As written in Toro's book[1] that eigenstructure of 3D Euler equation on Cartisan grid has been derived for 1D system. However, I haven't found any written formulas for 3D Euler equation of ideal gas. Some notes from [2] is helpful but unfortunately, internal energy is used instead of total energy. Therefore it is necessary to derive a similar eigenstructure for 3D Euler equation written in conservative form, with total energy as one of the conservative variables.
%--------------------------------------------------------------------------------------------- Notation: q=[rho,rho*u,rho*v rho*w E]'; where: -- rho:=density -- u,v,w:=velocities -- E:=total energy=p/(gamma-1)+0.5*rho*(u^2+v^2+w^2) F=[rho*u, rho*u^2+p, rho*v*u, rho*w*u, u*(E+p)] G=[rho*v, rho*u*v, rho*v^+p, rho*w*v, v*(E+p)] H=[rho*w, rho*u*w, rho*v*w, rho*w^2, w*(E+p)] %--------------------------------------------------------------------------------------------- Define R1,R2,R3 is the right eigenvectors of dF/dq, dG/dq, dH/dq, so similar to L1,L2,L3. The corresponding eigenvalue diagonal matrix is D1,D2,D3. Define v2 as (u^2+v^2+w^2)/2 ga=gamma; ga1=gamma-1; a=sqrt(gamma*p/rho); H=v2 + a^2/(gamma-1); After tedious derivation, we will have: % ============================= Right eigenvectors ========================== R1(1,:)=[1 1 1 0 0]; R1(2,:)=[u-a u u+a 0 0]; R1(3,:)=[v v v 1 0]; R1(4,:)=[w w w 0 1]; R1(5,:)=[H-u*a v2 H+u*a v w]; R2(1,:)=[1 1 1 0 0]; R2(2,:)=[u u u -1 0]; R2(3,:)=[v-a v v+a 0 0];% R2(4,:)=[w w w 0 -1];% R2(5,:)=[H-v*a v2 H+v*a -u -w];% R3(1,:)=[1 1 1 0 0]; R3(2,:)=[u u u 1 0]; R3(3,:)=[v v v 0 1]; R3(4,:)=[w-a w w+a 0 0]; R3(5,:)=[H-w*a v2 H+w*a u v]; % ============================= Left eigenvectors ========================== L1(1,:)=[H+a/ga1*(u-a) -(u+a/ga1) -v -w 1]; L1(2,:)=[-2*H+4/ga1*a^2 2*u 2*v 2*w -2]; L1(3,:)=[H-a/ga1*(u+a) -u+a/ga1 -v -w 1]; L1(4,:)=[-2*v*a^2/ga1 0 2*a^2/ga1 0 0]; L1(5,:)=[-2*w*a^2/ga1 0 0 2*a^2/ga1 0]; L1=L1*(ga1)/2/a^2; L2(1,:)=[H+a/ga1*(v-a) -u -(v+a/ga1) -w 1]; L2(2,:)=[-2*H+4/ga1*a^2 2*u 2*v 2*w -2]; L2(3,:)=[H-a/ga1*(v+a) -u -v+a/ga1 -w 1]; % L2(4,:)=[2*u*a^2/ga1 -2*a^2/ga1 0 0 0]; L2(5,:)=[2*w*a^2/ga1 0 0 -2*a^2/ga1 0]; L2=L2*(ga1)/2/a^2; L3(1,:)=[H+a/ga1*(w-a) -u -v -(w+a/ga1) 1]; L3(2,:)=[-2*H+4/ga1*a^2 2*u 2*v 2*w -2]; L3(3,:)=[H-a/ga1*(w+a) -u -v -w+a/ga1 1]; L3(4,:)=[-2*u*a^2/ga1 2*a^2/ga1 0 0 0]; L3(5,:)=[-2*v*a^2/ga1 0 2*a^2/ga1 0 0]; L3=L3*(ga1)/2/a^2; % ==================================================================== Reference: [1] Toro, Eleuterio F. Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Springer Science & Business Media, 2009. [2]Rohde, Axel. "Eigenvalues and eigenvectors of the Euler equations in general geometries." AIAA paper 2609 (2001): 2001. |
AuthorShaowu Pan Archives
December 2017
Categories
All
|