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Nonlocal theories and modelling of fully developed turbulence
MECHANICS RESEARCH COI4TONICATIONS Voi.15(4), 199-203, 1988. Printed in t h e USA 0093-6413/88 $3.00 + .00 Copyright (c) 1988 Pergamon Press plc
NONLOCAL THEORIES AND MODELLING OF FULLY DEVELOPED TURBULENCE G. Silber 1st Institute of Mechanics, Technical University of Berlin, Fed. Rep. Germany (Received 15 October 1987: accepted/or print 16 March 1988)
Introduction. In the 1960s an axiomatic phenomenological material description based on the principles of rational mechanics (determinism, material objectivity, local action) was created by the Truesdell-Noll school. This central branch of continuum mechanics contains guidelines for the generation of constitutive equations which only together with the universal balance equations lead to the necessary number of equations for unknown field quantities such as strains, displacements, density, etc. Such constitutive equations provide in general functional contexts between second order stress tensors and displacement vectors. The continuummechanical description of solids and fluids can now be fully accomplished using constitutive equations of "simple materials" [61. In this process the "physical reality" is approximated by means of a classical continuum model in which a single kinematical vector is assigned to each continuum point (for solid bodies the displacement vector, for fluids the velocity vector)i remote kinematical effects, however, remain restricted to an infinitesimal region in the sense of the principle of local action. In principle the stress tensors are thereby dependent on the first displacement resp. velocity gradient. Such constitutive equations are, however, no longer sufficient for explaining the behavior, say, of fluids with stationary fully developed turbulent flows in a continuum mechanical way [5]. The mapping of velocity profiles can be achieved (at least under exclusion of the area close to the wall)with the assistance of theories of nonlocal fluids [3,4]. These theories are characterized following [4] by the fact that the physical state of a continuum point is influenced by the kinematics of an enlarged area of influence corresponding to the "sensitivity" of the material. Now, in addition to the first velocity gradient, higher velocity gradients are also obtained as kinematical tensors in correspondence to the grade of the theory. Simultaneously, the condition of stress is no longer described by a single (second order) stress tensor, but rather by several higher order stress tensors, "dual 'T to the kinematical variables. These then form in a certain way a socalled overall stress tensor of second order. -
In particular, in order to verify now the occurrence of a inflection point at the velocity radient of a channel flow with a constant pressure gradient [2], a theory of nonlocal uids of third order is required [3].
199
200
G.
SILBER
Theory The theory of such a fluid is characterized (with restriction to locally isentropic processes [5]) by the kinematical variables C, ¥C and VVC - with the symmetric spatial velocity gradient C := (Vv + vV)/2 - and the three second, third and fourth order friction stress
(J)
tensors S R (j=2,3,4) dual to them [3]. Using these, the second order overall stress tensor is formed following (2)
(3)
(4)
SS = -pI + SR - V-SR + V-(V-SR)
(1)
in which p(p) denotes the static pressure as a function of the density p and V the spatial nabla operator. In addition to the balances of momentum and mass
V-SS + p k = p { '
, b +pV'v=O
(2),(3)
with the mass density vector k, the dissipation postulate in the form of non negative specific dissipation 13, is obtained as follows fi
(2)
(3)
(4)
= SR-- C + S R - - - (CV) + S R . . . . (CVV) > 0
(4)
Isotropic power series representations are taken as a basis for the isotropic representations
(J)
of the constitutive equations between the friction stress tensors SR 0=2,3,4) and the
(J)
(J)
kinematical tensors C, VC and VVC, which following SR = F (C,¥C,VVC) occur here as
(J) second, third and fourth order tensor functions F . In their linear versions the power series representations are as follows
(2)
SR = 3)
3
15
(4)(4)
(6)(6)
~ (2) Ak I k - - C + ~ (2)Ak I k .... (VVC) k--1 k=l 15
(6) (6)
(S R = ~ Ak Ik--- (¥C) (3) k---1 4)
15
(SR = ~ k=l
(5)
(6) (6)
105 ~
Ak Ik.- C + L (4) k=l
(6/ (8)(8) (4)
Ak Ik .... (¥VC)
(7)
NONLOCAL TURBULENCE MODELS
201
(2j) The 153 constant coefficients (i)Ak of the fourth, sixth and eighth order i8otropic tensors (21J k'~ (j=2,3,4) [3] characterize the material to be described and are therefore referred to as constitutive coefficients. This can be reduced to a total of 25 because of certain symmetries of the kinematical tensors and the friction stress tensors as well as by exclusion of
(j)
4 (j)
dissipation-neutral parts SILL with ~ SR± . . . . . . (CV V) = 0. j=2 ~ j-~ • j-2 t imes With equations (1) and (5) through (7) results the constitutive equation of the overall stress tensor of a NAVIER-STOKES fluid of grade three SE = - p I + 27tC + 272AC + 273AAC ,
(8)
Sp C = 0
provided that incompressibility is considered with V-v = 0. The 7j (j=1,2,3) represent linear combinations of the 25 constitutive coefficients mentioned above. Application to fully developed turbulence requires now, for stationary flows, that the velocity field v be understood as a quantity averaged at a location x over a certain period of time At
:=
t 0 +At 1 ~ v(x,t)dt
(9)
t=t 0 (The overbar indicating averaging is omitted in the following.) For plane flows in a channel, setting v(x) = v(y)e x - with the unit vector in the direction of flow ex, the cross-sectional coordinate y and the velocity coordinate v - insertation of (8) into (2), under consideration of conservative fields of the external body force k = -VU, leads to the ordinary differential equation of sixth order ¢1v +' dy2
2v + dy2
dY 2
= - - - - ( p + pU) =: A 73 dx
(10)
Ck := 7k/73 (k = 1,2) , A = const. In Order to determine the constant material coefficients Ck(material identification) a least square fit of the general solution of (10) with given velocity profile measurement of stationary fully developed turbulent Couette flow (A _= 0)(FIG.I, [1]) has been employed.
202
G.
SILBER
This lead to the numerical values of 01 = 0,38 cm -4 and ¢2 = - 1,2 cm -2. These results served for the subsequent verification of the velocity profile of channel flow with constant pressure gradient (A :/: 0)(FIG.2) given in [2].
9)
1.0
wu
WU
0.9
8
Theory of grade three
0.8
0.7
......
Theory of grade two
+
Measurement values
7-
/ 6~
0.6 S~ 0.5 45 0.4
3~
/
0.3
2
0.2
&
0. l
/
l
0.0 0.0
I 0.2
0.4
(a)
0.5
0.8
rl
1.0
0 0.0
o.~
o.i
o.~
o.8
(b)
,.o
q
FIG. 1 Couette flow: w U := v/U, U velocity of the moving boundaries, ~/:= y/(hc/2), h c channel width,
w U := dwu/d~/
The four illustrations show that the experimental results of both flow configurations with the strictly linear constitutive equations (5) through (8) for a nonlocal fluid of grade three can be satisfactorily illustrated qualitatively as well as quantitatively. FIG. 2b shows that at least one theory of third degree is necessary for the reproduction of the curvature change - even if the change is weak. Comparison of the wall slips calculated according to the theories of second and third degree (Wu(T/=l)=0,68 resp. Wv(~/=l)=0,61 and WU(T/=l)=0,75 resp. Wv(~/=1)=0,55) suggests the conjecture that these slips become smaller with increasing grade of the material. It remains an open question whether, with regard to the kinematical region of influence, the classical wall adhesion hypothesis (Wu(T/=I)=I resp. Wv(~/=l)=0) can be verified by a nonapproximated theory or at least
NONLOCAL TURBULENCE MODELS
203
by a materials model of grade N. .
1.05
0
~
'
'
i
i
i
0.6
0.8
i
i
w:l
WV I.OC
+
+
O. gS
-0.4
0.90
-0.6 0.85
-0.0 0.80
-I.0 0.'/5
-I .2 O, "/0 .
.
.
.
.
.
Tbeo~of ~*detwo t
+
Mewu'ement
0.0
1
according to REICHARDT
0.65
0.60
I\
values
o.2
o.~
(a)
o.~
0~
-1.4
-i .6
II
0.0
0.2
0.4
(b)
I.O
Tl
FIG. 2 Channel flow with constant pressure gradient: w V := v/V, V maximal velocity in A channel center, ~/:= y/(hk/2 ) hk channel width, w V := dwv/d7 / Reference~ [1] [2] [3] [4] [5] [6]
H. Reichhardt, ZAMM, Sonderheft, p 26 (1956) H. Reichhardt, ZAMM, Bd. 31, Nr. 7, p 208 (1951) G. Silber, Eine Systematik nichtlokaler kelvinhafter Fluide vom Grade drei auf der Basis eines klassischen Kontinuummodelles, VDI-Verlag, VDI-Fortschrittsberichte, Reihe 18, Nr. 26 (1986) R. Trostel, BeitrRge.zu den Ingenieurwissenschaften, S. 96, Universit~tsbibliothek der TU Berlin (1985) R. Trostel, Mechanik IV, Bd. 1, StrSmungsmechanik, Schriftenreihe Physikalische Ingenieurwissenschaft, Bd. 4, Universit~.tsbibliothek der TU Berlin (1985) C. Truesdell, W. Noll, The non-linear field theories of mechanics, Handbuch der Physik, Vol III/3, Springer (1965)
NONLOCAL THEORIES AND MODELLING OF FULLY DEVELOPED TURBULENCE G. Silber 1st Institute of Mechanics, Technical University of Berlin, Fed. Rep. Germany (Received 15 October 1987: accepted/or print 16 March 1988)
Introduction. In the 1960s an axiomatic phenomenological material description based on the principles of rational mechanics (determinism, material objectivity, local action) was created by the Truesdell-Noll school. This central branch of continuum mechanics contains guidelines for the generation of constitutive equations which only together with the universal balance equations lead to the necessary number of equations for unknown field quantities such as strains, displacements, density, etc. Such constitutive equations provide in general functional contexts between second order stress tensors and displacement vectors. The continuummechanical description of solids and fluids can now be fully accomplished using constitutive equations of "simple materials" [61. In this process the "physical reality" is approximated by means of a classical continuum model in which a single kinematical vector is assigned to each continuum point (for solid bodies the displacement vector, for fluids the velocity vector)i remote kinematical effects, however, remain restricted to an infinitesimal region in the sense of the principle of local action. In principle the stress tensors are thereby dependent on the first displacement resp. velocity gradient. Such constitutive equations are, however, no longer sufficient for explaining the behavior, say, of fluids with stationary fully developed turbulent flows in a continuum mechanical way [5]. The mapping of velocity profiles can be achieved (at least under exclusion of the area close to the wall)with the assistance of theories of nonlocal fluids [3,4]. These theories are characterized following [4] by the fact that the physical state of a continuum point is influenced by the kinematics of an enlarged area of influence corresponding to the "sensitivity" of the material. Now, in addition to the first velocity gradient, higher velocity gradients are also obtained as kinematical tensors in correspondence to the grade of the theory. Simultaneously, the condition of stress is no longer described by a single (second order) stress tensor, but rather by several higher order stress tensors, "dual 'T to the kinematical variables. These then form in a certain way a socalled overall stress tensor of second order. -
In particular, in order to verify now the occurrence of a inflection point at the velocity radient of a channel flow with a constant pressure gradient [2], a theory of nonlocal uids of third order is required [3].
199
200
G.
SILBER
Theory The theory of such a fluid is characterized (with restriction to locally isentropic processes [5]) by the kinematical variables C, ¥C and VVC - with the symmetric spatial velocity gradient C := (Vv + vV)/2 - and the three second, third and fourth order friction stress
(J)
tensors S R (j=2,3,4) dual to them [3]. Using these, the second order overall stress tensor is formed following (2)
(3)
(4)
SS = -pI + SR - V-SR + V-(V-SR)
(1)
in which p(p) denotes the static pressure as a function of the density p and V the spatial nabla operator. In addition to the balances of momentum and mass
V-SS + p k = p { '
, b +pV'v=O
(2),(3)
with the mass density vector k, the dissipation postulate in the form of non negative specific dissipation 13, is obtained as follows fi
(2)
(3)
(4)
= SR-- C + S R - - - (CV) + S R . . . . (CVV) > 0
(4)
Isotropic power series representations are taken as a basis for the isotropic representations
(J)
of the constitutive equations between the friction stress tensors SR 0=2,3,4) and the
(J)
(J)
kinematical tensors C, VC and VVC, which following SR = F (C,¥C,VVC) occur here as
(J) second, third and fourth order tensor functions F . In their linear versions the power series representations are as follows
(2)
SR = 3)
3
15
(4)(4)
(6)(6)
~ (2) Ak I k - - C + ~ (2)Ak I k .... (VVC) k--1 k=l 15
(6) (6)
(S R = ~ Ak Ik--- (¥C) (3) k---1 4)
15
(SR = ~ k=l
(5)
(6) (6)
105 ~
Ak Ik.- C + L (4) k=l
(6/ (8)(8) (4)
Ak Ik .... (¥VC)
(7)
NONLOCAL TURBULENCE MODELS
201
(2j) The 153 constant coefficients (i)Ak of the fourth, sixth and eighth order i8otropic tensors (21J k'~ (j=2,3,4) [3] characterize the material to be described and are therefore referred to as constitutive coefficients. This can be reduced to a total of 25 because of certain symmetries of the kinematical tensors and the friction stress tensors as well as by exclusion of
(j)
4 (j)
dissipation-neutral parts SILL with ~ SR± . . . . . . (CV V) = 0. j=2 ~ j-~ • j-2 t imes With equations (1) and (5) through (7) results the constitutive equation of the overall stress tensor of a NAVIER-STOKES fluid of grade three SE = - p I + 27tC + 272AC + 273AAC ,
(8)
Sp C = 0
provided that incompressibility is considered with V-v = 0. The 7j (j=1,2,3) represent linear combinations of the 25 constitutive coefficients mentioned above. Application to fully developed turbulence requires now, for stationary flows, that the velocity field v be understood as a quantity averaged at a location x over a certain period of time At
:=
t 0 +At 1 ~ v(x,t)dt
(9)
t=t 0 (The overbar indicating averaging is omitted in the following.) For plane flows in a channel, setting v(x) = v(y)e x - with the unit vector in the direction of flow ex, the cross-sectional coordinate y and the velocity coordinate v - insertation of (8) into (2), under consideration of conservative fields of the external body force k = -VU, leads to the ordinary differential equation of sixth order ¢1v +' dy2
2v + dy2
dY 2
= - - - - ( p + pU) =: A 73 dx
(10)
Ck := 7k/73 (k = 1,2) , A = const. In Order to determine the constant material coefficients Ck(material identification) a least square fit of the general solution of (10) with given velocity profile measurement of stationary fully developed turbulent Couette flow (A _= 0)(FIG.I, [1]) has been employed.
202
G.
SILBER
This lead to the numerical values of 01 = 0,38 cm -4 and ¢2 = - 1,2 cm -2. These results served for the subsequent verification of the velocity profile of channel flow with constant pressure gradient (A :/: 0)(FIG.2) given in [2].
9)
1.0
wu
WU
0.9
8
Theory of grade three
0.8
0.7
......
Theory of grade two
+
Measurement values
7-
/ 6~
0.6 S~ 0.5 45 0.4
3~
/
0.3
2
0.2
&
0. l
/
l
0.0 0.0
I 0.2
0.4
(a)
0.5
0.8
rl
1.0
0 0.0
o.~
o.i
o.~
o.8
(b)
,.o
q
FIG. 1 Couette flow: w U := v/U, U velocity of the moving boundaries, ~/:= y/(hc/2), h c channel width,
w U := dwu/d~/
The four illustrations show that the experimental results of both flow configurations with the strictly linear constitutive equations (5) through (8) for a nonlocal fluid of grade three can be satisfactorily illustrated qualitatively as well as quantitatively. FIG. 2b shows that at least one theory of third degree is necessary for the reproduction of the curvature change - even if the change is weak. Comparison of the wall slips calculated according to the theories of second and third degree (Wu(T/=l)=0,68 resp. Wv(~/=l)=0,61 and WU(T/=l)=0,75 resp. Wv(~/=1)=0,55) suggests the conjecture that these slips become smaller with increasing grade of the material. It remains an open question whether, with regard to the kinematical region of influence, the classical wall adhesion hypothesis (Wu(T/=I)=I resp. Wv(~/=l)=0) can be verified by a nonapproximated theory or at least
NONLOCAL TURBULENCE MODELS
203
by a materials model of grade N. .
1.05
0
~
'
'
i
i
i
0.6
0.8
i
i
w:l
WV I.OC
+
+
O. gS
-0.4
0.90
-0.6 0.85
-0.0 0.80
-I.0 0.'/5
-I .2 O, "/0 .
.
.
.
.
.
Tbeo~of ~*detwo t
+
Mewu'ement
0.0
1
according to REICHARDT
0.65
0.60
I\
values
o.2
o.~
(a)
o.~
0~
-1.4
-i .6
II
0.0
0.2
0.4
(b)
I.O
Tl
FIG. 2 Channel flow with constant pressure gradient: w V := v/V, V maximal velocity in A channel center, ~/:= y/(hk/2 ) hk channel width, w V := dwv/d7 / Reference~ [1] [2] [3] [4] [5] [6]
H. Reichhardt, ZAMM, Sonderheft, p 26 (1956) H. Reichhardt, ZAMM, Bd. 31, Nr. 7, p 208 (1951) G. Silber, Eine Systematik nichtlokaler kelvinhafter Fluide vom Grade drei auf der Basis eines klassischen Kontinuummodelles, VDI-Verlag, VDI-Fortschrittsberichte, Reihe 18, Nr. 26 (1986) R. Trostel, BeitrRge.zu den Ingenieurwissenschaften, S. 96, Universit~tsbibliothek der TU Berlin (1985) R. Trostel, Mechanik IV, Bd. 1, StrSmungsmechanik, Schriftenreihe Physikalische Ingenieurwissenschaft, Bd. 4, Universit~.tsbibliothek der TU Berlin (1985) C. Truesdell, W. Noll, The non-linear field theories of mechanics, Handbuch der Physik, Vol III/3, Springer (1965)