On the calculation of boundary-layer parameters from discrete data

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.

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:

$$\begin{aligned} \varepsilon _{\rm I}&= {} \frac{1}{(n+1)!}\frac{\partial ^{(n+1)} f}{\partial y^{(n+1)}}\int _{y_1}^{y_2}(\tilde{y}-y_1)(\tilde{y}-y_2) \hbox{d}\tilde{y} \\&= {} \frac{1}{(n+1)!}\frac{\partial ^{(n+1)}f}{\partial y^{(n+1)}}\left\{ -\frac{1}{6}(y_2-y_1)^3\right\} \\&= {} -\frac{\Delta y^3}{6(n+1)!}\frac{\partial ^{(n+1)}f}{\partial y^{(n+1)}} \end{aligned}$$

(16)

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:

$$\frac{\partial ^{2} f_{\delta ^{*}_i}}{\partial y^{2}}= \frac{\partial ^2}{\partial y^{2}}\left( 1-\frac{u}{u_{\infty }}\right) = -\frac{1}{u_{\infty }}\frac{\partial ^{2u}}{\partial y^{2}}$$

(17)

$$\begin{aligned} \frac{\partial ^2 f_{\theta _i}}{\partial y^2}&= {} \frac{\partial ^2}{\partial y^2}\left\{ \frac{u}{u_{\infty }}\left( 1-\frac{u}{u_{\infty }}\right) \right\} \\&= {} \frac{1}{u_{\infty }^2}\frac{\partial ^2}{\partial y^2}(uu_{\infty } - u^2) \\&= {} \frac{1}{u_{\infty }^2}\frac{\partial }{\partial y}\left( u_{\infty }\frac{\partial u}{\partial y}-2u\frac{\partial u}{\partial y}\right) \\&= {} \frac{1}{u_{\infty }^2}\left( (u_{\infty }-2u)\frac{\partial ^2 u}{\partial y^2} -2\left( \frac{\partial u}{\partial y}\right) ^2 \right) . \end{aligned}$$

(18)

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:

$$\delta ^{*}_{\rm i} =\frac{1}{u_{\infty }}\left( [u_{\infty }y]^{\infty }_0 - \left\{ [uy]_0^{\infty } - \int _0^{\infty } y\frac{\partial u}{\partial y} \hbox{d}y \right\} \right) .$$

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

$$\begin{aligned} \theta _i&= {} \frac{1}{u_{\infty }^2} \int _0^{\infty }\left( uu_{\infty } - u^2\right) \hbox{d}y \\&= {} \frac{1}{u_{\infty }^2} \left\{ [u_{\infty }uy]_0^{\infty } - \int _0^{\infty } u_{\infty }y\frac{\partial u}{\partial y} \hbox{d}y - \left( [u^2y]_0^{\infty } - 2\int _0^{\infty } yu\frac{\partial u}{\partial y} \hbox{d}y \right) \right\} . \end{aligned}$$

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.