Forced Convection: External Flow Over Flat Plates

Mastering forced convection over a flat plate is essential for designing everything from heat exchangers and electronic cooling systems to aerodynamic surfaces. At its core, this topic provides the predictive tools—empirical correlations—to calculate how much heat is transferred between a surface and a fluid flowing over it. Your ability to select and apply the correct correlation directly impacts the accuracy and efficiency of your thermal designs.

Fundamentals of the Boundary Layer

When a fluid flows over a surface, its velocity is zero at the wall due to the no-slip condition. The fluid velocity increases from zero to the free-stream velocity, $U_{\infty }$, over a small region adjacent to the plate called the velocity boundary layer. The thickness of this layer, $\delta$, grows along the length of the plate. Simultaneously, if the plate temperature $T_s$ differs from the free-stream temperature $T_{\infty }$, a thermal boundary layer develops. Its thickness, ${\delta }_t$, is the region where temperature gradients exist.

The behavior of these layers dictates the heat transfer rate. Within the boundary layer, heat is transferred primarily by conduction perpendicular to the surface. Outside of it, bulk fluid motion (advection) dominates. Therefore, the steepness of the temperature gradient at the wall ($y=0$) is the driving force for convective heat transfer, as described by Fourier's law: $q_s^{\prime \prime }=-k\frac{\partial T}{\partial y}{|}_{y=0}$.

Key Dimensionless Numbers and Their Physical Meaning

The correlations for flat plates are expressed using three critical dimensionless numbers. Understanding what they represent is key to applying the formulas correctly.

• Reynolds Number ($R{e}_L$ or $R{e}_x$): This ratio of inertial forces to viscous forces determines the flow regime. For a flat plate of length $L$, $R{e}_L=\frac{\rho U_{\infty }L}{\mu }=\frac{U_{\infty }L}{\nu }$. Flow is laminar (smooth, orderly) for $R{e}_x<5\times 10^5$ and transitions to turbulent (chaotic, mixing) at higher values. The local Reynolds number, $R{e}_x=\frac{U_{\infty }x}{\mu }$, uses the distance from the leading edge, $x$.

• Prandtl Number ($Pr$): This fluid property ratio compares momentum diffusivity (viscosity) to thermal diffusivity: $Pr=\frac{\nu }{\alpha }=\frac{c_p\mu }{k}$. A low $Pr$ (e.g., liquid metals) means heat diffuses quickly relative to momentum. A high $Pr$ (e.g., oils) means thermal diffusion is slow. It determines the relative thickness of the velocity and thermal boundary layers ($\delta$ vs. ${\delta }_t$).

• Nusselt Number ($N{u}_L$ or $N{u}_x$): This is the primary output of our correlations. It represents the enhancement of heat transfer through convection relative to conduction alone. The local Nusselt number is defined as $N{u}_x=\frac{h_{x}x}{k}$, where $h_x$ is the local convection coefficient. The average Nusselt number over a length $L$ is $N{u}_L=\frac{{\bar{h}}_{L}L}{k}$, where ${\bar{h}}_L$ is the average convection coefficient.

Laminar Flow Correlations (For $R{e}_x<5\times 10^5$)

For laminar flow with constant surface temperature, the boundary layer equations yield an exact similarity solution. The resulting correlations are straightforward power-law relationships.

Local Nusselt Number: The local heat transfer coefficient at any distance $x$ from the leading edge is given by: $$N{u}_x=\frac{h_{x}x}{k}=0.332R{e}_x^{1/2}P{r}^{1/3}\text{for}Pr\gtrsim 0.6$$ This shows $h_x$ is proportional to $x^{-1/2}$; the heat transfer coefficient decreases along the plate as the boundary layer thickens.

Average Nusselt Number: To find the average coefficient over a plate of length $L$ entirely in laminar flow, we integrate the local expression: $${\bar{h}}_L=\frac{1}{L}{\int }_0^L{h}_{x}dx$$ This integration results in the average correlation: $$N{u}_L=\frac{{\bar{h}}_{L}L}{k}=0.664R{e}_L^{1/2}P{r}^{1/3}$$ Notice the similarity to the local correlation; the average Nusselt number over length $L$ is simply twice the local Nusselt number evaluated at $x=L$: $N{u}_L=2N{u}_x$ at $x=L$.

Turbulent Flow Correlations (For $R{e}_x\gtrsim 5\times 10^5$)

Turbulent flow induces intense mixing, which dramatically enhances heat transfer compared to laminar flow. The correlations are based on empirical data and analogy with momentum transfer.

Local Nusselt Number: A widely used correlation for local turbulent flow from the leading edge is: $$N{u}_x=\frac{h_{x}x}{k}=0.0296R{e}_x^{4/5}P{r}^{1/3}$$ Valid for $0.6\lesssim Pr\lesssim 60$ and $5\times 10^5\le R{e}_x\le 10^7$. The exponent of $4/5$ on the Reynolds number is significantly larger than the laminar exponent of $1/2$, confirming the stronger dependence and higher heat transfer rates in turbulence.

Average Nusselt Number: For a plate with turbulent flow over its entire length, integration yields: $$N{u}_L=\frac{{\bar{h}}_{L}L}{k}=0.037R{e}_L^{4/5}P{r}^{1/3}$$ Again, note the relationship: for turbulent flow, $N{u}_L\approx 1.25N{u}_x$ at $x=L$.

Mixed Boundary Layer Analysis

Real engineering surfaces often experience a mixed boundary layer: laminar flow from the leading edge up to a critical distance $x_c$ (where $R{e}_{x_c}=5\times 10^5$), followed by turbulent flow thereafter. You cannot simply use the fully laminar or fully turbulent average correlations. The correct approach is to calculate the average convection coefficient by integrating the local contributions from each regime.

The average coefficient for a plate of length $L>x_c$ is found by piecewise integration: $${\bar{h}}_L=\frac{1}{L}\left({\int }_0^{x_c}{h}_{x,\text{lam}}dx+{\int }_{x_c}^L{h}_{x,\text{turb}}dx\right)$$

Assuming the transition occurs abruptly at $x_c$, performing this integration with the standard correlations leads to a practical formula for the average Nusselt number: $$N{u}_L=\frac{{\bar{h}}_{L}L}{k}=(0.037R{e}_L^{4/5}-A)P{r}^{1/3}$$ Where the constant $A$ accounts for the laminar region. For a standard critical Reynolds number of $R{e}_{x_c}=5\times 10^5$, the value is $A=871$.

Example Calculation: Air at 300 K ($Pr=0.71$, $\nu =1.6\times 10^{-5}{m}^2/s$) flows at $U_{\infty }=20m/s$ over a 1.5 m long isothermal plate. Find the average heat transfer coefficient.

1. Calculate $R{e}_L$: $R{e}_L=\frac{U_{\infty }L}{\nu }=\frac{20\times 1.5}{1.6\times 10^{-5}}=1.875\times 10^6$. This is > $5\times 10^5$, so the flow is mixed.
2. Apply the mixed flow correlation with $A=871$:

$$N{u}_L=(0.037(1.875\times 10^6{)}^{4/5}-871)\times (0.71{)}^{1/3}$$ $$N{u}_L=(0.037\times 1.105\times 10^5-871)\times 0.89$$ $$N{u}_L=(4089-871)\times 0.89=3218\times 0.89\approx 2864$$

1. Solve for ${\bar{h}}_L$: $k_{air}\approx 0.026W/m\cdot K$. ${\bar{h}}_L=\frac{N{u}_L\cdot k}{L}=\frac{2864\times 0.026}{1.5}\approx 49.6W/m^2\cdot K$.

Using the fully turbulent correlation here would have overestimated ${\bar{h}}_L$ by about 15%.

Common Pitfalls

1. Misapplying the Flow Regime: The most frequent error is using a laminar correlation for a high $R{e}_L$ flow or vice-versa. Always calculate $R{e}_L$ or $R{e}_x$ first to determine the correct regime. For $R{e}_L$ between $5\times 10^5$ and $10^7$, you must use the mixed boundary layer analysis unless the flow is intentionally tripped to be fully turbulent from the leading edge.

1. Confusing Local and Average Quantities: Using $h_x$ when you need ${\bar{h}}_L$ (or the corresponding $Nu$) will give you the heat transfer rate at a point, not the total for the plate. Remember the relationship: for laminar flow, ${\bar{h}}_L=2h_x$ at $x=L$; for turbulent flow, ${\bar{h}}_L\approx 1.25h_x$ at $x=L$.

1. Ignoring Property Evaluation Temperature: The fluid properties ($k,\mu ,\nu ,Pr$) are temperature-dependent. Correlations typically require properties to be evaluated at the film temperature, $T_f=(T_s+T_{\infty })/2$, which is an average of the surface and free-stream temperatures. Using properties at the wrong temperature can lead to significant error.

1. Overlooking the Prandtl Number Range: The standard correlations $Nu\propto P{r}^{1/3}$ are valid for gases and common liquids ($0.6\lesssim Pr\lesssim 60$). For extreme fluids like liquid metals ($Pr<<1$) or heavy oils ($Pr>>60$), specialized correlations with different $Pr$ dependencies must be used.

Summary

• Forced convection over a flat plate is governed by the development of velocity and thermal boundary layers, characterized by the Reynolds ($Re$) and Prandtl ($Pr$) numbers.
• The local Nusselt number for laminar flow varies as $N{u}_x\propto R{e}_x^{1/2}$, while for turbulent flow it varies more strongly as $N{u}_x\propto R{e}_x^{4/5}$, indicating more effective heat transfer.
• The average Nusselt number over a plate length $L$ is found by integrating the local expression. For fully laminar flow, $N{u}_L=0.664R{e}_L^{1/2}P{r}^{1/3}$.
• For practical flows with a mixed boundary layer (laminar then turbulent), the correct average correlation is $N{u}_L=(0.037R{e}_L^{4/5}-A)P{r}^{1/3}$, where $A$ accounts for the laminar starting length.
• Successful application requires carefully determining the flow regime, distinguishing between local and average values, and evaluating all fluid properties at the appropriate film temperature.