Boundary layer approximations

From Thermal-FluidsPedia

Jump to: navigation, search

If one assumes that the boundary layer thickness, δ, is very small compared to the characteristic dimension of the object, one can make the assumption that δ is significantly less than L (δ L), where L is the characteristic dimension of the object. Using scale analysis discussed in Chapter 1, for flow over a flat plate with constant free stream velocity, one can show, in a steady two-dimensional laminar boundary layer representation, that the following conditions are met within the boundary layer region, assuming there are no body forces:

\frac{{\partial ^2 u}}{{\partial y^2 }} \gg \frac{{\partial ^2 u}}{{\partial x^2 }}
u \gg v
\frac{{\partial p}}{{\partial x}} \approx \frac{{dp}}{{dx}}
\frac{{\partial p}}{{\partial y}} \approx 0
Boundary layer concept over a flat plate
Figure 1: Boundary layer concept over a flat plate.

A similar concept exists when there is heat and/or mass transfer between a fluid and the surface of an object. Again, the region in which the effect of temperature or concentration is dominant is, in the most practical case, a region very close to the surface, as shown in Fig. 4.4. In general, for the case of flow over a flat plate, one can expect three different boundary layer regions with thicknesses δ, δT, and δC, corresponding to momentum, thermal, and concentration boundary layers, respectively. δ, δT, and δC are not necessarily the same thickness and their values, as will be shown later, depend on the properties of the fluid such as kinematic viscosity ( ), thermal diffusivity (α = k/ρcp), specific heat, c, and mass diffusion coefficient, D. Using boundary layer approximation, scale analysis, and order of magnitude, one can show that similar approximations exist for thermal and mass concentration boundary layer analysis.

\frac{{\partial T}}{{\partial y}} \gg \frac{{\partial T}}{{\partial x}},\quad \frac{{\partial ^2 T}}{{\partial y^2 }} \gg \frac{{\partial ^2 T}}{{\partial x^2 }}
\frac{{\partial \omega }}{{\partial y}} \gg \frac{{\partial \omega }}{{\partial x}},\quad \frac{{\partial ^2 \omega }}{{\partial y^2 }} \gg \frac{{\partial ^2 \omega }}{{\partial x^2 }}

where ω is the mass fraction.