Fluid Mechanics and Soil Mechanics for Soil and Water Conservation

Flow is steady or unsteady, uniform or non-uniform, and is pictured by streamlines. Mass conservation for incompressible flow in a conduit is the continuity equation A₁V₁ = A₂V₂ = Q. Along a streamline of steady, incompressible, frictionless flow, energy per unit weight is conserved: p/(ρg) + V²/(2g) + z = constant — Bernoulli’s theorem, whose three terms are the pressure, velocity and elevation heads. For a real fluid a head-loss term h_L is added between two sections; the venturimeter and orifice meter use Bernoulli with a discharge coefficient to measure pipe flow, and the Pitot tube reads the velocity head directly.

Whether pipe flow is laminar or turbulent is decided by the Reynolds number Re = ρVD/μ = VD/ν: laminar below about 2000, turbulent above about 4000, transitional between. The head lost to friction in a pipe of length L and diameter D is given by the Darcy–Weisbach equation, h_f = fLV²/(2gD), where the friction factor is f = 64/Re in laminar flow and, in turbulent flow, depends on Re and the relative roughness (Moody diagram). For f = 0.02, L = 100 m, D = 0.1 m and V = 2 m/s, h_f = 0.02 × 1000 × 4/19.62 = 4.08 m. The empirical Hazen–Williams equation, V = 0.849 C R^0.63 S^0.54 in SI units (R the hydraulic radius, S the friction slope, C a roughness coefficient that is higher for smoother pipes), is used for water in pressure pipelines and is the usual basis for designing irrigation mains.

The V-notch is preferred for small field flows because its discharge rises as H^(5/2), so a small flow still gives a readable head. For a 90° notch with C_d = 0.6 and H = 0.2 m, Q = (8/15)(0.6)(1)(4.429)(0.2)^2.5 = 0.0254 m³/s = 25.4 L/s. In open channels under uniform flow, Manning’s equation V = (1/n)R^(2/3)S^(1/2) (and Chezy’s V = C√(RS)) relates velocity to the hydraulic radius R = A/P, the bed slope S and the roughness n; for n = 0.025, R = 0.5 m and S = 0.0004, V = 40 × 0.630 × 0.02 = 0.504 m/s. The Froude number Fr = V/√(gy) separates subcritical (Fr < 1) from supercritical (Fr > 1) flow.

Dimensional analysis (Buckingham’s π theorem: n variables in m fundamental dimensions form n − m independent dimensionless groups) and similitude let a model stand for a prototype. Similarity is geometric (all lengths in one ratio), kinematic (velocities in one ratio) and dynamic (force ratios equal), and dynamic similarity is achieved by matching the governing dimensionless number — the Reynolds number where viscosity dominates (pipes), the Froude number where gravity dominates (spillways, channels), and the Weber, Euler and Mach numbers where surface tension, pressure or compressibility do.

Soil is a three-phase system of solids, water and air. The definitions: void ratio e = V_v/V_s; porosity n = V_v/V = e/(1 + e); water content w = W_w/W_s (on the dry weight); degree of saturation S = V_w/V_v; specific gravity of solids G. They are tied by Se = wG and by the unit weights γ_d = Gγ_w/(1 + e) = γ/(1 + w) and γ_sat = (G + e)γ_w/(1 + e). A saturated soil at w = 20 per cent with G = 2.7 has e = 0.54; a soil of bulk unit weight 19.2 kN/m³ at w = 20 per cent has γ_d = 16.0 kN/m³.

Index properties classify a soil: grain-size distribution (sieve and hydrometer analysis; the coefficients of uniformity and curvature), and for fine soils the Atterberg limits — liquid limit LL, plastic limit PL, shrinkage limit — with the plasticity index PI = LL − PL. Water moves through soil by Darcy’s law, v = ki (k the coefficient of permeability, i the hydraulic gradient), and k is measured in the laboratory by the constant-head test k = QL/(Aht) for coarse soils and the falling-head test k = (aL/(At)) ln(h₁/h₂) for fine soils. Seepage under a structure is solved with a flow net: q = kH N_f/N_d per unit width, N_f flow channels and N_d equipotential drops. Where seepage flows upward, the effective stress vanishes at the critical hydraulic gradient i_c = (G − 1)/(1 + e), about 1 for most soils — the quick (boiling) condition that undermines the toe of dams and drop structures.

Compaction densifies soil by expelling air. The Proctor test compacts soil in a standard mould with a standard rammer and number of blows at a series of water contents; dry density rises to a peak, the maximum dry density, at the optimum moisture content (OMC), then falls as water takes the place of solids. The curve lies below the zero-air-voids line γ_d = Gγ_w/(1 + wG), which it can approach but never cross. Greater compactive effort (the modified test) raises the maximum dry density and lowers the OMC. Earthen dams and farm-pond embankments are specified as a percentage of Proctor maximum dry density at a moisture near optimum.

Shear strength follows the Mohr–Coulomb law τ_f = c + σ tan φ (in effective-stress terms c′ and φ′). On Mohr’s circle of stress the failure envelope is tangent to the circle at failure, and the failure plane is inclined at 45° + φ/2 to the major principal plane; for a cohesionless soil σ₁ = σ₃ tan²(45° + φ/2). Rankine’s earth pressure coefficients are K_a = (1 − sin φ)/(1 + sin φ) = tan²(45° − φ/2) for the active case (wall moving away from the soil) and K_p = 1/K_a for the passive case; the active thrust on a smooth vertical wall of height H in dry cohesionless soil is P_a = ½K_aγH², acting at H/3 above the base. For φ = 30°, K_a = 1/3, and a 6 m wall retaining soil of 18 kN/m³ carries P_a = 108 kN per metre. An infinite slope of dry cohesionless soil has a factor of safety FS = tan φ/tan β, independent of depth.

Terzaghi’s one-dimensional consolidation describes a saturated clay squeezing out water under load. The time factor is T_v = c_v t/d², with d the drainage path — half the layer thickness for double drainage, the full thickness for single drainage — and the average degree of consolidation U relates to T_v by T_v = (π/4)U² for U up to about 60 per cent. A 4 m clay layer drained at both faces (d = 2 m) with c_v = 1.0 m²/year reaches 50 per cent consolidation at t = (π/4)(0.25)(4)/1.0 = 0.785 year. The final settlement of a normally consolidated clay is S = C_cH/(1 + e₀) log₁₀((σ₀′ + Δσ)/σ₀′).