List of Symbols

c Jacobian of convective fluxes

v Jacobian of viscous fluxes

b constant depth of control volume in two dimensions

c speed of sound

cp specific heat coefficient at constant pressure

cv specific heat coefficient at constant volume

→ vector of characteristic variables

Cm molar concentration of species m (= ρYm/Wm)

CS Smagorinsky constant

v→ curl of →∇→×v→=∂w∂y−∂v∂z,∂u∂z−∂w∂x,∂v∂x−∂u∂y

d distance

D diagonal part of implicit operator

→ artificial dissipation

Dm effective binary diffusivity of species m

v→ divergence of →∇→⋅v→=∂u∂x+∂v∂y+∂w∂z

e internal energy per unit mass

E total energy per unit mass

f Fourier symbol of the time-stepping operator

→e vector of external volume forces

→ flux vector

¯¯ flux tensor

g amplification factor

→ grid velocity

h enthalpy

Δh local grid (cell) size

H total (stagnation) enthalpy

- Hessian matrix (matrix of second derivatives)

I imaginary unit (=−1)

 identity matrix

¯¯ unit tensor

h2h interpolation operator

h2h restriction operator

2hh prolongation operator

- system matrix (implicit operator)

J−1 inverse of determinant of coordinate transformation Jacobian

k thermal conductivity coefficient

K turbulent kinetic energy

Kf , Kb forward and backward reaction rate constants

lT turbulent length scale

L strictly lower part of implicit operator

Lij components of Leonard stress tensor

M Mach number

- mass matrix

→ unit normal vector (outward pointing) of control volume face

nx , ny , nz components of the unit normal vector in x-, y-, z-direction

N number of grid points, cells, or control volumes

NA number of adjacent control volumes

NF number of control volume faces

p static pressure

P production term of kinetic turbulent energy

- transformation matrix from primitive to conservative variables

-L,P-R left and right preconditioning matrix (Krylov-subspace methods)

Pr Prandtl number

˙h heat flux due to radiation, chemical reactions, etc.

Q source term

→ position vector (Cartesian coordinates); residual (GMRES)

→ij vector from point i to point j

R specific gas constant

Ru universal gas constant (= 8314.34 J/kg mol K)

→ residual, right-hand side

→* smoothed residual

- rotation matrix

Re Reynolds number

˙m rate of change of species m due to chemical reactions

→ face vector (n→ΔS)

Sij components of strain-rate tensor

Sx , Sy , Sz Cartesian components of the face vector

dS surface element

ΔS length/area of a face of a control volume

t time

tT turbulent time scale

Δt time step

T static temperature

- matrix of right eigenvectors

-−1 matrix of left eigenvectors

u, v, w Cartesian velocity components

uτ skin friction velocity =τw/ρ)

U general (scalar) flow variable

U strictly upper part of implicit operator

→ vector of general flow variables

→ velocity vector with the components u,v, and w

V contravariant velocity

Vr contravariant velocity relative to grid motion

Vt contravariant velocity of a face of the control volume

Wm molecular weight of species m

→ vector of conservative variables (= [ρ, ρu, ρv, ρw, ρE]T)

→p vector of primitive variables (= [p, u, v, w, T]T)

x, y, z Cartesian coordinate system

Δx cell size in x-direction

y+ nondimensional wall coordinate (= ρ yuτ/μw)

Ym mass fraction of species m

z Fourier symbol of the spatial operator

α angle of attack, inlet...