Foundations of Fluid Mechanics I
Foundations of Fluid Mechanics I M E 521
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Date Created: 11/01/15
The B nard Problem A Summary Author John M Cimbala Penn State University Latest revision 08 February 2008 1 Introduction and Problem Setup Consider two infinite stationary parallel at plates separated by distance d The T Anmld 2 012 lower plate is at temperature To and the upper plate is at temperature To AT d 1g 2L colder Initially the flow is at rest but this is an unstable situation since the warm uid on the bottom wants to rise We examine this problem using linear stability TU WW 2 quot12 analysis 2 Summary of Linear Stability Analysis applied to this problem The inclass analysis follows Kundu Section 123 closely filling in some of the details Step 0 Start with the Boussinesq equations NavierStokes equations for buoyant flows for total flow variables q 012 z 3931 0f 01 u 7 7 517aT7T 1 a 0 Mg 0 i This represents 5 equations and 5 unknowns Step 1 Generate the basic state equations 1b 2b and 3b Here we assume no flow so that U 0 P Pz d 0 airquot a f u 7c OxJOxJ at Bx vaxjaxj 3 hydrostatic pressure although P does not vary with z exactly linearly and Z Step 2 Add disturbances q Q q and plug into 1 2 and 3 This generates total equations 1t 2t and 3t Step 3 Subtract basic state equations from the total equations This generates disturbance equations 1d 2d and 3d Step 4 Linearize the disturbance equations to generate the linearized disturbance equations 11 21 and 31 au 1 0 0 0T7 BZTquot 11 7 pg53ocT39V 39 21 and 7l w 2c at p0 0xx x at 0x 0x equations and f unknowns now the disturbance vari bles are the unknowns since the basic state is known Step 5 Solve the linearized disturbance equations ll 2 and 3 After some algebraic manipulation we can eliminate pressure from the equations and rewrite the energy and z m omentum equations as follows T39 0 2 u 3l This still represents 5 V2141 gaVHZT39 VVAW 6 where E This set of two equations and two unknowns w and T is uncoupled from the other linearized disturbance equations SemiNormalization of the equations Following Kundu s notation normalize only the independent variables x and t I Z L 2 and 6 become if iw39V2T39 7 and V2 V2 VHZT39 8 at K Pr 0t V The boundary conditions for these two equatlons w1th respect to z are w 0 T39 0 at z i where this 2 1s def1ned Z as the original 2 divided by d znew zodgimld Method of Normal Modes Assume disturbances that are periodic in x and y but not growing or decaying in x or y but may be periodic and may be rowin or decaying in t temporal instability Let the disturbances be of the form and T39 TZ8 md m where V and f are complex amplitudes and k and l are the x andy components respectively of wavenumber vector 1 For temporal stability analysis both k and lmust be real but 039 the complex growth rate can be complex otherwise spatial instability would also be possible Plug these quot a 12 PI d rd4 Fdz d Rayleigh number Ra amp K k2 l2 and W E W defined for convenience Z K17 K We are now down to 2 odes and 2 unknowns into Eqs 7 and 8 to get 77032 71212 W 11 7D2 712 D2 7K2W7RaK2 where D The boundary conditions for Equations 11 and 12 are W DW T 0 at z i Step 6 Examine stability Finally we solve 11 and 12 for the case of marginal stability to be done in class B nard and Taylor Cells 1 B nard cells buoyancy driven convection cells Author John M Cimbala Penn State University Latest revision 15 February 2008 139 Buoyancydriven convection rolls Differential interferograms show side views of convective instability of silicone oil in a rectangular box of relative dimensions 1041 heath from below At the top is the classical Ray leigh Bonard situation uniform heating produces rolls parallel to the shorter side In the middle photograph the temperature cliflcrcnce and hence the amplitude of motion increase from right to left At the bottom the box is rotating about a vertical axis Canal 6 Kirchurt 1979 Oertel 19822 From Van Dyke M An Album afFluidMa an Stanford CA The Parabolic Press 1982 p 82 l 140 Circular buoyancydriven convection cells Sili cone oil containing aluminum powder is covered by a uniformly cooled glass plate which eliminates surface tensinn effects The circular boundary induces circular rolls In the left photograph the copper bottom is uni formly heated at 29 times the critical Rayleigh number giving regular rolls At the tight the bottom is hotter at the rim than at the center This induces an overall circula tion whichI superimposed on regular circular rolls pro duces alternately larger and smaller rolls Koschmietler 1974 1966 2 Taylor cells in the narrow gap between two concentric cylinders inner cylinder rotating 127 39 39 39 39 Taylor vnr re L39 quot pu dcl 39 39 a xe 39 39 one of relative radius 0727 The top and bottom plates are xed The rotation speed is 91 times that at which Taylor predicts the onset of the regularly spaced toroidal vottices seen here The ow is radially inward on the heavier dark hoti zontal rings and outward on the ner ones The motion was started impulsively giving narrower votticcs than would result mm a smooth start Burklialter E Kascllmledm 1974 a we 5 WM mm 125 Laminar Taylor vortices in a narrow gap A larger inner cylinder in the apparatus D the right gives a radius ratio of 0396 Again only the inner cylinder rotates The upper photograph shows the center section of axisymmertic vortices at 116 times the critical speed In the lower at 85 times the critical speed the ow is doubly riodic with SD waves around the circumference drifting with the rotation Koschmiedm 1979 From Van Dyke MAn Album afFluidJlIotion Stanford CA The Parabolic Press 1982 76 Elementary Planar Irrotational Flows in Complex Variables Author John M Cimbala Penn State University atest revision 17 October 2007 Note Consider steady incompressible irrotational Newtonian uid ow in which gravity is neglected The ow is assumed to be twodimensional in the x y or r Bplane Summary of the Equations Elementary Planar Irrotational Flows f rm stream in the x direct39 Uniform stream in an arbitrar direction C Line vortex at the origin
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