c Allerton Press, Inc., 2011. ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2011, Vol. 55, No. 9, pp. 80–82. c L.A. Aksent’ev and D.V. Maklakov, 2011, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2011, No. 9, pp. 95–98. Original Russian Text
An Integral Comparison Theorem for Cavitation Diagrams L. A. Aksent’ev* and D. V. Maklakov** Kazan (Volga Region) Federal University, ul. Kremlyovskaya 18, Kazan, 420008 Russia Received March 22, 2011; in final form, May 16, 2011
Abstract—In this paper we prove a comparison theorem for a functional whose positiveness is a necessary condition for the solvability of the problem of designing a hydrofoil by a given cavitation diagram. DOI: 10.3103/S1066369X11090106 Keywords and phrases: comparison theorem, cavitation diagram, hydrofoil, pressure envelope, solvability condition.
In hydrofoil design a cavitation diagram is the dependence F (α) of the minimum pressure coefficient Cp min taken with the opposite sign on the angle of attack α, i.e., F (α) = −Cp min (α) = 2
p∞ − pmin (α) , 2 ρv∞
where pmin is the minimum pressure on the contour profile, p∞ is the pressure at infinity, v∞ is the incident flow velocity, and ρ is the fluid density. The function F (α) is one of the main characteristics of hydrofoils, it allows one to predict the cavitation-free range of angles of attack. In [1–3] one has developed a method for designing hydrofoils whose cavitation diagram exactly coincides with a given function F (α). In order to solve this inverse problem, one conformally maps the flow domain on the outside of the unit circle so that points at infinity in the physical z- and parametric tplanes correspond to each other and the image of the trailing edge is the point t = 1 in the t-plane. This mapping represents the one-to-one correspondence between points of the parametric circumference t = eiγ (γ is the polar angle) and those of the profile surface, the correspondence being independent of the angle of attack α. According to papers [1–3] (see also [4]), one of the main step in solving the problem of hydrofoil design from a given cavitation diagram is the determination of a 2π-periodic continuous nonnegative function g(γ) (which does not vanish identically) from the equation γ − α = f (α), (1) max g(γ) cos γ∈R 2 where the function f (α) = 1 + F (α) is known. So, g(γ) ∈ G, where G = {g ∈ C(R) : g(γ) ≥ 0, g(γ + 2π) = g(γ), γ ∈ R} \ {0}. The difference between Eq. (1) and usual equations of the convolution type is the replacement of the integral sign with that of the maximum. See [5] for a comprehensive study of Eq. (1). Along with G we introduce the following class of functions: T = {f ∈ G : f (α) > 0, f (α) is trigonometrically convex}, i.e., T is the set of strictly positive π-periodic trigonometrically convex functions. Recall [7, 8] that the function f (α) is called trigonometrically convex if for any α1 and α2 , 0 < α2 − α1 < π, the following inequality holds: f (α) ≤ H(α), * **
α1 < α < α2 ,
E-mail:
[email protected]. E-mail:
[email protected].
80
(2)
AN INTEGRAL COMPARISON THEOREM FOR CAVITATION DIAGRAMS
81
where H(α) =
f (α1 ) sin(α2 − α) + f (α2 ) sin(α − α1 ) . sin(α2 − α1 )
Geometrically inequality (2) means that the graph of the function y = f (α) for [α1 , α2 ] lies not above the “trigonometric chord” determined by the equation y = H(α) = a cos α + b sin α and conditions H(α1 ) = f (α1 ) and H(α2 ) = f (α2 ). According to theorem 1 in [5], Eq. (1) is solvable in the class G if and only if f ∈ T . If f ∈ T , then the function f (α) gm (γ; f ) = min α∈R | cos(γ/2 − α)| belongs to the class G, is strictly positive, and represents a solution to Eq. (1). In addition, in the mentioned work a comparison theorem (theorem 3) is proved. Here we need the following assertion. Corollary ([5], theorem 3). If f and f ∗ belong to T , f (α) ≥ f ∗ (α), and f (α) ≡ f ∗ (α), then gm (γ; f ) ≥ gm (γ; f ∗ ) and gm (γ; f ) ≡ gm (γ; f ∗ ). In [5] Eq. is studied in detail, but this is not enough for the solvability of the initial hydrodynamic problem on the shape of a hydrofoil with a prescribed cavitation diagram. As is demonstrated in [1], the values of three constants, which are functionals of the function f (α) ∈ T , namely, π log gm (γ; f )dγ − 2π log 2, K0 [f ] = −π π π log gm (γ; f ) cos γdγ + π, K2 [f ] = log gm (γ; f ) sin γdγ, K1 [f ] = −π
−π
play an essential role in designing a physically realizable closed profile. positiveness of the nonlinear functional M[f ] = K0 [f ] − K1 [f ] + iK2 [f ]
In particular, the strict
is one of the most important necessary solvability conditions. In this paper we prove a comparison theorem for this functional. Theorem. If f and f ∗ belong to T , f (α) ≥ f ∗ (α), and f (α) ≡ f ∗ (α), then M[f ] > M[f ∗ ]. The proof is based on the following lemma. Lemma. Assume that g(x) is a continuous function on a segment [a, b]; c1 and c2 are real constants, D = c1 + ic2 ; a functional T[g] is defined by equalities b b ik(x) g(x)dx, P[g] = g(x)e dx + D , T[g] = P0 [g] − P[g], P0 [g] = a
a
where k(x) ∈ C[a, b] is a given function, k(x) ≡ const. If g and g∗ belong to C[a, b], g(x) ≥ g∗ (x), and g(x) ≡ g∗ (x), then T[g] > T[g∗ ]. Proof. We apply the triangle inequality to the difference of integrals: b b ik(x) ∗ ik(x) ≥ P[g] − P[g∗ ]. g(x)e dx + D − g (x)e dx + D a
(3)
a
Now we apply the Weierstrass formula ([9], P. 62) to the left-hand side of formula (3) written in the form b [g(x) − g∗ (x)]eik(x) dx: a
b
∗
[g(x) − g (x)]e
ik(x)
b
dx = Z
a
RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 55 No. 9 2011
a
[g(x) − g∗ (x)]dx.
(4)
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AKSENT’EV, MAKLAKOV
Here Z is the center of gravity of the system of masses m(x) = g(x) − g∗ (x) ≥ 0, m(x) ≡ 0, distributed along the arc of the circumference with the parametric equation z = eik(x) , x ∈ [a, b]. The equality |Z| = 1 is possible only if all masses are concentrated at one point, i.e., eik(x) = eiα , k(x) = α, the parameter α being a constant. Then Z = eiα . This is impossible if k(x) ≡ const. In all other cases Z satisfies the strict inequality |Z| < 1, because the center of gravity is located inside the convex hull of mass location points. Formulas (3) and (4) give the inequality b ∗ [g(x) − g∗ (x)]dx = P0 [g] − P0 [g∗ ], P[g] − P[g ] < a
which proves the statement of the lemma. Remark. The Weierstrass formula (4) is convenient for estimating complex and real integrals. Namely, b b b b i arg f (x) f (x)dx = |f (x)|e dx = |Z| |f (x)|dx ≤ |f (x)|dx a
a
a
a
with the equality sign for f (x) ≡ 0 only in the case arg f (x) ≡ const. For real integrals arg f (x) equals either 0 or π, and the equality sign in the estimate takes place only for arg f (x) ≡ 0 or arg f (x) ≡ π. The comparison theorem for the functional M[f ] immediately follows from the statement of the lemma with k(x) = x and the formulated above corollary of theorem 3 in the paper [5]. This theorem plays a principal role in solving extremal problems of the hydrofoil theory. First, the theorem simplifies the reasoning of the paper [2] which allows one to construct hydrofoils with optimal cavitation properties. Second, the theorem opens the way for designing optimal non-symmetrical profile shapes which are cavitation-free in the maximal range of angles of attack. REFERENCES 1. F. G. Avkhadiev and D. V. Maklakov, “A Solvability Criterion for a Problem of Profiles’ Design by Cavitation Diagram,” Izv. Vyssh. Uchebn. Zaved. Mat., 38 (7), 3–12 (1994) [Russian Mathematics (Iz. VUZ) 38 (7), 1–10 (1994)]. 2. F. G. Avkhadiev and D. V. Maklakov, “An Analytical Method of Constructing Hydrofoils by Given Pressure Envelopes,” Phys. Dokl. 343 (2), 195–197 (1995). 3. F. G. Avkhadiev and D. V. Maklakov, “A Theory of Pressure Envelope for Hydrofoils,” J. Ship Technology Research (Schiffstechnik) 42 (2), 81–102 (1995). 4. D. V. Maklakov, Nonlinear Problems of Hydrodynamics of Potential Flows with Unknown Boundaries (Yanus-K, Moscow, 1997) [in Russian]. 5. F. G. Avkhadiev and D. V. Maklakov, “New Equations of Convolution Type Obtained by Replacing the Integral by Its Maximum,” Matem. Zametki 71 (1), 18–26 (2002). 6. A. M. Elizarov, A. R. Kasimov, and D. V. Maklakov, Optimal Shape Design Problems in Aerohydrodynamics. (Fizmatlit, Moscow, 2008) [in Russian]. 7. B. Ya. Levin, Distribution of the Roots of Entire Functions (GITTL, Moscow, 1956) [in Russian]. ´ 8. G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities (University Press, Cambridge, 1934; Inost. Lit, Moscow, 1948). ´ Goursat, A Course in Mathematical Analysis (Ginn & Company, Boston, New York, 1916; GTTI, 9. E. Moscow-Leningrad, 1933), Vol. 2, Part 1.
Translated by D. V. Maklakov
RUSSIAN MATHEMATICS (IZ. VUZ) Vol. 55 No. 9 2011
