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Asymptotic Solution of the Flow Problem and Estimate of Delay of Cavitation Inception for a Hydrofoil with a Jet Flap

K.Rozhdestvensky, I.Belousov (St. Petersburg State Marine Technical University, Russia)

ABSTRACT

In the present paper the method of matched asymptotic expansions is applied to treat the flow problem for a hydrofoil of arbitrary aspect ratio with a jet flap in the case of small magnitudes of the jet momentum coefficient. In the nearfield, i.e. in the vicinity of the trailing edge with a jet flap, the problem is reduced to a two-dimensional one. The relevant system of integro-differential equations, governing the local flow, is solved by means of Carleman inversion formulae and Mellin transforms. It is shown that for practical case of small jet momentum coefficients the farfield asymptotics of the local solution contains a square root singularity of hydrodynamic loading at the trailing edge, which is compatible with the “saddle-backed ” character of the loading for a jet flapped wing observed in experiments. The matching principle not only prompts the increase of loading near the jet flapped trailing edge in inverse proportion to the square root of the distance from the edge, but also enables to express the strength of this singularity in terms of the jet deflection and momentum coefficient in corresponding cross sections of the wing spanwise. The outer description of the flow has been obtained through superposition of appropriate nonhomogeneous and homogeneous solutions of the integral equation of linear lifting surface theory. In case of the jet flap at the trailing edge the matching principle results in an additional condition, which plays the same role as a standard Kutta-Zhukovsky condition for a lifting surface without jet flap. Results are presented, illustrating dependence of the lift coefficient on the jet deflection angle and momentum coefficient, as well as upon the wing's aspect ratio. In order to evaluate possibility of delay of cavitation inception minimum pressure diagrams are plotted with “edge branches” obtained from the local flow problem for the rounded edge and “bottom branches”—from linear theory with account of both lifting and thickness effects. Results are presented which show that for the same magnitude of the lift coefficient cavitation inception occurs later for the case of the lift induced by the jet flap as compared to the case of the incidence induced lift.

1.
Introduction

The jet flap, formed by a high-speed jet of air or water injected from the trailing edge of the lifting surface, represents one of the effective devices of lift control accompanied by a drag reduction due to longitudinal component of the jet momentum, [1]. The possibility to increase lift by means of jet flap was demonstrated by Hagern and Ruden (1938) in Germany. However, this demonstration did not evoke any interest until more extensive research work had been made at the National Gas Turbine Institute. Comprehensive survey of the evolution of this technical concept in historical retrospective can be found in [2]. Other aspects of application of the jet flap have been revealed and reported, concerning the possibility to delay cavitation inception for hydrofoils. As indicated in [3], application of the jet flap may become practical in development of high speed hydrofoil ships of large displacement of the order of 1,000–1,300 tonnes designed in the range of cruise speeds of 65 –70 knots. The so called “law of square—cube” states that with increase of the ship's dimensions its weight, increases in proportion



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Twenty-First Symposium on NAVAL HYDRODYNAMICS Asymptotic Solution of the Flow Problem and Estimate of Delay of Cavitation Inception for a Hydrofoil with a Jet Flap K.Rozhdestvensky, I.Belousov (St. Petersburg State Marine Technical University, Russia) ABSTRACT In the present paper the method of matched asymptotic expansions is applied to treat the flow problem for a hydrofoil of arbitrary aspect ratio with a jet flap in the case of small magnitudes of the jet momentum coefficient. In the nearfield, i.e. in the vicinity of the trailing edge with a jet flap, the problem is reduced to a two-dimensional one. The relevant system of integro-differential equations, governing the local flow, is solved by means of Carleman inversion formulae and Mellin transforms. It is shown that for practical case of small jet momentum coefficients the farfield asymptotics of the local solution contains a square root singularity of hydrodynamic loading at the trailing edge, which is compatible with the “saddle-backed ” character of the loading for a jet flapped wing observed in experiments. The matching principle not only prompts the increase of loading near the jet flapped trailing edge in inverse proportion to the square root of the distance from the edge, but also enables to express the strength of this singularity in terms of the jet deflection and momentum coefficient in corresponding cross sections of the wing spanwise. The outer description of the flow has been obtained through superposition of appropriate nonhomogeneous and homogeneous solutions of the integral equation of linear lifting surface theory. In case of the jet flap at the trailing edge the matching principle results in an additional condition, which plays the same role as a standard Kutta-Zhukovsky condition for a lifting surface without jet flap. Results are presented, illustrating dependence of the lift coefficient on the jet deflection angle and momentum coefficient, as well as upon the wing's aspect ratio. In order to evaluate possibility of delay of cavitation inception minimum pressure diagrams are plotted with “edge branches” obtained from the local flow problem for the rounded edge and “bottom branches”—from linear theory with account of both lifting and thickness effects. Results are presented which show that for the same magnitude of the lift coefficient cavitation inception occurs later for the case of the lift induced by the jet flap as compared to the case of the incidence induced lift. 1. Introduction The jet flap, formed by a high-speed jet of air or water injected from the trailing edge of the lifting surface, represents one of the effective devices of lift control accompanied by a drag reduction due to longitudinal component of the jet momentum, [1]. The possibility to increase lift by means of jet flap was demonstrated by Hagern and Ruden (1938) in Germany. However, this demonstration did not evoke any interest until more extensive research work had been made at the National Gas Turbine Institute. Comprehensive survey of the evolution of this technical concept in historical retrospective can be found in [2]. Other aspects of application of the jet flap have been revealed and reported, concerning the possibility to delay cavitation inception for hydrofoils. As indicated in [3], application of the jet flap may become practical in development of high speed hydrofoil ships of large displacement of the order of 1,000–1,300 tonnes designed in the range of cruise speeds of 65 –70 knots. The so called “law of square—cube” states that with increase of the ship's dimensions its weight, increases in proportion

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Twenty-First Symposium on NAVAL HYDRODYNAMICS to the cube of its length, whereas for a given lift coefficient, the area of the lifting surfaces should be increased as a square of the ship's length. Therefore, to provide dynamic support of large hydrofoil ship one needs either to disproportionally increase the area of hydrofoils, or to increase the magnitude of the lift coefficient. The former measure results in heavy and “clumsy” hydrofoils whereas the latter measure normally requires an augmentation of the adjusted angle of attack and, consequently, leads to reduction of the speed range of noncavitating flow regime. Utilization of the jet flap, as follows from the article of Kaplan [3], allows to gain the same increment of lift as for the incidence induced lift case with diminution of the leading edge suction due to transition to the “saddle-backed” loading distribution. Kaplan and Goodman explored possibilities of utilization of jet flapped wings as anti- pitching devices, [4]. The same authors carried out experiments in water tunnel on symmetric hydrofoils with measurement of the lift and drag, estimation of required power and measurement of velocity distribution across the jet, [ 4]. In addition, Kaplan and Lehman conducted measurements of unsteady forces acting upon a wing due to unsteady variation of the jet deflection angle, in view of possible application to control surfaces of a submarine, [5]. One of the features of the flow problem for a lifting surface with a jet flap consists in a necessity to fulfil kinematic and dynamic boundary conditions upon the jet, the form of which is not known in advance. The important step in development of mathematical model of a jet flapped wing had been made by Spence in his research work [6], where the jet behind the wing was treated as a surface of tangential discontinuity, possessing a finite momentum and capable of withstanding pressure jump. In this and following publications [7], [8] ,[9] and [10]. Spence gave detailed analysis of the two-dimensional flow problem. Using linear theory, he replaced the wing and the jet by an equivalent vortex layer, distributed along the semi-infinite cut and subject to tangency condition on the foil as well as to kinematic and dynamic conditions upon the jet. The latter has been written as the requirement of proportionality of the pressure difference across the jet to its longitudinal curvature. The resulting mixed boundary problem had been solved by Spence, at first with use of Fourier expansions [ 6] and later on by means of the Mellin transform, [10]. Theoretical results of Spence are in fair agreement with experimental data up to sufficienly large magnitudes of the jet deflection angle of the order of 60°, which confirms the adequacy of the mathematical model of infinitely thin jet. Kuchemann [11] extended the approach of Spence to the case of the foil of small thickness and was able to calculate not only the lift, but also pressure distributions on the upper and lower surfaces of the winng with jet flap. His calculated results confirmed that a typical loading distribution along the wing with a jet flap has a “saddle-backed” character with maxima both near the leading edge and the jet flap. The said loading distribution is essentually different from that for the foil without jet flap at an angle of attack. For the latter case there is only one peak of hydrodynamic loading near the leading edge. Comparing loading distributions for a wing with and without jet flap, one can easily conclude that for the same magnitude of the lift coefficient the maximum suction at the leading edge is lower for the jet flapped wing. With use of Fourier method Spence had calculated 9 terms of the series decsribing the strength of the vortex layer with a square root singularity at he leading edge and logarithmic singularity at the trailing edge. In a work [10] with use of the method of integral transforms an asymptotic expansion of the flow problem solution had been constructed for the case of small magnitude of the jet momentum coefficient. As a result the lift coefficient was found in the form where To determine characteristics of a jet flapped wing Stratford [12] used an analogy between the wing with a jet flap and a wing with a mechanical flap. This approach resulted in an expression for the lift coefficient which for small values of the jet momentum coefficient was similar to that obtained by Spence in [6]. The work of Maskell and Spence [13] was the first where an attempt had been made to extend the approach of [6] to three-dimensional case. In [13] the flow problem for a wing of finite aspect ratio with a jet flap was formulated within linear theory of the lifting surface. Due to complexity of the problem the authors developed only approximate solution, based upon two-dimensional theory of Spence and assumption of uniform spanwise distribution of downwash which corresponds to elliptic loading along the span. As a matter of fact, the authors assumed that both local chord and momentum distribution along the span were elliptic for constant magnitudes of the incidence α and

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Twenty-First Symposium on NAVAL HYDRODYNAMICS jet deflection angle τ. Kerney [14] and Tokuda [15] considered the problem of the flow past a jet flapped wing of large aspect ratio by means of the method of matched asymptotic expansions. In Kerney's work the outer flow was modeled by a lifting line, and in the nearfield the problem was reduced to two-dimensional one, as treated by Spence. Matching of the farfield and nearfield descriptions of the flow gave the possibility to obtain closed expressions for coefficients of the lift, longitudinal moment and induced drag of the wing with a jet flap. Comparison with experimental data of Williams and Alexander, [16] for a flat wing of rectangular planform with no incidence and jet deflection angle τ=31.3° turned out to be quite good for a wing of aspect ratio λ=6.8 and satisfactory for a wing of moderate aspect ratio λ=2.75. It should be mentionned that, as pointed out by Tokuda [15], in Kerney's work no account had been taken of the fact that the lifting part 1 of the jet sheet, generally speaking, can extend into the outer flow region downstream of the trailing edge at distances of the order of O(1). However, in our opinion, for small magnitude of the quantity the assumption, that the outer part of the trailing jet sheet can be treated as nonlifting one, should be valid. Tokuda [15] had made an attempt to introduce corrections, accounting for the influence of the lifting part of the jet sheet in the outer flowfield upon aerodynamics of the jet flapped wing. As follows from comparison of solutions of Kerney and Tokuda, the correction introduced by the latter author into the expression of the lift coefficient has the order of Cj/λ. In the present paper there is considered a three-dimensional flow problem for a lifting surface of arbitrary planform and aspect ratio with a jet flap at the trailing edge. A linearized flow model is adopted in which for small incidences and jet defelection angles the system is supposed to produce small output of the same order. Such mathematical model, as shown through comparisons of existing experimental and theoretical data is valid for small angles of attack, but in surprisingly large range of the jet deflection angles. Such an “inequality” of parameters α and τ in what concerns correlation of calculated and measured results can be attributed to the fact that critical flow phenomena, such as separation from the leading edge, is much more sensitive to variation of the angle of attack than to variation of the jet deflection angle. The problem is solved by the method of matched asymptotic expansions [17], [18], [19]. with a small papameter ε defined as ε=Cjo/4, where Jo the jet momentum in a root chord cross-section of the wing, ρ- density of the fluid, Uo speed of the wing. Because in practical applications the jet momentum coefficient Cj does not exceed 2, the parameter ε≤1/2. According to the technology of the method of matched asymptotic expansions the flow field is subdivided into an “outer” region (outside of the trailing edge flow region with a longitudinal dimension of the order of O(ε)) and the “inner” region region (within the aforementionned local zone). Asymptotic dimension of the local (inner) region has been found by means of the least degeneracy principle as applied to the inner problem2. In the inner region, in terms of stretched coordinate =x/ε, the problem is reduced to a system of integro-differential equations to be solved upon a semi-infinite axis with respect to a function of hydrodynamic loading γi and ordinates of the surface of the jet sheet. This system has been solved with use of Carleman inversion formulae and Mellin integral transformation. Although, the inner solution reveals logarithmic singularity at the point of jet injection, its outer asymptotics upon the wing contains a square root singularity of the loading. Simultaneously, it has been found that that the slope of the jet sheet with respect to horizontal plane vanishes in downstream direction in inverse proportion to the square root of the distance from the trailing edge and becomes negligibly small at distances of the order of O(1) and small magnitudes of the jet momentum coefficient. It can be concluded from the aforementionned that for the case of small jet momentum coeficient the outer solution should contain a square root singularity of loading at the trailing edge, and the outer part of the jet sheet, situated downstream from the trailing edge, is practically nonlifting and “almost” coincides with a free vortex sheet emanating from the wing's trailing edge. The conclusions obtained so far, enable to considerably simplify the problem. The outer solution is represented by a set of admissible solution of the integral equation of linear lifting surface theory. In particular, part of the solution due to the angle of attack is represented by Prandtl-Birnbaum double series with a square root singularity at the leading edge and square root zeros at the trailing edge and side edges. Representation of contribution of the jet flap into the loading differs due to the fact in compliance with the outer asymptotics of the local solution, there should be introduced in the outer solution a square root singularity at the trailing edge with the strength determined in terms of the jet deflection angle and momentum coeffi- 1   That is the one with nonzero longitudinal curvature 2   Note, that as early 1956 in his investigation of the two-dimensional flow problem for the case of small jet momentum coefficient; Spence had found such a coordinate transformation x̄=x/Δ(Cj) (where Δ(Cj) is a stretching function) for which the integro-differential equation of the problem does not depend on the jet momentum coefficient. The stretching function derived in the above work turned out to be Δ (Cj)=Cj/4=ε.

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Twenty-First Symposium on NAVAL HYDRODYNAMICS cient at a given cross section by means of matching of the nearfield and farfield. Thickness of the wing is modeled in the outer region in the spirit of thin body theory by means of distributing upon an appropriate horizonal plane of a simple (sink-source) layer with the strength proportional the slope of thickness function in downstream direction. In order to evaluate the influence of the jet flap upon occurence of cavitation the minimum pressure diagrams have been calculated and plotted with use of the approach proposed in [20] and [21]. The “edge” branches of these diagrams, corresponding to inception of cavitation at the leading edge, can be obtained directly from the local problem for a flow past a parabola, approximating a rounded leading edge. The “bottoms” of minimum pressure buckets, corresponding to cavitation occurence in the mean part of the foil, can be determined from the outer solution. Note, that the method is readily extended to the case of a hydrofoil at finite submergences, if the corresponding outer solution accounts for the presence of the free surface. Similar approach can be applied to describe hydrodynamics of cascades and screw propellers with jet flaps. 2. Problem Formulation Consider an irrotational flow of ideal incompressible fluid past a hydrofoil of small thickness and finite aspect ratio λ with a jet flap at the trailing edge. For the purpose of simplification, let us assume in what follows the wing has a rectangular planform and symmetric foil section3. In what follows all functions and quantities are rendered nondimensional with half of the root chord co/2 as a characteristic length and the speed of translatory motion Uo as a characteristic velocity. An attached Cartesian coordinate system x,y,z is used with axis x directed downstream, axis y dfirected upwards and z—axis forming the left-hand side coordinate system with x and y. The wing has an angle of attack α, and its thickness distribution is described by a function yt(x,z)=±δft(x,z), x ∈ [–1,1], z ∈ [–λ,λ]. (1) The jet flap is activated by blowing the air from the trailing edge at an angle τ, the jet momentum coefficient being designated as Cj. Within the boundary problem formulation the velocity potential of the flow is governed by the three-dimensional Laplace equation and has to comply with kinematic boundary conditions of the flow tangency upon the upper and lower surfaces of the wing as well as upon the surface of the jet flap which is not known in advance. Additionally there should be satisfied upon the jet sheet a dynamic condition, according to which the pressure jump across the jet is proportional to the jet momentum coefficient Cj(z) and its longitudinal curvature at any cross section z=const. In linearized nondimensional form the said condition can be written down as (2) where Cp is a the pressure coefficient, γ is the density of the equivalent vortex layer which replaces the jet sheet, yj(x,z)—ordinates of the jet sheet. With account of the aforementionned the integral equation of the linearized problem for the flow past a wing with a jet flap and zero incidence4 will take the form (3) with β(x,z)=0 upon the wing |x|≤1, and upon the jet sheet upon the jet sheet x>1. Besides, in accordance with condition (2) the vortex density can be written as follows (4) and the condition of blowing at a given angle τ= τ(z), which can vary spanwise, has the form (5) 3. Local Flow Problem in the Vicinity of the Trailing Edge with a Jet Flap After introduction of stretching of local coordinates near the trailing edge (6) the local flow problem becomes two-dimensional in the plane z=const and the corresponding integral equation takes the form 3   The proposed approach is valid for a wing of arbitrary planform and with arbitrary distribution of curvature within the frame of linear theory of lifting surface 4   Due to the linearity of the flow problem the effects of the incidence and the jet flap can be studied separately, and their combined action can be estimated by means of superposition

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Twenty-First Symposium on NAVAL HYDRODYNAMICS (7) where negative semi-axis corresponds to the wing, and positive semi-axis corresponds to the jet. Applying the least degeneracy principle to the equation (7) with account of the formula (4) we can evaluate asymptotic dimension σ=σ(z) of characteristic flow region near the jet-flapped trailing edge from the order equation (8) wherefrom we can put σ(z)=ε(z) and the equation (7) can be written in form of the following system (9) (10) where differentiation in formulae (9) and (10) is performed with respect to stretched abscissa. Replacing =&#8722 and defining we can reduce (9),(10) to the equation set (11) (12) The equation of the form (13) can be reduced by means of change of variables to Carleman equation, solution of the latter being known. Therefore, the equation (13) can be written down as (14) where C is a constant. Utilizing the following easily provable relationship, (15) Substituting (15) into (14) we find (16) The equation (11) has the form (13). Then, using (16), we obtain 17 With the help of the residue theory one can derive the following result (18) Interchanging the order of integration in (17) we obtain with account of (18) (19) To perform calculation of the first integral in formula (12) replace by in (19) and accounting for the following result of the residue theory (20) we come to the following form of (12) (21) Due to the requirement that the magnitude of must be finite for → 0, we have to put K=0. Designating and f()=γi()/τ we come ot the following set of equations (22) (23) ω(0)=1 (24) ω(∞)=0 (25) 4. Solution of the Equation System Describing the Local Flow Problem Solution of the equation (23) had been derived by Lighthill [22] in his investigation of the two-dimensional problem with use of Mellin integral transform (26)

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Then for a real C such that C–1 is integrable with respect to from 0 to ∞ it follows from inverse transform theorem (27) (28) If the function SW(S) is regular within the strip C–1<ReS<C then the integration path in (28) can be shifted to ReS=C–1. Then, replacing S by S-1, we obtain (29) Applying equation (29) together with (27) to formula (23), Lighthill derived the following equation for W(S) and |ReS|<1/2 W(S)–(S–1) tan(πS)W(S–1)=0 (30) Introducing the function W1(S) through a relationship W(S)=Γ(S)W1(S) (31) where Γ(S) is the Euler Gamma function, and substituting (31) into (30) we come to the following equation for W1(S) W1(S)–tan(πS)W1(S–1)=0 (32) In order to comply with the condition ω(0)=1 we adopt 0<C<1/2 and W1(0)=1. The corresponding solution of the equation (30) was found by Lighthill in the form (33) Solution of the equation (32) can also be expressed by means of Alexeyevsky function (34) satisfying the equation H(1)=1, H(z+1)=Γ(z)H(z) (35) Making use of the latter definition (36) With account of the result found by Bornes for representation of Alexeyevsky function Lighthill obtained the following formula (37) He also derived an expression for W1(S) for |S|<1/2 (38) To obtain the function ω() we use the inverse Mellin transform (27). Then for ≤1 ω()is equal to the sum of residues of the function (39) at its poles located to the left of the integration path, and for is equal to minus sum of residues of the function Q(S) at its right-hand side poles. It follows from (33) that W1(S) has the poles of the order n at points S=–n, S=n–1/2, where n=1,2…. Gamma function Γ(S) is known to have the poles of the first order at points S= 0,S=–n. Thus, the function will contain terms of the form of and where n,m,p=0,1,2…and p,m≤n. Consider the first two terms of . The expansion of Γ(S) in the vicinity of the point α Γ(S+α)=Γ(α)(1+ψ(α)S)+O(S2) (40) where ψ(z) Gauss function which has the form (41) where γe≈0.577 is the Euler constant. Joint consideration of (40) and (41) leads to Γ(S+1)=1–γeS+O(S2) (42) (43) Besides, it follows from the expansion (38) W1(S)=1+S+O(S2), W1(S–1/2)=1–S+O(S2) Then, using (30) and a functional equation for Gamma function Γ(S+1)=SΓ(S), we obtain the following expression for Q(S) near the poles (44)

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Twenty-First Symposium on NAVAL HYDRODYNAMICS (45) (46) (47) Then the function can be written down as (48) (49) Function can be derived by means of differentiation of of equations (47) and (48) (50) (51) Now, let us find f(). In this connection we shall use the same Mellin integral transform (26). Applying (26) to (23) we have (52) where wherefrom (53) Performing integration in the right-hand side of the equation with the help of the residue theory and accounting for the formula (30), we derive from (53) (54) The function f() can be obtained in the same way as ω() with use of the inverse Mellin transform (55) Analysing (55) it is easy to see that the order of the poles of the function Q1(S) at points S=–n is equal to n+2, where n=0,1,2…. At points S=n, where n=1,2…the function Q1(S) has zeros of the order of n–1 for n>1. Thus, f() will contain terms of the form n lnm and –n+1/2 lnp , where n,m,p=0,1,2…, p≤, m≤n+1. Find the function f() for large and small . Using the formulae (44)–(46) and (55) one can derive the following expansions of Q1(S) near the poles (56) (57) (58) Then for small the function f() will be approximately equal to the residue of Q1(S) for S=0. For larger the function f() will be approximately equal to the sum of residues of Q1(S) at poits S=1/2 and S=3/2. Finally (59) (60) It follows from (59) that for → 0, (x → 1+0) the hydrodynamic loading behaves as ln (or as ln |x–1|) which is in proper accordance to the results of the linear theory due to the jump of downwash. In order to be able to carry out the asymptotic matching of the local solution with the far field it is necessary to find the first term of expansion of f() for → ∞. It can be seen from observation of (60) that one-term outer expansion of the inner solution has the form (61) where corresponds to magnitudes of the jet deflection angle and jet momentum coefficients at the root chord cross section,

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Twenty-First Symposium on NAVAL HYDRODYNAMICS is a function characterizing distribution of the jet flap parameters spanwise. Simultaneously, consideration of asymptotics of longitudinal curvature ω′ far from the place of blowing (see equation(51)) shows that for small magnitudes of the jet momentum coefficient Cj the pressure difference across the jet in the outer downstream region has the order of This means that for small Cj the contribution of the outer part of the jet sheetinto the lifting capacity of the jet flapped wing can be neglected. Equation (61) prompts that the outer solution of the linear lifting surface theory, describing loading distribution upon the jet flapped wing, should have the order of and square root singularities not only at the leading (shock entry) but also at the trailing edge. 5. Outer Solution of the Leading Order In the considered case of zero incidence and εo → 0 in the outer limit the longitudinal dimension of the region of the jet evolution tends to zero, and the problem is reduced to solution of the homogeneous integral equation of the lifting surface theory (62) with additional condition of asymptotic matching with the inner description of the loading near the trailing edge. Represent the outer loading in form of an asymptotic expansion (63) where with account of the expression (61) the structure of the function is adopted in terms of angular variables θ=arccos(–z/λ) and ψ= arccos(–x) with x ∈ [–1,1], z ∈ [–λ,λ] in the form (64) Matching of expressions (61) and (64) determines the scaling function and supplies the aforementionned additional condition at the trailing edge5 in the form 5   This condition replaces conventional Kutta-Zhukovsky condition for a wing with a jet flap (65) or, with account of the orthogonality of the system {sin nθ} upon the segment θ ∈ [0,π] (66) Corresponding contribution into a lift coefficient Cy can be found by means of the formula (67) Correspondingly, the expression for the longitudinal moment coefficient with respect to the mid-chord of the order of takes the form (68) In a more general case of the jet flapped wing at an angle of attack α, designating the parameter we can write the lift and moment coefficients in terms of hydrodynamic derivatives with respect to the incidence and characteristic similarity criteria of the jet flow qj as (69) Fig. 1. The ratio Cy/Cy2D versus aspect ratio for a rectangular wing with a jet flap. In Fig.1 there is presented a dependence of the ratio Cy/Cy2D where Cy and Cy2D represent correspondingly lift coefficients of the wing of finite and infinite span) upon the aspect ratio λ of the rectangular wing. The 2   Note, that as early 1956 in his investigation of the two-dimensional flow problem for the case of small jet momentum coefficient; Spence had found such a coordinate transformation x̄=x/Δ(Cj) (where Δ(Cj) is a stretching function) for which the integro-differential equation of the problem does not depend on the jet momentum coefficient. The stretching function derived in the above work turned out to be Δ (Cj)=Cj/4=ε.

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Twenty-First Symposium on NAVAL HYDRODYNAMICS lift coefficient of the infinite-aspect-ratio wing was obtained by the approach of this paper and can be shown to be This result coincides with the first term of the series solution by Spence, see Introduction of the present paper. For comparison in the same Figure some calculated results are plotted which are obtained from formulae of Tokuda [15] by putting Cj → 0. Fig.2 illustrates the lift coefficient of a rectangular jet flapped wing versus the angle of attack α for wing's aspect ratio λ=6.8, jet deflection angle τ°=31.3° and the jet momentum coefficient Cj=0.42. Calculated results are compared with experimental data of Williams and Alexander, [16] and theory of Tokuda. Fig. 2 The lift coefficient of a rectangular wing with a jet flap versus] angle of attack, λ= 6.8, τ°=31.3°, Cj=0.42. The above results have been obtained for the case of uniform distribution of the jet parameters spanwise. 6. Evaluation of the Possibility to Enlarge the Speed Range of Non-Cavitating Regime of the Flow Past a Hydrofoil of Finite Aspect Ratio by Means of the Jet Flap Now, we shall pass over to discussion of the limiting cavitation diagrams, characterizing the regimes of cavitation inception in the middle part and at the leading edge of the hydrofoil. We shall use a simplified approach of determining the diagrams (of minimum pressures) advocated in [20],[21] and consisting in separate determination of the “edge” branches (from the local flow problem at the rounded leading edge) and the “bottom” branches (from the outer linearized flow field) of the diagram. Let us derive the expression, describing the edge branch of the minimum pressure diagram. It can be shown that near a rounded leading edge the flow is close to that around a parabola in two-dimensions in the planes normal to the leading edge platform contour. At points of the surface of the wing near the rounded leading edge the velocity of the fluid can be found by means of the expression, [18]. (70) where X=x+1/δ2, δ is a maximum relative thickness of the wing, =ρle/δ2, ρle—radius of curvature of the wing's leading edge. Functions U1(z) and U2(z) are determined by means of matching of the local solution with the outer description of the velocity upon the wing's surface. It can be shown, [20] that the maximum magnitude of the fluid velocity near the leading edge is achieved at the point with magnitude of stretched abscissa (71) Substitution of (71) into (70) gives the following expression for minimum magnitudes of the pressure coefficient at the leading edge (72) Let us determine U1,2(z) and the equation of the “bottom” of the minimum pressure diagrams. The flow velocity upon the wing 's surface in its middle part is calculated with use of the outer linear solution in the form (73) where is a velocity component due to thickness of the wing, and γo is a velocity component due to effects of “asymmetry” (angle of attack, curvature, jet flap, etc.). Accounting for the fact that the vortex component of the flow velocity, generally speaking, has a square root singularity at the leading edge, that is for x → –1 (74) where are assumed to be of the order of O(1)6. Now, it is not difficult to perform matching of expressions (71) and (74) and find U1(z) and U2(z) in the form (75) 6   This is quite practical assumption, implying that all perturbation parameters are of the order of thickness

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Velocity component due to thickness is determined by means of thin body theory as (76) In the case under consideration (α=0) the function U2(z) is easily found with account of (64) in the form (77) Minimum pressure in the middle part of the wing can be calculated as (78) where xm is the abscissa of minimum pressure. It is easy to write down a functional dependence of upon the lift coefficient of the wing with a jet flap. For the “edge” branch of the diagram we have (80) For the “bottom” branch of the diagram (80) As an example, consider the simplest case of a wing of infinite aspect ratio. In this case the present approach enables to obtain the following equation of the “edge branch” of the minimum pressure diagram in the case of symmetric foil with a jet flap. (81) where, as earlier, δ is relative thickness of the foil, le=ρle/δ2, ρle is radius of curvature of the leading edge. In the case of small jet momentum coefficients. Cj the lift coefficient is For the purpose of comparison we can write corresponding expression for the case of the foil without jet flap at an angle of attack α (82) where Cy=Cy2D=2πα. Comparing the formulae (81) and (82), we can see that for the same magnitude of the lift coefficient Cy the cavitation inception at the leading edge can be expected to occur later for the foil with a jet flap than for that without the jet flap. The same conclusion can be deduced from comparison of the equations, describing “bottom” branches of the diagrams. These equations for the considered case of infinite-aspect-ratio wing have the form (83) (84) where Cpt(x) is a thickness contribution to the pressure coefficient of the foil, xm is the abscissa of the point of minimal pressure on the foil. Formulae (83) and (84) corrrespond respectingly to the cases of the foil with and without jet flap. Figures 3 and 4 show plotted minimum pressure diagrams for a NACA four-digit wing sections NACA 008 and NACA 0012. Fig. 3 Calculated minimum pressure diagrams for NACA 008 section for the case of the foil with and without jet flap. These diagrams clearly indicate that use of jet flapped wings can be expected to result in delay of cavitation inception both in the middle part of the foil and near the leading edge.

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Twenty-First Symposium on NAVAL HYDRODYNAMICS Fig. 4 Calculated minimum pressure diagrams for NACA 012 section for the case of the foil with and without jet flap. The authors would like to thank Dr. G.M.Fridman for his valuable comments in connection with the present paper. 7. REFERENCES 1. Thwaites, B., “Incompressible Aerodynamics”, Oxford at the Clarendon Press, 1960. 2. Davidson, I.M., J. Roy. Aero. Soc., 60, pp. 25–41, 1956. 3. Kaplan, P., “An Integrated Propulsion-Lift-Control Design for Large High Speed Hydrofoil Craft”, Proceedings of the Second International Conference on Fast Sea Transportation, Yokohama, Japan, 13–16 December 1993, pp. 501– 512. 4. Kaplan, P. and Goodman, T.R., “Use of Jet-Flapped Hydrofoils as Ship Antipitching Fins”, Journal of Aircraft, Vol. 4, No. 2 March–April, 1967. 5. Kaplan, P. and Lehman, A.F., “An Experimental and Analytical Study of Jet Flap Hydrodynamics for Application to Submarine Control Surfaces”, Oceanics Inc. Rpt No. 67–41, November 1967. 6. Spence, D.A. “The Lift Coefficient of Thin, Jet-Flapped Wing”, Proc. Roy. Soc., Series A, Mathematical and Physical Sciences, No. 1212, December 1956, Vol. A 238, 1956, pp. 46–68. 7. Spence, D.A., “Some Simple Results for Two-Dimensional Jet-Flapped Aerofoils”, The Aero nautical Quarterly, November, 1958, pp. 46–68. 8. Spence, D.A. Rep. Aero. Res. Coun., London, 20, 753, 1959. 9. Spence, D.A., “The Lift Coefficient of Thin Jet-Flapped Wing”, The Aeronautical Quarterly, October, 1960, pp. 97–118. 10. Spence, D.A., “A Solution of the Integro-Differential Equation for the Slope of the Jet”, Proc. Roy. Soc., Ser. A, Mathematical and Physical Sciences, No. 1304, April 1961, Vol. 261, pp. 97–118. 11. Kuchemann, D., “A Method for Calculating the Pressure Distribution of Jet Flapped Wings”, Gt. Britain Aero. Res. Council, Report & Memoranda 3036, 1956. 12. Statford, B.S. “Early Thoughts on the Jet Flap”, The Aeronautical Quarterly, Vol. VIII, Feb. 1956. 13. Maskell, E.C. and Spence, D.A., “A Theory of the Jet Flap in Three Dimensions ”, Proc. Roy. Soc., London, June 1959, Ser. A, 251, pp. 407–425. 14. Kerney, P.K., “An Asymptotic Theory of the High-Aspect-Ratio”, Ph.D. Thesis, Graduate School of Engineering, Cornell University, 1967. 15. Tokuda, N., “An Asymptotic Theory of the Jet Flap in Three Dimensions”, J. Fluid Mech., 1971, vol. 46, part. 4, pp. 705–726. 16. Williams, J., Alexander, A.J. Aeronautical Quarterly, v. 9, 1957, pp. 395–406. 17. Rozhdestvensky, K.V., “Matched Expansions Method in Hydromechanics of Lifting Bodies”, Book of Abstracts, The Third International Congress on Industrial and Applied Mathematics, Hamburg, 3–7 July, 1995, p. 144 (full text to appear in Zeitschrift fur Angewandte Mathematik und Mechanik, ZAMM) 18. Rozhdestvensky, K.V. Method of Matched Asymptotics in Hydromechanics of Wings, L., Sudostroenie, 1979, 208 p. 19. Van Dyke, M. “Perturbation Methods in Fluid Mechanics”, The Parabolic Press, Stanford, 1975. 20. Rozhdestvensky, K.V., Mishkevich, V.G. “Application of the Method of Matched Asymptotic Expansions to the Determination of the Flow Past a Thin Foil in Unsteady Motion”, Sbornik Trudov po Proektirovanyu Sudov, No. 37, pp. 60–73 (in Russian) 21. Mishkevich, V.G., Rozhdestvensky, K.V., Bubentsov, V.P. “Unsteady Flows Near Leading Edges of Lifting Surfaces: Pressures, Flow Separation, Cavities”, Proc. Symposium “Bubble Noise and Cavitation Erosion in Fluid Systems”, FED— Vol. 176, ASME 1993, pp.9–19.