BAIL 2006 Book of Abstracts - Institut für Numerische und ...
BAIL 2006 Book of Abstracts - Institut für Numerische und ... BAIL 2006 Book of Abstracts - Institut für Numerische und ...
J. HUSSONG, N. BLEIER, V.V. RAM: The structure of the critical layer of a swirling annular flow in transition ✬ ✫ where ω is the frequency, λx the disturbance wave number in the axial direction and nϕ the corresponding quantity in azimuthal direction which has to be an integer. nϕ = 0 represents an axisymmetric disturbance whereas nϕ �= 0 corresponds to helical disturbances. We have solved the linearised equations for the disturbance numerically by the spectral collocation method using Matlab over a range of parameters ɛR, Sa and Twi to determine the dependence of the critical Reynolds number on the parameters Sa and Twi from which the wave velocity and the location of the critical layer at this Reynolds number have been obtained. A sample of the computational results is presented in Fig. 1. c = omega / lambda x 0.3 0.29 0.28 0.27 0.26 0 0.1 0.2 S a 0.3 0.4 c = omega / lambda x 0.3 0.28 0.26 0.24 0.22 0 0.1 0.2 T wi 0.3 0.4 Figure 1: wave velocity for Twi = 0.1 (left) and Sa = 0.1 (right) An asymptotic analysis of the critical layer, conducted along the same lines as for the corresponding case in planar flow (see eg. [1], [4]), brings out the following points as the outstanding features of the critical layer under the influence of swirl: • Swirl exerts no influence on the scale of thickness of the critical layer through its axisymmetric disturbance mode, nϕ = 0 • For the nonaxisymmetric modes, nϕ �= 0, the thickness of the critical layer retains the same scale as in the case of classical planar flow , viz. 1 − O(λxRe) 3 , as long as the product of the transverse curvature and swirl 1 − parameters, ɛRSa, remains small to within O(Re 3 ) Results obtained from this asymptotic analysis of the critical layer are set against the computational results outlined above. References [1] Drazin, Reid, Hydrodynamic stability, Cambridge Univ. Pr., 1982. [2] Hinch, E. J., Perturbation Methods, Cambridge Univ. Pr., 1992. [3] Holmes, M. H., Introduction to Perturbation Methods, 20 in the Series Texts in Applied mathematics, Springer, 1995. [4] Maslowe, S. A., Critical layers in shear flows. In Annual Review of Fluid Mechanics, 18, 1986, 405-432. [5] Nayfeh, A. H., Perturbation Methods, John Wiley, 1973. [6] O Malley Jr., R. E., Introduction to Singular Perturbations, 14 in the Series Applied Mathematics and Mechanics, Academic Press, 1974. [7] Verhulst, F., Methods and Applications of Singular Perturbations, 50 in the Series Texts in Applied mathematics, Springer, 2000. 2 Speaker: RAM, V.V. 38 BAIL 2006 ✩ ✪
T.K. SENGUPTA, A. KAMESWARA RAO: Spatio-temporal growing waves in boundary-layers by Bromwich contour integral method ✬ ✫ ��Ô�ÖØÑ�ÒØÓ���ÖÓ×Ô���Ò��Ò��Ö�Ò�ÁÁÌÃ�ÒÔÙÖÍÈ ËÔ�Ø�ÓØ�ÑÔÓÖ�Ð�ÖÓÛ�Ò�Û�Ú�×�Ò�ÓÙÒ��ÖÝÐ�Ý�Ö×�Ý�ÖÓÑÛ�� ÌÃË�Ò�ÙÔØ��Ò��Ã�Ñ�×Û�Ö�Ê�Ó ÓÒØÓÙÖ�ÒØ��Ö�ÐÑ�Ø�Ó� ÔÐ�Ý×Ô�Ø��Ð�ÖÓÛØ��Ú�ÒÛ��ÒØ��×�×ÔÓ×���×�×Ô��Ø�Ñ���Ô�Ò��ÒØÔÖÓ�Ð�Ñ ÁØ�×Û�ÐÐ�ÒÓÛÒØ��ØÙÒ×Ø��Ð�Þ�ÖÓÔÖ�××ÙÖ��Ö����ÒØ�ÓÙÒ��ÖÝÐ�Ý�Ö��× ÁÒ��� �� �℄�ÓÖ×ÓÑ��ÒØ�ÖÒ�ÐÓÛ×�Ø��×Ó�Ø�Ò���ÒÓÒ��ØÙÖ��Ø��ØÛ��Ð�Ø��ÓÛ ��×ÔÐ�Ý×Ð�Ò��Ö×Ø���Ð�ØÝ�ÝÒÓÖÑ�ÐÑÓ���Ò�ÐÝ×�×ØÖ�Ò×�Ø�ÓÒÓÙÖ×�Ù�ØÓ×Ù Ô�ÖÔÓ×�Ø�ÓÒÓ�ÒÓÒÒÓÖÑ�Ð���Ý�Ò�ÑÓ��×Ø��Ø�Ò��×ÔÐ�ÝÚ�ÖÝ����ØÖ�Ò×��ÒØ �Ò�Ö�Ý�ÖÓÛØ��℄ÁÒØ��ÔÖ�×�ÒØÛÓÖ�Û�×�ÓÛ�Ý�ÖÓÑÛ��ÓÒØÓÙÖ�ÒØ��Ö�Ð Ñ�Ø�Ó�Ó�ÓÑÔÐ�Ø�×Ô�Ø�ÓØ�ÑÔÓÖ�Ð�ÔÔÖÓ���℄Ø��Ø×Ù�ØÖ�Ò×��ÒØ�ÖÓÛØ� Ó��Ò�Ö�Ý�Ð×ÓØ���×ÔÐ���ÓÖ�ØÔÐ�Ø��ÓÙÒ��ÖÝÐ�Ý�ÖÛ��Ò�ÐÐØ��ÒÓÖÑ�Ð ÑÓ��×��×ÔÐ�Ý×Ô�Ø��Ð×Ø���Ð�ØÝ Ø����×ØÙÖ��Ò�×ØÖ��Ñ�ÙÒØ�ÓÒ�Ý �Ø���Ý���ÖÑÓÒ�×ÓÙÖ��ØØ��Û�ÐÐ�Ø��ÖÙÐ�Ö�Ö�ÕÙ�ÒݬÁ�Û�Ö�ÔÖ�×�ÒØ Ï��ÒÚ�×Ø���Ø�Ø��Ö�×ÔÓÒ×�Ó��Þ�ÖÓÔÖ�××ÙÖ��Ö����ÒØ�ÓÙÒ��ÖÝÐ�Ý�Ö�Ü Û��Ö��Ö�Ò���Ø�Ø���ÖÓÑÛ��ÓÒØÓÙÖ×�ÓÐÐÓÛ���Ò�Ú�ÐÙ�Ø�Ò�Ø����ÓÚ��Ò Ø��Ö�Ð�ÒØ��ÓÑÔÐ�Ü«�Ò�¬ÔÐ�Ò�¬�ÔÔ��Ö�Ò�Ú��Ø��Û�ÐÐ�ÓÙÒ��ÖÝ �Ü�Ý�Ø����Ö�«�¬�Ý��«Ü ¬Ø�«�¬ Ì��ÔÓ×��ÔÖÓ�Ð�Ñ�××ÓÐÚ���ÝÐ�Ò��Ö�Þ�Ò�Ø��Æ�Ú��ÖËØÓ��×�ÕÙ�Ø�ÓÒ�ÒØ�� �℄�Ò��×ÓÒ�Ó�Ø���Ð���ÒØÛ�Ý×Ó�ØÖ��Ø�Ò��ÙÐÐ×Ô�Ø�ÓØ�ÑÔÓÖ�ÐÔÖÓ�Ð�Ñ× ×Ô�ØÖ�ÐÔÐ�Ò�×�Ò��ÜÔÖ�××�Ò��Ø�×Ø��ÇÖÖËÓÑÑ�Ö��Ð��ÕÙ�Ø�ÓÒ��Ú�Ò�Ý ÓÒ��Ø�ÓÒ�℄Ì��Ñ�Ø��Ñ�Ø��Ð��×�×Ó��ÖÓÑÛ��ÓÒØÓÙÖ�ÒØ��Ö�Ð�×��Ú�Ò�Ò ×Ô�Ø��ÐÒÓÖÑ�ÐÑÓ��×��Ú����Ò�Ú�ÐÙ�Ø���Ý�Ö��×��Ö�Ø��Ò�ÕÙ��ÑÔÐÓÝ�Ò� Ñ�ÒØØ���Ò�××À�Ö���Ð�×�Ù×�ÓÙÒ��ÖÝÐ�Ý�Ö�×ÓÒ×���Ö��Û�Ó×��Ü�×Ø�Ò� Û��Ö�ÍÝ�×Ø��Ñ��ÒÓÛ�Ò�Ø��Ê�ÝÒÓÐ�×ÒÙÑ��Ö�×��×��ÓÒ��×ÔÐ�� Í ¬�«� «� Í���«Ê���Ú «� «�� �ÒÌ��Ð� Ø��Ò�ÙØÖ�ÐÙÖÚ��Ò�Ø��ÓØ��Ö×�ØÓÖÖ�×ÔÓÒ��Ò�ØÓ¬������ÒØ�¬���×� ��ÑÔ��ÓÒ�×�ØÓÖÖ�×ÔÓÒ��Ò�ØÓ¬������ÒØ�¬���×�Ò� �ÒÌ��Ð� ÓÑÔÓÙÒ�Ñ�ØÖ�ÜÑ�Ø�Ó��℄�ÓÖÊ�� �ÓÖØ��×�Ô�Ö�Ñ�Ø�ÖÓÑ��Ò�Ø�ÓÒ×�Ü�×Ø�Ò�×Ô�Ø��ÐÑÓ��×�Ö��ÐÐ �Ò�¬����Ò����Ò���Ú�Ò ��Õ�Ò�� �××ÓÐÚ���ÐÓÒ�Ø���ÖÓÑÛ��ÓÒØÓÙÖ×�� �¬Ö��¬��� ×Ù�Ø��Ø��Ù×�Ð�ØÝÔÖ�Ò�ÔÐ��×ÒÓØ �«Ö��«�� �Ö���ÐÓÛ Ö�ÓÒ×ØÖÙØØ����×ØÙÖ��Ò�¬�Ð��Ò�Õ ÑÓ��×�××Ø��Ð�Ø��ÓÒØÓÙÖ�ÒØ��Ö�ÐÖ�×ÙÐØ×�Ò¬�ÙÖ�×�ÓÛ��ÐÓ�Ð×ÓÐÙØ�ÓÒ� Ú�ÓÐ�Ø���Ò�Û�Ú�×ØÖ�Ú�Ð�ÒØ��ÓÖÖ�Ø��Ö�Ø�ÓÒÇ�Ø��Ò���×�Ö�Ù×��ØÓ �Ò¬�ÙÖ��ÓÖØ���×�Ó�¬���Ï��Ð�Ø��Ø��Ð�×�ÓÛ×�ÐÐØ�Ö��×Ô�Ø��Ð Ø�Ñ�Ø�����Ý�Ò�Û�Ú�ÓÖÖ�×ÔÓÒ�×ØÓÑÓ����Ò�Ø��Ô���ØÓÖÖ�×ÔÓÒ�×ØÓ ������Ý�Ò�Û�Ú��Ò�����Ø�ÑÔÓÖ�ÐÐÝ�ÖÓÛ�Ò�Û�Ú�Ô���Ø��Ø�Ö×ÙÆ��ÒØ �Ò�Ø��Ú�ÐÓ�Øݬ�Ð��Ö��×ÔÐÓØØ�� Speaker: SENGUPTA, T.K. 39 BAIL 2006 ✩ ✪
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J. HUSSONG, N. BLEIER, V.V. RAM: The structure <strong>of</strong> the critical layer <strong>of</strong> a swirling<br />
annular flow in transition<br />
✬<br />
✫<br />
where ω is the frequency, λx the disturbance wave number in the axial direction<br />
and nϕ the corresponding quantity in azimuthal direction which has to be an<br />
integer. nϕ = 0 represents an axisymmetric disturbance whereas nϕ �= 0 corresponds<br />
to helical disturbances. We have solved the linearised equations for<br />
the disturbance numerically by the spectral collocation method using Matlab<br />
over a range <strong>of</strong> parameters ɛR, Sa and Twi to determine the dependence <strong>of</strong> the<br />
critical Reynolds number on the parameters Sa and Twi from which the wave<br />
velocity and the location <strong>of</strong> the critical layer at this Reynolds number have been<br />
obtained. A sample <strong>of</strong> the computational results is presented in Fig. 1.<br />
c = omega / lambda x<br />
0.3<br />
0.29<br />
0.28<br />
0.27<br />
0.26<br />
0 0.1 0.2<br />
S<br />
a<br />
0.3 0.4<br />
c = omega / lambda x<br />
0.3<br />
0.28<br />
0.26<br />
0.24<br />
0.22<br />
0 0.1 0.2<br />
T<br />
wi<br />
0.3 0.4<br />
Figure 1: wave velocity for Twi = 0.1 (left) and Sa = 0.1 (right)<br />
An asymptotic analysis <strong>of</strong> the critical layer, conducted along the same lines<br />
as for the corresponding case in planar flow (see eg. [1], [4]), brings out the following<br />
points as the outstanding features <strong>of</strong> the critical layer <strong>und</strong>er the influence<br />
<strong>of</strong> swirl:<br />
• Swirl exerts no influence on the scale <strong>of</strong> thickness <strong>of</strong> the critical layer<br />
through its axisymmetric disturbance mode, nϕ = 0<br />
• For the nonaxisymmetric modes, nϕ �= 0, the thickness <strong>of</strong> the critical<br />
layer retains the same scale as in the case <strong>of</strong> classical planar flow , viz.<br />
1 − O(λxRe) 3 , as long as the product <strong>of</strong> the transverse curvature and swirl<br />
1 − parameters, ɛRSa, remains small to within O(Re 3 )<br />
Results obtained from this asymptotic analysis <strong>of</strong> the critical layer are set<br />
against the computational results outlined above.<br />
References<br />
[1] Drazin, Reid, Hydrodynamic stability, Cambridge Univ. Pr., 1982.<br />
[2] Hinch, E. J., Perturbation Methods, Cambridge Univ. Pr., 1992.<br />
[3] Holmes, M. H., Introduction to Perturbation Methods, 20 in the Series Texts in Applied<br />
mathematics, Springer, 1995.<br />
[4] Maslowe, S. A., Critical layers in shear flows. In Annual Review <strong>of</strong> Fluid Mechanics, 18,<br />
1986, 405-432.<br />
[5] Nayfeh, A. H., Perturbation Methods, John Wiley, 1973.<br />
[6] O Malley Jr., R. E., Introduction to Singular Perturbations, 14 in the Series Applied Mathematics<br />
and Mechanics, Academic Press, 1974.<br />
[7] Verhulst, F., Methods and Applications <strong>of</strong> Singular Perturbations, 50 in the Series Texts<br />
in Applied mathematics, Springer, 2000.<br />
2<br />
Speaker: RAM, V.V. 38 <strong>BAIL</strong> <strong>2006</strong><br />
✩<br />
✪