Q: Rectangular notch — relationship between relative error in discharge and relative error in head measurement For a sharp-crested rectangular notch where Q ∝ H^(3/2), what is the ratio dQ/Q in terms of dH/H (assuming coefficient and width are constan…

Introduction / Context: Error propagation is crucial in open-channel flow measurements using sharp-crested rectangular notches (weirs). Because discharge Q depends on head H with a power law, a small error in head reading amplifies in Q. This question asks for the sensitivity dQ/Q due to dH/H. Given Data / Assumptions: Rectangular sharp-crested notch operating under free, fully aerated flow. Discharge relation: Q = K * H^(3/2), where K = (2/3) * C_d * b * sqrt(2g). Width b, coefficient C_d, and g treated as constants for small perturbations. Differentials are small (linearization valid). Concept / Approach: Use logarithmic differentiation of the discharge–head power law. When Q = K * H^n, the relative error relation is dQ/Q = n * dH/H, provided K is constant. Here, n = 3/2 for a rectangular notch. Step-by-Step Solution: Start with Q = K * H^(3/2).Take natural logs: ln Q = ln K + (3/2) ln H.Differentiate: (dQ/Q) = (3/2) * (dH/H).Hence, relative discharge error equals 1.5 times the relative head error. Verification / Alternative check: If the head H has a 2% reading uncertainty, the discharge uncertainty from head alone is approximately 3% (because 1.5 * 2% = 3%). This matches field experience where precise head measurement dominates uncertainty. Why Other Options Are Wrong: (1/2) or (2/3): underestimate sensitivity; would correspond to exponents 0.5 or 0.667, not 1.5. 3 or 2.5: overestimate sensitivity; these do not match the exponent in the standard formula. Common Pitfalls: Forgetting that C_d may vary slightly with H (submergence, aeration), but for first-order estimation we keep it constant. Also, ensure the nappe is properly ventilated to maintain the exponent. Final Answer: dQ/Q = (3/2) * (dH/H)

Question 1

Buoyancy — Definition Check The tendency of a liquid to uplift a fully or partially submerged body due to the fluid’s upward thrust is called “buoyancy.” Do you agree?

Accepted Answer

Introduction / Context: The concept of buoyancy explains why ships float, hot-air balloons rise, and hydrometers measure density. It is a cornerstone of fluid statics and engineering design related to floating and submerged structures. Given Data / Assumptions: Body is immersed in a fluid (fully or partially). Fluid is at rest; hydrostatic conditions apply. Gravity acts downward, creating a hydrostatic pressure field that increases with depth. Concept / Approach: Archimedes’ principle: a body wholly or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the displaced fluid. This “upward thrust” is precisely what is meant by buoyancy and produces the “tendency to uplift.” Step-by-Step Reasoning: Consider a submerged element. Pressure increases with depth: p = ρ g h. Bottom face of the element sees greater pressure than the top face. Net vertical force (integrated over the surface) acts upward; this is the buoyant force. Verification / Alternative check: For a floating body at equilibrium: buoyant force = weight of body. This balance is used to determine draft and stability, validating the definition operationally. Why Other Options Are Wrong: “Disagree” falsely denies the standard definition; buoyancy is universally defined via the upward hydrostatic resultant. Common Pitfalls: Confusing buoyancy (force from the fluid) with flotation (condition of force balance); ignoring that compressible fluids and temperature variations can alter density and thus buoyancy. Final Answer: Agree.

Question 2

Drain from a Basin — Identify the Flow Type Water draining through a hole at the bottom of a wash basin commonly exhibits which type of flow pattern?

Accepted Answer

Introduction / Context: Household sinks and basins often show a whirlpool-like motion as water drains. Classifying this vortex helps connect everyday observations with fluid mechanics concepts used in hydraulic engineering and mixing processes. Given Data / Assumptions: Draining through an orifice in a thin plate (bottom hole). No external device is forcing rotation (no agitator or rotating container). Viscous effects exist but the dominant rotation arises from circulation conservation. Concept / Approach: In a free vortex, fluid rotates due to conservation of angular momentum with essentially zero external torque. Tangential velocity varies as vθ ∝ 1/r and the free surface forms a dip. In contrast, a forced vortex requires an external agency to enforce solid-body rotation (vθ ∝ r). Step-by-Step Identification: Observe the whirlpool as water drains. No rotating walls or impellers impart torque; rotation develops naturally. Velocity profile and funnel-shaped free surface indicate a free vortex. Verification / Alternative check: If the container were physically rotated, a nearly solid-body rotation (forced vortex) would appear, contradicting the observed 1/r behavior of a natural drain whirl. Why Other Options Are Wrong: “Steady” and “uniform” describe time/space variation, not the rotational character; “forced vortex” needs external torque, which is absent here. Common Pitfalls: Assuming any rotation is “forced”; ignoring that small asymmetries or residual swirl seed a free vortex upon draining. Final Answer: Free vortex.

Question 3

Transition in Pipe Flow — Name the Velocity What is the term for the velocity at which laminar flow ceases (i.e., the upper limit of laminar regime before fully turbulent flow establishes)?

Accepted Answer

Introduction / Context: Flow in pipes is characterized by the Reynolds number Re = ρ V D / μ. As velocity increases, flow transitions from laminar to transitional to turbulent. Two commonly referenced thresholds bound the laminar and turbulent regimes for internal flows. Given Data / Assumptions: Constant properties and a circular pipe of diameter D. Laminar regime exists at sufficiently low Reynolds number. Definitions: lower and higher critical velocities bound transitional behavior in many texts. Concept / Approach: The lower critical velocity marks the onset of deviation from ideal laminar behavior (Re near about 2000 in smooth pipes). The higher critical velocity marks the end of the transitional regime (Re near about 4000), beyond which flow is fully turbulent. Thus, the velocity where laminar flow stops is the higher critical velocity in the sense that above this value, turbulence dominates. Step-by-Step Reasoning: Laminar for Re ≲ 2000 (below lower critical velocity). Transitional region roughly between Re ≈ 2000 and 4000. Fully turbulent beyond higher critical velocity (Re ≳ 4000) ⇒ laminar has ceased. Verification / Alternative check: Experimental pipe-flow maps show friction factor correlations deviating from laminar law f = 64/Re once Re exceeds the lower critical value and becoming fully turbulent past the higher threshold. Why Other Options Are Wrong: “Velocity of approach” is unrelated; “Lower critical velocity” denotes the start of transition, not the end; “None” contradicts standard terminology. Common Pitfalls: Treating the transition as a single sharp value; actual thresholds depend on disturbances, roughness, and entry effects. Final Answer: Higher critical velocity.

Question 4

Dimensional Analysis — Define the Froude Number The Froude number is defined as the ratio of which two characteristic effects in fluid motion?

Accepted Answer

Introduction / Context: Dimensionless numbers collapse complex physics into simple ratios. The Froude number Fr is essential in open-channel hydraulics, ship hydrodynamics, and free-surface flows because it compares inertia to gravity effects. Given Data / Assumptions: Characteristic velocity V and length L define inertia scaling. Gravitational acceleration g defines gravity scaling. Fr used for dynamic similarity in free-surface problems. Concept / Approach: Froude number is Fr = V / √(g L). Squaring gives Fr^2 = V^2 / (g L), which represents the ratio of inertia force per unit mass (∼ V^2 / L) to gravity acceleration g. Hence, Fr expresses the ratio of inertia force to gravity force, guiding whether waves and surface elevations dominate the behavior. Step-by-Step Interpretation: Identify inertia effect scale: a_inertia ∼ V^2 / L. Compare to gravitational acceleration: g. Therefore Fr^2 ∼ (inertia) / (gravity) ⇒ Fr compares these effects. Verification / Alternative check: In open channels, critical depth occurs at Fr = 1, separating subcritical (gravity-dominated) from supercritical (inertia-dominated) regimes—an operational confirmation of the definition. Why Other Options Are Wrong: “Disagree” would contradict standard dimensional analysis and textbook definitions. Common Pitfalls: Confusing Froude with Reynolds (inertia/viscous) or Weber (inertia/surface tension). Each has distinct applicability. Final Answer: Agree — the Froude number expresses the ratio of inertia force to gravity force.

Question 5

Hydrostatics — Centre of Pressure for a Vertically Immersed Plane For a vertically immersed plane surface with area A, centroid depth y_c below the free surface, and second moment of area about its centroid I_G, the distance of the centre of pressur…

Accepted Answer

Introduction / Context: The centre of pressure is the point of action of the resultant hydrostatic force on a submerged surface. Locating it is vital for gate design, retaining structures, and tank walls to compute moments and stability. Given Data / Assumptions: Plane surface of area A; centroid located at depth y_c from the free surface. Hydrostatic condition with fluid of constant density. I_G is the second moment of area about an axis through the centroid parallel to the free surface. Concept / Approach: The pressure varies linearly with depth: p = ρ g y. The resultant force acts at the centre of pressure located below the centroid for vertical planes. Applying the standard hydrostatics result gives y_cp = y_c + (I_G / (A y_c)). Step-by-Step Solution: Resultant hydrostatic force: F = ρ g A y_c. Taking moments about the free surface and using the pressure distribution: M = ρ g ∫ y dA * y = ρ g (I_0), with I_0 the second moment about the surface. Using the parallel-axis relation: I_0 = I_G + A y_c^2, then y_cp = M / F = (I_G + A y_c^2) / (A y_c) = y_c + I_G / (A y_c). Verification / Alternative check: Since I_G > 0, y_cp > y_c for a vertical plane, agreeing with physical intuition: pressure is greater at deeper points, pulling the resultant below the centroid. Why Other Options Are Wrong: Omitting y_c or using a minus sign contradicts the derivation; taking a square root ignores the linear depth relation. Common Pitfalls: Confusing I_G about the centroid with I_0 about the free surface; using the formula for an inclined plane without adjusting depths. Final Answer: y_cp = y_c + I_G / (A y_c).

Question 6

Power Transmission Through a Pipe — Efficiency Expression If the total supply head is H and the head lost due to friction is h_f, what is the efficiency of power transmission through the pipe?

Accepted Answer

Introduction / Context: When conveying water through long pipelines, frictional losses reduce the available head and thus the useful power at the delivery end. Efficiency quantifies how much of the supply head remains to do useful work (for example, to drive a turbine or supply pressure at the outlet). Given Data / Assumptions: Total head at inlet: H (measured above the delivery datum). Head loss due to pipe friction: h_f. Neglect minor losses for this expression or include them within h_f. Concept / Approach: Power in a steady incompressible flow is proportional to ρ g Q times the available head. Since discharge Q is the same throughout a single pipe, the ratio of output power to input power reduces to the ratio of available heads: (H − h_f) / H. Step-by-Step Derivation: Input power ∝ ρ g Q H. Useful power at delivery ∝ ρ g Q (H − h_f). Therefore efficiency η = (ρ g Q (H − h_f)) / (ρ g Q H) = (H − h_f) / H. Verification / Alternative check: If h_f = 0 (ideal frictionless pipe), η = 1; if h_f approaches H, η → 0, both consistent with physical expectations. Why Other Options Are Wrong: h_f / H measures loss fraction, not efficiency; squaring terms has no basis; square-root forms are unrelated to power proportionality. Common Pitfalls: Confusing head efficiency with mechanical efficiency of pumps; forgetting to include all losses in h_f when comparing to field data. Final Answer: η = (H − h_f) / H.

Question 7

Hydrostatic Resultant on a Vertical Wall — Per Metre Length A water tank contains water to a depth of 1.3 m. What is the hydrostatic force exerted by the water on one metre length of a vertical side wall?

Accepted Answer

Introduction / Context: Calculating the total hydrostatic thrust on vertical walls is fundamental for tank, dam, and gate design. For a rectangular vertical panel, the pressure varies linearly with depth, and the resultant equals the pressure at the centroid times the area. Given Data / Assumptions: Water depth h = 1.3 m; width considered = 1.0 m (per metre length). Density ρ ≈ 1000 kg/m^3; gravitational acceleration g ≈ 9.81 m/s^2. Hydrostatic (static fluid) conditions; atmospheric pressure cancels on both sides. Concept / Approach: For a vertical rectangle of height h and width b, the resultant hydrostatic force is F = ρ g A y_c, where A = b h and y_c = h/2 is the centroidal depth. This leverages the linear pressure distribution p = ρ g y. Step-by-Step Solution: Given h = 1.3 m, b = 1 m ⇒ A = 1.3 m^2. Centroid depth y_c = h/2 = 0.65 m. Compute F = ρ g A y_c = 1000 * 9.81 * 1.3 * 0.65 ≈ 8290 N = 8.29 kN. Verification / Alternative check: Average pressure on the panel = ρ g (h/2) = 9.81 * 0.65 ≈ 6.377 kPa; multiply by area 1.3 m^2 gives ≈ 8.29 kN, confirming the result. Why Other Options Are Wrong: 2.89 kN is far too small; 9.28 kN is too large for 1.3 m depth; 28.9 kN corresponds to much larger depths or widths. Common Pitfalls: Using p at the bottom times area (overestimates); forgetting that resultant acts at the centre of pressure, not at the centroid (important for moment calculations, though not needed for the magnitude here). Final Answer: 8.29 kN per metre length of wall.

Question 8

Capillarity Experiment — Why Use a Small-Diameter Glass Tube for Water? In a capillary-rise experiment with water, why is a small-diameter glass tube preferred?

Accepted Answer

Introduction / Context: Capillary phenomena are central to soil mechanics, ink pens, and microfluidics. Measuring capillary rise reliably requires careful choice of tube diameter and material to obtain a clear, measurable meniscus movement. Given Data / Assumptions: Working fluid: water (wetting fluid on clean glass). Tube is narrow and clean with a circular cross-section. Contact angle θ for water–glass is acute (cos θ ≈ positive). Concept / Approach: The capillary rise formula is h = 4 σ cos θ / (ρ g d), showing h ∝ 1/d for a given fluid–solid pair. Therefore, using a smaller internal diameter increases the rise, improving measurement accuracy and reducing relative reading error against the scale. Step-by-Step Reasoning: Identify variables: surface tension σ, density ρ, gravity g, contact angle θ, tube diameter d. Use relation h = 4 σ cos θ / (ρ g d). Smaller d ⇒ larger h for the same σ, θ, ρ, g ⇒ rise is easier to observe and quantify. Verification / Alternative check: Doubling d halves the capillary rise, often making readings less distinct; conversely, halving d doubles h, validating the experimental preference. Why Other Options Are Wrong: Contact angle is material-dependent and cleaning-dependent; small d does not force θ to zero. Metal tubes can exhibit capillarity with water depending on surface condition. Meniscus curvature is inherent and not eliminated by small d. Common Pitfalls: Using contaminated tubes (θ changes); ignoring temperature effects on σ; misreading the meniscus (bottom for water). Final Answer: Because capillary rise h is inversely proportional to tube diameter d, so a smaller d gives a larger, easier-to-measure rise.

Question 9

Pressure Measurement — Manometer Capability Can a manometer be configured to measure vacuum (negative gauge) pressures as well as positive gauge pressures?

Accepted Answer

Introduction / Context: Manometers are fundamental devices for measuring pressure differences using hydrostatic columns. With appropriate connection and reference, they can indicate pressures above or below atmospheric pressure (gauge positive or vacuum/negative gauge). Given Data / Assumptions: U-tube or inclined manometer with a suitable manometric fluid (e.g., mercury, water, oil). One leg can be open to atmosphere; the other connected to the system point. Hydrostatic equilibrium applies, neglecting capillary corrections at macroscales. Concept / Approach: The manometer measures pressure difference via Δp = ρ_m g Δh (adjusted for fluid columns). If the system pressure is below atmospheric, the meniscus moves opposite to the positive case, and the indicated height difference corresponds to a vacuum (negative gauge) value. Thus, by reading Δh and sign, vacuum is readily measured. Step-by-Step Reasoning: Connect one limb to the point of interest and the other to atmosphere. If p_system < p_atm, the column shifts to show a deficit; compute |p_atm − p_system| = ρ_m g Δh (with corrections if multiple fluids are present). Report vacuum magnitude and, if needed, convert to absolute pressure p_abs = p_atm − |p_vac|. Verification / Alternative check: Industrial vacuum lines routinely use mercury or oil manometers and Bourdon gauges scaled for negative gauge pressure, confirming practicality. Why Other Options Are Wrong: “Disagree” ignores common practice and the symmetric operation of manometers around atmospheric reference. Common Pitfalls: Misreading sign conventions; using an inappropriate manometric fluid (too light or hazardous); neglecting capillary or temperature corrections in precise work. Final Answer: Agree — manometers measure both vacuum and positive gauge pressures.

Question 10

Rectangular notch — relationship between relative error in discharge and relative error in head measurement  For a sharp-crested rectangular notch where Q ∝ H^(3/2), what is the ratio dQ/Q in terms of dH/H (assuming coefficient and width are constan…

Accepted Answer

Introduction / Context: Error propagation is crucial in open-channel flow measurements using sharp-crested rectangular notches (weirs). Because discharge Q depends on head H with a power law, a small error in head reading amplifies in Q. This question asks for the sensitivity dQ/Q due to dH/H. Given Data / Assumptions: Rectangular sharp-crested notch operating under free, fully aerated flow. Discharge relation: Q = K * H^(3/2), where K = (2/3) * C_d * b * sqrt(2g). Width b, coefficient C_d, and g treated as constants for small perturbations. Differentials are small (linearization valid). Concept / Approach: Use logarithmic differentiation of the discharge–head power law. When Q = K * H^n, the relative error relation is dQ/Q = n * dH/H, provided K is constant. Here, n = 3/2 for a rectangular notch. Step-by-Step Solution: Start with Q = K * H^(3/2).Take natural logs: ln Q = ln K + (3/2) ln H.Differentiate: (dQ/Q) = (3/2) * (dH/H).Hence, relative discharge error equals 1.5 times the relative head error. Verification / Alternative check: If the head H has a 2% reading uncertainty, the discharge uncertainty from head alone is approximately 3% (because 1.5 * 2% = 3%). This matches field experience where precise head measurement dominates uncertainty. Why Other Options Are Wrong: (1/2) or (2/3): underestimate sensitivity; would correspond to exponents 0.5 or 0.667, not 1.5. 3 or 2.5: overestimate sensitivity; these do not match the exponent in the standard formula. Common Pitfalls: Forgetting that C_d may vary slightly with H (submergence, aeration), but for first-order estimation we keep it constant. Also, ensure the nappe is properly ventilated to maintain the exponent. Final Answer: dQ/Q = (3/2) * (dH/H)

Question 11

Turbulent flow — which force dominates over viscosity in defining turbulence?

Accepted Answer

Introduction / Context: In fluid mechanics, the nature of flow (laminar vs. turbulent) is determined by the balance of forces acting on the fluid. Turbulence is characterized by chaotic eddies and strong mixing caused by inertial effects being much larger than viscous damping. Given Data / Assumptions: Incompressible Newtonian fluid for conceptual discussion. Characteristic velocity U, length scale L, and kinematic viscosity ν. High Reynolds number regime (Re = U * L / ν ≫ 1). Concept / Approach: The Reynolds number represents the ratio of inertia to viscous forces: Re ∝ Inertia/Viscous. Turbulent flow occurs when inertia forces dominate viscous forces (high Re), leading to instability of laminar layers and energy cascade from large to small scales. Step-by-Step Solution: Identify governing nondimensional group: Re = U * L / ν.Interpretation: Large Re → inertia force ≫ viscous force → turbulent flow.Therefore, the dominating force in turbulent flow is the inertia force. Verification / Alternative check: Classic experiments (e.g., pipe flow) show transition around Re ≈ 2300 for smooth pipes; well above this, turbulence appears as inertial effects overpower viscous smoothing. Why Other Options Are Wrong: Elastic and surface tension forces are relevant in compressible and capillary phenomena respectively, not the primary cause of turbulence. Viscous force dominates in laminar (low Re) flows. Common Pitfalls: Confusing the trigger of turbulence (instability) with effects such as pressure fluctuations; also misstating that viscosity is irrelevant—viscosity still dissipates energy but does not dominate the dynamics in turbulence. Final Answer: Inertia force

Question 12

Definition check — vacuum pressure vs. atmospheric pressure  When pressure intensity at a point is lower than local atmospheric pressure, the difference (p_atm − p_local) is called vacuum pressure. Do you agree?

Accepted Answer

Introduction / Context: Pressure terminology in hydraulics distinguishes among absolute pressure, gauge pressure, and vacuum pressure. Understanding these definitions is essential for instrumentation, cavitation analysis, and pump selection. Given Data / Assumptions: Local atmospheric pressure head h_atm is known. Measured local absolute pressure head h_abs may be above or below h_atm. We adopt standard sign conventions used in hydraulics. Concept / Approach: Gauge pressure p_gauge = p_abs − p_atm. When p_abs < p_atm, p_gauge is negative. The magnitude of this negative gauge pressure is termed vacuum pressure: p_vacuum = p_atm − p_abs. In head terms: h_vacuum = h_atm − h_abs. Step-by-Step Solution: State definition: Vacuum pressure is the shortfall of local absolute pressure below atmospheric pressure.Express: p_vacuum = p_atm − p_abs (or h_vacuum = h_atm − h_abs).Thus, whenever the local pressure intensity is less than atmosphere, the difference is vacuum pressure. Verification / Alternative check: Manometers with one leg open to atmosphere read vacuum as an upward deflection on the vacuum side; the magnitude equals p_atm − p_abs. This matches the definition above. Why Other Options Are Wrong: Disagree / qualifiers: Vacuum pressure is not limited to gases or cavitation; any fluid at p_abs < p_atm qualifies. Common Pitfalls: Confusing absolute and gauge scales; forgetting that absolute pressure cannot be negative (lower bound is zero absolute). Final Answer: Agree

Question 13

Identify the single incorrect statement about common hydraulic concepts:

Accepted Answer

Introduction / Context: This item checks conceptual clarity on energy composition, venturimeter geometry, Pitot tube use, and the meaning of two-dimensional flow. Only one statement is intentionally incorrect. Given Data / Assumptions: Inviscid-energy picture for Bernoulli's components (pressure, velocity head, elevation). Standard venturimeter with smooth convergent and longer diffuser. Pitot tube measures stagnation and static pressure for local velocity. Two-dimensional flow refers to dependence on two spatial coordinates. Concept / Approach: Two-dimensional (2D) flow is defined by velocity components u(x,y,t), v(x,y,t) with no variation in the third coordinate. Streamlines may be curved or straight in 2D; curvature does not make the flow 3D. Venturimeters use a longer diffuser to limit adverse pressure gradients and suppress separation. Step-by-Step Solution: Evaluate (a): Correct — Bernoulli energy per unit weight is pressure head + velocity head + elevation head.Evaluate (b): Correct — divergent section is longer/shallower to minimize losses.Evaluate (c): Correct — Pitot tube provides local velocity via dynamic pressure.Evaluate (d): Incorrect — 2D flow can have curved streamlines; curvature does not imply 3D. Verification / Alternative check: Classic 2D potential flows (e.g., around a cylinder) have strongly curved streamlines yet remain 2D (no spanwise variation). Why Other Options Are Wrong: (a), (b), (c) are standard and widely taught facts. (e) claims all are correct, which is false because (d) is incorrect. Common Pitfalls: Equating streamline curvature with three-dimensionality; assuming equal convergent and divergent lengths in a venturimeter. Final Answer: Two-dimensional flow means streamlines are necessarily straight lines; curved streamlines cannot be two-dimensional.

Question 14

Discharge through an external mouthpiece — practical formula  Let a be the cross-sectional area of the external mouthpiece and H the head above its centerline. The discharge Q is approximately:

Accepted Answer

Introduction / Context: Mouthpieces are short tubes fitted at tank orifice openings. For a well-designed external mouthpiece running full, empirical studies give a practical coefficient close to 0.855 for the discharge relation. Given Data / Assumptions: External (protruding) mouthpiece of short length. Steady incompressible flow; fully aerated jet; negligible approach velocity. Coefficient of discharge C_d ≈ 0.855 (typical value). Concept / Approach: Ideal discharge from Bernoulli/continuity is Q_ideal = a * sqrt(2 * g * H). Real discharge is Q = C_d * Q_ideal with C_d < 1 to account for contraction and losses through the mouthpiece. Step-by-Step Solution: Write ideal flow: Q_ideal = a * sqrt(2 g H).Apply coefficient: Q = C_d * a * sqrt(2 g H).Use C_d ≈ 0.855 for external mouthpiece → Q ≈ 0.855 a sqrt(2 g H). Verification / Alternative check: Compare to sharp-edged orifice (C_d ≈ 0.62): the mouthpiece tends to have a higher C_d when running full because it controls contraction and reduces separation. Why Other Options Are Wrong: 1.855: too high; implies C_d > 1. 0.585: too low for a full, well-formed mouthpiece. 0.855 * a * (2 g H): missing the square root, wrong dimensions. Coefficient = 1: assumes ideal flow without losses. Common Pitfalls: Mixing orifice and mouthpiece coefficients; forgetting the square-root dependence on head. Final Answer: Q = 0.855 * a * sqrt(2 * g * H)

Question 15

Critical depth definition — specific energy minimum in open channel flow  The depth of water at which the specific energy is minimum is called the critical depth. Do you agree?

Accepted Answer

Introduction / Context: Specific energy E for open channel flow (per unit weight) is defined as E = y + v^2/(2g), where y is flow depth and v is mean velocity. The depth that minimizes E for a given discharge is the critical depth, a cornerstone concept in gradually varied flow and control transitions. Given Data / Assumptions: Steady, prismatic channel with discharge per unit width q (for simplicity). Negligible energy losses across the control section. Hydrostatic pressure distribution. Concept / Approach: For given q, E(y) = y + q^2/(2 g y^2) (rectangular channel). The minimum of E occurs where dE/dy = 0, yielding the critical condition. The associated Froude number Fr = v/√(g y) equals 1 at critical depth. The definition (depth at minimum E) is general and underpins control sections like weirs and flumes. Step-by-Step Solution: Write E(y) and set derivative to zero to find y_c.For rectangular channels, y_c = (q^2/g)^(1/3), and Fr = 1 at y = y_c.Thus, the depth corresponding to minimum specific energy is, by definition, critical depth. Verification / Alternative check: Plotting E vs. y shows a distinct minimum at y_c. Alternately, using Fr = 1 criteria confirms the same critical condition. Why Other Options Are Wrong: Limiting to rectangular or wide channels is unnecessary; the definition of critical depth via minimum specific energy extends with appropriate geometric relations. Fr = 0 never occurs for flowing water; critical depth corresponds to Fr = 1, not zero. Common Pitfalls: Confusing normal depth (from uniform flow) with critical depth; they coincide only under special conditions. Final Answer: Agree

Question 16

Mouthpieces and pressure head — absolute pressure at vena contracta  In external or internal mouthpieces, when the atmospheric pressure head is equivalent to 10.3 m of water, the absolute pressure head at the vena contracta can drop to zero under su…

Accepted Answer

Introduction / Context: In short-tube (mouthpiece) flows, the pressure at the vena contracta can fall below atmospheric. When the vacuum equals the local atmospheric head (~10.3 m of water at sea level), the absolute pressure approaches zero, which triggers a change in flow regime (e.g., mouthpiece running full) or cavitation limits. Given Data / Assumptions: Water at standard conditions; atmospheric head ≈ 10.3 m of water. Well-formed vena contracta occurs inside the mouthpiece. Neglecting dissolved gases and vapor pressure nuances for the conceptual limit. Concept / Approach: Absolute pressure head h_abs = h_atm − h_vac. If h_vac → h_atm (≈ 10.3 m), then h_abs → 0. At or below this limit, cavitation (or flow alteration) occurs; practical operation avoids approaching the absolute zero because vapor pressure is reached earlier (~0.2–0.3 m head for water at room temperature). Step-by-Step Solution: Recognize that p_abs ≥ 0 by definition; negative absolute pressure is nonphysical.As driving head increases, local pressure at vena contracta drops.At the limiting case where vacuum head = 10.3 m, p_abs → 0 head of water.This condition qualitatively explains transitions like mouthpiece running full. Verification / Alternative check: Engineering texts describe that when absolute pressure at the vena contracta is reduced sufficiently, separation patterns change and full running occurs; also, vapor pressure is reached prior to zero absolute in practice. Why Other Options Are Wrong: “Incorrect” conflicts with the conceptual limit statement. Absolute pressure cannot be negative; that option is physically impossible. Limiting to gas flow or sharp-edged orifices is unnecessary; the concept is general. Common Pitfalls: Equating the practical cavitation onset (at vapor pressure) with zero absolute pressure; zero is a theoretical limit, cavitation starts earlier. Final Answer: Correct

Question 17

Classifying pipe flow at constant rate along a long, prismatic pipe  A flow through a long pipe at a constant volumetric rate is best described as:

Accepted Answer

Introduction / Context: Flow classification depends on time variation (steady vs. unsteady) and spatial variation along the streamline (uniform vs. non-uniform). Understanding these terms helps interpret measurements and apply the correct governing equations and loss models. Given Data / Assumptions: Constant volumetric flow rate Q (no time variation). Long, prismatic pipe of constant diameter and roughness. Single-phase incompressible liquid; temperature changes negligible. Concept / Approach: Steady means properties at a fixed point do not change with time (∂()/∂t = 0). Uniform means properties do not change along the direction of flow at a given instant. In a constant-diameter pipe with constant Q, the average velocity V = Q/A is constant along the pipe; though pressure drops due to friction, velocity magnitude remains spatially uniform. Step-by-Step Solution: Since Q is constant in time → steady.Area A is constant (prismatic) → V = Q/A is the same at all sections.Therefore, the flow is steady (no time variation) and uniform (no spatial variation in mean velocity). Verification / Alternative check: Bernoulli with head loss explains pressure drop along the pipe without change in average speed for a constant area, consistent with “uniform.” Why Other Options Are Wrong: Steady non-uniform: would require area change (nozzle/diffuser) or imposed acceleration. Unsteady options: contradict constant-rate condition. Common Pitfalls: Confusing uniformity of velocity with uniformity of pressure. Uniform flow does not imply constant pressure; it implies constant velocity magnitude along the streamline for constant area. Final Answer: Steady uniform flow

Question 18

Cipolletti (trapezoidal) weir — Francis discharge formula  For a Cipolletti weir of crest length L and head H, which expression gives the discharge Q?

Accepted Answer

Introduction / Context: The Cipolletti weir is a trapezoidal sharp-crested weir with side slopes chosen so that the discharge coefficient is nearly constant and the end-contraction effect is compensated. Francis provided a convenient empirical discharge relation widely used in practice. Given Data / Assumptions: Free, fully aerated nappe; sufficient approach condition corrections applied. Head H measured above crest; crest length L between nappe ends. Standard Cipolletti slope (1 horizontal : 4 vertical). Concept / Approach: For the Cipolletti weir, Francis-type formula is Q = C * L * H^(3/2) with C ≈ 1.84 (SI units when Q in m^3/s, L and H in m). The exponent 3/2 arises from the velocity head integration across the depth over the crest. Step-by-Step Solution: Recognize trapezoidal weir with compensating side slopes.Select Francis relation: Q = 1.84 L H^(3/2).Confirm unit consistency and applicability (free nappe, ventilated). Verification / Alternative check: Limit case: If side slopes vanish (rectangular weir), the constant differs and end-contraction corrections are needed; Cipolletti geometry removes the correction, keeping 1.84 applicable. Why Other Options Are Wrong: H^(1/2), H, H^(5/2): incorrect exponent. Divide by L: in a weir, discharge grows with crest length, not inversely. Common Pitfalls: Using the rectangular weir coefficient directly for Cipolletti; measuring H too close to the crest (should be upstream at a recommended distance). Final Answer: Q = 1.84 * L * H^(3/2)

Question 19

Definition check — turbulent flow and particle paths  A flow in which each liquid particle does not have a definite smooth path and the paths appear to intermix or cross due to eddies is termed turbulent flow. Do you agree?

Accepted Answer

Introduction / Context: Turbulence is characterized by irregular, three-dimensional, time-dependent motion with eddies and intense mixing. Unlike laminar flow where particles follow smooth, non-intersecting paths, turbulent motion yields fluctuating velocities and seemingly intersecting tracks when viewed over time. Given Data / Assumptions: Newtonian fluid; discussion in kinematic terms. High Reynolds number environment where inertia dominates. Qualitative description of particle trajectories (pathlines/streamlines). Concept / Approach: In laminar flow, neighboring layers slide smoothly with well-defined streamlines; instantaneous pathlines do not cross each other at the same instant. In turbulent flow, strong fluctuations cause rapid changes in the local velocity direction and magnitude, so over time the observed paths appear to intermix and “cross,” reflecting eddy motion and mixing. Step-by-Step Solution: Define laminar: orderly, layered motion with predictable paths.Define turbulent: random, eddying motion with fluctuating components u', v', w'.Hence, the descriptive statement corresponds to turbulence, not laminarity. Verification / Alternative check: Dye injection in a turbulent pipe shows rapid dispersion and irregular streaklines compared to straight streamlines in laminar flow. Why Other Options Are Wrong: “Disagree” or constraints to low Re/compressibility are incorrect: turbulence is high-Re and occurs in liquids and gases. Calling it laminar contradicts standard definitions. Common Pitfalls: Over-interpreting “crossing” literally at the same instant; in strict kinematics, instantaneous streamlines do not intersect, but the time-evolving paths give the observed intermixing. Final Answer: Agree

Question 20

Fluid mechanics – Euler’s equation of motion (assumptions) Euler’s equation for inviscid flow is derived under several idealizing assumptions. Which of the following correctly list those assumptions used for the motion of liquids?

Accepted Answer

Introduction / Context: Euler’s equation is a cornerstone of ideal fluid mechanics. It expresses Newton’s second law for a fluid particle when viscosity is neglected. Understanding the underlying assumptions is essential because they define when Euler’s equation (and its integrated form, the Bernoulli equation) can be applied reliably in engineering problems. Given Data / Assumptions: Ideal fluid model: inviscid, homogeneous, frequently treated as incompressible for liquids. Flow field that is steady for the common derivation used in hydraulics. Equation written along a streamline for general three-dimensional flows. Cross-sectional uniformity is often invoked when reducing to 1-D forms. Concept / Approach: Euler’s differential momentum equation comes from applying Newton’s second law to a fluid element while neglecting viscous stresses. In typical liquid-flow applications, incompressibility and steadiness simplify terms. When engineers move to a 1-D treatment, they additionally assume sectionally uniform velocity to relate pointwise variables to section-averaged values. Step-by-Step Solution: State inviscid assumption → shear stresses are zero, leaving only pressure and body forces.Adopt incompressibility for liquids → density treated as constant (homogeneous fluid).Assume steady flow along a streamline → time derivatives vanish for steady case; streamline direction simplifies the vector form.For 1-D reduction → assume uniform velocity over a section to use averaged quantities. Verification / Alternative check: If viscosity were significant (e.g., near walls, in laminar layers), Navier–Stokes terms with viscous stresses would be required. Euler’s equation is then insufficient, confirming the necessity of the inviscid assumption. Why Other Options Are Wrong: Options A, B, and C are each individually true within the standard derivations; the comprehensive choice is “All of the above”. “None of the above” contradicts classical fluid mechanics. Common Pitfalls: Blindly applying Bernoulli/Euler to real, highly viscous, or unsteady flows without checking assumptions such as inviscid behavior, steadiness, and streamline alignment. Final Answer: All of the above

Hydraulics and Fluid Mechanics Questions