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Prévia do material em texto

Appendix A
General constitutive relation
In this appendix� the reasoning leading to non�linear constitutive relations is sum�
marized� The derivation of a general constitutive relation which consists of the work
done by Lumley ����� Pope ���� and others is discussed� The development method for
a general constitutive relation proposed by Shih and Lumley �	� �
� is shown� Sub�
sequently� the relation between this general form and the general form used in this
work is determined�
The basic assumption of an eddy�viscosity hypothesis is that the Reynolds�stresses
are uniquely related to the rates of strain and local scalar quantities� The assumption
of the locally determined Reynolds�stresses is in contrast with the exact Reynolds�
stress equations� where convection terms are present� Convection is neglected� which
is strictly only correct when homogeneity of the rates of strain occurs� However� in
nearly homogeneous �ows� where local e�ects dominate transport e�ects� an e�ective
viscosity hypothesis may provide an adequate representation of the Reynolds�stresses�
Lumley ���� discussed the possible constitutive relations for turbulent correlations�
One of his conclusions was that turbulent constitutive relations may exist for situa�
tions in which the length and time�scales of turbulence are smaller than those in the
mean �ow 
eld so that the e�ect of initial and boundary conditions on the relations is
not signi
cant� However� in most practical situations� the scales of turbulence are of
the same order of magnitude as those of the mean �ow 
eld� hence a general turbulent
relation should contain time�history and spatial information� As an engineering ap�
proach� the time and spatial e�ects are neglected and the relationship at the present
time and the local point is considered as the 
rst order approximation in the time
and spatial expansions of the functional form�
On dimensional grounds� at least two scaling parameters are needed to relate the
Reynolds�stresses to the rates of strain� These can be chosen as a velocity scale v
and a time�scale � � v� provides the dimensions of stress and � can be used to non�
dimensionalize the rate of strain�
A convenient choice is that of k and � ��t �
k
�
��
���
���
By applying dimensional analysis� imposing invariance under co�ordinate transforma�
tions and exploiting the tensor properties of �ui
�xj
and uiuj� the form of the general
stress�strain relation can be deduced�
The number of independent invariants depends on the number of independent tensors
which can be formed� Shih and Lumley �	� showed a procedure for determining the
seventeen independent second rank tensors which can be formed by two general sec�
ond rank tensors using generalized Caley�Hamilton formulas �Rivlin ���	���
Assuming the Reynolds�stresses to be a function of mean velocity gradient� turbulent
kinetic energy k and turbulent dissipstion rate �� means writing�
uiuj � Fij�
�ui
�xj
� k� �� �A���
Non�dimensionalization results in regrouping the arguments as
Aij � �
�ui
�xj
and
uiuj
�k
� Fij�Aij�
where � is a turbulent time�scale�
Since A is a general tensor� its transpose B �� A �Bij � ATij � �
�uj
�xi
�� In order to
obtain a general relationship� a general tensorial form of Fij�Aij� Bij� is 
rst sought�
Subsequently� Bij is replaced by ATij� Shih and Lumley �	� have shown that the
independent tensors formed by A and B are the following seventeen tensors�
A�A�� B�B�� AB�BA�AB�� A�B�BA�� B�A�A�B�� B�A��
ABA�� B�AB�AB�A�� B�A�B�ABA�B��
To determine the form of the tensor function Fij� an invariant basis is 
rst formed
using two arbitrary non�dimensional vectors �i and �i as follows �Lumley �	����
�ij�i�j� Aij�i�j� ���� B
�
ij�i�j� ����AB�ij�i�j� ���� �AB
��ij�i�j� ����
�A�B��ij�i�j� ���� �ABA
��ij�i�j� ���� �AB
�A��ij�i�j� ���� �ABA
�B��ij�i�j�
Then uiuj�i�j
�k
is required to be a linear function of the above invariants� This is because
�i and �i are the arbitrary vectors� and
uiuj�i�j
�k
is bilinear in �i�j� therefore� the form
of the function should also be bilinear in �i and �j�
�	� A� General constitutive relation
Finally the following relation is obtained�
uiuj
�k
� a��ij � a�A� a�B � a�A� � a�B� � a�AB � a�BA
� a�AB� � a	A�B � a�
BA� � a��B�A� a��A�B�
� a��B�A� � a��ABA� � a��B�AB � a��AB�A�
� a��B�A�B � a���ABA�B� �A�B�AB� �A���
where a��a�� are in general scalar functions of all invariants of the tensors in question�
Note that the two tensors in the last term can be combined as they are not linearly
independent� however the current form is more convenient to satisfy the symmetry
condition which will be imposed on �A����
Using the conditions� uiuj � ujui� uiui � �k and Bij � ATij � Aji� the following
relations between the coe�cients are obtained�
a� � a� � a� � a� � a� � a	 � a�
 � a�� � a�� � a�� � a�� � a��
and
a� �
�
�
h
�� �a�Aii � �a�AikAki � �a� � a��AikBki � ��a� � a�
�AikB
�
ki
� �a�� � a���A
�
ikB
�
ki � �a��AikBklA
�
li � �a��AikB
�
klA
�
li
� �a��AikBklA
�
lmB
�
mi
i
�A���
where A�ij � AikAkj or u
�
i�j � ui�kuk�j� where ui�j is used as a shorthand for
�ui
�xj
�
Therefore we obtain�
uiuj
k
�
�
�
�ij � �a�� �ui�j � uj�i �
�
�
uk�k�ij�
� �a��
��u�i�j � u
�
j�i �
�
�
���ij� � �a��
��ui�kuj�k �
�
�
���ij�
� �a��
��uk�iuk�j �
�
�
���ij� � �a��
��ui�ku
�
j�k � u
�
i�kuj�k �
�
�
���ij�
� �a�
�
��uk�iu
�
k�j � uk�ju
�
k�i �
�
�
���ij�
� �a���
��u�i�ku
�
j�k �
�
�
���ij� � �a���
��u�k�iu
�
k�j �
�
�
���ij�
� �a���
��ui�kul�ku
�
l�j � uj�kul�ku
�
l�i �
�
�
���ij�
� �a���
��ui�ku
�
l�ku
�
l�j � uj�ku
�
l�ku
�
l�i �
�
�
���ij�
� �a���
��ui�kul�ku
�
l�mu
�
j�m � uj�kul�ku
�
l�mu
�
i�m �
�
�
���ij� �A�
�
�	�
where
�� � ui�kuk�i��� � ui�kui�k��� � ui�ku
�
i�k��� � u
�
i�ku
�
i�k�
�� � ui�kul�ku
�
l�i��� � ui�ku
�
l�ku
�
l�i��� � ui�kul�ku
�
l�mu
�
i�m �A���
It can be seen that the 
rst two terms on the right hand side of �A�
� represent the
standard k�� eddy�viscosity model� and that the 
rst 
ve terms of �A�
� are similar
to the models derived from both the two�scale DIA approach �Yoshizawa ������ and
the RNG method �Rubinstein and Barton ������� The relation �A�
� is the most
general model for uiuj under the assumption of �A���� It contains eleven undetermined
coe�cients which are� in general� scalar functions of various invariants of the tensors
in question� The detailed forms of these scalars must be determined by other model
constraints �e�g� realizability� and experimental data� Equation �A�
� contains twelve
terms� however� in practice� one may not need all of these terms�
In this work� the constitutive relation is restricted to cubic terms� and is systematically
written using shear and rotation components instead of strains� In the remaining of
this appendix� the relation between the general cubic constitutive relation used in this
work �A�	�� and the general constitutive relation �A�
� that was developed by Shih
and Lumley �	�� is derived�
�bij �
u�iu
�
j
k
�
�
�
�ij
� ��c�� �Sij �
�
�
�ij �Sll�
�c�� �Sik �Skj �
�
�
�ij �Slk �Skl� � c����ik �Skj � �Sik ��kj�
�c����ik ��kj �
�
�
�ij ��lk ��kl�
�c����ik �Skl �Slj � �Sik �Skl��lj�
�c����ik ��kl �Slj � �Sik ��kl ��lj � ��lk ��kl �Sij �
�
�
��kl �Slm��mk�ij�
�c�� �Slk �Skl �Sij� � c����lk ��kl �Sij�� �A�	�
Using the de
nitions of the components of the shear and rotation tensors�
Sij �
�
�
�ui�j � uj�i� and �ij �
�
�
�ui�j � uj�i� �
the strains appearing in relation �A�
� can be written as
ui�j � Sij � �ij� uj�i � Sji � �ji � Sij � �ij � �A���
�	� A� General constitutive relation
Using equations �A���� the following relations can be obtained�
ui�j � uj�i �
�
�
uk�k�ij
� Sij �
�
�
�Skk � �kk��ij
� Sij �
�
�
�Skk��ij
�� � ui�kuk�i
� �Sik � �ik��Sik ��ik� � SikSik � Sik�ik � �ikSik � �ik�ik
� SklSlk � �kl�lk
u�i�j � u
�
j�i �
�
�
���ij
� �Sik � �ik��Skj � �kj� � �Sjk � �jk��Ski � �ki� �
�
�
���ij
� SjkSki � Sjk�ki � �jkSki � �jk�ki
�SikSkj � Sik�kj � �ikSkj � �ik�kj �
��
���ij
� �SikSkj � ��ik�kj �
�
�
�SklSlk � �k�l�l�k��ij
�� � ui�kui�k
� �Sik � �ik��Sik � �ik� � SikSik � Sik�ik � �ikSik � �ik�ik
� SklSlk � �kl�lk � ��klSlk
ui�kuj�k �
�
�
���ij
� �Sik � �ik��Sjk � �jk��
�
�
���ij
� SikSjk � Sik�jk � �ikSjk � �ik�jk �
�
�
���ij
� SikSkj � Sik�kj � �ikSkj � �ik�kj �
�
�
�SklSlk � �kl�lk � ��klSlk��ij
�	�
uk�iuk�j �
�
�
���ij
� �Ski � �ki��Skj � �kj��
�
�
���ij
� SkiSkj � Ski�kj � �kiSkj � �ki�kj �
�
�
���ij
� SikSkj � Sik�kj � �ikSkj � �ik�kj �
�
�
�SklSlk � �kl�lk � ��klSlk��ij
�� � uk�luk�mum�l
� �Skl � �kl��Skm � �km��Slm � �lm�
� �SklSkm � Skl�km � �klSkm � �kl�km��Slm � �lm�
� SklSkmSlm � Skl�kmSlm � �klSkmSlm � �kl�kmSlm
�SklSkm�lm � Skl�km�lm � �klSkm�lm � �kl�km�lm
� SklSlmSmk � �kl�lm�mk � SklSlm�mk � �klSlmSmk
��klSlm�mk � Skl�lmSmk � Sk�lm�mk � �kl�lSmk
� SklSlmSmk � �kl�lm�mk � �klSlm�mk � �klSlmSmk
ui�ku
�
j�k � u
�
i�kuj�k �
�
�
���ij
� �Sik � �ik��Sjl � �jl��Slk � �lk�
��Sil � �il��Slk � �lk��Sjk � �jk��
�
�
���ij
� �Sik � �ik��SjlSlk � Sjl�lk � �jlSlk � �jl�lk�
��SilSlk � Sil�lk � �ilSlk � �il�lk��Sjk � �jk��
�
�
���ij
� SikSjlSlk � SikSjl�lk � Sik�jlSlk � Sik�jl�lk
��ikSjlSlk � �ikSjl�lk � �ik�jlSlk � �ik�jl�lk
�SilSlkSjk � Sil�lkSjk � �ilSlkSjk � �il�lkSjk
�SilSlk�jk � Sil�lk�jk � �ilSlk�jk � �il�lk�jk �
�
�
���ij
� SikSklSlj � SilSlkSkj � �ik�kl�lj � �il�lk�kj
�Sik�klSlj � SikSkl�lj � Sik�kl�lj � �ikSklSlj
��ik�klSlj � �ikSkl�lj � Sil�lkSkj � �ilSlkSkj
��il�lkSkj � SilSlk�kj � Sil�lk�kj � �ilSlk�kj �
�
�
���ij
� �SikSklSlj � ��ikSklSlj � �SikSkl�lj � ��ikSkl�lj
�
�
�
�SklSlmSmk � �kl�lm�mk � �klSlm�mk � �klxlmSmk��ij
�	
 A� General constitutive relation
uk�iu
�
k�j � uk�ju
�
k�i �
�
�
���ij
� �Ski � �ki��Skl � �kl��Slj � �lj�
��Skj � �kj��Skl � �kl��Sli � �li��
�
�
���ij
� �Sik � �ik��SklSlj � Skl�lj � �klSlj � �kl�lj�
��Sjk � �jk��SklSli � Skl�li � �klSli � �kl�li��
�
�
���ij
� �SikSklSlj � SikSkl�lj � Sik�klSlj � Sik�kl�lj�
���ikSklSlj � �ikSkl�lj � �ik�klSlj � �ik�kl�lj�
SjkSklSli � SjkSkl�li � Sjk�klSli � Sjk�kl�li
��jkSklSli � �jkSkl�li � �jk�klSli � �jk�kl�li �
�
�
���ij
� �SikSklSlj � ��ikSklSlj � �SikSkl�lj � ��ikSkl�lj
�
�
�
�SklSlmSmk � �kl�lm�mk � �klSlm�mk � �klSlmSmk��ij
Using the previous relations� the restriction of the general constitutive relation �A�
�
to terms up to third order� can be rewritten as�
uiuj
k
�
�
�
�ij
� �a��
�
Sij �
�
�
�Skk�
�
� �a��
�
�
�SikSkj � ��ik�kj �
�
�
�SklSlk � �k�l�l�k��ij
�
� �a��
� �SikSkj � Sik�kj � �ikSkj � �ik�kj
�
�
�
�SklSlk � �kl�lk � ��klSlk��ij
�
� �a��
� �SikSkj � Sik�kj � �ikSkj � �ik�kj
�
�
�
�SklSlk � �kl�lk � ��klSlk��ij
�
� �a��
� ��SikSklSlj � ��ikSklSlj � �SikSkl�lj � ��ikSkl�lj
�
�
�
�SklSlmSmk � �kl�lm�mk � �klSlm�mk � �klSlmSmk��ij
�
� �a�
�
� ��SikSklSlj � ��ikSklSlj � �SikSkl�lj � ��ikSkl�lj
�
�
�
�SklSlmSmk � �kl�lm�mk � �klSlm�mk � �klSlmSmk��ij
�
�	�
This last relation is equivalent to the relation �A�	�� if the coe�cients are related
to each other in the following way �using the relation SikSklSmk �
�
�
SklSlmSmk �
�
�
SklSklSij� obtained by using the Cayley�Hamilton theorem��
�a� � ��a� � a�
�� �Slk �Skl � ��c� � c���lk ��kl � c� �Slk �Skl � c���lk ��kl�
��a� � a� � a�� � c��
�a� � a�� � c��
��a� � a� � a�� � c��
��a� � a�
� � c��
�a� � a�
� � c�

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