1 / 14

Boundary Layer on a Flat Plate: Blasius Solution

Boundary Layer on a Flat Plate: Blasius Solution. from Kundu’s book. Assuming displacement of streamlines is negligible → u = U = constant everywhere, as if the boundary didn’t exist. H. z. The irrotational flow, according to Euler’s equation:. = 0 @ u = constant.

mihaly
Download Presentation

Boundary Layer on a Flat Plate: Blasius Solution

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Boundary Layer on a Flat Plate: Blasius Solution from Kundu’s book Assuming displacement of streamlines is negligible →u = U = constant everywhere, as if the boundary didn’t exist H z The irrotational flow, according to Euler’s equation: = 0 @ u = constant

  2. The complete set of equations for Boundary Layer are: from Kundu’s book H z

  3. The velocity profile in the boundary layer can be obtained with a SIMILARITY SOLUTION – following Blasius, a student of Prandtl Velocity distributions at various x can collapse into a single curve if the solution has the form For similarity solution, use streamfunction: from Kundu’s book Using similarity form above: H z Using the definition:

  4. Applying streamfunction to:

  5. f and its derivatives do not explicitly depend on x : initial and boundary conditions: Can be valid only if: Blasius equation

  6. % uses Matlab ODE45 - Runge-Kutta method ti = 0.0; % start of integration tf = 7.0; % final value of integration bcinit = [0.0 0.0 0.33206]; % initial values [eta f] = ode45('state',[titf],bcinit); ================== function stst = state(eta,f) stst = [ f(2) , f(3) , -0.5*f(1)*f(3)]';

  7. Boundary Layer Thickness Distance η where u = 0.99 U η = 4.9 Rex

  8. ν= 1×10-6 m2/s; U = 1 m/s

  9. displacement thickness momentum thickness

  10. Skin Friction Local wall shear stress using: @z = 0

  11. Skin Friction Local wall shear stress Wall shear stress then changes as x -½, i.e., decreases with increasingx

  12. τ decreases because of thickening of δ

  13. Local shear stress at wall can be expressed in terms of the local drag coefficient and the drag force per unit width of plate of length L So the drag force is proportional to the 3/2 power of velocity (U 2/U 1/2) For high Re the drag force is proportional to the square of velocity Now, the overall drag coefficient is defined as: overall drag coefficient is average of local drag coefficient

  14. http://www.symscape.com/node/447 Breakdown of Blasius solution Transition from laminar to turbulent region occurs at Recr(~106) Transition depends on a) surface roughness and b) shape of leading edge Boundary layer grows faster in the turbulent region because of macroscopic eddies

More Related