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 (= ?Ym/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

?h 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?S)

Sij components of strain-rate tensor

Sx , Sy , Sz Cartesian components of the face vector

dS surface element

?S length/area of a face of a control volume

t time

tT turbulent time scale

?t time step

T static temperature

- matrix of right eigenvectors

--1 matrix of left eigenvectors

u, v, w Cartesian velocity components

ut skin friction velocity =tw/?)

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 (= [?, ?u, ?v, ?w, ?E]T)

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

x, y, z Cartesian coordinate system

?x cell size in x-direction

y+ nondimensional wall coordinate (= ? yut/µw)

Ym mass fraction of species m

z Fourier symbol of the spatial operator

a angle of attack, inlet...