An investigation of computational methods
for all speed compressible flow field for an application of
the preliminary study in aeroacoustics
An investigation of computational methods
for all speed compressible flow field for an application of
the preliminary study in aeroacoustics
Student
Chung-Gang Li
Advisor
Wu-Shung Fu
A Thesis
Submitted to Department of Mechanical Engineering College of Engineering National Chiao Tung University
in partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy in
Mechanical Engineering June 2011
i
Preconditioning
dissipative model
OpenMp GPU(Graphics processing unit)
An investigation of computational methods
for all speed compressible flow field for an application of the preliminary study in aeroacoustics
Abstract
With the improvement of the life quality, the noise induced by aeroacoustic becomes one of the most important and complicated problems of the noise control. Before resolving the problems of the aeroacoustics, due to the necessary of defining the aeroacoustic source, the aim of this study will develop a computational method for all speed regions. In order to obtain the accurate information of the aeroacoustic source, the preconditioning method is adopted to make the program is suitable for all speed regions. Besides, due to the serious effect of the boundary conditions on fluid and acoustic fields, the computational method should be matched with the non-reflection boundary condition. With the non-reflection boundary condition, the accuracy can be increased and the efficiency can be improved. In the turbulence modeling, the dissipative model is created to generalize the computational method. The extra equations for the motions of small scales can be elimated and the efficiency can be improved. Besides, this model is also available in the curvilinear coordinate without any modifications so it can be used more general in real applications. Finally, because the tremendous computational time is demanded, parallel computation by OpenMp and GPU are built to improve the efficiency. Therefore, the computational method developed in this study can be potentially a basement in the computational aeroacoustic.
iii
i ii iii iv v vi viii 1 6 LES 19 23 51 58 62 135 137 1 143
v 1. (4-71) (4-72) 41 2. Khanafer Vafai [40] 75 3. 0.005 86 4. 0.5 93 5. Dissipative model 101 6. 131
1. 8 2. 12 3. 15 4. 18 5. 27 6. L1 L2 L3 L4 L5 47 7. Roe scheme [20] 52
8. Roe scheme [20] Reynolds stress 53
9. Roe scheme [20] 54 10. filter 57 11. CUDA 61 12 local 63 13 Ra 64 14. 4 10 Ra= 67 15. 72 16. Khanafer Vafai [41] 74 17. 4 10 Ra= 77 18 Isothermal surfaces 81 19 z x=W x=W / 2 x=0 local 83 20 Rayleigh 84 21. 87 22. φ=40% 89
23. LES Smagorinsky model [20] 90
24. LES Smagorinsky model [20] Reynolds stress 91
25. LES Smagorinsky model [20] 92
26. LES Smagorinsky model [20] Reynolds stress 94
27. LES Smagorinsky model [20] 95
vii
29 LES Smagorinsky model [45] 97
30 (0.5 ) 98
31. ILES dissipative model [20] 102
32. ILES dissipative model [20] Reynolds stress 103
33. ILES dissipative model [20] 104
34. ILES dissipative model [46] 105
35. ILES dissipative model [46] Reynolds stress 106
36. ILES dissipative model [46] 107
37. ILES dissipative model [47] 108
38. ILES dissipative model [47] Reynolds stress 109
39. ILES dissipative model [47] 110
40. ILES dissipative model [47] 111
41. ILES dissipative model [47] Reynolds stress 112
42. ILES dissipative model [47] 113
43. ILES dissipative model [48] 114
44. ILES dissipative model [48] Reynolds stress 115
45. ILES dissipative model [48] 116
46. xy xz yz 118 47. Isothermal surfaces Reτ =180 500 940 119 48. ( xz ) (xz ) 121 49. 122 50. 124 51. 125 52. 126 53. spectra(dB) 127 54 130 55 132 56 [51] 133 57 [51] 134
c original acoustic wave speed (m s/ ) ER Energy ratio , , F G H flux vectors g acceleration of gravity ( 2 / m s ) ( ) G φ Filter function , , i j k Location index J Jacobian of transformation k Thermal conductivity (w m K/ ⋅ ) l Length (m) . Ma Mach number
Nu Local Nusselt numbers defined in Eq. (7-1), (7-11) and (7-18)
Nu Average Nusselt numbers defined in Eq. (7-3), (7-6), (7-7), (7-8),
(7-19) and (7-20)
Nu Average Nusselt numbers defined in Eq. (7-13) and (7-15)
P Pressure ( Pa ) Pr Prandtl number
rms Root mean square
R Gas constant (J kg K ) / /
Ra Rayleigh number defined in Eq. (7-2), (7-4), (7-5), (7-9) and (7-10)
Ra Rayleigh number defined in Eq. (7-14)
*
Ra Rayleigh number defined in Eq. (7-12)
Re Reynolds number based on characteristic velocity Reτ Reynolds number based on friction velocity
S Source term vector
t Physical time (s)
T Temperature(K )
U conserved variable vector
p
U primitive variable vector
uτ Friction velocity τ ρw/ (m s/ ) , ,
x y z Cartesian axes ( m )
ix
Greek symbols
ρ density ( 3
/
kg m )
γ Specific heat ratios Cp/Cv
µ viscosity ( 2
/
N s m⋅ )
w
τ Wall shear stress ( Pa )
β Thermal expansion coefficient (1/ K )
Γ Preconditioning matrix
ε Coefficient of dissipative model τ Artificial time (s)
Mathematical Operators
(CEN) 2001 125
(structure-borne) (Structure-borne noise) (air-borne)
2
Lighthill [1] Lighthill
Lighthill’s analogy Williams [2] Lighthill’s analogy Lighthill
Navier-Stokes Ffowcs Williams-Hawkings ( FW-H ) Lighthill 1992 William [3] CAA(Computational AeroAcoustic) CFD CAA Lighthill’s analogy CFD
Ohnishi [4] Lighthill’s analogy
Kenji [5]
Lighthill Prieur [6]
FW-H Kirchhoff
FW-H [2] Kirchhoff
Lighthill’s analogy Liner Euler Equation(LEE) Addad [7] Star-CD
LEE (vortex) (acoustic source)
[8] Euler Equation Bogey [9]
( 0.3 ) ( 0.3 )
Navier-Stokes
CFL stiff
Briley [10] preconditioning
Navier-Stokes Turkel[11] preconditioning Choi [12]
stiff factorization error
preconditioning 0.05
Choi [12] [13] preconditioning
Roe[14] averaged
variables Roe
Weiss [15] Choi Roe
preconditioning N-S dual time stepping
density based pressure based
Patankar Spalding[16] SIMPLE(Semi-Implicit Method for Pressure Linked Equations) pressure based
pressure based
4
Rudy [17]
Poinsot [18] Navier-Stokes characteristics boundary conditions(NSCBC)
Freund [19]
0.3
DNS(Direct Numerical Simulation) RANS(Reynolds Averaged Navier-Stokes) LES(Large Eddy Simulation)
DNS DNS Benchmark Kim [20] RANS LES DNS RANS DNS RANS LES
(large scale) (small scale)
LES LES (ILES)
LES Smagorinsky dynamic
Boris [21] Grinstein [22]
(compressible turbulent flow) LES preconditioning
Lessani [23] LES preconditioning
Xu [24] LES pipe DNS Alkishriwi [25] LES preconditioning
multigrid Reτ =590 Re=3900 Runge-Kutta 4-60 Bossinesq 30 [26] correlation DNS
6 2.1 1 l0 w 2 l 1 l l3 Th 298.06 c T = K P0 =1atm (101300Pa) (1) (2) (3) N-S U F G S t x y ∂ +∂ +∂ = ∂ ∂ ∂ (2-1) P=ρRT (2-2) U F G S
u U v E ρ ρ ρ ρ = 2 xx xy xx xy u u P F uv T Eu Pu k u v x ρ ρ τ ρ τ ρ τ τ + − = − ∂ + − − − ∂ 2 yy yx yy v vu G v P T Ev Pv k u v y ρ ρ ρ τ ρ τ τ = + − ∂ + − − − ∂ 0 0 0 ( ) 0 ( ) g S gu ρ ρ ρ ρ − − = − − 2 2 1 ( ) ( 1) 2 p E u v ρ γ = + + − Sutherland’s law 3 0 2 0 0 110 ( ) ( ) 110 T T T T T µ =µ + + (2-4) ( ) ( ) ( 1) Pr T R k T µ γ γ = − (2-5) 3 0 1.18kg m/ ρ = 2 9.81 / g= m s µ0 =1.85 10× −5N s m⋅ / 2 T0 =298.06K 1.4 γ = R=287 /J kg K/ Pr=0.72 (2-3)
8 1.
Outlet
Non-reflecting
Inlet
Non-reflecting
A
B
C
D
E
F
A
2.2 2 11W 2W W 2W 2W Th 298.06 c T = K P0 =1atm (1) (2) (3) N-S H + U F G S t x y z ∂ +∂ +∂ ∂ = ∂ ∂ ∂ ∂ (2-6) P=ρRT (2-7)
10 U F G H S u U v w E ρ ρ ρ ρ ρ = 2 xx xy xz xx xy xz u u P uv F uw T Eu Pu k u v w x ρ ρ τ ρ τ ρ τ ρ τ τ τ + − − = − ∂ + − − − − ∂ 2 yx yy yz yx yy yz v uv v P G vw T Ev Pv k u v w y ρ ρ τ ρ τ ρ τ ρ τ τ τ − + − = − ∂ + − − − − ∂ 2 zx zy zz zx zy zz w uw vw H w P T Ew Pw k u v w z ρ ρ τ ρ τ ρ τ ρ τ τ τ − − = + − ∂ + − − − − ∂ 0 0 0 0 ( ) 0 ( ) S g gv ρ ρ ρ ρ = − − − − 2 2 2 1 ( ) ( 1) 2 p E u v w ρ γ = + + + − Sutherland’s law 3 0 2 0 0 110 ( ) ( ) 110 T T T T T µ =µ + + (2-9) (2-8)
( ) ( ) ( 1) Pr T R k T µ γ γ = − (2-10) 3 0 1.18kg m/ ρ = ,g =9.81 /m s2,µ0 =1.85 10× −5N s m⋅ / 2,T0 =298.06K,γ =1.4, 287 / / R= J kg K and Pr=0.72.
12 2(a).
2(b).
2.3 LES 3 x y z u v w l1 2 l l3 (1) (2) Trong [27] 0 U F G H t x y z ∂ +∂ +∂ +∂ = ∂ ∂ ∂ ∂ (2-11) u U v w E ρ ρ ρ ρ ρ = ɶ ɶ ɶ ɶ (2-12) 2 ( 2 )[2 ( )] 3 ( 2 )[ ] ( 2 )[ ] ( 2 ) 2 { [2 ( )] [ ] [ ]} 3 t t t t i u u uu P V x u v F uv y x u w uw z x T Eu Pu k x u u v u w T u V v w k x y x z x x ρ ρ µ ρν ρ µ ρν ρ µ ρν ρ µ ρν ∂ + − + − ∇ ⋅ ∂ ∂ ∂ = − + + ∂ ∂ ∂ ∂ − + + ∂ ∂ ∂ + − − + ∂ ∂ ∂ ∂ ∂ ∂ ∂ − ∇⋅ + + + + − ∂ ∂ ∂ ∂ ∂ ∂ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶɶ ɶɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ (2-13) Pɶ =ρRTɶ (2-14)
bar filter tilde filter
14 Sutherland’s law 3 0 2 0 0 110 ( ) ( ) 110 T T T T T µ =µ + + ɶ ɶ ɶ (2-15) 5 2 0 1.85 10 N s m/ µ = × − ⋅ , 0 298.06 T = K Smagorinsky νt 2 t C S ν = ∆ ɶ (2-16) 1 3 1 2 3 ( ) ∆ = ∆ ∆ ∆ , Sɶ = 2S Sɶ ɶij ij , 1( ) 2 j j ij i i u u S x x ∂ ∂ = + ∂ ∂ ɶ ɶ ɶ
(2-16) C Van Driest damping function
0.01 1 exp[ ( )] 25 d C + = − − (2-17) u d d ρ τ µ + = u τ τ ρw/ τw d
3.
16 2.4 Lighthill 4 Navier-Stokes l1 2 l R 10m s/ R 3000 l1/R=50 l2/R=120 Buffer zone 10R (1) (2) (3) N-S U F G S t x y ∂ +∂ +∂ = ∂ ∂ ∂ (2-18) P=ρRT (2-19) U F G S u U v E ρ ρ ρ ρ = 2 xx xy xx xy u u P F uv T Eu Pu k u v x ρ ρ τ ρ τ ρ τ τ + − = − ∂ + − − − ∂ 2 yy yx yy v vu G v P T Ev Pv k u v y ρ ρ ρ τ ρ τ τ = + − ∂ + − − − ∂ (2-20)
2 2 1 ( ) ( 1) 2 p E u v ρ γ = + + − Sutherland’s law 3 0 2 0 0 110 ( ) ( ) 110 T T T T T µ =µ + + (2-21) ( ) ( ) ( 1) Pr T R k T µ γ γ = − (2-22) 3 0 1.18kg m/ ρ = 2 9.81 / g= m s µ0 =1.85 10× −5N s m⋅ / 2 T0 =298.06K 1.4 γ = R=287 /J kg K/ Pr=0.72 Lighthill 1 60R 150R 2 2 2 2 2 2 2 2 0 2 ( 2 2 ) 2 2 2 u v uv c t x y x y x y ρ′ ρ′ ρ′ ρ ρ ρ ∂ − ∂ +∂ =∂ +∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ (2-23)
18 4. Navier-Stokes Buffer zone Buffer zone Lighthill
LES
LES LES
LES (large-scale) (subgrid scale
(SGS)) LES Navier-Stokes
(subgrid-scale stress models) Smagorinsky
LES Deardorff[28]
LES tensor Kolmogorov
(filter) filter function ( , ) ( ) ( , ) U r t =U r +U r t′ (3-1) (wave number) U r t′( , ) 3 1 ˆ ( , ) ( , ) (2 ) ikr U r t U r t e dk π ∞ ′ =
∫
′ (3-2) ( , ) f r t f r t( , ) ( , ) ( , ) ( , ) f r t G r r f r t dr ∞ ′ ′ ′ =∫
G r r( , )′ filter function (3-3) filter function20 (1) Gussian filter function
( , ) 1 exp 6( 2 ) 6 i i i i i i x x G x x π − ′ ′ = × − ∆ ∆ (3-4)
(2) Top hat filter function 1 i ∆ 2 2 i i i i i x −∆ <x′< +x ∆ G x x( ,i i′ =) (3-5) o otherwise
(3) sharp cutoff filter function
( ) 2 ( , ) ( ) i i in i i i i i x x S G x x x x π π − ′ − ∆ ′ = ′ − (3-6) ∆i filter width f r( )= f r( )+ f r′( ) filter Navire-stokes 0 i i u x ∂ = ∂ (3-7) 2 2 1 ij i i i j j i j j u u P u u v t x x x x τ ρ ∂ ∂ + ∂ = − + ∂ − ∂ ∂ ∂ ∂ τij =u ui j−u ui j =Lij+Cij+Rij (3-8) ij L (Leonard) u ui j−u ui j
filter function sharp cutoff filter function
ij
modeling filtering
ij
L Cij Lij Cij Rij order Lij+Cij
ij
R (Reynold ) u ui′ ′j
SGS(subgrid scale) model Reynolds stress s ( ) 3 j k k i ij i j ij t j i u u u u R u u v x x ς ′ ′ ∂ ∂ ′ ′ = − = − + ∂ ∂ (3-9) ij R u uk′ ′k P p P 1 3 k k P= +p ρu u′ ′ (3-10) 1 2 2 1 2 ( ) ( ) 2 j i t s j i u u v C x x ∂ ∂ = ∆ + ∂ ∂ 1 3 1 2 3 ( ) ∆ = ∆ ∆ ∆ (3-11) 1 3 G p P K ρ = + 2 2 ( ) t G v v K C = ∆ Cv =0.094 s C 0.1 0.25 0.15 ∆ (3-4) (3-6) filter Navier-Stokes Favre-filtered(density-weighted) f filtered f Favre-averaged fɶ 1 V f GfdV V =
∫
f ρf ρ = ɶ G filtering Navier-Stokes 0 k k u t x ρ ρ ∂ + ∂ = ∂ ∂ ɶ (3-12)22 0 k kl kl k l l k l l u p u u t x x x x ρ ρ τ σ ∂ + ∂ + ∂ −∂ +∂ = ∂ ∂ ∂ ∂ ∂ ɶ ɶ ɶ ɶ (3-13) ( ) 0 t kl l t l l k kl k v l l l l l l l E q E u pu u k T u C t x x x x x x x ρ τ σ ∂ + ∂ + ∂ − ∂ − ∂ ∂ + ∂ + ∂ = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ɶ ɶ ɶ ɶ ɶ ɶ ɶ (3-14) p=ρRTɶ (3-15) Navier-Stokes (3-13) kl σ ( ) kl uu u uk l σ =ρ − ɶ ɶ (3-16)
LES (close system) (3-16)
Moin (3-16) 2 1 1 2 2 ( ) 3 3 kl C S Skl Smm kl q kl σ = − ρ∆ ɶ ɶ − ɶ δ + δ (3-17) 2 ii q =σ 1 ( ) 2 k l kl l k u u S x x ∂ ∂ = + ∂ ∂ ɶ ɶ ɶ Sɶ = 2S Sɶ ɶkl kl Smagorinsky model 2 t C S ν = ∆ (3-18) 1 3 ( x y z) ∆ = ∆ ∆ ∆ S = 2S Sij ij C C Germano [29]
dynamic SGS model Trong[26]
(Euler equation) Riemann Roe (flux) preconditioning (implicit) (explicit) Navier-Stokes 0 U F G H t x y z ∂ +∂ +∂ +∂ = ∂ ∂ ∂ ∂ (4-1) u U v w E ρ ρ ρ ρ ρ = 2 2 V E= +e e ( 1) p e ρ γ = − 2 2 2 V =u + +v w 2 xx xy xz xx xy xz u u p vu F wu T Eu pu k u v w x ρ ρ τ ρ τ ρ τ ρ τ τ τ + − − = − ∂ + − − − − ∂ 2 yx yy yz yx yy yz v uv v p G wu T Ev pv k u v w y ρ ρ τ ρ τ ρ τ ρ τ τ τ − + − = − ∂ + − − − − ∂
24 2 zx zy zz zx zy zz w uw vw H w p T Ew pw k u v w z ρ ρ τ ρ τ ρ τ ρ τ τ τ − − = + − ∂ + − − − − ∂ inviscid viscid F =F +F = 2 0 xx xy xz xx xy xz u u p uv uw T u v w Eu pu k x ρ ρ τ ρ τ ρ τ τ τ τ ρ + − ∂ + + + + − ∂ inviscid viscid G=G +G = 2 0 yx yy yz yx yy yz v vu v p vw T u v w Ev pv k y ρ ρ τ ρ τ ρ τ τ τ τ ρ + − ∂ + + + + − ∂ inviscid viscid H =H +H = 2 0 zx zy zz zx zy zz w wu wv w p T u v w Ew pw k z ρ ρ τ ρ τ ρ τ τ τ τ ρ − + ∂ + + + + − ∂
Euler equation U Finviscid Ginviscid Hinviscid 0
t x y z ∂ ∂ ∂ ∂ + + + = ∂ ∂ ∂ ∂ u U v w E ρ ρ ρ ρ ρ =
2 inviscid u u p uv F uw T Eu pu k x ρ ρ ρ ρ ρ + = ∂ + − ∂ 2 inviscid v vu v p G vw T Ev pv k y ρ ρ ρ ρ ρ + = ∂ + − ∂ 2 inviscid w wu wv H w p T Ew pw k z ρ ρ ρ ρ ρ = + ∂ + − ∂
(hyperbolic) (conservative type)
(piecewise) (Riemann) 0 u u a t x ∂ + ∂ = ∂ ∂ a Jacobian (4-2) 0 ( , 0) ( ) u x =u x = L R u u 0 0 x x < > (4-3) (4-2) 0 t x U +AU = (4-4) A 1 A= ΛK K− Λ 1 ... 0 0 ... 0 : : : 0 ... m λ λ Λ = (1) ( ) [ ,..., m ] K = K K AK( )i =λiK( )i
26 W (characteristic varibales) ( 1) W =K − U U =KW t t U =KW Ux=KWx (4-4) 0 t x KW +AKW = 0 t x W + ΛW = (4-5) (4-5) Canonical form Characteristic form
0 i i i w w t λ x ∂ + ∂ = ∂ ∂ 1 1 1 2 2 3 3 ... 0 0 ... 0 0 : : : : : 0 ... m t x w w w w w w λ λ + = (4-5) 0 ( , ) ( ) i i i w x t =w x−λt = i i α β 0 0 i i x t x t λ λ − < − > (4-6) i α βi U =KW 0 ( ) ( , ) ( ) m i i i i u x t =
∑
w x−λt K 5y
x<0 x=0 x>0 x
28 ( ) ( ) 1 1 ( , ) p m i i i i i i p u x t α K βK = = + =
∑
+∑
(4-7) ( , ) u x t jump ∆u ( ) 1 m i R L i i u u u αK = ∆ = − =∑
ɶ (4-8) i i i α β αɶ = − exact solution(approximation Riemann problem)
exact solution Roe[14]
0 U F t x ∂ +∂ = ∂ ∂ (4-9) chain rule (4-9) 0 U F U t U x ∂ + ∂ ∂ = ∂ ∂ ∂ ( ) F A U U ∂ = ∂ (4-9) ( ) 0 U U A U t x ∂ + ∂ = ∂ ∂ (4-10) ( ) A U Jaconian
Roe Jaconian A U( ) Jaconian (constant Jaconian matrix) A U Uɶ( L, R) ( ) 0 U U A U t x ∂ + ∂ = ∂ ɶ ∂ (4-11) ( , 0) U x = L R U U 0 0 x x < > (4-11)
Roe Jaconian Jaconian
(4-9)
Jaconian Roe
2. UR−UL →U A U Uɶ( L, R)→A U( ) F A U ∂ = ∂ 3. A Uɶ( L−UR)=FL−FR 4. Aɶ Roe Jaconian
3. conservation law Rankine-Hugoniot (4-6) (4-8) Ui+1/ 2( / )x t ( ) 1/ 2 0 ( / ) i i i L i U x t U K λ α + < = +
∑
ɶ (4-12) ( ) 1/ 2 0 ( / ) i i i R i U x t U K λ α + > = −∑
ɶ (4-13) 1/ 2 i+ (face) ( ) 0 U F Q t x ∂ +∂ = ∂ ∂ ɶ ɶ (4-11) Fɶ = AUɶ ( R) ( L) ( R) ( L) F Uɶ −F Uɶ =F U −F U (4-14) 1/ 2(0) i U+ (flux) 1/ 2 ( 1/ 2(0)) ( ) ( ) i i R R F+ =F Uɶ + −F U −F Uɶ (4-15) Fɶ = AUɶ 1/ 2 1/ 2(0) ( ) i i R R F+ =AUɶ + −F U −AUɶ (4-16) (4-12) (4-13) 1 ( ) ( ) 0 1 2 ( ) ( ) i m i i R i R i i i i F F U A K F U K λ α λ α + + = − ɶ∑
> ɶ = −∑
= ɶ ɶ ɶ (4-17) ( ) ( ) 1 0 1 2 ( ) ( ) i m i i R i L i i i i F F U A K F U K λ α λ α − + = + ɶ∑
> ɶ = +∑
= ɶ ɶ ɶ (4-18)30 (4-17) (4-18) λɶi− λɶi+ 1 2 i F + ( ) 1 1 2 1 ( ) ( ) 2 m i R L i i i i F F U F U λ αK + = = + −
∑
ɶ ɶ (4-19) (4-7) 1 2 i F + 1 2 1 ( ) ( ) 2 R L i F F U F U A U + = + − ∆ɶ (4-20) R L U U U ∆ = − 1 Aɶ = Aɶ+−Aɶ− = ΛKɶ ɶ ɶK− 1 2 ( , ,..., m) diag λ λ λ Λ =ɶ ɶ ɶ ɶ ( ) 0 t x U +F U = (4-21) 1 2 u U u u ρ ρ = = 1 2 2 2 f u F f u a p ρ ρ = = + a (4-21) jacobian 2 2 0 1 ( ) 2 F A U a u u U ∂ =∂ = − (4-22) 1 u a λ = − λ2 = +u a (1) 1 K u a = − (2) 1 K u a = + parameter vector 1 2 q U Q q u ρ ρ ρ = = (4-23) F U Q 2 1 1 1 2 1 2 u q U q Q u q q = = = (4-24) 1 1 2 2 2 2 2 2 1 f q q F f q a q = = + (4-25)U ∆ ∆F averaged vectorQɶ 1 2 1 1 ( ) 2 2 L R L R L L R R q Q Q Q q u u ρ ρ ρ ρ + = = + = + ɶ ɶ ɶ (4-26) ( ) Bɶ =B Qɶ ɶ Cɶ =C Qɶ ɶ( ) U B Q ∆ = ∆ɶ ∆ = ∆F C Qɶ (4-27) (4-27) 1 ( ) F CB− U ∆ = ɶ ɶ ∆ (4-28) 3 Jcaobian 1 Aɶ =CBɶ ɶ (4-29) − (4-29) 1 2 1 2q 0 B q q = ɶ ɶ ɶ ɶ 2 1 2 2 1 2 2 q q C a q q = ɶ ɶ ɶ ɶ ɶ (4-30) (4-29) 2 2 0 1 2 A a u u = − ɶ ɶ ɶ (4-31)
uɶ Roe averaged velocity
L L R R L R u u u ρ ρ ρ ρ + = + ɶ (4-32) preconditioning Weiss Smith preconditioning [15] 0 U F G H t x y z ∂ +∂ +∂ +∂ = ∂ ∂ ∂ ∂ (4-33) (4-33) (conserved variables) (primitive variables) 0 p U F G H M t x y z ∂ +∂ +∂ +∂ = ∂ ∂ ∂ ∂ (4-34) [ ]T p U = P u v w T M
32 0 0 0 0 0 0 0 0 0 1 p T p T p T p p T p T p u u U v v M U w w H u v w H C ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ∂ = = ∂ − + (4-35) p p ρ ρ =∂ ∂ T T ρ ρ =∂ ∂ (4-35) K 2 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 ( ) 0 0 0 T p u v K w H V C ρ ρ − − = − − − (4-36) K M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 p T p KM C ρ ρ ρ ρ ρ ρ = − (4-37) (4-37) (4-34) ( ) 0 p p u v w t x y z ρ ρ ρ ρ ∂ +∂ +∂ +∂ = ∂ ∂ ∂ ∂ (4-38) (4-38) 2 ( ) 0 p u v w C t x y z γ ∂ +∂ρ +∂ρ +∂ρ = ∂ ∂ ∂ ∂ (4-39) C (4-39) ρp (4-38) 0 u v w x y z ρ ρ ρ ∂ +∂ +∂ = ∂ ∂ ∂ (4-40) (4-37) ρp (local velocity) (order) CFL
θ ρp 2 1 1 ( ) r p U TC θ = − (4-41) ε×Umax if u < ×ε C Ur = u if ε× < <C u C (4-42) C if u >C ε 5 10− (stagnation point) (singular point) Ur
(local diffusion velocity) Ur
max( , ) r r U U x ν = ∆ θ (4-37) Γnc 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 T nc p C θ ρ ρ ρ ρ ρ Γ = − (4-43) (4-34) ( ) 0 p nc U F G H K t x y z ∂ ∂ ∂ ∂ Γ + + + = ∂ ∂ ∂ ∂ (4-44) (4-44) 1 K− 1 ( nc) p 0 U F G H K t x y z −Γ ∂ +∂ +∂ +∂ = ∂ ∂ ∂ ∂ (4-45) (4-45)
34 1 0 0 0 0 0 0 0 0 0 1 nc p T u u T v K v T w w T H u v w H C T ρ θ ρ θ ρ ρ θ ρ ρ θ ρ ρ θ ρ ρ ρ ρ − − − − Γ = Γ = − − − + (4-46) 0 p U F G H t x y z ∂ ∂ ∂ ∂ Γ + + + = ∂ ∂ ∂ ∂ (4-47) Roe (4-20) Fi+1/ 2 1( ( ) ( )) 2 F UR +F UL artificial viscosity term 1
2 A U∆
ɶ preconditioning
artificial viscosity term
1 1 1 0 ( ) 0 ( ) 0 ( ) 0 p p p p p p p U F G H t x y z U F G H t x y z U U U U A B C t x y z U U U U AM BM CM t x y z − − − ∂ ∂ ∂ ∂ Γ + + + = ∂ ∂ ∂ ∂ ∂ + Γ ∂ +∂ +∂ = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + Γ + + = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + Γ + + = ∂ ∂ ∂ ∂ p U M U ∂ = ∂
artificial viscosity terms 1 1 2 1 1 ( ) ( ) 2 2 n R L R L i F F F − AM U U + = + − Γ Γ + (4-48) 1 1 AM KA DA KA − − Γ = × × (4-47) Navier-Stokes dual time
stepping[15]
Navier-Stokes artificial time
term 0 U U F G H t x y z τ ∂ +∂ +∂ +∂ +∂ = ∂ ∂ ∂ ∂ ∂ (4-49) τ artificial time t physical time
artificial time term preconditioning
0 p U U F G H t x y z τ ∂ ∂ ∂ ∂ ∂ Γ + + + + = ∂ ∂ ∂ ∂ ∂ (4-50)
artificial time term physical time term
1 1 1 1 1 1 1 1 1 , , , , , , , , 2 2 2 2 2 2 3 4 2 1 1 1 ( ) ( ) ( ) 0 k k k n n p p k k k k k k i j k i j k i j k i j k k k U U U U U t F F G G H H τ ξ η ζ + + − ′ ′ ′ ′ ′ ′ + − + − + − − − + Γ + ∆ ∆ + − + − + − = ∆ ∆ ∆ (4-51) bar 2 k′ =k
Runge Kutta k′ = +k 1 LUSGS
k′ n 1 k k p p U U τ + − Γ ∆ (4-51) N-S Runge Kutta 1 3 2 1 1 3 3 3 2 2 1 3 3 3 ( ) 2 3 1 1 3 ( ) 4 4 4 2 1 2 2 3 ( ) 3 3 3 2 k k k p p k k k k p p p k k k k p p p U U M R t U U U M R t U U U M R t τ τ τ + + + + + + + ∆ = + Γ + ∆ ∆ = + + Γ + ∆ ∆ = + + Γ + ∆ (5-52) p U M U ∂ = ∂
36 1 1 1 1 1 1 1 , , , , , , , , 2 2 2 2 2 2 3 4 ( ) 2 1 1 1 [ ( ) ( ) ( )] n n i j k i j k i j k i j k k k U U U R t F F G G H H ξ η ζ Φ − Φ Φ Φ Φ Φ Φ Φ + − + − + − − + = − ∆ − − + − + − ∆ ∆ ∆ 1 2 , , 3 3 k k k Φ = + + LUSGS 1 k U + k 1 F + 1 k k p U + =U + ∆M U (4-53) p U M U ∂ = ∂ 1 k k p p p U U + U ∆ = − 1 k k p p F + =F + ∆A U (4-54) 1 k p p F A U ∂ = ∂ Jacobian k p p G B U ∂ = ∂ k p p H C U ∂ = ∂ 1 k G + Hk (4-53) (4-54) (4-51) 1 3( ) 4 2 ( ) ( ) ( ) 0 k n n p p k k k p p p p p p U U M U U U t F A U G B U H C U ξ η ζ τ δ δ δ − ∆ + ∆ − + Γ + ∆ ∆ + + ∆ + + ∆ + + ∆ = (4-55) ξ δ δη δς (4-55) 3 [ ( )] 2 k p p p p I M A B C U R t δξ δη δζ τ Γ + + + + ∆ = ∆ ∆ (4-56) 1 3 4 ( ) ( ) 2 k n n k U U U k k k R F G H t δξ δη δζ − − + = − − + + ∆ I Γ 1 3 1 1 1 1 { [ ( ) ( ) ( )]} 2 k p p p p I M A B C U R t δξ δη δζ τ + Γ− + Γ− + Γ− + Γ− ∆ = Γ− ∆ ∆ (4-57) (4-57) Yoon Jameson[30] LUSGS
1 1 1 k p p k p p k p p A A B B C C − − − = Γ = Γ = Γ ɶ ɶ ɶ (4-58) p Aɶ Bɶp Cɶp p p p p p p p p p A A A B B B C C C + − + − + − = + = + = + ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ (4-59) 1 ( ) 2 1 ( ) 2 1 ( ) 2 p p A p p B p p C A A I B B I C C I λ λ λ ± ± ± = ± = ± = ± ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ (4-60) A λɶ λBɶ λCɶ Aɶp Bɶp Cɶp (4-57) 1 3 1 [ ( ) ( ) ( )] 2 k p p p p p p p I M A A B B C C U R t δξ δη δζ τ + Γ− + ++ − + ++ − + ++ − ∆ = Γ− ∆ ∆ ɶ ɶ ɶ ɶ ɶ ɶ (4-61) (Ap Ap) ξ δ ɶ++ ɶ− , , 1 , 1 , (Ap Ap) Ap Ap Ap i Ap i Ap i Ap i ξ ξ ξ δ δ δ ξ ξ + + − − − + ++ − = − ++ + − = − + − ∆ ∆ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ (4-62) p Aɶ+ δξ− Aɶp− δξ+ (4-62) (4-61) , , 1 , 1 , 1 , , 1 , 1 , , , 1 , 1 , 1 3 [ 2 ] p i p i p i p i p j p j p j p j p k p k p k p k k p A A A A I M t B B B B C C C C U R τ ξ ξ η η ζ ζ + + − − − + − + + − − + + − − − + − + − − − + Γ + + ∆ ∆ ∆ ∆ − − − − + + + + ∆ = Γ ∆ ∆ ∆ ∆ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ ɶ (4-63) (4-63)
38 1 (L+ +D U)∆Up = Γ−Rk (4-64) 1, , , 1, , , 1 1 1 1 ( p )i j k ( p )i j k ( p )i j k L A B C ξ + − η + − ζ + − = − + + ∆ ∆ ∆ ɶ ɶ ɶ (4-65) 1 , , , , 3 1 (( ) ( ) ) 2 p i j k p i j k I D M A A t τ − ξ + − = + Γ + − ∆ ∆ ∆ ɶ ɶ , , , , , , , , 1 1 ((Bp )i j k (Bp )i j k) ((Cp )i j k (Cp )i j k) η + − ζ + − + − + − ∆ ɶ ɶ ∆ ɶ ɶ (4-66) 1, , , 1, , , 1 1 1 1 ( p )i j k ( p )i j k ( p )i j k U A B C ξ − + η − + ζ − + = + + ∆ ∆ ∆ ɶ ɶ ɶ (4-67) (4-61) 1 3 4 ( ) ( ) 2 k n n k U U U k k k R F G H t δξ δη δζ − − + = − − + + ∆ (4-1) F 2 inviscid u u p uv F uw T Eu pu k x ρ ρ ρ ρ ρ + = ∂ + − ∂ (4-68) Roe scheme (4-20)
{
1}
1 , 2 1 1 ( ) 2 R L 2 p P inviscid i F F F ε − A U + = + − Γ ∆ (4-67) Roe scheme DNS ε 1 ε ( , , , )x y z tε ε= Roe upwinding dissipation term 1
{
1}
2ε Ap UP−
Abalakin [31] MUSCL (4-67) P U ∆ 1/ 2 1/ 2 L R P i i U u+ u+ ∆ = − (4-68) 1/ 2 1/ 2 1/ 2 L L i i i u+ = +u ∆u+ (4-69) 1/ 2 1/ 2 1/ 2 R R i i i u+ = −u ∆u+ (4-70) 1/ 2 (1 )( 1 ) ( 1) L i i i i i u+ β u+ u β u u− ∆ = − − + − 1 1 2 2 1 1 ( 3 3 ) ( 3 3 ) c d i i i i i i i i u u u u u u u u θ − + + θ − − + + − + − + + − + − + (4-71) 1/ 2 (1 )( 1 ) ( 2 1) R i i i i i u+ β u+ u β u+ u+ ∆ = − − + − 1 1 2 1 2 3 ( 3 3 ) ( 3 3 ) c d i i i i i i i i u u u u u u u u θ − + + θ + + + + − + − + + − + − + (4-72) β θc θd 1 4 2 8 1 8 1 2 ( ) 12 i i i i u u u u u o x x x − − − + + − + ∂ = + ∆ ∂ ∆ (4-73) ( , , )ξ η ζ ( ) / ( ) / ( ) / k x y z k x y z k x y z F F G H J G F G H J H F G H J ξ ξ ξ η η η ς ς ς = + + = + + = + + (4-74) J Jacobian 1 3 4 ( ) { [( ) / ] 2 k n n k x y z U U U R F G H J t δ ξξ ξ ξ − − + = − − + + + ∆ [( xF yG zH) / ]J [( xF yG zH) / ]}J η ζ δ η +η +η +δ ς +ς +ς (4-74) (4-64) k R ( 1, , ) ,( 1, , ) ( , 1, ) ,( , 1, ) ( , , 1) ,( , , 1) ( , , ) ,( , , ) 1 ( 1, , ) ,( 1, , ) ( , 1, ) ,( , 1, ) ( , , 1) ,( , , 1) ( , , ) k k k k i j k p i j k i j k p i j k i j k p i j k i j k p i j k k k k k i j k p i j k i j k p i j k i j k p i j k i j k L U L U L U D U U U U U U U R − − − − − − − + + + + + + ∆ + ∆ + ∆ + ∆ + ∆ + ∆ + ∆ = Γ (4-75)
40 Yoon Jameson[30] (4-64)
1 1
(L+D D) − (D U+ )∆Upk = Γ−Rk
1. (4-71) (4-72) β θc θd Order 1/3 1/3 1/3 1/3 0 -1/6 0 -1/10 0 0 -1/6 -1/15 3 4 4 5
42 (4-76) 1. forward sweep * 1 ( ) k p L+D ∆U = Γ−R (4-77) * 1 ( ) k p p U D− D U U ∆ = + ∆ (4-77) * * 1 k p p L U∆ + ∆D U = Γ−R (4-78) D U∆ *p = Γ−1Rk− ∆L U*p (4-79) * p U ∆ * 1 1 * ( k ) p p U D− − R L U ∆ = Γ − ∆ . (4-80) (4-80) * 1 * ,( , , ) ( , , )[ ( , , ) ( 1, , ) ,( 1, , ) k p i j k i j k i j k i j k p i j k U D − R L − U − ∆ = Γ − ∆ − L( ,i j−1, )k ∆U*p i j,( , −1, )k −L( , ,i j k−1)∆U*p i j k,( , , −1)] (4-81) 1 i= j=1 k=1 ∆U*p 2. backward sweep k p U ∆ (4-77) ( ) k p p D U+ ∆U = ∆D U ∗ (4-82) (4-82) 1 k k p p p U U ∗ D U U− ∆ = ∆ − ∆ (4-83) k p U ∆ * 1 ,( , , ) ,( , , ) ( , , )[ ( 1, , ) ,( 1, , ) k k p i j k p i j k i j k i j k p i j k U U D− U + U + ∆ = ∆ − ∆ + ( , 1, ) ,( , 1, ) ( , , 1) ,( , , 1)] k k i j k p i j k i j k p i j k U + ∆U + +U + ∆U + (4-84) 1 i= j=1 k=1 ∆U*p 3. k 1 p U + 1 k k k p p p U + =U + ∆U (4-85)
4. 1~3 1 0 k k p p U U τ + − ≈ ∆ 1 1 n k p p U + =U + Unp+1 LUSGS ( , 0, ) ( ,1, ) ( , 0, ) ( ,1, ) ( , 0, ) ( ,1, ) ( , 0, ) ( ,1, ) ( , 0, ) ( ,1, ) P i k P i k u i k u i k v i k v i k w i k w i k T i k T i k = = − = − = − = (4-86) ( , , 0) ( , ,1) ( , , 0) ( , ,1) ( , , 0) ( , ,1) ( , , 0) ( , ,1) ( , , 0) ( , ,1) P i j P i j u i j u i j v i j v i j w i j w i j T i j T i j = = − = − = − = (4-87) ( , 0, ) ( ,1, ) ( , 0, ) ( ,1, ) ( , 0, ) ( ,1, ) ( , 0, ) ( ,1, ) ( , 0, ) 2 h ( ,1, ) P i k P i k u i k u i k v i k v i k w i k w i k T i k T T i k = = − = − = − = − (4-88) 0 1
Poinsot Lele [18] LODI(local one-dimensional inviscid relations)
preocnditioning
LODI Navier-Stokes
44 0 p U F x τ ∂ ∂ Γ + = ∂ ∂ (4-89) 1 − Γ 1 0 p U F x τ − ∂ + Γ ∂ = ∂ ∂ (4-90) 1 F x − ∂ Γ ∂ 1 1 p 1 p p p U U F F A x U x x − ∂ − ∂ ∂ − ∂ Γ = Γ = Γ ∂ ∂ ∂ ∂ (4-91) (4-91) (4-89) primitive form 1 0 p p p U U A x τ − ∂ ∂ + Γ = ∂ ∂ (4-92) 1 p A − Γ 1 1 p A K Kλ − − Γ = (4-93) K λ Γ−1Ap Dennis [32] 1 2 3 4 5 u u u u c u c λ λ λ λ λ λ = = ′ ′+ ′ ′− (4-94) ( 1) 2 u u′ = Θ + 2 2 2 ( 1) 4 2 u c c′ = Θ − + Θ Θ ≈100M2 1 Up L K x λ − ∂ = ∂ (4-95) L
1 2 3 4 5 1 ( ) ( ) ( )[ ( ) ] ( )[ ( ) ] T P P u x x x w L u x L v L L u x L P u u c u c u L x x P u u c u c u x x γ ργ ρ ρ ∂ ∂ ∂ + − ∂ ∂ ∂ ∂ ∂ ∂ = = − ∂ ∂ ∂ ′+ ′ − ′− −′ ∂ ∂ ∂ ∂ ′ ′ ′ ′ − − + − ∂ ∂ (4-96) L (4-94) L1 L2 L3 L4 L5 u u u u′+c′ u′−c′ (4-95) (4-92) 0 p U KL τ ∂ + = ∂ (4-97) (4-97) 4 5 4 5 3 2 1 4 5 1 [ ( ) ( )] 0 2 1 ( ) 0 2 0 0 1 1 1 [ ( ) ( )] 0 2 p L u c u L u c u c u L L c v L w L T L L u c u L u c u c τ τ ρ τ τ γ τ ρ γ ∂ + ′+ − −′ ′− −′ = ′ ∂ ∂ + − = ′ ∂ ∂ − = ∂ ∂ + = ∂ ∂ + + − ′+ − −′ ′− −′ = ′ ∂ (4-98) (4-98) 1 4 5 1 4 5 1 3 1 2 1 1 1 [ ( ) ( )] 2 ( ) 2 1 1 ( ) k k k k k k k k k k k k t p p L u c u L u c u c t u u L L c v v L t w w L t T T L t p p ρ ρ γ ρ γ + + + + + + ∆ ′ ′ ′ ′ = − + − − − − ′ ∆ = − − ′ = + ∆ = − ∆ + = − ∆ + − (4-99) 6(a) L1 L2 L3 L4 (4-96) L L L L u′−c′
46 5 L 0 p τ ∂ = ∂ (4-98) 4 5 1 [ ( ) ( )] 0 2c′ L u′+ − −c′ u L u′− −c′ u = (4-100) 5 L 5 4 ( ) ( ) u c u L L u c u ′+ −′ = ′− −′ (4-101) 6(b) L4 (4-98) 1 L L2 L3 0 L5 (4-101)
(a) (b) 6. L1 L2 L3 L4 L5
aperture
channel
surroundings
fluid velocity →
aperture
channel
surroundings
fluid velocity ←
48 (0, , ) ( , , ) (0, , ) ( , , ) (0, , ) ( , , ) (0, , ) ( , , ) (0, , ) ( , , ) P j k P nx j k u j k u nx j k v j k v nx j k w j k w nx j k T j k T nx j k = = = = = (4-102) ( 1, , ) (1, , ) ( 1, , ) (1, , ) ( 1, , ) (1, , ) ( 1, , ) (1, , ) ( 1, , ) (1, , ) P nx j k P j k u nx j k u j k v nx j k v j k w nx j k w j k T nx j k T j k + = + = + = + = + = (4-103) 0 (ghost cell) nx+1 1 nx driving fore ( , , ) p( , , ) P i j k =βx+P i j k (4-104) β Pp βx source term (0, , ) ( , , ) p p P j k =P nx j k (4-105) ( 1, , ) (1, , ) p p P nx+ j k =P j k (4-106) β β Xu [24]
1 1 0 1 [( ) 2( ) ( ) ] n n m m n m n t Ac Ac Ac β + =β − − + − ∆ ɺ ɺ ɺ (4-107) Ac mɺ ∆t ()0 ()n n ()n+1 1 n+ 1. 2. 3. MUSCL (4-71) (4-72) FR FL ∆Up 4. ∆Up (4-67) Roe Finviscid 5. 6. LUSGS (4-76) Upk+1 7. 2 6 1 3 1 10 n n n φ φ φ + − + − < , , , , p u v w T φ = 1 3 1 10 k k k φ φ φ + − + − < k 1 p U + p U (2-23) 2 2 2 2 2 2 2 2 0 2 ( 2 2 ) 2 2 2 u v uv c t x y x y x y ρ′ ρ′ ρ′ ρ ρ ρ ∂ − ∂ +∂ =∂ +∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ (4-108) Lighthill (4-108) ρ 2 2 1 1 2 2 2 ( ) i i i o x x x ρ ρ ρ ρ′ ′+ − ′+ ′− ∂ = + ∆ ∂ ∆ (4-109) 2 1, 1 1, 1 1, 1 1, 1 2 2 ( ) ( ) ( ) ( ) ( , ) 4 i j i j i j i j uv uv uv uv uv o x y x y x y ρ ρ ρ ρ ρ ′ + + − ′ + − − ′ − + + ′ − − ∂ = + ∆ ∆ ∂ ∂ ∆ ∆ (4-110) 2 2 t ρ′ ∂ ∂
50 2 1 2 2 2 2 ( ) t t t o t t t ρ ρ ρ ρ′ ′− ′− + ′− ∂ = + ∆ ∂ ∆ (4-111) (4-109)~(4-111) (4-108) (4-109) (4-110) ρt′−1 ρt′−2 ρ′ 2 0 P′=c ρ′ (4-112)
(Sound Pressure Level, SPL) dB
10 0 20log P p L p = (4-113) 0 p 2 10 Pa× −5
dissipative model
dissipative model (4-67) Roe upwinding
dissipation term 1
{
1}
2ε Ap UP − Γ ∆ DNS Roe scheme(ε =1) 7 9Reynolds stress Reτ =180 Kim [20]
DNS DNS
Trong[27] Roe scheme Smagorinsky model
damping preconditioning ε ε 0.1 0.1 ε = 0.1 ε = Truncated Navier-Stokes (TNS) TNS under resolved DNS
Domaradzki [33] (filter)
52 7. Roe scheme [20]
DNS by Kim et al. [20] The original Roe scheme
8. Roe scheme [20] Reynolds stress DNS by Kim et al. [20] The original Roe scheme
54 9. Roe scheme [20]
DNS by Kim et al. [20]
Roe upwinding dissipation term
{
1}
1 2ε Ap UP − Γ ∆ TNS Roeupwinding dissipation term
ε ε [34] dissipative model dissipative model ε [34] ER 3 2 1 3 2 1 ( ) ( ) (2 ) ˆ ( ) i i i i i i u u I ER I u u = = − ∆ = = ∆ −
∑
∑
(5-1) (5-1) ui TNS ui (filter width) ∆ uˆi 2∆(5-1) filter secondary filter primary filter box filter Jeanmart Winckelmans[35] ∆ 2∆ box filter
1 1 ( n) 0.5[ ( n ) ( n) ( n )] u y u y u y u y + − − + + − + − ∆ ∆ = + + ∆ + ∆ ∆ + ∆ (5-2) 1 1 ˆ( n) ( n ) ( n ) u y u y u y + − − + + − + − ∆ ∆ = + ∆ + ∆ ∆ + ∆ (5-3) 1 n y − yn yn+1 ∆ =+ yn+1−yn ∆ =− yn−yn−1
Approximate deconvolution[36] primary filter approximate deconvolution
1 ( ) N v N Q G− I G = ≈ =
∑
− (5-4)56
N
Q primary filter G approximate deconvolution G box filter secondary filter ( ( )) i N i u = Q G ∆ ∗u (5-5) ˆi ( N (2 )) i u = Q G ∆ ∗u (5-6) 10 filter kc∆ =π/∆ LES
nominal cutoff wave number N =5 secondary filter
2∆ ∆ 2∆
(5-5) (5-6) (5-1) ER
primary filter secondary
filter [34] ER 0.007 0.01 ε ER 0.01 ε ER 0.007 ε ER ε 0.007≤ER≤0.01 ε ER 0.1 ε = ER 0.01 ε 0.5 DNS (ε =1) 0.1 0.5 ER 0.01 ε 0.05 ε ER ( , , , )x y z t ε ε= ( , , , )x y z t 0.5 ε = , if ER>0.01 ( , , , )x y z t 0.1 ε = , if 0.007≤ER≤0.01 (5-7) ( , , , )x y z t 0.05 ε = , if ER<0.007
10. filter Primary filter, Secondary filter, Secondary filter, Spectral filter
58
MPI (Message Passing Interface) OpenMP (Open Multi-Processing)
(Animation)
(Video Card) (Graphics processing unit GPU)
(Central Process Unit CPU)
CUDA Compute Unified Device Architecture
(Nvidia) GPU
GPU (Stream Processor
SP) CUDA CUDA CUDA CUDA 11 (Host) (CPU) (Device) CPU (serial
code) CUDA GPU
(kernel) kernel (Grid) (Block)
(Thread) kernel
(Stream Processor SP) kernel (serial code) CPU
CUDA
GPU (Stream
(Stream Processor SP) SM 8 SP (shared memory)
CUDA kernel
SM SM
(warp)
Tesla C1060 warp 32 SM
32 warp( 1024 ) warp CUDA
SM 512 warp ID ID 0~31 32~63 1~512 32 warp CUDA
GPU (Block) (Thread)
CPU
(i,j,k)
Device ( assign numbers ) = Host ( i ny nz× × + × +j nz k ) (6-1)
nx ny nz x y z
GPU
(Stream Processor SP)
CUDA (Single Instruction, Multiple Thread SMIT)
60 kernel CUDA (shared memory) (Barrier)
62 7.1 Prandtl 0.72 1atm 298.06K 5 10 Ra=
Paillere [37] local Nusselt number Rayleigh number
0 [ ( ) ] ( h c) w L T Nu k T k T T y ∂ = − ∂ (7-1) 2 3 0 2 0 ( ) Pr ( ) h c g T T L Ra T T ρ µ − = (7-2) 0 101325 P = Pa T0 =600K ρ0 =P0/(RT0) k0 T0 =600K 606K
594K、L 12 local Nusselt number
Paillere [37]
606K Gray Giorgini[26] Bossinesq
30K
Churchill Chu[38] Fu Huang[39] [38] [39] Boussinesq
1 l l1/ 2 =81 l0/l2 =102 w l/ 2 =2 Fu Huang[39]
Bossinesq Fu Huang[39]
10K [39]
13 Churchill Chu[38] Fu Huang[39] Rayleigh number
12. local Paillere et al. [37] Present results
64
13. Ra
Fu and Huang [39]
Churchill and Chu [38]
2 2 2 [ ] ( h c) w l l T Nu dx l T T y ∂ = − ∂
∫
(7-3) 2 3 0 2 2 ( ) Prg Th T lc Ra ρ β µ − = (7-4) β Bossinesq Bossinesq 30K Rayleigh 2 3 0 2 2 0 ( ) Pr ( ) h c g T T l Ra T T ρ µ − = (7-5) 14 Ra=104 0 t> h T l l1/ 2 =19 l0/l2 =25 w l/ 2 =2 110 h c T T T K ∆ = − = 14(1) t =0.0005s 14(1)c 14(1)a 0.02 t = s 14(2)a 14(2)b 14(2)c 0.05s 14(3)a 0.1s 14(4)b66 1s 14(5)a Dyer [40] Bernoulli [40] 15 Rayleigh Rayleigh 3 2 2 3 2 0 [ ( ) ] ( ) l l l w h c l L T Nu k T dx l k T T y + ∂ = − ∂
∫
(7-6) Rayleigh 2 0.213 0.794 l Nu = ×Ra (7-7) (7-7) Rayleigh
0.005
t = s
14(1). Ra=104
68 0.02
t= s
14(2). Ra=104
0.05
t= s
14(3). Ra=104
70 0.1
t= s
14(4). Ra=104
1
t= s
14(5). Ra=104
72 15.
