Question

Consider laminar flow over a flat plate. Will the friction coefficient change with distance from the leading edge? How about the heat transfer coefficient?

Short answer

Answer: Yes, both the friction coefficient and the heat transfer coefficient change with the distance from the leading edge in laminar flow over a flat plate.

Step-by-step solution

Defining Friction Coefficient for Boundary Layer Flow

Friction coefficient (\(C_f\)) is defined as the ratio of wall shear stress (\(\tau_w\)) to the dynamic pressure (\(\frac{1}{2}\rho U_\infty^2\)) in incompressible boundary layer flows over a flat plate: \(C_f=\frac{\tau_w}{\frac{1}{2}\rho U_\infty^2}\) Here \(\tau_w\) represents the wall shear stress, \(\rho\) denotes the fluid density, and \(U_\infty\) is the free-stream velocity of the fluid approaching the flat plate. For laminar boundary layer flow, we can derive the friction coefficient as a function of Reynolds number (\(Re_x\)), where \(x\) is the distance from the leading edge.

Deriving Friction Coefficient Dependence on Reynolds Number and Distance

For laminar flow over a flat plate, the Reynolds number at any distance x along the surface of the plate can be defined as: \(Re_x=\frac{\rho U_\infty x}{\mu}\) Here, \(\mu\) is the fluid's dynamic viscosity. By using the Blasius solution for laminar boundary layer flow over a flat plate (which is applicable for \(Re_x < 5\times10^5\)), we can express the friction coefficient as a function of \(Re_x\): \(C_f=\frac{1.328}{\sqrt{Re_x}}\) Since \(Re_x\) depends on \(x\): \(Re_x \propto x\), we conclude that the friction coefficient changes with the distance from the leading edge.

Defining Heat Transfer Coefficient for Boundary Layer Flow

The heat transfer coefficient (\(h\)) is defined as the proportionality constant between the local heat flux (\(q_w\)) and the temperature difference between the free-stream fluid temperature (\(T_\infty\)) and the wall temperature (\(T_w\)): \(h=\frac{q_w}{T_w-T_\infty}\) We will now evaluate if \(h\) changes with the distance from the leading edge for laminar boundary layer flow over a flat plate.

Determining Heat Transfer Coefficient's Dependence on Reynolds Number and Distance

For laminar flow over a flat plate, we can apply the analogy between momentum transfer (wall shear stress) and heat transfer through Reynolds Analogy, which states that the Nusselt number (\(Nu_x\)) is related to \(Re_x\) as: \(Nu_x=\frac{h x}{k}=\frac{C_{fx} Re_x Pr}{2}\) Here, \(k\) represents the thermal conductivity of the fluid, and \(Pr\) is its Prandtl number. Since both \(Nu_x\) and \(Re_x\) depend on distance \(x\): \(Nu_x \propto x\) and \(Re_x \propto x\), we conclude that the heat transfer coefficient changes with the distance from the leading edge for laminar boundary layer flow over a flat plate. In summary, both the friction coefficient and the heat transfer coefficient change with the distance from the leading edge in laminar flow over a flat plate.

Key concepts

Friction Coefficient

The friction coefficient, denoted as \(C_f\), is an important parameter when analyzing boundary layer flow, especially over flat surfaces like plates. It is calculated as the ratio of wall shear stress \(\tau_w\) to the dynamic pressure \(\frac{1}{2}\rho U_\infty^2\). This coefficient helps us understand how much force opposes the motion of the fluid as it travels along the surface. In a laminar flow over a flat plate, the friction coefficient is influenced by the Reynolds number \(Re_x\), which is a function of the distance \(x\) from the leading edge of the plate.

This relationship implies that as you move further from the leading edge, which increases \(x\) and therefore \(Re_x\), the friction coefficient decreases because it is inversely proportional to the square root of \(Re_x\) (\(C_f = \frac{1.328}{\sqrt{Re_x}}\)). This means the resistance to motion, or the frictional forces, lessen the further the fluid travels over the plate.

Heat Transfer Coefficient

The heat transfer coefficient, symbolized as \(h\), is a measure of how efficiently heat is transferred between a fluid and a surface. It is defined by the relationship \(h = \frac{q_w}{T_w - T_\infty}\), where \(q_w\) is the local heat flux and \(T_w - T_\infty\) is the temperature difference between the wall and the free stream fluid.

In the context of laminar boundary layer flow over a flat plate, this coefficient also varies with distance from the leading edge, similar to the friction coefficient. Using the Reynolds Analogy, which relates momentum transfer and heat transfer, the Nusselt number \(Nu_x\) can help in expressing \(h\) in terms of the Reynolds number \(Re_x\).

The Nusselt number, \(Nu_x = \frac{h x}{k}\), implies that the heat transfer coefficient \(h\) decreases with increasing \(x\), as both \(Nu_x\) and \(Re_x\) are proportional to \(x\). This phenomenon means that the rate of heat exchange reduces as the fluid traverses the surface of the plate.

Reynolds Number

Reynolds number \(Re_x\) is a dimensionless quantity that describes the flow characteristics of a fluid. For flow over a flat plate, it's specifically defined as \(Re_x = \frac{\rho U_\infty x}{\mu}\), where \(\rho\) is the fluid density, \(U_\infty\) is the free-stream velocity, \(x\) is the distance from the leading edge, and \(\mu\) is the dynamic viscosity of the fluid.

As \(x\) changes, so does \(Re_x\). In the scope of laminar flow, the Reynolds number is crucial because it helps predict the transition from laminar to turbulent flow. For laminar flow to be maintained, \(Re_x\) must generally remain below a value of around \(5 \times 10^5\).

Therefore, the Reynolds number is not just a measure but also acts as a threshold indicator for the type of flow and is directly linked to other flow characteristics, such as the friction coefficient and heat transfer efficiency.

Boundary Layer Flow

Boundary layer flow refers to the thin region adjacent to the surface of an object where the effects of viscosity are significant. In fluids moving past a flat plate in laminar flow, a boundary layer develops from the leading edge and increases in thickness with distance \(x\) along the plate.

Within this boundary layer, the velocity of the fluid changes from zero at the surface of the plate (due to the no-slip condition) to the free-stream velocity \(U_\infty\) as one moves away from the plate. This gradual velocity change within the boundary layer is where shear stresses play a significant role, influencing parameters like the friction coefficient.

The nature of boundary layer flow, particularly in laminar conditions, impacts both momentum and heat transfer characteristics. Understanding the boundary layer is essential for predicting how friction and thermal energy are distributed along the surface, which is vital in designing efficient thermal systems or predicting aerodynamic properties.