Analysis of perturbations in boundary layers - from receptivity to multimode decomposition</head> <body bgcolor="#ffffe6"> <font color="blue"> <h1>Analysis of perturbations in boundary layers - from receptivity to multimode decomposition</h1> <h3>Anatoli Tumin</a></h3> </font><br> University of Arizona<br> <p> Abstract- <br> The solution of linearized Navier-Stokes equations can be presented as an expansion into normal modes of the discrete and continuous spectra. In the case of an initial-value problem, the normal modes are associated with the temporal stability theory, whereas in the case of a signaling problem, they are associated with the spatial stability theory. Because the operator is not self-adjoint, the normal modes are non-orthogonal. However, there is an orthogonality (biorthogonality) condition for modes of the direct and adjoint problems. The latter can serve as a filter to find out the amplitude and phase of a mode in a perturbation field. <br> The biorthogonal eigenfunction system may serve as a tool for the analysis of perturbations. Particularly, one can easily find the amplitude of an instability mode generated by an oscillator in a boundary layer, or the amplitude of a mode of the continuous spectrum generated by a hump on the wall. Other applications are associated with the decomposition of a perturbation field in a computational study. If some a priori information about the perturbation is available, the method can be used for the decomposition of experimental data when only partial measurements (for example, one velocity component) are provided. The discussion is accompanied by illustrations for low- and high-speed boundary layers. <br> The figure illustrates a theoretical flow-field downstream of a hump. The solution is comprised of normal modes of the continuous spectrum. Vectors represent perturbations of the spanwise and the normal, , velocity components, whereas the contours represent perturbation of the streamwise velocity, u. One can see the wake region behind the hump , and the high speed streaks, , at the sides of the hump ( mm). </p> <p><img src="Tumin_image.jpg"></p> <p> <hr> <table width=800> <tr> <td> <address> <A HREF="http://www.galcit.caltech.edu/"> <IMG align=middle SRC="http://www.galcit.caltech.edu/pics/smallrelief.gif"> GALCIT Home Page</A> </address> </td> <td> <table> <tr> <td> <A HREF= "index.html"> <font size=4>2004-2005 Fluids Seminar Page </font></A> </td> </tr> </table> </td> </tr> </table> <p> <hr> <address>Maintained by: James Faddy <address>EMail: <a href="mailto:faddy@caltech.edu">James Faddy</a></address>  Last modified: October 25, 2004  </body> </html>