Boundary layer theory

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Unless the geometry is very simple, it is difficult to solve for the complete viscous fluid flow around a body. A full domain numerical solution is time consuming and impractical, because one needs to solve the full Navier-Stokes equations in the full domain, which are nonlinear, elliptic, and complex.

In 1904, Prandtl discovered that for most practical applications, the influence of viscosity is observed in a very thin domain close to the object, as shown in Fig.1. Outside this region one can assume the flow is inviscid (µ = 0) .

Viscous or momentum boundary layer
Figure 1: Viscous or momentum boundary layer.

The thin region where the effect of viscosity is dominant is called the momentum or viscous boundary layer. The solution of boundary layer analysis can be simplified due to the fact that its thickness is much smaller than the characteristic dimension of the object. The fluid adjacent to the surface of the body has zero relative velocity; ufluid – usurface = 0. This is also called the no slip boundary condition.

One of the assumptions of boundary layer approximations is that the fluid velocity at the wall is at rest relative to the surface. This is true except when the fluid pressure is very low, and therefore, the Kundsen number, Kn= λ/L, of the fluid molecules is much larger than 1. For external flow, the flow next to an object can be divided into two parts. The larger part is related to a free stream of fluid, in which the effect of viscosity is negligible (potential flow theory). The smaller region is a thin layer next to the surface of the body, in which the effects of molecular transport (such as viscosity, thermal conductivity and mass diffusivity) are very important.

Boundary layer concept over a flat plate
Figure 1: Boundary layer concept over a flat plate.

Potential flow theory neglects the effect of viscosity, and therefore, significantly simplifies the Navier-Stokes equations, which provides the solution for the velocity distribution. A disadvantage of the potential theory is that since second order terms are neglected, the effect of viscosity, no slip, and impermeability boundary conditions at the surface cannot be accounted for. In general, potential flow theory predicts the free stream field accurately, despite its simplicity. Boundary layer thickness is defined as the distance within the fluid in which most of the velocity change occurs. This thickness is usually defined as the thickness in which the velocity reaches 99% of the free stream velocity, u = 0.99U.

Figure 2 illustrates how the momentum boundary layer thickness changes along the plate. Flow is laminar at relatively small values of x, where it is shown as the laminar boundary layer region. As x increases, the fluid motion begins to fluctuate. This is called the transition region. The boundary layer may be either laminar or turbulent in this region. Finally, above a certain value of x, the flow will always be turbulent. There is a very thin region next to the wall in the turbulent region where the flow is still laminar, called the laminar sublayer.

For flow over a flat plate, experimental data indicates

Rex ≤ 2×105 the flow is laminar
2×105 <Rex <3×106 the flow is in transition
Rex ≥ 3×106 the flow is turbulent


\operatorname{Re} _x  = \frac{{\rho U_\infty  x}}
{\mu }