# fsd weighted (vn+sd) correlations

[u, v, w, y]

c(latex="c") = 1.0 - y
wrate (latex="\dot{\omega}") = rxn_rate(c)
fsd (latex="\Sigma_f'")= sqrt (sqr(ddx(c)) + sqr(ddy(c)) + sqr(ddz(c)))

lapc(latex="\Delta c") = d2dx(c) + d2dy(c) + d2dz(c)
sd(latex="S_d") = ($rod * lapc + wrate) / fsd

nx(latex="n_x") = - ddx(c) / fsd
ny(latex="n_y") = - ddy(c) / fsd
nz(latex="n_z") = - ddz(c) / fsd

unx = u * nx
vny = v * ny
wnz = w * nz
vn(latex="v_n") = unx + vny + wnz

divn(latex="\nabla \cdot \vec{n}") = ddx(nx) + ddy(ny) + ddz(nz)

divn2 = sqr(divn)

# grad(n) : grad(n)
gradn_gradn(latex="\nabla \vec{n} : \nabla \vec{n}") = sqr(ddx(nx)) + sqr(ddy(ny)) + sqr(ddz(nz)) + 2*(ddx(ny)*ddy(nx)) + 2*(ddx(nz)*ddz(nx)) + 2*(ddy(nz)*ddz(ny))

absk = abs(divn)

vnsd = vn+sd

divn_nx = divn*nx

gradn_vnsd (latex="\nabla_{\mathrm{n}}(v_n+S_d)") = ddx(vnsd) * nx + ddy(vnsd) * ny + ddz(vnsd) * nz

gradt_vnsd1 (latex="\nabla_{\top,x}(v_n+S_d)")= ddx(vnsd) - nx * gradn_vnsd

divn2_dndn = divn2 - gradn_gradn

sign_divn(latex="\mathrm{sign}(\nabla \cdot \vec{n})") = $dsign(1.0, divn)

sign_divn2_dndn = sign_divn * divn2_dndn

vnsd_divn = vnsd * divn

vnsd_divn_cor = vnsd' * divn'

vnsd_divn_nx_cor = vnsd' * divn_nx'

vnsd_divn2_dndn_cor = vnsd' * divn2_dndn'

vnsd_sign_divn2_dndn_cor = vnsd' * sign_divn2_dndn'

vnsd_divn_nx = vnsd * divn_nx

vnsd_divn2_dndn = vnsd * divn2_dndn

vnsd_sign_divn2_dndn = vnsd * sign_divn2_dndn

vnsd_divn_nxfluc = vnsd * divn * nx'

vnsd_divn_divnfluc = vnsd * divn * divn'

vnsd_divn_abskfluc = vnsd * divn * absk'

vnsd_dndn = vnsd * gradn_gradn

sign_vnsd_dndn = sign_divn * vnsd * gradn_gradn

avg { c, u, v, w, fsd, lapc, ddx_c, ddy_c, ddz_c, d2dx_c, d2dy_c, d2dz_c }

avg fsd { u, v, w, nx, ny, nz, absk, divn,
unx, vny, wnz,
vn, sd,
gradt_vnsd1,
vnsd,
vnsd_divn,
vnsd_divn_cor,
     divn_nx,
vnsd_divn_nx,
vnsd_divn_nx_cor,
     divn2_dndn,
vnsd_divn2_dndn,
vnsd_divn2_dndn_cor,
     sign_divn2_dndn,
vnsd_sign_divn2_dndn,
vnsd_sign_divn2_dndn_cor,

vnsd_divn_nxfluc,
vnsd_divn_divnfluc,
vnsd_divn_abskfluc,
vnsd_dndn,
sign_vnsd_dndn

}