A smooth flat plate \(2.4 \mathrm{~m}\) long and \(900 \mathrm{~mm}\) wide moves lengthways at \(6 \mathrm{~m} \cdot \mathrm{s}^{-1}\) through still atmospheric air of density \(1.21 \mathrm{~kg} \cdot \mathrm{m}^{-3}\) and kinematic viscosity \(14.9 \mathrm{~mm}^{2} \cdot \mathrm{s}^{-1}\). Assuming the boundary layer to be entirely laminar, calculate the boundary layer thickness (i.e. the position at which the velocity is \(0.99\) times the free-stream velocity) at the trailing edge of the plate, the shear stress half-way along and the power required to move the plate. What power would be required if the boundary layer were made turbulent at the leading edge?

The Reynolds number is a crucial factor in determining the flow characteristics over a surface. It is a dimensionless number that predicts whether the flow will be laminar or turbulent. The formula to calculate the Reynolds number is: Re = \frac{U \times L}{u} where \(U\) is the velocity, \(L\) is the characteristic length, and \(u\) is the kinematic viscosity. In our example, using the values provided, the Reynolds number would be approximately 966,442. When the Reynolds number is low, the flow tends to be laminar. High Reynolds numbers indicate turbulence. Understanding the Reynolds number helps in predicting the behavior of different types of flows in various engineering applications.

Laminar flow occurs when a fluid flows in parallel layers with no disruption between them, resulting in a smooth motion. In this state, the fluid moves in orderly straight lines, and this generally happens at lower velocities. For our example, we assumed the boundary layer to be entirely laminar, meaning that the flow over our plate is smooth and predictable.

In laminar flow, the viscosity of the fluid plays a significant role in determining the force required for movement. This is in contrast to turbulent flow, where the motion is chaotic and the forces are higher.
Understanding laminar flow helps in various applications from designing vehicle aerodynamics to predicting weather patterns.

Shear stress is the force per unit area exerted by the fluid parallel to the surface. It is an essential factor in calculating friction drag and understanding how fluids interact with solid surfaces. In laminar flow, the shear stress \( \tau\) can be determined using the formula: \tau = 0.332 \times \rho \times U^2 \times \left(\frac{u}{Ux}\right)^{0.5} where \(\rho\) is the density, \(U\) is the velocity, \(u\) is the kinematic viscosity, and \(x\) is the position along the plate. In our solved exercise, we calculated the shear stress halfway along the plate as approximately 0.36 Pa. This value tells us how much force the moving fluid exerts on the plate, playing an important role in determining energy losses and mechanical design criteria.

The boundary layer thickness represents the distance from the solid surface to the point in the fluid where the velocity reaches a significant fraction, generally 99%, of the free-stream velocity. For laminar flow, the boundary layer thickness \( \delta (x)\) at a given distance \( x \) from the leading edge of the plate is given by: \delta (x) = 5.0 \times \sqrt{\frac{x}{Re_x}} In our exercise, we found the boundary layer thickness at the trailing edge of the plate to be approximately 0.00768 meters. This thin layer of fluid slows down due to viscosity, creating a gradient from the no-slip boundary condition at the surface to almost the free-stream velocity.

Understanding boundary layer thickness is critical for predicting aerodynamic drag, heat transfer rates, and in setting up lubrication in mechanical systems.

Calculating the power required to move a plate through a fluid involves understanding the force exerted by the flow and the velocity of movement. The power required can be determined using the formula: P = \tau \times A \times U where \( \tau\) is the shear stress, \(A\) is the surface area, and \(U\) is the velocity. In our given exercise, with an area of 2.16 \(m^2\), a velocity of 6 m/s, and a shear stress of 0.36 Pa, the power required was calculated to be approximately 4.67 W.

When the flow becomes turbulent, the power required generally increases due to higher shear stress values. Turbulent flow induces more friction drag, which directly impacts the energy required to maintain motion. Understanding power calculations is important for designing efficient systems in fields such as automotive, aerospace, and marine engineering.