Hi, everyone.

I have one question about tensor notation.

du_i / (dx_j dx_k)

'd' means partial derivative

and i,j,k are index notation.

above tensor notation showed in epsilon equation of k-e turbulence mode. (Yang and Shih,1993)

I am very qurious about developing tensor notaton to cartesian form.

thanks in advance.

I think your epsilon eq. is not in the tensor notation. Rather it is in the indicial notation. In complete tensor notation you should have the symbolic form of the metrics multiplied to the different terms. The form of the metric is different in different spaces. Regarding the derivative you posted, I think it is already in the cartesian form. If you are assuming a 3D system then i,j,k independently take the values 1,2,3. 'x_1' means 'x', 'x_2' means 'y' and 'x_3' means 'z'. Further, 'u_1' means 'u', 'u_2' means 'v' and 'u_3' means 'w'. Since epsilon is a scalar therefore each term in the epsilon eq. must contain the indices i, j and k twice. When a index occurs twice in a term then you have to invoke the summation convention.