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Past a Rigid Boundary Abstract In this paper, stream function and velocity potential for the flow field under the condition of irrotationality in two-dimensional motion are considered. Then complex potential and complex velocity are described. Some elementary solutions corresponding to the uniform flow, source, sink, doublet and vortex in two-dimensional motion are presented. Blasius theorem which is useful to find the hydrodynamic force and the hydrodynamic moment acting on the body is investigated. The flow past a cylinder without circulation and flow past a cylinder with circulation are expressed. For the flow past a cylinder, hydrodynamic force and the hydrodynamic moment acting on the cylinder are derived by using the Blasius theorem. 1. Fundamental Concepts to Two-Dimensional Flow 1.1 Stream function The continuity equation, in Cartesian coordinates, for the flow field under consideration is EMBED Equation.DSMT4 0, (1) where u, v are components of the fluid velocity. Now introduce a function ((x, y) that is defined as follows u EMBED Equation.DSMT4 , v EMBED Equation.DSMT4 . (2) With this definition, the continuity equation is satisfied identically for all functions (. The function ( is called the stream function. The condition of irrotationality is EMBED Equation.DSMT4 . Substituting for u and v from (1) and (2) show that ( must satisfy the following equation EMBED Equation.DSMT4 . (3) That is, the stream function (, like the velocity potential (, must satisfy the Laplaces equation. The flow lines that correspond to ( constant are the stream lines of the flow field. To show this, it is noted that ( is a function of both x and y in general so that the total variation in ( associated with a change in x and a change in y may be calculated from the expression d( EMBED Equation.DSMT4 ( v dx u dy. Then the equation of the line ( constant will be ( vdx u dy 0 or EMBED Equation.DSMT4 . Hence the lines corresponding to ( constant are the streamlines, and each value of the constant defines a different stream line. It is this property of the function ( that justifies the name stream function. Finally, it should be noted that the streamlines ( constant and the lines ( constant, which are called equipotential lines, are orthogonal to each other. This may be shown by noting that if ( depends upon both x and y, the total change in ( associated with changes in both x and y will be d( EMBED Equation.DSMT4 udx vdy. Then the lines corresponding to ( constant will be defined by udx vdy 0, or EMBED Equation.DSMT4 . That is, EMBED Equation.DSMT4 . This means that, the slope of the lines ( constant is the negative reciprocal of the slope of the lines ( constant, so that these lines must be orthogonal. 1.2 Complex potential and complex velocity The velocity components u and v may be expressed in terms of either the velocity potential or the stream function. These expressions are u EMBED Equation.DSMT4 , v EMBED Equation.DSMT4 . That is, the function ( and ( are related by the expressions EMBED Equation.DSMT4 , EMBED Equation.DSMT4 . But these will be recognized as the Cauchy-Riemann equations for the functions ((x, y) and ((x, y).Then consider the complex potential F(z), which is defined as follows F(z) ((x, y) i((x, y), (4) where z x iy, ( and ( are called conjugate functions. That is, for every analytic function F(z) the real part is automatically a valid velocity potential and the imaginary part is a valid stream function. Another quantity of prime interest, apart from the complex potential F(z), is the derivative of F(z) with respect to z. Since F(z) is supposed to be analytic, EMBED Equation.DSMT4 will be a point function whose value is independent of the direction in which it is calculated. Then, denoting this derivative by W, its value will be given by W(z) EMBED Equation.DSMT4 EMBED Equation.DSMT4 , W(z) u ( iv, (5) that is, W(z) EMBED Equation.DSMT4 , where use has been made of (4), EMBED Equation.DSMT4 and (2). W(z) is called the complex velocity and its complex conjugate EMBED Equation.DSMT4 Then EMBED Equation.DSMT4 (u ( iv)(u iv) u2 v2 q2. The significance of this result is that the quantity EMBED Equation.DSMT4 u2 v2 appears in the Bernoulli equation. The complex velocity may be readily obtained in cylindrical coordinates by converting the Cartesian components of the velocity vector (u,v) to cylindrical components (uR, u(). Figure 1 shows a velocity vector OP decomposed into its Cartesian components and also its cylindrical components. Thus, u uR cos ( ( u( cos EMBED Equation.DSMT4 uR cos ( ( u( sin (, v uR sin ( u( sin EMBED Equation.DSMT4 uR sin ( u( cos (. Substituting these expressions into (5) gives the expression for the complex velocity W in terms of uR and u( . Figure 1. W(z) (uR cos ( ( u(sin () ( i(uR sin( u( cos() uR(cos ( ( i sin () ( i u( (cos( ( i sin (). That is, W (uR ( iu()e(i(. (6) 1.3 Uniform flows The simplest analytic function of z is proportional to z itself, and the corresponding flow fields are uniform flows. First, consider F(z) to be proportional to z where the constant of proportionality is real. That is F(z) cz, where c is real. Then from (5), W(z) u ( iv c. Then, by equating real and imaginary parts of this equation, the velocity components corresponding to this complex potential are u c and v 0. But this is the velocity field for a uniform rectilinear flow as shown in Figure 2(a). Thus the complex potential for such a flow whose velocity magnitude is U in the positive x direction will be F(z) Uz. (7) Next consider the complex potential to be proportional to z with an imaginary constant of proportionality. Then F(z) ( icz, where c is real. The minus sign has been included to make the velocity component positive when c is positive. For this complex potential W(z) u ( iv ( ic so that the velocity components are u 0, v c. (a) (b) (c) Figure 2. This is a uniform vertical flow as shown in Figure 2(b). Then the complex potential for such a flow whose velocity magnitude is V in the positive y direction will be F(z) ( iVz. (8) Finally, consider a complex constant of proportionality so that F(z) ce(i(z, where c and ( are real. For this complex potential W(z) u ( iv c(cos( ( i sin(). Hence, the velocity components of the flow field are u c cos( and v c sin(. This corresponds to a uniform flow inclined at an angle ( to the x-axis as shown in Figure 2(c). Hence the complex potential for such a flow whose velocity magnitude is V will be F(z) Vze(i(. (9) This last result, of course, contains the two previous results as special cases corresponding to ( 0 and ( EMBED Equation.DSMT4 . 1.4 Source, sink and vortex flows Complex potentials that correspond to the flow fields generated by sources, sinks and vortices are obtained by considering F(z) to be proportional to log z. Consider, first, the constant of proportionality to be real. Then F(z) c log z, F(z) c log(Rei(), ( i( c log R ic(. Hence, ( c log R and ( c(. This gives a flow field as shown in Figure 3(a) in which the stream lines are shown solid and the direction of the flow is shown for c 0. The direction of the flow is readily confirmed by evaluating the velocity components so that W(z) EMBED Equation.DSMT4 . (a) (b) Figure 3. Streamlines (shown solid) and equipotential lines (shown dashed). (a) source flow (b) vortex flow in the positive sense. Comparison with (6), shows that the velocity components are uR EMBED Equation.DSMT4 and u( 0. The flow field indicated in Figure 3(a) is called a source. Sources are characterized by their strength, denoted by m, which is defined as the volume of fluid leaving the source per unit time per unit depth of the flow field. Then m EMBED Equation.DSMT4 2(c. Then, c may be replaced by EMBED Equation.DSMT4 , giving the following complex potential for a source of strength m F(z) EMBED Equation.DSMT4 log z. Then the complex potential for a source of strength m located at the point z z0 will be F(z) EMBED Equation.DSMT4 log(z ( z0). (10) Clearly, the complex potential for a sink, which is a negative source, is obtained by replacing m by ( m in (10). Now, consider the constant proportionality in the logarithmic complex potential to be imaginary. That is, consider F(z) (ic logz, where c is real and the minus is included to give a positive vortex. Then, using cylindrical coordinates, F(z) (ic log(Rei() c( ( ic log R. Then, from (4), the velocity potential and the stream function are ( c(, ( ( c log R. The velocity components may be evaluated by use of the complex velocity W(z) EMBED Equation.DSMT4 . Comparison with (6) shows that the velocity components are uR 0 and u( EMBED Equation.DSMT4 . Hence, the direction of the flow is positive for c 0, and the resulting flow field is called a vortex. A vortex is characterized by its strength, which may be measured by the circulation ( associated with it. The circulation ( associated with the singularity at the origin is ( EMBED Equation.DSMT4 EMBED Equation.DSMT4 2(c. Then, c may be replaced by EMBED Equation.DSMT4 , giving the following complex potential for a positive vortex of strength (, F(z) (i EMBED Equation.DSMT4 log z. Then the complex potential for a positive vortex located at z z0 will be F(z) (i EMBED Equation.DSMT4 log(z ( z0). (11) 1.5 Flow due to a doublet The function EMBED Equation.DSMT4 has a singularity at z 0 and in the context of complex potentials, this singularity is called a doublet. However, it turns out that the doublet may be considered to be the coalescing of a source and a sink and the required complex potential may be obtained through a limiting procedure that uses this fact. Referring to the geometry indicated in Figure 4(a), consider a source of strength m and a sink of strength (m, each of which is located on the real axis at a small distance ( from the origin. The complex potential for such a configuration is, from (10) F(z) EMBED Equation.DSMT4 log(z () ( EMBED Equation.DSMT4 log (z ( (), EMBED Equation.DSMT4 log EMBED Equation.DSMT4 EMBED Equation.DSMT4 log EMBED Equation.DSMT4 . If the nondimensional distance EMBED Equation.DSMT4 is considered to be small, the argument of the logarithm may be expressed as follows F(z) EMBED Equation.DSMT4 EMBED Equation.DSMT4 . The logarithm is now in the form log (1 (), where ( 1, so that the equivalent expression ( O((2) may be used. Then F(z) EMBED Equation.DSMT4 . It is now propose to let ( ( 0 and m ( ( in such a way that EMBED Equation.DSMT4 , where ( is a constant. Then the complex potential for a doublet becomes F(z) EMBED Equation.DSMT4 . Thus the complex potential EMBED Equation.DSMT4 may be thought of as being the equivalent of the superposition of a very strong source and a very strong sink that are very close together. The stream function will be established as follows, F(z) EMBED Equation.DSMT4 EMBED Equation.DSMT4 . (a) (b) Figure 4. Then, ( EMBED Equation.DSMT4 . Thus, the equation of the streamlines ( constant is x2 y2 EMBED Equation.DSMT4 0 or x2 EMBED Equation.DSMT4 But this is the equation of the circle of radius EMBED Equation.DSMT4 whose center is located at y ( EMBED Equation.DSMT4 . This gives the streamlines pattern shown in Figure 4(b). The complex velocity for this complex potential is W(z) ( EMBED Equation.DSMT4 ( EMBED Equation.DSMT4 ( EMBED Equation.DSMT4 . Hence, the velocity components are uR ( EMBED Equation.DSMT4 cos ( and u( ( EMBED Equation.DSMT4 sin (. The flow field illustrated in Figure 4(b) is called a doublet flow, and the singularity that is at the heart of the flow field is called a doublet. Then the complex potential for a doublet of strength ( that is located at z z0 is F(z) EMBED Equation.DSMT4 . (12) 2. Flow Past a Rigid Cylinder 2.1 Flow past a circular cylinder without circulation Consider the superposition of a uniform rectilinear flow and a doublet at the origin. Then, from (7) and (12), the complex potential for the resulting flow field will be F(z) Uz EMBED Equation.DSMT4 . (13) On the circle R a, the value of z is aei(, so that the complex potential on this circle is F(z) Uaei( EMBED Equation.DSMT4 EMBED Equation.DSMT4 . Thus, the value of the stream function on the circle R a is ( EMBED Equation.DSMT4 . For general values of (, ( is clearly variable, but if we choose the strength of the doublet to be ( Ua2, then ( 0 on R a. The flow pattern for this doublet strength is shown in Figure 5(a). For R ( a, the flow field due to the doublet of strength Ua2 and the uniform rectilinear flow of magnitude U give the same flow as that for a uniform flow of magnitude U past a circular cylinder of radius a. The latter flow is shown in Figure 5(b). (a) (b) Figure 5. Then the complex potential for a uniform flow of magnitude U past a circular cylinder of radius a is F(z) U EMBED Equation.DSMT4 . (14) 2.2 Flow past a circular cylinder with circulation If a vortex was added at the origin to the flow around a circular cylinder, as described in the previous section, the fact that the circle R a was a streamline would be unchanged. Thus, from (14) and (11), z0 being zero in the latter, the complex potential for the flow around a circular cylinder with a negative bound vortex around it will be F(z) U EMBED Equation.DSMT4 EMBED Equation.DSMT4 . In order to evaluate the constant c, the value of the stream function on the circle R a will be computed. Then, putting z aei(, the complex potential becomes F(z) EMBED Equation.DSMT4 , F(z) 2Uacos( ( EMBED Equation.DSMT4 . Hence on the circle R a the value of ( is indeed constant, and by choosing c ( EMBED Equation.DSMT4 , the value of this constant will be zero. With this value of c, the complex potential becomes F(z) U EMBED Equation.DSMT4 EMBED Equation.DSMT4 , (15) which describes a uniform rectilinear flow of magnitude U approaching a circular cylinder of radius a that has a negative vortex of strength ( around it. The corresponding velocity components will be evaluated from the complex velocity. W(z) U EMBED Equation.DSMT4 , W(z) EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4 . Hence, by comparison with (6), the velocity components are uR EMBED Equation.DSMT4 , (16) u( EMBED Equation.DSMT4 . (17) On the surface of the cylinder, R a, we have uR 0, u( ( 2U sin ( ( EMBED Equation.DSMT4 . The fact that uR 0 on R a is to be expected, since this is the boundary condition. For this flow field the stagnation points are defined by sin(s ( EMBED Equation.DSMT4 , (18) where (s is the value of ( corresponding to the stagnation point. For ( 0, sin(s 0, so that (s 0 or (, which agree with Figure 5(b), for the circular cylinder without circulation. For nonzero circulation, the value of (s clearly depends upon the magnitude of the parameter EMBED Equation.DSMT4 and it is convenient to discuss (18) for different ranges of this parameter. First, consider the range 0 EMBED Equation.DSMT4 1. Here sin (s 0, so that (s must lie in the third and fourth quadrants. There are two stagnation points. When EMBED Equation.DSMT4 1. Here sin (s ( 1, so that (s EMBED Equation.DSMT4 . The corresponding flow configuration is shown in Figure 6(b). (a) (b) (c) Figure 6. Finally, consider the case where EMBED Equation.DSMT4 1. Since it seems likely that any stagnation points there may be will not lie on the surface of the cylinder. Then, if Rs and (s are the cylindrical coordinates of the stagnation points, it follows from (16) and (17) that Rs and (s must satisfy the equations EMBED Equation.DSMT4 , (19) EMBED Equation.DSMT4 . (20) Since it is assumed that the stagnation points are not on the surface of the cylinder, it follows that Rs ( a, so that (19) requires that (s EMBED Equation.DSMT4 or EMBED Equation.DSMT4 . For these values of (s, (20) becomes EMBED Equation.DSMT4 . (21) The equation for Rs now becomes EMBED Equation.DSMT4 or Rs2 ( EMBED Equation.DSMT4 a2 0. Hence, Rs EMBED Equation.DSMT4 or EMBED Equation.DSMT4 EMBED Equation.DSMT4 , EMBED Equation.DSMT4 EMBED Equation.DSMT4 , where the dots indicate terms of order EMBED Equation.DSMT4 or smaller. So that the coordinates of the stagnation point in the fluid outside the cylinder are (s EMBED Equation.DSMT4 , and EMBED Equation.DSMT4 . (22) This gives a single stagnation point below the surface of the cylinder. The corresponding flow configuration is shown in Figure 6(c). 3. Application of Blasius Formula to the Flow Past a Rigid Cylinder 3.1 Blasius formula The components of the volume flow that pass through this element of surface are also indicated. Then the Newtonian mechanics for the x-direction may be expressed to the following equation ( X ( EMBED Equation.DSMT4 Also, the mass efflux across the element of the surface C0 is ((udy ( vdx), so that the product of this quantity and the x component of velocity, when integrated around the surface C0, gives the net increase in the x-component of momentum. Figure 7. The net external force acting in the positive y direction must equal the net rate of increase of the y component of the momentum yields the equation ( Y EMBED Equation.DSMT4 So that, X EMBED Equation.DSMT4 Y EMBED Equation.DSMT4 . The pressure may be eliminated from these equations by use of the Bernoulli equation, which for the case under consideration, may be written in the form p EMBED Equation.DSMT4 ((u2 v2) B, where B is the Bernoulli constant. Then by eliminating the pressure p, the expressions for X and Y become X EMBED Equation.DSMT4 Y EMBED Equation.DSMT4 where the fact that EMBED Equation.DSMT4 for any constant B around any closed contour C0 has been used. Consider the following complex integral involving the complex velocity W EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4 X ( iY. That is, the complex force X ( iY may be evaluated from X ( iY EMBED Equation.DSMT4 where W(z) is the complex velocity for the flow field and C0 is any closed contour that encloses the body under consideration. The quantity M is the moment acting on the body about its center of gravity. Then, taking clockwise moments as positive, moment equilibrium of the fluid enclosed between C0 and Ci requires that (M EMBED Equation.DSMT4 . The hydrodynamic moment M gives M EMBED Equation.DSMT4 Substituting p B ( EMBED Equation.DSMT4 ((u2 v2), from the Bernoulli equation gives M EMBED Equation.DSMT4 where the fact has been used that EMBED Equation.DSMT4 for any constant B and any closed contour C0. The moment M may be put in the following form, M EMBED Equation.DSMT4 Consider the real part of the following complex integral Re EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4 ( M. That is, the hydrodynamic moment acting on a body is given by M ( EMBED Equation.DSMT4 where W(z) is the complex velocity for the flow field and C0 is any closed contour which encloses the body. 3.2 Force and moment on a circular cylinder without circulation A force exists on a circular cylinder that is immersed in a uniform flow. The magnitude of this force may now be evaluated. The complex potential for a circular cylinder of radius a in a uniform rectilinear flow of magnitude U is F(z) U EMBED Equation.DSMT4 . Then the complex velocity for this flow field is W(z) U EMBED Equation.DSMT4 . Therefore, W2(z) EMBED Equation.DSMT4 . From the Blasius integral law, X ( iY EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4 0. So, Y 0. In order to evaluate the hydrodynamic moment M acting on the cylinder, the quantity zW2 must be evaluated zW2(z) U2z ( EMBED Equation.DSMT4 But from the Blasius integral law M ( EMBED Equation.DSMT4 , M EMBED Equation.DSMT4 0. There is no hydrodynamic moment acting on the cylinder. 3.3 Force and moment on a circular cylinder with circulation The complex potential for a circular cylinder of radius a in a uniform rectilinear flow of magnitude U and having a bound vortex of magnitude ( in the negative direction is F(z) U EMBED Equation.DSMT4 EMBED Equation.DSMT4 . Then the complex velocity for this flow field is W(z) U EMBED Equation.DSMT4 . Therefore, W2(z) EMBED Equation.DSMT4 . But, from the Blasius integral law, X ( iY EMBED Equation.DSMT4 EMBED Equation.DSMT4 . Hence, the residue of W2(z) at z 0 is EMBED Equation.DSMT4 . Then the value of complex force is X ( iY EMBED Equation.DSMT4 ( i(U(. The value of the lift force is Y (U(. In order to evaluate the hydrodynamic moment M acting on the cylinder, the quantity zW2 must be evaluated. From the expression for W2(z) that was established above, zW2(z) EMBED Equation.DSMT4 But from the Blasius integral law M EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4 0. There is no hydrodynamic moment acting on the cylinder. 3.4 Cylinder in a tunnel If F(z) ( ((, z ic cot EMBED Equation.DSMT4 (, then ( ( (( , ( ( ((, and ( is constant when ( constant, while ( decreases by 2(( when we go round one of the circles ( constant. The potential F(z) represents the flow due to the circulation 2(( about a cylinder ( (1 enclosed within a cylinder ( (2 (cylinder in a tunnel). Eliminating (, we obtain z (ic cot EMBED Equation.DSMT4 , F(z) 2 EMBED Equation.DSMT4 the complex velocity for this motion is W EMBED Equation.DSMT4 . Thus, by the Blasius formula, the force on the inner cylinder is given by X ( iY EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4 , X ( iY ( EMBED Equation.DSMT4 . Therefore X ( EMBED Equation.DSMT4 , Y 0, and the resultant fluid thrust therefore tends to increase the distance between the axis of the cylinder and the axis of the tunnel. Figure 8. An interesting case occurs when the radius of the circle (2 becomes infinite, so that this circle coincides with the radical axis. We have then the case of a cylinder whose axis is parallel to the wall. The cylinder is urged towards the wall with the force EMBED Equation.DSMT4 . Figure 9. Drawing the tangent OP, we have, since A is a limiting point, c2 OA2 OP2 OC12 ( a2 h2 ( a2, where h is the distance of the axis from the wall and a is the radius. Hence the force is EMBED Equation.DSMT4 REFERENCES 1 Chorlton, F., Textbook of Fluid Dynamics, D.Van Nostrand Company Ltd., London, 1967. 2 Currie, I.G., Fundamental Mechanics of Fluids, Third Edition, Marcel Dekker, Inc., New York, 2003. 3 Milne-Thomson, L.M., Theoretical Hydrodynamics, Fifth Edition, Macmillan and Co. Ltd., London, 1968. BLASIUS FORMULA AS APPLIED TO THE FLOW PAST A RIGID BOUNDARY Tin Min Naing PhD.(Res)-Math-5 Paper presented for the PhD. Candidature in Mathematics University of Mandalay May, 2017 PAGE PAGE 24 u y O v uR u( x P ( x y O x y ( O x y O y x ( ( ( m m ( ( O x y C0 X Y M udy vdx dx dy x y O A c1 c2 ( ( ( (1 (2 O A c1 ( ( (1 P Zffa)S5.DU. x3 uwh qBDlJtR,bi 4 YI.hG M3PPsUjApkWs2 0gjm2C/2 XNy0 V
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