Proeedings of the 7 th International Symposium on Cavitation CAV9 Paper No. ## August 7-, 9, Ann Arbor, Mihigan, USA Cavity flow with surfae tension past a flat plate Yuriy Savhenko Institute of Hydromehanis of the NAS Kiev, Ukraine Yuriy Semenov Institute of Hydromehanis of the NAS Kiev, Ukraine ABSTRACT The effet of surfae tension fores on the avity flow parameters is onsidered for steady flow past a rounded-edge plate perpendiular to the inident flow. The fluid is assumed to be ideal, weightless, and inompressible. The problem is solved in a parameter region by finding analytial expressions for the flow potential and the funtion that onformally maps the parameter region into the flow region in the physial plane. The dynami boundary ondition inludes the surfae tension fore, whih is proportional to the free boundary urvature, and allows one to obtain an integral equation in the veloity magnitude on the free boundary. The integral equation is solved numerially by the method of suessive approximations. The results of alulations of the effet of the Weber number and the plate edge radius on the geometry of the avity and the drag oeffiient of the plate are presented. INTRODUCTION The surfae tension fore arises at the phase interfae as a result of the work done by the reversible isothermokineti proess of free boundary formation. Aording to the Young Laplae equation, a pressure ump proportional to the surfae urvature develops aross the phase interfae. This ompliates the dynami boundary ondition beause the veloity on the avity boundary depends on the avity urvature. A signifiant diffiulty in aounting for surfae tension stems from the fat that, aording to the theory of ets in ideal fluids, in the limiting ase of the absene of surfae tension the free surfae urvature at the point of flow separation from a sharp edge beomes infinite. Introduing even a small surfae tension oeffiient into suh a model results in an infinite fore and the absene of avitation. Attempts to aount for the surfae tension fore in avity flow problems have been made in Refs. [ 4]. In Ref. [3], the ase of small surfae tension oeffiients is onsidered using the method of mathed asymptoti expansions. In this ase, the free boundary urvature at the separation point is equal to the urvature of the plate. Suh a formulation of the problem gives rise to waves on the free surfae. As shown in Ref. [4], these waves are unfeasible beause they would require energy supply from infinity. In Refs. [ ] it is assumed that at the flow separation point the slope of the flow boundary is a disontinuous funtion. Aording to the theory of ets in ideal fluids, that assumption results in an infinite veloity and urvature at the separation point. This gives no way of obtaining a ontinuous transition to the limiting ase of the absene of surfae tension, for whih the veloity at the separation point is finite. In this work we show that the point of flow separation from a smooth surfae in the presene of surfae tension is determined by the Brillouin Villat ondition in exatly the same manner as for neglet of surfae tension. A sharp edge is onsidered as the limiting ase when the urvature of the rounded plate edge tends to infinity. In this formulation, no physial ontraditions our in the solution of the problem, and the flow with zero surfae tension fore results as a limiting ase. MATHEMATICAL FORMULATION AND THE METHOD OF SOLUTION We onsider symmetri flow of an ideal weightless fluid past a urved, in the general ase, obstale with the formation of a avity downstream of the body. A shemati of the flow is shown in Figure. The wedge-shaped body of vertex angle α has a retilinear part, whih smoothly hanges into a irular ar of radius R. The avity detahes from the rounded part of the body at point O at angle β and loses on the urvilinear ontour CDB, on whih the veloity distribution is speified. Point C, whih is the infletion point on the ontour OCB, orresponds to the end of the avity and the start of the losing ontour. The maximum width H of the body is hosen as a harateristi dimension. On the avity boundary, the surfae tension fore gives rise to a pressure ump, whih is determined from the Young Laplae equation P =τχ ( P
where P is the pressure on the fluid of the avity boundary, P is the pressure in the avity, τ is the surfae tension oeffiient, χ = dγ ds is the avity urvature, and γ is the slope of the avity ontour. In the neighborhood of the separation point O, the avity urvature beomes negative beause for the hosen diretion of inreasing oordinate s its value dereases when moving along the avity ontour. As follows from Eq. (, surfae tension redues the pressure in the fluid on the avity boundary. dw, and the derivative of the omplex potential, dw, in the region of the parametri variable ς. If these expressions are found, the orrespondene between the parameter region and the physial flow region is given by the mapping funtion ς dw z ( ς = z. (4 dw A method for the onstrution of expressions for the omplex veloity and the derivative of the omplex potential for a mixed boundary value problem is reported in Ref. [8]. One has to find the funtion dw that satisfies the following boundary onditions dw β ( ξ, arg =, dw = v(, < ξ <, < ξ <, < <, =, =, ξ =. (5 Figure : Shemati of avity flow past urvilinear ontour: а physial plane; b parameter region The Bernoulli equation for the irrotational flow under onsideration V P V = ρ P ρ gives the veloity on the avity boundary, whih, in view of Eq. (, an be written in dimensionless form as follows ( H v = V V = σ χ (3 We ρv H P P where We = is the Weber number and σ = is τ ρv the avitation number. Sine the flow is irrotational, we an introdue its potential φ and the stream funtion ψ, whih is the harmoni onugate of φ, both defined in a parameter region, for whih the first quadrant is hosen (Figureb. By the onformal mapping theorem, the images of three points an be fixed arbitrarily; let the points ς =, ς =, and ς = in the parameter region orrespond to points O, A, and B in the physial plane. The mathematial formulation of the problem is as follows: find the omplex flow potential w = φ iψ that meets the nonpassing ondition on the body and the dynami boundary ondition (3 on the free boundary. Following Zhukovsky s [5] and Chaplygin s [6 7] methods, the problem is solved by onstruting expressions for the omplex veloity, Here, β (ξ is the slope in the physial plane as a funtion of the real-axis oordinate in the parameter region. This angle dereases when moving along the body from point O to point A, β ( = α ; v ( is the veloity magnitude on the free boundary as a funtion of the imaginary-axis oordinate in the parameter region. An expression for the omplex veloity that satisfies the above boundary onditions (5 has the form dw = v ς ς α ς ξ exp ln ς ξ i d ln v i ς ln i ς When ς = ξ and ς = i are substituted in turn into Eq. (6, it is apparent that the boundary onditions (5 for the omplex veloity are satisfied. The derivative of the omplex potential is found by Chaplygin s singularity method [7] (6 dw = Kς (7 where K is a sale fator, whih will be determined below. Substituting Eqs. (6 and (7 into Eq. (4 allows one to find the derivative of the mapping funtion z = z(ς as well as the dependenes s = s(ξ (the oordinate along the body ontour and s = s( (the oordinate along the free boundary:
Kς ς = v ς α i exp d ln v i ς ln i ς, (8 ς ξ ln ς ξ dw s( = d v( v( = = K ξ ς = i s( ξ = ξ d. ς = ξ ς = i The sale fator K appearing in Eqs. (9 is determined from the ondition for the length of the wetted part of the body ξ =, (9 S w = s( ξ. ( Cavity losure sheme. The losure of the avity is speified impliitly by speifying the veloity distribution along the losing ontour CDB. The veloity on the ontour CD dereases from the value v = σ χh / We at point C down to the value v at point D. The veloity distribution along the segment CD an be speified by the linear funtion v ( s = v ( s v s, ( where s = ( s s / SCD and s is the oordinate of the avity losure point C determined from the avity losure ondition, whih has the form Im Im i d = = = res ς, ς 4 ς= ς = ς= ( where ς =/ ς. Taking the integral in Eq. ( along an infinitely large ontour with the help of the residue theorem, we obtain the following equation d ln v ξ = α. (3 The veloity distribution ( v [ s( ] v = along the avity losure ontour CD must satisfy Eq. (. This is ahieved with the help of the oordinate s in Eq. (, whih affets the veloity distribution along the free boundary. Condition for avity detahment from the rounded edge. The point of flow separation from the rounded edge is determined from the Brillouin Villat ondition [7], whih is equivalent to the extremum ondition for the veloity magnitude funtion on the ontour of the body d ln v lim =. (4 s ds d ln v d ln v Using the relation =, differentiating the ds ds funtion v ( ξ = dw, and substituting the result into ς = ξ Eq. (4, we obtain the following equation d ln v = α. (5 ξ Eq. (5 is an equation in the length S w of the wetted part of the body beause this quantity affets the funtion β ( ξ = β[ s( ξ ], < s < S w. Boundary onditions and orresponding integrodifferential equations. For the known shape of the urved setion of the body given by the funtion β (s, the funtion β (ξ is determined by the numerial solution of the integro-differential equation ds = (6 ds or α Kξ ξ ξ ξ = χ[ s( ξ ] exp ln v ξ ξ ξ d ln v tan ξ where χ (s is the urvature of the body given as a funtion of the ar length s. In order to derive an integral equation in the funtion d (ln v, we first need to get an expression for the urvature χ along the avity boundary, i.e. along the imaginary axis of the parameter region. The slope of the free surfae is determined from equation (6 as follows dw α β ( = arg = tan α d ln v ln tan ξ Differentiating the above equation and taking into aount equation (9, we find the urvature of the free surfae 3
dγ χ = ds v α = K ξ ξ d ln v (7 The behavior of the funtion way d β is determined in a similar ds ds Kξ K = = χ( ξ = χ( ξ, ~ χ ξ, ( ds v( ξ v By substituting equation (7 into the dynami boundary ondition (3, we obtain the following integral equation where d ln v α / ξ = ξ WeK ( σ v Rv( d ln v v ( = v exp and v = σ / We. (8 Curvature of the free surfae at the point of separation. In order to evaluate the urvature of the free surfae at the separation point, where at = equation (7 has an indeterminate form, we differentiate the numerator and denominator of equation (7 to obtain: χ = lim χ( = lim K dv ξ ξ d ln v v lim K ( ξ d ln v ( ξ ( (9 The first limit in the above equation equals zero due to the flow separation ondition (5. In order to evaluate the seond term ontaining the singular integrals, we have to estimate the behavior of the funtions d ln v at and d β at ξ. From the Brillouin-Villat ondition (4 it follows d ln v d ln v ds = =. ds In view of equation (9, it an be seen that for ds = K v( ~. Therefore, from equation (4 it follows d ln v lim =, d ln v ~ α,, α. (9 where v = v ξ = lim lim ( dw, and χ ξ ξ ς = ξ is the urvature of the body at the point of separation. By substituting the estimates ( and ( of the funtions d ln v and d β into the orresponding integrals in equation (9 and taking into aount that lim ( = and ξ lim ( ξ =, 4 we an find that the urvature of the free surfae at the point of flow separation is equal to the urvature of the body, i.e. χ = lim ( χ = χ. ( NUMERICAL METHOD The body shape is hosen to be a irular ylinder of radius R, the length of the losure interval being related to R as S CD / R =. In disrete form, the solution is sought on a fixed set of points ξ, =,..., M distributed along the real axis of the parameter region and on a fixed set of points, =,..., N distributed along its imaginary axis. The total number of points was hosen in the range N = 5, and the total number of points ξ was hosen in the range M = 3 N to hek the onvergene of the solution proedure. The points ξ are distributed so as to provide a higher density of the points s = s( ξ at points O and A, at whih the derivative ds d ξ = has singularities. The distribution ς = ξ of the points is hosen so as to provide a higher density of the points s = s( on the free surfae near the point of flow separation. The solution of equations (8 was found by the method of suessive approximations applying the Hilbert transform to determine the (k -th approximation d(lnv F ( k ( k ( 4 = F ( k = WeK ( k R v ( ( ( k ( k ( σ ( v, < <C, (3 ξ ξ, (4 4
where v ( k d(lnv ( = v exp. ( k In eah iteration step k the integro-differential equation (6 in the funtion d β was solved using the inner iteration proedure ( l ( l ( l K ξ ξ = χ[ s ( ξ] v ξ ( l ( k ξξ d lnv exp ln tan ξ ξ ξ (.6 (l In eah iteration step l the parameters K, C and S w were alulated from equation (, (3 and (5, respetively. The integrals appearing in the system of equations were evaluated using the linear interpolation of the funtions β (ξ and d (ln v on the intervals ξ, ξ and,, ( ( respetively. In the first iteration the funtions β (ξ and ( d ln v were given as β ( ξ and v ( v. The onvergene of the inner iteration proedure required 5 iterations to reah the ondition ( β ξ l d d ( l < ε. The onvergene of the outer iterations required from several hundreds to several thousands of iterations, and it was obtained applying the under-relaxation method. The value of the relaxation parameter depends on the number N of nodes along the free surfae. With inrease in the number of nodes, the relaxation parameter should be hosen smaller to provide the onvergene of the solution. This is due to the singularity in the integral in (3, whih is evaluated using the linear interpolation of the funtion F ( on the intervals,. ( / / REZULTS Figure shows the alulated results for avity flow past a plate with a rounded edge of radius R =. 5H. The avity ontour is depited by a thin line, and the losing ontour is depited by a thik one. It an be seen that at large Weber numbers the flow parameters are little affeted by surfae tension. The effet of surfae tension beomes notieable at We <, and in this ase the avity length and height derease. It may be expeted that with a further derease in the Weber number the flow, in the limit, beomes free of avitation. Figure 3 shows the angle of flow turn on the rounded edge (the angle δ is shown in Figure versus the Weber number for different values of the edge radius. At large Weber numbers when the effet of surfae tension on the avity flow parameters an be negleted, the flow turn angle depends on the edge radius as follows: the smaller the edge radius, the smaller the flow turn angle. In the limiting ase of zero edge radius (a Y/H -.5 - -.5 - -.5 X/H - 3 4 5 6 R/H=.5 σ =. 7 We= 3 9 Figure : Effet of the Weber number on the avity ontour (thin lines for avity flow past a plate with edge radius R =. 5H at avitation number σ =.. The thik lines orrespond to the avity losure ontours. Figure 3: Angle of flow turn on the plate edge versus the Weber number for different edge radii at avitation number σ =. sharp edge, the flow turn angle tends to zero, whih orresponds to the flow separating tangentially to the retilinear part of the body. At Weber numbers smaller than, the effet of surfae tension beomes notieable; the smaller the edge radius, the greater the Weber number at whih this effet omes into play. It an be seen from Eq. ( that the pressure drop aross the avity boundary in the neighborhood of the separation point inreases with the edge urvature. Two harateristi regions an be distinguished in the graph: a region of strong dependene of the flow turn angle δ on the Weber number ( We < and a region of weak dependene ( We >. Figure 4 shows the angle of flow turn on the edge versus the Weber number at different avitation numbers. With inreasing avitation number the flow turn angle dereases, i.e. 3 the effet of the Weber number diminishes. For We >, the flow turn angle depends almost not at all on the Weber number or the avitation number. 5
Figure 4: Effet of the avitation number on the Weber number dependene of the flow turn angle for edge radius R / H =. Figure 6: Relative hange of the flow parameters at Weber numbers We = and We = versus the avitation number: the maximum avity width Δ h / h, the length Δ L / L orresponding thereto, and the drag oeffiient Δ C / x C x Figure 6 shows the effet of the Weber number in the range to on the avity flow parameters for entrane edge radius R / H =.. Shown in the figure are the relative hanges of the maximum avity width, Δ h / h, and the orresponding distane to the plate, Δ L / L, as well as the relative hange of the drag oeffiient, Δ C x / Cx, versus the avitation number. It an be seen that with dereasing avitation number the effet of surfae tension inreases, and the hange of the above flow parameters orrelates with the hange of the angle of flow turn on the entrane edge. Figure 5: Effet of the avitation number on the drag oeffiient for edge radius R / H =. Figure 5 shows the drag oeffiient versus the avitation number for different Weber numbers at edge radius R / H =.. The drag oeffiient dereases somewhat as the Weber number dereases. Figure 6 shows the effet of the Weber number in the range to on the avity flow parameters for entrane edge radius R / H =.. Shown in the figure are the relative hanges of the maximum avity width, Δ h / h, and the orresponding distane to the plate, Δ L / L, as well as the relative hange of the drag oeffiient, Δ C x / Cx, versus the avitation number. It an be seen that with dereasing avitation number the effet of surfae tension inreases, and the hange of the above flow parameters orrelates with the hange of the angle of flow turn on the entrane edge. CONCLUSION The results presented above show that the effet of surfae tension on the avity flow parameters beomes notieable at Weber numbers less that 3. For superavitating axisymmetri flows, a greater effet of surfae tension on the flow parameters may be expeted due to a greater urvature of the axisymmetri avity, whih has both a meridional and a radial omponent. Besides, the effet of surfae tension inreases with dereasing avitator diameter, whih appears in the Weber number as a harateristi dimension. REFERENCES [] Vanden-Broek, J.-M. (4, Nonlinear apillary freesurfae flows. J. of Engineering Mathematis, 5, 45 46. [] Vanden-Broek, J.-M. (983 The influene of surfae tension on avitating flow past a urved obstales. J. of Fluid Meh., 33, 55 64. 6
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