(a) Assuming the law for the velocity distribution in the boundary layer as v/V = 2(y/(δ)) - (y/(δ))², derive an expression for…

(a) Boundary layer thickness: derivation and value at 2.5 m (15 marks)

With u/V = 2η − η² and η = y/δ, the momentum-integral equation for a flat plate (zero pressure gradient) is τ₀ = ρV² dθ/dx, where θ is the momentum thickness.

Momentum thickness.

θ = ∫₀^δ (u/V)(1 − u/V) dy = δ ∫₀¹ (2η − η²)(1 − 2η + η²) dη = δ ∫₀¹ (2η − 5η² + 4η³ − η⁴) dη

θ = δ (1 − 5/3 + 1 − 1/5) = 2δ/15

Wall shear stress. τ₀ = μ (du/dy) at y = 0 = μ (V/δ)(2 − 2η) at η = 0 = 2μV/δ.

Equating. 2μV/δ = ρV² (2/15) dδ/dx, so

δ dδ = (15μ/ρV) dx

Integrating from the leading edge with δ = 0 at x = 0: δ²/2 = 15μx/ρV, so

δ = √(30 μx/ρV) = 5.48 x/√Re_x

Value at x = 2.5 m. With ρ = 1000 kg/m³ (water), V = 0.3 m/s and μ = 0.001 N·s/m²:

Re_x = 1000 × 0.3 × 2.5/0.001 = 7.5 × 10⁵

δ = 5.48 × 2.5/√(7.5 × 10⁵) = 13.7/866 = 0.0158 m ≈ 15.8 mm

The derivation assumes a laminar layer. At this Re_x, above the usual transition value of 5 × 10⁵, the layer may in practice already be turbulent, so 15.8 mm is the value on the laminar assumption asked for.

(b) Channel slope classification and zones of gradually varied flow (15 marks)

The section is triangular with side slope 1:1, so A = y², T = 2y, P = 2√2 y and R = y/(2√2) = 0.3536y.

Normal depth. Manning's equation gives

Q = (1/n) A R^(2/3) S₀^(1/2)

0.2 = (1/0.015) × y² × (0.3536y)^(2/3) × 0.001^(1/2) = 1.054 y^(8/3)

y^(8/3) = 0.1897, so y_n = 0.536 m.

Critical depth. Q²/g = A³/T gives Q²/g = y⁶/2y = y⁵/2, so

y_c⁵ = 2Q²/g = 2 × 0.04/9.81 = 0.008155, so y_c = 0.382 m.

Since y_n (0.536 m) is greater than y_c (0.382 m), the slope is mild. The zones for the type 1, 2 and 3 curves are then:

• Zone 1 (M1 curve): y > y_n, that is y > 0.536 m.
• Zone 2 (M2 curve): y_c < y < y_n, that is 0.382 m < y < 0.536 m.
• Zone 3 (M3 curve): y < y_c, that is y < 0.382 m.

(c)(i) Total passive pressure and point of application (10 marks)

The wall is smooth and vertical, so Rankine's passive theory applies, with φ′ = 20°, c′ = 35 kPa, γ = 19.5 kN/m³, H = 5 m and surcharge q = 60 kPa.

K_p = tan²(45° + φ′/2) = tan² 55° = 2.04, and √K_p = 1.428.

Passive pressure at the top: σ_p,top = qK_p + 2c′√K_p = 60 × 2.04 + 2 × 35 × 1.428 = 122.4 + 100.0 = 222.3 kPa

Passive pressure at the base: σ_p,base = 222.3 + γHK_p = 222.3 + 19.5 × 5 × 2.04 = 222.3 + 198.9 = 421.2 kPa

The diagram is a rectangle of intensity 222.3 kPa plus a triangle rising to 198.9 kPa. For a metre run of wall:

P₁ (rectangle) = 222.3 × 5 = 1111.7 kN/m, acting 2.5 m above the base

P₂ (triangle) = ½ × 198.9 × 5 = 497.2 kN/m, acting 5/3 = 1.667 m above the base

Total P_p = 1608.9 kN/m ≈ 1609 kN per metre run.

Height of the resultant above the base: ȳ = (1111.7 × 2.5 + 497.2 × 1.667)/1608.9 = 3608/1608.9 = 2.24 m, a little below mid-height, because the cohesion and surcharge terms are uniform with depth.

(c)(ii) Earthquake effects on the stability of earth dams (10 marks)

An earthquake affects an earth dam in four main ways:

• Inertia forces. The horizontal and vertical ground accelerations add inertia forces on the embankment, taken in the pseudo-static method as α_h W and α_v W with seismic coefficients from IS 1893 (Part 5). These add to the driving force on potential slip surfaces and reduce the factor of safety of the slopes, especially the downstream slope in the reservoir-full condition.
• Pore-pressure build-up and liquefaction. Cyclic shear raises pore water pressure in loose saturated cohesionless soils in the shell or foundation. Effective stress and strength fall, and in the extreme the soil liquefies, which can cause flow slides and a sudden loss of the dam.
• Deformation. Slumping, settlement and loss of freeboard, longitudinal and transverse cracks at the crest and abutment contacts, which can start piping through the crack, and sliding of the dam along a weak foundation layer.
• Hydrodynamic effects. Added dynamic water pressure on the upstream face, and waves or a seiche in the reservoir that can overtop the crest.

The measures to control the effects are flatter slopes, a wider crest and extra freeboard, dense compaction of the shell, removal or densification of liquefiable foundation soil, a well-graded filter and chimney drain to keep the phreatic surface low, crack-resistant transition zones, and design checks by pseudo-static and, for important dams, dynamic analysis.