CFD Assignment 1

Problem 1 :- Temperature distribution in one dimensional rod

Problem is solved using BTCS scheme by using TDMA algorithm (a) Dirichlet boundary condition on both ends

Comment :- (a) As we are increasing number of grid points, error increases at each grid points . This may be due to error accumulation for large grid points.

(b) We observe from graph that at N=12 we get minimum error and we are not getting any advantage by increasing number of grid points, so for this problem N=12 is best grid

resolution .

Nitish KumarSC08B086

Analytical and numerical solution are plotted and linear variation of temperature is observed as expected with an error shown above.

(b) Dirichlet and Neumann boundary condition on left and right ends respectively

Comment :- (a) As we are increasing number of grid points, maximum error increases. (b) Error for each N changes randomly.

Analytical and numerical solution are plotted and linear variation of temperature is observed as expected with an error shown above.

(c) Neumann boundary condition on both endsSince only flux value is given at the ends of rod, unique solution will not exists. Our solution will depend on initial value of temperature assumed at one end. If we assume temperature at starting end to be 283K, solution will be same as case(b).

Problem 2 :- Solution of 2nd order ODE

(a) 1st order approximation of fluxWe observe that we get a large error whose variation is linear.

(a) 2nd order approximation of flux

When we use 2nd order approximation, we observe that error reduces drastically and its variation is almost linear. Also analytical and numerical solution are plotted

Problem 3 :- Wall Temperature distribution

(a) Diffusion Number d=0.4(I) FTCS Scheme

At time t=0.1hr to 0.5 hr, analytical and numerical solutions are plotted at an interval of 0.1hr.Also error for each time step is plotted.

(II) BTCS Scheme

(b) Diffusion Number d=0.6(I) FTCS

(II) BTCS

Problem 4 :-Convection of Triangular wave

comment