in and ). Neither one of these assumptions is true for the general turbulent boundary layer case so care must be exercised in applying this formula. Displacement thickness The displacement thickness, or , is the normal distance to a reference plane representing the lower edge of a hypothetical inviscid fluid of uniform velocity that has the same flow rate as occurs in the real fluid with the boundary layer. The displacement thickness essentially modifies the shape of a body immersed in a fluid to allow, in principle, an inviscid solution if the displacement thicknesses were known a priori. The definition of the displacement thickness for compressible flow, based on mass flow rate, is where is the density. For incompressible flow, the density is constant so the definition based on volumetric flow rate becomes For turbulent boundary layer calculations, the time-averaged density and velocity are used. For laminar boundary layer flows along a flat plate that behave according to the Blasius solution conditions, the displacement thickness is where is constant. The displacement thickness is not directly related to the boundary layer thickness but is given approximately as . It has a prominent role in calculating the Shape Factor. It also shows up in various formulas in the Moment Method. Momentum thickness The momentum thickness, or , is the normal distance to a reference plane representing the lower edge of a hypothetical inviscid fluid of uniform velocity that has the same momentum flow rate as occurs in the real fluid with the boundary layer. The momentum thickness definition for compressible flow based on the mass flow rate is For incompressible flow, the density is constant so that the definition based on volumetric flow rate becomes where are the density and is the 'asymptotic' velocity. For turbulent boundary layer calculations, the time averaged density and velocity are used. For laminar boundary layer flows along a flat plate that behave according to the Blasius solution conditions, the momentum thickness is where is constant. The momentum thickness is not directly related to the boundary layer thickness but is given approximately as . It has a prominent role in calculating the Shape Factor. A related parameter called the Energy Thickness is sometimes mentioned in reference to turbulent energy distribution but is rarely used. Shape factor A shape factor is used in boundary layer flow to help to differentiate laminar and turbulent flow. It also shows up