Question

 For the laminar flow of water over a sphere, the drag coefficient C_F is defined as C_F = F / (ρU²D²), where F is the drag force, ρ is the fluid density, U is the fluid velocity and D is the diameter of the sphere. The density of water is 1000 kg/m³. When the diameter of the sphere is 100 mm and the fluid velocity is 2 m/s, the drag coefficient is 0.5. If water now flows over another sphere of diameter 200 mm under dynamically similar conditions, the drag force (in N) on this sphere is ________.

Answer 1

Concept:

From Buckingham’s π – theorem,

\(\frac{F}{{\rho {U^2}{D^2}}} = \phi \left( {\frac{{\rho UD}}{\mu }} \right) = \phi \left( {Re} \right) = {C_F}\)

For dynamic similarity, Re₁ = Re₂

\(\frac{{{\rho _1}{U_1}{D_1}}}{{{\mu _1}}} = \frac{{{\rho _2}{U_2}{D_2}}}{{{\mu _2}}};\;{\rho _1} = {{\rm{\rho }}_2}\;\& \;{\mu _1} = {\mu _2}\)

U₁D₁ = U₂D₂

2 × 100 = U₂ × 200

U₂ = 1 m/s

\(\frac{{{F_1}}}{{{\rho _1}U_1^2D_1^2}} = \frac{{{F_2}}}{{{\rho _2}U_2^2D_2^2}} \Rightarrow \frac{{{F_1}}}{{U_1^2D_1^2}} = \frac{{{F_2}}}{{U_2^2D_2^2}} \Rightarrow \frac{{{F_1}}}{{{2^2} \times {{100}^2}}} = \frac{{{F_2}}}{{{1^2} \times {{200}^2}}}\)

F₁ = F₂

\({F_1} = {C_F} ~\rho _1U_1^2D_1^2 = 0.5 \times 1000 \times {2^2} \times {\left( {0.1} \right)^2}\)

F₁ = 20 N ⇒ F₂ = 20 N

Drag force on second sphere of 200 mm diameter is 20 N