When tested in water at 20^∘ C flowing at 2 m / s, an 8-cm-...

When tested in water at $20^{\circ} \mathrm{C}$ flowing at $2 \mathrm{m} / \mathrm{s},$ an $8-\mathrm{cm}-$ diameter sphere has a measured drag of 5 N. What will be the velocity and drag force on a 1.5 -m-diameter weather balloon moored in sea-level standard air under dynamically similar conditions?

Key Concepts

Scaling Laws in Fluid Mechanics

Scaling laws in fluid mechanics help predict how different physical quantities, such as force and velocity, change with the size of the object. These laws are derived from the non-dimensional analysis of the governing equations and indicate how drag scales with object dimensions and velocity. They are crucial in engineering applications where experiments on small-scale models are used to predict full-scale behavior.

Drag Coefficient

The drag coefficient is a dimensionless number that characterizes the drag per unit area of an object in a fluid flow. It encapsulates the effects of the object's shape, surface roughness, and flow conditions, and remains approximately constant under dynamically similar conditions. When moving between scales, if the Reynolds number is kept constant, the drag coefficient is expected to remain the same, which simplifies the prediction of drag forces.

Reynolds Number

The Reynolds number is a dimensionless parameter defined as the ratio of inertial forces to viscous forces in a fluid flow. It is given by Re = (density × velocity × characteristic length) / viscosity. Matching the Reynolds number in different situations is critical in achieving dynamic similarity, as it ensures that the flow patterns and drag coefficients are comparable between scaled models and full-scale applications.

Dynamic Similarity

Dynamic similarity refers to the condition where two different flow situations are governed by the same non-dimensional parameters, such as the Reynolds number. This condition allows one to infer the behavior of one system from the other because the flow fields (when properly non-dimensionalized) are identical. In practice, ensuring dynamic similarity means that experiments conducted in one fluid or at one scale can be applied to predict behavior in another, provided the key dimensionless quantities match.

Drag Force

The drag force is the aerodynamic or hydrodynamic force experienced by an object moving through a fluid, acting opposite to the direction of motion. It is generally modeled by the equation F_D = (1/2) × fluid density × velocity^2 × reference area × drag coefficient. This relation shows that drag force is dependent on both the size of the object (through the reference area) and the square of the velocity, making it a key factor in scaling problems.

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