Fluid Mechanics Interview Questions

Reynolds number (Re) is a dimensionless quantity that represents the ratio of inertial forces to viscous forces in a fluid flow. It is calculated as Re = ρVD/μ or Re = VD/ν, where ρ is density, V is velocity, D is characteristic length, μ is dynamic viscosity, and ν is kinematic viscosity. Reynolds number is significant because it predicts whether flow will be laminar (Re < 2300 for pipe flow) or turbulent (Re > 4000), which affects pressure drop, heat transfer, and mixing characteristics.

Dynamic viscosity (μ) measures a fluid's internal resistance to flow when an external force is applied, with units of Pa·s or Poise. Kinematic viscosity (ν) is the ratio of dynamic viscosity to fluid density (ν = μ/ρ), with units of m²/s or Stokes. Dynamic viscosity is used when dealing with forces and stresses, while kinematic viscosity is preferred in equations involving gravity-driven flows or when density effects are considered separately.

Bernoulli's equation states that for steady, incompressible, inviscid flow along a streamline: P₁/ρg + V₁²/2g + z₁ = P₂/ρg + V₂²/2g + z₂, where P is pressure, V is velocity, z is elevation, ρ is density, and g is gravitational acceleration. The key assumptions are: (1) flow is steady, (2) flow is incompressible, (3) flow is inviscid (frictionless), (4) flow is along a streamline, and (5) no work is done by or on the fluid between the two points.

Laminar flow is characterized by smooth, orderly fluid motion in parallel layers with no mixing between layers, occurring at low Reynolds numbers (Re < 2300 in pipes). Turbulent flow has chaotic, irregular motion with significant mixing, eddies, and fluctuations, occurring at high Reynolds numbers (Re > 4000). The transition region exists between Re 2300-4000. Turbulent flow has higher energy losses due to increased friction but provides better heat and mass transfer due to enhanced mixing.

The continuity equation states that for steady flow, mass flow rate remains constant throughout a flow system: ρ₁A₁V₁ = ρ₂A₂V₂. For incompressible flow, this simplifies to A₁V₁ = A₂V₂, meaning velocity increases when cross-sectional area decreases and vice versa. Practical applications include designing nozzles and diffusers, calculating flow velocities in variable-area ducts, sizing pipes for required flow rates, and analyzing venturi meters and flow measurement devices.

Newtonian fluids have a constant viscosity that remains unchanged regardless of shear rate, following the relationship τ = μ(du/dy). Examples include water, air, light oils, and glycerin. Non-Newtonian fluids have viscosity that varies with shear rate or applied stress. Types include: shear-thinning (paint, blood, ketchup), shear-thickening (cornstarch in water), Bingham plastics (toothpaste, drilling mud), and viscoelastic fluids (polymer solutions). Understanding fluid behavior is crucial for pump selection and process design.

Pascal's law states that pressure applied to an enclosed fluid is transmitted equally in all directions throughout the fluid and acts perpendicular to all surfaces in contact with the fluid. Mathematically: ΔP = F₁/A₁ = F₂/A₂. Applications include hydraulic lifts and jacks (force multiplication), hydraulic brakes in vehicles, hydraulic presses for metal forming, hydraulic actuators in construction equipment, and aircraft control systems. The mechanical advantage is the ratio of output area to input area.

A venturi meter measures fluid flow rate by utilizing Bernoulli's principle. It consists of a converging section, throat (minimum area), and diverging section. As fluid enters the converging section, velocity increases and pressure decreases according to Bernoulli's equation. The pressure difference between the inlet and throat is measured using a differential pressure gauge. Flow rate is calculated as Q = Cd × A₂ × √(2ΔP/ρ(1-(A₂/A₁)²)), where Cd is the discharge coefficient (typically 0.95-0.99). Venturi meters have low permanent pressure loss compared to orifice meters.

A centrifugal pump converts mechanical energy to hydraulic energy using a rotating impeller. Fluid enters the pump through the suction eye at the impeller center. As the impeller rotates (typically 1450-2900 RPM), centrifugal force accelerates the fluid radially outward through the impeller vanes. The fluid gains velocity, and this kinetic energy is converted to pressure energy in the volute casing or diffuser. The curved volute gradually increases in area, causing velocity to decrease and pressure to increase according to Bernoulli's principle.

Absolute pressure is measured relative to a perfect vacuum (zero reference), while gauge pressure is measured relative to local atmospheric pressure. The relationship is: P_absolute = P_gauge + P_atmospheric. At sea level, atmospheric pressure is approximately 101.325 kPa or 14.7 psi. Gauge pressure can be negative (vacuum condition) when below atmospheric pressure. Absolute pressure is used in gas law calculations and compressible flow analysis, while gauge pressure is commonly used in industrial applications for pump discharge, tank pressures, and piping systems.

Specific gravity (SG) is the ratio of a fluid's density to the density of a reference substance (water at 4°C for liquids, air at standard conditions for gases). Being dimensionless, SG = ρ_fluid/ρ_reference. For water at 4°C, SG = 1.0. Applications include: calculating fluid density (ρ = SG × 1000 kg/m³), determining buoyancy forces, selecting appropriate materials for tanks and pipes, sizing pumps for different fluids, and converting between mass and volume flow rates. Common values: gasoline (0.72-0.78), crude oil (0.8-0.95), glycerin (1.26).

Archimedes' principle states that a body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the body: F_b = ρ_fluid × V_displaced × g. A body floats when its weight equals the buoyant force. Applications include: designing ships and submarines (hull displacement calculations), hydrometers for measuring liquid density, hot air balloons, determining purity of metals, floatation separation in mineral processing, and analyzing stability of floating structures like offshore platforms.

Head represents energy per unit weight of fluid and has three main forms: (1) Pressure head (P/ρg) - energy due to fluid pressure, measured in meters of fluid column. (2) Velocity head (V²/2g) - kinetic energy of flowing fluid. (3) Elevation head (z) - potential energy due to height above a reference datum. The total head is the sum of all three: H_total = P/ρg + V²/2g + z. In pump systems, we also consider friction head (energy lost to friction) and pump head (energy added by pump).

Incompressible flow assumes constant fluid density regardless of pressure changes, valid when density variation is less than 5% (Mach number < 0.3). Liquids are typically treated as incompressible. Compressible flow involves significant density changes with pressure, important for gases at high velocities. The criterion is Mach number: M = V/c, where c is speed of sound. At M < 0.3, compressibility effects are negligible. Compressible flow analysis is essential for aircraft design, gas pipelines, steam turbines, and high-speed nozzles.

Surface tension is the property of liquid surfaces to behave like elastic membranes due to cohesive forces between surface molecules. It is measured as force per unit length (N/m). Capillary action occurs due to the interplay between adhesive forces (liquid-solid) and cohesive forces (liquid-liquid). When adhesive forces dominate (water in glass tube), liquid rises; when cohesive forces dominate (mercury in glass), liquid depresses. Capillary rise is h = 4σcosθ/(ρgd), where σ is surface tension, θ is contact angle, and d is tube diameter.

A pump curve (H-Q curve) shows the relationship between head developed and flow rate for a specific pump at constant speed, typically decreasing as flow increases. The system curve represents total head required by the piping system versus flow rate, including static head and friction losses (which increase with flow squared). The operating point is where these curves intersect, indicating the actual head and flow rate the pump will deliver in that system. If the operating point is not optimal, impeller trimming, speed variation (VFD), or pump selection revision may be needed.

NPSH (Net Positive Suction Head) is the absolute pressure available at pump suction minus vapor pressure of the liquid. NPSH_available = (P_atm - P_vapor)/ρg + h_s - h_f, where h_s is suction head and h_f is friction loss. NPSH_required is specified by pump manufacturer. Cavitation occurs when NPSH_A < NPSH_R, causing vapor bubble formation and collapse, damaging impeller. Prevention methods: increase suction tank level, reduce suction pipe losses, use larger suction pipes, reduce liquid temperature, select pump with lower NPSH_R, or use a booster pump.

The Darcy-Weisbach equation calculates pressure drop due to friction in pipes: h_f = f × (L/D) × (V²/2g), where f is Darcy friction factor, L is pipe length, D is diameter, and V is velocity. For laminar flow (Re < 2300): f = 64/Re. For turbulent flow, use Colebrook-White equation: 1/√f = -2log₁₀(ε/3.7D + 2.51/Re√f) or Moody diagram. Relative roughness (ε/D) depends on pipe material: commercial steel (0.046mm), PVC (0.0015mm), cast iron (0.26mm). This equation is fundamental for pipe sizing and pump head calculations.

Minor losses (local losses) occur at fittings, valves, bends, expansions, and contractions due to flow separation and turbulence. They are calculated using loss coefficients (K-factors): h_L = K × V²/2g. Typical K-values: 90° elbow (0.3-0.9), gate valve fully open (0.2), globe valve (6-10), sudden expansion ((1-A₁/A₂)²), sudden contraction (0.5), entrance sharp-edged (0.5), exit (1.0). For long pipelines, minor losses are typically 5-10% of friction losses, but in compact systems with many fittings, they can dominate total losses.

Pump affinity laws relate pump performance at different speeds or impeller diameters: (1) Flow: Q₂/Q₁ = (N₂/N₁) or (D₂/D₁), (2) Head: H₂/H₁ = (N₂/N₁)² or (D₂/D₁)², (3) Power: P₂/P₁ = (N₂/N₁)³ or (D₂/D₁)³. Applications include: predicting performance when using VFD speed control, estimating effect of impeller trimming, scaling from test data to actual conditions. Important limitation: these laws are approximate and assume geometrically similar flow conditions; significant deviations occur at extreme speed changes or major impeller trim.

Centrifugal pumps use kinetic energy transfer via rotating impeller; flow varies with system resistance while head is relatively constant. Best for high flow, low-medium viscosity fluids. Positive displacement (PD) pumps trap fixed volumes and push them through discharge; flow is nearly constant regardless of pressure, ideal for high viscosity, accurate metering, and high-pressure applications. PD types include reciprocating (piston, diaphragm), rotary (gear, screw, lobe). Selection criteria: centrifugal for clean, low-viscosity, high-flow; PD for viscous fluids, precise dosing, or high pressure.

When fluid flows over a flat plate, viscous effects create a boundary layer where velocity varies from zero at the surface (no-slip condition) to freestream velocity at boundary layer edge (δ). Initially, flow is laminar with parabolic velocity profile; boundary layer thickness grows as δ ∝ √(νx/U). At critical Reynolds number (Re_x ≈ 5×10⁵), transition to turbulent boundary layer occurs with fuller velocity profile and faster growth. Turbulent boundary layer has higher skin friction but better momentum transfer. Understanding boundary layers is crucial for drag prediction, heat transfer, and flow separation analysis.

Both measure flow using differential pressure from Bernoulli's principle. Orifice meter: simple flat plate with hole, low cost ($100-500), easy installation, but high permanent pressure loss (40-80% of differential), Cd ≈ 0.6-0.65. Venturi meter: converging-diverging design, higher cost ($1000-5000), requires more space, but low permanent loss (10-20%), Cd ≈ 0.95-0.99. Venturi better for large pipes, abrasive fluids, and when pumping cost is significant. Orifice preferred for small pipes, clean fluids, and limited budget. Both require straight pipe runs upstream (15-40D) for accuracy.

Energy Grade Line (EGL) represents total energy head at any point: EGL = P/ρg + V²/2g + z. It decreases in flow direction due to friction losses and has sudden drops at minor losses. Hydraulic Grade Line (HGL) represents pressure head plus elevation: HGL = P/ρg + z = EGL - V²/2g. The vertical distance between EGL and HGL equals velocity head. HGL can rise or fall depending on velocity changes (pipes expanding/contracting). If HGL falls below pipe centerline, pressure becomes subatmospheric, risking cavitation. These lines are essential for pump placement and pipeline design.

Water hammer is a pressure surge caused by sudden velocity changes in liquid flow, typically from valve closure or pump shutdown. Pressure rise is ΔP = ρ×c×ΔV, where c is wave speed (1000-1400 m/s in water). This can cause pipe rupture, joint failure, and equipment damage. Prevention methods: slow valve closure (closure time > 2L/c), surge tanks or accumulators, air chambers, pressure relief valves, controlled pump startup/shutdown using VFDs, non-return valves with dashpots, proper pipeline anchoring, and avoiding air pockets that can accelerate water columns.

Pelton turbine: impulse type, uses high-velocity water jets hitting buckets on runner, suitable for high head (300-2000m) and low flow, efficiency 85-90%. Francis turbine: reaction type, water flows radially inward through guide vanes to runner then axially out, suitable for medium head (40-600m) and flow, most common type, efficiency 90-95%. Kaplan turbine: reaction type with adjustable runner blades, water flows axially, suitable for low head (10-70m) and high flow (run-of-river plants), efficiency 90-93%. Selection based on specific speed, head, and flow characteristics of the site.

CFD solves three fundamental conservation equations: (1) Continuity equation (mass conservation): ∂ρ/∂t + ∇·(ρV) = 0. (2) Momentum equations (Navier-Stokes): ρDV/Dt = -∇P + μ∇²V + ρg, representing Newton's second law for fluids with pressure, viscous, and body forces. (3) Energy equation: ρD(e)/Dt = -P(∇·V) + ∇·(k∇T) + Φ, where Φ is viscous dissipation. These partial differential equations are discretized using finite difference, finite volume, or finite element methods and solved iteratively. Additional equations are needed for turbulence (RANS, LES) and multiphase flows.

Reynolds-Averaged Navier-Stokes (RANS) decomposes flow variables into mean and fluctuating components (Reynolds decomposition), averaging equations over time. This introduces Reynolds stresses that require closure models. Common models: (1) k-ε standard: robust, widely used for fully turbulent flows, struggles with separation. (2) k-ω: better near-wall behavior, sensitive to freestream values. (3) k-ω SST: combines k-ε far-field with k-ω near-wall, excellent for adverse pressure gradients. (4) Spalart-Allmaras: one-equation model for aerospace. Model selection depends on flow physics: separated flows need SST or RSM, jets/mixing need k-ε realizable.

A Pitot tube measures fluid velocity by sensing stagnation and static pressures. The Pitot tube opening faces the flow, bringing fluid to rest (stagnation point) and measuring total pressure (P_total = P_static + ρV²/2). A separate static pressure tap measures ambient pressure. Applying Bernoulli's equation: V = √(2(P_total - P_static)/ρ). For compressible flows, temperature correction is needed. Pitot-static tubes combine both measurements in one probe. Applications: aircraft airspeed indicators, duct velocity measurement, and point velocity surveys in pipes. Accuracy depends on alignment with flow direction (typically ±15° tolerance).

Specific speed (Ns) is a dimensionless parameter characterizing pump type and impeller geometry: Ns = N√Q/H^(3/4), where N is RPM, Q is flow (m³/s or GPM), and H is head (m or ft). Different Ns ranges correspond to different impeller types: low Ns (500-1000 US units): radial flow, high head; medium Ns (1000-4000): mixed flow; high Ns (4000-15000): axial flow, high flow/low head. Specific speed helps select appropriate pump type for given duty, predict efficiency, and scale performance. Pumps with similar Ns have similar efficiency curves and hydraulic characteristics.

Manometers measure pressure difference using liquid column height. Types: (1) Simple U-tube: measures gauge or differential pressure, ΔP = ρgh. (2) Inclined manometer: increases sensitivity for low pressures by angling tube. (3) Differential manometer: measures pressure difference between two points in a system. (4) Inverted manometer: uses lighter fluid (air) above manometric liquid for small pressure differences. (5) Micromanometer: precision instrument for very small pressures. Selection criteria: measurement range, required accuracy, fluid compatibility. Mercury used for high pressures (high density), water or oil for low pressures.

The linear momentum equation states: ΣF = d(mV)/dt = ṁ(V_out - V_in), where ṁ is mass flow rate. For steady flow through a control volume, forces include pressure forces, weight, and reaction forces from solid boundaries. For a pipe bend: F_x = P₁A₁ - P₂A₂cosθ + ṁ(V₁ - V₂cosθ) and F_y = P₂A₂sinθ + ṁV₂sinθ. These forces must be resisted by pipe supports or anchors. Applications include: designing pipe anchors and supports, calculating thrust blocks for bends, analyzing jet forces on turbine blades, and sizing reaction forces in nozzles.

Flow separation occurs when boundary layer detaches from a surface due to adverse pressure gradient (pressure increasing in flow direction). The boundary layer loses momentum to overcome pressure rise, velocity near wall reverses, and flow separates forming recirculation zones. Effects include increased drag, pressure loss, vibration, and reduced efficiency. Prevention methods: streamlined body shapes with gradual area changes, boundary layer suction, vortex generators to energize boundary layer, slot or blown jets to add momentum, dimples (golf ball effect) to trip turbulent boundary layer earlier, and maintaining favorable pressure gradients in diffuser design (expansion angle < 7°).

Series operation: pumps connected sequentially, flow remains constant while heads add. Used when single pump cannot provide required head (e.g., multistage pumps, booster applications). Combined curve shows same flow with doubled head. Parallel operation: pumps share common suction and discharge headers, heads equalize while flows add. Used to increase flow capacity or provide redundancy. Combined curve shows doubled flow at same head. Parallel operation requires matching pump characteristics; pumps with significantly different curves may cause one pump to operate inefficiently or even run out (low flow, high temperature). VFDs help balance parallel pump operation.

Key mesh quality metrics: (1) Skewness: deviation from ideal element shape (target < 0.85, ideal < 0.5). (2) Aspect ratio: longest to shortest edge ratio (target < 5:1 for tets, can be higher for boundary layers). (3) Orthogonality: angle between face normal and line connecting cell centers (target > 0.1, ideal > 0.3). (4) Smoothness: gradual size transition between adjacent cells (growth rate < 1.2). (5) Y+ value: dimensionless wall distance for turbulence modeling (y+ ≈ 1 for resolving viscous sublayer, y+ > 30 for wall functions). Poor mesh quality causes convergence issues, numerical diffusion, and inaccurate results.

DNS (Direct Numerical Simulation) resolves all turbulent scales without modeling, requiring mesh size smaller than Kolmogorov scale (η = (ν³/ε)^(1/4)). Grid points scale as Re^(9/4), making it computationally prohibitive for industrial Re (>10⁶). Used for research and model validation. LES (Large Eddy Simulation) resolves large energy-containing eddies while modeling subgrid scales using SGS models (Smagorinsky, dynamic). Requires mesh resolving 80% of turbulent kinetic energy. 10-100x cheaper than DNS but still expensive. RANS averages all turbulence, solving time-averaged equations with turbulence models. 1000x cheaper than LES, suitable for industrial design. Hybrid DES/SAS methods combine RANS near walls with LES in separated regions.

Shock waves are discontinuous changes in flow properties occurring when supersonic flow encounters disturbance. Across a normal shock: velocity decreases (supersonic to subsonic), pressure and temperature increase abruptly, entropy increases (irreversible), and total pressure decreases. Rankine-Hugoniot relations connect upstream (1) and downstream (2) properties: ρ₂/ρ₁ = (γ+1)M₁²/((γ-1)M₁²+2), P₂/P₁ = (2γM₁²-(γ-1))/(γ+1), T₂/T₁ = P₂ρ₁/(P₁ρ₂). Oblique shocks form at angle β to flow direction, with downstream flow deflected by angle θ. Detached bow shocks form when θ exceeds maximum for given M₁. Understanding crucial for supersonic inlets, nozzles, and high-speed aerodynamics.

Multiphase flow modeling depends on phase fractions and interface resolution needs: (1) Euler-Euler: treats both phases as interpenetrating continua with separate conservation equations; includes Volume of Fluid (VOF) for immiscible fluids with sharp interface tracking, Mixture model for homogeneous flows, and Eulerian model for dispersed phases. (2) Euler-Lagrange: continuous phase solved on Eulerian grid while dispersed phase (particles, droplets, bubbles) tracked as Lagrangian particles; suitable for dilute dispersions (<10% volume fraction). (3) Interface tracking: Level-Set and VOF for accurate surface tension and interface dynamics. Model selection depends on coupling strength, phase fraction, size distribution, and computational resources. Applications: fluidized beds, sprays, boiling, sedimentation.

Pump recirculation occurs at low flows when fluid recirculates within impeller passages, causing suction and discharge recirculation vortices. Suction recirculation causes pre-rotation, cavitation damage, and pressure pulsations. Discharge recirculation creates backflow at impeller discharge, reducing efficiency and causing vibrations. Minimum continuous stable flow (MCSF) is typically 25-50% of BEP flow depending on specific speed and suction energy level. Determination methods: acoustic monitoring for onset of recirculation noise, vibration analysis showing broadband increase, manufacturer's recommendations based on Ns and suction specific speed, thermal considerations (temperature rise = P(1-η)/(ṁCp)). Protection via minimum flow bypass with control valve or orifice.

Method of Characteristics (MOC) solves hyperbolic partial differential equations governing unsteady pipe flow by transforming them into ordinary differential equations along characteristic lines. The governing equations (continuity and momentum) propagate as waves at speed a (acoustic velocity) relative to fluid velocity. Characteristic lines: C+ with slope dx/dt = V+a carries Riemann invariant J+ = V + (g/a)H + (fV|V|/2D)Δt, C- with slope dx/dt = V-a carries J- = V - (g/a)H - (fV|V|/2D)Δt. Solution marches in time, computing pressure and velocity at grid intersections from upstream conditions. Boundary conditions handled by combining characteristic equations with device-specific relationships (valves, pumps, reservoirs). Essential for water hammer analysis, surge protection design, and pipeline integrity assessment.

Vortex shedding occurs when flow separates alternately from sides of a bluff body, creating periodic pressure fluctuations. Strouhal number St = fD/V relates shedding frequency (f) to flow velocity (V) and characteristic dimension (D). For circular cylinders, St ≈ 0.2 over wide Re range (300-10⁵). When shedding frequency approaches structure's natural frequency, resonance causes large-amplitude vortex-induced vibrations (VIV). Prevention strategies: helical strakes disrupt vortex correlation along span, shrouds reduce approaching velocity, fairings streamline body to prevent separation, mass dampers absorb energy, stiffening raises natural frequency above shedding range, galloping analysis for non-circular sections. Critical for offshore risers, heat exchanger tubes, chimneys, and bridge cables.

Hydraulic jump is an abrupt transition from supercritical (Fr>1) to subcritical (Fr<1) flow in open channels, with significant energy dissipation. Applying momentum equation to control volume across jump (ignoring friction): ρQ(V₂-V₁) = ½ρg(y₁²-y₂²)b, where y is depth and b is channel width. Using continuity Q = V₁y₁b = V₂y₂b and Froude number Fr₁ = V₁/√(gy₁): y₂/y₁ = ½(√(1+8Fr₁²)-1). This conjugate depth relationship shows downstream depth increases with upstream Froude number. Energy loss: ΔE = (y₂-y₁)³/(4y₁y₂). Jump classification by Fr₁: weak (1.7-2.5), oscillating (2.5-4.5), steady (4.5-9), strong (>9). Applications: stilling basins, energy dissipators downstream of spillways, flow measurement.

Discretization schemes approximate convective terms in transport equations. First-order upwind: uses upstream cell value, unconditionally stable but introduces numerical diffusion that smears gradients, order O(Δx). Central differencing: uses average of neighboring cells, second-order accurate O(Δx²) but can produce unbounded oscillations (wiggles) when Peclet number Pe = ρVΔx/Γ > 2. Second-order upwind: quadratic interpolation using two upstream cells, reduces numerical diffusion, O(Δx²). QUICK (Quadratic Upstream Interpolation): third-order scheme using quadratic profile through two upstream and one downstream cell, O(Δx³), good balance of accuracy and stability. TVD schemes (Total Variation Diminishing) with flux limiters combine accuracy in smooth regions with stability near discontinuities. Selection based on flow physics, mesh quality, and accuracy requirements.

Rotordynamic analysis predicts pump rotor vibration behavior under operating conditions. Key aspects: (1) Lateral analysis: calculates critical speeds (natural frequencies), mode shapes, unbalance response, and stability; must avoid operating near critical speeds (separation margin >15%). (2) Stiffness and damping: from bearings (sleeve, tilting pad), seals (wear rings, balance drum), and fluid interaction. (3) Cross-coupled stiffness from annular seals and impellers can cause subsynchronous instabilities when destabilizing forces exceed damping. Lomakin effect in wear rings provides stiffness proportional to pressure drop. (4) API 610/617 require rotordynamic analysis for between-bearing pumps. Stability analysis using eigenvalue approach predicts log decrement (>0.1 required). Mitigation: swirl brakes, anti-swirl devices, optimized seal geometry, damper bearings.

Two-phase pressure drop is significantly higher than single-phase due to interfacial friction and flow pattern effects. Methods: (1) Homogeneous model: treats mixture as single fluid with averaged properties, reasonable for dispersed bubble/mist flow; ΔP = f(ρ_m)L V_m²/(2D). (2) Separated flow models: Lockhart-Martinelli correlation uses parameter X² = (ΔP_L/ΔP_G) and multiplier φ² where ΔP_TP = φ_L²ΔP_L; accounts for flow patterns through C parameter. (3) Mechanistic models (Beggs-Brill, OLGA): predict flow pattern then apply pattern-specific correlations; most accurate for vertical and inclined pipes. (4) CFD with VOF or Euler-Euler for detailed analysis. Flow pattern maps (Baker, Taitel-Dukler) required for model selection based on superficial velocities and fluid properties.

Potential flow theory assumes inviscid, irrotational (∇×V=0) flow where velocity derives from scalar potential: V = ∇φ. Satisfies Laplace equation ∇²φ = 0, allowing superposition of elementary solutions: uniform flow, source/sink, doublet, vortex. Superposition creates complex flows: cylinder with circulation (Magnus effect), Rankine bodies, airfoil flows. Stream function ψ defines streamlines (ψ = constant). Limitations: cannot predict drag on symmetric bodies (d'Alembert's paradox), ignores viscous effects and boundary layers, cannot capture flow separation or wakes, misses rotational flows in turbomachinery. Practical use: initial estimates of pressure distribution, panel methods for external aerodynamics, combined with boundary layer analysis, starting point for wing design. Must be supplemented with viscous corrections for accurate force predictions.

Viscous fluids significantly reduce centrifugal pump performance compared to water. Hydraulic Institute method provides correction factors: C_Q (flow), C_H (head), C_η (efficiency) based on fluid viscosity and pump specific characteristics. High viscosity reduces efficiency more than head or flow. Effects: increased friction losses, reduced head coefficient, slip in clearances, higher power consumption. Selection approach: (1) Calculate required water-equivalent performance: Q_w = Q/C_Q, H_w = H/C_H. (2) Select pump for water duty. (3) Apply corrections to get viscous performance. (4) Verify NPSH_R increase (typically 1.5-2x for viscous fluids). For viscosity >500 cSt, consider positive displacement pumps. Temperature control may reduce viscosity during operation. Verify motor sizing for viscous power requirement at startup temperature (cold, high viscosity).

Verification confirms equations are solved correctly (solving equations right); validation confirms correct equations are solved (solving right equations). Error sources: (1) Modeling errors: turbulence model inadequacy, simplified geometry, boundary condition assumptions. (2) Discretization errors: truncation from finite differencing, addressed by grid independence study using Richardson extrapolation or GCI (Grid Convergence Index). (3) Iteration errors: incomplete convergence of nonlinear solver, monitored by residuals and integral quantities. (4) Round-off errors: finite precision arithmetic, usually negligible. Validation involves comparison with experimental data, quantified using metrics like error norms, drag/lift coefficients. Best practices: systematic grid refinement (factor 1.3-2), multiple turbulence models, sensitivity studies on boundary conditions, uncertainty quantification (UQ). ASME V&V 20 provides guidelines.

Flow through porous media (packed beds, filters, catalysts) is governed by resistance from viscous and inertial effects. Darcy's law for low Re (creeping flow): ΔP/L = μV/K, where K is permeability (m²), valid when Re_p < 1 based on pore dimension. For higher velocities, Forchheimer equation adds inertial term: ΔP/L = μV/K + βρV², where β is inertial resistance factor. Combined Darcy-Forchheimer: ΔP/L = (μ/α)V + C₂(½ρV²), where α is viscous permeability and C₂ is inertial factor. Ergun equation for packed beds: ΔP/L = 150μ(1-ε)²V/(ε³d_p²) + 1.75ρ(1-ε)V²/(ε³d_p), where ε is void fraction and d_p is particle diameter. Applications: oil reservoir simulation, groundwater flow, filtration system design, catalyst bed pressure drop.

Velocity triangles relate absolute velocity (C), relative velocity (W), and blade velocity (U) at inlet and outlet of turbomachine rotors: C = W + U (vector addition). Components: C_u (tangential/whirl) and C_m (meridional/flow). Euler's turbomachinery equation from angular momentum: specific work w = U₂C_u2 - U₁C_u1, positive for compressors/pumps (energy added), negative for turbines (energy extracted). For pumps, head H = (U₂C_u2 - U₁C_u1)/g. Design parameters: flow coefficient φ = C_m/U, head coefficient ψ = gH/U², degree of reaction R = (W₂²-W₁²)/(C₂²-C₁²+W₂²-W₁²). Blade angles determined from triangles: tan β = C_m/(U-C_u). Slip factor σ = C_u2_actual/C_u2_ideal accounts for imperfect guidance. Understanding triangles essential for blade design, off-design analysis, and performance prediction.