Non-Dimensional Fluid-Structure Equations

Chapter II: Leakage Flow Model

2.3 Fluid-Structure Coupling

2.3.6 Non-Dimensional Fluid-Structure Equations

C_si =











c0

b fori =0 0 fori =[1,n]

, (2.101)

K_si =











k0

b fori =0

E I2D d⁴

dx⁴g_i(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.

M_i^∗ = 2 ρ_fL²

Msi+2M_fi

= M_si^∗ + M_fi^∗,

C_i^∗ = 2 ρ_fqx0

h¯ L

2C_fi+C_si

=C_fi^∗ +C_si^∗,

K_i^∗ = 2 ρ_f

h¯² q²_x0

2K_fi+Ksi

= K_fi^∗ +K_si^∗,

T_f^∗ = 2 ρ_f

h¯² q_x0

2T_f

.

(2.105)

We examine the definition of the fluid coefficients in more detail by aggregating parts of equations in appendix A, where we can separate viscous, geometrical, and boundary terms from one another. Looking closely at equations A.1 - A.8, they consist of integral and differential operators of the channel geometry, he and g_i, often multiplied by either ξ_x, f₀, η, ζ_in, or ζ_out. We will define viscous terms as those integral and differential operators with a factor of either f₀ or η (subscript v), geometrical or nonlinear as those with a factor of ξ_x or only a function of the geometry (subscript g), and boundary terms those with factors of either ζ_in or ζ_out (subscript b). For disambiguation, we will list the parameter dependence, before the semi-colon, as each term is defined. Non-dimentionalizing the mass term from equation A.1, as defined in equation 2.105,

M_fi^∗ = 4

ρ_fL²M_fi = 4M_gi^∗(x^∗). (2.106) Similarly, the damping term from equation A.2,

C_fi^∗ = 4 ρ_fq_x0

h¯ LCfi

=4

C_gi^∗(ξx;x^∗)+C_bi^∗(ζout;x^∗) +4

L h¯

f₀

2 + q_x0η 4

C_vi^∗(x^∗),

(2.107)

the stiffness term from equation A.3, K_fi^∗ = 4

ρ_f h¯² q²_x0K_fi

=4

K_gi^∗(ξ_x;x^∗)+K_bi^∗(ζ_in, ζ_out;x^∗) +4

L h¯

3f₀ 4

K_vi^∗(x^∗),

(2.108)

and the flow rate forcing term in equation A.4, T_f^∗ = 4

ρ_f h¯² q_x0Tf

=4

T_g^∗(ξ_x;x^∗)+T_b^∗(ζin, ζout;x^∗) +4

L h¯

f0

2 + qx0η 4

T_v^∗(x^∗).

(2.109)

We must also non-dimentionalize the ODE 2.65 forQ¯_x1, d ¯Q^∗_x1

dt^∗ =

∞

Õ

i=0

B^∗_qid²a_i

dt^∗2 +D^∗_qida_i

dt^∗ +E_qi^∗a_i

+G^∗_qQ¯^∗_x1. (2.110)

where the coefficients are, for acceleration in equation A.5 B_qi^∗ = 1

hL¯ Bqi= B_qgi^∗ (x^∗), (2.111) for velocity in equation A.6,

D_qi^∗ = 1 q_x0D_qi

=

D^∗_qgi(ξ_x;x^∗)+D^∗_qbi(ζ_out;x^∗)

− L

h¯ f₀

2 + q_x0η 4

D^∗_qvi(x^∗),

(2.112)

for displacement in equation A.7, E_qi^∗ = hL¯

q²_x0E_qi

=

E_qgi^∗ (ξ_x;x^∗)+E_qbi^∗ (ζ_in, ζ_out;x^∗)

− L

h¯ 3f₀

4

E_qvi^∗ (x^∗),

(2.113)

for the forcing flow rate in equation A.8, G^∗_q= hL¯

q_x0Gq

=

G^∗_qg(ξ_x;x^∗)+G^∗_qb(ζ_in, ζ_out;x^∗)

− L

h¯ f₀

2 + q_x0η 4

G^∗_qv(x^∗).

(2.114)

Equations 2.106 - 2.109 and 2.111 - 2.114 show that the hydrodynamic boundary and geometrical terms are not impacted by the length scale of the gaph, but rather¯ the “shape” of the channel gap along the axial direction. However, by changing the relative scaling of the channel, _L^h¯, we affect the viscous term. Looking specifically

at the laminar flow case as defined in equation 2.29, and the definition in equation 2.49,

η = df dQ_x

Qx=qx0

= − 48 Reh¯

1

qx0 = − f₀ qx0

, (2.115)

the coefficients of all viscous terms simplify to 3

4f0

L h¯ = 36

Reh¯

L h¯ ∼

ε²_hReL

⁻1

, (2.116)

as defined in the lubrication closure in equation 2.18. Once again, the parameter ε²_hRe_L appears an the dominant viscous ratio, rather thanRe_L orε_hRe_L = Re_h. As ε²_hRe_L increases, two important things occur:

1. Viscous effects diminish, as all the viscous coefficients for equations 2.106 - 2.109 and 2.111 - 2.114 become small;

2. The hydrodynamic model with the lubrication closure becomes less repre- sentative of the flow physics since inertial terms in equation 2.17 become important, as noted in condition 3 in section 2.2.1.

We now examine the relevance of structural non-dimensional parameters. First considering the elastic-translating boundary conditions from equation 2.70 using the non-dimensional structural terms in equation 2.105,

M_si^∗ = 2

ρ_fL²M_si =











 2

1 ρ_fL²

m0

b h¯ L

fori =0 2

_ρ

sh_b ρfL

h¯ L

g_i^∗(x^∗) fori =[1,n]

, (2.117)

C_si^∗ = 2 ρ_fq_x0

h¯ LC_si =











 2 _¯

ρ_fhqx0

c0

bL h¯ L

fori = 0

0 fori = [1,n]

, (2.118)

K_si^∗ = 2 ρ_f

h¯² q²_x0K_si =











 2

_¯

h² ρfq²_x0

k0

b h¯ L

fori =0 2 _¯

h²E ρ_fq²_x0

I2D

L³ h¯ L

d⁴

dx^∗4g_i^∗(x^∗) fori =[1,n]

. (2.119)

The parameters for the clamped-free boundary condition in equations 2.70 are the elements of equations 2.117, 2.118, and 2.119 applicable fori =[1,n]. Writing the coupled system as proportional to the non-dimensional order of magnitude for each term for the beam, where we aggregate with like terms parameters inF^∗ 1, we have

h¯ L

"

ρ_sh_b ρ_fL

F_sM^∗ + h¯²E ρ_fq²_x0

I_2D L³

! F_sK^∗

#

∼ F_fg^∗ +F_fb^∗ +

"

h¯ L

2

Re_L

#⁻1

F_fv^∗, (2.120)

and similarly for the elastic boundary condition,

h¯ L

"

1 ρ_fL²

m₀ b

F_sbM^∗ + h¯

ρ_fqx0

c₀ bL

F_sbC^∗ + h¯² ρ_fq²_x0

k₀ b

! F_sbK^∗ + ρ_sh_b

ρ_fL

F_sM^∗ + h¯²E ρ_fq²_x0

I_2D L³

! F_sK^∗

#

∼ F_fg^∗ +F_fb^∗ +

"

h¯ L

2

Re_L

#−1

F_fv^∗. (2.121)

The four non-dimensional parameter that appear in the clamped-free system are shown in table 2.3. An additional three pertain only to the moving elastic boundary condition, shown in table 2.4. Equations 2.120 and 2.121 illustrate the general effect of each non-dimensional parameter on individual terms in the equations of motion.

Clearly hˆ scales the influence of structure as compared with the fluid right-hand- side, indicating that as it diminishes, the fluid terms will dominate the dynamics.

As mentioned, hReˆ _h dictates the fluid viscous effects. Once the elastic boundary condition is included, its structural parameters then play a role on the overall system dynamics are well.

The total number of non-dimensional parameters with the elastic-translating bound- ary conditions is 7. Not all parameters in table 2.1 have been used in the fluid- structure portion of the analysis, namely those relating to electrical quantities. We also note that h_t, x_t, R,α_i, and α_o are parameters used to describe the shape of the channel, all either with units of l or non-dimensional. Once the gap is normalized by h, then the remaining¯ l dimensional parameters will also be normalized by h,¯ and their contribution embedded into theFterms in equations Equations 2.120 and

1Subscripts s for structure, f for fluid, b for boundary, M for mass, C for damping, K for stiffness, and as previously defined, g for geometrical, v for viscous. The point of this exercise is not to define the non-dimensional expressions for F^∗, but to show the coefficients that scale as a function of dimensional parameters forming non-dimensional groups.

Table 2.3: Table of clamped-free fluid structure non-dimensional parameters.

Variable Expression Description mˆ ^ρ_ρ^shb

fL mass ratio

kˆ _ρ^h¯2E

fq²_x0 I2D

L³ stiffness ratio hˆ ^h_L^¯ =ε_h gap or throat ratio hReˆ _h ε²_hRe_L = _¯

h L

2

Re_L viscous parameter

Table 2.4: Table of elastic-translating fluid structure non-dimensional parameters.

Variable Expression Description mˆbc 1

ρfL² m0

b boundary mass ratio kˆbc h¯²

ρ_fq²_x0 k0

b boundary stiffness ratio ˆ

c_{bc ρ} ^h¯

fqx0L c0

b boundary damping ratio

2.121. The span bwill be discussed further in the next section, as we derive the quasi-1D model that includes spanwise leakage flow.