1 計算流體力學〈一〉1).pdf · 2018. 10. 26. · 1 計算流體力學〈一〉 〈...
TRANSCRIPT
1
計算流體力學〈一〉
〈Computational Fluid Dynamics〉
方富民 教授 〈Dr. Fuh-Min Fang〉
Department of Civil Engineering
National Chung Hsing University
September 2017
2
Numerical Strategy 〈A〉 Domain (Geometry) 〈B〉 Governing Equations (O.D.E., P.D.E.) & B.C.’s 〈C〉 Difference Equations (Discretization) & B.C.’s 〈D〉 Finite Difference / Finite Element or other methods * Importance of boundary conditions and initial conditions
〈1〉If not exist, only general solutions can be obtained. 〈2〉For practical applications, they are necessary. * Types of problems
〈1〉Elliptic：e.g.
0yx 2
2
2
2
=∂φ∂
+∂φ∂ (Laplace equation)
〈2〉Parabolic：boundary-layer flows (upstream-dominant) 〈3〉Hyperbolic：
∂∂
+∂∂
=∂∂
2
2
2
2
yC
xCk
tC
* Uniqueness and existence of particular solution. * Types of boundary conditions：
〈I〉Dirichlet problems：given values at boundaries 〈II〉Neumann problems：given normal derivatives at boundaries
3
Finite Difference Method in Solving Differential Equations
〈I〉Laplace equation
〈II〉Mass Transport equation
i-1 i i+1x∆ x∆
x
〈A〉Forward (downwind) Difference
xFF
xF i1i
i ∆−
=
∂∂ +
* By Taylor’s expansion
( )xxFF 1j ∆+=+
( ) ( )......
!3x
xF
!2)x(
xF
xxF
xF3
x3
32
x2
2
x
+∆
∂∂
+∆
∂∂
+∆
∂∂
+= 1 E
( ) ( )[ ]
( ) ( ) [ ]xOx
xFxxFxF
]x[OxFxxFxxF
x
2
x
∆+∆
−∆+=
∂∂
∆+−∆+=∆
∂∂
〈B〉Backward (upwind) Difference
xFF
xF 1ii
i ∆−
=
∂∂ −
* By Taylor’s expansion
)xx(FF 1i ∆−=−
( ) ( ) ( ) ( )......
!3x
xF
!2x
xF
xxF
xF3
3
32
x2
2
x
+∆−
∂∂
+∆−
∂∂
+∆−
∂∂
+= A2 E
4
( ) ( )[ ]
( ) ( ) [ ]xOx
xxFxFxF
]x[OxxFxFxxF
x
2
x
∆+∆
∆−−=
∂∂
∆+∆−−=∆
∂∂
〈C〉Central Difference
x2FF
xF 1i1i
i ∆−
=
∂∂ −+
* A1 E
A－A2 E
( ) ( ) ( )
( ) ( ) ]x[Ox2
xxFxxFxF
]x[Ox2xFxxFxxF
2
x
3
x
∆+∆
∆−−∆+=
∂∂
∆+∆
∂∂
=∆−−∆+
Comments：
(i) Better accuracy
(ii) Worse space resolution
(iii) Gradient independency of local values
5
* Example 1： （Dirichlet Type）
0yx 2
2
2
2
=∂φ∂
+∂φ∂
0
1
1x
50=φ
40=φ
0=φ
100=φy
B F
j+1
i+1ii-1
j
j-1
=> xx
j,ij,1iF
j,i ∆φ−φ
=
∂φ∂ +
xxx
x
BF
j,i2
2
∆
∂φ∂
−
∂φ∂
=
∂φ∂
xxj,1ij,i
B
j,i ∆φ−φ
=
∂φ∂ − 2
j,1ij,ij,1i
x2∆
φ+φ−φ⇒ −+
Similarly, 21j,ij,i1j,i
j,i2
2
y2
y ∆
φ+φ−φ=
∂φ∂ −+
0=φ
400→=φ 800 →=φ
8040→=φ
6
By substituting into the O.D.E., one gets
0y
2x
22
1j,ij,i1j,i2
j,1ij,ij,1i =∆
φ+φ−φ+
∆φ+φ−φ −+−+ (difference equation)
If one takes ,yx ∆=∆=∆ then
)(41 n
1j,in
1j,in
j,1in
j,1i1n
j,i −+−++ φ+φ+φ+φ=φ (Jacobi scheme)
j+1
i+1ii-1
j
j-1
Procedure：
(1) Make initial guesses at all interior points. (2) Iterate until convergence is reached.
* Example 2：（Dirichlet + Neumann）
0yx 2
2
2
2
=∂φ∂
+∂φ∂
0
1
1x
0=∂∂
xφ
0=∂∂
xφ
0=φ
100=φy
0=φ
1000→=φ
10070→=φ
0x=
∂φ∂
7
Image conditions at phantom points：
j+1
10-1
j
j-1
i=
0=∂∂
xφ
Exterior InteriorBoundary
Noted that at i = 0 , φ’s are unknowns.
( )n1j,0
n1j,0
nj,1
nj,1
1nj,0 4
1+−−
+ φ+φ+φ+φ=φ
nj,1
nj1,- take, 0iat 0
x φ=φ==∂φ∂
( )n1j,0
n1j,0
nj,1
1nj,0 2
41
+−+ φ+φ+φ=φ
8
Approaches
1) Stochastic
2) Deterministic
（I）Watershed Run-off Model
（i）Unit Hydrograph
Q(t)
Watershed
Given rainfall data and watershed characteristics, runoff in the watershed can be obtained as a function of time. The continuity equation is the major basis.
Q
Run-off
tTp
unit rainfall
peak
Assume that the system is linear, the actual hydrograph is equal to unit hydrograph multiplied by the actual amount of rainfall.
9
＃ Forms of unit hydrograph
A1 E
A Continuous function, u(t) A2 E
A Tabulated u
tt∆
u1u2
u3
u4
u5
u6
Noted that 1utN
1ii =∑⋅∆
=
（II）Conceptual Model
Assume that the watershed is a linear reservoir
storage:S tdSdQI
equation Continuity From
=−
For linear reservoirs, QkS = ( k is a constant to be calibrated)
Then (*).....................................................IQtdQdk =+
where ( )
>=∞
=0t ,00t ,
tI and 1tI =∆⋅
s
I(t) (Given)
Q(t)
S
10
For I = constant, the solution is
−=
−kt
e1IQ
For a unit hydrograph,
kt
ek1)t(uQ
−
==
u(t)
k1
t
The trend of the exponential decay implies that this approach is not good.
Improvement
n-reservoir model (assume k is the same for all the reservoirs)
Q1(=I2)
I
Q2(=I3)
I2
Q
In
Qn-1
kt
ek
−=
1
2IQ =
( )Idt)t(u thatso 0 =∫∞
11
( )kt
ekt
!1nk1u
1n
n−
−
−
= where k and n need to be calibrated.
un(t)
t
(un)max
Tp
To obtain Tp and (un)max , 0tu=
∂∂
.
The (theoretical) results is
( )
( ) ( )( ) 1n
1n
maxn
p
e!1nk1nu
k1nT
−
−
−−
=
−=
* Anther possibility is to choose different values of k for different reservoirs.
（III）Parallel Reservoirs (k varies)
more flexible but more complicated.
SSARR model.
12
Maskingum Method
It is similar to reservoir models, but is more like a channel flow condition.
I S O
For equation)ontinuity (storage/c tdSdOI =−
One may take ( )[ ]O1IkS α−+α= It is still a linear reservoir but depends on both the inflow (I) and outflow (O). Numerical Application
Qi Qi+1
ii+1
For tdSdOI =− (Superscripts denote time steps.)
( )
∆−
α−+∆−
α=+
−+ +
++
++++
+
tQQ1
tQQk
2QQ
2QQ n
1i1n
1ini
1ni
1n1i
n1i
1ni
ni
13
t∆
x∆
n+1
n
i i+1
t
x
Initial Condition
Then n
i1n
in
1i1n
1i QcQbQaQ ++= ++
++ (explicit)
where ( )( )
( )
( )t
1k2t
k2
t1k2t
k2
t1k2
11c
11b
111k2a
∆α−
∆α
∆α−
∆α
∆α−
++
=
+−
=
−−α−
=
One needs to determine 4 parameters (Δx , Δt , k , α) in order to calibrate the model .
Question:
a + b + c = ?
14
Kinematic Wave Method
I O
x∆ S0 1(bed slope)
B
dy
A
The continuity (storage) equation is tdSdOI =−
with QI =
) x , t Function(A ,x A S , xxQOI
expansion)order -1s(Taylor' xxQQO st
=∆=∆∂∂
−=−
∆∂∂
+=
Then, for a 1-D flow, (choose fixed xΔ )
( )
xtx
xAx
tA
tdxAdx
xQ
∆⋅∂∂
∂∂
+∆⋅∂∂
=
∆=∆
∂∂
−
Q , A : unknowns 2 unsteady) , D-(1 0xQ
tA gets one
, 0x A ,x smallafor Assume
=∂∂
+∂∂
→∂∂
∆
. equation 2 theneeds One
.flow lateral L
xQ
tA
:form General *
nd→
→=∂∂
+∂∂
15
Manning’s Equation
)onlyy(fASRn49.1Q 2
132
0 →== valid for steady flows
Assume this equation is also valid for gradually-varying flows ) xQ small (
∂∂
The hydraulic radius PAR = ; where P = wetted perimeter.
For dyBdA ≅ 0xy
ydfd
tyB =
∂∂
+∂∂
* Noted that B is actually a function of x and even t .
Or 0xy
B'f
ty
=∂∂
+∂∂
………………………………………………(1)
Since the elevation of water surface y = y(x, t), tdxd
xy
ty
tdyd
∂∂
+∂∂
=
If one sets B'f
tdxd=
then equation (1) becomes 0tdyd= , (O.D.E.)
and UB'f
tdxd
== (kinematic wave speed)
If U = constant, then the water surface is preserved. When U varies with x (mostly), the shape of the water surface will change.
y
x
tU∆tU∆
tU∆
16
For a wide channel (B >> y) with an approximately rectangular cross-section, from Manning’s equation (for a fixed So)
yBy2
yBPAR
ASRn49.1Qf 2
132
o
≅+
≅=
==
with an average velocity 21
32
oSRn49.1
AfV ==
The kinematic wave speed
]S)yB(yn49.1[
dyd
B1
B'fU 2
1
o32
==
]SyBn49.1[
35
B1 2
1
o32
= V35
⇒
Since the average velocity 32
32
y~R~V , and , the largest velocity occurs at the peak (with a maximum value of water depth y).
Accordingly, a wave shape tends to become steeper as it goes downstream.
In reality, a surface shape should become flatter due to diffusion (or dispersion). This implies that the use of kinematic wave model may lead to problems.
Arguments
A1 E
A Validity of using Manning’s equation for unsteady flows.
A2 E
A Slope “So” and Manning’s “n” varies with x .
Gravity effect should be included.
B
y
V~U
17
Diffusive Wave Model
Choose )S,y(fQ e= , where Se is the effective slope.
For uniform flows, xy S S oe ∂
∂−==
For non-uniform flows, 2
2
foe yu
gxy SS S
∂∂ν
−∂∂
−=+=
From the storage (continuity) equation 0xQ
tyB =
∂∂
+∂∂ , one has
0xS
Sf
xy
yf
tyB e
e
=∂∂
∂∂
+∂∂
∂∂
+∂∂
Or 0xy
BSf
xy
Byf
ty
2
2e =∂∂∂
∂
−∂∂∂
∂
+∂∂
Compared with the form 0xyD
xyU
ty
2
2
=∂∂
−∂∂
+∂∂ (Diffusive wave model)
yf
B1U∂∂
= ; 0Sf
B1D
e
>∂∂
= (diffusive coefficient)
For a wide channel with an approximately rectangular cross-section
ee
ee
oee
S2yV
SB2QD
SQ
21
Sf
)SS(yBSyn49.1fQ
V35U
21
32
≅≅
≅∂∂
>>≅=
≅
18
Numerical Diffusion in Kinematic Wave Method
(1) Utx , 0
tdyd
=∆∆
=
t
x
(B.C.)
(I.C.)y=
0
0
0
0
0 0 0 0 0 0
0 5 10 5 0 0
0 5 10 5 0
0 5 10 5 0
5 10 5characteristic line
Wave shape remains unchanged.
(2) t) Ux ( U2tx , 0
tdyd
∆>∆=∆∆
=
t
xy=
0
0
0
0
0 0 0 0 0 0
0 2.5 7.5 7.5 2.5 0
0 1.25 5 7.5 5
0 6.25 3.125 6.25 6.25
5 10 5
0 0
0
3.125
1.25
6.25 0
0
Peak decays and the wave shape becomes flatter.
* As t U x ∆>∆ , or Utx>
∆∆ , numerical diffusion exists. Also, the
solution is stable.
0.63 0.63
19
(3) t)U x( 5.1
Utx , 0
tdyd
∆<∆=∆∆
=
t
xy=
0
0
0
0
0 0 0 0 0 0
0 -2.5 2.5 12.5 7.5 0
5 10 5
< Extrapolation >
UNSTABLE SOLUTION !
t
xy=
0
0
0
0
0 0 0 0 0 0
0 0 2.5 7.5 7.5 2.5
5 10 5
0 0
0 0 0 1.25 5 7.5 5 1.25
Unstable Solution!
< Interpolation >
If interpolation is used, however, the result is still stable.
Courant stability criterion:
1tU
x≥
∆∆
Improvement
20
Schemes of Finite Differences
i-1 i i+1x∆ x∆
By Taylor’s expansion
( ) ( ) ( )
(B)......!3
xxy
!2x
xyx
xyy
......!3x
xy
!2x
xyx
xyyy
(A)......!3
xxy
!2x
xyx
xyyy
3
i3
32
i2
2
ii
3
i3
32
i2
2
ii1i
3
i3
32
i2
2
ii1i
+∆
∂∂
−∆
∂∂
+∆
∂∂
−=
+∆−
∂∂
+∆−
∂∂
+∆−
∂∂
+=
+∆
∂∂
+∆
∂∂
+∆
∂∂
+=
−
+
To representxy
∂∂ :
(1) Backward difference (Upwind scheme)
From (B),
( )
[ ]xOxyy
xy
]x[Oyyxxy
1ii
21ii
∆+∆−
=∂∂
∆+−=∆
∂∂
−
−
(2) Forward Difference (Downwind scheme)
From (A)
( )
[ ]xOx
yyxy
]x[Oyyxxy
i1i
2i1i
∆+∆−
=∂∂
∆+−=∆
∂∂
+
+
21
(3) Central Difference
From (A) - (B)
]x[Ox2yy
xy
]x[Oxxy2yy
21i1i
31i1i
∆+∆−
=∂∂
∆+∆
∂∂
=−
−+
−+
* Central difference yields 2nd-order errors. Problem of central difference:
Oscillation is due to unreasonably small dependency on local characteristics at the previous time step.
Example :
y= 5 10 5 10 5 10 5x
The above is a solution for 0xy=
∂∂ if central difference is applied.
However, this solution is obviously unacceptable.
* Although central difference leads to smaller errors, the resolution of grid system becomes worse. (Instead of Δx, 2Δx is the actual grid resolution.)
22
From Taylor’s expansion,
t
xR
P
A
t∆
( ) ( )
12
A2
2RA
A
32
A2
2
AAR
................... ]'x[O2
'xxy
'xyy
xy
]'x[O!2
'xxy'x
xyyy
∆∆+∆
∂∂
+∆−
=
∂∂
∆+∆−
∂∂
+∆−
∂∂
+=
For 0tdyd= ,
if one takes tyy
ty RP
∆−
=∂∂ (Forward difference, no choice)
then
0tdxd
xy
ty
=∂∂
+∂∂
0]'x[O]2
'xxy
'xyy[U
tyy 2
2
2RARP =∆+
∆∂∂
−∆−
+∆−
Thus for tU'x ∆=∆ 0xy
2tU
xyyU
tyy
2
22RARP ≅
∂∂∆
−∆−
+∆−
Compared to the diffusive wave model equation (see pp.17)
2tUD0
xyD
xyU
ty 2
2
2 ∆=⇒=
∂∂
−∂∂
+∂∂
Strategy: Noted that Δt is limited by Courant condition for numerical stability
and has to be adjusted to fit the field data. Also, 'x∆ (= U Δt) is calculated to form a non-uniform grid system at the new time step.
U
tU'x ∆=∆
23
Method of Characteristics
In the kinematic wave model
(1) Explicit (conditionally stable)
0xQ
B1
ty
=∂∂
+∂∂ with )Q(Fy = (for example, Manning’s equation)
Eliminating y yields
0xQ
B1
tQ
QdFd
=∂∂
+∂∂
Or 0xQ
'FB1
tQ
=∂∂
+∂∂ (
'FB1 is the kinematic wave speed “U”)
In a fixed grid system (Δx and Δt are fixed)
'x∆
t
x
n+1
n
i i+1
t∆
x∆
i,n+1
i,n
i+1,n+1
i+1,nU
1
)QQ(x
tUQQ
0x
QQUtQQ
ni
n1i
n1i
1n1i
ni
n1i
n1i
1n1i
−∆∆
−=
=∆−
+∆−
++++
++++
where U)-1(U U n1i
ni +η+η= with
xx' ∆∆
=η (by linear interpolation)
24
The finite difference form becomes
n1i
ni
1n1i Q)1(QQ +++ γ−+γ=
where x
tU∆∆
=γ , is the Courant number.
To have numerical stability, only interpolation is applied.
(2) Explicit (an alternative scheme)
Choose 1010
≤β≤≤α≤
( ) ( )
1n1i
1ni
n1i
ni
1ni
1n1i
ni
n1i
n1i
1n1i
ni
1ni
U, U, U, Uamong average weightedSomeU
0xQQ1
xQQU
tQQ1
tQQ
++
++
+++++
++
+
=
=
∆−
β−+∆−
β+
∆−
α−+∆−
α
One gets
∆β
+∆α
+
∆
β−+
∆α−
+
∆β
−∆α−
=
∆β−
+∆α− +
+++ x
Ut
Qx
)1(Ut
Qx
Ut
1Qx
1Ut
1Q ni
1ni
n1i
1n1i
1∆ 2∆ 3∆ 4∆
For ni
1ni
n1i
1n1i QcQbQaQ ++= +
+++
1
4
1
3
1
2 c,b,a∆∆
=∆∆
=∆∆
=
* Compared to Maskingum method results on pp.13.
when Uxk
21 ∆
=α=α=β
(i+1,n+1)
(i,n) (i+1,n)
25
Suggested procedure :
A1 E
A For numerical stability, choose an appropriate α (suggested 0.3)
A2 E
A From field data, calibrate k (or Ux∆
).
x
I.C. Final profile
A3 E
A Calibrate the magnitude of diffusion from field data.
Select Δt based on Courant condition, find Δx .
t = t0 t = tf
y
x
26
Dynamic Model (for 1-D flows on a 2-D vertical plane)
Basic assumption: gradually-varying flows
→ Streamlines are straight and parallel. Therefore, pressure is hydrostatically distributed.
* Local “A” remains roughly unchanged (with respect to x).
s0 1x
y1 y2
F1 F2
F3
x∆
FrictionF4
From the Newton 2nd law, amF =
Along the x-direction 4321 FFFFF −+−=
where
Axxyx
xyy
x2y
x2y
2yFF
222
21
∆
∂∂
γ−=∆∂∂
γ−=
∆⋅
γ∂∂
+γ
−γ
=−
( )1yA ⋅=
f4
03
SAxFSAxF⋅⋅∆γ=⋅⋅∆γ=
=
=
line)energy of (slope SSor
FFflows uniformin that Note
fo
43
control volume
g
27
Mass of the control volume xAm ∆ρ=
and xVV
tVa
∂∂
+∂∂
= (1-D approach)
By substitution, the momentum equation becomes
0)SS(gxyg
xVV
tV
0f =−+∂∂
+∂∂
+∂∂ ……………………………..……(1)
Meanwhile, the storage (continuity) equation is
0 xV
BA
xyV
ty or
0xVA
xAV
tyB
=∂∂
+∂∂
+∂∂
=∂∂
+∂∂
+∂∂
…………………………….(2)
It is noted that
(1) & (2) → 2 equations , 2 unknown : (V & y)
For cases with approximately constant “A” and “B” along x
tA
AQ
tQ
A1)
AQ(
ttV
2 ∂∂
−∂∂
=∂∂
=∂∂
xQ
AQ
x
)AQ(
A1
xQ
AV
x)VQ(
A1
xV
AQ
xVV 2
2
∂∂
−∂
∂=
∂∂
−∂
∂=
∂∂
=∂∂
Equation (1) becomes
( ) ( ) 0SSAgx
)BA(
g)xQ
tA(
AQ]
xQV
tQ[
A1
of2 =−+∂
∂+
∂∂
+∂∂
−∂
∂+
∂∂
Or ( ) (1a)................0SSAgxA
BAg
xAQ
tQ
of
2
=−+∂∂
+∂
∂
+∂∂
incorporated with
0xQ
tA
=∂∂
+∂∂
…………….………..…………… (2a)
28
Examine the equation of motion in the dynamic model
( ) 0SSgxyg
xVV
tV
0f =−+∂∂
+∂∂
+∂∂
A1 E
A A2 E
A A3 E
A A4 E
A1 E
A +A2 E
A → inertia terms (subject to acceleration)
A3 E
A → gravitational effect
A4 E
A → kinematic wave model term (A) For A4 E
A = 0 → kinematic wave model
No back-water effect Dominated by upstream condition
(B) For A3 E
A + A4 E
A = 0 → diffusive wave model
( ) 20f
1
.......................0SSgxyg
............................ 0xQ
tyB
∆=−+∂∂
∆=∂∂
+∂∂
If Manning’s equation is used,
2/1fS)y(kQ = where )y(A)]y(R[
n49.1)y(k 3/2=
Or 2f )
kQ(S =
x2
∂∆∂ yields 0]
xy
ydkd
kQ2)
xQ(
kQ2
xy[g 3
2
22
2
=∂∂
−∂∂
+∂∂ (For 0
xS0 =∂∂ )
Substitution of tyB
xQ
∂∂
−=∂∂ yields
0xy
QB2k
xy
ydkd
kBQ
ty
0xy
ydkd
kQ2
tyB
kQ2
xy
2
22
3
2
22
2
=∂∂
−∂∂
+
∂∂
=∂∂
−
∂∂
−+∂∂
29
Then the kinematic wave speed
ydkd
kBQU =
The diffusive coefficient
QB2kD
2
=
Recall in the diffusive wave model (pp. 17)
ydkd
kBQ
BkQ
ydkd
B
Sydkd
ByQ
U
2/1f
===∂∂
= (O.K.)
QB2k
)kQ(B2
kSB2
kBS
k21
BSQ
D2
2/1f
2/1ff ====
∂∂
= (O.K.)
30
Characteristic Equations in Dynamic Model
L1 : ( ) 0SSgxyg
xVV
tV
of =−+∂∂
+∂∂
+∂∂
L2 : 0xV
BA
xyV
ty
=∂∂
+∂∂
+∂∂ (2 P.D.E.’s)
Introduce a parameter λ , so that 0LL 21 =λ+ yields
( ) 0SSgxygV
ty
xV
BAV
tV
of =−+
∂∂
λ
++∂∂
λ+
∂∂
λ++
∂∂
Let gVBAV
tdxd
λ+=λ+=
Then, one gets ( ) 0SSgtdyd
tdVd
of =−+λ+ (O.D.E.)
where Cg
ABg
±=±=λ (C is wave speed)
then CVtdxd
±= and BAgC =
Noted that for a channel with an approximately rectangular cross-section
ygBAgC ≈= → gravitational wave speed
There exist 2 sets of characteristic equations：
−++
+=+
)SS(gtdyd
Cg
tdVd
CVtdxd
:Cof
−+−
−=−
)SS(gtdyd
Cg
tdVd
CVtdxd
:Cof
31
c c
V
* V > C → super-critical flows
V < C → sub-critical flows
Regarding the characteristic diagram:
Sub-critical flow cases： (V < C)
S"" at )S-(SgG 2*.......0GtΔyy
Cg
tΔVV:C
R"" at )S-(SgG 1*.....0GtΔyy
Cg
tΔVV:C
ofSSSP
C
SP
ofRRRP
C
RP
==+−
−−
==+−
+−
−
+
There are 2 unknowns : VP and yP
2 equations
t∆
t
x
P
C+
RA C S BC-
V+C C-V
1 1
( ) tVC ∆+
tVC ∆−
V-C
32
( ) ( )
( ) ( )RSC
SRCSR
P
SRRSC
SRP
GGg2
tCVVg2
C2
yyy
GG2tyy
C2g
2VVV
−∆
+−++
=
+∆
−−−+
=
where velocities (V) and depths (y) at points “R” and “S” are interpolated from values at points “A” , “B” and “C” .
Comments:
〈1〉 One may use 2
CCC CP += in equation (*1)
and 2
CCC CP += in equation (*2)
to establish the finite difference equation . However, since CP is unknown, it yields an implicit scheme. Iterations are therefore needed.
Supercritical flow cases：(V > C)
t∆
t
x
P
C+
A C BC-
V+CC-V
11
( ) tCV ∆+
( ) tCV ∆− This is an upstream-dominant (parabolic) problem. Values at point “B” are useless in the predictions.
33
Boundary Conditions
〈I〉Upstream Boundary
〈i〉Sub-critical flows
P
C+(?)
A S B
C-
t
x
For 1-D flows, there are 2 unknowns, (y, V) or (A , Q) .
Only 1 equation ( −C ) is available => need 1 more.
Suppose one uses Q = Qo(t) , then at a time, t
)t(Q)y(AV oPP =⋅
Incorporated with the −C equation
0Gtyy
Cg
tVV
SSP
S
SP =+∆−
−∆−
where VS and yS are interpolated from points A & B ; CS and GS are known.
Elimination of VP yields Function(yP) = 0 → non-linear One can use Newton-Raphson method to obtain the values of yP .
34
For F(yP) = 0, supposed an initial guess, y0, is made and results in
0)y(F o ≠ ,
a “small” adjustment of yo value is needed, say , yyy oP −=∆
so that 0)yy(F)y(F 0P =∆+=
By Taylor’s expansion (for small Δy)
0]yΔ[OyΔyF)y(F)yΔy(F 2
yoo
o
=+
∂∂
+=+
then oyy
Fo
)()y(Fy
∂∂−≅∆
Noted that since 1st order Taylor’s expansion is used (linear extrapolation), Δy has to be small to achieve convergence. That is, the initial guess (yo) should be good enough so that Δy is small.
Try-and-Error procedure:
(1) Make an initial guess , yo , (yo = ya or yS is suggested)
(2) For F(yP) = 0 , find )y(funyF
p=∂∂
(3) Substitute yo into F and yF
∂∂
(4) 2nd guess oyy
Fo
01)y(Fyy
∂∂
−=
* Generally, nyy
Fn
n1n )()y(Fyy
∂∂+ −=
(5) Proceed iteration until
n1n yy −+ < some prescribed criterion
35
〈ii〉Super-critical flows
No equation is available (neither +C nor −C ) for the solution of the 2 unknowns. => Need 2 conditions.
C+(?)
C-
t
x
(?)
USE
〈II〉Downstream Boundary
〈i〉Sub-critical flows
t
x
P
C+ C-(?)
DSE
Only 1 equation (C+) is available => Need 1 more condition.
Given Q=QP(t) at the downstream end, Newton-Raphson method can be used to determine the yP values of the 1-D flows . The procedure of Newton-Raphson method is the same as that in the upstream boundary case.
36
〈ii〉Super-critical flows
t
x
P
C+ C-
DSE
As the 2 equations are both available, there is NO need of any other conditions.
〈III〉Boundary Specifications at Joints
4
5
1
2
3 6
Presumably, y and V at points 4 , 5 , 6 are known. At the joint, there are 6 unknowns for the 1-D flows (VP1 , VP2 , VP3 , yP1 , yP2 , yP3).
For a sub-critical flow
(A) Characteristic (momentum) equations −++321 C,C,C (3 equations)
(B) Continuity equation (assume storage change is insignificant)
321 QQQ =+
or 33P22P11P AVAVAV =+ (1 equation)
37
(C) Energy Equation (2 equations)
3
23P
1
23P
3P
21P
1P1 zg2
Vkg2
Vyg2
Vyz +++=++
3
23P
2
23P
3P
22P
2P2 zg2
Vkg2
Vyg2
Vyz +++=++
1∆ and 2∆ are energy losses at the joint.
* k1 and k2 are energy loss coefficients. They can be determined based on local geometries. z’s are bed elevations.
6 unknowns, 6 equations => (O.K.)
* For cases with a small Froude Number
<< y
g2V2
(C) => 3P32P2
3P31P1
yzyzyzyz
+=++=+
1∆
2∆
38
Surge/Hydraulic Jump Shock Fitting method:
V2
V1
W
y1
y2
Using a moving coordinate system that goes with the surge,
V2+W
V1+W
Super-critical Sub-critical
11
11
ygCwhere
CWV
=
>+
22
22
ygCwhere
CWV
=
<+
Noted that 2211 VCWVC −<<− (see proofs on pp. 40-41)
An unsteady surge looks like a hydraulic jump on a coordinate moving at a speed of the surge.
Note: The gradually varing flow assumption does not hold in the vicinity of the surge.
39
On the characteristic diagram
t
x
1
t∆
2
C1+
C1-
C2-
tW∆
( ) tVC ∆−1
( ) tVC ∆−2
tW∆
* Noted that the C2+ equation is not available since it crosses the gap (region
of the surge), where the gradually-varying-flow assumption does not hold.
Unknowns : (5)
W , y1 , V1 , y2 , V2
Need 5 equations:
+1C , −
1C , −2C (3)
Continuity:
2211 y)WV(y)WV( +=+ (1)
From the impulse/momentum theorem
)]WV()WV[(y)WV()yy(2 1211
22
21 +−++ρ=−
γ
)VV(y)WV( 1211 −+ρ= (1)
40
Supplemental proof：
Momentum equation
)VV(y)WV()yy(2 1211
22
21 −+ρ=−
γ …………… (1)
Continuity equation
2211 y)WV(y)WV( +=+ ………..……………… (2)
From equation (2)
( )
−+=−
−+=
1yyWVVV
1yyW
yyVV
2
1112
2
1
2
112
Substitution into equation (1) yields
)yy(yy)WV()yy()yy(
2 212
1212121 −+ρ=−+
γ ……… (3)
)yy1(
yyy
2g)WV(
1
2
1
21
21 +=+
)yy1(
yy
2yg
1
2
1
21 +=
2
)yy1(
yy
C 1
2
1
2
21
+= …………...……….. (4)
For 1,yy 112 >∆> therefore 11 CWV >+ (super-critical)
or 11 VCW −>
From equations (3) and (2) 2
1
2221
2
12121 ]
yy)WV[()yy(
yy)yy()yy(
2+−ρ=−+
γ
)yy(yy)WV( 21
1
222 −+ρ=
1∆
41
2
)yy1(
yy
C)yy1(
yy
2yg)WV( 2
1
2
1
22
2
1
2
1222
+=+=+
For 1,yy 212 <∆> therefore 22 CWV <+ (sub-critical)
or 22 VCW −<
* For surges (or hydraulic jumps), energy equation is seldom use in the analysis. Since it involves a huge energy loss, the energy loss coefficient is difficult to estimate.
2∆
42
Stability Consideration in Physical Sense
〈1〉Courant–Friedrich–Lewy Condition
x
A P
C
( ) tCV ∆+
R B S D
x∆( ) tCV ∆−
If Δt is large such that xtCV ∆>∆+ , it leads to divergence if
extrapolation is used.
To ensure numerical stability,
xtVC
xtCV
∆≤∆−
∆≤∆+
or
1x
tCV≤
∆∆±
(CFL condition)
* Noted that if values of y and V at point A are known (so that values at
point C can be interpolated from points A and B), the numerical results are converged (disregard the numerical accuracy).
43
〈2〉Kolen’s stability criterion (due to non-linearity)
In the dynamic model,
0)SS(gxyg
xVV
tV
of =−+∂∂
+∂∂
+∂∂
Since Sf depends on the local values of y and V, strictly, the momentum
equation contains 2 non-linear terms (xVV
∂∂
and Sf ).
To achieve a stable scheme
o
oVSg
o
o
F1F21
t−+
≤∆ (Kolen’s criterion)
where o
oo C
VF = (Froude Number) ; oo ygC =
* Noted that if the flow is very slow (F0→0), then Δt has to be extremely small (not economical) in order to obtain numerical stability.
x∆
t∆
CV ±
1
( )maxt∆
( )maxx∆
stable
Courant
Kolen
Combination of the 2 criteria yields
)F1()1F21(y
)x(Soo
o
maxo +−+≤∆
44
Schemes of the Dynamic Model
〈A〉 Explicit (conditionally stable)
0xQ
B1
ty
=∂∂
+∂∂
( ) 0SSgxyg
xVV
tV
of =−+∂∂
+∂∂
+∂∂
where
)t,x(GtV
)t,x(Fty
=∂∂
=∂∂
A general form for ty
∂∂ is
( )t
]1y[yty 2
yyni
1ni
n1i
n1i
∆α−+α−
=∂∂ +− ++
* For
ty
ty
0
2
yy1ni
n1i
n1i
∆
−=
∂∂
=α
+− ++ (Diffusive Scheme)
x2VV
xV n
1in
1i
∆−
=∂∂ −+ (Central Difference)
Note： For
tyy
ty
1ni
1ni
∆−
=∂∂
=α+
Numerical result shows divergence.
t
n+1
n
i i+1 x
i-1
45
〈B〉Implicit (unconditionally stable)
xVV)1(
xVV
xV
tyy)1(
tyy
ty
ni
n1i
1ni
1n1i
n1i
1n1i
ni
1ni
∆−
β−+∆−
⋅β=∂∂
∆−
α−+∆−
⋅α=∂∂
+++
+
+++
+
Unknowns: (4)
1n1i
1ni
1n1i
1ni V,V,y,y +
+++
++
At the updated time step, one interior node has two equations in the forms of
0JVIVGyHyF0EVDVByCyA
1n1i
1ni
1n1i
1ni
1n1i
1ni
1n1i
1ni
=++++
=++++++
+++
+
++
+++
+
(4 unknowns, 2 equations), * A ~ J are “some” coefficients.
For ”I-1” interior nodes in the grid at the updated time step , there are 2I corresponding equations and 2(I+1) unknowns . The remaining 2 conditions (to solve for the set of equations) must come from the boundary conditions at the upstream and/or downstream ends. (USBC and DSBC).
For sub-critical flows, at the (n+1) time step：
.......................................
....................................... (5) ....... 0J VIyHVGy F (4) ....... 0E VDyCVByA (3) ....... 0J VIyHVGyF(2) ....... 0E VDyCVByA
(1) ....... 0C Vbya
232322222
232322222
121211111
121211111
11111
=++++=++++=++++=++++=++
DSBC： ……(2I+2) The matrix is called a banded matrix.
0Vbya 1I1I1I1I =+ ++++
USBC：
(downwind scheme)
46
Solution Procedure of Tri-Diagonal Matrix Algorithm (TDMA) 〈1〉 From equations (1) , (2) and (3), one can get
)(.......0CVbya 122222 ∆=++⇒ 〈2〉 From equations (Δ1), (4) and (5)
)(.......0CVbya 233333 ∆=++⇒ 〈3〉 Eventually one gets
)(.......0CVbya I'
1I1I'
1I1I'
1I ∆=++ +++++ 〈4〉 Incorporated with the DSBC, equation (2I+2), 1Iy + and 1IV + can
be solved. 〈5〉Back substitution to get 11II )V,y(,,)V,y(,)V,y( →→ − .
47
Comparison of Models
The continuity (storage equation) is the one to be used in all models.
The equation of motion for dynamic models is (see also pp. 30 for Eq. L1)
0)SS(xy
tDVD
g1
of =−+∂∂
+
〈1〉 〈2〉 〈3〉
where 〈1〉→ Sa： slope of acceleration (inertia)
〈2〉→ Sw： slope of water surface
〈3〉→ For a uniform flow Sf = So
〈A〉If So >> Sa and Sw , → kinematic wave model 〈B〉If Sw ~ So >> Sa , → diffusive wave model 〈C〉If Sa ~ Sw ~ So , → dynamic model
Example of a rectangular channel：
A crude estimate is (q is the volumetric flow rate per unit width)
31
C )qg(V = ,
Q(t)
yc (critical depth)
So=10-3
48
Then tdqd
q1g
31
tdVda 3/2
3/1C ==
32
)qg(3tdqd
gaSa ==
For a gradually-varying flow with q = 1 ft3/s, in order to have Sa ~ So (so that the dynamic model has to be used)
s/ft/cfs1004.3)12.32)(3)(10(tdqd 23/23 −− ×=×≈
Supposed, one defines that Sa > 10% So is the criterion to use dynamic model, then when
s/ft/cfs1004.3tdqd 3−×>
the dynamic model should be used.
49
Fundamental Equations
Continuity：
0)u(t
=ρ•∇+∂ρ∂
Momentum：
Tf)uutu( •∇+ρ=∇•+∂∂
ρ
where u•∇
k
kji
j
i
i
jjiji x
u')xu
xu
(pT∂∂
µδ+∂∂
+∂∂
µ+δ−=
stress due to compressibility, µ=µ32'
For a Newtonian fluid with a constant viscosity (compressible flows), the form of Navier-Stokes equation is
upftdud 2∇µ+∇−ρ=ρ
* For incompressible flows, (ρ = constant → given)
0t=
∂ρ∂
the continuity equation becomes
0u =•∇ (Elliptic)
* For compressible flows,
0t≠
∂ρ∂
Time domain → parabolic
Space domain → elliptic Hyperbolic⇒
τ
50
Equations in a Conservative Form (Laminar case)
Continuity: 0zw
yv
xu
t=
∂ρ∂
+∂ρ∂
+∂ρ∂
+∂ρ∂
(valid for compressible flows)
Momentum: ( Ω∇=g ) ( D2I)u( µ+•∇λ=τ )
0)wu(z
)vu(y
)pu(xt
uzxyxxx
2 =τ−ρ∂∂
+τ−ρ∂∂
+Ωρ+τ−+ρ∂∂
+∂ρ∂
0)wv(z
)pv(y
)vu(xt
vzyyy
2yx =τ−ρ
∂∂
+Ωρ+τ−+ρ∂∂
+τ−ρ∂∂
+∂ρ∂
0)pw(z
)wv(y
)wu(xt
wzz
2zyzx =Ωρ+τ−+ρ
∂∂
+τ−ρ∂∂
+τ−ρ∂∂
+∂ρ∂
For 2-D incompressible flows
=ρ=
∂∂
= ttancons,0z
,0w
0yv
xu
=∂∂
+∂∂
0)vu(y
)pu(xt
u yxxx2 =ρ
τ−
∂∂
+Ω+ρ
τ−
ρ+
∂∂
+∂∂
0)pv(y
)vu(xt
v yy2yx =Ω+ρ
τ−
ρ+
∂∂
+ρ
τ−
∂∂
+∂∂
If z = constant (no gravity effect) , ν = constant
(3)........yv
xv
yp1
yvv
xvu
tv
(2)........yu
xu
xp1
yuv
xuu
tu
1)(...........................................................0yv
xu
2
2
2
2
2
2
2
2
∂∂
+∂∂
ν=∂∂
ρ+
∂∂
+∂∂
+∂∂
∂∂
+∂∂
ν=∂∂
ρ+
∂∂
+∂∂
+∂∂
=∂∂
+∂∂
51
Vorticity-Stream Function Approach (in 2-D incompressible flows)
Let yu
xv
∂∂
−∂∂
=ζ (vorticity) ………. (*)
For an incompressible flow, there exist a stream function ψ such that
vx
uy
=∂y∂
−
=∂y∂
(right-hand rule) ……….(#)
By substituting into the continuity equation, one gets
0xyyx
22
≡∂∂y∂
−∂∂y∂
* Noted that also from equation (*), Poisson equation is obtained as
ζ−=∂y∂
+∂y∂
2
2
2
2
yx ……………………. (4)
ζ and ψ are related based on this equation.
The result of
∂∂
−∂∂
y)2(
x)3( yields
)yx
(y
vx
ut 2
2
2
2
∂ζ∂
+∂ζ∂
ν=∂ζ∂
+∂ζ∂
+∂ζ∂
……………………. (5)
<storage> <convection> <diffusion> ⇒ vorticity transport equation
Strategy：
〈1〉Guess initial conditions (t = 0), solve for ζ from equation (5) at t = Δt .
〈2〉At t = Δt , solve for ψ based on equation (4)
〈3〉Obtain u , v (as functions of x , y) from ψ based on equations (#)
〈4〉Proceeds to the next time step, solve for ζ from equation (5) at t = t+Δt.
If irrotational, ζ = 0 => Laplace equation
52
* What about the solution of pressure?
∂∂
+∂∂
y)3(
x)2( yields
)xv
yu
yv
xu(2p2
∂∂
∂∂
−∂∂
∂∂
ρ=∇
Solution Procedure:
〈1〉Specify initial values of ζ and ψ from initial flow fields (u , v , p) at the interior and boundary nodes.
〈2〉Based on equation (5), update ζ at next time step with prescribed
boundary conditions. (pp. 55) 〈3〉Based on equation (4) , calculate ψ based on new values of ζ . (pp. 53) 〈4〉According to the solution and definition of ψ , find u and v . (# on
pp.51) 〈5〉Solve equation (6) for pressure, p . (pp. 52) 〈6〉Determine the boundary condition of ζ for next update. (pp. 54 for solid
boundaries; Upstream end: given; Downstream end: zero gradient.) 〈7〉Proceed iteration by repeating steps 〈2〉to〈6〉.
53
Explicit Method for the solution of ψ (equation (4) on pp. 51)
ζ−=∂y∂
+∂y∂
2
2
2
2
yx (Poisson equation)
Take a difference form as (see also pp. 6)
j,i21j,ij,i1j,i
2j,1ij,ij,1i
y2
x2
ζ−=∆
y+y−y+
∆y+y−y −+−+
j+1
i+1ii-1
j
j-1
y
x
For convenience, take Δx = Δy = Δ 2
j,ij,i1j,i1j,ij,1ij,1i 4 ∆ζ−=y−y+y+y+y −+−+
)(41 2
j,i1j,i1j,ij,1ij,1ij,i ∆ζ+y+y+y+y=y −+−+
* In general, if Δ x ≠ Δ y, one has
fedcba j,i1j,i1j,ij,1ij,1i =y−y+y+y+y −+−+
At a specific time step,
J
I32
3
2
J+1
i=1 I+1j=1
Lx
Ly
yJLxIL
y
x
∆⋅=∆⋅=
At each time step, iteration has to be conducted until convergence is reached.
Unknowns: (I+1)(J+1) Equations: - (I-1)(J-1) Number of interior points
2(I+J) Number of conditions needed (Number of boundary points)
54
At a solid boundary,
y∆1j =
2j =
3j =
y
xconst→=y After ψ has been solved, one needs to find the ζ field. To do so, ζ’s at the (solid) boundary are needed.
Since at (i , 1) , 0uy 1,i ==
∂y∂ , from Taylor’s expansion
])y[(O!2)y()
y()y()
y( 3
2
2,i2
2
1,i1,i2,i ∆+∆
∂y∂
+∆∂y∂
+y=y
Also, v = 0 at y = 0 for all x’s ,
1,i2
2
1,i1,i1,i )y
()yu()
xv(
∂y∂
−=∂∂
−∂∂
=ζ (no-slip)
Therefore,
221
2,i1,i1,i
21,i1,i2,i
)y(
)y(21
∆y−y
=ζ
∆ζ−y=y
55
Alternating Direction Implicit Method (ADI Method)
To solve the transport equation of ζ numerically,
)yx
(y
vx
ut 2
2
2
2
∂ζ∂
+∂ζ∂
ν=∂ζ∂
+∂ζ∂
+∂ζ∂
2 steps:
〈I〉At 2ttt ∆
+= , solve along x-direction implicitly.
∆ζ−ζ+ζ
+∆
ζ−ζ+ζν=
∆ζ−ζ
+∆ζ−ζ
+∆ζ−ζ
−+++
−++
−++−
++
+
2
nj,i
n1j,i
n1j,i
2
nj,i
nj,1i
nj,1i
n1j,i
n1j,in
j,i
nj,1i
nj,1in
j,i21
nj,i
nj,i
)y(2
)x(2
y2v
x2u
t
21
21
21
21
21
21
〈II〉At ttt ∆+= solve along y-direction implicitly.
∆ζ−ζ+ζ
+∆
ζ−ζ+ζν=
∆ζ−ζ
+∆ζ−ζ
+∆ζ−ζ
++−
++
++−
++
+−
+++
+−
+++
++
2
1nj,i
1n1j,i
1n1j,i
2
nj,i
nj,1i
nj,1i
1n1j,i
1n1j,in
j,i
nj,1i
nj,1in
j,i21
nj,i
1nj,i
)y(2
)x(2
y2v
x2u
t
21
21
21
21
21
21
21
21
Note :
〈I〉Values at the (n+1/2) step are solved implicitly. 〈II〉Values at the (n+1) step are also solved implicitly.
56
Weakly-Compressible-Flow Method (WCF Method)
The conservative form is 0HtG
=•∇+∂∂
(storage) (fluxes)
For a laminar and inviscid (non-viscous) flow, Continuity equation：
0)uρ(tρ
=•∇+∂∂ ................................................................ (1)
Momentum equation：
xp
ρ1u)u(
tu
∂∂
−=∇•+∂∂ .................................................. (2)
yp
ρ1v)u(
tv
∂∂
−=∇•+∂∂ .................................................. (3)
zp
ρ1w)u(
tw
∂∂
−=∇•+∂∂ ................................................. (4)
Based on barotropic assumption (ρ is a function of p only),
tp
pdd
t ∂∂ρ
=∂ρ∂
Then, equation (1) becomes
1*
0)u(d
pdtp
=ρ•∇ρ
+∂∂
.................................................. (1a)
Or
2*
0ρdpduρu
ρdpdρ
tp
=
∇•−
•∇+
∂∂
........................ (1b)
* For a perfect gas under isentropic condition,
γρ= oCp (Co is a constant and v
p
CC
=γ ; Cp and Cv are specific heats.)
57
1oC
dpd −γργ=ρ
u
xu
xuC
x)1(Cu
1*2*
xux
10
20
ρ≅
∂∂
ρ+∂ρ∂
ργ
ρ
∂ρ∂
−γγρ⇒
∂∂∂ρ∂
−γ
−γ
Take the x-momentum equation for example,
∂∂
ρ−=+
∂∂
+∂∂
+∂∂ρ
xp1
yuv
xuu
tu
u
one has
∂∂
∂∂
∂∂
∂∂
ρtp
u1
)(p
u1 O
xp
u1 O
xu
2ux2
From equation (1),
0z
)w(y
)v(x
)u(t
=∂ρ∂
+∂ρ∂
+∂ρ∂
+∂ρ∂
∂∂
∂∂ρ
∂ρ∂
∂ρ∂
⇒∂ρ∂
+∂∂
ρtp
a1 O
tp
pdd
t
xu
xu
xu
2
Thus, , ]M[OO1*2* 2
tp
u1
tp
a1
2
2
=
=
∂∂
∂∂
where M is the Mach number.
Accordingly, with the error of an order or M2, equation (1a) can be approximated as
0uρdpdρ
tp
=
•∇+
∂∂
* For perfect gases, dp/dρ = a2 . (“a” is sound speed.)
58
* For liquids and low-Mach-number gases, dp/dρ ∼ a2
With an error of order [M2], the continuity equation for WCF method becomes
0)uaρ(tp 2 =•∇+
∂∂ .......................................................... (1c)
Noted that the definition of bulk modulus of elasticity (k) is
k/dpd
−=∀∀
(∀ is the volume of fluid)
For mass conservation, =∀ρ constant or ∀ρ=ρ∀ dd .
Therefore, ρρ−=∀∀ /d/d
Then, k/d
pd−=
ρρ−
Or 2akd
pd≈
ρ=
ρ 2ak ρ≈
* For a low-mach-number flow, k is large and approximately remains constant.
Finally, the continuity equation for WCF method is
0)uk(tp
=•∇+∂∂ .............................................................. (1d)
Now, go back to the momentum equation. Again, take the x-momentum equation for example,
7* xp
ρ1uu)uu(
tu
x
)(p
x)(
uu)uu(tu
6 * 5 * 8*
ρ1
ρp
∂∂
−=•∇−•∇+∂∂
∂
∂+
∂
∂−=•∇−•∇+
∂∂
................. (2a)
59
][MOa1uO
pdρd
ρpO
)(p
)(p
7*6*
22
2
xp
ρ1
xp
pdρd
ρ1
xp
ρ1
xρ
ρ1
22
=
=
=
−=
−=
∂∂
∂∂−
∂∂
∂∂−
Examine term *8 from the continuity equation
0)uρ(tρ
=•∇+∂∂
or 0uρρutρ
=•∇+∇•+∂∂
Then tp
a1
ρuρu
ρuuu
10 * 9 * 8 *
2 ∂∂
−∇•−=•∇
Term *9 is like xρ
ρu2
∂∂ , then
]M[OauO
7*9* 2
2
2
xρ
ρ1
xp
pdρd
ρu 2
=
==
∂∂∂∂
( For a small “M”, O[*8] ∼ O[*10] .)
As the pressure moves at a sound speed,
]M[OauO
a
7*10*
xp
ρ1
xp
a1
ρu
2
=
==
∂∂
∂∂
Therefore, in equation (2a), by neglecting terms *8 (O[M]) and *6 (O[M2]), the momentum equations become
x)/p()uu(
tu
∂ρ∂
−=•∇+∂∂
y)/p()uv(
tv
∂ρ∂
−=•∇+∂∂
z)/p()uw(
tw
∂ρ∂
−=•∇+∂∂
60
If the flow is viscous (laminar), the approximate form of equations (2) to (4)
becomes
uρp)uu(
tu 2∇ν+
−∇=•∇+
∂∂ .................................. (2b)
where ν is the kinematic viscosity.
In cases of turbulent flows, through a process of space-averaging (“” and “ ’ ” denote “averaged” and “fluctuating quantities), the governing equation of WCF method are represented as
0)ua(xt
pj
2
j
=ρ∂∂
+∂∂
............................................................................ (1e)
...... (2c)
By adopting Reynold’s averaging assumption as
0)uuuu(uuuu jijijiji =−+′−′ ............................................ (5)
Equation (2c) becomes
∂∂
ν+δ′′−′′−∂∂
+ρ∂
∂−=
∂∂
+∂∂
j
ijijiji
j
*
jj
jii
xu)uu
31uu(
x)p(
xxuu
tu (2d)
where jiδ is the Kronecker delta function; jiji* uu
3pp δ′′ρ+= 。
As the subgrid-scale terms are modeled by
jitjijiji S)uu31uu( ν=δ′′−′′− .............................................. (6)
where )xu
xu
(Sj
i
i
jji ∂
∂+
∂∂
= and tν is the turbulent viscosity coefficient.
∂∂
ν+−−′−′′−′′−∂∂
+
∂ρ∂
=∂∂
+∂∂
j
ijijijijiji
j
jj
jii
xu)uuuu(uuuuuu
x
x)/p(
xuu
tu
61
Equation (2d) can be rewritten as
)(x
)p(xx
uut
u ji
j
*
jj
jii
ρ
τ
∂∂
+ρ∂
∂−=
∂∂
+∂∂
........................................ (2e)
where jiτ is the combination of the viscous and turbulent stresses.
The subgrid-scale turbulent diffusivity, corresponding the total stress ( jiτ ),
can be expressed in the form suggested by Smagorinsky (1963) as
2/12ji2
st )2
S()C( ∆=ν .......................................................................... (7)
where ∆ denotes characteristic length of the computational mesh; Cs is the Smagorinsky constant (0.23 for 2-D and 0.10 for 3-D computations).
Finally for WCF method, the averaged forms of the governing equations, equations (1e) and (2e), are all in a conservative form as
0Ft
Gi
i =•∇+∂∂
3,2,1i = ............................................. (8)
where the scalar Gi and vector iF can be defined as the rows of the
following matrices: (2-D form)
=
vup
G ；
ρτ−+ρτ−ρτ−ρτ−+=
/)p(v/vu/vu/)p(u
vkukF
yy*2
xy
yxxx*2 ........... (9)
The computation can proceed using an integration over a specific volume ∀ as (finite-volume method)
∫ ∫∀ ∀=∀•∇+∀
∂∂ 0dFd
tG
........................................................... (10)
By the divergence theorem, one has
∫ •∀
−=∂∂
sm dSFn~1
tG ....................................................... (11)
62
where Gm ( ∀∀
= ∫ ∀ dG1 ) represents the mean quantity referred to the
geometric center of the volume ∀ ; n~ is the normal vector of control surface (S). The integral form of equation (11) can be used to calculate the change of Gm within an elapsed period ( t∆ ), and thus update the Gm values for the next time step. Noted that in equation (11)：
⇒∂∂
tGm change (with respect to time) of storage within ∀ .
⇒•∀
− ∫sdSFn~1 sum of fluxes through the (control) surfaces of ∀ .
63
Stability Criteria for compressible flows
〈1〉Inviscid flows 1
222I)z()y()x(
az
wy
vx
umint−
∆+∆+∆+
∆+
∆+
∆≤∆
〈2〉Viscous flows
∆+∆σ
≤∆Re
2I
V 1t
t
where σ: safety factor (0.8 ~ 0.9 suggested)
]Re,Re,Re[minRe zyx ∆∆∆∆ =
ν∆
=∆
xuRe X
ν∆
=∆
yvRe Y
ν∆
=∆
zwRe Z
Difficulty:
For viscous cases, in the near-wall region ∆Re is small (due to small grid
size). The allowable Δt has to be rather small for numerical stability.
64
1-D Analysis of WCF Method (regarding the specifications of penetrable boundaries)
Taking the inviscid 1-D case for example,
L1 : 0xPk
tP
=∂∂
+∂∂
( 2ak ρ= )
L2 : 0xP1
xUU
tU
=∂∂
ρ+
∂∂
+∂∂
The linear combination L1 + λ L2 = 0 is
0xUUk
tU
xP
tP
=
∂∂
+
λ+
∂∂
λ+
∂∂
ρλ
+∂∂
Set the characteristic speed as
λρ
+=ρλ
=2aU
tdxd
The solution of 0aU 222 =ρ−λρ−λ is ]a4UU[2
22 +±ρ
=λ
+±=+±=
ρλ
=4
M12Ma]a4UU[
21
tdxd 2
22 where aUM =
For all M > 0 ,
04
M12Ma
tdxd:C
04
M12Ma
tdxd:C
2
2
<
+−=
>
++=
−
+
For the 1-D flow, there are 2 unknowns (P and U) at each spatial location. The characteristic diagram is as follows:
65
+C −C +Ct∆
X
t
x∆ knownnlSo ′
Upstreamend
Downstreamend
−C
At the upstream and downstream ends, there is only 1 equation available ( −C at the upstream and C+ at the downstream) to determine the solution at the next time step). Therefore, one more specification is needed at each end.
Suggestion for penetrable boundary specifications:
Upstream boundary：Prescribe velocity to define the flow condition.
Downstream boundary：Set reference pressure.
Solutions are known.
66
Boundary Specifications at Non-penetrable (Solid) Boundaries
〈A〉 Viscous calculations (use no-slip condition)
〈B〉 Inviscid calculations (use slip condition)
IIP Vn2VV •−=
What about specification of pressures ( P )？
* For flat walls, R = ∞ pP = pI
• P
I
)(R +↑
r∆
rR
VPP I
Ip ƥ+=
2
ρ
I
P Phantom point
Interior
I
P Phantom point
Interior
IP VV −=
rR
Vpp
2
IIP ∆×ρ+=
∆r
67
Vertically Integrated (Depth-Average) Method
The equations of motions are integrated along the direction of Z . Thus a 3-D problem can be converted into a 2-D one. However, the solution provides information independent of Z. This method is usually used in free-surface flows (channel flows, lake or sea).
Domain of interest: ZB < z < ho
where h = ho﹣ZB (ho = h + ZB)
* Noted that both the free-surface and bottom are streamlines.
〈i〉 0tDhD o =
or 0yhv
xhu
th ooo =
∂∂
+∂∂
+∂∂
〈ii〉Non-penetrable condition 0Zu B =∇•
or
Assume streamlines are straight and parallel, then the pressure is hydro-statically distributed.
)zh(gpp oatm −ρ=−
h
AZ
Bh
•
BZ
)Z(n B
)t,y,x(hZ 0=
)y,x(ZZ B=
z
x
y ν
u
ω
0y
Zvx
Zu BB =∂∂
+∂∂
hA
w v
68
Introduce depth averaged variables
∫=+
=)h(hZ
Z
0B
Bdz)t,z,y,x(u
h1U
fun(x , y , t) uUu ∆+=
∫= 0
B
h
Zdz)t,z,y,x(v
h1V vVv ∆+=
Integrate the continuity with respect to z from ZB to ho
0wh1dz
yv
h1dz
xu
h1
0dzzw
yv
xu
h1
hZ
Z
hZ
Z
hZ
Z
h
Z
B
B
B
B
B
B
0
B
=+∂∂
+∂∂
=
∂∂
+∂∂
+∂∂
+++
∫∫
∫
1 E
A A2 E
A A3 E In A1 E
A => From Liebnitz’s rule
∂
∂−
∂+∂
++∂∂
=∂∂
∫∫++
x)Z()Z(u
x)hZ()hZ(u
h1dz
xu
h1dzu
xh1 B
BB
B
hZ
Z
hZ
Z
B
B
B
B
Thus, ( ) ( )
−+
∂
∂+
∂+∂
−
∂∂
=∂∂
∫
∫∫
+∂∂
+
++
hZ
Z2yh
BZ
BhZ
hZ
Z
hZ
Z
B
B
BB
B
B
B
B
dzuh
h1
yZu
xhZudzu
h1
ydz
xu
h1
Similarly,
+
∂
∂+
∂+∂
+−
∂∂
=∂∂
∫
∫∫
+∂∂
++
hZ
Z2yh
BB
BB
hZ
Z
hZ
Z
B
B
B
B
B
B
dzvh
h1
y)Z()Z(v
y)hZ()hZ(vdzv
h1
ydz
yv
h1
69
Using the kinematic conditions at the free surface and bottom, one has
0yh
hV
xh
hU
th
h1
yV
xU
=∂∂
+∂∂
+∂∂
+∂∂
+∂∂
or
0)hV(y
)Uh(xt
h=
∂∂
+∂∂
+∂∂
The x-momentum equation becomes (S : surface , B : bottom)
∫∫++
−−ρ∂∂
ρ−−ρ
∂∂
ρ−
ρτ−τ
+∂τ∂
ρ+
∂τ∂
ρ+
∂+∂
−=∂∂
+∂∂
+∂∂
B
B
B
B
Zh
Z
Zh
Z
2
BxsxxyxxB
dz)Uv()Uu(yh
1dz)Uu(xh
1
hy)h(
h1
x)h(
h1
x)Zh(g
yUV
xUU
tU
where h
Zy
Zhx
Zhh
xzBBxyBBxxBBx −∂∂τ
+∂∂τ
=τ
−
Δ1 and Δ2 are “dispersion terms” due to velocity non-uniformity
“” stands for depth average
The y-momentum equation is
∫∫++
−ρ∂∂
ρ−−−ρ
∂∂
ρ−
ρτ−τ
+∂τ∂
ρ+
∂τ∂
ρ+
∂+∂
−=∂∂
+∂∂
+∂∂
B
B
B
B
Zh
Z
2Zh
Z
BysyyyxyB
dz)Vv(yh
1dz)Vv()Uu(xh
1
hy)h(
h1
x)h(
h1
y)Zh(g
yVV
xVU
tV
(∆1) (∆2)
70
Considerations of Large-Reynolds-Number Flows
Problems encountered under the following conditions:
〈i〉 Boundary layer becomes thin, which leads to a need of small meshes in computations.
〈ii〉Boundary layer separation results in unsteadiness (vortex shedding).
〈iii〉Small-scale turbulence modeling requires extremely small grid sizes.
Outer (Inviscid)-Inner (viscous) Approach
Outer
WakeInner
Strategy
Outer flow is assumed inviscid (non-viscous).
The boundary applies slip condition
Inner flow is viscous.
Boundary layer equation is used.
The boundary applies no-slip condition.
71
Turbulence Modeling
〈A〉Prediction of a turbulent flows * Turbulent flow is highly complex, unsteady and three-dimensional. * A complete (direct) numerical solution of the turbulent flows (governed by
the Navier-Stokes equation) is virtually beyond the reach of present-day computations, because ingredients of turbulent flows are contained in eddies, which are only a few millimeters in size in a flow domain of several meters. Even a simple problem would require an order of 109 grid point typically. The computing time needed to predict all the random motions on this grid would be truly excessive.
* The engineering interest is mostly in the averaged (time or space) behavior
of the flow.
〈B〉Empirical and semi-empirical methods * Empirical relationships, such as the friction factor for fully-developed pipe
flows, can be obtained from experimental data. They are useful only for interpolating experimental results, and are limited to simple situations involving only a few parameters.
* Another approach, due to Reynolds, is to write the time-averaged equations
and employ empirical input for turbulent transport terms to “close” the system of equations. Such methods are called semi-empirical.
〈C〉Turbulence model
Definition: A set of equations that determine the turbulent transport terms in the averaged system of equations.
* Turbulence model is an approximation. A given model, with a particular set of constants, is usually satisfactory for only a certain class of flows.
72
* A good model is capable of describing well a wide range of turbulent flows with a single set of constants .If the set of constants must change from flows to flows, the usefulness of the model is limited.
* A very widely applicable model is usually not quite economical. Practically,
the level of the model should be chosen to balance economy and generality. 〈D〉Derivation of the time-averaged equations For
'↓↓
φ+φ=φ
mean fluctuation
where '' and noise) (white 0' 212121 φφ+φφ=φφ=φ
The time-averaged form of the equation
Sxx
)u(x
)(t ii
ii
+
∂φ∂
Γ∂∂
=φρ∂∂
+φρ∂∂
is
φφ +
φρ−
∂φ∂
Γ∂∂
=φρ∂∂
+φρ∂∂ S''u
xx)u(
x)(
t iii
ii
where ''ui φρ− is the turbulent diffusion flux.
The time-averaged momentum equation is
S'u'uxu
x)uu(
x)u(
t jii
j
iji
ii +
ρ−
∂∂
µ∂∂
=ρ∂∂
+ρ∂∂
The quantities 'u'u jiρ− are turbulent (Reynolds) stresses. In most of the
flow regions (except where it is very near wall), the turbulent stresses are much larger than the laminar stresses.
* The goal of a turbulence model is to provide a means of calculating turbulent stresses 'u'u jiρ− and fluxes ''ui φρ− .
Source term
73
〈E〉Eddy Viscosity Boussinesq (1877) expressed the turbulence stresses by analogy with their laminar counterparts as
jii
j
j
itji k
32
xu
xu'u'u δρ−
∂∂
+∂∂
µ=ρ−
where
k (turbulence kinetic energy) 'u'u21
ii=
δij (kronecker delta function)
ji,1ji,0
=≠
=
The term jik32
δρ− can be absorbed in the pressure gradient term by
re-definig pressure as ρ+= k32pP
* The eddy (turbulent) viscosity is not a property of fluid (different from μ) but a property of flows. In most regions (except near a solid boundary), μt >> μ .
* The eddy viscosity hypothesis is the basis of many simple and advanced turbulence models.
〈F〉Eddy Diffusivity Let the turbulent diffusion flux be
∂φ∂
Γ−=φρ−i
ti x''u
where Γ is the turbulent (eddy) diffusivity. By Reynolds analogy,
t
tt σ
µ=Γ , where σt is the turbulent Prandtl/Schmidt number.
* Mostly, ≈σ t constant .
74
Free jets and wakes: 5.0t =σ (suggested)
Wall bounded layers: 9.0t =σ (suggested)
〈G〉Physical interpretation of eddy viscosity Just as the laminar viscosity resulting from the momentum exchange by colliding molecules, the eddy viscosity is regarded to be caused by the turbulent eddies.
µ t = (Constant) ρ V t L
where Vt and L are the characteristic velocity and length.
75
Algebraic (Zero-Equation Model)
〈A〉Constant eddy viscosity model
μt = a constant independent of space
〈B〉Parndtl’s free-shear-layer model
Let ( ) ttminmaxt VcuuxV δρ=µ−=
where δ is the thickness of free-shear layer.
tu99.0
δ
Jets014.0c:plane =
)uu(99.0 t0 −
δ
0uWakes
026.0c:plane =
1u
2u
δ )uu(99.0 21 −
LayersMixing01.0=c
〈C〉Mixing Length Model (Prandtl, 1952)
21
i
i
i
j
j
i2mt x
uxu
xu
∂∂
∂∂
+∂∂
ρ=µ
Free shear layers：
=δ
wakeplane 0.16 jet round 0.075
jet plane 0.09layer mixing 0.07
m
76
Wall Bounded Layers：
δκλ
>δλ
δκλ
≤κ=
y,
y,ym
Pantankar suggests 435.0=κ (usually 0.41) ; 09.0=λ .
m
y
Fully-developed pipe flows： (R is the pipe radius)
)(Nikuradse Ry106.0
Ry108.014.0
R
42
m
−−
−−=
Equilibrium boundary layers：
( ) uyy,2526A
e1y
*
mAy
ν==
−κ=
++
− ++
77
Regions Near A Wall
* Accurate solution of the near-wall region requires many grid points because of the steep gradients.
* For economical reasons (or under certain special situations), “wall function” can be applied. This universal function essentially relates the wall shear stress to the flow condition outside the viscous layer.
* In turbulent flows, teff µ+µ=µ
For most part of a shear layer, μt >> μ , or teffective µ≈µ .
In regions very near wall, μ is not negligible.
For example, based on logarithmic law (valid for equilibrium boundary layer)
νκ= *
*
uyEln1uU
E = 9 for a smooth turbulent boundary layer.
Remarks on Mixing Length Models
* The model is limited to simple flows. It cannot handle rapid changing flow, re-circulating flow, effect of free-stream turbulence, and so on.
* By using this model, μt becomes zero where yu
∂∂ is zero. It prevents
turbulent heat flux across planes of zero velocity gradients.
0=∂∂
yu
Hot plate
Cool plate
78
* Even for simple flows, the distribution of m is not universal. Difficulties
occur when an initial mixing layer becomes a jet, a jet meets a wall boundary layer, and so on.
* For most complex flows, there is no hope of prescribing a useful distribution of m .
〈D〉One-Equation Model
The most meaningful scale for velocity fluctuations is 21k , take
Lk 21
t ρ=µ where turbulence kinetic energy 2/'w'v'uk 222 ++=
The transport equation of k in high-Reynolds-number cases
ερ−+=ρ∂∂
+ρ∂∂ PD)ku(
x)k(
t jj
where
k of diffusion:Dk of ndissipatio viscous:ewher
xu'u'uP
j
iji
ε
∂∂
ρ−=
Based on eddy-viscosity hypothesis
NumberSchmidt turbulent: xk
xD k
jk
t
j
σ
∂∂
σµ
∂∂
= of k
If ε is assumed to depend on k and L , dimensional homogeneity leads to
L
kC2
3
D=ε (an inviscid estimate) where CD is a constant.
The final form of the k transport equation is
LkC)
xu()
xu
xu()
xk(
x)ku(
x)k(
t
2/3
Dj
i
i
j
j
it
jk
t
jj
j
ρ−
∂∂
∂∂
+∂∂
µ+∂∂
σµ
∂∂
=ρ∂∂
=ρ∂∂
where 1k ≈σ ; CD = 0.09 .
79
Relation to mixing length model
If all terms except the production (P) and dissipation (ε) terms in the k transport equation are considered negligible, a local equilibrium given by
ερ=P must prevail.
Thus, for simple flows
L
kCyu 2
3
D
2
t ρ=
∂∂
µ (e.g., for 2-D nearly parallel flows)
Furthermore, since Lk 23
t ρ=µ for the one-equation model
length) (Mixing yu
yuLC 2
m2
Dt2
1
∂∂
ρ⇒∂∂
ρ=µ −
(Incidentally , 4
1
Dm C
L= )
Remarks on One-Equation Model
* Turbulence models employing the transport equation of k and algebraic equation of L are called one-equation models .
* 1-equation models are able to account for free-stream turbulence, and they
do not lead to μt = 0 at the location where 0yu=
∂∂ .
* However, they share with the difficulties of the mixing length model about prescribing L . The length scale (L) cannot be given for complex flows, and even for simple flows. The empirical constants must change from flows to flows.
80
〈E〉Two-Equation Models
* The difficulties in the algebraic specification of L have led to the use of a differential equation for L , or for a related quantity.
* The available 2-equation models employ differential equations for k and Z,
where Z ~ k L
popularmost on)(dissipati Lk
Lk
)(vorticity Lk
2
23
21
→
The k-ε model : (High-turbulence-Reynolds-number form)
( )
εερ−+
∂ε∂
σµ
∂∂
=ερ∂∂
+ερ∂∂
ερ−+
∂∂
σµ
∂∂
=ρ∂∂
+ρ∂∂
ε kCPC
xx)u(
x)(
t
Pxk
x)ku(
x)k(
t
21j
t
jj
j
jk
t
jj
j [ ε
ρ=µ µ
2
tkC ]
where
3.10.192.1C44.1C09.0C
k
2
1
=σ=σ==
=
ε
µ
Remarks on 2-Equation Model
* A 2-equation model can predict the distribution of L , which has to be assumed in the algebraic or one-equation model . Therefore, any complex flow, at least in principle, can be handled.
* An early success of the 2-equation model was in explaining why the mixing length model constant for free shear layers has to change from flows to flows. The solution from 2-equation models shows that the predicted length scales are indeed different for mixing layers, jets and wakes.
81
Wall Function (for equilibrium turbulent boundary layers)
Mixing lengths in the wall region
〈I〉Inertia sublayer (logarithmic region)
ydudu'v'u t
2* ν==−
The logarithmic-law leads to
yu
ydudandCyln1uln1uu
Cuyln1uu
***
*
*
κ=
+
κ+
νκ=
+νκ
=
Since in the inertia sublayer 2*
*t u
yu'v'u =κ
ν= , one gets yu*t κ=ν
Also, from the mixing-length hypothesis,
yu
ydud *2
m2
mt κ==ν
Therefore, ym κ= (valid in wall region)
What does 'v'u− behave?
dyx'u'v
0y'v
x'u
y
0∫ ∂∂
−=
=∂∂
+∂∂
y)t,x(fy)t,x(f'u Expand 221 ++= (at y = 0, 0'u = , fo = 0)
]y[Oxfy
31
xfy
21 dyy
xfy
xf'v 42312y
0
221 +∂∂
−∂∂
−=
+
∂∂
+∂∂
−= ∫
82
]y[Oxff
2y
xf
3y
xf
2y)yfyf('v'u 41
1
32
31
22
21 +∂∂
=
+
∂∂
+∂∂
++=−
For fully developed flows, 0xff
xff 2
21
1 ⇒∂∂
=∂∂ ,
then 4y~'v'u− 〈II〉Viscous sublayer
ν=
ν=
ν=
2*
2*
*
*
uydud,uyu
uyuu
From the mixing-length concept
2m
422
*2m
22
m
y~
y~uydud'v'u
∴
ν
=
=−
* From the inviscid estimate
LkC
23
µ=ε ( L = (constant) m⋅ , see pp. 79)
∞→ε→←0y
lim , 0yAt
* Dissipation is infinitive at the wall.
At the wall, numerical specification on ε is not possible.
Strategy
In solving the ε-transport equation, the value of ε is specified (not solved) based on the log-law at the first interior point. At the 1st interior point, (in the logarithmic region)
LkC
23
µ=ε
83
where m4/1
DCL = (pp. 79： µ= CCD ; pp.81： ym κ= )
yC 4/1 κ= µ
ykC 2
34
3
κ=ε⇒ µ
This is based on the assumption that the length scale L is proportional to y. * The wall function can be applied where turbulent boundary layers are
equilibrium (log-law hold). It is not applicable to predict transition to turbulence, nearly separating flows, re-laminarization and so on.
88
第十五章 CFD 應用與計算風工程
方富民 國立中興大學土木工程學系 教授
本章摘要
隨著計算軟硬體之快速進步，在分析風工程問題時，數值模擬方法可以和風洞模型試驗交
相配合，以達相輔相成之效。此將單純的流場動力數值解析(CFD)延伸到與風工程相關問題
的動力學計算模擬，即發展成為計算風工程(CWE)的領域。除了 CWE 的基本概念與執行程
序外，本章亦針對風場模擬中涉及之控制方程式與紊流模型作了回顧。最後，文中再列舉
了一些典型鈍體流風場模擬分析實例，藉自數值預測而得全場、全時段所有相應動力變數
動態資料中的部份擷取和圖繪呈現，以突顯 CWE 在問題細部檢視與分析上之優勢。
本章關鍵字：計算流體動力學，計算風工程、數值模擬
1. 前言
89
「計算風工程」(computational wind engineering；CWE)始見於第一屆國際計算風工程研討會
(First International Symposium on Computational Wind Engineering)。為了要因應「計算流體動
力學」(computational fluid dynamics； CFD)在風工程領域中之快速發展，並突顯數值模擬
(numerical simulation)計算在風工程問題解析上漸形重要之大勢所趨，於 1992 年日本東京舉
辦的研討會中，邀集了世界各國之學者專家，除就 CWE 相關之研究成果作經驗之交換外，
並對 CWE 未來之應用與展望作一確立。
2. 計算流體力學(CFD)與計算風工程(CWE)之基本異同
一、專注對象與範疇
CFD與CWE都是應用數值解析的方法探討工程問題的途徑。但在專注的範疇上，CFD著重
於所有流體動力學(航空、熱傳、化工、空氣動力、水力等)問題之解析，對象為流體，屬科
研與應用與並重；CWE 則偏重於與風力相關實際工程問題之探討，對象不僅限於流體
(風)，甚而包括流體(風)與結構振動互制效應(interaction)等相關動力學問題之探討。在現階
段，CWE 應較著重於工程應用。
二、流體特性上之假設
不似 CFD 面對的是所有流體動力學的問題， CWE 在目標上是圍繞著所有與風相關的議
題。在以「風」為主題之前提下，氣流之壓縮性常被忽略，而多採用不可壓縮、紊流之流況
(incompressible turbulent flows)進行問題之剖析。
三、流場分析空間與形態
在 CFD 的探討中可為如管流(pipe flow)等之內部流場(internal flows)或是開放空間之外部流
場(external flows)問題。另一方面，除了如室內通風(ventilation) 外，CWE 大多探討在大氣
邊界層(atmospheric boundary layer)中的外部流場問題。因此，鈍體流(bluff-body flows)常成
為 CWE 之典型流況形態。
3. CFD/CWE 在工程分析方法上之比較
一般而言，在分析流體動力學相關問題常用的方法有三，即數學解析法 (analytical
method)、實驗法(experimental method)與數值模擬法(numerical simulation method)三大系
統。
在電腦尚未被廣泛應用前，數學解析的方法常與實驗方法並列為研究方法的兩大主流。對
一個流場問題而言，其求解係以一個足以正確描述流場變數(流速、壓力、溫度、濃度等)對
90
時間與空間變化的微分方程式(亦稱控制方程式；governing equation)為本，配合以適當的起
始與邊界條件 (initial/boundary condition)後，應用數學工具推衍出相應之流場數學解
(mathematical solution)。然而，一個動力問題相應的控制微分方程式往往是非線性
(non-linear)的，對此類問題數學解之尋求並非易事。更何況在實際問題中，隨著流場區域
邊界幾何或動力條件複雜程度之增加，欲獲取一個精確的流場結果往往有頗高的難度。因
此，在古典的解析方法中，常將區域邊界予以簡化以利求解，甚至沿用簡化但對真實流況
描述能力較差之控制方程式以獲得流場的結果。在採取此雙重簡化的情況下，雖然達成了
問題解析的目的，但是結果的真確度卻可能大打折扣。有幸，隨著近年來電子計算機軟、
硬體方面的快速進步，以往在數學解析方面可能遭遇的各種困難，在運用數值解析
(numerical analysis)的方法後，多已迎刃而解(此即所謂 CFD 的方法)。值得一提的是，在古
典數學解析的方法中，流場結果是以時間與空間函數的數學形式表示出來，其在時空方面
之變化均具有連續性；而在數值解析方法中，則為離散化(discretized)數值形態時空分佈的
流場結果。
另一方面，實驗方法包括實場量測(field measurement)與模型試驗(model experiment)兩類。
通常實驗方法在執行上所費不貲，除了因設施之建構以及量測儀器與設備之整合上需要相
當之空間與經費外，相對地亦會耗費人力與時間。尤其在鈍體流模型試驗研究中，因縮尺
效應(scale effect)對結果可能會造成相當程度之誤差。因此，在近期風工程的研究中，常以
風洞試驗量測(wind tunnel measurement)配合風場數值模擬，交相進行問題之探討。
4. CWE 與風洞試驗方法的互補關係
在近代風工程的研究中，風洞試驗與數值模擬已成為最常被使用的兩種解析工具。前者可
提供局部量測結果以作為數值模式比較與驗證之依據，待後者模式確立後則可依數值計算
所有變數、全區、全時之預測結果進行問題之細部檢視與分析。在應用的考慮上，茲將二
者之優缺點說明如後。
一、經濟的考量
一般而言，以試驗量測的方法需要較高之成本(設施、設備、人力、電力等)且較為耗時。另
一方面，數值模擬則較利於系統化之風場分析，分析者可以輕易地改變流場邊界的幾何與
動力條件，以得到相應之結果。
二、需要的硬體設備
試驗量測需要的硬體設備包括設施本體(風洞)、模型、量測儀器(視量測項目而定，如測速
計、壓力計、平衡儀、放煙設備、攝影與視化設備等)，並應配合以足夠大的試驗場地與量
測空間。相對地，在執行數值模擬時所需要的實質空間需求並不大，但最重要的是要有運
91
算快速且具大量資料儲存之電腦硬體設備。以一個三維(three-dimensional)真實流的模擬為
例，通常要考慮使用工作站級以上或高速運算之電腦，以滿足大尺度計算之需求。
三、複雜幾何情況之處理
在實際問題中，其相應之幾何情況往往是複雜的。這在模型試驗中並不會造成太大的技術
性問題，但在進行數值模擬時則經常會造成困擾。例如，人行道的植樹可以縮尺模型置於
風洞內並進行風況量測，以評估其對人行道風(pedestrian wind)之減風效果，然欲以數值模
擬來直接解析此類問題，現階段尚有一些困難有待克服。
四、結果之完整性
常受限於現有量測設備之數量或能力，或為了避免對流場之干擾，於試驗中往往僅能針對
有限之變數作局部性之量測。尤其在處理非恆定(unsteady)的時變問題時，其困難度更形增
加。相對地，應用數值模擬的方法，可以在不造成量測儀器干擾的情況下獲取所有變數整
場的時空變化結果，提供設計者詳盡的流場資料以供細部分析之用。
五、結果之真確性
在從事試驗量測時，除了可能因為量測儀器對流場之干擾致使結果產生誤差外，模型試驗
既有之縮尺效應(scale effect)對量測結果正確性之影響亦可能顯著。一般而言，風域中建物
縮尺模型之製作常取決雷諾數(Reynolds number)，為確保模型與實場二者之風況滿足相似
(similarity)的條件，理論上二者相應之雷諾數應一致。但事實上，在實場之雷諾數常在 107
或 108以上，而在模型試驗中往往最高僅能達到 104至 105。儘管在雷諾數夠大的情況下，無
因次(dimensionless)之流場結果可能已不再受雷諾數改變之影響(即已發生所謂的高雷諾數之
不變性；high-Reynolds-number invariance)，然在雷諾數差距三個量級(order)以上的情況
下，模型中流場之紊流結構(turbulence structure)是否確實與實場相似，則成為獲得試驗結果
準確與否之關鍵。另一方面，應用數值模擬的方法進行流場解析則無縮尺效應的問題，然
其預測結果亦有誤差產生的可能。諸如離散化計算引致的截斷誤差(truncation error)以及對
複雜幾何邊界簡化處理所造成之誤差等，均須於模擬中使用相當緻密的網格系統方能予以
消弭；但另一方面，則可能大幅增加計算量與所需之資料儲存空間。此外，為了要正確地
反應出真實風場中的紊流效應 (turbulence effect)，數值模擬中使用紊流模型(turbulence
model)的適切與否，更是影響結果精確程度的重要因素。以目前的情況而言，使用時間平
均 (time average)的 ε−k 紊流模型或空間平均 (space average)的大渦流模擬 (large eddy
simulation；LES)對簡單或中度複雜的風況已有許多成功的模擬實例。但在複雜程度較高的
流場模擬，此二紊流模型之應用在提供高精確度風場結果上仍有相當待努力的空間。
由前述之比較可見，CWE 與風洞試驗都有一些限制與缺憾，但從另一個角度來看，兩者間
實有互補之關係。以現階段之情況而論，若能將這兩個方法作適當地配合，在分析上常能
92
達到相輔相成之功效。
5. 數值解析方法的基本執行步驟
針對一個流體動力學的問題，典型數值解析方法的執行概分為如後四個步驟：
一、計算區域的選定
儘管在真實的情況中流場涵蓋的空間幅員可能是無限大的，然因電子計算機在運算與資料
處理上能量與儲存空間之限制，基於經濟上的考量，數值計算往往僅能針對有限的空間中
進行。誠然，當選取的區間(domain)愈小時愈能降低運算所需之儲存空間、能量與時間，但
另一方面則往往導致計算結果誤差的增加。因此，在計算效率與誤差的雙重考慮下，如何
就計算區間作一個最佳的選定(妥協)，則取決於執行者的專業判斷(professional judgment)。
二、控制方程式的決定
針對待解的風場相關變數(速度、壓力、溫度、濃度等)，下一步則為相應動力方程式的決
定。在風工程的範疇中，可以選擇的例如有簡化線性的拉普拉斯方程式(Laplace equation)、
忽略流體黏滯性(viscosity)效應的尤拉方程式(Euler equation)與較複雜但較接近於真實流況
的那維爾−史脫克斯方程式(Navier-Stokes equation)等。若想要將風場作更精確的描述或模
擬，則須採用較高層次的控制方程式，而其耗費的電腦能量亦大，取捨之間執行者必須再
次作專業性的判斷。目前就電腦可以提供的能量現況而言，絕大多數風工程的問題已使用
基於那維爾−史脫克斯方程式的數值解析，並配合以紊流模型之應用，風場預測的結果一般
皆有不錯之真確性。
三、數值離散化(discretization)
如前所述，數值方法係數學解析法的延伸。為了要應用計算機進行數值運算，就必須要將
原有具時空連續性的問題轉化為離散化的狀況。換言之，在進行數值解析時，必須要把計
算區域作網格形式的劃分，並針對所有網格(grid)幾何中心或網格節點(node)相應的離散位
置進行流場變數之計算，以獲致風場的近似解(approximated solution)。理論上，如果取用的
網格為無限小時，其計算的結果就會與真實的結果一致。此時，流場之模擬亦無需使用紊
流模型。但事實上，由於此將會使得計算量變得過大而顯得極不經濟。故而，在從事數值
計算時，應在確保計算結果精準度與節省計算量兩者間作一個適當的取捨，據以決定出計
算網格的細密程度。
四、數值計算之執行
一旦網格系統建立後，接下來的工作則是數值計算方法的選取與執行。常見的數值計算方
93
法如有限元素法(finite element method)、有限差分法(finite difference method)、有限體積法
(finite volume method)等等。以有限差分法為例，則須先將連續性的微分方程式(differential
equation)轉化為離散化的差分方程式(difference equation)，繼而進行流場近似解的計算與求
取。
6. CWE 的基本控制方程式
6.1 層流情況之模擬
CWE 主要是針對因風的流動而引致問題之解析。由於大多專注於低馬赫數(Mach number)流
的情況，故氣流之壓縮性應可忽略，一般多採用不可壓縮、紊流之流況(incompressible
turbulent flows)進行問題之剖析。
在不可壓縮流的假設下，基於質量守恆(conservation of mass)定律，風場相應之連續方程式
(continuity equation)為如後之橢圓形形式(elliptic form)：
0u =⋅∇ (15-1)
其中，流速 )w,v,u(u = 。
於層流(laminar flow)情況下，針對非恆定(unsteady)、黏滯性(viscous)的旋性流場(rotational
flow)，相應之動量方程式(momentum equation)一般多採用如後雙曲線形式(hyperbolic form)
的那維爾−史脫克斯方程式：
upuutu 2∇+
∇−=∇⋅+
∂∂ ν
ρ
(15-2)
其中，p、ρ、ν分別為壓力(pressure)、空氣密度(density)與運動黏滯度(kinematic viscosity)。
值得一提的是，在處理溫度差異不明顯的問題時，由於空氣的密度低，相對的重力效應可
以忽略，故上式中並未含重力加速度(gravitational acceleration)。
在一個三維空間中，(15-1)與(15-2)式，共計四組微分方程式，則成為描述氣流運動之控制
方程式，據此可以聯立解析式中風速(包括 u, v, w 三個變數)與壓力(p)之空間與時間分佈。
94
6.2 質量傳輸
在風工程範疇中，濃度擴散(diffusion)亦常為關注之問題之一。在不考慮浮力(buoyancy)與
其他如化學反應之情況下，描述風場中濃度質量傳輸(mass transport)之典型方程式為：
CCutC 2∇=∇⋅+∂∂ Γ (15-3)
其中，C 為濃度；Γ為擴散係數(diffusion coefficient)。
一般而言，若浮力效應不明顯時，風場受濃度場變化而產生之影響極微，通常無需將(15-3)
式併入與(15-1)式、(15-2)式之聯立求解以減省計算量。換言之，在數值模擬過程中可以先
由(15-1)式、(15-2)式計算出風場結果(u, v, w)，再依據(15-3)式求得相應之瞬間濃度場(C)空
間分佈。
6.3 紊流情況之模擬
在實際的情況中，絕大多數之氣流流動均為紊流(turbulent flow)之形態。由於紊流中有渦漩
(eddy)之存在而具有高度(但非全然)之散漫特性，故較難以掌握。
若將速度(u, v, w)、壓力(p)與濃度(C)分別以一個均值(mean)和一個擾動量(fluctuation)的和表
示：
'uuu += ; 'vvv += ; 'www += ; 'ppp += ; 'CCC += (15-4)
繼而代入(15-1)式至(15-3)式後會發生因未知變數數目多於方程式數目而無法解析的問題
(closure problem)。直觀上，此可經由直接模擬(direct numerical simulation; DNS)的方法進行
風場計算，但是計算中使用網格之尺寸必須小於紊流中渦漩最小的 Kolmogorov 尺寸
(Kolmogorov scale)，而導致計算量驟增。就目前電腦的能量而言，仍然是不經濟且難以負
荷的。
現階段常採用的處理方式是將方程式作時間或空間平均，而基於雷諾平均原則(Reynolds
averaging principle)可得如後之指標(index notation)表示式：
95
0xu
i
i =∂∂
(15-5)
)uuxu
(xx
p1xu
ut
u 'j
'i
j
i
jjj
ij
i −∂∂
∂∂
+∂∂
−=∂∂
+∂∂
νρ
(15-6)
)'CuxC(
xxCu
tC '
ijjj
j −∂∂
∂∂
=∂∂
+∂∂ Γ
(15-7)
依據 Boussinesq 渦漩−黏滯度假說 (eddy-viscosity hypothesis)， (15-6)式中的紊流應力
(turbulent stress)項可以表示為：
jii
j
j
itji k
32)
xu
xu
(uu δν −∂
∂+
∂∂
=− (15-8)
其中，νt為渦漩或紊流黏滯度(eddy/turbulent viscosity)；k ( 2/'w'v'u 222 ++= )為紊流動能
(turbulent kinetic energy)，δij 為 kronecker delta 函數。
同理，(15-7)式中的紊流通量(turbulent flux)亦可表示為：
jt
t
jt
'i x
CxC'Cu
∂∂
=∂∂
=−σν
Γ (15-9)
其中，Γt 為紊流擴散係數 (turbulent diffusivity)，σt 為紊流史密特數 (turbulent Schmidt
number)。相關試驗顯示，一般σt約略為常數。
至此，(15-6)與至(15-7)式則成為：
96
]xu
)[(xx
p1xu
ut
u
j
it
jjj
ij
i
∂∂
+∂∂
+∂∂
−=∂∂
+∂∂
ννρ
(15-10)
]xC)[(
xxCU
tC
jt
jjj ∂
∂+
∂∂
=∂∂
+∂∂ ΓΓ
(15-11)
其中，νt 與Γt 需藉紊流模型(turbulence model)獲得。而(15-5)式、(15-10)式與(15-11)式則為
紊流中解析變數均值(u 、 v、 w與C )之控制方程式。
值得一提的是，除了在極端接近固體表面處之外，紊流黏滯度(νt)都遠大於流體之運動黏滯
度(ν)，且紊流擴散係數(Γt) 都遠大於流體之擴散係數(Γ)。因此，(15-10)式與(15-11)式中
可視模擬情況而考慮將ν與Γ予以省略。
6.4 紊流模型
在(15-10)式與(15-11)式中，ν與Γ屬流體之特性，在溫度差異不顯著時一般可視為固定值。
另一方面，νt 與Γt 則深受流場紊亂之程度而定(與流體之特性無關)，因此是隨著流場在空間
與時間之變化而改變的。
6.4.1 平均之概念
目前在平均上多採用時間平均與空間平均兩種概念，前者係針對空間各位置相應之變數，
於足夠長的時段中予以平均；後者則針對某一瞬間，就各空間網格(grid)內變數作平均。時
間平均的方法發展較早，驗證實例較多。另一方面，空間平均方法發展的起步較晚，然文
獻中顯示[1]在鈍體流流場預測之準確度上，其表現較時間平均的 k-ε方法佳。
在時間平均下，相應於某位置流場變數(M)之平均值可依據後式而得(參見圖 15-1)：
97
T
( )'tM
2Tt − t 2
Tt + 't
圖 15-1 時間平均示意圖
'dt)'t(uT1u
i
2/Tt
2/Tti ∫+
−= (15-12)
其中，T 為時間平均之時段。依據統計原理，T 應愈長愈佳。然於時間變化急促之流況時，
則有牴觸統計學理之虞。
另一方面，在空間平均的概念中，某一計算網格中流場變數(M)之平均值可依據後式而得
(參見圖 15-2)：
r
圖 15-2 空間平均示意圖
∫ ∫ ∫∞
∞−−= ξξξ d)t,(M)r(G)t,r(M (15-13)
其 中 ， )r(G ξ− 為 濾 函 數 (filter function) 或 權 重 函 數 (weighing function) ， 且
98
∫ ∫ ∫ =∞
∞−1rd)r(G ，以反映在空間平均中因與網格幾何中心(平均值相應點)距離之增
大而呈現影響式微之趨勢。由於使用濾函數將導致計算量之明顯增加，使得空間平均模擬
更為耗時。然而，當計算網格之尺寸足夠小時，使用不同濾函數形式造成之差異已不顯
著，故可以省略濾函數之使用以減省計算所需要之時間。值得一提的是，若網格之尺寸小
於流場中之 Kolmogorov 長度尺寸時，紊流模型已無必要，實際上的計算已經是在執行直接
模擬(DNS)了。
6.4.2 典型之紊流模型
一、時間平均概念之零方程式模型(zero-equation model)
所謂零方程式紊流模型意指在獲得νt之形式前無需解析任何與紊流(turbulence)相關變數之傳
輸方程式。此類紊流模型除了在流場預測有相對較大之誤差外，由於νt 之決定係經實驗結
果率定(calibration)而得，故無從獨立進行。此外，在不同之流況下，νt值亦不相同。
最簡單表徵紊流黏滯度的方式是將νt 視為不隨時間與空間改變之定值，此即固定紊流黏滯
度模型(constant viscosity model)。儘管此與紊流之理論相違，但的確省時省工。
另一個零方程式模型源自於混合長度 (mixing length)的概念。於二維近平行流 (nearly
parallel flow)中，在等向性紊流(isotropic turbulence)的假設前提下， Prandtl(1925)依據時間
平均的概念提出：
yu2
t ∂∂
= ν (15-14)
其中， 為混合長度，即一個標徵(nominal)渦漩的尺度。同樣地， 之分佈需經試驗值率
定，且隨著流況之不同而異。
二、時間平均概念之壹方程式模型(one-equation model)
在時間平均的概念下，模式中先求解紊流動能之傳輸方程式。在等向(isotropic)與均勻
(homogeneous) 紊 流 以 及 忽 略 層 流 黏 滯 項 後 ， 可 得 如 後 高 紊 流 雷 諾 數
(high-turbulece-Reynolds-number)情況與忽略溫差效應下 k 之傳輸方程式：
99
LkC
xu
xu
xu
xk
xxku
tk 2/3
Dj
i
i
j
j
it
jk
t
jjj +
∂∂
∂
∂+
∂∂
+
∂∂
∂∂
=∂∂
+∂∂
νσν (15-15)
其中， 0.1k =σ ， 09.0CD = 。L 之概念與混合長度相仿，亦需經試驗值率定而得，且隨
著流況之不同而異。
待獲得 k 之分佈後，νt 則依據後式獲得：
Lk'Ct µν = (15-16)
其中，98'C =µ 。
三、時間平均概念之貳方程式模型(two-equation model)
為了要從事獨立之紊流流場數值模擬(即免除需經試驗值率定之束縛)，除了求得紊流動能傳
輸方程式之解外，另外需尋求一足以表徵紊流特性變數之傳輸方程式。據此，較為著名者
為紊流動能−消散率模型(turbulence kinetic energy -dissipation model；k-ε model)。
在高紊流雷諾數(high-turbulence-Reynolds-number)情況與忽略溫差效應下，紊流消散率(ε)之
傳輸方程式為：
kC
kxu
uuCxxx
ut
2
2i
j'j
'i1
j
t
jjj
εεεσνεε
εεε
−∂
∂−
∂∂
∂∂
=∂∂
+∂∂ (15-17)
其中， =εσ 1.3； =ε1C 1.44； =ε2C 1.92。
待分別獲得 k 與ε之分佈後，νt 則依據後式獲得：
εν µ
2
tkc= (15-18)
其中， =µC 0.09。
100
四、空間平均概念之大渦流模擬
在空間平均大渦流模擬的概念中，大於計算網格尺寸渦漩之運動係採直接模擬，小於計算
網格尺寸渦漩之運動則以次網格紊流模型(subgrid-scale turbulence model)模擬之。
在典型之次網格模型中，Smagorinsky[2]提出紊流黏滯度的形式為：
ijij2
St SS2)C( ∆ν = (15-19)
其中，CS為 Smagorinsky 常數，在三為紊流流場計算之建議值為 0.1；∆為數值計算網格之
平均尺寸； )xu
xu
(21S
i
j
j
iij ∂
∂+
∂∂
= 。
在近期文獻中[3]另有動態次網格紊流模型(dynamic subgrid-scale turbulence model)之提出，
CS 並非固定值，其在各瞬間之空間分佈係經由粗與細兩套網格系統獲得之流場結果間比較
而得。
7. CWE 模擬實例
如後列舉幾個典型數值模擬的 CWE 應用實例。在這些例子中，紊流流場模擬係採用微可壓
縮流的方法(weakly compressible flow method；WCF[4])，配合以 Smagorinsky[2]之次網格紊
流模型。其中，在 7-2 節中的案例為風與結構(橋體)振動的流場誘發振動(flow-induced
vibration)問題，模擬中涉及流體動力與結構動力兩組控制方程式，係以交替解析的方式以
反映出流場與結構振動間之互制效應(interaction)。
值得一提的是，儘管部份列舉問題涉及二維之基本流態，然因紊流屬三度空間之表現，故
流場之模擬仍於三維空間中進行(第三個方向應選取足夠之長度)。
7.1 典型鈍體流風場模擬
7.1.1 均勻來流通過二維方柱/矩柱
依據數值模擬結果，圖 15-3 顯示方柱之無因次流場結果(雷諾數 Re=UD/ν=107，U 為來流風
速)。由圖 15-3(a-c)之瞬間渦度場(vorticity field)可見，當流體通過方柱時，於柱體之兩個前
緣(leading edges)處發生了分流(separation)，上下兩條分流線續朝下游延伸，兩側之渦流並
於中央對稱線區域產生相互干擾、襲捲(swirl)與剝離(detachment)的現象，繼於柱後尾流區
101
(wake)內產生了非恆定之渦散(vortex shedding)。由方柱順風(along-wind)向阻力(drag)係數
(CD)與橫風(acrosswind)向升力(lift)係數(CL)的歷時圖(圖 15-3d)可見，二者約略呈週期性之變
化，且在升力的變化量( 'LC )明顯地大於阻力的變化量( '
DC )。此外，長時段平均流場相應之
速度等值線分佈呈現上下對稱但左右不對稱(因有尾流區之存在)之現象(圖 15-3e)，故相應
之平均升力係數( LC )為零，但平均阻力係數( DC )不為零。
至於在相對的矩柱(B/D=4)情況中(參見圖 15-4)，非恆定之渦散現象亦於柱後發生。與方柱
情況不同的是，自柱體前緣引發之分流線在柱體兩側發生了再接觸現象(re-attachment)，使
得矩柱後尾流區之橫寬變窄。儘管相應之平均升力係數仍為零，但其平均阻力係數則較方
柱情況為小。
7.1.2 邊界層來流通過地面二維方柱
在同等雷諾數下邊界層來流(boundary layer approaching flow)通過地面二維方柱情況中(參見
圖 15-5)，分流線自方柱迎風面上緣向下游襲捲，產生了順時
(a)
(a)
(b)
(b)
(c)
(c)
D
B
102
(d)
tU/D200 400
-4
0
4
-3
0
3
CD CL
CD
CL
(d)
tU/D200 400
-4
0
4
-3
0
3
CD CL
CD
CL
(e)
1
1
1.3
1.1
1.1
1.31.2
1.2
0.60.8 0.9
1
0.91
0.9 0.7
(e) 1
1
0.9
1.1
1.21.3
1.1
1.21.3
1
0.1
0.4 0.6 0.70.8 0.9
0.81
0.9
圖 15-3 均勻流通過二維方柱結果 圖 15-4 均勻流通過二維矩柱結果
針方向的渦流列(圖 15-5a 至圖 15-5c)。長時段平均流場相應之速度等值線分佈結果顯示(圖
15-5d)，柱後低風速尾流區內存在一個常駐之分流泡(separation bubble)，但此和各瞬間呈現
的流況是不盡然相同的。
(a)
(a)
(b)
(b)
103
(c)
(c)
(d)
1
0.9
0.80.70.1
1
0.1 0.2
0.50.60.70.80.9
0.40.30.20.1
(d) 1
1
0.9
0.8
1.1
1.1
1.21.3
1.21.3
1.2
1.2
0.1 0.3 0.5 0.6 0.7
0.80.9
11.1
0.80.9
1
1.1
圖 15-5 邊界層來流通過地面二維方柱 圖 15-6 均勻來流通過二維直列雙
結果 方柱結果
7.1.3 均勻來流通過二維直列雙方柱
當均勻來流通過直列(tandem arrangement)雙方柱時，由於後柱存在於前柱的尾流區內，對
前柱分流線向下游之延伸以及渦流之剝離與襲捲造成了相當程度之影響(圖 15-6a 至圖
15-6c)，使得流場之變化機制更形複雜。由長時段平均流場相應之速度等值線分佈(圖 15-6d)
可見，較之於單柱情況(圖 15-3e)，後柱下游處尾流區之橫寬明顯增大。
7.1.4 均勻來流通過二維直列雙方柱
當均勻來流通過橫列(side-by-side arrangement)雙方柱時(圖 15-7)，上柱下方與下柱上方產生
之渦流會產生相互之影響，使得流場之時變程度增強。圖 15-7d顯示，儘管長時段平均流場
呈現上下對稱之情形，但此與各瞬間相應之流場情況(圖 15-7a 至圖 15-7c)顯有不同。
7.1.5 邊界層來流通過地面三維單方柱
數值結果顯示(圖 15-8)，在邊界層來流通過地面上高寬比為 10 之三維單方柱的流況中，由
於受到柱頂與柱底處產生之頂渦流(tip vortex)與馬蹄形渦流(horse-shoe vortex)之影響，使得
流場呈現強烈的三維特性。於此較大的方柱高寬比情況中，長時段平均流場結果顯示，在
相應於柱體半高的水平面上(圖 15-8a)，其平均速度等值圖之形態類似於均勻流通過單一二
104
維方柱之結果(圖 15-3e)；另一方面，相應於垂直對稱面上之平均速度分佈形態(圖 15-8b)亦
與邊界層來流通過地面單一柱體(如見圖 15-5d)雷同。
7.2 均勻來流中橋體受風作用之互制振動模擬
相對於一般橋梁，懸索橋或斜張橋有著輕軟之結構特性，儘管其上部結構較具抗震性，然
在抗風方面則存在著敏感的氣動力不穩定性(aerodynamic instability)考量，而此流場誘發振
動問題則依橋梁斷面幾何形狀等因素之改變而呈現不同之氣動力行為。
當風吹襲非流線外形之橋體時，於下風處將因橋體兩側產生渦流之交互作用而引致渦散
(vortex shedding)現象之發生，而流場中非恆定的外力將使得橋體產生振動。當橋體振動顯
著時，其運動將對鄰近風場造成影響進而導致橋體受風效應之二次改變，此風與橋體間的
互制效應即反映出橋體之氣彈力行為(aero-elastic behavior)。
(a) 0.9
1
0.8
0.9
0.6
0.7 0.80.7
0.8
0.50.3
(b)
10.9
0.8
0.7
0.90.8
0.70.60.5
0.1
0.3
0.9
0.8
圖 15-8 邊界層來流通過地面三維單方柱平均流場結果
105
模擬中需處理描述流體動力與結構動力的兩組控制方程式，乃以交替解析的方式以反映出
流場與結構振動間之互制效應。換言之，在初始預設橋體幾何與動力條件下先進行瞬間風
場之模擬計算，以獲得橋體之瞬間受力，進而求得橋體在下一個瞬間之振動反應。至於在
下一個瞬間之風場模擬中，則將橋體之振動(位移與速度)納入流場計算中邊界條件之考量，
依此交替解析。
在一個均勻來風中梯形斷面橋體的問題中(圖 15-9)[5]，表 15-1 列舉相關之橋體特性。案例
中乃以此斷面模型(section model)之風洞試驗資料為基準，平行地進行在各風速(U)下的數值
模擬動力計算。
依據數值計算結果，圖 15-10 顯示在零風攻角(attack angle)時橋體質心在垂直(heaving)與橋
面在扭轉(torsional)兩個方向之均方根值振動量( 'Vy 與
'α 結果)。其中，Ur 為以無因次形式
表示的約化風速(reduced velocity；Ur = fVD)。當風速(U)自低速漸增時，流場對橋體作用力
之時變量亦增，導致橋體之均方根值振動量變大。另一方面，渦散頻率(shedding frequency)
也會隨著來流風速之增加而增加，而當 U 達到約 7.4 與 10.6 m/s 時因渦散頻率分別與橋體垂
直向與扭轉向之基本頻率(fundamental frequency)相同，致使橋體振動量發生局部尖峰(peak)
值。然而，當此共振(resonance)發生時，振動行為之總阻尼(結構阻尼值與因互制效應引致
氣動力阻尼值(aerodynamic damping ratio)之和)仍為正值，儘管橋體振動量大，但仍為有限
值。
圖 15-9 均勻流通過梯形斷面橋體簡示圖
表 15-1 橋體斷面特性
Mass ( kg / m)
Moment of inertia
( kg-m2 / m)
Fundamental frequency ( Hz )
Damping ratio ( % )
Heaving ( fv )
Torsional ( ft )
Heaving ( ξv )
Torsional ( ξt )
0.472 6.31×10-4 14.8 28.4 0.80 0.70
B=4D
D 45∘
U
106
(a) 垂直向
Ur
U ( m/s )
y V'/
D
0 2 4 6 8 10
0 5 10 15 20
0.0
0.5
1.0
1.5
2.0
2.5
ExperimentalCalculated
10.67.4
4.773.33
(b) 扭轉向
Ur
U ( m/s )
α '(d
egre
e)
0 2 4 6 8 10
0 5 10 15 20
0.0
0.5
1.0
1.5
2.0
2.5
ExperimentalCalculated
10.67.4
4.773.33
圖 15-10 橋體均方根值振動量結果
當 U 自共振風速繼續增加時，橋體振動量除了在初期因脫離了鎖定(lock-in)風速範圍而有稍
許降低外，大致上仍呈現遞增之趨勢。數值結果顯示，當U達到約21.6 m/s時，橋體振動量
呈現出急遽增加之發散現象，此乃因振動行為之總阻尼值已由正轉負而發生了顫振(flutter)
現象之故。至於在從事風洞試驗時，為避免因振動量過大而導致模型之毀損，並未進行在
此高風速下之量測，因此在圖 17-10 中並未呈現相應之試驗結果。
8. 參考文獻
[1] Murakami, S., Mochida, A. and Hayashi, Y., “Examining the k-ε Model by Means of a
Wind Tunnel Test and Large-Eddy Simulation of the Turbulence Structure around a Cube,”
Journal of Wind Engineering and Industrial Aerodynamics, Vol. 35, pp. 87-100, 1990.
[2] Smagorinsky, J., “General Circulation Experiments with Primitive Equations,” Month
Weather Review, Vol. 91, No. 3, pp. 99-164, 1963.
[3] Germano, U., Piomelli, P. and Cabot, W.H., “A Dynamic Subgrid-scale Eddy Viscosity
Model,” Physics of Fluids, Vol. A, No. 3, pp. 1760-1765, 1991.
[4] Song, C. and Yuan, M., “A Weakly Compressible Flow Model and Rapid Convergence
Method,” Journal of Fluids Engineering, ASME, Vol. 110, No. 4, pp.441-455, 1988.
[5] Fang, Fuh-Min, Li, Yi-Chao, Chen, Chu-Chang, Liang, Tsung-Chi and Chen, Jwo-Hua,
“Numerical Predictions on the Dynamic Response of a Suspended Bridge with a Trapezoidal
Cross-section,” Journal of the Chinese Institute of Engineers, Vol. 28, No. 2, pp. 281-291,
2005.