Question: How do researchers determine the complicated, nonlinear correlations for experimental data?

I'm currently reading a copy of Pipe Flow, by Donald Rennels, and on the section of loss coefficients relating dynamic pressure to head loss for a fluid flowing in a pipe

$$h_L=K\frac {V^2}{2g}$$

The author mentions several estimations for head loss coefficients $K_o$ for a variety of orifices as a function of $\beta=\frac {d_o}{d_2}$, where $d_o$ is the orifice diameter and $d_2$ is the downstream pipe diameter, and $\frac {t}{d_o}$, where $t$ is the thickness of the orifice. Some examples I've included are

$$K_o= 0.0696(1-\beta^5)\lambda^2+(\lambda-1)^2+(1-\beta^2)^2+f_o\left(\frac t{d_o}-1.4\right)\qquad\frac t{d_o}\geq1.4$$

$$K_o=0.0696(1-\beta^5)\lambda^2+C_{\text{th}}\left[1-\left(\frac {d_o}{d_2}\right)^2\right]^2+(1-C_{\text{th}})\left[(\lambda-1)^2+\left(1-\left(\frac {d_o}{d_2}\right)^2\right)^2\right]$$

$$K_o=0.0696(1-\beta^5)\lambda^2+(\lambda-1)^2\left[1-\left(\frac {d_o}{d_2}\right)^2\right]^2+f_o\left(\frac t{d_o}-1.4\right)$$


$$C_{\text{th}}=\left[1-\frac 12\left(\frac t{1.4d_o}\right)^{2.5}-\frac 12\left(\frac t{1.4d_o}\right)^3\right]^{4.5}$$


My question is simply on how these equations were derived. For dimensional terms, I am aware of Buckingham Pi Theorem to come up with relations between a set of parameters, but that does not work with non-dimensional coefficients such as the loss coefficient $K$ or discharge coefficient $C_d$. Is there a software that automatically determines the nonlinear lines of best fit given a certain set of parameters? I can't imagine a team of researchers deciding that adding a $\lambda^2(1-\beta^5)$ term to their equation would be the perfect addition to improving their equation's correlation.

I know that there is empirical data the equations are curve-fitted on, but how the form and terms of that equation determined is what absolutely confuses me.

  • $\begingroup$ Why not check when Buckingham did his work? Or Rayleigh or many others… check which version of Windows was around or which CFD… Also check out Joseph Fourier… Software is not the holy grail of solutions - so much was done prior. $\endgroup$
    – Solar Mike
    Commented Mar 3, 2023 at 5:44
  • $\begingroup$ Are you most interested in this specific case? Then Solar Mike's comment is very apt. Or are you interested in methods of producing such correlations in general? Then search out such things as generalized least squares, semi-empirical, and so on. $\endgroup$
    – Boba Fit
    Commented Mar 3, 2023 at 17:50
  • $\begingroup$ @BobaFit In general. I am very unfamiliar with regressions other than least squares, so I will give the list you mentioned a try $\endgroup$
    – Frank W
    Commented Mar 4, 2023 at 1:19
  • $\begingroup$ First few terms of suitable series expansion is one of the tricks. $\endgroup$ Commented Mar 5, 2023 at 18:38


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