NUMERICAL SOLUTION OF COUETTE FLOW USING CRANK NICOLSON TECHNIQUE

MANOJKUMAR MAURYA

M.E. 13906

DEFINATION

• It is a flow between two parallel plates in which the lower plate is at rest while the upper plate is moving.

• The plates are considered to be infinitely long• The flow field is driven by the shear stress exerted on fluid due to the

movement of the upper plate.

Analytic method

• The governing equation for the flow is

• The exact analytic solution for the velocity profile of couette flow is

• The result is the exact linear profile

Crank Nicolson technique

• We assume that the velocity profile is not linear.• Let us assume a velocity profile defined as

u=0 for 0 y < D u= for y = D

• we consider this to be initial condition at t = 0• We step up a time marching solution and expect

to see the changes in velocity profile and after enough time step velocity profile will approach in steady flow state

Governing equation• The governing equation for unsteady, incompressible, couette flow is

which is a parabolic partial differential equation

• For the convenience, the above equation is converted to non-dimensional form by defining following non-dimensional variables

• By substituting in above equation we get the following equation in a non-dimensional form or

• For the above equation we find the numerical solution

• For the convenience purpose we drop the primes, thus the above equation becomes

Crank Nicolson technique, the finite difference representation

• The crank nicolson technique, the finite difference representation is

which can be simplified as

where

• We get different equations as we apply this equation at different grid points

• The order of the matrix depends upon the number of grid points taken into consideration

• The matrix thus obtained is tridiagonal matrix which is converted into bidiagonal form using Thomas algorithm or Gauss elimination method

Stability criteria

• The main advantage of implicit method is that we can take a large value of time step

• Also crank nicolson technique is unconditionally stable

• For explicit method, the stability criteria is

• With above equation, for the implicit method we calculate as

Representation of tridiagonal matrix

• = where subscript denotes the grid points while the superscript denotes the time step

• For solving the above matrix we have to convert it into bidiagonal matrix using gauss elimination or Thomas algorithm

Thomas algorithm• Consider a system of N linear, simultaneous

equations with N unknowns, , ,...., given in the form

+ = + = + =

= =

• This is a tridiagonal system, with finite coefficients only on main diagonal, lower diagonal , and upper diagonal

• Thomas algorithm is essentially the result of applying gaussian elimination to the tridiagonal system of equations

• By applying Gaussian elimination we realize that except for the first equation in all the other equations we drop the first term and replace the coefficient of the main-diagonal term by

i=2,3,...,Nand replace the term on right by

i=2,3,....,Nthus we get a upper bidiagonal form of equations given by

= = = = =

In the last equation we have only one unknown which can be found by

The value of can be found by following equation = The above equation can be generalized as followed

Using above equation we can calculate the velocities at different grid points and we can repeat the same procedure for different time step.

This is the velocity how the velocity profile changes at different time step.