Turbulent Flow in Fluids

Dimensional Analysis of Reynolds Number

Conditions: Incompressible fluid flow in a straight cylindrical pipe

• Assume critical velocity (v) depends on fluid density (ρ), dynamic viscosity (η), and pipe diameter (d).
• Write the dimensional equation: [v] = [ρ]^a [η]^b [d]^c.
• Substitute dimensions: [LT^-1] = [ML^-3]^a [ML^-1T^-1]^b [L]^c.
• Equate powers of M, L, and T: For M: a + b = 0; For L: -3a - b + c = 1; For T: -$b = -1$.
• Solve for constants: b = 1, $a = -1$, $c = -1$.
• Substitute back to get the proportionality: v ∝ η / (ρd), which rearranges to (ρvd)/(η) = constant.

Result: Re = \frac{\rho v d}{\eta} = \text{constant}

Head Loss in Turbulent Pipe Flow (Darcy-Weisbach Relation)

Conditions: Fully developed turbulent flow in a constant diameter horizontal pipe

• Consider shear stress at the pipe wall (τ_w) balancing the pressure drop (ΔP).
• Express pressure force as (ΔP)(πd²)/4 and frictional force as τ_w(πd)L.
• Equate the forces: Δ$P = 4(\tau_{w}L)/d$.
• Introduce the Darcy friction factor f where τ_w = (f/8)ρv².
• Substitute τ_w into the pressure drop equation to obtain head loss h_f.

Result: $h_f = f \frac{L}{d} \frac{v^2}{2g}$

Reynolds' Dye Experiment

Draw: A schematic showing a glass pipe with water flow from a reservoir, injecting a fine filament of colored dye at the entrance. At low flow, the dye forms a straight line; at high flow, it diffuses completely.

Interpret: Visualizes the transition from laminar flow (straight dye streak) to turbulent flow (dispersed, mixing dye streak) as velocity increases.

Exam use: Frequently asked conceptual questions regarding the critical velocity and visual confirmation of turbulence.

Friction Factor vs Reynolds Number (Moody Diagram Sketch)

Draw: Log-log plot of friction factor f against Reynolds number Re. Shows a downward sloping curve for laminar region ($f = 64/Re)$ and distinct curves for varying relative roughness in the turbulent zone.

Interpret: Demonstrates that in turbulent flow, friction factor depends on both Reynolds number and pipe wall roughness, unlike laminar flow.

Exam use: Interpreting parameter dependencies in fluid friction problems.

Velocity Distribution Profile

Draw: Comparison of parabolic velocity profile (laminar) versus a flattened, blunt velocity profile across a pipe diameter for turbulent flow.

Interpret: Turbulent flow exhibits a much more uniform velocity distribution across the central core of the pipe due to rapid momentum mixing by eddies.

Exam use: MCQs contrasting laminar and turbulent velocity profiles.

Water at 20°C flows through a pipe of diameter 0.05 m. If the density of water is 1000 kg m⁻³ and dynamic viscosity is 1.0 × 10⁻³ Pa s, what is the maximum velocity for laminar flow? (Assume critical Reynolds number is 2000)

• Identify given parameters: d = 0.05 m, $\rho = 1000 kg m⁻³$, $\eta = 1.0 × 10⁻³ Pa s$, Re = 2000.
• Use the critical velocity formula: $v_c = (Re * \eta) / (\rho * d)$.
• Substitute the values: $v_c = (2000 * 1.0 × 10⁻³) / (1000 * 0.05)$.
• Calculate the result: $v_c = 2 / 50 = 0.04 m s⁻¹$.

Answer: 0.04 m s⁻¹

Air with density 1.2 kg m⁻³ and kinematic viscosity 1.5 × 10⁻⁵ m² s⁻¹ flows through a large ventilation duct of diameter 2.0 m. Calculate the Reynolds number when the average flow velocity is 5 m s⁻¹ and determine the flow regime.

• Identify parameters: v = 5 m s⁻¹, d = 2.0 m, $\nu = 1.5 × 10⁻⁵ m² s⁻¹$.
• Use the Reynolds number formula involving kinematic viscosity: R$e = (v * d) / \nu$.
• Substitute values: R$e = (5 * 2.0) / (1.5 × 10⁻⁵) = 10 / (1.5 × 10⁻⁵)$.
• Compute value: Re ≈ 6.67 × 10⁵.
• Compare with threshold: Since Re >> 3000, the flow is highly turbulent.

Answer: Re = 6.67 × 10⁵ (Fully turbulent flow)

Oil of density 900 kg m⁻³ and dynamic viscosity 0.1 Pa s flows through a pipe of diameter 0.1 m at a velocity of 2 m s⁻¹. Find the head loss due to friction over a length of 50 m using the friction factor relationship $f = 64/Re$ for laminar, or appropriate turbulent correlation if applicable.

• Calculate Reynolds number: R$e = (\rho * v * d) / \eta = (900 * 2 * 0.1) / 0.1 = 1800$.
• Since Re = 1800 (< 2000), the flow is actually laminar. Use laminar friction factor $f = 64 / Re$.
• Calculate f: $f = 64 / 1800 = 0.03556$.
• Apply Darcy-Weisbach equation for head loss: $h_f = (f * L * v²) / (2 * g * d)$.
• Substitute values ($g = 9.8 m s⁻²): h_f = (0.03556 * 50 * 2²) / (2 * 9.8 * 0.1) = 7.112 / 1.96 = 3.63 m$.

Answer: 3.63 m (Note: Flow verified as laminar despite turbulent context)