by There are some advantages to also include descriptions of moments of the boundary layer profile derivatives with respect to the height above the wall. Consider the first derivative velocity profile central moments given by where the first derivative mean location is the displacement thickness . Finally the second derivative velocity profile central moments are given by where the second derivative mean location, , is given by where is the viscosity and where is the wall shear stress. The mean location, , for this case is formally defined as ue(x) divided by the area under the second derivative curve. The above equations work for both laminar and turbulent boundary layers as long as the time-averaged velocity is used for the turbulent case. With the moments and the mean locations defined, the boundary layer thickness and shape can be described in terms of the boundary layer widths (variance), skewnesses, and excesses (excess kurtosis). Experimentally, it is found that the thickness defined as where , tracks the very well for turbulent boundary layer flows. Taking a cue from the boundary layer momentum balance equations, the second derivative boundary layer moments, track the thickness and shape of that portion of the boundary layer where the viscous forces are significant. Hence the moment method makes it possible to track and quantify the laminar boundary layer and the inner viscous region of turbulent boundary layers using moments whereas the boundary layer thickness and shape of the total turbulent boundary layer is tracked using and moments. Calculation of the 2nd derivative moments can be problematic since under certain conditions the second derivatives can become positive in the very near-wall region (in general, it is negative). This appears to be the case for interior flow with an adverse pressure gradient (APG). Integrand values do not change sign in standard probability framework so the application of the moment methodology to the second derivative case will result in biased moment measures. A simple fix is to exclude the problematic values and define a new set of moments for a truncated second derivative profile starting at the second derivative minimum. If the width, , is calculated using the minimum as the mean location, then the viscous boundary layer thickness, defined as the point where the second derivative profile becomes negligible above the wall, can be properly identified with this modified approach. For derivative moments whose integrands do not change sign,