Chapter II: Leakage Flow Model
2.3 Fluid-Structure Coupling
2.3.5 Fluid-Structure Coupled Operator
In the current formulation, we disregard the spanwise direction in the Euler-Bernoulli beam equation,
ρshb ∂2
∂t2δ(x,t)+ ∂2
∂x2
E I2D
∂2
∂x2δ(x,t)
= Pbot(x,t) −Ptop(x,t), (2.85) where
I2D = h3b
12(1−ν2) (2.86)
is the cross-section area moment of inertia (in two Dimensions), andPbot(x,t)and Ptop(x,t) are the pressure operators for the bottom and top channels respectively.
Figure 2.5 illustrates the force balance as a function of beam shape. To connect the pressures in the top and bottom channels, we write the geometrical constraint,
htop0 (x) −δtop(x,t)+
δbot(x,t) −hbot0 (x)
= h0top(x) −hbot0 (x), (2.87)
where we recover the simple relationship
δ(x,t)= δtop(x,t)= −δbot(x,t). (2.88) The geometrical constraint for the total gap size of the two channels means that we only have one shape of the beam,δ, that relates the two pressures.
Figure 2.5: Illustration hydrodynamic force balance as a function of beam shape.
Substituting the linear expansion forδ andPinto equation 2.85, and assuming that Eand Iare spatially uniform in x,
E I2D
d4
dx4δ0(x)+ε
ρshb
∂2
∂t2δ1(x,t)+E I2D
∂4
∂x4δ1(x,t)
= pbot0 −ptop0 +ε
pbot1 −ptop1
. (2.89)
Here we note that the left-had-side of equation 2.89 depends only on δ, as it rep- resents the internal stresses and acceleration of the beam. The right-hand-side, however, depends on the channel gap size, which includes h0 and δ (he). Equat- ing εcoefficients, we can solve for the equilibrium beam shape δ0by solving the nonlinear ODE at0thorder,
E I2D d4
dx4δ0(x)= pbot0 −ptop0 , (2.90)
wherepbot0 and ptop0 are the equilibrium pressures in equation 2.58 evaluated athbot0 and htop0 , respectively. Here we note that if htop0 = −hbot0 , making the system sym- metric, the right-hand-side of equation 2.90 is zero, and hence a sufficient condition for the equilibrium beam shapeδ0to also equal zero. Due to the nonlinear nature of p0, and the broad definition of h0(x), other equilibria may exist. Although we will focus our analysis on the stability of δ0 = 0 case, equation 2.90 gives the means to tunehe to yield different equilibria and perhaps different system behaviors. The linear term atε1follows,
ρshb ∂2
∂t2δ1(x,t)+E I2D ∂4
∂x4δ1(x,t)= pbot1 − ptop1 , (2.91) where
pbot1 − ptop1 =Tfbot(x)Q¯botx1(t) −
∞
Õ
i=0
Mfibot(x) Üai(t)+Cfibot(x) Ûai(t)+Kfibot(x)ai(t)
−
"
Tftop(x)Q¯topx1(t)+
∞
Õ
i=0
Mfitop(x) Üai(t)+Cfitop(x) Ûai(t)+Kfitop(x)ai(t)
# , (2.92) are the first order pressure perturbations from equation 2.64 evaluated at the the top and bottom channels, and includes the substitution of equation 2.88 and the basis function expansion forδ1. By expandingδ1on left-hand-side of equation 2.91 using the same basis functions, we obtain a coupled, linear ODE forai,
∞
Õ
i=0
Msi(x) Üai(t)+Csi(x) Ûai(t)+Ksi(x)ai(t)
=Tfbot(x)Q¯botx1(t) −Tftop(x)Q¯topx1(t) −
∞
Õ
i=0
h
Mfibot(x)+Mfitop(x)
aÜi(t)+
Cfibot(x)+Cfitop(x) aÛi(t)+
Kfibot(x)+Kfitop(x)
ai(t) i,
(2.93) where
Msi = ρshbgi(x), Ksi = E I2D d4
dx4gi(x), (2.94)
and Csi is the damping term associated with the moving boundary condition, to be defined in the next sub-section, in our current formulation. We note that Csi
can also be associated with internal damping models of the beam if any are in- cluded in equation 2.85, although none currently are. Equation 2.93 represents the fully-coupled linear fluid-structure operator, and can account for two independent channels. Looking at the fundamental structure of equations 2.90 and 2.93, in order, allows us to understand the dynamics of the system. To construct a discrete opera- tor that can be evaluated, we carry out the Galerkin projection of equation 2.93 on a subset of basis functionsgj for j ∈Z:[0,n]to formulate its weak form,
∫ L
0
Õn
i=0
Msi(x) Üai(t)+Csi(x) Ûai(t)+Ksi(x)ai(t)
gj(x)dx =
∫ L
0
Tfbot(x)Q¯botx1(t) −Tftop(x)Q¯topx1(t)
gj(x)dx−
∫ L 0
n
Õ
i=0
h
Mfibot(x)+Mfitop(x)
aÜi(t)+
Cfibot(x)+Cfitop(x) aÛi(t)+
Kfibot(x)+Kfitop(x) ai(t)i
gj(x)dx.
(2.95)
Coupled with equation 2.65, we can rewrite the system of equations in matrix form as the evolution of statesai,aÛi, andQ¯x1, by solving foraÜi,
Û a0
...
Û an
aÜ0 ...
aÜn
Û¯ Qbotx1
Û¯ Qtopx1
=
0 1 0 0
M−1K M−1C M−1Tbot M−1Ttop
−[Eqbot+Bqbot(M−1K)] −[Dbotq +Bbotq (M−1C)] −Bbotq (M−1Tbot) [Gbotq −Bqbot(M−1Ttop)]
Eqtop+Bqtop(M−1K) Dqtop+Btopq (M−1C) Gtopq +Bqtop(M−1Tbot) Btopq (M−1Ttop)
a0
...
an
aÛ0 ...
aÛn
Q¯botx1 Q¯topx1
,
(2.96) where the following exist inRn+1×n+1,
Mi j = 1 Ni j
∫ L 0
Msi(x)+ Mfibot(x)+Mfitop(x)
gj(x)dx, Ci j = − 1
Ni j
∫ L 0
Csi(x)+Cfibot(x)+Cfitop(x)
gj(x)dx, Ki j = − 1
Ni j
∫ L
0
Ksi(x)+Kfibot(x)+Kfitop(x)
gj(x)dx,
(2.97)
where the norm is defined as,
Ni j =
∫ L
0
gi(x)gj(x)dx. (2.98) The following exist inRn+1×1,
Tjbot = 1 Ni j
∫ L 0
Tfbot(x)gj(x)dx
Tjtop =− 1 Ni j
∫ L 0
Tftop(x)gj(x)dx.
(2.99)
The vectors Bq, Dq, and Eq exist inR1×n+1, and are the evaluated coefficients in equation 2.65 for their respective channels. The state vector is comprised of the coefficients of the expansionai, their time derivativesaÛi and the forcing flow rate from the top and bottom channels,Q¯topx1 andQ¯botx1. Equation 2.96 is the linear time evolution of the state vector, and by evaluating it at an equilibrium point, calculated by solving the nonlinear equation 2.90, its eigenvalues and eigenvectors dictate what rate and shape a small disturbance will grow or decay. The part that remains is defining the structural boundary conditions and an expansion basisgithat satisfy them.
Structural Constants for Elastic-Translating Boundary Condition
When applying the elastic translating boundary conditions with basis functions in equation 2.84, we augment the structural constants to include those in the boundary equation 2.74,
Msi =
m0
b fori =0
ρshbgi(x) fori =[1,n]
, (2.100)
Csi =
c0
b fori =0 0 fori =[1,n]
, (2.101)
Ksi =
k0
b fori =0
E I2D d4
dx4gi(x) fori =[1,n]
. (2.102)
When definitions in equations 2.100 - 2.102 are considered in equations 2.95, 2.97, and 2.99, then equation 2.96 becomes the linear map including a translating bound- ary condition in data 2.70 defined by 2.71 as a0(t) and aÛ0(t) states. We will re- fer to these equations are the quasi-1D model with either clamped-free or elastic- translating/moving boundary conditions in the subsequent text.