Boundary Layer Equations and Different Boundary Layer Thickness

Boundary Layer Equations and Different Boundary Layer Thickness

Topics Covered

• Different Boundary Layer Thickness
  • Nominal Thickness
  • Displacement Thickness
  • Momentum Thickness
  • Energy Thickness
• Equations for different BL thickness
• Boundary Layer Equations
• Assumptions

Nominal Thickness

Nominal thickness of the boundary layer is defined as the thickness of zone extending from solid boundary to a point where velocity is 99% of the free stream velocity (U). This is arbitrary, especially because transition from 0 velocity at boundary to the U outside the boundary takes place asymptotically. It is based on the fact that beyond this boundary, effect of viscous stresses can be neglected.

Many other definitions of boundary layer thickness has been introduced at different times and provide important concepts based on mathematical calculations and logic

These definitions are

• Displacement Thickness (d*)
• Momentum Thickness (q)
• Energy Thickness (de)

Displacement Thickness

Presence of boundary layer introduces a retardation to the free stream velocity in the neighborhood of the boundary. This causes a decrease in mass flow rate due to presence of boundary layer. A “velocity defect” of (U-u) exists at a distance y along y axis. Displacement thickness may be thought of as the distance (measured perpendicular to the boundary) with which the boundary may be imagined to have been shifted such that the actual flow rate would be the same as that of an ideal fluid (with slip) flowing around the displaced boundary. This may be imagined in as explained in figures on next page.

Momentum Thickness

Retardation of flow within boundary layer causes a reduction in the momentum flux too. So similar to displacement thickness, the momentum thickness (q) is defined as the thickness of an imaginary layer in free stream flow which has momentum equal to the deficiency of momentum caused to actual mass flowing inside the boundary layer

Energy Thickness

Similarly Energy thickness (de) is defined as the thickness of an imaginary layer in free stream flow which has energy equal to the deficiency of energy caused to actual mass flowing inside the boundary layer

Boundary Layer Assumptions

Following assumptions are made for the analysis of the boundary layer

• It is assumed (also observed to great extend) that Reynolds number of flows are large and the thickness of boundary layer are small in comparison with any characteristic dimension of the boundary
• The boundary is streamlined so that the flow pattern and pressures determined by ideal flow theory are accurate
• It is possible to treat the flow at constant density and isothermal conditions prevail so that viscosity is also constant

Coordinate System

Last approximations allows us to choose coordinate system with x-axis along the curved body and y axis along normal to boundary as shown below

Strictly speaking this coordinate system is curvilinear but is expected to behave like a rectangular system in the thin region of the boundary layer

Continuity Equation

Only steady two dimensional flow is considered for simplicity

Continuity Equation in 2D is

refer ppt

Where u and v are velocity components in x and y axes

Momentum Equation

Since velocity component in y direction is negligibly small so momentum equation is considered only in x direction

Considering a small control volume of sides Dx and Dy and thickness in the third direction as unity is considered as shown on next page.

refer ppt

Summation of Forces

Neglecting component of gravity in x-direction, only pressure and shear stress as shown on control volume are considered

There would be a negative shear stress on lower face because layer below is trying to retard the motion of particles within control volume. Similarly shear stress on top surface would be positive

for more details refer ppt below