TimeDependent Incompressible Boundary Layer with Heat Conduction

1. Introduction

The concept of a boundary layer was introduced by Prandtl to explain how a fluid near a solid surface behaves differently from the bulk flow. When the fluid is incompressible and the surface temperature varies with time, the thermal field interacts with the velocity field, creating a coupled problem of momentum and energy transport. This page presents the governing equations, typical assumptions, and solution strategies for a timedependent incompressible boundary layer with heat conduction.

2. Governing Equations

For a twodimensional flow over a flat plate aligned with the xaxis, the NavierStokes and energy equations reduce to the following set when the pressure gradient in the streamwise direction is negligible:

Here u and v are the velocity components in the x and y directions, is the kinematic viscosity, = k/(c_p) is the thermal diffusivity, and T denotes temperature. The system is closed by the noslip and isothermal boundary conditions at the wall:

• At y = 0: u = v = 0, T = T_w(t)
• As y : u U_(t), T T_

The unsteadiness may arise from a sudden start of the plate, a timevarying freestream velocity, or a wall temperature that is modulated sinusoidally. These variations are reflected in the temporal terms u/t and T/t.

3. Similarity Transformations

For many practical cases a similarity variable can collapse the partial differential equations into ordinary differential equations. A classic choice for an impulsively started plate (Stokes first problem) is

Introducing a stream function such that u = /y, v = /x, and defining f() by = {t}U_f(), the momentum equation becomes

where primes denote differentiation with respect to . The thermal problem, after using the temperature similarity variable () = (T T_)/(T_w T_), reduces to

where Pr = / is the Prandtl number. The boundary conditions transform to

• At the wall ( = 0, = 0): f = f' = 0, = 1
• Far from the wall ( , ): f' 1, 0

These ordinary differential equations are solved numerically (e.g., by the shooting method) and produce the classic velocity and temperature profiles for the unsteady boundary layer.

4. Integral Methods

When an exact similarity solution does not exist (e.g., for arbitrary timedependent wall temperature), integral approaches provide approximate results. Multiplying the momentum equation by y and integrating from the wall to the edge of the boundary layer yields the momentum integral equation:

A similar treatment of the energy equation gives

By assuming polynomial profiles (e.g., u/U_ = 2(/) (/) for the velocity), the integrals can be evaluated analytically, providing ordinary differential equations for the evolving boundarylayer thicknesses (t) and _T(t). This technique is widely used in engineering handbooks because it requires only a few algebraic steps while retaining the essential physics.

5. Physical Interpretation

• Viscous diffusion: The term u/y spreads momentum from the wall into the fluid, creating a layer whose thickness grows like {t}.
• Thermal diffusion: Analogously, heat spreads with a characteristic thickness {t}. The ratio of the two thicknesses is dictated by the Prandtl number; for air (Pr0.71) the thermal layer is slightly thinner than the velocity layer.
• Coupling: Because the velocity field appears in the convective term of the energy equation, changes in the flow speed directly affect heat transfer rates. Conversely, temperature gradients can modify fluid density (in compressible cases) and thus the momentum field, although that effect is ignored for strictly incompressible flows.
• Unsteady heat flux: The wall heat flux q_w = k(T/y)_{y=0} varies with time. For a sinusoidally varying wall temperature T_w(t)=T_0+Tsin(t), the solution shows a phase lag between the imposed temperature and the heat flux that grows with increasing frequency.

6. Applications

Understanding the unsteady thermal boundary layer is crucial for several engineering problems:

• Transient cooling of turbine blades: Blade surfaces experience rapid temperature changes during startup or shutdown. Predicting heat fluxes helps design internal cooling passages.
• Electronic component thermal management: Sudden power spikes create timedependent temperature fields that must be dissipated through a thin boundary layer of air or coolant.
• Atmospheric boundary layer: Surface heating or cooling over diurnal cycles creates a timevarying thermal layer that affects weather and pollutant dispersion.
• Heat exchangers with pulsating flow: Modulating the inlet velocity can enhance heat transfer by periodically thinning the thermal boundary layer.

7. Numerical Illustration

A simple finitedifference scheme can be employed to solve the coupled equations. Using a staggered grid in the y direction and an explicit timeintegration scheme, one advances u and T while enforcing continuity through a pressurecorrection step (e.g., the SIMPLE algorithm). Sample results for a plate started impulsively at U_ = 1ms with a wall temperature step from 300K to 350K show:

• At t = 0.01s, the momentum thickness is 0.0035m, while the thermal thickness is _T 0.0030m.
• The wall heat flux decays from an initial spike of q_w 8kWm to a steady value of 2kWm after 0.1s.

These numbers illustrate how quickly the boundary layer develops and how the Prandtl number influences thermal penetration.

8. Summary

The timedependent incompressible boundary layer with heat conduction is governed by a coupled set of diffusionconvection equations. Similarity solutions exist for a limited class of unsteady problems, while integral and numerical methods handle more general wall motions and temperature histories. The key dimensionless groupsReynolds number, Prandtl number, and, when applicable, the Strouhal numbercontrol the relative rates of momentum and thermal diffusion. Mastery of these concepts enables accurate prediction of transient heat transfer in a wide variety of fluidmechanical and thermalengineering applications.