<s>
In	O
order	O
to	O
simplify	O
the	O
existing	O
EASM	O
and	O
to	O
achieve	O
an	O
efficient	O
numerical	O
implementation	O
the	O
underlying	O
tensor	B-Device
basis	O
plays	O
an	O
important	O
role	O
.	O
</s>
<s>
The	O
five-term	O
tensor	B-Device
basis	O
that	O
is	O
introduced	O
here	O
tries	O
to	O
combine	O
an	O
optimum	O
of	O
accuracy	O
of	O
the	O
complete	O
basis	O
with	O
the	O
advantages	O
of	O
a	O
pure	O
2d	O
concept	O
.	O
</s>
<s>
Based	O
on	O
that	O
the	O
new	O
model	O
is	O
designed	O
and	O
validated	O
in	O
combination	O
with	O
different	O
eddy-viscosity	O
type	O
background	O
models	O
.	O
</s>
<s>
Due	O
to	O
numeric	O
requirements	O
an	O
explicit	O
formulation	O
based	O
on	O
a	O
low	O
number	O
of	O
tensors	B-Device
is	O
desirable	O
and	O
was	O
already	O
introduced	O
originally	O
most	O
explicit	B-Algorithm
algebraic	I-Algorithm
stress	I-Algorithm
models	I-Algorithm
are	O
formulated	O
using	O
a	O
10-term	O
basis	O
:	O
</s>
<s>
The	O
reduction	O
of	O
the	O
tensor	B-Device
basis	O
however	O
requires	O
an	O
enormous	O
mathematical	O
effort	O
,	O
to	O
transform	O
the	O
algebraic	O
stress	O
formulation	O
for	O
a	O
given	O
linear	O
algebraic	O
RSTM	O
into	O
a	O
given	O
tensor	B-Device
basis	O
by	O
keeping	O
all	O
important	O
properties	O
of	O
the	O
underlying	O
model	O
.	O
</s>
<s>
This	O
transformation	O
can	O
be	O
applied	O
to	O
an	O
arbitrary	O
tensor	B-Device
basis	O
.	O
</s>
<s>
In	O
the	O
present	O
investigations	O
an	O
optimum	O
set	O
of	O
basis	O
tensors	B-Device
and	O
the	O
corresponding	O
coefficients	O
is	O
to	O
be	O
found	O
.	O
</s>
<s>
The	O
projection	B-Algorithm
method	O
was	O
introduced	O
to	O
enable	O
an	O
approximate	O
solution	O
of	O
the	O
algebraic	O
transport	O
equation	O
of	O
the	O
Reynolds-stresses	O
.	O
</s>
<s>
In	O
contrast	O
to	O
the	O
approach	O
of	O
the	O
tensor	B-Device
basis	O
is	O
not	O
inserted	O
in	O
the	O
algebraic	O
equation	O
,	O
instead	O
the	O
algebraic	O
equation	O
is	O
projected	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
chosen	O
basis	O
tensors	B-Device
does	O
not	O
need	O
to	O
form	O
a	O
complete	O
integrity	O
basis	O
.	O
</s>
<s>
However	O
,	O
the	O
projection	B-Algorithm
will	O
fail	O
if	O
the	O
basis	O
tensor	B-Device
are	O
linear	O
dependent	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
complete	O
basis	O
the	O
projection	B-Algorithm
leads	O
to	O
the	O
same	O
solution	O
as	O
the	O
direct	O
insertion	O
,	O
otherwise	O
an	O
approximate	O
solution	O
in	O
the	O
sense	O
is	O
obtained	O
.	O
</s>
<s>
In	O
order	O
to	O
prove	O
,	O
that	O
the	O
projection	B-Algorithm
method	O
will	O
lead	O
to	O
the	O
same	O
solution	O
as	O
the	O
direct	O
insertion	O
,	O
the	O
EASM	O
for	O
two-dimensional	O
flows	O
is	O
derived	O
.	O
</s>
<s>
In	O
two-dimensional	O
flows	O
only	O
the	O
tensors	B-Device
are	O
independent	O
.	O
</s>
<s>
The	O
projection	B-Algorithm
leads	O
then	O
to	O
the	O
same	O
coefficients	O
.	O
</s>
<s>
For	O
example	O
the	O
shear	O
stress	O
variation	O
in	O
a	O
rotating	O
pipe	O
cannot	O
be	O
predicted	O
with	O
quadratic	O
tensors	B-Device
.	O
</s>
<s>
Hence	O
,	O
the	O
EASM	O
was	O
extended	O
with	O
a	O
cubic	O
tensor	B-Device
.	O
</s>
<s>
In	O
order	O
to	O
do	O
not	O
affect	O
the	O
performance	O
in	O
2D	O
flows	O
,	O
a	O
tensor	B-Device
was	O
chosen	O
that	O
vanish	O
in	O
2d	O
flows	O
.	O
</s>
<s>
A	O
cubic	O
tensor	B-Device
,	O
which	O
vanishes	O
in	O
3d	O
flow	O
is	O
:	O
</s>
<s>
The	O
projection	B-Algorithm
with	O
tensors	B-Device
T(1 )	O
,	O
T(2 )	O
,	O
T(3 )	O
and	O
T(5 )	O
yields	O
then	O
the	O
coefficients	O
of	O
the	O
EASM	O
.	O
</s>
<s>
Since	O
in	O
the	O
projections	B-Algorithm
to	O
determine	O
the	O
EASM	O
coefficients	O
the	O
complexity	O
is	O
reduced	O
by	O
neglecting	O
higher	O
order	O
invariants	O
.	O
</s>
