Civil Engineering 2022 Paper I 50 marks Solve

Q6

(a) The velocity profile in a laminar boundary layer on a flat plate is modelled by the cubic expression μ/μ₁ = a₀ + a₁y + a₂y² + a₃y³ μ = velocity at a distance y from the surface of the plate μ₁ = main stream velocity Evaluate all the constants in terms of boundary layer thickness. Draw the velocity distribution and stress distribution curves. Indicate the application and significance of boundary layer. If the plate is moving with a velocity of 2 m/s in positive x-direction, what will be the velocity distribution curve? (15 marks) (b) Plot the variations of (i) total pressure, (ii) neutral stress and (iii) effective stress for a fine sand deposit, having a porosity of 40% and specific gravity of 2.7, extending to a depth of 10 m below the ground surface. The groundwater table is 5 m below the ground surface and the sand is saturated by capillary water up to a height of 1 m above the water table. The degree of saturation of the first 4 m of moist soil below the ground surface is 10%. Take the unit weight of water as 10 kN/m³. (20 marks) (c) Reservoir A (elevation 65 m) is filling reservoir B (elevation 110 m) and reservoir C (elevation 90 m) by a pump and pipe system. The discharge to reservoir C is 0.10 m³/s. If the efficiency of the pump is 0.70, calculate the required power of the pump. The physical characteristics of the pipe system are given in the figure below. Neglect the minor losses. Draw the HGL and EGL : (10 marks)

हिंदी में प्रश्न पढ़ें

(a) एक चपटी प्लेट पर स्तरीय परिसीमा परत में वेग परिछेदिका (प्रोफाइल) को निम्न घनीय व्यंजक द्वारा निर्देशित किया गया है : μ/μ₁ = a₀ + a₁y + a₂y² + a₃y³ μ = प्लेट की सतह से y दूरी पर वेग μ₁ = मुख्य धारा वेग सभी नियतांकों का मान परिसीमा परत की मोटाई के रूप में ज्ञात कीजिए। वेग वितरण और प्रतिबल वितरण वक्रों को आरेखित कीजिए। परिसीमा परत का उपयोग और महत्व बताइए। यदि प्लेट 2 m/s के वेग से धनात्मक x-दिशा में चल रही है, तो वेग वितरण वक्र क्या होगा? (15 अंक) (b) धरातल से 10 m तक की गहराई वाले एक महीन रेत के निक्षेप, जिसकी संरचना 40% और जिसका विशिष्ट घनत्व 2.7 है, के लिए (i) कुल दाब, (ii) उदासीन प्रतिबल तथा (iii) प्रभावी प्रतिबल के परिवर्तन को आरेखित कीजिए। भूमजल स्तर धरातल से 5 m नीचे है और रेत केशिकीय जल द्वारा भूमजल स्तर से 1 m ऊपर तक संतृप्त है। धरातल की नीचे की प्रथम 4 m की नम मृदा की संतृप्ति मात्रा 10% है। जल का एकक भार 10 kN/m³ लीजिए। (20 अंक) (c) एक पंप और पाइप तंत्र द्वारा जलाशय A (ऊँचाई 65 m), जलाशय B (ऊँचाई 110 m) और जलाशय C (ऊँचाई 90 m) को भर रहा है। जलाशय C में निस्सरण 0.10 m³/s है। यदि पंप की दक्षता 0.70 है, तो पंप की आवश्यक शक्ति की गणना कीजिए। पाइप तंत्र के भौतिक अभिलक्षण नीचे चित्र में दिए गए हैं। लघु हानियों की उपेक्षा कीजिए। एच० जी० एल० और ई० जी० एल० को आरेखित कीजिए : (10 अंक)

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Approach

Begin with a brief introduction acknowledging the three distinct domains: fluid mechanics boundary layer theory, geotechnical stress distribution, and hydraulic machinery with pipe networks. Allocate approximately 33% effort to part (a) due to its conceptual depth and drawing requirements, 44% to part (b) as it carries the highest marks with detailed plotting, and 23% to part (c) focusing on energy equation application and HGL/EGL construction. Present each part sequentially with clear sub-headings, showing all derivations and calculations before concluding with integrated insights on civil engineering applications.

Key points expected

  • For (a): Apply boundary conditions (no-slip at y=0, u=u₁ at y=δ, du/dy=0 at y=δ, and d²u/dy²=0 at y=0 for cubic profile) to determine constants a₀=0, a₁=3/(2δ), a₂=0, a₃=-1/(2δ³), yielding u/u₁ = (3/2)(y/δ) - (1/2)(y/δ)³
  • For (a): Sketch velocity profile showing S-shape with inflection point and shear stress distribution linearly decreasing from maximum at wall to zero at δ; for moving plate, superpose plate velocity on relative velocity profile
  • For (b): Calculate unit weights: γ moist = 18.9 kN/m³ (0-4m), γ sat = 19.8 kN/m³ (4-5m), γ sat = 19.8 kN/m³ below GWT; account for capillary saturation zone with negative pore pressure
  • For (b): Plot total stress increasing linearly from 0 to 189 kPa at 10m; neutral stress zero to -10 kPa (capillary), then 0 to 50 kPa; effective stress showing discontinuity handling at interfaces with proper capillary tension effects
  • For (c): Apply energy equation between A and B (pumped system) and A to C (gravity-assisted), determine flow to B using continuity, compute total dynamic head, and calculate pump power P = ρgQH/η ≈ 85-90 kW
  • For (c): Draw HGL and EGL showing: EGL starting at A, jumping at pump, gradual slope in pipes, drops at junction; HGL parallel below by velocity head, with proper elevation references to reservoirs B and C

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness22%10Correctly applies all four boundary conditions for cubic profile in (a); accurately identifies capillary zone mechanics and effective stress principle in (b); properly recognizes parallel pipe network with different outlet elevations in (c); cites Blasius solution or von Kármán momentum integral for boundary layer significanceIdentifies basic boundary conditions but misses one condition (typically inflection point) in (a); calculates unit weights correctly but mishandles capillary pore pressure sign or magnitude in (b); treats system as simple series pipe or ignores flow distribution in (c)Fundamental errors in boundary condition application; confuses total and effective stress concepts; treats pump head as static lift only without considering friction losses and velocity heads
Numerical accuracy24%11Precise calculation: (a) exact constants a₀=0, a₁=3/(2δ), a₂=0, a₃=-1/(2δ³); (b) γ moist=18.9 kN/m³, γ sat=19.8 kN/m³, correct stress values at all depths with capillary tension -10 kPa; (c) flow distribution Q_B=0.14-0.15 m³/s, total head ~60-65m, power ~85-92 kW with η=0.70Correct methodology but arithmetic errors in constants or unit weights; minor errors in stress calculations at layer interfaces; reasonable pump head but incorrect flow split or power calculation off by 10-20%Major calculation errors: wrong boundary layer constants; order-of-magnitude errors in unit weights (e.g., using γ_w instead of γ_sat); completely wrong pump power or failure to account for efficiency
Diagram quality20%9Clear labeled sketches: (a) velocity profile with proper S-curve shape, linear shear stress decay, and transformed profile for moving plate; (b) three separate stress plots or combined plot with distinct lines showing discontinuities at 4m, 5m, and capillary zone; (c) HGL/EGL to scale with pump head jump, proper slopes, and reservoir levelsRecognizable diagrams with most features present but poor labeling, missing key features (inflection point, capillary zone detail), or incorrect relative positioning of curves; HGL/EGL shows general trend but not to scaleMissing diagrams or unrecognizable sketches; confused velocity and stress distributions; HGL below HGL or both above EGL; no indication of scale or key points
Step-by-step derivation18%8Systematic derivation: (a) explicit statement of four boundary conditions, solution of 4×4 system for constants; (b) phase-by-phase unit weight calculation, incremental stress computation with clear depth intervals; (c) Bernoulli equation setup for both branches, simultaneous solution for unknowns, explicit power formula substitutionShows main steps but skips intermediate algebra (jumping to final constants); presents stress values without showing incremental calculation; applies energy equation but skips continuity equation or assumes equal flowsNo derivations shown—only final answers; or completely wrong approach with nonsensical steps; confuses derivations between parts
Practical interpretation16%7Explicit applications: (a) drag reduction, aircraft wing design, wind tunnel corrections, sports ball aerodynamics relevant to Indian context (cricket ball swing); (b) foundation design in Ganga basin alluvial deposits, excavation stability, liquefaction potential; (c) multi-reservoir water supply systems like Chennai or Bangalore metropolitan schemes, energy optimizationGeneric mention of boundary layer importance, soil mechanics relevance, or pump applications without specific engineering context or Indian examplesNo interpretation provided; or completely irrelevant applications (confusing boundary layer with turbulent mixing, effective stress with consolidation settlement only)

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