Abstract
An investigation of the errors inherent in the calculation of integral boundary-layer parameters from discrete datasets has been carried out. The primary errors examined were those due to discretization of the velocity profile, distance of the first data location from the wall, and uncertainty in the floor location. A range of turbulent velocity profiles with different shape factors from analytical models and published DNS investigations has been examined. This analysis demonstrates that the spacing of the first measurement point from the floor is by far the most critical error source. Furthermore, the error is shown to be a function of boundary-layer shape factor, and therefore, a correction factor chart has been derived. Two alternative methods of estimating integral boundary-layer parameters have been examined: wall modeling and a gradient-based formulation. These have both been shown to generate smaller errors than the basic integration approach, although both are susceptible to external influences.
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Notes
It was found that plotting \(H_{\rm iCALC}\) against \(H_{\rm i}\) produced curves which were too closely spaced to be of practical use.
While the determination of the boundary-layer thickness is not crucial for the integral parameters, its accurate determination is certainly useful for other comparative purposes.
In the analysis, the number of samples above \(y_1\) was set to 600 which approximates a continuous profile.
Note that the additional wall-modeled point is not included in the definition of the measurement resolution, N.
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The authors would like to acknowledge the financial support provided by the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom (UK). The authors would also like to thank Todd Davidson and Nina Siu for thoroughly proofreading this article prior to publication.
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Appendix
Appendix
1.1 Numerical integration errors
In Sect. 2.2, the variation in error incurred due to numerical integration is shown as a function of wall-normal height. For each y, this has been estimated by considering the error in the integral between two points around y, \(y_1 =y-\Delta y/2\) and \(y_2 =y+\Delta y/2\). Let f be a smooth function between the two points \(y_1\) and \(y_2\). The value of the integral \(I = \int _{y_1}^{y_2}f \hbox{d}y\) is estimated by numerical integration which involves interpolating between the end points using a curve of order n. The general formula for the error in the resulting value of the integral is given by:
For details, the reader should refer to Milne-Thomson (1951), Delves and Walsh (1974), Nonweiler (1984). In the present paper, the trapezium rule has been used, which interpolates with straight lines, i.e., \(n=1\). Equation 16 therefore shows that the error will be proportional to the second derivative of f. For the boundary-layer integral parameters:
Inserting Eqs. 17 and 18 into Eq. 16 recovers the quoted formulae, Eqs. 7 and 8.
1.2 Gradient-based integral parameters
The integral boundary-layer parameters may be redefined in terms of velocity gradients using integration by parts. Starting with the kinematic displacement thickness, \(\delta ^{*}_{\rm i}\), Eq. 4: integrating the first term directly and the second term by parts gives:
Inserting the boundary conditions, \(u(y\rightarrow \infty ) = u_{\infty }\) and \(u(y=0) = 0\), enables the first two terms to cancel, leaving the result quoted in Eq. 14 (Sect. 4.2). Similarly for the kinematic momentum thickness, \(\theta _{\rm i}\), integrating both terms in the integrand of Eq. 5 by parts yields
Again, using the boundary conditions, the terms \([u_{\infty }uy]_0^{\infty }\) and \([u^2y]_0^{\infty }\) cancel. Some minor rearrangement leads directly to Eq. 15 in Sect. 4.2.
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Titchener, N., Colliss, S. & Babinsky, H. On the calculation of boundary-layer parameters from discrete data. Exp Fluids 56, 159 (2015). https://doi.org/10.1007/s00348-015-2024-5
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DOI: https://doi.org/10.1007/s00348-015-2024-5