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Velocity and Thermal Boundary Layers for Fluid Dynamics

Key Takeaways

  • The region defined by the velocity gradient where the flow velocity is distributed among the different fluid layers is called the velocity boundary layer. 

  • The thermal boundary layer is the region of fluid flow defined by the temperature gradient formed due to the thermal energy exchange among the adjacent layers. 

  • The study of thermal and momentum diffusivity facilitates understanding of the relationship between frictional resistance of the fluid and heat transfer.

Velocity and thermal boundary layer analysis in a fluid system

Velocity and thermal boundary layer analysis in a fluid system

In the discussion of boundary conditions in fluid dynamics, the fundamental concept lies in the interaction of velocity and temperature layers in the flow region near the boundaries. Fluid systems exhibit unique boundary condition attributes, and designers must have an understanding of these interactions to properly design a fluid system and analyze its characteristics in laminar or turbulent conditions. In this article, we will discuss the velocity boundary layer, thermal boundary layer, the relationship between thermal and fluid friction measurements, and boundary-layer analysis.

Velocity Boundary Layer

Let us consider a fluid flow over a stationary surface. Irrespective of the flow behavior, the fluid particles stick to the surface due to the viscous effect and the velocity of the fluid touching the solid surface is brought to rest due to the shear stress. This shearing stress also creates a velocity gradient in the direction normal to the flow. This region where the flow velocity is distributed among the different fluid layers is called the velocity boundary layer.

Outside the velocity boundary layer, the shear stress is negligible due to the absence of a velocity gradient. Over time, the boundary layers can develop laminar or turbulent attributes, and can be determined by analyzing the Reynolds number for each flow type. Usually, the turbulent boundary layer has larger shear stress and exhibits higher heat transfer rates. 

Thermal Boundary Layer

Similar to the velocity boundary layer, as we consider a fluid flow over a solid surface, the thermal boundary layer forms. If the fluid with temperature ‘T’ is flowing over a flat plate with temperature ‘Twall,’ the boundary layer fluid particle that comes in immediate contact with the surface will achieve a thermal equilibrium at the surface temperature. The energy flow at this stage is due to conduction. However, among the adjacent fluid layers, conduction and diffusion facilitate an energy exchange to form a thermal gradient, which has a similar attribute as that of the velocity gradient near the boundary layer. The region of fluid flow with the temperature gradient is called the thermal boundary layer.

The thickness of the thermal boundary layer, ઠₜ, is the distance from the surface boundary to the point where the temperature of the flow has reached 99% as the free-stream temperature. For laminar flow, the thermal boundary layer thickness can be expressed as:

Thermal boundary layer thickness for laminar flow

Note that:

Pr is the Prandtl number

ઠᵥ is velocity boundary layer thickness

v is the kinematic viscosity

x is the distance measured downstream from the start of the boundary layer

u。is the freestream velocity

Similarly, for the turbulent boundary layer: 

 Thermal boundary layer thickness formula for turbulent flow

Note that Reₓ is the Reynolds number. 

Thermal Diffusivity

In heat transfer problems, the thermal diffusivity is better explained with the Prandtl number. The Prandtl number is simply the ratio of momentum and thermal diffusivity and is expressed as:

Prandtl number

Note that Cp is the specific heat capacity, µ is the dynamic viscosity, and K is the thermal conductivity.

Thermal diffusivity explains how fast heat diffusion occurs through a given material. Generally, the smaller the Prandtl number, i.e., Pr<<1, the more thermal diffusivity controls the flow behavior. 

The Relationship Between Thermal and Fluid Friction Measurements

Since both thermal and momentum diffusivity are observed during flow and boundary layer analysis, it is ideal to understand the relationship between frictional resistance of the fluid and the heat transfer. The Reynold-Colburn analogy can ideally relate heat transfer and friction factor coefficients, which can be expressed as:

Reynold-Colburn analogy

Note that:

F is the friction factor

Nu is the Nusselt number

Re is the Reynolds number

Pr is the Prandtl number

Pr is the Stanton number, which in the boundary layer represents the change in thermal energy due to the transfer of heat from the surface wall.

For laminar flow on a flat plate, the above formula can be expressed as:

Elaboration of Reynold-Colburn analogy on laminar flow on a flat plate

However, in the case of turbulent flow on a flat plate, three regions can be observed:

  1. The laminar sublayer, where a thin layer near the surface exhibits laminar behavior.
  2. The buffer layer at distance y from the plate surface, where the turbulence, viscosity, and heat conduction all take effect.
  3. The fully turbulent layer.

On a flat plate, the friction factor can be determined with Reynold-Colburn analogy as:

Elaboration of Reynold-Colburn analogy on turbulent flow on a flat plate

For a laminar-turbulent boundary layer, the average heat transfer can be expressed using the Reynold-Colburn analogy as:

 Average heat transfer equation for laminar-turbulent boundary layer

Boundary-Layer Analysis on a CFD Platform

The boundary-layer analysis for laminar or turbulent flow requires system designers to understand the complete picture of the momentum and thermal transfer within a flow system. This is facilitated through computational fluid dynamics (CFD) solvers, which allow identification of factors that affect analysis such as shear stress, diffusivity, and friction. The CFD tools from a platform such as Omnis facilitate numerical analysis of the governing equations to ideally predict the profile for velocity and thermal boundary layers. With an advanced CFD platform, designers can be assured of high-quality modeling and accurate simulations for all kinds of boundary layer analysis.

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