Interdisciplinary Applied Mathematics

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k’j


Pr


. &L


dx J


/


0



1


d


,, (pu , dv 4


P!dy ^ dx>


+ Re


dy


2    /o dv du


3    P’^dy dx)


vi


P (2


dv


dy


du 5 dx )


■V + Pi^ + Ш)-


u +


k’j


Pr


dT dy )

where k is the thermal conductivity and 7 is the ratio of specific heats. The unknowns are the conservative variables, i.e.,


(p, pu, pv, E).


The energy is defined as


E = p[T +1/2(u2 + v2)],


and the pressure p is obtained from the equation of state


P = (Y — 1)PT.


The nondimensionalization is done with respect to reference velocity, density, and    length    scales (i.e.,    U0,    p0,    l0),    and    the    reference temperature    is


chosen as T0 = U2/Cv; here, Cv is the const ant-volume specific heat. The dynamic viscosity p is related to temperature by Sutherland’s law


h_ = fT3/2T0 + S ho To) T + S


where p0    is    the    viscosity    at    the    reference    temperature    T0,    and Si is a


constant, which for air is


Si = 110K.


It is convenient to simplify this equation to a simpler power law of the form


— = ( Jj- ) , with 0.5 < ш < 1.

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