Question

For steady two-dimensional flow, what are the boundary layer approximations?

Short answer

Answer: The main approximations made in boundary layer theory for steady two-dimensional flow are: 1. The flow is steady, meaning fluid properties do not change with time. 2. The flow is two-dimensional, with changes in fluid properties occurring only along the surface and normal to the surface. 3. Incompressible flow, with constant density throughout the flow. 4. The pressure is constant across the boundary layer and varies only in the direction along the surface. 5. Negligible viscous forces in the direction along the surface compared to pressure forces and inertial forces.

Step-by-step solution

Understand the concept of boundary layer

A boundary layer is a region close to a solid surface (boundary) where the effects of fluid viscosity become significant. In this region, fluid velocity changes from zero at the solid surface (due to the no-slip condition) to a value near the free-stream velocity away from the surface.

Identify the governing equations for fluid motion

The Navier-Stokes equations and the continuity equation govern fluid motion. The Navier-Stokes equations describe the relationship between fluid velocity, pressure, and viscous forces, while the continuity equation describes the conservation of mass for an incompressible flow. These equations need to be simplified in the context of the boundary layer to make them more manageable.

Boundary layer approximations for steady two-dimensional flow

For steady two-dimensional flow in boundary layer theory, the following approximations are made: 1. The flow is steady: This means that fluid properties (velocity, pressure, etc.) do not change with time. Mathematically, the time derivatives of these properties are considered to be zero. 2. The flow is two-dimensional: This implies that all fluid properties are invariant in one spatial dimension, meaning that changes in fluid properties occur only in the two other dimensions. In the case of boundary layer flow, these dimensions are along the surface and normal to the surface. 3. Incompressible flow: The density of the fluid is assumed to remain constant throughout the flow. In practice, this assumption is valid for most liquid flows and low-speed gas flows where the Mach number is less than approximately 0.3. 4. The pressure is constant across the boundary layer: The pressure does not change significantly in the direction normal to the surface and can be assumed to vary only in the direction along the surface. Essentially, any pressure changes across the boundary layer can be neglected in comparison to those along the surface. 5. Negligible viscous forces in the direction along the surface: Viscous forces acting in the direction along the surface have much smaller magnitudes than pressure forces and inertial forces and can be neglected. By incorporating these approximations into the Navier-Stokes equations and the continuity equation, we can obtain a simplified set of governing equations for steady two-dimensional boundary layer flow. Note that the assumptions described in the step 3 are specific to the steady two-dimensional flow. Depending on the problem at hand, these assumptions may need to be adjusted. For example, for unsteady or three-dimensional flow, additional considerations and approximations must be made.