901 Words4 Pages

The second order upwind discretization model was used in this problem. The truncated error due to selected terms in the Taylor series expansion is reduced and a more accurate solution is implied. Fewer grid points are necessary to give the same level of accuracy.

∂u/∂x+u=0,〖(du/dx)〗_i + ui =0 u_(i-1)=u_i-∆x(du/dx)_i+(∆x^2)/2 ((d^2 u)/(dx^2 ))_i-(∆x^3)/6 ((d^3 u)/(dx^3 ))_i+0(∆x^4) u_(i-1)=u_i+∆x(du/dx)_i+(∆x^2)/2 ((d^2 u)/(dx^2 ))_i-(∆x^3)/6 ((d^3 u)/(dx^3 ))_i+0(∆x^4)

The central differencing method is used to find an expression for d2u/dx2 in the form ui-1 +ui+1 u_(i-1)+u_(i+1)=〖2u〗_i+∆x^2 ((d^2 u)/(dx^2 ))_i+O(∆x^4) ((d^2 u)/(dx^2 ))_i=((u_(i-1) )+(u_(i-1) )-(2u_i))/(∆x^2 )

This is the discretised expression for d2u/dx2 and the truncation error is above O(∆x^2)

Q6a. You were instructed to use an inviscid flow model. Justify the use of that model for this calculation. (2 marks)

Assuming no viscous forces present an inviscid model has been used for the calculations. Also from the equation of the Reynolds number Re=ρvl/μ due to Re being really big rearranging and assuming v and l to be constant the viscous force μ =ρvl/Re becomes negligible.

Q6b. Write down the simplified Cartesian Navier-Stokes momentum equations that you think are the closest representation to the equations you actually solved in Fluent. State why these aren’t actually the equations you solved. (10 marks)

Navier stokes momentum

∂u/∂x+u=0,〖(du/dx)〗_i + ui =0 u_(i-1)=u_i-∆x(du/dx)_i+(∆x^2)/2 ((d^2 u)/(dx^2 ))_i-(∆x^3)/6 ((d^3 u)/(dx^3 ))_i+0(∆x^4) u_(i-1)=u_i+∆x(du/dx)_i+(∆x^2)/2 ((d^2 u)/(dx^2 ))_i-(∆x^3)/6 ((d^3 u)/(dx^3 ))_i+0(∆x^4)

The central differencing method is used to find an expression for d2u/dx2 in the form ui-1 +ui+1 u_(i-1)+u_(i+1)=〖2u〗_i+∆x^2 ((d^2 u)/(dx^2 ))_i+O(∆x^4) ((d^2 u)/(dx^2 ))_i=((u_(i-1) )+(u_(i-1) )-(2u_i))/(∆x^2 )

This is the discretised expression for d2u/dx2 and the truncation error is above O(∆x^2)

Q6a. You were instructed to use an inviscid flow model. Justify the use of that model for this calculation. (2 marks)

Assuming no viscous forces present an inviscid model has been used for the calculations. Also from the equation of the Reynolds number Re=ρvl/μ due to Re being really big rearranging and assuming v and l to be constant the viscous force μ =ρvl/Re becomes negligible.

Q6b. Write down the simplified Cartesian Navier-Stokes momentum equations that you think are the closest representation to the equations you actually solved in Fluent. State why these aren’t actually the equations you solved. (10 marks)

Navier stokes momentum

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