Stability of Couette flow in the wide gap of two circular concentric cylinders with rotating inner cylinder and finite growth rate

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1 Arch. Mech, 4&, 5, pp. &35-S4, Warsawa 996 Stability of Couette flow in the wide gap of two circular concentric cylinders with rotating inner cylinder and finite growth rate H.S. TAKHAR (MANCHESTER), M.A. ALl (BAHRAIN) and V.M. SOUNDALGEKAR (THANE) A FINITE-DIFFERENCE sournon for the stability of flow of a viscous fluid in an annular wide-gap space is carried out by taking into account the effects of finite growth rate of the amplification factor 7. The numerical values of the minimum Taylor number Tarn and the critical Thylor number The for different values of J and for 7 セ P@ and 7 = 0, respectively, arc derived and tabulated. le effects of 7 and 7 o n the radial component of disturbance and on the cell patterns are shown. It is observed that for increasing 7(> 0), the cell patterns arc reduced in sie, while for decreasing 7( < 0) they arc enlarged.. Introduction STABILITY OF AISYMMETRIC FLOW of a viscous incompressible fluid between two concentric rotating cylinders has been investigated by TAYLOR [6], CHANDRASEK- HAR ( ), HARRIS and REro (2), W ALOWIT, TSAO and D PRIMA [7), SOUNDALGEKAR et al. [4], who used difterent methods and difterent boundary conditions. The usual mathematical procedure of the stability analysis is to assume that small disturbances are superimposed on the steady motion. These disturbances are assumed to be periodic in the -direction and proportional to e 7, where a is the amplification parameter or growth rate factor. Then the parameter which governs the stability of the motion is the exponential time factor a, the motion being stable or unstable according to whether the real part of a is less than or greater than ero, and when a = 0, it is known as the marginal state. Almost all the papers in this field applying the linear stability analysis dealt with a = 0, i.e. the marginal state of stability. RoBERTS [3] was the first to study the effects of non-ero values of the growth rate on the Taylor number in the wide-gap Couette flow, and computed the smallest characteristic values of the Taylor number for a given wave number a and the growth rate factor a, by employing the numerical method given by Harris and Reid. However, the eftects of a on the radial velocity perturbation and on the cell pattern have not been studied or shown graphically. Hence it is now proposed to solve the eigen-value problem of ROBERTS [3] by using a finite-difierence technique and to derive the minimum values of the Taylor number, Tarn, for difierent values of ±a of the radial velocity perturbations and the cell-patterns. In the next section, a short account of the finite-difterence method is given and numerical values of Tam are tabulated for different values of

2 836 II.S. TAJ,IIAil,.tii.A. ALI AND V..t. Sour-<DALGCKAR ±a and wave nu mber a. In order to verify our results, we have also computed the numerical values of Tac, the critical Taylor number fo r a = 0 and critical wave number ac. 2. Solution to the axisymmeh ic pr oblem a t finite growth r ate The axisymmetric linear stability of Couette flow at finite growth rate, wher. the outer cylinder is at rest and the two cylinders are separated by a wide gap, ca be shown to be governed by the following system of sixth order (e.g. SOUNDALGEKAR et al. [5]). (2.) (2.2) (2.3) (DD*- a 2 - a)v = L, u = v =Du= 0 at x = 0,. Here u, v are the radial and aimuthal components of the disturbances, a the axial wave number, a is the growth rate and Ta is the T<tylor number. The) are defined as follows: (2.4) d D = - dx' x = cl D* =!!_ dx = - x, ] + ( -J).l:, Ta = 4?d4 ( f2t) //, where R and R 2 are the radii of the inner and outer cyl inders, respectively, and.0 is the rate of rotation of the inner cylinder. Our Ta is equivalent to 2Tan where Thn is the Taylor number defined by Roberts or Chandrasekhar, as 'PJ n = 2?d4 (EJ.) 2 - '/ 2 V 3. Method of solution I3y solving the above eigenvalue problem defin ed by Eqs. (2.)- (2.3), W { determine the smallest characteristic value of the Titylor number, denoted by fam, for given wave number a and a. To so lve Eqs. (2.)- (2.2) by the finite-difter!nce technique, we first expand these as follows: (3.) [n 4 + 2kD 3 - (3.: 2 + 2a 2 +a) D 2 + (3.: 3-2al.:- al.:) D +(2a 2 k 2-3!.: 4 + a 4 + a(k 2 + a 2 )] u = -a 2 Ta g (r ) v, (3.2) [n 2 + kd - (k 2 + a 2 + a)] V = u,.: = 7.

3 S T ABI L ITY O r C O UET TE F'LO W 837 We write the d erivatives in terms o f central di fterences and rearrange the terms. Then E qs. (3. ) and (3.2) reduce to (3.3) m ;Ui+2 + mui+ l + m3 U; + n4ui- l + msui-2 = - h 4 a 2 Td g (x ) V;, (3.4) c v;.+l + c2 v;. + C3 vi- = h 2 Ui, where m = + hk, m = - 4-2hk - h 2 (3k 2 + 2a 2 + a) + 2 h3 (3k 3-2a 2 k - ak), (3.5) m3 = 6 + 2h 2 (3k 2 + 2a 2 + a )+ h 4 (2a 2 k 2-3k 4 + a 4 + a k 2 + a a 2 ), m4 = hk- h 2 (3k 2 + 2a 2 +a) - 2_h 3 (3k 3-2a 2 k - ak), m 5 = - hk, c = + -hk 2, C = -2- h 2 (k 2 + a 2 + a ), c3 = - 2 hk. The suft i sta nds fo r the pivotal point under consideration. T he step length h = /N, wh ere N is the number of interv als into which the range [0, ] is divided. The boundary conditions (2.3) imply that (3.6) Equations (3.3) and (3.4) with conditio ns (3.6) can be wri tten in matrix no tation as (3.7) A U = Ta D \, A 2 V = Ti, where A, A 2 and B are the coefllcient matrices of order n x n, and n = N -. E quatio ns (3.7) can be combined into an eigenvalue equation of the form (3.8) ( C - Ta I) V = 0. The eigenvalues are computed by using the QR algorithm for a = 0, ± 0.5, ±, ±.5, ± 2.0 and are listed in Table for 7 = 0.85, 0.5, 0. which corresponds to

4 838 H.S. TAI-: H An., M.A. Au AND V.M. SoUNDALGEI<AR the wide gap case. For a = 0, the marginal state, the eigenvalues Tac are known as critical values of the Taylor number which corresponds to lowest values of the wave number a and are denoted by ac, the critical wave number. We observe from Table that for the marginal state of stability (a = 0), the critical values of the Taylo r number and wave number increase by increasing the width of the gap between two concentric cylinders. H owever, when the amplification factor a (> 0) is increasing, there is also an increase in the minimum values of Taylor number, and opposite is the case when the amplification factor a(< 0) is decreasing; then the minimum values of Ta vi. Tam also decrease. Table. Values of ac, Tac (a = 0) and am, Ta,,, (a =I 0). TJ ac Tac Tac(Jl) TJ/a CL m Tam am Tam CL m Tam U(x) 0.8 '7 = 0.5 'r------\----- '7 = a= 0 a = a = P N P ᆬ M MLMLMNM MLM M M FIG.. Radial velocity U(x).

5 STABILITY OF COUETTE FLOW 839 The radial component of velocity disturbance U(x ) is shown in Fig. l for TJ = 0.5, 0. and for a = 0, ±2.0. It is observed that U( x) increases near the inner cylinder and decreases near the outer cylinder when a = 2.0 as compared to that at the onset of instability (a = ), and opposite is the case when a = The cell patterns are shown for TJ = 0.5 and 0. for a = 0, ± 2.0 in Figs It is observed from these figures that the cells get reduced in sie for a = 2.0 and get enlarged in sie for a = -2.0 as compared to those at the onset of instability FIG. 2. Cell patterns at the onset of instabil ity for 7 = 0.5 and a=, /J = U(x )cosaz F IG. 3. Cell pattern instability for 7 = 0.5 and a = 2.0.

6 @Pセ o.4 o.g o.a.0 F IG. 4. Cell pattern instability for 7 = 0.5 and a = H F IG. 5. Cell patterns at the o nset of instabi lity fo r 7 = 0. and a = 0. 0, '/; = U (x ) cm az F IG. 6. Ce ll patterns for = 0. and a = 2.0. (840]

7 STABTLITY OF COVETTE FLOW F rg. 7. Cell pattern for 77 = 0. and a = References ケャゥョ イセᄋL@. S. CuANDRASEKIIAR, The stability ofviscotl' flow between rotating cylimlen, If, Proc. Roy. Soc. London, 268, 45-52, D.L HARRIS and W.H. REID, On the.l tability of viscous flmv between rotating Pt 2. Numerical analysis, J. Fluid Mcch., 20, 95-0, R.J. DONNELLY and K.. W. SCIIWAHZ, Experim ents 0 the stability of viscous flow between rotating cylinders, VI. Finite amplitude experiments, P.H. ROilEI<rS, (Append ix), Proc. Roy. Soc. London, A 283, , V.M. SOUNDALCEKAH, H.S. TAKIIAR and T.J. SMITH, Effects of radial temperature gradient on the stability of viscou. flow in an annulus with a rotating inner cylinder, w and Stoffiibcrtragung, 5, , V.M. SOUNDALCEIV\R, H.S. TAKitAR and M.A. Au, Sabiliy of flo ov bemeen rowting cylinders- Wide-gap problem, ASME J. Fluids Engng.,,97-77, G.!. TAYLOR, Swbility of a viscous li id CUIIIained ben.-een /IVo ro/{/ing cylinder:,, Phi!. Trans. Roy. Soc. London, A22J, , J. W ALOwrr, S. TSAO and R.C. D PRIMA, Swbility of flow between arbitrarily spaced concenuic cylidlical surfaces including the effect of a radial empemture gmdient, ASME J. Applied Mcch., 3, , 964. MANCIIESTER SCIIOOL m ENCINEERING, UN IVERSITY OF MANCIIESfER, MANCH ESTER, ENGl. \ ND; DEPARTMEr-IT OF MAT IIF:MAT ICS, UNIVERISTY OF BAIIRAIN, UAIIRAIN (MIDDLE EAST) BRINDAVAN SOCIETY, TIIANE, IND IA. Received December 8, 995.