Fluid Mechanics and Heat Transfer: Flow Over a Flat Plate

Vocabulary and correlations for laminar flow, heat transfer, and boundary layer theory over flat plates.

Local Drag Coefficient ($C_{f,x}$)

For laminar flow over a flat plate, it is defined by the correlation $C_{f,x} = 0.664 Re_x^{-1/2}$.

Average Drag Coefficient ($\bar{C}_f$)

For laminar flow over the entire length of a flat plate, it is defined as $\bar{C}_f = 1.328 Re_L^{-1/2}$.

Wall Shear Stress ($\tau_w$)

The stress exerted on the fluid by the wall, calculated as $\tau_w = \frac{1}{2} \rho u^2 \bar{C}_f$.

Power (Drag)

The energy per unit time required to overcome fluid resistance, expressed as $\text{Power} = \text{force} \times \text{velocity}$ or $P = 0.664 \rho u^{5/2} \nu^{1/2} L^{1/2}$.

Film Temperature ($T_f$)

The temperature used to evaluate fluid properties in the boundary layer, calculated as the average of the wall temperature and the free stream temperature: $T_f = \frac{T_w + T_{\infty}}{2}$.

Average Nusselt Number ($Nu$)

A dimensionless parameter for convective heat transfer across the plate length $L$, given by the correlation $Nu = \frac{\bar{h}L}{k} = 0.664 Re_L^{1/2} Pr^{1/3}$.

Heat Flux ($q''$)

The rate of heat transfer per unit area, defined as $q'' = \bar{h} \Delta T$.

Local Nusselt Number ($Nu_x$)

The dimensionless heat transfer coefficient at a specific point $x$, defined as $Nu_x = 0.332 Pr^{1/3} Re_x^{1/2}$.

Average vs. Local Convective Coefficient

For laminar flow over a flat plate, the average convective heat transfer coefficient ($\bar{h}$) is twice the local coefficient ($h_x$) at the end of the plate: $\bar{h} = 2h_x$.

Critical Reynolds Number ($Re_{critical}$)

The threshold value marking the transition from laminar to turbulent flow on a flat plate, typically taken as $Re_x < 5 \times 10^5$ for laminar conditions.

Boundary Layer Thickness ($\delta$)

The distance from the wall where the fluid velocity reaches 99% of the free stream velocity, given in the notes as $\delta = 4.92 x Re_x^{-1/2}$.

Modified Blasius Equation

The governing equation for boundary layer flow over a surface with a power-law velocity distribution ($U_{\infty} = Cx^m$), expressed as $2f''' + (m+1)ff'' + 2m[1 - (f')^2] = 0$.

Similarity Variable ($\eta$)

A dimensionless coordinate used to transform partial differential equations into ordinary differential equations, defined as $\eta = y \sqrt{\frac{Re_x}{x}}$.