KIL 3002 Fluid Mechanics 2 Notes (2024) PDF

APPROXIMATE SOLUTIONS OF NAVIER-STOKES EQUATIONS 1. Flow applications – solving Continuity and Navier- Stokes equations. 2. Analysis of a common flow case – Fully developed Couette flow. Similar flow case will be solved numerically in CFD cha...

APPROXIMATE SOLUTIONS OF NAVIER-STOKES EQUATIONS 1. Flow applications – solving Continuity and Navier- Stokes equations. 2. Analysis of a common flow case – Fully developed Couette flow. Similar flow case will be solved numerically in CFD chapter. 3. Example – No-slip boundary conditions (solid/wall). DETERMINATION OF PRESSURE FIELD FROM KNOWN VELOCITY Analysis Simple flow case as velocity field is provided Previous example: Velocity field: Consider a steady, two-dimensional, incompressible velocity ! = %& + ( ) + −%+ + ,& - field with ! = #! , #" = %& + ( ) + −%+ + ,& -, where %, ( and , are constants: % = 0.05 2 #$, ( = 1.5 5⁄2 and , = 0.35 Assumptions: 2 #$. Determine the pressure as a function of & and +. Flow is steady and incompressible. Flow is laminar and fluid has constant properties. Flow is two-dimensional. No gravity in both x and y directions.  DETERMINATION OF PRESSURE FIELD FROM KNOWN VELOCITY Simplifications Steady and no gravity, For a two-dimensional flow in x and y directions, the incompressible Navier- #& = 0, Stokes equations: #! (&), #" (&, +) & − >?@A,=?BC: 0 0 0 0 0 0 9: 9 %#! 9 %#! 9 %#! 9#! 9#! 9#! 9#! 9: 9 %#! 9#! 78! − +; + + = 7 + # + # + # =; − 7 #! 9& 9& % 9+ % 9< % 9= ! 9& " 9+ & 9< 9& 9& % 9& no gravity steady + − >?@A,=?BC: 0 0 0 0 9: 9 %#" 9 %#" 9 %#" 9#" 9#" 9#" 9#" 9: 9 %#" 9 %#" 9#" 9#" 78" − +; + + = 7 + # + # + # =; + − 7 #! + #" 9+ 9& % 9+ % 9< % 9= ! 9& " 9+ & 9< 9+ 9& % 9+ % 9& 9+ DETERMINATION OF PRESSURE FIELD FROM KNOWN VELOCITY Velocity components ! – direction: Pressure fields: #! = %& + ( ! – direction: 9: 9 %#! 9#! =; − 7 #! 9#! 9& 9& % 9& =% 9& 9: 9 %#" 9 %#" 9#" 9#" 9 %#! " – direction: =0 =; + − 7 #! + #" 9& % 9+ 9& % 9+ % 9& 9+ " – direction: Substituting the velocity fields into the pressure fields: #" = −%+ + ,& In this exercise, the pressure 9: field will be determined from ! – direction: = 7 −%%& − %( 9#" 9#" 9& from x – direction component =, = −% 9+ 9& " – direction: 9: = 7 −(, + %%+ 9 %#" 9 %#" 9+ =0 =0 9+ % 9& % DETERMINATION OF PRESSURE FIELD FROM KNOWN VELOCITY Notes When performing a partial )# To find # $, & , from the differential equation: = * −,! $ − ,- )$ integration, add a variable with a function of the other variable(s) Integrating the differential equation: # $, & = ( )# = ( * −,! $ − ,- )$ New unknown function/variable ,! $ ! has appeared # $, & = * − − ,-$ + ℎ(&) 2 )# 3ℎ 3ℎ To find the new variable, differentiating = = * −-4 + ,! & )& 3& 3& #($, &) with respect to &: From the pressure field, the pressure at any point in the system can be ,! & ! & calculated. Integrating the differential equation to find ℎ(&): ! ℎ & = ( * −-4 + , & 3& = * −-4& + +5 2 ,! $ ! ,! & ! The final pressure field: # $, & = * − − ,-$ + * −-4& + +5 2 2 $ ! ! ! ! 0.05 $ 0.05 & # $, & = * − − (0.05)(1.5)$ + * −(1.5)(0.35)& + +5 2 2 PROBLEM SETUP 1. PROBLEM SETUP Sketch Sketch the geometry the geometry and identify and identify all all relevant relevant dimensions dimensions 4. INTEGRATION and parameters. and parameters. Integrate equations motions leading to 2. ASSUMPTIONS & BC’S one or more constant of List all appropriate integration. assumptions and specify boundary conditions related to the problem. 5. SOLVE Apply the BC’s to solve for the 3. EQUATION OF MOTIONS constants of integration. Check or List all relevant verify the results. differential equations and perform simplifications as much as possible  Stea d y sta te a n d In com pr essi bl e fl ow Continuity Equation: 0 )* ) ) ) + (*& #" = + O$& + O% ; 2;