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Unformatted text preview: OVERVIEW BALANCE EQUATIONS ISOTHERMAL SINGLEPHASE FLOW AND
TURBULENCE MODELS André Bakker CONTENTS
2 OVERVIEW EQUATIONS FOR TIME AVERAGED INCOMPRESSIBLE STEADY STATE FLOW
3 CONTINUITY EQUATION
MOMENTUM BALANCE
MEAN MOTION  CONTINUITY
MEAN MOTION — MOMENTUM
4 DEFINITIONS REYNOLDS AVERAGING
5 REYNOLDS STRESSES  ANALYTICAL EQUATION (incompressible flow)
6 TURBULENT KINETIC ENERGY  ANALYTICAL EQUATION (Incompressible flow)
TURBULENT KINETIC ENERGY — MODEL EQUATION (Incompressible flow)
7 ENERGY DISSIPATION RATE — ANALYTICAL EQUATION
ENERGY DISSIPATION RATE AT HIGH REYNOLDS NUMBERS
8 HIGH REYNOLDS NO. k—e MODEL (incompressible flow)
LOW REYNOLDS N0. k—e MODEL (incompressible flow)
9 MODEL EQUATIONS FULL RSMMODEL (incompressible flow)  PAGE A
10 MODEL EQUATIONS FULL RSMMODEL (incompressible flow)  PAGE B
11 MODEL EQUATIONS ASM—MODEL (incompressible flow)
12 VECTOR NOTATION OVERVIEW EQUATIONS FOR TIME AVERAGED INCOMPRESSIBLE STEADY STATE FLOW The continuity equation: ))
V.U = 0
The momentum balance: _) v.(ﬁﬁ) = it :1“,
=lw + ) T The molecular stress tensor: FILL a
a
I + vﬁﬁ + (66?) 'DI’U The Reynolds stress tensor: 9 _
—) )) T = uu k—a model:
3 3 ) —)
nT = — g k I + vt(vﬁ + (3U)T) Turbulent viscosity: _ 2
vt — c“ k /2 Model equation for k: )) ) >
U.Vk = V.((v + v /ok) Vk) + Pk  a t 9; )—> Pk =  uu:[VU) Model equation for e: —)—> _) —) g _ g—
U.Vs — V.((v + vt/ce) V8) + 081 E Pk 082 k
Algebraic stress model:
P
k
9 9 (1—c )—— 1—c
71T=§kf1—P +P 2 §[§—§Pki]
_§(1—c1) _§(1—c1)
c c
9 __ __
4
P: names) + (aa.(aanT] CONTINUITY EQUATION 6 ‘ “ _
a—Xj(P U1)  O + MOMENTUM BALANCE A 6U 3 “‘ 6 ‘“ “ 6P 6 1 ——(pU.) + ———(pU.U ) = —~__ ___ [ #(___ 6t 1 axk 1 k 6xi 6xk axk 2 aUm 3 5;; 61k = 0 for 1ncompres51ble flows
MEAN MOTION  CONTINUITY 6p 6 _ 5? + 6x.(p U1) ‘ 0 For incompressible flows: 6Ui
at” 1 MEAN MOTION — MOMENTUM
6U
6 6 _ _6P 6 k _
5E(pui) + 6x.(pUin) ' 6x. 52‘[ “52‘
J 1 k k For incompressible flows:
“_1 U “_1=_1£+_8_[Vf’f’1 _uu
at J axj p 6xi axk 6xk 1 k DEF I N I T IONS REYNOLDS AVERAG I NG U.=U +u.
1 .6 >1
II
C! Fl
II
C 0 > '6' > II II I61 '6‘
+
'8 e 
I
o b ’I
II
b "6)
H
"U
+
"d REYNOLDS STRESSES — ANALYTICAL EQUATION (incompressible flow) a a an 8Uj
5E(uiuj) + Uk 5;;(uiuj) = — [ u.uk 5;; + uiuk 5;; ] L—————*— (A) *~—————J a [ — ——— u.u.u
Bxk 1 J k axk i J L—————————————————(B) ————————————————————J _ a ”—‘“ P
v ~——(u u.) + 5 (uiajk + uJBik) ] au. au. Bu. au. +2(_1._J)_2,,_1_J p 6xj 8xi 6xk axk L—— (c) ——J L——— (D) ——J (A) Rate of generation of u u. by the effects of the mean strain. 1 J
Transfer of energy from the mean flow to turbulence. Symbol Pi j. (B) "Diffusion" Dij of uiu\j in three components:
(1) "turbulent" diffusion involving triple products.
(2) "viscous" diffusion written in this way for convenience. (3) "pressure" diffusion. (C) The pressurestrain correlation. Symbol Wij. Appears by rearrangement of the pressure terms: This decomposition into pressure diffusion plus pressure strain is
convenient because the second component has zero trace on contraction. It is not unique. (D) Viscous destruction of uiuj. Symbol sij. Appears by rearrangement
of the viscous terms.
2 2 2 au. au.
vu. 9—;(Ui+ ui) + vui§—§(U.+ u.) = v§—§(uiu.) — 2125;3 5;;
J Bxk axk J J axk J k k Convenient because the first component has the form of viscous gradient diffusion of uiuj. The second component has useful properties. TURBULENT KINETIC ENERGY  ANALYTICAL EQUATION (Incompressible flow) _ 1
k — Z uiui
.a_k+U..a_k_=—'_‘1uk____aui—..._a [luu2_v_a_k_+2u]_v_a_u_iil:j;
at k axk 1 axk Bxk 2 k 1 axk p k axk axk
L—— (A) —J l (B) J I (c) —J (A) Production rate of k by interaction with the mean strain rate. Symbol Pk. (B) "Diffusion" of k: (1) “Turbulent”, (2) "Viscous", (3) "Pressure" (C) Viscous destruction of k, symbol 8. TURBULENT KINETIC ENERGY  MODEL EQUATION (Incompressible flow) 8k 8k 6 [ 6k
—— + U ——~ = ——— (v v /o_) ——— ] + P — e
at k axk axk t k 6xk k
6Ui P = — u.u ——— k 1 k 6xk Vt = c kZ/e c = 0.09 u
ok = 1.0 This model equation is used in both the low Reynolds no. and the high Reynolds
no. k—e eddy viscosity model, as well as in higher order closure models like the ASM model and full Reynolds stress models. ENERGY DISSIPATION RATE — ANALYTICAL EQUATION Bu1 Bu.
8=v—__—
6xk axk
a_e+u £=_3_[Wkaﬁi‘ii_+2292_au_k_ 6;]
at k axk axk 6X1 8xl p 6x1 6x1 axk
{1} {1} {Ref}
__ _____ 2
6xk 6x1 6x1 6x1 Bxk k 6x1 axkaxl
{Re'l/Z} {Red/2}
l 1
—————————— 2
Bu. 6u. 6n 6 u. 2
 2 v ——£ ——i ——E  2 [v 1 ]
axk 6X1 6x1 axkaxl
{Re+1/2} {Re+1/2}
l 1
ENERGY DISSIPATION RATE AT HIGH REYNOLDS NUMBERS
2
.6_>~:+U grim BLELZML‘E 2v“_ia_“gi“5_2[v“ ]2
at k axk axk k 6X1 6xl p 6x1 6x1 axk 6x1 6x1 axkaxl
L—————~————— (A) —————————————J L———— (B) ———J L—— (C) ——J Terms (B) and (C) individually vary as Rei/z, i.e. become infinite, but when
taken together are of order unity. When these terms are modelled they must be
modelled together! However, the 8 equation is in general modelled using a standard type of convection/diffusion equation assuming isotropy: 2
68 88 _ 6 as c _ e
73t+uk6x ‘ 6x[(V+Vt/o)ax ]+°e1EPk 082k—
k k k
c = 1 3 c = 1.45 c = 1.92 0 = 1 3
e 1 22 e HIGH REYNOLDS NO. k—s MODEL (incompressible flow) 6Ui an 2
“iuj = "’t [6x. + 5x. ] + 5 613 k
J 1
2
vt — c k /8
6k 6k 3 [ k
—— + U ——— = ——— (v v /¢ ) ——— ] + P  e
at k axk axk t k axk k
68 as a as e 82
5? + Uk 6x = 6 [(v + vt/cs) 6x ] + C81 E Pk _ Caz E—
k k k
6U.
P = — u.u ——i
k 1 k Bxk
c = 0.09 c = 1.44 c = 1.92 o = 1.0 o = 1.3
p 81 82 k 8 LOW REYNOLDS NO. ke MODEL (incompressible flow) <1
I  0 1372
u 0
II 0.09 exp[—3.4/(1 + R /50)2)] t 2
c82 — 1.92 (1 — 0.3 exp(Rt))
Rt = ka/(VE) The other equations are the same as in the high Reynolds no. k—e model. MODEL EQUATION EQUATIONS FULL RSMMODEL (incompressible flow)  PAGE A (Reynolds stress model) %E(uiuj) + Uk 52;(E;53) = Pij + Dij + Wij — sij
Pij Production by mean strain rate, requires no modelling:
_ an an
Pij — — [ ujuk 5;; + uiuk 5;; ]
Dij Diffusion, can be modelled using a gradient diffusion approximation:
Dij = 5:; [03 g GEE; 521(uiuj)] A model constant c5 = 0.22 has been introduced. The following simpler model is also used: Instead of Vt only, (v + Vt) can also be used. Wij The pressure—strain relation can be modelled by:
wiJ = — c1 E [ uluJ — g k aij ]  C2 [ P13 — % Pk 6ij ]
Pk =épii eij Viscous destruction, can be modelled by (assuming isotropy):
eij = g sij 8 MODEL EQUATION EQUATIONS FULL RSMMODEL (incompressible flow)  PAGE B The turbulent kinetic energy is calculated using the standard model equation: The energy dissipation rate can be calculated using the model equation from
the k—c model: or, using the information about the anisotropy in the flow, the following equation can be used: The model constants are: c = 0.22
s c = 1.80
1 c = 0.60
2 c = 0.18
s c = 1.44
81 c = 1.92
82 10 MODEL EQUATION EQUATIONS ASMMODEL (incompressible flow) (Algebraic stress turbulence model) 8 6 —— __ _
—t(uiuj) + Uk aihliuj) — Pij + 131‘j + Wij Sij This equation can be simplified by combining the convection and diffusion terms: a a —— a a _ _
53(uiuj) + [ Uk 5;;(uiuj) + 5;; [Vt 5;;(uiuj)] ] — Pij + wij eij which leads to: Q: ——(u.u.) + T.. = P.. + W..  8..
t 1 J 13 lJ 13 13 This equation can be simplified using Rodi’s approximation. It is assumed that the transport of uiuj is proportional to the transport of k, with the ratio uiuj/k as proportionality factor: a? + Uk 5;; — D1ff(k) _a_ _ uiuj [6k 6k ] uiuj C
uu.=gk5 1—£— +——— P..—§Pa.+—~3A..
3 1J 3 1c2 1J The term Aij is the socalled added convection term, which is zero when the
equations are written out in cartesian coordinates but plays an important role when the equations are written in cylindrical—polar coordinates. The equations used for calculating k and 8 are the same as in the Reynolds stress model. The model constants are: c 2.50
c = 0.55 11 VECTOR NOTATION 9) X.Y = Xiyi (Scalar/dot product) a
.9 9—;
XY = 2 z. (Dyadic product) >~L vb
53¢ ~L
II
OJ: iv
0 A28 = a..b..
1 J J 1
(Double dotted product) lj ij aik U 12 I:
E
El ...
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 Fall '10
 N/A
 Fluid Dynamics, Incompressible Flow, turbulent kinetic energy

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