Question

Which of these formulae is correct to find the gradient of the element ‘k’?

(a) ∇Φk = \(\frac{1}{V_k}(\Sigma_{n\leftarrow f<k}\Phi_n\vec{S_n}-\Sigma_{n\leftarrow f>k} \Phi_n\vec{S_n})\)

(b) ∇Φk = \(\frac{1}{V_k}(-\Sigma_{n\leftarrow f}\Phi_n \vec{S_n})\)

(c) ∇Φk = \(\frac{1}{V_k}(\Sigma_{n\leftarrow f}\Phi_n \vec{S_n})\)

(d) ∇Φk = \(\frac{1}{V_k}(- \Sigma_{n\leftarrow f<k}\Phi_n\vec{S_n}+\Sigma_{n\leftarrow f>k} \Phi_n\vec{S_n})\)

I got this question by my college professor while I was bunking the class.

I'm obligated to ask this question of FVM topic in section Finite Volume Methods of Computational Fluid Dynamics

Answer 1

Correct option is (d) ∇Φk = \(\frac{1}{V_k}(- \Sigma_{n\leftarrow f<k}\Phi_n\vec{S_n}+\Sigma_{n\leftarrow f>k} \Phi_n\vec{S_n})\)

For explanation: The formula for the gradient, in general, is the summation of the product of the flow variables and the area of the faces. As the direction matters here, the summation will be negative till ‘n’ reaches ‘k’. Therefore,

∇Φk = \(\frac{1}{V_k}(- \Sigma_{n\leftarrow f<k}\Phi_n\vec{S_n}+\Sigma_{n\leftarrow f>k} \Phi_n\vec{S_n})\).