाह क्षेत्र V = 2xy î + (x² – y²) ĵ द्वारा दिया गया है, जहाँ î और ĵ क्रमशः x और y अक्षों के साथ एकक सदिश हैं । निम्नलिखित के उत्तर दीजिए : (i) x = 3 m और y = 1 m पर वेग सदिश का परिमाण और इसके द्वारा x-अक्ष के साथ बनाए जाने वाले कोण का निर्धारण कीजिए । (ii) क्या प्रवाह भौतिक रूप में संभव है ? यदि हाँ, तो धारा फलन का व्यंजक निर्धारित कीजिए । (iii) (1, 0) और (0, 1) से गुजरने वाली धारा रेखाओं के बीच निस्सरण क्या है ? (iv) क्या प्रवाह अघूर्णी है ? उचित कारणों के साथ अपने उत्तर का औचित्य सिद्ध कीजिए । 15 एक प्रतिधारक भित्ति नीचे चित्र में दर्शाई गई है : z 3 m γ = 17 kN/m³ φ' = 28° C = 0 परत ① भौम जल स्तर 4 m γsat = 20 kN/m³ φ' = 35° C = 0 परत ② यह मानते हुए कि भित्ति का प्रारंभ पयांस रूप से हो सकता है, रैंकिन का सक्रिय बल भित्ति की प्रति एकक लंबाई पर ज्ञात कीजिए और क्रिया की परिणामी रेखा की स्थिति भी ज्ञात कीजिए ।

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Approach

Solve this multi-part numerical problem by allocating approximately 35% time to the open channel flow sub-parts (i)-(ii), 40% to the fluid mechanics sub-parts (iii)-(iv), and 25% to the retaining wall problem. Begin with clear identification of given data, apply relevant governing equations (specific energy, continuity, stream function theory, Rankine earth pressure), show all computational steps, and conclude with physical interpretation of results including flow classification and pressure distribution diagrams.

Key points expected

  • For (i): Calculate critical depth (yc = 0.972 m) and specific energy (E = 2.028 m), then solve for conjugate depths using specific energy equation at constriction, identifying supercritical and subcritical alternatives
  • For (ii): Classify the GVF profile as M1 curve with proper justification based on normal depth > critical depth and mild slope conditions upstream of constriction
  • For (iii)-(iv): Verify continuity equation (∂u/∂x + ∂v/∂y = 0), derive stream function ψ = x²y - y³/3, calculate velocity magnitude (6.32 m/s) and angle (18.43°), compute discharge between streamlines (0.333 m³/s/m), and check irrotationality via vorticity (ωz = 0)
  • For retaining wall: Calculate active earth pressure coefficients (Ka1 = 0.361, Ka2 = 0.271), determine effective stresses at layer interfaces and groundwater table, compute resultant force per unit length (≈ 180-200 kN/m), and locate centroid of pressure distribution
  • Present clear free-body diagrams for the retaining wall showing pressure distribution with hydrostatic component below GWT and effective stress above

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly applies specific energy principles for open channel flow including Froude number analysis; properly identifies stream function derivation and irrotationality conditions; accurately uses Rankine active earth pressure theory with effective stress concept for submerged soilApplies basic formulas correctly but confuses critical/supercritical flow conditions or makes minor errors in earth pressure coefficient application; partial understanding of stream function propertiesFundamental conceptual errors such as using static pressure instead of specific energy, confusing stream function with velocity potential, or applying total stress instead of effective stress for submerged soil
Numerical accuracy20%10All calculations precise to 3 significant figures; correct identification of both possible depths in constriction (0.685 m and 1.343 m); accurate discharge value of 1/3 m³/s/m; resultant force calculation within 2% of standard solutionMinor arithmetic errors in intermediate steps but correct final approach; one incorrect depth value or small error in stream function evaluation; reasonable force magnitude with calculation errorsSignificant numerical errors (>10% deviation), wrong units, or failure to solve quadratic equations for conjugate depths; incorrect discharge calculation or order-of-magnitude errors in earth pressure
Diagram quality15%7.5Clear specific energy diagram showing E-y curve with critical point and conjugate depths; neat streamlines/sketch of flow field; properly dimensioned retaining wall with linear pressure distribution diagram showing kink at GWT and layer interfaceBasic diagrams present but missing labels or incorrect proportions; pressure diagram shows correct trend but imprecise values; flow sketch lacks detailMissing essential diagrams, unrecognizable sketches, or diagrams that contradict calculated values; no indication of pressure distribution with depth
Step-by-step derivation25%12.5Explicit derivation of critical depth formula, quadratic solution for conjugate depths with both roots analyzed; complete stream function integration showing constant of determination; full Rankine pressure integration with moment calculations for resultant locationKey steps shown but skips algebraic manipulation or assumes intermediate results; partial derivation of stream function; correct pressure formulas but abbreviated integration for force and momentJumps directly to final answers without derivation; no working shown for quadratic solutions; missing integration steps or uses incorrect formulas without justification
Practical interpretation20%10Physically interprets which depth occurs in constriction based on upstream momentum and energy considerations; explains M1 profile significance for canal design; discusses engineering implications of irrotational flow; relates retaining wall results to stability analysis and Indian Standard code provisions (IS 456, IS 6403)Basic physical interpretation of flow classification; mentions practical relevance without elaboration; standard conclusion on wall stability without code referenceNo physical interpretation of numerical results; fails to classify flow profile or explain pressure distribution significance; purely mathematical treatment without engineering context

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