Correlation between Pressure Minima and Cavitation Inception Numbers: Fundamentals and Hydrofoil Flows

Abstract

Cavitation inception predetermines a jump of noise radiated from marine vehicles. Usually marine propellers are the main sources of such a noise. In the situation of cavitation inception near the blade’s leading edge, its prediction remains a challenge. Though contemporary CFD tools for fully turbulent flows satisfactorily predict pressure distribution around cavitation-free blades and with cavities of length comparable with the blade size, analysis of blade cavitation inception is a difficult task for these tools. On the other hand, there are validated computational tools for 2D multizone flows capable of predicting cavitation inception. There is the possibility of considering the real 3D flow around the leading edges of blades as a 2D flow with the known pressure distribution along the blade section; the cavitation inception number is computed in this 2D cavitating flow, and correlations between this number and the pressure minimum in cavitation-free flow around the same section are determined. Such a correlation would be usable with any tool for cavitation-free flow. The issue of their applicability to arbitrary blades can be solved with the employment of asymptotic solutions for the pressure around contours with rounded leading edges.

1. Introduction

The main incentive for studies of cavitation inception has been the accompanying jump of flow-induced noise. Most frequently, cavitation inception first occurs on marine propellers. Prediction of cavitation inception in marine engineering was initially associated with the necessity to enhance the silent operation speeds of vehicles, whereas the interest in the environmental impact of such a jump appeared more recently. Determination of these speeds has been traditionally based on model tests in hydrodynamic facilities (mainly in water tunnels) followed by extrapolations to full-scale conditions.

Extrapolation of model test results has been complicated by several circumstances. First of all, it is impossible to simultaneously maintain the full-scale advance ratio and load in the tests of models that are much smaller than the full-size propeller. This impossibility amplifies the demand for computational predictions of cavitation inception. It is clear that such a prediction must take into account the effects of the Reynolds number, Re. Moreover, as shown in Figure 1 with the experimental data of [1], the types of cavities that appear can be different for the same propeller model for the various values of its advance ratios J = V_a/nD; here V_a is the inflow speed, and the propeller cavitation inception number is defined as ${\sigma }_n=2\left(P_0-P_c\right)/\left(\rho n^2{D}^2\right)$, where D is the propeller diameter, n is its rotation frequency, P₀ and P_c are inflow pressure and the pressure in the cavity, respectively, and ρ is water density.

According to the archaic point of view, cavitation inception should appear at the point of pressure minima when such a minimum drops down to the vapor pressure P_V. This point of view was refuted many decades ago by various comparisons of measured cavitation inception numbers, σ_I and σ_n, with reliable computations or with measurements of these minima. Two examples of such comparisons are presented in Figure 2. Shown in the upper part of this Figure is the difference between –min{Cp} and cavitation inception number σ_I (Re) $=\max \left\{2\left(P_0-P_c\right)/\left(\rho V_a^2\right)\right\}$ for a 15% thickness hydrofoil (after [2]). In the right part of this figure, this difference for a marine propeller is plotted after [3]. One can also see in the review in [4] that the difference D_σ = |min{Cp}| − σ_I becomes negligible at the local Reynolds number Re_S ~ 10⁷. However, although cavitation inception over propeller blades takes place at Re_S < 10⁷, even in full-scale conditions, D_σ remains significant for them (especially for design off conditions). Thus, the prediction of cavitation inception cannot be completed using the computation of pressure in cavitation-free flow only.

However, the successful experience of numerical investigation of cavitating flows taking into account the viscous (Re-dependent) effects already exists. Two main paths of such an investigation must be pointed out. The first path is a multi-zone model (MZM). This model takes into account the existence of recirculating flow zones with their specific velocity profiles (wall functions) around the attached cavity, as well as the existence of a laminar boundary layer (with a possible transition) upstream of the cavity. Surface tension effects are also included in that model. Cavitation inception in MZM studies is defined as the maximum value of the cavitation number corresponding to steady cavities of decreasing size. The MZM results are in good agreement with experimental data for cavitation inception in various 2D and axisymmetric flows, as was reported in [5,6,7,8,9]. However, MZM studies have involved the viscous–inviscid interaction procedure with integral methods for boundary layers, as the numerical tools and analysis of 3D flows, is a challenge for them.

The second path is based on the models developed for fully turbulent flow (MFTF) of a homogeneous gas–liquid mixture. MFTF has been broadly employed after publications [10,11,12,13,14,15]. Diverse turbulence equations were used in these publications (and even diverse definitions of the mixture density). Moreover, there are no certain cavity surfaces in all versions of MFTF, whereas regions with a high void fraction are accepted as the cavities when the mixture pressure in such regions goes down to vapor pressure and may be considered as $P_c$. The maximum of cavitation number calculated with its use are accepted as cavitation inception number. In the last two decades, computational studies of hydrofoil and propeller cavitation were carried out mainly using MFTF. Such studies relate to prediction of hydrofoil cavitation inception in particular [16,17,18]. Evaluations [19,20] of the results of MFTF for marine propellers by the recognized experts were skeptical a decade ago. Nevertheless, more recently there appeared quite successful numerical studies of propeller cavitation accomplished with this flow model [1,21,22,23,24,25,26]. Note that the studies [1,25,26] relate in particular to the vortex cavitation inception. Oppositely, prediction of blade cavitation inception using MFTF remains difficult. In particular, RANS computations of blade cavitation in [1,2] did not provide cavitation inception numbers at all. The main challenge is to accurately predict the flow characteristics near the blade leading edges, where blade cavitation starts. As manifested in [17] and illustrated in Figure 3, computational tools developed for fully turbulent flows inaccurately predict pressure in the vicinity of hydrofoil/blade leading edges, even in cavitation-free conditions. As one can understand from Figure 3, the real pressure along the separation zone is substantially different from predictions for fully turbulent flow; in that particular case |Cp| in this zone constitutes around 2/3|minCp| and around 0.8|Cp| in the point where laminar boundary layer separation criterion predict it without taking into account the viscous–inviscid interaction effects. Moreover, pressure in the appearing cavity would be very close to the pressure in the separation zone. Thus, analysis of cavitation-free flow cannot be suitable for prediction of cavitation inception, necessitating use of the multi-zone analysis therein. One may also recall that RANS solvers inaccurately describe velocity profiles inside the cavities, as is well illustrated in [27], but as was recently manifested in [28], the assumptions of the cavity content do not affect computations of pressure distributions (at least in steady inflows).

On the other hand, the satisfactory predictions of cavitation inception on marine propeller blades were obtained in [29,30] using the perturbation method. This method employed ideal fluid theory for pressure distributions over cavitation-free propeller blades as the 3D unperturbed solutions. Further cavitation is analyzed as a perturbation of such ideal fluid flows. A 2D MZM of viscous fluid flows was used for the ensemble of blade sections. This perturbation approach was successful for the initial stages of cavitation because the difference in invicsid flow velocities in cavitating and cavitation-free flows is much smaller than the inflow speed.

However, the numerical tools employed three decades ago to find the unperturbed solutions are not popular now. At present, engineers use mainly RANS and LES solvers for determination of pressure over the blades. Such solvers can provide a pressure minimum in the cavitation-free flow with acceptable accuracy, but determination of cavitation inception numbers remains too difficult for them. Hypothetically, these solvers may deliver unperturbed solutions for the above-mentioned perturbation method, but the employed MZM solvers are neither based on simple algorithms nor are publicly available in their existing 2D versions. Fortunately, the situation is not hopeless because, firstly, smooth marine propeller blades always have rounded leading edges. Therefore, as shown in [31,32,33], there are only four parameters predetermining pressure distribution in the vicinity of these edges for the entire diversity of such blades. Secondly, cavitation inception over the blades usually occurs just near these edges. In this situation it looks reasonable to compute the differences D_σ as functions of these parameters. Such functions can be employed in the future independently of the kind of tools used for the determination of min{Cp} over the blade. Thus, the objective of computation of such a difference consists of enhancing the capabilities of the broadly employed CFD tools by a relatively simple additional analysis of their results. This objective looks to be practically important and doubtless achievable.

This paper aims at describing the physical background for determination of these differences (correlations) and their validation for 2D flows with the use of well-known experimental data. The main physical features of cavitation inception over hydrofoils and blades coincide. That is why sometimes twisted hydrofoils are used instead of blades for their analysis [34,35]. However, 2D flows are more convenient for their explanation and analysis. In particular, analytical methods still remain relevant there as they oftentimes result in formulae explicitly providing insight into influence of various factors upon the phenomena under study, thus greatly facilitating engineering predictions. In addition, because of the necessity to validate computational results with the experimental data of other researchers, the various aspects related to experimental determination of cavitation inception and data interpretation (such as water quality effects [36]) are discussed here.

Let us recall that there are two procedures of determination of cavitation inception in experiments. First, one can reduce pressure or increase the inflow speed in the facility, starting from a cavitation-free flow and recording a randomly appearing cavity. The corresponding value of the cavitation number is known as the cavitation inception number. Second, one can increase pressure or reduce speed there, starting from a partially cavitating flow and keeping this procedure till the collapse of the cavity (or cavities). The maximal of the corresponding cavitation numbers is defined as the cavitation desinence number. Nevertheless, there is no distinction of these numbers in the substantial fraction of the reported experiments. The second procedure is more popular because it is both easier technically and has lower discrepancies of the results. The presented computations will relate to it.

2. Computation of Pressure around the Leading Edge

The objective of this section is to describe the asymptotic solution for the leading edges of hydrofoils or blades. Verification of these solutions with the precise numerical solutions for 2D ideal fluid flows will also be presented here.

In its asymptotic part, this paper treats the corresponding ideal fluid flow problem as two-dimensional. The region of the leading edge is under scrutiny through stretching of the local coordinates and addressing the flow past osculating parabola obtained by superposition of the symmetric and circulating (asymmetric) components. Lighthill [37] and Van Dyke [38] were the first to apply singular perturbation approaches (method of matched asymptotic expansions and method of deformed coordinates) to treat the local flow problem for the rounded leading edge of a thin foil. The resulting expressions for full velocity U is as follows:

$$U\left(X\right)=\frac{\left(U_2\pm U_1\sqrt{X}\right)}{\sqrt{X+r/2}}$$

(1)

and pressure coefficient Cp = 1 − U² along the upper and lower contours of the parabola can be obtained [31] in a simple and universal form. Here, $X=\left(1+x\right)/\left(rC{\delta }^2\right)$ is the auxiliary variable, r = r_le/(Cδ²), r_le is the radius of the rounded leading edge, C is the hydrofoil chord, δ is its thickness and the abscissa x varies from −1 to 1. The sign minus in Equation (1) must be used for the hydrofoil pressure side.

The matched asymptotic expansion method enables one to derive closed form expressions for the relative (full) velocity and pressure coefficient. Differentiating the formula for the former with respect to $X$ and equating the result to zero, we can obtain [32] and determine the abscissa $X_{max}$ of the point of maxium velocity (or after substitution of Equation (1) into the Bernoulli equation that of minimum pressure) and the abscissa of stagnation point $X_{stag}.$

The parameters U₁ and U₂ can be determined through asymptotic matching with the linear (outer) solution for a thin slightly curved foil at a small angle of attack. The latter provides the following asymptotic expression for a full velocity on the upper (+) and lower (−) sides of the foil, related to the speed of the flow in upstream infinity U₀

$${\upsilon }^o\left(x\right)=1+\frac{\delta }{\pi }v.p.{{\displaystyle \int }}_{-1}^1\frac{d{f}_t}{d\xi }\left(\xi \right)\frac{d\xi }{x-\xi }\pm \frac{\alpha }{\pi }\sqrt{\frac{1-x}{1+x}}\pm \frac{{\delta }_c}{\pi }\sqrt{\frac{1-x}{1+x}}v.p.{{\displaystyle \int }}_{-1}^1\frac{d{f}_c}{d\xi }\left(\xi \right)\sqrt{\frac{1+\xi }{1-\xi }}d\xi ,$$

(2)

where ${\delta }_c$—the relative camber of the foil, α—the angle of attack in radians. Thickness $f_t\left(x\right)$ and camberline $f_c\left(x\right)$ functions, as well as parameters $\bar{\alpha }=\alpha /\delta ,{\bar{\delta }}_c={\delta }_c/\delta$ are all of the order of $O\left(1\right).$ Note that the integrals in Equation (2) are defined in the Cauchy principal value sense and can be reduced to quadrature for practically any foil having analytical description. The corresponding lift coefficient obtained within assumptions of linear theory is as follows:

$$C_L=2{{\displaystyle \int }}_{-1}^1\left({\delta }_c\frac{d{f}_c}{d\xi }-\alpha \right)\sqrt{\frac{1+\xi }{1-\xi }}d\xi =C_L^{\alpha }\alpha +C_L^{\delta }\delta =C_L^{\alpha }\left(\alpha -{\alpha }_0\right)$$

(3)

where α₀ is the angle of zero lift. The unknown parameters U₁ and U₂ entering Equation (1) are derived through asymptotic matching of Equation (1) with Equation (2) in the region of mutual validity in the following form:

$$U_1=1+\delta u_{\delta }$$

(4)

$$U_2=\frac{\sqrt{2}}{\delta }\alpha +\frac{{\delta }_c}{\delta }{u}_c$$

(5)

Note that the parameters $u_{\delta }$ and $u_c$ depend only on the foil shape and are readily found from:

$$u_{\delta }=\frac{1}{\pi }\underset{x\to -1}{\lim }v.p.\underset{-1}{\overset{1}{{\displaystyle \int }}}\frac{d{f}_t}{d\xi }\frac{d\xi }{x-\xi }$$

(6)

$$u_c=\frac{1}{\pi }\underset{x\to -1}{\lim }v.p.\underset{-1}{\overset{1}{{\displaystyle \int }}}\frac{d{f}_c}{d\xi }\frac{d\xi }{x-\xi }$$

(7)

The following representations [39],

$$f_c\left(x\right)={\left(1-x\right)}^{\lambda 1}{\left(1+x\right)}^{\mu 1}{\displaystyle \sum }_{n=0}^M{a}_n^{\left(1\right)}{P}_n^{\left(\lambda 1,\mu 1\right)}\left(x\right)$$

(8)

$$f_t\left(x\right)={\left(1-x\right)}^{\lambda 2}{\left(1+x\right)}^{\mu 2}{{\displaystyle \sum }}_{n=0}^M{a}_n^{\left(2\right)}{P}_n^{\left(\lambda 1,\mu 1\right)}\left(x\right),$$

(9)

of the functions$f_t\left(x\right)$ and $f_c\left(x\right)$ are useful for a wide class of practically used foils and for providing convenient calculations of the parameters $u_{\delta },u_c$. Here, λ₁, μ₁, λ₂, μ₂ are the constants depending on the foil shape; $P_n^{\left(\lambda , \mu \right)}$ are Jacobi polynomials [40], and the coefficients in the series and the necessary numbers of their terms also depend on the foil shape. For example, for the rounded leading edge and sharp trailing edge, one has λ₁ = μ₁ = 1.0; λ₂ = 1.0, μ₂ = 0.5. For the simplest case of uncurved elliptic foil of thickness $\delta$ and angle of attack α, one has $f_t\left(x\right)=\sqrt{1-x^2}$, and the general formulae yield$U_1=1+\delta$ and$U_2=\sqrt{2}\overline{\alpha }$. Some other foils particulars are given in

Table 1. Coefficients of asymptotic (1) calculated using ideal fluid theory.

Foil Type	${\mathit{u}}_{\mathit{\delta }}$	${\mathit{u}}_{\mathit{c}}$	R
Ellipse	1	0	1
Zhukovsky foil	2.228	0	2.35
NACA66012	0.817	0	0.874
Gö-7 Walchner foil	0.919	0.246	0.964
NACA4412	2	0.296	1.026
NACA66m (a = 0.8)	0.75	0.208	0.848

.

Although the outer flow around blades is much more complex than Equation (2) describes, the above approximation of the velocity around the leading edges by parabolas remains acceptable with the use of other methods of computation of the outer flow. According to the asymptotic analysis, the point of the velocity maximum is linked with the abscissa of the stagnation point by the equation $X_{max}{X}_{stag}=r^2/4,$ where $r=2{\delta }^{-2}\sqrt{\left(1+x_{max}\right)\left(1+x_{stag}\right)}$ for any method of determination of U(x). Moreover, there are the conditions $U_1\sqrt{X_{max}}+U_2=U_{max}\sqrt{r+X_{max}}$ and $\left[\frac{R}{\sqrt{X_{max}}}\right]U_1-U_2=0$ to determine $U_1=U_{max}\sqrt{X_{max}/\left(r+X_{max}\right)}$ and $U_2=U_{max}r/\sqrt{\left(r+X_{max}\right)}$. Further, the arc abscissa counted downstream from the stagnation point can be found by integration of the equation $\frac{dS}{dX}=\pm \sqrt{\left(X+0.5\right)/X}$. Thus, the results of various computations can be easily approximated by Equation (1) because $U_{max}$, $x_{max}$ and $x_{stag}$ can be found from distributions of pressure obtained using any numerical method.

The differences between −min{Cp} and cavitation inception number can be represented by the function D_σ{δ, r, U₁, U₂}. This function would be seen as the universal correction (allowance) compatible with the various tools employed for cavitating-free hydrofoils and the rotating blade. The first step in verification of the possibility of such a correction is an evaluation of the difference between numerically determined and asymptotic pressure distributions for several hydrofoils. The most accurate verification of Equation (1) seems to be carried out using exact analytical solutions for 2D ideal fluid problems. However, as manifested about a century ago [41], the ideal fluid theory gives a much better agreement to the experimental data when the measured dependency of the lift coefficient C_L on the angle of attack α° is used instead of the C_L value found using the condition of Kutta–Joukovskii at the foil trailing edge. The comparisons in Figure 4 and Figure 5 cover ranges of {α°, δ, δ_c} for four hydrofoils. Selection of these hydrofoils was made because the detailed descriptions of the experiment on determination of cavitation inception numbers were published [2,42,43] for three of them. Therefore, the actual dependencies C_L(α°, Re) are known (and consequently, the data from these hydrofoils will be useful for the further validation of the suggested computational technique).

Generally, the asymptotic accurately describes pressure from the leading edges to the point of the pressure minima and along the arcs of approximately the same lengths downstream of the minima. At greater distances downstream from this arc, the accuracy of the asymptotic gradually drops. However, for the thicknesses and angles of attack inherent in the contours of propeller blade sections, cavitation inception takes place just in the vicinity of these minima. In addition, as shown in [44], BEM and RANS computations of pressure over the leading part of the propeller blades give close pressure distributions.

3. Model of 2D Cavitating Flows

The objective of this section is to give a brief description of the employed MZM for 2D cavitating flows. The recently published descriptions [8,9] of MZM include numerous equations and conditions for matching the flow in diverse zones. Therefore, here it is acceptable to point out the physics behind MZM, omitting the majority of the related mathematical aspects.

The attached blade cavitation can be studied as a special kind of separated flows. The corresponding flow scheme is shown in Figure 6. There are two recirculation zones around the cavity. The boundary layer separates upstream of the first zone under the combined influence of the unperturbed pressure gradient and of the meniscus, taking apart the cavity leading part from the surrounding water. The second zone downstream of the cavity is associated with the pulsating reentrant jet in the cavity tail. The separated boundary layer becomes turbulent over the cavity, and its reattachment takes place where the product ${\delta }^{*}d{C}_p/ds$ exceeds an empirically found constant. Pressure in the cavity equals the pressure in the middle between points X₁ and X₂.

The viscous–inviscid interaction computational procedure employed in MZM is generally based on two effects. The first one is that the pressure gradient influences the thicknesses of boundary layers. The second one is that the pressure perturbation is due to displacement of inviscid flow from the body surface. As is usual in such procedures, the boundary between viscous and inviscid parts of flow depends on the thickness displacement δ*, but the specifics of the cavitating flows is that δ* between X₀ and X₂ is counted from the zero-friction line or from the cavity surface (not from the body surface). This thickness can be determined by solving the von Karman equation. As one can find in [45], for 2D unsteady flows, this momentum equation has the form

$$\frac{\partial {\delta }^{**}}{\partial x}+\frac{\partial {\delta }^{*}}{U\partial t}=\frac{\partial U}{\partial x}\frac{\left(\widehat{\delta }-{\delta }^{*}-2{\delta }^{**}\right)}{U}+\frac{\left(\widehat{\delta }-{\delta }^{*}\right)}{U^2}\frac{\partial U}{\partial t}+\frac{C_f}{2}$$

(10)

Here, $\widehat{\delta },{\delta }^{**}$ are boundary layer thickness and momentum thickness, respectively, and C_f is the friction coefficient. In the different zones of viscous flow, the velocity profiles u(x, y) are different. Usually, such a profile is two-parametric, and an additional equation is required (for example, mass conservation law integrated across the layer). In addition, it is assumed that no friction exists on the cavity surface, but the velocity u(x) on this surface must go to zero by the reattachment point (this is one of conditions used for determination of the cavity length).

In the considered perturbation method, the pressure is time-dependent because the angles of attack of the blade sections usually are time-dependent. However, the contribution of the terms proportional to derivatives by time is relatively small near the leading edges because there $\left[\frac{\partial {\delta }^{*}}{U\partial t}-\frac{\left(\widehat{\delta }-{\delta }^{*}\right)}{U^2}\frac{\partial U}{\partial t}\right]{\left[\frac{\partial U}{\partial x}\frac{\left(\widehat{\delta }-{\delta }^{*}-2{\delta }^{**}\right)}{U}\right]}^{-1}~\frac{L}{D}$, where L is the cavity length. Therefore, the instant values of U(t) can be used in computations of boundary layers and separated flows.

The cavity content is not taken into account during the described computations. However, this account is important for validation of computational results with the experimental data. As was noted [42,46] more than a half century ago, ${\sigma }_{vapor}$ can substantially exceed σ. One can estimate the difference of these numbers $d\sigma ={\sigma }_{vapor}-\sigma$ using the formula

$$d\sigma =\frac{2P_V}{{\rho }_w{U}_0^2}\left[1-\Omega -m\left(\frac{\rho {\Lambda }^{1-1/\kappa }}{{\rho }_{at}}{\left(\frac{P_{at}}{P_V}\right)}^{1/\kappa }-\Omega \right)\right]$$

(11)

derived in [8]. Here,$\Omega =1+b{(\lambda -1)}^{\kappa }$, κ = 1.4, $P_V$ is vapor pressure, m is gas mass concentration in inflow, $\lambda =P_0/P_V$.

As one can see in Figure 7, the coefficient b may vary from 0.02 to 0.03. The correction $d\sigma$ will be applied to the experimental data when the information on m is available. The correction determined by Equation (11) is important because employment of vapor cavitation number (very usual in experiments) instead of actual cavitation number leads to overestimations of σ_i.

The computational procedure is based on the variation of the couple {X₁, X₂} for the given Re and cavitation-free distribution Cp(x). For some couples, a solution may not exist. The corresponding values of σ and Weber number We (or of the chord C) will be found in iterations. The result of this variation will be the maximum of σ for the fixed {Re, C}, and this maximum will be considered as the cavitation inception number.

4. Computations of Hydrofoil Cavitation Inception

The objective of this section is to show how accurately the described method for the computation of cavitation inception numbers works for 2D hydrofoil flows. The computations were carried out with employment of both numerically and asymptotically determined pressure distributions in the vicinities of the leading edges.

The employed numerical method is a boundary element method (BEM) based on the solution of the following integral equation:

$$-\frac{q\left(x,y\right)}{2}+\frac{1}{2\pi }\left({{\displaystyle \oint }}^{ }\frac{R_x{N}_x+R_y{N}_y}{R^2}ds+\underset{0}{\overset{C}{{\displaystyle \int }}}\gamma \frac{R_x{N}_y-R_y{N}_x}{R^2}dl\right)=N_x$$

(12)

for densities q(s) of sources/sinks distributed on the hydrofoil contour. Here, R = [(R_x)² + (R_y)²]^0.5, R_y = y − η, R_x = x − ξ, x and y are cartesian coordinates of the flow domain, 𝜉 and 𝜂 are coordinates of the integration contours, vector N = {N_x, N_y} is the outer normal to the hydrofoil contour S. The second integral in Equation (12) is defined along the hydrofoil camber line, and the function $\gamma$ depends on a coefficient that may be either found from the Kutta–Joukovskii condition or fitted to the measured lift coefficient. Inflow is directed along the x-axis. The velocity U is the gradient of a potential defined with employment of functions q and $\gamma .$

The comparisons of computed and measured cavitation inception numbers are provided for several tested hydrofoils here. Computations of cavitation inception number for the hydrofoil E817 in Figure 8 were carried out using the measured [47] dependency C_L(α°) in the above BEM for determination of cavitation-free pressure distribution. The BEM solvers generally allow for taking into account the effect of water tunnel walls as well. No corrections of the inflow air content were made for this hydrofoil.

Computations of the cavitation inception number for the NACA0015 hydrofoil presented in Figure 9 were carried out using numerically determined pressure distribution for the conditions of the experiment described in [2]. In addition, the computational results for the tenfold higher Reynolds number were obtained with the same C_L(α°). One can see that for these relatively thick hydrofoils at moderate angles of attack, the scale effect on cavitation for the same load is very moderate.

The values of D_σ gradually increased with the angle of attack, as one can see in Figure 10 for another thick hydrofoil. At a relatively small value of C_L/δ (such as is inherent in NACA0015 at α = 4°), there were no sharp velocity maxima near leading edges, and cavitations appeared in the middle of the hydrofoil at cavitation numbers very close to |min{Cp}|. As one can see in Figure 4 and Figure 5, at the very small lift coefficients, the asymptotic looked nice, but there was no edge cavitation at such coefficients. Comparison of computational results with experimental data is not a trivial task. One may see that the experimental data in Figure 10 do not coincide with the values of vapor cavitation numbers reported in [42]. Note that these data were corrected using Equation (2) and the information on inflow air content from [42].

In Figure 11, the vapor cavitation inception number from [48] is plotted together with the cavitation inception number corrected using Equation (11). This figure presents an example of the scale of such a correction. Comparisons of cavitation inception numbers computed with both numerical and asymptotical pressure distribution for substantially thinner hydrofoils (δ = 0.06) are given in Figure 12. The results for the Walchner hydrofoil are compared with the experimental data from [43]. These data were not corrected for the air content impact, whereas the measured dependency of the lift coefficient on the angle of attack was used here. The difference between cavitation inception number computed using two pressure distributions was incomparably smaller that the difference between them and –min{Cp}. Thus, the provided comparisons show the generally good agreement of MZM theory with the experimental data for both methods of determination of Cp at the sufficiently high ratios of C_L/δ.

In addition, it was proven that cavitation-free Cp computed using a RANS code can be successfully used for the prediction of cavitation inception at the leading edges. In particularly, for the propeller DTMB 4381 at the advance ratio of 0.75, the computed cavitation inception number σ_n = 4.0 (and the correlation in this case was about 48% of the value calculated using min{Cp} only), whereas the measured [1] vapor cavitation inception number at the inflow speed 4 m/s was 4.75. However, for such small inflow speeds, the difference between these two numbers turns out to be between 15% and 20% (see Figure 7).

The provided comparisons sufficiently support the idea of computing D_σ as a function of the coefficients of the employed asymptotic with the goal of further using such correlations for the prediction of cavitation inception on 3D blades.

5. Correlations as Functions of the Coefficients of the Asymptotic

The objective of this section is to determine the general trends in the dependencies of correlations on the coefficients of the described asymptotic and on the hydrofoil thickness. Herein, the structure of asymptotic formulae is used as a calibration basis for available data. One can find the coefficient from the above-described examples with the determination of {r, U₁, U₂} from the numerical analysis in

Table 2. Coefficients asymptotics (1) calculated using numerical tools.

	δ	α0	r	U1	U2
Walchner	0.06	2	3.16	0.911	1.89
		3	3.16	0.866	2.42
		5	3.16	0.713	3.4
NACA0006	0.06	3	4.1	1.108	1.6
		5	4.1	1.134	2.56
NACA4412	0.12	8	5.13	0.509	3.77
		10	5.13	0.488	4.24
NACA0015	0.15	6	1.602	1.114	1.523
		8	1.602	1.081	1.927

. Some very general trends are seen. For example, in the situations with sharp U maxima, the discrepancy in the determination of r is ±3%, and its value may be averaged. The values of U₂ increase with these maxima and the ratio α°/δ, whereas variations of U₁ are much smaller. Furthermore, one may return to Table 1 to see that r obtained using linear theory for the outer flow is five times smaller, and U₁ is two times larger than their values obtained with the use of the measured lift coefficient.

However, the set of parameters presented in Table 2 is insufficient to understand the sensitivity of D_σ to these parameters. On the other hand, the ranges of their variation must be limited. As one can see in [49], the typical range of α° for a marine propeller blade varies from −4° to 4° near the propeller axis and from −1.5° to 3° near the blade tip, whereas the variation of δ between these sections is approximately 0.08 down to 0.04. In addition, as noted in the compendium [50], the cavitation inception number at the propeller leading edge insignificantly depends on the Reynolds number when the blade loading is kept constant.

An analysis of the influence of parameters presented in Figure 13, Figure 14, Figure 15 and Figure 16 relates to the quite close ranges of them. The influences of diverse parameters on the correlation look qualitatively different.

One can see in Figure 14 and Figure 15 that the correlation gradually increases with U₂ for fixed r but has a minimum inside of the range of r variation. Such a minimum cannot be predicted simply from Equation (1). As seen in Figure 16, there is a maximum of correlation inside the interval of thickness variation. In addition, the correlation decreases with the increase of U₁ and with other parameters of the asymptotic kept constant.

Tables/nomograms presenting the correlations within the ranges of their parameters realistic for typical sections of propeller blades would be useful. The authors hardly see a possibility to reduce the computation volume by expressing one of parameters through others. On the other hand, there also exist combinations of the parameters describing pressure distributions that do not result in the leading edge cavitation inception. Moreover, some sets of parameters would not reveal any signs of cavitating flow at all.

However, the above-mentioned nomograms could rather be the object of future work together with the validation of correlations for several tested cavitating propellers.

6. Conclusions

Cavitation inception predetermines a jump of noise radiated from marine vehicles. Usually, marine propellers are the main sources of this noise. There are diverse forms of cavities appearing around propellers. Cavitation inception on the leading edges of blades remains a challenge for computational tools.

Moreover, currently there are various well-validated computational tools (including commercially available solvers) that allow for sufficiently accurate prediction of pressure over the blades in cavitation-free flows (in particular, for prediction of pressure minima). In addition, the satisfactory theory of cavitation inception near hydrofoil leading edges has already been developed. The described paper shows the path of the combined use of the above-mentioned achievements to meet the mentioned challenge.

The combined employment of this 2D theory with the asymptotic description of the pressure in the vicinity of the blade leading edge allows for derivation of correlations between pressure minima in 3D flows around propeller blades and cavitation inception numbers as functions of the Reynolds number and coefficients of these correlations. Such correlations can supplement existing CFD codes for cavitation-free flows and allow for computation of the cavitation inception number without modification of these codes.

This paper describes the procedure of determination of these coefficients as well as the validation of their use in computations of cavitation inception by comparisons with the experimental data for various two-dimensional hydrofoils. Such comparisons are always a difficult task, which is why some descriptions of the accurate comparison procedures are also included herein. The sensitivity of these correlations to their coefficients is also investigated.