- Email: [email protected]
Numerical solution of the problem of the rotation of a cylinder in a flow of a viscous incompressible fluid
178
V. A. Lyul’ka REFERENCES
1.
GOL’DENVEIZER, A. L., Theory of elastic thin shells (Teoriya uprugikh tonhkh obolochek), Gostekhizdat, Moscow, 1953.
2.
KLABUKOVA, L. S., An approximate method of solving mixed boundary value problems of the theory of momentless elastic spherical shells. Zh. vj%hisl.Mat. mat. Fiz., 15, 1, 148-162, 1975.
3.
ABRAMOV, A. A., On the transfer of the condition of boundedness for some systems of ordinary linear differential equations. Zh. vghisl. Mat. mat. Fiz., 1,4,733-737, 1961.
4.
ABRAMOV, A. A., On the transfer of boundary conditions for systems of ordinary linear differential equations (a version of the dispersive method). Zh. v&hisl Mat. mat. Fiz., 1, 3,542-545, 1961.
5.
KLABUKOVA, L. S. and STADNIKOVA, N. A., Analysis of a conical body acted upon by a load concentrated at the apex. Zh. vj%hisl.Mat. mat. Fiz., 14,4,955-969, 1974.
NUMERICAL SOLUTION OF THE PROBLEM OF THE ROTATION OF A CYLINDER IN A FLOW OF A VISCOUS INCOMPRESSIBLE FLUID* V. A. LYUL’KA
(Received 23 September
1975)
A NUMERICAL study is made of the Magnus effect for an infinite cylinder. The aerodynamic characteristics of a rotating cylinder are obtained. The formation of a lifting force when a rotating cylinder or sphere moves in a viscous fluid is known as the Magnus effect. To explain this phenomenon the classical theory of the circulatory flow of an ideal fluid round the cylinder is usually used [ 1,2,3] . This theory connects the formation of the lifting force with the value of the circulation of velocity existing in the fluid, and within certain limits qualitatively gives correctly the direction of the lifting force acting on a rotating body in a flow [3]. To obtain the value of the lifting force Bernoulli’s theorem on the contour of the cylinder is used and the pressure distribution on it is calculated. After this the lifting force acting on the cylinder is found [l] . To explain the formation of the lifting force and to estimate its value for the rotation of a body in a flow of an incompressible viscous fluid, described by the Navier-Stokes equations, the classical theory of circulatory flow is useless, since Bernoulli’s theorem is not satisfied in the case of the flow of a viscous fluid and there is no way of connecting the circulatory velocity with the angular velocity of rotation of the cylinder [2,3]. It must also be added that the theory of ideal streamlining does not take into account the contribution of the viscosity forces to the lifting force, which for low Reynolds numbers, according to our data, may be extremely significant - more than 20%. Another deficiency of the classical theory is the fact that the resisting force acting on the rotating cylinder in the flow is assumed to equal zero, and for the flow past a rotating cylinder of a viscous fluid it is of course non-zero. In this paper the problem of the rotation of a circular cylinder in the flow of a viscous incompressible fluid, possessing a constant velocity at infinity and directed along the x-axis, is solved numerically for Reynolds numbers Re = 1, 5,7.5, 10 and angular velocities in the interval from 0 to 5. The angular velocity was measured in rad/sec. Previously this problem was solved in [4] for small velocities of rotation of the cylinder and in [5] . A non-monotonic scheme was used in the latter paper with the consequence that the results for a cylinder at rest differed by 30-50% from the results of other authors. For a rotating sphere the similar problem was solved analytically for low Reynolds numbers in [6] . *Zh. vj%hisl Mat. mat. Fiz., 17,2,470-480,
1977.
Rotation of a cylinder in a flow of a viscous incompressible fluid
179
1. Statement of the problem and the method of solution We consider a cylinder of radius a = 1 rotating with angular velocity $2 in an incident flow of a viscous incompressible fluid. We take the system of Naviet-Stokes equations, describing the motion of a viscous ~compressible fluid, in the form expressed in terms of the stream function $ and the vortex of velocity o: do _d..p+ at
a+ do (___%E dy ax dy
dx
1,
A$=-0.
(1)
Here Re=atr,/p is the Reynolds number, A is Laplace’s operator, urn is the velocity of the flow at infmity, and 1-1is the viscosity of the fluid. We will change to cylindrical logarithmic coordinates by the formulas x=65+ cos rrrt. where $==n-’ In (&ty’)“,
O
O
y=eXE sin ztt),
G9
Then Eq. (1) becomes
(3) Here A=~2/~~2+P/&~z is the Laplace operator in 6 and n coordinates. For the numerical solution the infinite domain is replaced by an annular domain with internal radius a = 1 and external radius R (in our calculations R=cz’.~~--111.3). On transformation of the coordinates by formulas (2) the annular domain becomes the rectangle O&r@!, Often-’ In R. We determine the boundary values for the functions $ and w on the boundary of the rectangular domain. For g=O the value of the stream function $ is unknown and must be found during the solution; we describe the method of fiding it below. The value of the velocity vortex on this part of the boundary is obtained by using the same method as was used to obtain the local boundary condition (see [7] ):
Here A!$ is the step of the difference mesh along the .&xis. For r) = 0 and 2 it is necessary to satisfy the periodicity conditions for the functions Q[ E,
01=4EE,21 $=e”& sin 3rq
and w[E, O]=o[& 21. For t=n-’ and o = 0 are established.
In R the conditions of uniform flow
To determine the values of the stream function on the surface of the cylinder at w = 0 we use the pressure-periodicity condition, that is, p [ 0,01 = p IO, 2 1. Solving the problem completely for a definite specified value of the stream function on the surface of the cylinder, we determine. at the points with coordinates (0,O) and (0,2) the values of the pressure, which in general will be different. Giving a small increment e to the stream function
180
V.A. Lyul’ka
on the surface and again solving the problem completely, we obtain new values of the pressure at the same points. After this we use Newton’s method of tangents as was done in [8]. The new value of the stream function on the surface of the cylinder will be
where Go is the initial value of the stream function for E = 0, E is the increment of the value of the stream function for t = 0, qPO is the difference between the values of the pressure at the points with coordinates (0,2) and (0,O) for $[ 0,111 =I&, Apt isthe difference between the values of the pressure at the same points for I+[0, ~1 =go-t-a, and 0.25G~GO.5 is a coefficient introduced for the convergence of the method. This process of finding the values of the stream function on the boundary was continued until the inequality 1pf0, 01 -p[O, 21 1 G.l.005.was satisfied. In [4] a different method of finding the boundary value of the stream function was used. An implicit symmetrical scheme of the method of alternating directions was used to approximate the differenti~ equations (3) by difference equations. ‘Themesh steps taken were Az=O.O5 and Aq=O.O5 for 30 X 40 mesh points. To obtain a numerical solution of system (3) a “non-stationary method” was used; at each time step the following systems of agebraic equations were solved:
(4)
at the first half-step and
at the second half-step, where 6’/@“, fi’lfirl’. h/6& S/fir are the difference analogs of the second and first derivatives of the corresponding functions, expressed in terms of central differences. On each time layer the systems were solved by pivotal ~ndensation, at this time step T and t were in general taken to be different. On the stationary solution this scheme approximates the Navier-Stokes equations with accuracy U(h2).
Rotation of a cylinder in a flow of a viscous incompressible jluid
181
2. Results of the calculations In the numerical
calculation
it was observed that the results obtained
depend on the accuracy
with which the difference equations
(4) and (5) are solved. In our calculations
such that the maximum
in these equations
discrepancy
greater degree the results of the calculations,
the accuracy was
did not exceed 2.5 X 10P4. To a much
and in particular,
the drag coefficient
depended on
R: Cx for Re = 1 varied by 10% when R changed from 20 to 111 bores.
5
FIG. 1.
FIG. 2.
X
FIG. 3. Figures 1 and 2 show the flow fields obtained by numerical calculation for Re = 1 and 10, and R = 1 and 3. The form of the stream for viscous flow differs essentially from ideal flow past a rotating cylinder [l] . For any angular velocity of rotation of the cylinder, as a consequence of the adhesion of the viscous fluid there exists a layer of fluid rotating together with the cylinder. The fact of the vanishing of the stagnation zone behind the cylinder for Re = 10 and a = 1 and 3, although for Re = 10 and R = 0 the stagnation zone existed, and should be noted, see Fig. 3.
182
V. A. Lyulka
In [4] the stagnation zone was detected for Re = 20 and 52 = 0.025. It is possible that at larger Reynolds numbers and the same velocities of rotation the trace behind the cylinder also will not vanish.
FIG. 4.
FIG. 5.
The layer of fluid rotating together with the cylinder is contained between the mean stream line (see Figs. 1 and 2) and the cylinder, The mean stream line is a line of separation: the stream lines which begin at infinity above it pass round the cylinder above it, and those which begin below pass round the cylinder below it. Within the layer of fluid rotating with the cylinder the stream lines are closed. For a given Re the thickness of this layer increases as the speed of rotation increases and decreases as Re increases for a given speed of rotation. The absence of symmetry of the flow with respect to the planes Oxy and Oyz is striking. In the circulatory flow of an ideal fluid round a cylinder there is symmetry of the flow about the plane Oyz [l] . The absence of symmetry of the flow with respect to these planes is the cause of the formation of the lifting force and the resisting force acting on the cylinder. The most important effect on the rotation of a cylinder in the flow of a viscous fluid is the formation of the lifting force acting on the cylinder. The coefficient of the lifting force is calculated by the formula
=c,,+c,,. The first term of the sum is the contribution of the pressure forces to the lifting force, and the second is the contribution of the viscous forces. Here - (do/%) CT and -mCT are the derivative of the vortex and the vortex at the surface of the cylinder respectively.
Rotation of a cylinder in a frow of a viscous i~co~pre~~bie j&id
183
Figure 4 shows the relations between the modulus of the lifting force coefficient and n for various Reynolds numbers. It is seen from the graphs that the coefficient of the lifting force increases steadily as the angular velocity of rotation increases for all the Re numbers studied. The value of the coefficient increases as the Reynolds number increases for a fixed angular velocity of rotation of the cylinder. The sign of the lifting force is the same as the sign of the force given by the classical theory of the circulation ffow round the cylinder, The greater contribution to the value of the lifting force is given by the pressure forces Cyp, appro~mately 77% for Re = 1 and 90% for Re = 10. It is characteristic that Cvr, the projection of the viscous forces on the y-axis, has the same sign as the contribution of the pressure forces to the lifting force in the Re range studied. It is interesting to note that the contribution of the viscous forces to the lifting force decreases as the Reynolds number increases in comparison with the contribution of the pressure forces. Table 1 gives the values of Cyp and CY,.for the given Re and various angular velocities. TABLE 1
1
2
1
3.2 1.1
4.5
1.9
7.3 2.7
.5
” 1 iii
4.7 0.9
7.5 I.4
7.5
2.1 cf.3
4.6 0.7
7.5 1.3
4. 1 0..a ”
45 0.6
i.3 1.n
IO
4
5
10.3 3.4
43.5 4.0
TABLE 2
RC
C.Y
cXP
3.51 3.39
3.41 3.32
3.23 3.42
2.77 3.89
5
2::
1.58 1.18
I.36 1.28
1.37
t .4-i
I.‘7 1.6
7.5
CX,, c,,
I.$4 t.04
I.46 1.04
1.46 1.l.T
1.7
c,,,
I.31
1.37
1.54 1.01
2.1.; 1.18
1
IO
cx,
C \T
(i.ST
0.89
I.33
2.23 3.95
125 4.18
184
% A. Lyul'ka
For a rotating sphere in the flow of a viscous fluid, by (61, the lifting force depends linearly on the angular velocity of rotation and is ~de~ndent of the viscosity of the fluid. In contrary to the theory of the ideal flow round a cylinder the drag force acting on a rotating cylinder in a viscous fluid is non-zero. The relation between the drag coefficient C, of the cylinder and the velocity of rotation for various Re is shown in Fig. 5. The value of C, was calculated by the formula
The first term of the sum is the part of the drag force due to the pressure forces, and the second term is the part of the drag force due to the viscous forces. The deviation of the calculated values of C, in this paper from the results of other authors for 52,= 0 does not exceed 6% ]7,9, IO]. It is obvious from Fig. 5 that C’ decreases as the velocity of rotation increases for Re = I, and for Re = 5,7.5, 10 the drag coefficient increases as the velocity of rotation increases over the range of variation of 51investigated. We studied in greater detail the behaviour of C, in the range of velocities of rotation from 0 to 1 for given Reynolds numbers. It was established that here the nature of the behaviour of C, is the same as.for the whole curve, that is, C, for Re = 1 decreases as 52increases, and for Re = 7.5 and 10 it increases as Sz increases. The decrease of the drag coefficient for Re = 1 as the angular velocity of rotation increases is unexpected, since when a cylinder rotates in a viscous fluid there is a layer rotating with the cylinder, which leads to an increase of the effective cross-section of the cylinder, and should lead to an increase and not a decrease in the drag coefficient. As seen from Table 2, for Re = 1 there is a decrease in C, due to a decreased cont~bution of the pressure forces to the drag force. For Re = 5 the value of CXPalso decreases, and C,, increases, the drag coefficient of the cylinder slowly increasing with increase of the angular velocity of rotation of the cylinder. It is possible that over the range of Re numbers from 1 to 5 the drag coefficient does not behave as a monotonic function of the velocity of rotation of the cylinder; however we have not studied this range. For Re = 7.5 and 10 and velocities of rotation from 0 to 3 the nature of the behaviour of the drag coeffcient is the same: it increases as the angular velocity of rotation of the cylinder increases. Here both CXPand C,. increase simultaneously. In [6], where an analytic solution of the problem of the rotation of a sphere in a viscous fluid is given, an expression is obtained for the drag coefficient which is independent of the speed of rotation and increases linearly with the Reynolds number. The variation of the value of the moment acting on a cylinder from the viscous fluid, as a function of s1, is shown in Fig. 6. The value of the moment was calculated by the formula
Rotation of a cylinder in a flow of a viscous incompressible fluid
185
The value of the moment obtained in our paper is a strictly linear function of Cl and is inversely proportional to Re.
FIG. 6.
FIG. 7.
In [6] for a rotating sphere it was shown that the moment acting on the sphere is also directly proportional to the angular velocity of rotation and inversely proportional to Re. Figures 7 and 8 give the distribution of the vortex velocity A$= --w on the surface of the cylinder for various angular velocities of rotation and Re = 1 and 7.5 respectively. For a velocity of rotation of the cylinder equal to zero the distribution of the vorticity agrees well with the results of [9, lo]. The pressure distribution over the surface of the cylinder is given in Figs. 9 and 10 for various values of S2and Re = 1 and 7.5 respectively. The pressure was calculated by the formula
where p. is the pressure at the trailing stagnation point. For 52=0 the pressure distribution agrees well with the results of [9]. It is seen from the graphs that the behaviour of the pressure as the angular velocity increases is the same for various values of Re: the maximum and minimum pressures are increased in absolute value, the maximum and minimum values and also the value of the pressure equal to zero being attained at approximately identical points of the cylinder in the range of Re investigated.
186
K A.
plO,eJ 5
t
Q
-5 I
IFIG. 8.
FIG. 10.
FIG. 9.
FIG. 11.
9
II
-5
-‘!l
Since in a viscous fluid a large part of the value of the lifting force is due to the pressure forces, it is interesting to compare the pressure distribution given by the classical theory of the circulatory flow past a cylinder with the pressure distribution obtained in our paper for the same value of the circulation of velocity in the neighborhood of the cylinder. Figure 11 gives the pressure distribution
187
Rotation of a cylinder in a flow of a viscous incompressible fluid
on the surface of the cylinder for an ideal flow with the velocity circulation I’ = 6n, for an angular velocity of rotation of the cylinder equal to 3, in a viscous flow, and for various Re, s1= 3. For viscous flow the pressure was calculated by Eq. (6), and for ideal flow by the formula
It is seen from the graph that the pressure distribution in an ideal fluid differs from the pressure distribution in a viscous fluid and hence the theory of the ideal streamlining of a cylinder can only very crudely predict the value of the lifting force. It must also be mentioned that the curve of pressures corresponding to Re = 10 and 52 = 3, is situated practically between the curves giving the pressure distribution for Re = 1 and 5. This kind of behaviour of the pressure is not understood.
3. Conclusion The problem of the steady motion of a rotating cylinder is reduced to the problem of the flow around the cylinder with boundary conditions on the boundary of the cylinder different from zero. Therefore, in this problem we have the case of the motion of a solid for which there exists an interaction between two degrees of freedom, via a non-linear medium whose motion is described by the Navier-Stokes equations. From a consideration of the results obtained it is obvious that this motion possesses a much greater variety of phenomena than the simple flow past a cylinder. Of the most interesting phenomena of the flow around a rotating cylinder we must mention the behaviour of the drag coefficient of the cylinder as a function of 0 for various values of Re (Fig. 5), and also the coefficient of the lifting force (Fig. 4). The classical theory of the circulatory flow round a cylinder cannot explain, as already indicated, either the behaviour of C,, or that of Cr. The theory of these phenomena can be constructed if it is possible to find the functionals which would “measure” the asymmetry of the flow with respect to the planes Oyz and Oxz, since the formation of the aerodynamic forces is connected with it. It is interesting to note that the quantity
(the integral is taken over the surface of the cylinder) to a high degree of accuracy remains constant in the range of angular velocities of rotation and Reynolds numbers investigated. We emphasize also that the behaviour of the pressure for the same velocity of rotation, but different Reynolds numbers (Fig. 11) is not understood, since the curve of pressures for Re = 10 lies practically between the curves for Re = 1 and 5. It is extremely probable that in the range of velocities of rotation and Reynolds numbers investigated transitional modes exist which require a more detailed study. In conclusion we point out that the calculation of one version on the BESM-6 computer took from 2 to 8 hours. Translated by
J. Berry.
if. I. Avertsev and Yti. I, ~5k~R
188
REFEREN&ES 1‘
LOiTSY ANSKII, L. G., ~ec~a~~cs of a it&id and gas (Me~~~a 1959.
2.
BATCHELOR, G., Introduction to the dynamicsof a liquid (Vvedenie v dinamiku zhidkosti), “Mir”, Moscow, 1973.
3.
BIRKHOFF, G.,~yd~dynam~~s ~G~drod~arn~a~, fid-vo in. lit., Moscow, 1963.
4.
SfMUNI, L. M., Sofution of some problems of the motion of a V~SCWS &id, connectedwith the Bow around a cyimder and sphere. Izv. SO Akud. Nauk SSSR. Ser. Tekhn., 8,2,X3-27,196?.
5.
KORYAVOV. P. P. and PAVLQVSKII, Yu. N., Numerical solution of the problem of the motion of a circular cylinder in the flow of a viscous fluid. In: Problems of applied mathematicsand tiechanics (Probl. prikl. matem. i mekhan.) 247-262, “Nauka”, Moscow, 1971.
6.
RUBINOV, S. I. and KELLER, J. B., The transverse force on a spinning sphere moving in a viscous fluid. J. Fl~~d~ec~., 1, ?,447-459, 1961.
7.
DENNIS, S. C. R. and CHANG, GAU-ZU., Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100, J. ~l~~~e~~., 42,3,471-489, 1970.
8.
LYUL’KA, V. A., The ~teraction of solid particles dragged by a viscous ~~ornpre~jbIe Mat. mat. Fit., fO, 4, I&%4--1049,197O.
9.
KELLER, N. B. and TAKAMI, H., Numerical studies of steady viscous flow about cylinders. In: Numet. solutions of nonlinear differential equations, 115-140, Wiley, New York, 1966.
zhidkosti i gaza), Fizmatgiz, Moscow,
Liquid. 2!r. vj&isi
10. BABENKO, K. I., VVEDENSKAYA, N. D. and ORLOVA, M. G., Results of the calculation of the flow of a viscous fluid around an infinite circular cylinder. Preprint No. 38, IPM, Moscow, 1971.
AL~O~TH~
FOR THE CONTROL OF ~R~~~~
A~T~~ATA~
V, 1.AVERTSEVand Yu. I. MOKIN Moscow (Received 19 September 1975)
AN ANALYTIC expression is obtained for the area of a small contact area. An approximate formula for estimating its size is given. An asymptotic formula is derived for the range D of control of the drive of a grinding wheef, suitable for engineering cakufations. In the process of grinding in mass production one of the chief problems is the st~~isation of the modes of operation of machine tools in the periods between readjustments. In the grinding of bearing rings scores of items are ground by one grinding wheel. The diameter of a grinding wheel is reduced to approximately half in the processing of a batch of rings, which causes a decrease of the area S of the small contact area. Until recently the variation of the area of contact of the wheel with the component has been disregarded, and it has had a considerable influence on the temperature state of the grinding, the durability of the wheel, the quality of the processing of the component etc. At the present time the question has been posed of the introduction of automatic control systems for controlling grinding machines f 1,2] f ensuring stable quality of component processing and high produ~ti~ty inde~~dently of the degree of wear of the wheel. Therefore an urgent problem is the determination of the size of the small contact area for various grinding methods and various profiles of the grinding wheel. A valid estimate of 5’is of great value for the mathematical description of the grinding process allowing for heat effects and the conditions of the shaping of the macrogeometry and microgeometry of the components.
5%. ~j&isI.Mat. mat. Fis., 17, 2,48f-489,
1977.
V. A. Lyul’ka REFERENCES
1.
GOL’DENVEIZER, A. L., Theory of elastic thin shells (Teoriya uprugikh tonhkh obolochek), Gostekhizdat, Moscow, 1953.
2.
KLABUKOVA, L. S., An approximate method of solving mixed boundary value problems of the theory of momentless elastic spherical shells. Zh. vj%hisl.Mat. mat. Fiz., 15, 1, 148-162, 1975.
3.
ABRAMOV, A. A., On the transfer of the condition of boundedness for some systems of ordinary linear differential equations. Zh. vghisl. Mat. mat. Fiz., 1,4,733-737, 1961.
4.
ABRAMOV, A. A., On the transfer of boundary conditions for systems of ordinary linear differential equations (a version of the dispersive method). Zh. v&hisl Mat. mat. Fiz., 1, 3,542-545, 1961.
5.
KLABUKOVA, L. S. and STADNIKOVA, N. A., Analysis of a conical body acted upon by a load concentrated at the apex. Zh. vj%hisl.Mat. mat. Fiz., 14,4,955-969, 1974.
NUMERICAL SOLUTION OF THE PROBLEM OF THE ROTATION OF A CYLINDER IN A FLOW OF A VISCOUS INCOMPRESSIBLE FLUID* V. A. LYUL’KA
(Received 23 September
1975)
A NUMERICAL study is made of the Magnus effect for an infinite cylinder. The aerodynamic characteristics of a rotating cylinder are obtained. The formation of a lifting force when a rotating cylinder or sphere moves in a viscous fluid is known as the Magnus effect. To explain this phenomenon the classical theory of the circulatory flow of an ideal fluid round the cylinder is usually used [ 1,2,3] . This theory connects the formation of the lifting force with the value of the circulation of velocity existing in the fluid, and within certain limits qualitatively gives correctly the direction of the lifting force acting on a rotating body in a flow [3]. To obtain the value of the lifting force Bernoulli’s theorem on the contour of the cylinder is used and the pressure distribution on it is calculated. After this the lifting force acting on the cylinder is found [l] . To explain the formation of the lifting force and to estimate its value for the rotation of a body in a flow of an incompressible viscous fluid, described by the Navier-Stokes equations, the classical theory of circulatory flow is useless, since Bernoulli’s theorem is not satisfied in the case of the flow of a viscous fluid and there is no way of connecting the circulatory velocity with the angular velocity of rotation of the cylinder [2,3]. It must also be added that the theory of ideal streamlining does not take into account the contribution of the viscosity forces to the lifting force, which for low Reynolds numbers, according to our data, may be extremely significant - more than 20%. Another deficiency of the classical theory is the fact that the resisting force acting on the rotating cylinder in the flow is assumed to equal zero, and for the flow past a rotating cylinder of a viscous fluid it is of course non-zero. In this paper the problem of the rotation of a circular cylinder in the flow of a viscous incompressible fluid, possessing a constant velocity at infinity and directed along the x-axis, is solved numerically for Reynolds numbers Re = 1, 5,7.5, 10 and angular velocities in the interval from 0 to 5. The angular velocity was measured in rad/sec. Previously this problem was solved in [4] for small velocities of rotation of the cylinder and in [5] . A non-monotonic scheme was used in the latter paper with the consequence that the results for a cylinder at rest differed by 30-50% from the results of other authors. For a rotating sphere the similar problem was solved analytically for low Reynolds numbers in [6] . *Zh. vj%hisl Mat. mat. Fiz., 17,2,470-480,
1977.
Rotation of a cylinder in a flow of a viscous incompressible fluid
179
1. Statement of the problem and the method of solution We consider a cylinder of radius a = 1 rotating with angular velocity $2 in an incident flow of a viscous incompressible fluid. We take the system of Naviet-Stokes equations, describing the motion of a viscous ~compressible fluid, in the form expressed in terms of the stream function $ and the vortex of velocity o: do _d..p+ at
a+ do (___%E dy ax dy
dx
1,
A$=-0.
(1)
Here Re=atr,/p is the Reynolds number, A is Laplace’s operator, urn is the velocity of the flow at infmity, and 1-1is the viscosity of the fluid. We will change to cylindrical logarithmic coordinates by the formulas x=65+ cos rrrt. where $==n-’ In (&ty’)“,
O
O
y=eXE sin ztt),
G9
Then Eq. (1) becomes
(3) Here A=~2/~~2+P/&~z is the Laplace operator in 6 and n coordinates. For the numerical solution the infinite domain is replaced by an annular domain with internal radius a = 1 and external radius R (in our calculations R=cz’.~~--111.3). On transformation of the coordinates by formulas (2) the annular domain becomes the rectangle O&r@!, Often-’ In R. We determine the boundary values for the functions $ and w on the boundary of the rectangular domain. For g=O the value of the stream function $ is unknown and must be found during the solution; we describe the method of fiding it below. The value of the velocity vortex on this part of the boundary is obtained by using the same method as was used to obtain the local boundary condition (see [7] ):
Here A!$ is the step of the difference mesh along the .&xis. For r) = 0 and 2 it is necessary to satisfy the periodicity conditions for the functions Q[ E,
01=4EE,21 $=e”& sin 3rq
and w[E, O]=o[& 21. For t=n-’ and o = 0 are established.
In R the conditions of uniform flow
To determine the values of the stream function on the surface of the cylinder at w = 0 we use the pressure-periodicity condition, that is, p [ 0,01 = p IO, 2 1. Solving the problem completely for a definite specified value of the stream function on the surface of the cylinder, we determine. at the points with coordinates (0,O) and (0,2) the values of the pressure, which in general will be different. Giving a small increment e to the stream function
180
V.A. Lyul’ka
on the surface and again solving the problem completely, we obtain new values of the pressure at the same points. After this we use Newton’s method of tangents as was done in [8]. The new value of the stream function on the surface of the cylinder will be
where Go is the initial value of the stream function for E = 0, E is the increment of the value of the stream function for t = 0, qPO is the difference between the values of the pressure at the points with coordinates (0,2) and (0,O) for $[ 0,111 =I&, Apt isthe difference between the values of the pressure at the same points for I+[0, ~1 =go-t-a, and 0.25G~GO.5 is a coefficient introduced for the convergence of the method. This process of finding the values of the stream function on the boundary was continued until the inequality 1pf0, 01 -p[O, 21 1 G.l.005.was satisfied. In [4] a different method of finding the boundary value of the stream function was used. An implicit symmetrical scheme of the method of alternating directions was used to approximate the differenti~ equations (3) by difference equations. ‘Themesh steps taken were Az=O.O5 and Aq=O.O5 for 30 X 40 mesh points. To obtain a numerical solution of system (3) a “non-stationary method” was used; at each time step the following systems of agebraic equations were solved:
(4)
at the first half-step and
at the second half-step, where 6’/@“, fi’lfirl’. h/6& S/fir are the difference analogs of the second and first derivatives of the corresponding functions, expressed in terms of central differences. On each time layer the systems were solved by pivotal ~ndensation, at this time step T and t were in general taken to be different. On the stationary solution this scheme approximates the Navier-Stokes equations with accuracy U(h2).
Rotation of a cylinder in a flow of a viscous incompressible jluid
181
2. Results of the calculations In the numerical
calculation
it was observed that the results obtained
depend on the accuracy
with which the difference equations
(4) and (5) are solved. In our calculations
such that the maximum
in these equations
discrepancy
greater degree the results of the calculations,
the accuracy was
did not exceed 2.5 X 10P4. To a much
and in particular,
the drag coefficient
depended on
R: Cx for Re = 1 varied by 10% when R changed from 20 to 111 bores.
5
FIG. 1.
FIG. 2.
X
FIG. 3. Figures 1 and 2 show the flow fields obtained by numerical calculation for Re = 1 and 10, and R = 1 and 3. The form of the stream for viscous flow differs essentially from ideal flow past a rotating cylinder [l] . For any angular velocity of rotation of the cylinder, as a consequence of the adhesion of the viscous fluid there exists a layer of fluid rotating together with the cylinder. The fact of the vanishing of the stagnation zone behind the cylinder for Re = 10 and a = 1 and 3, although for Re = 10 and R = 0 the stagnation zone existed, and should be noted, see Fig. 3.
182
V. A. Lyulka
In [4] the stagnation zone was detected for Re = 20 and 52 = 0.025. It is possible that at larger Reynolds numbers and the same velocities of rotation the trace behind the cylinder also will not vanish.
FIG. 4.
FIG. 5.
The layer of fluid rotating together with the cylinder is contained between the mean stream line (see Figs. 1 and 2) and the cylinder, The mean stream line is a line of separation: the stream lines which begin at infinity above it pass round the cylinder above it, and those which begin below pass round the cylinder below it. Within the layer of fluid rotating with the cylinder the stream lines are closed. For a given Re the thickness of this layer increases as the speed of rotation increases and decreases as Re increases for a given speed of rotation. The absence of symmetry of the flow with respect to the planes Oxy and Oyz is striking. In the circulatory flow of an ideal fluid round a cylinder there is symmetry of the flow about the plane Oyz [l] . The absence of symmetry of the flow with respect to these planes is the cause of the formation of the lifting force and the resisting force acting on the cylinder. The most important effect on the rotation of a cylinder in the flow of a viscous fluid is the formation of the lifting force acting on the cylinder. The coefficient of the lifting force is calculated by the formula
=c,,+c,,. The first term of the sum is the contribution of the pressure forces to the lifting force, and the second is the contribution of the viscous forces. Here - (do/%) CT and -mCT are the derivative of the vortex and the vortex at the surface of the cylinder respectively.
Rotation of a cylinder in a frow of a viscous i~co~pre~~bie j&id
183
Figure 4 shows the relations between the modulus of the lifting force coefficient and n for various Reynolds numbers. It is seen from the graphs that the coefficient of the lifting force increases steadily as the angular velocity of rotation increases for all the Re numbers studied. The value of the coefficient increases as the Reynolds number increases for a fixed angular velocity of rotation of the cylinder. The sign of the lifting force is the same as the sign of the force given by the classical theory of the circulation ffow round the cylinder, The greater contribution to the value of the lifting force is given by the pressure forces Cyp, appro~mately 77% for Re = 1 and 90% for Re = 10. It is characteristic that Cvr, the projection of the viscous forces on the y-axis, has the same sign as the contribution of the pressure forces to the lifting force in the Re range studied. It is interesting to note that the contribution of the viscous forces to the lifting force decreases as the Reynolds number increases in comparison with the contribution of the pressure forces. Table 1 gives the values of Cyp and CY,.for the given Re and various angular velocities. TABLE 1
1
2
1
3.2 1.1
4.5
1.9
7.3 2.7
.5
” 1 iii
4.7 0.9
7.5 I.4
7.5
2.1 cf.3
4.6 0.7
7.5 1.3
4. 1 0..a ”
45 0.6
i.3 1.n
IO
4
5
10.3 3.4
43.5 4.0
TABLE 2
RC
C.Y
cXP
3.51 3.39
3.41 3.32
3.23 3.42
2.77 3.89
5
2::
1.58 1.18
I.36 1.28
1.37
t .4-i
I.‘7 1.6
7.5
CX,, c,,
I.$4 t.04
I.46 1.04
1.46 1.l.T
1.7
c,,,
I.31
1.37
1.54 1.01
2.1.; 1.18
1
IO
cx,
C \T
(i.ST
0.89
I.33
2.23 3.95
125 4.18
184
% A. Lyul'ka
For a rotating sphere in the flow of a viscous fluid, by (61, the lifting force depends linearly on the angular velocity of rotation and is ~de~ndent of the viscosity of the fluid. In contrary to the theory of the ideal flow round a cylinder the drag force acting on a rotating cylinder in a viscous fluid is non-zero. The relation between the drag coefficient C, of the cylinder and the velocity of rotation for various Re is shown in Fig. 5. The value of C, was calculated by the formula
The first term of the sum is the part of the drag force due to the pressure forces, and the second term is the part of the drag force due to the viscous forces. The deviation of the calculated values of C, in this paper from the results of other authors for 52,= 0 does not exceed 6% ]7,9, IO]. It is obvious from Fig. 5 that C’ decreases as the velocity of rotation increases for Re = I, and for Re = 5,7.5, 10 the drag coefficient increases as the velocity of rotation increases over the range of variation of 51investigated. We studied in greater detail the behaviour of C, in the range of velocities of rotation from 0 to 1 for given Reynolds numbers. It was established that here the nature of the behaviour of C, is the same as.for the whole curve, that is, C, for Re = 1 decreases as 52increases, and for Re = 7.5 and 10 it increases as Sz increases. The decrease of the drag coefficient for Re = 1 as the angular velocity of rotation increases is unexpected, since when a cylinder rotates in a viscous fluid there is a layer rotating with the cylinder, which leads to an increase of the effective cross-section of the cylinder, and should lead to an increase and not a decrease in the drag coefficient. As seen from Table 2, for Re = 1 there is a decrease in C, due to a decreased cont~bution of the pressure forces to the drag force. For Re = 5 the value of CXPalso decreases, and C,, increases, the drag coefficient of the cylinder slowly increasing with increase of the angular velocity of rotation of the cylinder. It is possible that over the range of Re numbers from 1 to 5 the drag coefficient does not behave as a monotonic function of the velocity of rotation of the cylinder; however we have not studied this range. For Re = 7.5 and 10 and velocities of rotation from 0 to 3 the nature of the behaviour of the drag coeffcient is the same: it increases as the angular velocity of rotation of the cylinder increases. Here both CXPand C,. increase simultaneously. In [6], where an analytic solution of the problem of the rotation of a sphere in a viscous fluid is given, an expression is obtained for the drag coefficient which is independent of the speed of rotation and increases linearly with the Reynolds number. The variation of the value of the moment acting on a cylinder from the viscous fluid, as a function of s1, is shown in Fig. 6. The value of the moment was calculated by the formula
Rotation of a cylinder in a flow of a viscous incompressible fluid
185
The value of the moment obtained in our paper is a strictly linear function of Cl and is inversely proportional to Re.
FIG. 6.
FIG. 7.
In [6] for a rotating sphere it was shown that the moment acting on the sphere is also directly proportional to the angular velocity of rotation and inversely proportional to Re. Figures 7 and 8 give the distribution of the vortex velocity A$= --w on the surface of the cylinder for various angular velocities of rotation and Re = 1 and 7.5 respectively. For a velocity of rotation of the cylinder equal to zero the distribution of the vorticity agrees well with the results of [9, lo]. The pressure distribution over the surface of the cylinder is given in Figs. 9 and 10 for various values of S2and Re = 1 and 7.5 respectively. The pressure was calculated by the formula
where p. is the pressure at the trailing stagnation point. For 52=0 the pressure distribution agrees well with the results of [9]. It is seen from the graphs that the behaviour of the pressure as the angular velocity increases is the same for various values of Re: the maximum and minimum pressures are increased in absolute value, the maximum and minimum values and also the value of the pressure equal to zero being attained at approximately identical points of the cylinder in the range of Re investigated.
186
K A.
plO,eJ 5
t
Q
-5 I
IFIG. 8.
FIG. 10.
FIG. 9.
FIG. 11.
9
II
-5
-‘!l
Since in a viscous fluid a large part of the value of the lifting force is due to the pressure forces, it is interesting to compare the pressure distribution given by the classical theory of the circulatory flow past a cylinder with the pressure distribution obtained in our paper for the same value of the circulation of velocity in the neighborhood of the cylinder. Figure 11 gives the pressure distribution
187
Rotation of a cylinder in a flow of a viscous incompressible fluid
on the surface of the cylinder for an ideal flow with the velocity circulation I’ = 6n, for an angular velocity of rotation of the cylinder equal to 3, in a viscous flow, and for various Re, s1= 3. For viscous flow the pressure was calculated by Eq. (6), and for ideal flow by the formula
It is seen from the graph that the pressure distribution in an ideal fluid differs from the pressure distribution in a viscous fluid and hence the theory of the ideal streamlining of a cylinder can only very crudely predict the value of the lifting force. It must also be mentioned that the curve of pressures corresponding to Re = 10 and 52 = 3, is situated practically between the curves giving the pressure distribution for Re = 1 and 5. This kind of behaviour of the pressure is not understood.
3. Conclusion The problem of the steady motion of a rotating cylinder is reduced to the problem of the flow around the cylinder with boundary conditions on the boundary of the cylinder different from zero. Therefore, in this problem we have the case of the motion of a solid for which there exists an interaction between two degrees of freedom, via a non-linear medium whose motion is described by the Navier-Stokes equations. From a consideration of the results obtained it is obvious that this motion possesses a much greater variety of phenomena than the simple flow past a cylinder. Of the most interesting phenomena of the flow around a rotating cylinder we must mention the behaviour of the drag coefficient of the cylinder as a function of 0 for various values of Re (Fig. 5), and also the coefficient of the lifting force (Fig. 4). The classical theory of the circulatory flow round a cylinder cannot explain, as already indicated, either the behaviour of C,, or that of Cr. The theory of these phenomena can be constructed if it is possible to find the functionals which would “measure” the asymmetry of the flow with respect to the planes Oyz and Oxz, since the formation of the aerodynamic forces is connected with it. It is interesting to note that the quantity
(the integral is taken over the surface of the cylinder) to a high degree of accuracy remains constant in the range of angular velocities of rotation and Reynolds numbers investigated. We emphasize also that the behaviour of the pressure for the same velocity of rotation, but different Reynolds numbers (Fig. 11) is not understood, since the curve of pressures for Re = 10 lies practically between the curves for Re = 1 and 5. It is extremely probable that in the range of velocities of rotation and Reynolds numbers investigated transitional modes exist which require a more detailed study. In conclusion we point out that the calculation of one version on the BESM-6 computer took from 2 to 8 hours. Translated by
J. Berry.
if. I. Avertsev and Yti. I, ~5k~R
188
REFEREN&ES 1‘
LOiTSY ANSKII, L. G., ~ec~a~~cs of a it&id and gas (Me~~~a 1959.
2.
BATCHELOR, G., Introduction to the dynamicsof a liquid (Vvedenie v dinamiku zhidkosti), “Mir”, Moscow, 1973.
3.
BIRKHOFF, G.,~yd~dynam~~s ~G~drod~arn~a~, fid-vo in. lit., Moscow, 1963.
4.
SfMUNI, L. M., Sofution of some problems of the motion of a V~SCWS &id, connectedwith the Bow around a cyimder and sphere. Izv. SO Akud. Nauk SSSR. Ser. Tekhn., 8,2,X3-27,196?.
5.
KORYAVOV. P. P. and PAVLQVSKII, Yu. N., Numerical solution of the problem of the motion of a circular cylinder in the flow of a viscous fluid. In: Problems of applied mathematicsand tiechanics (Probl. prikl. matem. i mekhan.) 247-262, “Nauka”, Moscow, 1971.
6.
RUBINOV, S. I. and KELLER, J. B., The transverse force on a spinning sphere moving in a viscous fluid. J. Fl~~d~ec~., 1, ?,447-459, 1961.
7.
DENNIS, S. C. R. and CHANG, GAU-ZU., Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100, J. ~l~~~e~~., 42,3,471-489, 1970.
8.
LYUL’KA, V. A., The ~teraction of solid particles dragged by a viscous ~~ornpre~jbIe Mat. mat. Fit., fO, 4, I&%4--1049,197O.
9.
KELLER, N. B. and TAKAMI, H., Numerical studies of steady viscous flow about cylinders. In: Numet. solutions of nonlinear differential equations, 115-140, Wiley, New York, 1966.
zhidkosti i gaza), Fizmatgiz, Moscow,
Liquid. 2!r. vj&isi
10. BABENKO, K. I., VVEDENSKAYA, N. D. and ORLOVA, M. G., Results of the calculation of the flow of a viscous fluid around an infinite circular cylinder. Preprint No. 38, IPM, Moscow, 1971.
AL~O~TH~
FOR THE CONTROL OF ~R~~~~
A~T~~ATA~
V, 1.AVERTSEVand Yu. I. MOKIN Moscow (Received 19 September 1975)
AN ANALYTIC expression is obtained for the area of a small contact area. An approximate formula for estimating its size is given. An asymptotic formula is derived for the range D of control of the drive of a grinding wheef, suitable for engineering cakufations. In the process of grinding in mass production one of the chief problems is the st~~isation of the modes of operation of machine tools in the periods between readjustments. In the grinding of bearing rings scores of items are ground by one grinding wheel. The diameter of a grinding wheel is reduced to approximately half in the processing of a batch of rings, which causes a decrease of the area S of the small contact area. Until recently the variation of the area of contact of the wheel with the component has been disregarded, and it has had a considerable influence on the temperature state of the grinding, the durability of the wheel, the quality of the processing of the component etc. At the present time the question has been posed of the introduction of automatic control systems for controlling grinding machines f 1,2] f ensuring stable quality of component processing and high produ~ti~ty inde~~dently of the degree of wear of the wheel. Therefore an urgent problem is the determination of the size of the small contact area for various grinding methods and various profiles of the grinding wheel. A valid estimate of 5’is of great value for the mathematical description of the grinding process allowing for heat effects and the conditions of the shaping of the macrogeometry and microgeometry of the components.
5%. ~j&isI.Mat. mat. Fis., 17, 2,48f-489,
1977.