# Analytical computation of five unresolved integrals in the linear theory of partially cavitating hydrofoils

Jean-Baptiste Leroux, Matthieu Sacher

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Jean-Baptiste Leroux, Matthieu Sacher. Analytical computation of five unresolved integrals in the linear theory of partially cavitating hydrofoils. Results in Applied Mathematics, 2025, 27, 10.1016/j.rinam.2025.100599. hal-05379620

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# Analytical computation of five unresolved integrals in the linear theory of partially cavitating hydrofoils

Jean-Baptiste Leroux*, Matthieu Sacher

ENSTA, Institut Polytechnique de Paris, UMR CNRS 6027, IRDL, 29200 Brest, France

## ARTICLE INFO

**Keywords:**

Analytical integration

Theoretical hydrodynamics

Cavitation theory

Airfoil integrals

## ABSTRACT

Five non-tabulated integrals are analytically calculated. These integrals emerge from the linear theory of partially cavitating hydrofoils and propeller blades. They appear in a series of weight functions involved in the determination of the cavitation number and cavity shape. The present analytical results eliminate the need for unnecessary numerical integrations, which could be beneficial in reducing computational costs and improving the robustness of numerical models.

Linear theory of thin wing sections is conventionally applied to perform the geometric design of thickness and camber distributions for hydrofoils and propeller blade sections. This approach leads to well-known singular integral equations of the first kind with a Cauchy kernel, which must be solved, in both direct or inverse design processes. This eventually requires the calculation of the principal value of singular integrals, some of which have been tabulated for practical applications (see Van Dyke [1], Breslin and Andersen [2], Katz and Plotkin [3] for examples).

Following earlier works by Tulin [4], Acosta [5], Wade [6] and others, Kinnas [7,8,9] extends this linear theory to cavitating hydrofoils. Solving of the problem and determining the cavitation number and cavity shape ultimately requires computing two weight functions. The definition of these functions involves evaluating the five integrals given in Eqs. (1) and (2), wherein $s_0$ and $s$ are real numbers and $0 < s_0 < s$:

$$F_k(s_0, s) = \int_{s_0}^{s} \sqrt{\frac{s-\eta}{\eta-s_0} \frac{\eta^k}{1+\eta^2}} d\eta \quad k \in \{0, 1\} \qquad (1)$$

$$G_k(s_0, s) = \int_{s_0}^{s} \sqrt{\frac{s-\eta}{\eta-s_0} \frac{\eta^k}{(1+\eta^2)^2}} d\eta \quad k \in \{0, 1, 2\} \qquad (2)$$

To the authors' knowledge, closed form formulas for these integrals are known to exist when $s_0 = 0$ (Persson [10], transcribed in Breslin and Andersen [2] p. 526). However, when dealing with cavitation inception or partial cavitation cases, $s_0$ is a sensitive, strictly positive parameter, which influences the accuracy of the theoretical results, with regard to experiences or other models (see Kinnas [7,8]). For $s_0 > 0$, these integrals remain un-tabulated. In his work, Kinnas [8] opted for numerical integration in order to compute $F_k(s_0, s)$ and $G_k(s_0, s)$.

In this work, to enhance computational efficiency and the robustness of numerical models for evaluating or optimizing the cavitating behavior of hydrofoils in design processes, the following analytical formulas are proposed for efficient direct calculations. Considering Eq. (3):

$$A = \left( \frac{1 + s^2}{1 + s_0^2} \right)^{\frac{1}{4}} \quad \text{and} \quad \gamma = \frac{\arctan s - \arctan s_0}{2} \qquad (3)$$

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: jean-baptiste.leroux@ensta.fr (J.-B. Leroux).</footnote>
The five un-tabulated integrals are evaluated using Eqs. (4) and (5):

$$F_0(s_0, s) = \pi A \sin \gamma, \quad F_1(s_0, s) = \pi (A \cos \gamma - 1) \quad (4)$$

$$G_0(s_0, s) = -\frac{\pi}{4} \frac{s - s_0}{(1 + s^2)^2} A^3 [(s^2 - 1) \cos 3\gamma - 2s \sin 3\gamma] + \frac{\pi}{2} A \sin \gamma \quad (5)$$

$$G_1(s_0, s) = \frac{\pi}{4} \frac{s - s_0}{(1 + s^2)^2} A^3 [2s \cos 3\gamma + (s^2 - 1) \sin 3\gamma] \quad (5)$$

$$G_2(s_0, s) = \frac{\pi}{4} \frac{s - s_0}{(1 + s^2)^2} A^3 [(s^2 - 1) \cos 3\gamma - 2s \sin 3\gamma] + \frac{\pi}{2} A \sin \gamma$$

In the following, the main computational steps are first presented, followed by a numerical verification of the results. This verification is carried out using an in-house adaptative numerical integration procedure with error control, based on Gauss-Kronrod nested rules and the error estimate of Laurie [11].

For the calculation, through elementary algebra, the problem is first reduced to the calculation of two complex-valued integrals F and G, defined by Eq. (6):

$$F = \int_{s_0}^{s} \sqrt{\frac{s-\eta}{\eta-s_0}} \frac{1}{\eta+i} d\eta \quad \text{and} \quad G = \int_{s_0}^{s} \sqrt{\frac{s-\eta}{\eta-s_0}} \frac{1}{(\eta+i)^2} d\eta \quad (6)$$

Indeed, through elementary calculations, it is straightforward to derive the relations in Eq. (7):

$$F_0 = -\mathfrak{F}(F), \quad F_1 = \mathfrak{R}(F) \\ G_0 = -\frac{1}{2}\mathfrak{R}(G) + \frac{1}{2}F_0, \quad G_1 = -\frac{1}{2}\mathfrak{F}(G) \quad \text{and} \quad G_2 = \frac{1}{2}\mathfrak{R}(G) + \frac{1}{2}F_0 \quad (7)$$

Next, the variable transformation, given by Eq. (8) is applied:

$$u = \sqrt{\frac{s-\eta}{\eta-s_0}} \Rightarrow \eta = \frac{s+s_0 u^2}{1+u^2} \quad \text{and} \quad d\eta = -2 \frac{(s-s_0)u}{(1+u^2)^2} du \quad (8)$$

And let Eq. (9) be noted:

$$\omega = \frac{s_0 + i}{s + i} = \rho e^{i\alpha} \quad (9)$$

where $\rho = |\omega| = \sqrt{(1+s_0^2)/(1+s^2)}$ and $\alpha = \arg \omega = \arctan s - \arctan s_0$. Then, it follows that F and G can be rewritten as Eq. (10):

$$F = -\pi + 2I \quad \text{and} \quad G = 2 \frac{s - s_0}{(1 + s^2)^2} (s^2 - 1 - i 2s)J \quad (10)$$

where $I = \int_{0}^{+\infty} \frac{1}{1 + \omega u^2} du$ and $J = \int_{0}^{+\infty} \frac{u^2}{(1 + \omega u^2)^2} du$.

The computation of I and J is finally performed using the residue theorem, in order to integrate $f(z) = 1/(1 + \omega z^2)$ and $g(z) = z^2/(1 + \omega z^2)^2$, along the path $\Gamma_R = \{(x, y) / x \in [-R; R], y = 0\} \cup \{(x, y) / x^2 + y^2 = R^2, y > 0\}$, taking the limit of the results for $R \to +\infty$ and using parity of the integrands.

$$f(z) = \frac{z_0}{2} \left[ \frac{-1}{z - z_0} + \frac{1}{z + z_0} \right] \\ g(z) = \frac{z_0^3}{4} \left[ \frac{1}{z - z_0} + \frac{-1}{z + z_0} \right] + \frac{z_0^4}{4} \left[ \frac{1}{(z - z_0)^2} + \frac{1}{(z + z_0)^2} \right] \quad (11)$$

The partial fraction decompositions of f and g are given in Eq. (11), where $z_0 = i e^{-i\frac{\alpha}{2}} / \sqrt{\rho}$ is the root of $1 + \omega z^2 = 0$ located in the semi-plane $y > 0$. From Eq. (11), the only pole surrounded by $\Gamma_R$ for f and g is then $z_0$ and their residues at $z_0$ are $\text{Res}(f)|_{z_0} = -z_0/2$ and $\text{Res}(g)|_{z_0} = z_0^3/4$. This leads to the final results in Eq. (12), which conclude the calculations.

$$I = \frac{\pi}{2\sqrt{\rho}} e^{-i\frac{\alpha}{2}} \quad \text{and} \quad J = \frac{\pi}{4\sqrt{\rho}} e^{-i\frac{3\alpha}{2}} \quad (12)$$

A final verification step of the formulas in Eqs. (4) and (5) is then provided. Prior to numerical testing, the apparent singularity in $F_k(s_0, s)$ and $G_k(s_0, s)$ is removed by the variable transformation $\eta = s_0 + (s - s_0) \sin^2 \theta / 2$, which formally leads to:

$$\int_{s_0}^{s} \sqrt{\frac{s-\eta}{\eta-s_0}} h(\eta) d\eta = \int_{0}^{\pi} (s-s_0) h(\eta) \cos^2 \theta / 2 d\theta$$

For a pair $(s_0, s)$, considering an $n_g$ points Gauss-Legendre rule and its Kronrod extention, $F_k(s_0, s)$ and $G_k(s_0, s)$ are iteratively calculated using the bisection algorithm, which is processed until the absolute error estimate $err$ falls below a specified absolute tolerance $tol$. In the end, the number of integrand evaluations is given by $n_{eval} = (2n_g + 1)(2n_{sub} + 1)$, where $n_{sub}$ is the number of bisection performed. These numerical results are then compared to the analytical ones, computed directly using elementary intrinsic functions.
<table><caption>**Table 1**
Verification of formulas in Eqs. (4) and (5), where $s_0 = 1/\sqrt{2}$, $s = \sqrt{7}$, $n_g = 5$ and $tol = 10^{-14}$.</caption><thead><tr><th></th><th>(Result type)</th><th>Numerical value</th><th>err</th><th>n<sub>eval</sub></th></tr></thead><tbody><tr><td rowspan="2">F<sub>0</sub></td><td>(numerical)</td><td>1.397 064 665 594 5 7 91</td><td rowspan="2">6.2E-15</td><td rowspan="2">77</td></tr><tr><td>(analytical)</td><td>1.397 064 665 594 5 7 86</td></tr><tr><td rowspan="2">F<sub>1</sub></td><td>(numerical)</td><td>1.423 611 104 316 6 7 41</td><td rowspan="2">6.7E-16</td><td rowspan="2">77</td></tr><tr><td>(analytical)</td><td>1.423 611 104 316 6 7 41</td></tr><tr><td rowspan="2">G<sub>0</sub></td><td>(numerical)</td><td>0.727 153 557 594 9 86</td><td rowspan="2">3.2E-15</td><td rowspan="2">99</td></tr><tr><td>(analytical)</td><td>0.727 153 557 594 9 81</td></tr><tr><td rowspan="2">G<sub>1</sub></td><td>(numerical)</td><td>0.667 341 398 142 0 4 33</td><td rowspan="2">5.9E-15</td><td rowspan="2">99</td></tr><tr><td>(analytical)</td><td>0.667 341 398 142 0 4 32</td></tr><tr><td rowspan="2">G<sub>2</sub></td><td>(numerical)</td><td>0.669 911 107 999 5 8 08</td><td rowspan="2">5.2E-15</td><td rowspan="2">77</td></tr><tr><td>(analytical)</td><td>0.669 911 107 999 5 8 05</td></tr></tbody></table>

An example of the results is provided in Table 1. The numerical values confirm the validity of the formulas in Eqs. (4) and (5), with the differences $|numerical - analytical|$ being smaller than the estimated error err once convergence is reached. The number of integrand evaluations $n_{eval}$, required to reach the prescribed tolerance, also illustrates the gain in number of operations provided by the analytical formulas compared to a numerical integration procedure.

The interest of formulas in Eqs. (4) and (5) becomes clear for certain applications, particularly when it comes to limiting the number of functions evaluations or eliminating the need for multiple precision control steps. This work thus serves as a foundation for the implementation of an efficient cavitation model in the in-house CFD code PUFFIn (Perali et al. [12]), which is dedicated to general simulations of unsteady flow around hydrofoils (Faye et al. [13]).

Appendix provides five additional similar integrals formulas, directly derived from this work, which have been verified using the same approach.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Appendix. Table of additional similar integrals

$$F'_{k}(s_{0}, s) = \int_{s_{0}}^{s} \sqrt{\frac{z - s_{0}}{s - z}} \frac{z^{k}}{1 + z^{2}} dz \quad k \in \{0, 1\} \qquad (A.1)$$

$$G'_{k}(s_{0}, s) = \int_{s_{0}}^{s} \sqrt{\frac{z - s_{0}}{s - z}} \frac{z^{k}}{(1 + z^{2})^{2}} dz \quad k \in \{0, 1, 2\} \qquad (A.2)$$

$$B = 1/A = [(1 + s_0^2)/(1 + s^2)]^{1/4} \quad \text{and} \quad \gamma = (\arctan s - \arctan s_0)/2 \qquad (A.3)$$

$$F'_{0}(s_{0}, s) = \pi B \sin \gamma, \quad F'_{1}(s_{0}, s) = \pi(1 - B \cos \gamma) \qquad (A.4)$$

$$G'_{0}(s_{0}, s) = -\frac{\pi}{4} \frac{s - s_{0}}{(1 + s_{0}^{2})^{2}} B^{3} [(s_{0}^{2} - 1) \cos 3\gamma + 2s_{0} \sin 3\gamma] + \frac{\pi}{2} B \sin \gamma \\ G'_{1}(s_{0}, s) = \frac{\pi}{4} \frac{s - s_{0}}{(1 + s_{0}^{2})^{2}} B^{3} [2s_{0} \cos 3\gamma - (s_{0}^{2} - 1) \sin 3\gamma] \qquad (A.5)$$

$$G'_{2}(s_{0}, s) = \frac{\pi}{4} \frac{s - s_{0}}{(1 + s_{0}^{2})^{2}} B^{3} [(s_{0}^{2} - 1) \cos 3\gamma + 2s_{0} \sin 3\gamma] + \frac{\pi}{2} B \sin \gamma$$

## Data availability

No data was used for the research described in the article.

## References

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[2] Breslin JP, Andersen P. Hydrodynamics of ship propellers. Cambridge University Press; 1994.

[3] Katz J, Plotkin A. Low-speed aerodynamics. 2nd Ed.. Cambridge University Press; 2001.

[4] Tulin MP. Linearized theory for flows about lifting foils at zero cavitation number. Tech. Rep. C-638, David Taylor Model Basin; 1955.

[5] Acosta AJ. A note on partial cavitation of flat plate hydrofoils. Rep. E-19.9, California Institute of Technology; 1955.

[6] Wade RB. Linearized theory of a partially cavitating plano-convex hydrofoil including the effects of camber and thickness. J Ship Res 1967;11(1):20-7.
[7] Kinnas SA. Non-linear corrections to the linear theory for the prediction of the cavitating flow around hydrofoils. [PhD thesis], Massachusetts Institute of Technology; 1985.

[8] Kinnas SA. Leading-edge corrections to the linear theory of partially cavitating hydrofoils. J Ship Res 1991;35(1):15-27.

[9] Kinnas SA. Inversion of the source and vorticity equations for supercavitating hydrofoils. J Eng Math 1992;26(2):349-61.

[10] Persson B. Theoretical study of cavitation on flat plate. Tech. Rep. 78/440, DnV; 1978.

[11] Laurie DP. Practical error estimation in numerical integration. J Comput Appl Math 1985;12:425-31.

[12] Perali P, Sacher M, Leroux J-B, Wackers J, Augier B, Hauville F, Bot P. Performance prediction of a hydrofoil near the free surface using low (BEM) and high (RANS) fidelity methods. Appl Ocean Res 2024;151:104157.

[13] Faye A, Perali P, Augier B, Sacher M, Leroux J-B, Nême A, Astolfi J-A. Fluid-structure interactions response of a composite hydrofoil modelled with 1D beam finite elements. J Sailing Technol 2024;9(1):19-41.