Fluid Mechanics – Dimensional Analysis & Model Testing

Vocabulary flashcards covering key terms from the Fluid Mechanics lecture on dimensional analysis, similarity, and model testing.

Dimensional Analysis

A mathematical technique that employs fundamental dimensions (mass, length, time, etc.) to derive relationships, check dimensional homogeneity, and design hydraulic models.

Primary (Basic) Dimensions

Independent physical quantities such as Mass (M), Length (L), Time (T), Temperature (Θ), Electric Current (I), Luminous Intensity (J), and Amount of Substance (N).

Derived (Secondary) Dimensions

Quantities formed from primary dimensions, e.g., Force (MLT⁻²), Pressure (ML⁻¹T⁻²), and Density (ML⁻³).

Dimensionless Term (π-term)

A group of variables combined so their overall dimensions cancel to unity, used extensively in similarity and model testing.

Dimensional Homogeneity

The requirement that every additive term in a physical equation has identical dimensions.

Rayleigh’s Method

An empirical technique that assumes a functional form with unknown exponents and determines them by equating dimensions on both sides of an equation.

Buckingham π Theorem

A formal method stating that any physically meaningful equation involving n variables and m fundamental dimensions can be rewritten as (n − m) independent, dimensionless π-terms.

Model (Fluid Mechanics)

A scaled representation of a prototype built to study behavior under controlled conditions.

Prototype

The full-scale structure or machine that the model represents.

Similarity (Similitude)

The condition where a model and prototype share geometric, kinematic, and dynamic likeness.

Geometric Similarity

Similarity where model and prototype are the same shape; all linear dimensions differ by a constant scale factor.

Kinematic Similarity

Similarity in which corresponding points of model and prototype have proportionally equal velocities and accelerations.

Dynamic Similarity

Similarity that requires the ratio of corresponding forces in model and prototype to be constant.

Scale Ratio (s)

The ratio of a characteristic length of the prototype to that of the model, used to relate areas (s²) and volumes (s³).

Reynolds Number (Re)

Dimensionless ratio of inertia to viscous forces, Re = ρVL/μ, governing laminar–turbulent transition.

Froude Number (Fr)

Dimensionless ratio comparing inertia to gravity forces, Fr = V/√(gL), important in free-surface flows and ship design.

Mach Number (Ma)

Dimensionless ratio of flow velocity to the speed of sound, Ma = V/c, governing compressibility effects in gas dynamics.

Weber Number (We)

Dimensionless ratio of inertia to surface-tension forces, We = ρV²L/σ.

Euler Number (Eu)

Dimensionless pressure coefficient, Eu = Δp/(ρV²), relating pressure forces to inertia forces.

Prandtl Number (Pr)

Dimensionless ratio of momentum diffusivity to thermal diffusivity, Pr = ν/α.

Drag Coefficient (C_D)

Dimensionless number expressing drag force, CD = FD/(½ρV²A).

Power Coefficient (C_P)

Dimensionless parameter for turbomachinery performance, C_P = P/(ρN³D⁵).

Dynamic Viscosity (μ)

Fluid property measuring resistance to shear stress; dimensions ML⁻¹T⁻¹.

Kinematic Viscosity (ν)

Ratio of dynamic viscosity to density, ν = μ/ρ; dimensions L²T⁻¹.

Density (ρ)

Mass per unit volume; dimensions ML⁻³.

Specific Weight (γ)

Weight per unit volume, γ = ρg; dimensions ML⁻²T⁻².

Bulk Modulus (K)

Measure of a fluid’s resistance to uniform compression; dimensions ML⁻¹T⁻².

Surface Tension (σ)

Force per unit length acting along a liquid surface; dimensions MT⁻².

Shear Stress (τ)

Force per unit area acting tangentially within a fluid, τ = μ(du/dy); dimensions ML⁻¹T⁻².

Volumetric Flow Rate (Discharge, Q)

Volume of fluid passing a section per unit time; dimensions L³T⁻¹.

Head (H)

Energy per unit weight of fluid, often expressed as height of a fluid column; dimensions L.

Drag Force (F_D)

Resistance force exerted by a fluid on a moving body; dimensions MLT⁻².

Inertia Force

Product of mass and acceleration representing resistance to change in motion; dimensions MLT⁻².

Gravity Force

Force due to weight of the body, F = mg; dimensions MLT⁻².

Elastic Force

Restoring force in a deformable medium proportional to elastic stress; dimensions MLT⁻².